Subgroups — C₆₇×#60254 #30739505

Order 17152.

1658 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2987	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2986	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2985	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2985	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2984	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2983	C₂	1	2	yes	maximal, complemented
8	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2962	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2962	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2961	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2925	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#2925	C₂	1	2	yes	maximal
16	C₆₇×#2923	C₂	1	2	yes	maximal
17	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic
18	Q₈²×C₆₇	C₄	1	1	yes	characteristic
19	#557×C₆₇	C₄	1	1	yes	characteristic
20	#557×C₆₇	C₄	1	2	yes	—
21	#554×C₆₇	C₄	1	2	yes	—
22	#554×C₆₇	C₄	1	1	yes	characteristic
23	#554×C₆₇	C₄	1	4	yes	—
24	#554×C₆₇	C₄	1	2	yes	—
25	D₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
26	D₄×Q₈×C₆₇	C₄	1	2	yes	—
27	#533×C₆₇	C₂²	1	1	yes	characteristic
28	#533×C₆₇	C₂²	1	2	yes	—
29	#533×C₆₇	C₂²	1	2	yes	—
30	#533×C₆₇	C₂²	1	4	yes	—
31	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×#41×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
35	#446×C₆₇	C₂²	1	1	yes	characteristic
36	#446×C₆₇	C₂²	1	1	yes	characteristic
37	#446×C₆₇	C₂²	1	2	yes	—
38	#446×C₆₇	C₂²	1	1	yes	characteristic
39	#446×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	2	yes	—
41	#446×C₆₇	C₂²	1	2	yes	—
42	#446×C₆₇	C₂²	1	2	yes	—
43	#446×C₆₇	C₂²	1	4	yes	—
44	#446×C₆₇	C₂²	1	4	yes	—
45	#446×C₆₇	C₂²	1	4	yes	—
46	#446×C₆₇	C₂²	1	4	yes	—
47	#446×C₆₇	C₂²	1	2	yes	—
48	#446×C₆₇	C₂²	1	4	yes	—
49	#446×C₆₇	C₂²	1	4	yes	—
50	#446×C₆₇	C₂²	1	4	yes	—
51	#446×C₆₇	C₂²	1	4	yes	—
52	#446×C₆₇	C₂²	1	2	yes	—
53	#446×C₆₇	C₂²	1	4	yes	—
54	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
55	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
56	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
57	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
58	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
59	#433×C₆₇	C₂²	1	2	yes	—
60	#433×C₆₇	C₂²	1	2	yes	—
61	#433×C₆₇	C₂²	1	4	yes	—
62	#433×C₆₇	C₂²	1	4	yes	—
63	#433×C₆₇	C₂²	1	4	yes	—
64	#433×C₆₇	C₂²	1	4	yes	—
65	#433×C₆₇	C₂²	1	4	yes	—
66	#432×C₆₇	C₂²	1	4	yes	—
67	#432×C₆₇	C₂²	1	4	yes	—
68	#432×C₆₇	C₂²	1	4	yes	—
69	#432×C₆₇	C₂²	1	2	yes	—
70	#431×C₆₇	C₂²	1	2	yes	—
71	#431×C₆₇	C₂²	1	2	yes	—
72	#431×C₆₇	C₂²	1	2	yes	—
73	#431×C₆₇	C₂²	1	4	yes	—
74	#424×C₆₇	C₂²	1	1	yes	characteristic
75	#424×C₆₇	C₂²	1	1	yes	characteristic
76	#424×C₆₇	C₂²	1	2	yes	—
77	#424×C₆₇	C₂²	1	2	yes	—
78	#423×C₆₇	C₂²	1	1	yes	characteristic
79	#423×C₆₇	C₂²	1	2	yes	—
80	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
81	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
82	#408×C₆₇	C₂²	1	1	yes	characteristic
83	#408×C₆₇	C₂²	1	1	yes	characteristic
84	#408×C₆₇	C₂²	1	2	yes	—
85	#405×C₆₇	C₂²	1	2	yes	—
86	#405×C₆₇	C₂²	1	2	yes	—
87	#405×C₆₇	C₂²	1	2	yes	—
88	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
89	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
90	C₂×#97×C₆₇	C₂²	1	2	yes	—
91	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
92	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
93	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
95	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
98	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
101	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
103	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
104	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
105	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
107	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
108	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
109	#125×C₆₇	C₂×C₄	1	2	yes	—
110	#125×C₆₇	C₂×C₄	1	2	yes	—
111	#125×C₆₇	C₂×C₄	1	2	yes	—
112	#125×C₆₇	C₂×C₄	1	4	yes	—
113	#125×C₆₇	C₂×C₄	1	4	yes	—
114	#125×C₆₇	C₂×C₄	1	4	yes	—
115	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
116	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
117	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
118	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
119	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
120	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
121	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
122	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
123	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
124	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
125	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
126	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
127	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
128	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
129	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
130	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
131	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
132	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
133	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
134	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
135	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
136	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
137	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
138	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
139	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
140	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
141	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
142	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
143	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
144	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
145	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
146	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
147	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
148	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
149	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
150	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
151	#117×C₆₇	C₂³	1	1	yes	characteristic
152	#117×C₆₇	C₂³	1	2	yes	—
153	#117×C₆₇	C₂³	1	2	yes	—
154	#117×C₆₇	C₂³	1	4	yes	—
155	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
156	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
157	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
158	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
159	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
160	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
161	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
162	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
163	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
164	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
165	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
166	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
167	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
168	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
169	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
170	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
171	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
172	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
173	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
174	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
175	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
176	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
177	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
178	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
179	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
180	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
181	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
182	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
183	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
184	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
185	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
186	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
187	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
188	#97×C₆₇	C₂³	1	2	yes	—
189	#97×C₆₇	C₂³	1	2	yes	—
190	#97×C₆₇	C₂³	1	2	yes	—
191	#97×C₆₇	C₂³	1	2	yes	—
192	#97×C₆₇	C₂³	1	4	yes	—
193	#97×C₆₇	C₂³	1	2	yes	—
194	#97×C₆₇	C₂³	1	4	yes	—
195	#97×C₆₇	C₂³	1	4	yes	—
196	#97×C₆₇	C₂³	1	4	yes	—
197	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
198	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
199	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
200	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
201	C₄×C₈×C₆₇	C₂³	1	2	yes	—
202	C₄×C₈×C₆₇	C₂³	1	4	yes	—
203	C₄×C₈×C₆₇	C₂³	1	4	yes	—
204	C₄×C₈×C₆₇	C₂³	1	4	yes	—
205	C₄×C₈×C₆₇	C₂³	1	4	yes	—
206	#41×C₆₇	—	2	2	no	—
207	#41×C₆₇	—	2	2	no	—
208	C₂×Q₈×C₆₇	—	4	2	no	—
209	C₂×Q₈×C₆₇	—	4	4	no	—
210	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
211	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
212	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
213	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
214	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
215	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
216	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
217	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
218	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
219	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
220	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
221	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
222	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
223	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
224	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
225	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
226	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
227	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
228	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
229	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
230	M₁₆×C₆₇	—	2	2	no	—
231	M₁₆×C₆₇	—	2	2	no	—
232	M₁₆×C₆₇	—	2	4	no	—
233	C₂×C₈×C₆₇	—	2	2	no	—
234	C₂×C₈×C₆₇	—	2	4	no	—
235	C₂×C₈×C₆₇	—	2	2	no	—
236	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
237	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
238	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
239	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
240	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
241	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
242	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
243	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
244	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
245	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
246	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
247	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
248	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
249	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
250	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
251	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
252	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
253	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
254	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
255	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
256	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
257	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
258	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
259	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
260	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
261	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
262	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
263	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
264	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
265	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
266	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
267	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
268	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
269	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
270	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
271	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
272	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
273	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
274	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
275	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
276	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
277	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
278	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
279	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
280	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
281	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
282	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
283	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
284	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
285	C₄²×C₆₇	C₂⁴	1	2	yes	—
286	C₄²×C₆₇	C₂⁴	1	2	yes	—
287	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
288	C₄²×C₆₇	C₂⁴	1	2	yes	—
289	C₄²×C₆₇	C₂⁴	1	4	yes	—
290	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
291	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
292	Q₈×C₆₇	—	2	2	no	—
293	Q₈×C₆₇	—	2	2	no	—
294	Q₈×C₆₇	—	2	2	no	—
295	Q₈×C₆₇	—	2	4	no	—
296	Q₈×C₆₇	—	4	2	no	—
297	Q₈×C₆₇	—	4	4	no	—
298	Q₈×C₆₇	—	4	2	no	—
299	Q₈×C₆₇	—	4	4	no	—
300	D₄×C₆₇	—	2	4	no	—
301	D₄×C₆₇	—	2	2	no	—
302	C₂×C₄×C₆₇	—	2	2	no	—
303	C₂×C₄×C₆₇	—	2	4	no	—
304	C₂×C₄×C₆₇	—	2	2	no	—
305	C₂×C₄×C₆₇	—	2	2	no	—
306	C₂×C₄×C₆₇	—	4	1	no	—
307	C₂×C₄×C₆₇	—	4	2	no	—
308	C₂×C₄×C₆₇	—	4	2	no	—
309	C₂×C₄×C₆₇	—	4	4	no	—
310	C₂×C₄×C₆₇	—	4	2	no	—
311	C₂×C₄×C₆₇	—	4	4	no	—
312	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
313	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
314	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
315	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
316	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
317	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
318	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
319	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
320	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
321	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
322	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
323	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
324	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
325	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
326	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
327	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
328	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
329	C₈×C₆₇	—	2	1	no	—
330	C₈×C₆₇	—	2	1	no	—
331	C₈×C₆₇	—	2	2	no	—
332	C₈×C₆₇	—	2	2	no	—
333	C₈×C₆₇	—	2	1	no	—
334	C₈×C₆₇	—	2	1	no	—
335	C₈×C₆₇	—	2	2	no	—
336	C₈×C₆₇	—	2	4	no	—
337	C₈×C₆₇	—	2	2	no	—
338	C₂²×C₆₇	—	2	1	no	—
339	C₂²×C₆₇	—	2	1	no	—
340	C₂²×C₆₇	—	2	2	no	—
341	C₂²×C₆₇	—	2	2	no	—
342	C₂²×C₆₇	—	2	2	no	—
343	C₂²×C₆₇	#519	1	2	yes	—
344	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
345	C₄×C₆₇	—	2	2	no	—
346	C₄×C₆₇	—	2	1	no	—
347	C₄×C₆₇	—	2	2	no	—
348	C₄×C₆₇	—	2	2	no	—
349	C₄×C₆₇	—	2	2	no	—
350	C₄×C₆₇	—	2	4	no	—
351	C₄×C₆₇	—	2	1	no	—
352	C₄×C₆₇	—	2	2	no	—
353	C₄×C₆₇	—	2	1	no	—
354	C₄×C₆₇	—	2	2	no	—
355	C₄×C₆₇	—	2	2	no	—
356	C₄×C₆₇	—	2	4	no	—
357	C₄×C₆₇	#519	1	2	yes	—
358	C₄×C₆₇	#568	1	2	yes	—
359	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
360	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
361	#60254	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
362	C₂×C₆₇	—	2	2	no	—
363	C₂×C₆₇	—	4	2	no	—
364	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
365	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
366	C₂×C₆₇	#3573	1	1	yes	characteristic
367	Q₈×#41	C₂×C₆₇	1	1	yes	characteristic
368	#2987	C₂×C₆₇	1	4	yes	complemented
369	#2986	C₂×C₆₇	1	2	yes	complemented
370	#2985	C₂×C₆₇	1	2	yes	complemented
371	#2985	C₂×C₆₇	1	4	yes	complemented
372	#2984	C₂×C₆₇	1	2	yes	complemented
373	#2983	C₂×C₆₇	1	2	yes	complemented
374	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
375	#2962	C₂×C₆₇	1	1	yes	characteristic, complemented
376	#2962	C₂×C₆₇	1	2	yes	complemented
377	#2961	C₂×C₆₇	1	2	yes	complemented
378	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
379	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
380	#2925	C₂×C₆₇	1	1	yes	characteristic
381	#2925	C₂×C₆₇	1	2	yes	—
382	#2923	C₂×C₆₇	1	2	yes	—
383	#2922	C₂×C₆₇	1	1	yes	characteristic
384	C₆₇	#60254	1	1	yes	characteristic, direct product factor, Sylow
385	Q₈²	C₄×C₆₇	1	1	yes	characteristic
386	#557	C₄×C₆₇	1	1	yes	characteristic
387	#557	C₄×C₆₇	1	2	yes	—
388	#554	C₄×C₆₇	1	2	yes	—
389	#554	C₄×C₆₇	1	1	yes	characteristic
390	#554	C₄×C₆₇	1	4	yes	—
391	#554	C₄×C₆₇	1	2	yes	—
392	D₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
393	D₄×Q₈	C₄×C₆₇	1	2	yes	—
394	#533	C₂²×C₆₇	1	1	yes	characteristic
395	#533	C₂²×C₆₇	1	2	yes	—
396	#533	C₂²×C₆₇	1	2	yes	—
397	#533	C₂²×C₆₇	1	4	yes	—
398	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
399	C₄×#41	C₂²×C₆₇	1	2	yes	—
400	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
401	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
402	#446	C₂²×C₆₇	1	1	yes	characteristic
403	#446	C₂²×C₆₇	1	1	yes	characteristic
404	#446	C₂²×C₆₇	1	1	yes	characteristic
405	#446	C₂²×C₆₇	1	2	yes	—
406	#446	C₂²×C₆₇	1	1	yes	characteristic
407	#446	C₂²×C₆₇	1	2	yes	—
408	#446	C₂²×C₆₇	1	2	yes	—
409	#446	C₂²×C₆₇	1	2	yes	—
410	#446	C₂²×C₆₇	1	4	yes	—
411	#446	C₂²×C₆₇	1	4	yes	—
412	#446	C₂²×C₆₇	1	4	yes	—
413	#446	C₂²×C₆₇	1	4	yes	—
414	#446	C₂²×C₆₇	1	2	yes	—
415	#446	C₂²×C₆₇	1	4	yes	—
416	#446	C₂²×C₆₇	1	4	yes	—
417	#446	C₂²×C₆₇	1	4	yes	—
418	#446	C₂²×C₆₇	1	4	yes	—
419	#446	C₂²×C₆₇	1	2	yes	—
420	#446	C₂²×C₆₇	1	4	yes	—
421	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
422	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
423	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
424	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
425	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
426	#433	C₂²×C₆₇	1	2	yes	—
427	#433	C₂²×C₆₇	1	2	yes	—
428	#433	C₂²×C₆₇	1	4	yes	—
429	#433	C₂²×C₆₇	1	4	yes	—
430	#433	C₂²×C₆₇	1	4	yes	—
431	#433	C₂²×C₆₇	1	4	yes	—
432	#433	C₂²×C₆₇	1	4	yes	—
433	#432	C₂²×C₆₇	1	4	yes	—
434	#432	C₂²×C₆₇	1	4	yes	—
435	#432	C₂²×C₆₇	1	4	yes	—
436	#432	C₂²×C₆₇	1	2	yes	—
437	#431	C₂²×C₆₇	1	2	yes	—
438	#431	C₂²×C₆₇	1	2	yes	—
439	#431	C₂²×C₆₇	1	2	yes	—
440	#431	C₂²×C₆₇	1	4	yes	—
441	#424	C₂²×C₆₇	1	1	yes	characteristic
442	#424	C₂²×C₆₇	1	1	yes	characteristic
443	#424	C₂²×C₆₇	1	2	yes	—
444	#424	C₂²×C₆₇	1	2	yes	—
445	#423	C₂²×C₆₇	1	1	yes	characteristic
446	#423	C₂²×C₆₇	1	2	yes	—
447	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
448	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
449	#408	C₂²×C₆₇	1	1	yes	characteristic
450	#408	C₂²×C₆₇	1	1	yes	characteristic
451	#408	C₂²×C₆₇	1	2	yes	—
452	#405	C₂²×C₆₇	1	2	yes	—
453	#405	C₂²×C₆₇	1	2	yes	—
454	#405	C₂²×C₆₇	1	2	yes	—
455	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
456	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
457	C₂×#97	C₂²×C₆₇	1	2	yes	—
458	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
459	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
460	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
461	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
462	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
463	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
464	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
465	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
466	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
467	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
468	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
469	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
470	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
471	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
472	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
473	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
474	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
475	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
476	#125	C₂×C₄×C₆₇	1	2	yes	—
477	#125	C₂×C₄×C₆₇	1	2	yes	—
478	#125	C₂×C₄×C₆₇	1	2	yes	—
479	#125	C₂×C₄×C₆₇	1	4	yes	—
480	#125	C₂×C₄×C₆₇	1	4	yes	—
481	#125	C₂×C₄×C₆₇	1	4	yes	—
482	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
483	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
484	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
485	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
486	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
487	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
488	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
489	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
490	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
491	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
492	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
493	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
494	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
495	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
496	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
497	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
498	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
499	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
500	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
501	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
502	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
503	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
504	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
505	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
506	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
507	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
508	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
509	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
510	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
511	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
512	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
513	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
514	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
515	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
516	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
517	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
518	#117	C₂³×C₆₇	1	1	yes	characteristic
519	#117	C₂³×C₆₇	1	2	yes	—
520	#117	C₂³×C₆₇	1	2	yes	—
521	#117	C₂³×C₆₇	1	4	yes	—
522	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
523	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
524	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
525	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
526	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
527	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
528	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
529	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
530	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
531	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
532	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
533	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
534	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
535	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
536	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
537	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
538	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
539	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
540	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
541	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
542	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
543	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
544	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
545	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
546	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
547	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
548	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
549	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
550	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
551	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
552	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
553	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
554	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
555	#97	C₂³×C₆₇	1	2	yes	—
556	#97	C₂³×C₆₇	1	2	yes	—
557	#97	C₂³×C₆₇	1	2	yes	—
558	#97	C₂³×C₆₇	1	2	yes	—
559	#97	C₂³×C₆₇	1	4	yes	—
560	#97	C₂³×C₆₇	1	2	yes	—
561	#97	C₂³×C₆₇	1	4	yes	—
562	#97	C₂³×C₆₇	1	4	yes	—
563	#97	C₂³×C₆₇	1	4	yes	—
564	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
565	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
566	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
567	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
568	C₄×C₈	C₂³×C₆₇	1	2	yes	—
569	C₄×C₈	C₂³×C₆₇	1	4	yes	—
570	C₄×C₈	C₂³×C₆₇	1	4	yes	—
571	C₄×C₈	C₂³×C₆₇	1	4	yes	—
572	C₄×C₈	C₂³×C₆₇	1	4	yes	—
573	#41	—	2	2	no	—
574	#41	—	2	2	no	—
575	C₂×Q₈	—	4	2	no	—
576	C₂×Q₈	—	4	4	no	—
577	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
578	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
579	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
580	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
581	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
582	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
583	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
584	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
585	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
586	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
587	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
588	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
589	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
590	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
591	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
592	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
593	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
594	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
595	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
596	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
597	M₁₆	—	2	2	no	—
598	M₁₆	—	2	2	no	—
599	M₁₆	—	2	4	no	—
600	C₂×C₈	—	2	2	no	—
601	C₂×C₈	—	2	4	no	—
602	C₂×C₈	—	2	2	no	—
603	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
604	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
605	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
606	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
607	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
608	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
609	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
610	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
611	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
612	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
613	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
614	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
615	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
616	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
617	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
618	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
619	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
620	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
621	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
622	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
623	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
624	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
625	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
626	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
627	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
628	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
629	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
630	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
631	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
632	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
633	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
634	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
635	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
636	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
637	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
638	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
639	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
640	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
641	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
642	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
643	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
644	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
645	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
646	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
647	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
648	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
649	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
650	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
651	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
652	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
653	C₄²	C₂⁴×C₆₇	1	2	yes	—
654	C₄²	C₂⁴×C₆₇	1	4	yes	—
655	C₄²	C₂⁴×C₆₇	1	2	yes	—
656	C₄²	C₂⁴×C₆₇	1	2	yes	—
657	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
658	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
659	Q₈	—	2	2	no	—
660	Q₈	—	2	2	no	—
661	Q₈	—	2	2	no	—
662	Q₈	—	2	4	no	—
663	Q₈	—	4	2	no	—
664	Q₈	—	4	4	no	—
665	Q₈	—	4	2	no	—
666	Q₈	—	4	4	no	—
667	D₄	—	2	4	no	—
668	D₄	—	2	2	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	4	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	4	2	no	—
674	C₂×C₄	—	4	4	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	2	no	—
677	C₂×C₄	—	4	2	no	—
678	C₂×C₄	—	4	4	no	—
679	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
680	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
681	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
682	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
683	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
684	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
685	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
686	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
687	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
688	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
689	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
690	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
691	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
692	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
693	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
694	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
695	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
696	C₈	—	2	1	no	—
697	C₈	—	2	1	no	—
698	C₈	—	2	2	no	—
699	C₈	—	2	2	no	—
700	C₈	—	2	1	no	—
701	C₈	—	2	1	no	—
702	C₈	—	2	2	no	—
703	C₈	—	2	2	no	—
704	C₈	—	2	4	no	—
705	C₂²	—	2	1	no	—
706	C₂²	—	2	1	no	—
707	C₂²	—	2	2	no	—
708	C₂²	—	2	2	no	—
709	C₂²	—	2	2	no	—
710	C₂²	#519×C₆₇	1	2	yes	—
711	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
712	C₄	—	2	1	no	—
713	C₄	—	2	2	no	—
714	C₄	—	2	2	no	—
715	C₄	—	2	2	no	—
716	C₄	—	2	4	no	—
717	C₄	—	2	1	no	—
718	C₄	—	2	2	no	—
719	C₄	—	2	1	no	—
720	C₄	—	2	2	no	—
721	C₄	—	2	2	no	—
722	C₄	—	2	4	no	—
723	C₄	—	2	2	no	—
724	C₄	#519×C₆₇	1	2	yes	—
725	C₄	#568×C₆₇	1	2	yes	—
726	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
727	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
728	C₂	—	2	2	no	—
729	C₂	—	4	2	no	—
730	C₂	C₂×#519×C₆₇	1	1	yes	characteristic
731	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
732	C₂	C₆₇×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	C₂×#41×C₆₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2987	C₂	semidirect
3. C₆₇×#2986	C₂	semidirect
4. C₆₇×#2985	C₂	semidirect
5. C₆₇×#2985	C₂	semidirect
6. C₆₇×#2984	C₂	semidirect
7. C₆₇×#2983	C₂	semidirect
8. Q₈×M₁₆×C₆₇	C₂	semidirect
9. C₆₇×#2962	C₂	semidirect
10. C₆₇×#2962	C₂	semidirect
11. C₆₇×#2961	C₂	semidirect
12. C₆₇×#2960	C₂	semidirect
13. C₂×#446×C₆₇	C₂	semidirect
361. #60254	C₆₇	direct (N × H)
368. #2987	C₂×C₆₇	semidirect
369. #2986	C₂×C₆₇	semidirect
370. #2985	C₂×C₆₇	semidirect
371. #2985	C₂×C₆₇	semidirect
372. #2984	C₂×C₆₇	semidirect
373. #2983	C₂×C₆₇	semidirect
374. Q₈×M₁₆	C₂×C₆₇	semidirect
375. #2962	C₂×C₆₇	semidirect
376. #2962	C₂×C₆₇	semidirect
377. #2961	C₂×C₆₇	semidirect
378. #2960	C₂×C₆₇	semidirect
379. C₂×#446	C₂×C₆₇	semidirect
384. C₆₇	#60254	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×#41×C₆₇	2
C₆₇×#2987	2
C₆₇×#2986	2
C₆₇×#2985	2
C₆₇×#2984	2
C₆₇×#2983	2
Q₈×M₁₆×C₆₇	2
C₆₇×#2962	2
C₆₇×#2961	2
C₆₇×#2960	2
C₂×#446×C₆₇	2
C₆₇×#2925	2
C₆₇×#2923	2
C₆₇×#2922	2
#60254	67

Subgroups — C67×#60254 #30739505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60254 #30739505