Subgroups — C₆₇×#34934 #30714185

Order 17152.

2580 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2554	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2552	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2541	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2538	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2512	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2490	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2445	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2426	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2392	C₂	1	4	yes	maximal, complemented
12	C₄×#127×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2293	C₂	1	2	yes	maximal, complemented
16	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#127×C₆₇	C₂²	1	2	yes	—
20	#529×C₆₇	—	2	16	no	—
21	C₂×#126×C₆₇	C₂²	1	4	yes	—
22	C₂×#125×C₆₇	C₂²	1	2	yes	—
23	C₂×#124×C₆₇	C₂²	1	2	yes	—
24	C₂×#123×C₆₇	C₂²	1	4	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#121×C₆₇	C₂²	1	2	yes	—
27	C₂×#121×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
37	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
38	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
39	C₂×#117×C₆₇	C₂²	1	4	yes	—
40	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
41	#398×C₆₇	C₂²	1	4	yes	—
42	#397×C₆₇	C₂²	1	4	yes	—
43	#396×C₆₇	C₂²	1	4	yes	—
44	#396×C₆₇	C₂²	1	4	yes	—
45	#394×C₆₇	C₂²	1	4	yes	—
46	#394×C₆₇	C₂²	1	4	yes	—
47	#394×C₆₇	C₂²	1	4	yes	—
48	#392×C₆₇	C₂²	1	4	yes	—
49	#390×C₆₇	C₂²	1	2	yes	—
50	#390×C₆₇	C₂²	1	2	yes	—
51	#390×C₆₇	C₂²	1	2	yes	—
52	#390×C₆₇	C₂²	1	2	yes	—
53	#390×C₆₇	C₂²	1	2	yes	—
54	#390×C₆₇	C₂²	1	2	yes	—
55	#390×C₆₇	C₂²	1	2	yes	—
56	#390×C₆₇	C₂²	1	4	yes	—
57	#390×C₆₇	C₂²	1	4	yes	—
58	#390×C₆₇	C₂²	1	2	yes	—
59	#389×C₆₇	C₂²	1	2	yes	—
60	#389×C₆₇	C₂²	1	2	yes	—
61	#389×C₆₇	C₂²	1	2	yes	—
62	#388×C₆₇	C₂²	1	4	yes	—
63	#388×C₆₇	C₂²	1	4	yes	—
64	#388×C₆₇	C₂²	1	4	yes	—
65	#386×C₆₇	C₂²	1	4	yes	—
66	#384×C₆₇	C₂²	1	2	yes	—
67	#384×C₆₇	C₂²	1	1	yes	characteristic
68	#380×C₆₇	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
72	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
78	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
82	C₄³×C₆₇	C₂²	1	1	yes	characteristic
83	C₄³×C₆₇	C₂²	1	2	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
86	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
87	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
88	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
89	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
91	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
92	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
93	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
94	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
95	#127×C₆₇	—	2	8	no	—
96	#127×C₆₇	—	2	4	no	—
97	#126×C₆₇	—	2	16	no	—
98	#125×C₆₇	—	2	8	no	—
99	#124×C₆₇	—	2	8	no	—
100	#123×C₆₇	—	2	16	no	—
101	C₂²⋊Q₈×C₆₇	—	2	8	no	—
102	#121×C₆₇	—	2	8	no	—
103	#121×C₆₇	—	2	8	no	—
104	C₄×Q₈×C₆₇	—	2	4	no	—
105	C₄×Q₈×C₆₇	—	2	4	no	—
106	C₄×Q₈×C₆₇	D₄	1	8	yes	complemented
107	C₄×D₄×C₆₇	—	2	4	no	—
108	C₄×D₄×C₆₇	—	2	8	no	—
109	C₄×D₄×C₆₇	—	2	16	no	—
110	C₄×D₄×C₆₇	—	2	4	no	—
111	C₄×D₄×C₆₇	—	2	8	no	—
112	C₄×D₄×C₆₇	—	2	4	no	—
113	C₄×D₄×C₆₇	—	4	8	no	—
114	C₄×D₄×C₆₇	—	4	4	no	—
115	#117×C₆₇	—	2	16	no	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
127	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
128	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
133	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
135	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
136	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
137	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
138	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
139	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
141	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
142	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
143	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
144	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
145	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
146	C₂×C₄²×C₆₇	—	2	8	no	—
147	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
148	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
149	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
152	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
153	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
154	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
155	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
156	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
157	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
158	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
159	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
160	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
161	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
162	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
163	#95×C₆₇	C₂³	1	2	yes	—
164	#95×C₆₇	C₂³	1	2	yes	—
165	#95×C₆₇	C₂³	1	2	yes	—
166	#95×C₆₇	C₂³	1	4	yes	—
167	#95×C₆₇	C₂³	1	4	yes	—
168	#95×C₆₇	C₂³	1	2	yes	—
169	#95×C₆₇	C₂³	1	2	yes	—
170	#95×C₆₇	C₂³	1	2	yes	—
171	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
172	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
173	C₂×Q₈×C₆₇	—	2	2	no	—
174	C₂×Q₈×C₆₇	—	2	2	no	—
175	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
176	C₂×D₄×C₆₇	—	2	8	no	—
177	C₂×D₄×C₆₇	—	2	8	no	—
178	C₂×D₄×C₆₇	—	2	8	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	2	2	no	—
181	C₂×D₄×C₆₇	—	2	2	no	—
182	C₂×D₄×C₆₇	—	2	4	no	—
183	C₂×D₄×C₆₇	—	2	8	no	—
184	C₂×D₄×C₆₇	—	2	4	no	—
185	C₂×D₄×C₆₇	—	2	4	no	—
186	C₂×D₄×C₆₇	—	2	4	no	—
187	C₂×D₄×C₆₇	—	2	4	no	—
188	C₂×D₄×C₆₇	—	2	4	no	—
189	C₂×D₄×C₆₇	—	4	4	no	—
190	C₂×D₄×C₆₇	—	4	2	no	—
191	C₂×D₄×C₆₇	—	4	4	no	—
192	C₂×D₄×C₆₇	—	4	2	no	—
193	C₂×D₄×C₆₇	#41	1	4	yes	—
194	C₂²×C₄×C₆₇	—	2	4	no	—
195	C₂²×C₄×C₆₇	—	2	4	no	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	2	4	no	—
198	C₂²×C₄×C₆₇	—	2	8	no	—
199	C₂²×C₄×C₆₇	—	2	4	no	—
200	C₂²×C₄×C₆₇	—	2	4	no	—
201	C₂²×C₄×C₆₇	—	2	8	no	—
202	C₂²×C₄×C₆₇	—	2	4	no	—
203	C₂²×C₄×C₆₇	—	2	4	no	—
204	C₂²×C₄×C₆₇	—	2	8	no	—
205	C₂²×C₄×C₆₇	—	4	2	no	—
206	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
207	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
208	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
212	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
216	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
218	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
219	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
220	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
221	C₄⋊C₄×C₆₇	—	2	8	no	—
222	C₄⋊C₄×C₆₇	—	2	8	no	—
223	C₄⋊C₄×C₆₇	—	2	4	no	—
224	C₄⋊C₄×C₆₇	—	2	8	no	—
225	C₄⋊C₄×C₆₇	—	2	2	no	—
226	C₄⋊C₄×C₆₇	—	2	2	no	—
227	C₄⋊C₄×C₆₇	—	2	2	no	—
228	C₄⋊C₄×C₆₇	—	2	4	no	—
229	C₄⋊C₄×C₆₇	—	2	4	no	—
230	C₄⋊C₄×C₆₇	—	2	4	no	—
231	C₄⋊C₄×C₆₇	—	2	2	no	—
232	C₄⋊C₄×C₆₇	—	4	4	no	—
233	C₄⋊C₄×C₆₇	—	4	2	no	—
234	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
235	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
236	C₄⋊C₄×C₆₇	#41	1	4	yes	—
237	C₂²⋊C₄×C₆₇	—	2	4	no	—
238	C₂²⋊C₄×C₆₇	—	2	8	no	—
239	C₂²⋊C₄×C₆₇	—	2	8	no	—
240	C₂²⋊C₄×C₆₇	—	2	8	no	—
241	C₂²⋊C₄×C₆₇	—	2	8	no	—
242	C₂²⋊C₄×C₆₇	—	2	8	no	—
243	C₂²⋊C₄×C₆₇	—	2	4	no	—
244	C₂²⋊C₄×C₆₇	—	2	8	no	—
245	C₂²⋊C₄×C₆₇	—	2	4	no	—
246	C₂²⋊C₄×C₆₇	—	2	8	no	—
247	C₂²⋊C₄×C₆₇	—	4	4	no	—
248	C₂²⋊C₄×C₆₇	—	4	4	no	—
249	C₂²⋊C₄×C₆₇	—	4	4	no	—
250	C₂²⋊C₄×C₆₇	—	4	4	no	—
251	C₄²×C₆₇	—	2	4	no	—
252	C₄²×C₆₇	—	2	4	no	—
253	C₄²×C₆₇	—	2	4	no	—
254	C₄²×C₆₇	—	2	2	no	—
255	C₄²×C₆₇	—	2	4	no	—
256	C₄²×C₆₇	—	2	2	no	—
257	C₄²×C₆₇	—	2	4	no	—
258	C₄²×C₆₇	—	2	4	no	—
259	C₄²×C₆₇	—	2	2	no	—
260	C₄²×C₆₇	—	2	4	no	—
261	C₄²×C₆₇	—	4	1	no	—
262	C₄²×C₆₇	—	4	2	no	—
263	C₄²×C₆₇	C₂×D₄	1	4	yes	—
264	C₄²×C₆₇	C₂×D₄	1	8	yes	—
265	C₄²×C₆₇	#41	1	4	yes	—
266	C₄²×C₆₇	#41	1	4	yes	—
267	C₂³×C₆₇	—	2	2	no	—
268	C₂³×C₆₇	—	2	2	no	—
269	C₂³×C₆₇	—	2	2	no	—
270	C₂³×C₆₇	—	2	2	no	—
271	C₂³×C₆₇	—	2	2	no	—
272	C₂³×C₆₇	—	2	2	no	—
273	C₂³×C₆₇	—	2	4	no	—
274	C₂³×C₆₇	—	2	4	no	—
275	C₂³×C₆₇	—	2	4	no	—
276	C₂³×C₆₇	—	2	4	no	—
277	C₂³×C₆₇	—	2	4	no	—
278	C₂³×C₆₇	—	2	4	no	—
279	C₂³×C₆₇	—	2	4	no	—
280	C₂³×C₆₇	C₂×#41	1	4	yes	—
281	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
282	Q₈×C₆₇	—	2	8	no	—
283	D₄×C₆₇	—	2	8	no	—
284	D₄×C₆₇	—	4	16	no	—
285	D₄×C₆₇	—	4	16	no	—
286	D₄×C₆₇	—	4	8	no	—
287	D₄×C₆₇	—	4	8	no	—
288	D₄×C₆₇	—	4	8	no	—
289	C₂×C₄×C₆₇	—	2	1	no	—
290	C₂×C₄×C₆₇	—	2	2	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	2	1	no	—
293	C₂×C₄×C₆₇	—	2	2	no	—
294	C₂×C₄×C₆₇	—	2	2	no	—
295	C₂×C₄×C₆₇	—	2	1	no	—
296	C₂×C₄×C₆₇	—	2	2	no	—
297	C₂×C₄×C₆₇	—	2	1	no	—
298	C₂×C₄×C₆₇	—	2	1	no	—
299	C₂×C₄×C₆₇	—	2	1	no	—
300	C₂×C₄×C₆₇	—	2	2	no	—
301	C₂×C₄×C₆₇	—	2	2	no	—
302	C₂×C₄×C₆₇	—	2	2	no	—
303	C₂×C₄×C₆₇	—	2	2	no	—
304	C₂×C₄×C₆₇	—	2	4	no	—
305	C₂×C₄×C₆₇	—	2	1	no	—
306	C₂×C₄×C₆₇	—	2	2	no	—
307	C₂×C₄×C₆₇	—	2	1	no	—
308	C₂×C₄×C₆₇	—	2	2	no	—
309	C₂×C₄×C₆₇	—	2	2	no	—
310	C₂×C₄×C₆₇	—	2	2	no	—
311	C₂×C₄×C₆₇	—	2	4	no	—
312	C₂×C₄×C₆₇	—	2	2	no	—
313	C₂×C₄×C₆₇	—	2	2	no	—
314	C₂×C₄×C₆₇	—	2	1	no	—
315	C₂×C₄×C₆₇	—	2	2	no	—
316	C₂×C₄×C₆₇	—	2	4	no	—
317	C₂×C₄×C₆₇	—	2	2	no	—
318	C₂×C₄×C₆₇	—	2	2	no	—
319	C₂×C₄×C₆₇	—	4	8	no	—
320	C₂×C₄×C₆₇	—	4	8	no	—
321	C₂×C₄×C₆₇	—	4	8	no	—
322	C₂×C₄×C₆₇	—	4	16	no	—
323	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
324	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
325	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
326	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
327	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
328	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
329	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
330	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
331	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
332	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
333	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
334	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
335	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
336	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
337	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
338	C₂²×C₆₇	—	2	4	no	—
339	C₂²×C₆₇	—	2	4	no	—
340	C₂²×C₆₇	—	2	4	no	—
341	C₂²×C₆₇	—	4	2	no	—
342	C₂²×C₆₇	—	4	2	no	—
343	C₂²×C₆₇	—	4	2	no	—
344	C₂²×C₆₇	—	4	2	no	—
345	C₂²×C₆₇	—	4	4	no	—
346	C₂²×C₆₇	—	4	4	no	—
347	C₂²×C₆₇	—	4	4	no	—
348	C₂²×C₆₇	—	4	4	no	—
349	C₂²×C₆₇	—	4	4	no	—
350	C₂²×C₆₇	—	4	4	no	—
351	C₂²×C₆₇	—	4	4	no	—
352	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
353	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
354	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
355	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
356	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
357	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
358	C₂²×C₆₇	#585	1	1	yes	characteristic
359	C₄×C₆₇	—	2	4	no	—
360	C₄×C₆₇	—	2	2	no	—
361	C₄×C₆₇	—	2	2	no	—
362	C₄×C₆₇	—	2	4	no	—
363	C₄×C₆₇	—	2	2	no	—
364	C₄×C₆₇	—	2	4	no	—
365	C₄×C₆₇	—	4	2	no	—
366	C₄×C₆₇	—	4	2	no	—
367	C₄×C₆₇	—	4	2	no	—
368	C₄×C₆₇	—	4	4	no	—
369	C₄×C₆₇	—	4	2	no	—
370	C₄×C₆₇	—	4	1	no	—
371	C₄×C₆₇	#529	1	4	yes	complemented
372	C₄×C₆₇	#532	1	4	yes	—
373	C₄×C₆₇	#550	1	4	yes	—
374	#34934	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
375	C₂×C₆₇	—	4	4	no	—
376	C₂×C₆₇	—	8	4	no	—
377	C₂×C₆₇	C₂×#529	1	1	yes	characteristic
378	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
379	C₂×C₆₇	#3460	1	1	yes	characteristic
380	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
381	C₂×C₆₇	D₄×#41	1	1	yes	characteristic
382	C₂×C₆₇	#3480	1	1	yes	characteristic
383	C₂×C₆₇	#3481	1	1	yes	characteristic
384	C₂×#529	C₂×C₆₇	1	2	yes	complemented
385	#2554	C₂×C₆₇	1	4	yes	complemented
386	#2552	C₂×C₆₇	1	1	yes	characteristic, complemented
387	#2548	C₂×C₆₇	1	1	yes	characteristic, complemented
388	#2541	C₂×C₆₇	1	1	yes	characteristic
389	#2538	C₂×C₆₇	1	1	yes	characteristic, complemented
390	#2512	C₂×C₆₇	1	2	yes	complemented
391	#2490	C₂×C₆₇	1	2	yes	complemented
392	#2445	C₂×C₆₇	1	2	yes	complemented
393	#2426	C₂×C₆₇	1	2	yes	complemented
394	#2392	C₂×C₆₇	1	4	yes	complemented
395	C₄×#127	C₂×C₆₇	1	2	yes	complemented
396	C₄×#127	C₂×C₆₇	1	1	yes	characteristic, complemented
397	C₄×#121	C₂×C₆₇	1	2	yes	complemented
398	#2293	C₂×C₆₇	1	2	yes	complemented
399	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
400	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
401	C₆₇	#34934	1	1	yes	characteristic, direct product factor, Sylow
402	C₂×#127	C₂²×C₆₇	1	2	yes	—
403	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
404	#529	—	2	16	no	—
405	C₂×#126	C₂²×C₆₇	1	4	yes	—
406	C₂×#125	C₂²×C₆₇	1	2	yes	—
407	C₂×#124	C₂²×C₆₇	1	2	yes	—
408	C₂×#123	C₂²×C₆₇	1	4	yes	—
409	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
410	C₂×#121	C₂²×C₆₇	1	2	yes	—
411	C₂×#121	C₂²×C₆₇	1	2	yes	—
412	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
413	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
414	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
415	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
416	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
417	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
418	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
419	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
420	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
421	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
422	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
423	C₂×#117	C₂²×C₆₇	1	4	yes	—
424	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
425	#398	C₂²×C₆₇	1	4	yes	—
426	#397	C₂²×C₆₇	1	4	yes	—
427	#396	C₂²×C₆₇	1	4	yes	—
428	#396	C₂²×C₆₇	1	4	yes	—
429	#394	C₂²×C₆₇	1	4	yes	—
430	#394	C₂²×C₆₇	1	4	yes	—
431	#394	C₂²×C₆₇	1	4	yes	—
432	#392	C₂²×C₆₇	1	4	yes	—
433	#390	C₂²×C₆₇	1	2	yes	—
434	#390	C₂²×C₆₇	1	4	yes	—
435	#390	C₂²×C₆₇	1	2	yes	—
436	#390	C₂²×C₆₇	1	2	yes	—
437	#390	C₂²×C₆₇	1	4	yes	—
438	#390	C₂²×C₆₇	1	2	yes	—
439	#390	C₂²×C₆₇	1	2	yes	—
440	#390	C₂²×C₆₇	1	2	yes	—
441	#390	C₂²×C₆₇	1	2	yes	—
442	#390	C₂²×C₆₇	1	2	yes	—
443	#389	C₂²×C₆₇	1	2	yes	—
444	#389	C₂²×C₆₇	1	2	yes	—
445	#389	C₂²×C₆₇	1	2	yes	—
446	#388	C₂²×C₆₇	1	4	yes	—
447	#388	C₂²×C₆₇	1	4	yes	—
448	#388	C₂²×C₆₇	1	4	yes	—
449	#386	C₂²×C₆₇	1	4	yes	—
450	#384	C₂²×C₆₇	1	2	yes	—
451	#384	C₂²×C₆₇	1	1	yes	characteristic
452	#380	C₂²×C₆₇	1	2	yes	—
453	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
454	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
455	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
456	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
457	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
458	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
459	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
460	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
461	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
462	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
463	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
464	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
465	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
466	C₄³	C₂²×C₆₇	1	1	yes	characteristic
467	C₄³	C₂²×C₆₇	1	2	yes	—
468	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
469	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
470	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
471	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
472	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
473	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
474	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
475	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
476	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
477	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
478	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
479	#127	—	2	8	no	—
480	#127	—	2	4	no	—
481	#126	—	2	16	no	—
482	#125	—	2	8	no	—
483	#124	—	2	8	no	—
484	#123	—	2	16	no	—
485	C₂²⋊Q₈	—	2	8	no	—
486	#121	—	2	8	no	—
487	#121	—	2	8	no	—
488	C₄×Q₈	—	2	4	no	—
489	C₄×Q₈	—	2	4	no	—
490	C₄×Q₈	D₄×C₆₇	1	8	yes	complemented
491	C₄×D₄	—	2	4	no	—
492	C₄×D₄	—	2	8	no	—
493	C₄×D₄	—	2	16	no	—
494	C₄×D₄	—	2	4	no	—
495	C₄×D₄	—	2	8	no	—
496	C₄×D₄	—	2	4	no	—
497	C₄×D₄	—	4	4	no	—
498	C₄×D₄	—	4	8	no	—
499	#117	—	2	16	no	—
500	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
501	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
502	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
503	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
504	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
505	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
506	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
507	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
508	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
509	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
510	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
511	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
512	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
513	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
514	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
515	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
516	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
517	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
518	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
519	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
520	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
521	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
522	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
523	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
524	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
525	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
526	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
527	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
528	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
529	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
530	C₂×C₄²	—	2	8	no	—
531	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
532	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
533	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
534	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
535	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
536	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
537	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
538	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
539	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
540	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
541	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
542	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
543	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
544	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
545	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
546	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
547	#95	C₂³×C₆₇	1	2	yes	—
548	#95	C₂³×C₆₇	1	2	yes	—
549	#95	C₂³×C₆₇	1	2	yes	—
550	#95	C₂³×C₆₇	1	4	yes	—
551	#95	C₂³×C₆₇	1	4	yes	—
552	#95	C₂³×C₆₇	1	2	yes	—
553	#95	C₂³×C₆₇	1	2	yes	—
554	#95	C₂³×C₆₇	1	2	yes	—
555	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
556	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
557	C₂×Q₈	—	2	2	no	—
558	C₂×Q₈	—	2	2	no	—
559	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
560	C₂×D₄	—	2	8	no	—
561	C₂×D₄	—	2	8	no	—
562	C₂×D₄	—	2	8	no	—
563	C₂×D₄	—	2	4	no	—
564	C₂×D₄	—	2	2	no	—
565	C₂×D₄	—	2	2	no	—
566	C₂×D₄	—	2	4	no	—
567	C₂×D₄	—	2	8	no	—
568	C₂×D₄	—	2	4	no	—
569	C₂×D₄	—	2	4	no	—
570	C₂×D₄	—	2	4	no	—
571	C₂×D₄	—	2	4	no	—
572	C₂×D₄	—	2	4	no	—
573	C₂×D₄	—	4	4	no	—
574	C₂×D₄	—	4	2	no	—
575	C₂×D₄	—	4	4	no	—
576	C₂×D₄	—	4	2	no	—
577	C₂×D₄	#41×C₆₇	1	4	yes	—
578	C₂²×C₄	—	2	4	no	—
579	C₂²×C₄	—	2	4	no	—
580	C₂²×C₄	—	2	4	no	—
581	C₂²×C₄	—	2	4	no	—
582	C₂²×C₄	—	2	8	no	—
583	C₂²×C₄	—	2	4	no	—
584	C₂²×C₄	—	2	4	no	—
585	C₂²×C₄	—	2	8	no	—
586	C₂²×C₄	—	2	4	no	—
587	C₂²×C₄	—	2	4	no	—
588	C₂²×C₄	—	2	8	no	—
589	C₂²×C₄	—	4	2	no	—
590	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
591	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
592	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
593	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
594	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
595	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
596	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
597	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
598	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
599	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
600	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
601	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
602	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
603	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
604	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
605	C₄⋊C₄	—	2	8	no	—
606	C₄⋊C₄	—	2	8	no	—
607	C₄⋊C₄	—	2	4	no	—
608	C₄⋊C₄	—	2	8	no	—
609	C₄⋊C₄	—	2	2	no	—
610	C₄⋊C₄	—	2	2	no	—
611	C₄⋊C₄	—	2	2	no	—
612	C₄⋊C₄	—	2	4	no	—
613	C₄⋊C₄	—	2	4	no	—
614	C₄⋊C₄	—	2	4	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	—	4	4	no	—
617	C₄⋊C₄	—	4	2	no	—
618	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
619	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
620	C₄⋊C₄	#41×C₆₇	1	4	yes	—
621	C₂²⋊C₄	—	2	4	no	—
622	C₂²⋊C₄	—	2	8	no	—
623	C₂²⋊C₄	—	2	8	no	—
624	C₂²⋊C₄	—	2	8	no	—
625	C₂²⋊C₄	—	2	8	no	—
626	C₂²⋊C₄	—	2	8	no	—
627	C₂²⋊C₄	—	2	4	no	—
628	C₂²⋊C₄	—	2	8	no	—
629	C₂²⋊C₄	—	2	4	no	—
630	C₂²⋊C₄	—	2	8	no	—
631	C₂²⋊C₄	—	4	4	no	—
632	C₂²⋊C₄	—	4	4	no	—
633	C₂²⋊C₄	—	4	4	no	—
634	C₂²⋊C₄	—	4	4	no	—
635	C₄²	—	2	4	no	—
636	C₄²	—	2	4	no	—
637	C₄²	—	2	4	no	—
638	C₄²	—	2	2	no	—
639	C₄²	—	2	4	no	—
640	C₄²	—	2	2	no	—
641	C₄²	—	2	4	no	—
642	C₄²	—	2	4	no	—
643	C₄²	—	2	2	no	—
644	C₄²	—	2	4	no	—
645	C₄²	—	4	1	no	—
646	C₄²	—	4	2	no	—
647	C₄²	C₂×D₄×C₆₇	1	4	yes	—
648	C₄²	C₂×D₄×C₆₇	1	8	yes	—
649	C₄²	#41×C₆₇	1	4	yes	—
650	C₄²	#41×C₆₇	1	4	yes	—
651	C₂³	—	2	2	no	—
652	C₂³	—	2	2	no	—
653	C₂³	—	2	2	no	—
654	C₂³	—	2	2	no	—
655	C₂³	—	2	2	no	—
656	C₂³	—	2	2	no	—
657	C₂³	—	2	4	no	—
658	C₂³	—	2	4	no	—
659	C₂³	—	2	4	no	—
660	C₂³	—	2	4	no	—
661	C₂³	—	2	4	no	—
662	C₂³	—	2	4	no	—
663	C₂³	—	2	4	no	—
664	C₂³	C₂×#41×C₆₇	1	4	yes	—
665	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
666	Q₈	—	2	8	no	—
667	D₄	—	2	8	no	—
668	D₄	—	4	16	no	—
669	D₄	—	4	16	no	—
670	D₄	—	4	8	no	—
671	D₄	—	4	8	no	—
672	D₄	—	4	8	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	1	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	2	2	no	—
679	C₂×C₄	—	2	1	no	—
680	C₂×C₄	—	2	2	no	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	2	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	4	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	4	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	2	2	no	—
698	C₂×C₄	—	2	1	no	—
699	C₂×C₄	—	2	2	no	—
700	C₂×C₄	—	2	4	no	—
701	C₂×C₄	—	2	2	no	—
702	C₂×C₄	—	2	2	no	—
703	C₂×C₄	—	4	8	no	—
704	C₂×C₄	—	4	8	no	—
705	C₂×C₄	—	4	8	no	—
706	C₂×C₄	—	4	16	no	—
707	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
708	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
709	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
710	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
711	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
712	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
713	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
714	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
715	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
716	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
717	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
718	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
719	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
720	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
721	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
722	C₂²	—	2	4	no	—
723	C₂²	—	2	4	no	—
724	C₂²	—	2	4	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	2	no	—
727	C₂²	—	4	2	no	—
728	C₂²	—	4	2	no	—
729	C₂²	—	4	4	no	—
730	C₂²	—	4	4	no	—
731	C₂²	—	4	4	no	—
732	C₂²	—	4	4	no	—
733	C₂²	—	4	4	no	—
734	C₂²	—	4	4	no	—
735	C₂²	—	4	4	no	—
736	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
737	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
738	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
739	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
740	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
741	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
742	C₂²	#585×C₆₇	1	1	yes	characteristic
743	C₄	—	2	4	no	—
744	C₄	—	2	2	no	—
745	C₄	—	2	2	no	—
746	C₄	—	2	4	no	—
747	C₄	—	2	2	no	—
748	C₄	—	2	4	no	—
749	C₄	—	4	2	no	—
750	C₄	—	4	2	no	—
751	C₄	—	4	2	no	—
752	C₄	—	4	4	no	—
753	C₄	—	4	2	no	—
754	C₄	—	4	1	no	—
755	C₄	#529×C₆₇	1	4	yes	complemented
756	C₄	#532×C₆₇	1	4	yes	—
757	C₄	#550×C₆₇	1	4	yes	—
758	C₂	—	4	4	no	—
759	C₂	—	8	4	no	—
760	C₂	C₂×#529×C₆₇	1	1	yes	characteristic
761	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
762	C₂	C₆₇×#3460	1	1	yes	characteristic
763	C₂	C₂×#550×C₆₇	1	1	yes	characteristic
764	C₂	D₄×#41×C₆₇	1	1	yes	characteristic
765	C₂	C₆₇×#3480	1	1	yes	characteristic
766	C₂	C₆₇×#3481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₆₇×#2541	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#529×C₆₇	C₂	semidirect
2. C₆₇×#2554	C₂	semidirect
3. C₆₇×#2552	C₂	semidirect
4. C₆₇×#2548	C₂	semidirect
6. C₆₇×#2538	C₂	semidirect
7. C₆₇×#2512	C₂	semidirect
8. C₆₇×#2490	C₂	semidirect
9. C₆₇×#2445	C₂	semidirect
10. C₆₇×#2426	C₂	semidirect
11. C₆₇×#2392	C₂	semidirect
12. C₄×#127×C₆₇	C₂	semidirect
13. C₄×#127×C₆₇	C₂	semidirect
14. C₄×#121×C₆₇	C₂	semidirect
15. C₆₇×#2293	C₂	semidirect
16. C₄²×Q₈×C₆₇	C₂	semidirect
17. C₄²×D₄×C₆₇	C₂	semidirect
106. C₄×Q₈×C₆₇	D₄	semidirect
371. C₄×C₆₇	#529	semidirect
374. #34934	C₆₇	direct (N × H)
384. C₂×#529	C₂×C₆₇	semidirect
385. #2554	C₂×C₆₇	semidirect
386. #2552	C₂×C₆₇	semidirect
387. #2548	C₂×C₆₇	semidirect
389. #2538	C₂×C₆₇	semidirect
390. #2512	C₂×C₆₇	semidirect
391. #2490	C₂×C₆₇	semidirect
392. #2445	C₂×C₆₇	semidirect
393. #2426	C₂×C₆₇	semidirect
394. #2392	C₂×C₆₇	semidirect
395. C₄×#127	C₂×C₆₇	semidirect
396. C₄×#127	C₂×C₆₇	semidirect
397. C₄×#121	C₂×C₆₇	semidirect
398. #2293	C₂×C₆₇	semidirect
399. C₄²×Q₈	C₂×C₆₇	semidirect
400. C₄²×D₄	C₂×C₆₇	semidirect
401. C₆₇	#34934	direct (N × H)
490. C₄×Q₈	D₄×C₆₇	semidirect
755. C₄	#529×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₆₇×#2554	2
C₆₇×#2552	2
C₆₇×#2548	2
C₆₇×#2541	2
C₆₇×#2538	2
C₆₇×#2512	2
C₆₇×#2490	2
C₆₇×#2445	2
C₆₇×#2426	2
C₆₇×#2392	2
C₄×#127×C₆₇	2
C₄×#121×C₆₇	2
C₆₇×#2293	2
C₄²×Q₈×C₆₇	2
C₄²×D₄×C₆₇	2
#34934	67

Subgroups — C67×#34934 #30714185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34934 #30714185