Subgroups — C₆₇×#22529 #30701780

Order 17152.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 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	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1995	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1994	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1986	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1986	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1806	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1806	C₂	1	2	yes	maximal
10	C₆₇×#1759	C₂	1	2	yes	maximal
11	C₄×#104×C₆₇	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#557×C₆₇	C₄	1	2	yes	complemented
14	#557×C₆₇	C₄	1	1	yes	characteristic, complemented
15	#554×C₆₇	—	2	2	no	—
16	#554×C₆₇	—	2	1	no	—
17	#554×C₆₇	C₄	1	1	yes	characteristic, complemented
18	#554×C₆₇	C₄	1	2	yes	complemented
19	D₄×Q₈×C₆₇	—	2	1	no	—
20	D₄×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#533×C₆₇	—	2	1	no	—
22	#533×C₆₇	—	2	2	no	—
23	#533×C₆₇	C₂²	1	1	yes	characteristic
24	#533×C₆₇	C₂²	1	2	yes	—
25	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×#41×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	#446×C₆₇	—	2	1	no	—
30	#446×C₆₇	—	2	2	no	—
31	C₈×Q₈×C₆₇	—	2	1	no	—
32	#423×C₆₇	C₂²	1	1	yes	characteristic
33	#423×C₆₇	C₂²	1	2	yes	—
34	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#104×C₆₇	C₂²	1	2	yes	—
36	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	—	2	1	no	—
40	#389×C₆₇	—	2	1	no	—
41	#387×C₆₇	—	2	2	no	—
42	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
44	#376×C₆₇	C₂²	1	2	yes	—
45	C₄³×C₆₇	C₂²	1	1	yes	characteristic
46	#339×C₆₇	C₂²	1	1	yes	characteristic
47	#339×C₆₇	C₂²	1	2	yes	—
48	#339×C₆₇	C₂²	1	2	yes	—
49	#339×C₆₇	C₂²	1	2	yes	—
50	#339×C₆₇	C₂²	1	2	yes	—
51	#339×C₆₇	C₂²	1	2	yes	—
52	#339×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₆₇	—	2	1	no	—
55	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
57	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₇	—	2	1	no	—
60	C₄⋊Q₈×C₆₇	—	2	2	no	—
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	—	2	1	no	—
63	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
67	#125×C₆₇	—	2	2	no	—
68	#125×C₆₇	—	2	2	no	—
69	#125×C₆₇	—	2	2	no	—
70	#125×C₆₇	C₂×C₄	1	2	yes	—
71	#125×C₆₇	C₂×C₄	1	2	yes	—
72	#125×C₆₇	C₂×C₄	1	2	yes	—
73	C₂²⋊Q₈×C₆₇	—	2	1	no	—
74	C₂²⋊Q₈×C₆₇	—	2	2	no	—
75	C₂²⋊Q₈×C₆₇	—	2	1	no	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	—	2	1	no	—
78	C₂²⋊Q₈×C₆₇	—	2	2	no	—
79	C₂²⋊Q₈×C₆₇	—	2	2	no	—
80	C₂²⋊Q₈×C₆₇	—	2	1	no	—
81	C₂²⋊Q₈×C₆₇	—	2	2	no	—
82	C₂²⋊Q₈×C₆₇	—	2	2	no	—
83	C₂²⋊Q₈×C₆₇	—	2	2	no	—
84	C₄×Q₈×C₆₇	—	2	1	no	—
85	C₄×Q₈×C₆₇	—	2	2	no	—
86	C₄×Q₈×C₆₇	—	2	1	no	—
87	C₄×Q₈×C₆₇	—	2	1	no	—
88	C₄×Q₈×C₆₇	—	2	2	no	—
89	C₄×Q₈×C₆₇	—	2	2	no	—
90	C₄×Q₈×C₆₇	—	2	1	no	—
91	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
95	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
98	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
100	C₄×Q₈×C₆₇	D₄	1	2	yes	—
101	C₄×D₄×C₆₇	—	2	1	no	—
102	C₄×D₄×C₆₇	—	2	2	no	—
103	C₄×D₄×C₆₇	—	2	1	no	—
104	C₄×D₄×C₆₇	—	2	2	no	—
105	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
109	#117×C₆₇	—	2	2	no	—
110	#117×C₆₇	—	2	1	no	—
111	#117×C₆₇	C₂³	1	1	yes	characteristic
112	#117×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
118	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	—	2	1	no	—
120	C₂×C₄²×C₆₇	—	2	2	no	—
121	C₂×C₄²×C₆₇	—	2	1	no	—
122	C₂×C₄²×C₆₇	—	2	1	no	—
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
128	C₄⋊C₈×C₆₇	—	2	1	no	—
129	C₄⋊C₈×C₆₇	—	2	2	no	—
130	C₄⋊C₈×C₆₇	—	2	2	no	—
131	C₄⋊C₈×C₆₇	—	2	1	no	—
132	#104×C₆₇	—	2	2	no	—
133	#104×C₆₇	—	2	4	no	—
134	#104×C₆₇	Q₈	1	4	yes	complemented
135	#97×C₆₇	—	2	2	no	—
136	#97×C₆₇	—	2	1	no	—
137	C₄×C₈×C₆₇	—	2	1	no	—
138	C₄×C₈×C₆₇	—	2	2	no	—
139	#95×C₆₇	—	2	2	no	—
140	#95×C₆₇	—	2	2	no	—
141	#95×C₆₇	—	2	1	no	—
142	#95×C₆₇	—	2	2	no	—
143	#41×C₆₇	—	2	1	no	—
144	#41×C₆₇	—	2	1	no	—
145	#41×C₆₇	C₂×Q₈	1	2	yes	—
146	#41×C₆₇	#41	1	2	yes	—
147	C₂×Q₈×C₆₇	—	2	2	no	—
148	C₂×Q₈×C₆₇	—	2	1	no	—
149	C₂×Q₈×C₆₇	—	2	2	no	—
150	C₂×Q₈×C₆₇	—	2	1	no	—
151	C₂×Q₈×C₆₇	—	4	2	no	—
152	C₂×Q₈×C₆₇	—	4	4	no	—
153	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
154	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
155	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
156	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂×D₄×C₆₇	—	2	1	no	—
158	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	1	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	1	no	—
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	—	2	2	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	M₁₆×C₆₇	—	2	1	no	—
173	M₁₆×C₆₇	—	2	1	no	—
174	M₁₆×C₆₇	—	2	2	no	—
175	M₁₆×C₆₇	—	2	2	no	—
176	M₁₆×C₆₇	—	2	1	no	—
177	M₁₆×C₆₇	C₂×Q₈	1	2	yes	complemented
178	C₂×C₈×C₆₇	—	2	1	no	—
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	1	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	1	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	4	2	no	—
196	C₄⋊C₄×C₆₇	—	4	4	no	—
197	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
198	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
200	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
201	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
202	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
203	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
204	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
205	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
206	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
207	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
208	C₂²⋊C₄×C₆₇	—	2	1	no	—
209	C₂²⋊C₄×C₆₇	—	2	2	no	—
210	C₂²⋊C₄×C₆₇	—	2	1	no	—
211	C₂²⋊C₄×C₆₇	—	2	2	no	—
212	C₂²⋊C₄×C₆₇	—	2	2	no	—
213	C₂²⋊C₄×C₆₇	—	2	1	no	—
214	C₄²×C₆₇	—	2	1	no	—
215	C₄²×C₆₇	—	2	2	no	—
216	C₄²×C₆₇	—	2	2	no	—
217	C₄²×C₆₇	—	2	1	no	—
218	C₄²×C₆₇	—	2	1	no	—
219	C₄²×C₆₇	—	2	2	no	—
220	C₄²×C₆₇	—	2	1	no	—
221	C₄²×C₆₇	—	2	1	no	—
222	C₄²×C₆₇	—	2	2	no	—
223	C₄²×C₆₇	—	2	1	no	—
224	C₄²×C₆₇	—	2	2	no	—
225	C₄²×C₆₇	—	4	2	no	—
226	C₄²×C₆₇	—	4	2	no	—
227	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
228	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
229	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
230	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
231	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
232	C₄²×C₆₇	C₂×D₄	1	2	yes	—
233	C₄²×C₆₇	C₂×D₄	1	2	yes	—
234	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₆₇	C₂×Q₈	1	2	yes	complemented
236	C₂³×C₆₇	—	2	1	no	—
237	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
238	Q₈×C₆₇	—	2	1	no	—
239	Q₈×C₆₇	—	2	2	no	—
240	Q₈×C₆₇	—	2	2	no	—
241	Q₈×C₆₇	—	4	2	no	—
242	Q₈×C₆₇	—	4	4	no	—
243	Q₈×C₆₇	—	4	2	no	—
244	Q₈×C₆₇	—	8	1	no	—
245	Q₈×C₆₇	—	8	2	no	—
246	Q₈×C₆₇	C₄×Q₈	1	2	yes	—
247	D₄×C₆₇	—	2	2	no	—
248	D₄×C₆₇	—	2	2	no	—
249	D₄×C₆₇	—	2	1	no	—
250	D₄×C₆₇	C₄×Q₈	1	2	yes	—
251	C₂×C₄×C₆₇	—	2	1	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	1	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	4	1	no	—
262	C₂×C₄×C₆₇	—	4	2	no	—
263	C₂×C₄×C₆₇	—	4	2	no	—
264	C₂×C₄×C₆₇	—	4	4	no	—
265	C₂×C₄×C₆₇	—	4	1	no	—
266	C₂×C₄×C₆₇	—	4	1	no	—
267	C₂×C₄×C₆₇	—	4	1	no	—
268	C₂×C₄×C₆₇	—	4	1	no	—
269	C₂×C₄×C₆₇	—	4	2	no	—
270	C₂×C₄×C₆₇	—	4	2	no	—
271	C₂×C₄×C₆₇	—	4	2	no	—
272	C₂×C₄×C₆₇	—	4	2	no	—
273	C₂×C₄×C₆₇	—	8	1	no	—
274	C₂×C₄×C₆₇	—	8	2	no	—
275	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
278	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
280	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
286	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
288	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
289	C₈×C₆₇	—	2	2	no	—
290	C₈×C₆₇	—	4	2	no	—
291	C₈×C₆₇	—	4	1	no	—
292	C₂²×C₆₇	—	2	1	no	—
293	C₂²×C₆₇	—	2	1	no	—
294	C₂²×C₆₇	—	2	2	no	—
295	C₂²×C₆₇	—	4	1	no	—
296	C₂²×C₆₇	—	4	1	no	—
297	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
298	C₂²×C₆₇	#548	1	1	yes	characteristic
299	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
300	C₄×C₆₇	—	2	2	no	—
301	C₄×C₆₇	—	2	1	no	—
302	C₄×C₆₇	—	2	2	no	—
303	C₄×C₆₇	—	2	2	no	—
304	C₄×C₆₇	—	2	1	no	—
305	C₄×C₆₇	—	2	2	no	—
306	C₄×C₆₇	—	2	1	no	—
307	C₄×C₆₇	—	2	2	no	—
308	C₄×C₆₇	—	4	2	no	—
309	C₄×C₆₇	—	4	2	no	—
310	C₄×C₆₇	—	4	2	no	—
311	C₄×C₆₇	—	4	1	no	—
312	C₄×C₆₇	—	4	2	no	—
313	C₄×C₆₇	—	4	1	no	—
314	C₄×C₆₇	—	8	2	no	—
315	C₄×C₆₇	#421	1	2	yes	—
316	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
317	C₄×C₆₇	#519	1	1	yes	characteristic
318	C₄×C₆₇	#548	1	1	yes	characteristic
319	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
320	#22529	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
321	C₂×C₆₇	—	2	1	no	—
322	C₂×C₆₇	—	2	1	no	—
323	C₂×C₆₇	—	4	2	no	—
324	C₂×C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
325	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
326	C₂×C₆₇	#2901	1	1	yes	characteristic
327	Q₈×#41	C₂×C₆₇	1	1	yes	characteristic
328	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
329	#2295	C₂×C₆₇	1	1	yes	characteristic, complemented
330	#1995	C₂×C₆₇	1	2	yes	complemented
331	#1994	C₂×C₆₇	1	1	yes	characteristic, complemented
332	#1986	C₂×C₆₇	1	1	yes	characteristic, complemented
333	#1986	C₂×C₆₇	1	2	yes	complemented
334	#1806	C₂×C₆₇	1	1	yes	characteristic
335	#1806	C₂×C₆₇	1	2	yes	—
336	#1759	C₂×C₆₇	1	2	yes	—
337	C₄×#104	C₂×C₆₇	1	1	yes	characteristic
338	C₆₇	#22529	1	1	yes	characteristic, direct product factor, Sylow
339	Q₈²	C₄×C₆₇	1	1	yes	characteristic, complemented
340	#557	C₄×C₆₇	1	2	yes	complemented
341	#557	C₄×C₆₇	1	1	yes	characteristic, complemented
342	#554	—	2	2	no	—
343	#554	—	2	1	no	—
344	#554	C₄×C₆₇	1	1	yes	characteristic, complemented
345	#554	C₄×C₆₇	1	2	yes	complemented
346	D₄×Q₈	—	2	1	no	—
347	D₄×Q₈	C₄×C₆₇	1	1	yes	characteristic, complemented
348	#533	—	2	1	no	—
349	#533	—	2	2	no	—
350	#533	C₂²×C₆₇	1	1	yes	characteristic
351	#533	C₂²×C₆₇	1	2	yes	—
352	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
353	C₄×#41	C₂²×C₆₇	1	2	yes	—
354	C₂×C₄×Q₈	—	2	1	no	—
355	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
356	#446	—	2	1	no	—
357	#446	—	2	2	no	—
358	C₈×Q₈	—	2	1	no	—
359	#423	C₂²×C₆₇	1	1	yes	characteristic
360	#423	C₂²×C₆₇	1	2	yes	—
361	C₂×#104	C₂²×C₆₇	1	1	yes	characteristic
362	C₂×#104	C₂²×C₆₇	1	2	yes	—
363	C₂×#104	C₂²×C₆₇	1	1	yes	characteristic
364	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
365	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
366	#391	—	2	1	no	—
367	#389	—	2	1	no	—
368	#387	—	2	2	no	—
369	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
370	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
371	#376	C₂²×C₆₇	1	2	yes	—
372	C₄³	C₂²×C₆₇	1	1	yes	characteristic
373	#339	C₂²×C₆₇	1	2	yes	—
374	#339	C₂²×C₆₇	1	2	yes	—
375	#339	C₂²×C₆₇	1	2	yes	—
376	#339	C₂²×C₆₇	1	1	yes	characteristic
377	#339	C₂²×C₆₇	1	2	yes	—
378	#339	C₂²×C₆₇	1	2	yes	—
379	#339	C₂²×C₆₇	1	1	yes	characteristic
380	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
381	C₂²×Q₈	—	2	1	no	—
382	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
383	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
384	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
385	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
386	C₄⋊Q₈	—	2	1	no	—
387	C₄⋊Q₈	—	2	2	no	—
388	C₄⋊Q₈	—	2	2	no	—
389	C₄⋊Q₈	—	2	1	no	—
390	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
391	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
392	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
393	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
394	#125	—	2	2	no	—
395	#125	—	2	2	no	—
396	#125	—	2	2	no	—
397	#125	C₂×C₄×C₆₇	1	2	yes	—
398	#125	C₂×C₄×C₆₇	1	2	yes	—
399	#125	C₂×C₄×C₆₇	1	2	yes	—
400	C₂²⋊Q₈	—	2	1	no	—
401	C₂²⋊Q₈	—	2	2	no	—
402	C₂²⋊Q₈	—	2	1	no	—
403	C₂²⋊Q₈	—	2	2	no	—
404	C₂²⋊Q₈	—	2	1	no	—
405	C₂²⋊Q₈	—	2	2	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	1	no	—
408	C₂²⋊Q₈	—	2	2	no	—
409	C₂²⋊Q₈	—	2	2	no	—
410	C₂²⋊Q₈	—	2	2	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×Q₈	—	2	2	no	—
413	C₄×Q₈	—	2	1	no	—
414	C₄×Q₈	—	2	1	no	—
415	C₄×Q₈	—	2	2	no	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	1	no	—
418	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
419	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
420	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
421	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
422	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
423	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
424	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
425	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
426	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
427	C₄×Q₈	D₄×C₆₇	1	2	yes	—
428	C₄×D₄	—	2	1	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	1	no	—
431	C₄×D₄	—	2	2	no	—
432	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
433	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
434	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
435	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
436	#117	—	2	2	no	—
437	#117	—	2	1	no	—
438	#117	C₂³×C₆₇	1	1	yes	characteristic
439	#117	C₂³×C₆₇	1	2	yes	—
440	C₂×C₄⋊C₄	—	2	2	no	—
441	C₂×C₄⋊C₄	—	2	1	no	—
442	C₂×C₄⋊C₄	—	2	2	no	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
445	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	—	2	1	no	—
447	C₂×C₄²	—	2	2	no	—
448	C₂×C₄²	—	2	1	no	—
449	C₂×C₄²	—	2	1	no	—
450	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
452	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
453	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
455	C₄⋊C₈	—	2	1	no	—
456	C₄⋊C₈	—	2	2	no	—
457	C₄⋊C₈	—	2	2	no	—
458	C₄⋊C₈	—	2	1	no	—
459	#104	—	2	2	no	—
460	#104	—	2	4	no	—
461	#104	Q₈×C₆₇	1	4	yes	complemented
462	#97	—	2	2	no	—
463	#97	—	2	1	no	—
464	C₄×C₈	—	2	1	no	—
465	C₄×C₈	—	2	2	no	—
466	#95	—	2	2	no	—
467	#95	—	2	2	no	—
468	#95	—	2	1	no	—
469	#95	—	2	2	no	—
470	#41	—	2	1	no	—
471	#41	—	2	1	no	—
472	#41	C₂×Q₈×C₆₇	1	2	yes	—
473	#41	#41×C₆₇	1	2	yes	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	2	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	—	4	2	no	—
479	C₂×Q₈	—	4	4	no	—
480	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
481	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
482	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
483	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	2	no	—
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
496	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
499	M₁₆	—	2	1	no	—
500	M₁₆	—	2	1	no	—
501	M₁₆	—	2	2	no	—
502	M₁₆	—	2	2	no	—
503	M₁₆	—	2	1	no	—
504	M₁₆	C₂×Q₈×C₆₇	1	2	yes	complemented
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	4	2	no	—
523	C₄⋊C₄	—	4	4	no	—
524	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
525	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
526	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
527	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
528	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
529	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
530	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
531	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
532	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
533	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
534	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
535	C₂²⋊C₄	—	2	1	no	—
536	C₂²⋊C₄	—	2	2	no	—
537	C₂²⋊C₄	—	2	1	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	2	no	—
543	C₄²	—	2	2	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	2	2	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	2	no	—
552	C₄²	—	4	2	no	—
553	C₄²	—	4	2	no	—
554	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
555	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
556	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
557	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
558	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
559	C₄²	C₂×D₄×C₆₇	1	2	yes	—
560	C₄²	C₂×D₄×C₆₇	1	2	yes	—
561	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
562	C₄²	C₂×Q₈×C₆₇	1	2	yes	complemented
563	C₂³	—	2	1	no	—
564	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
565	Q₈	—	2	1	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	2	2	no	—
568	Q₈	—	4	2	no	—
569	Q₈	—	4	4	no	—
570	Q₈	—	4	2	no	—
571	Q₈	—	8	1	no	—
572	Q₈	—	8	2	no	—
573	Q₈	C₄×Q₈×C₆₇	1	2	yes	—
574	D₄	—	2	2	no	—
575	D₄	—	2	2	no	—
576	D₄	—	2	1	no	—
577	D₄	C₄×Q₈×C₆₇	1	2	yes	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	4	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	8	1	no	—
601	C₂×C₄	—	8	2	no	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
603	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
604	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
605	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
606	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
607	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
608	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
609	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
610	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
611	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
612	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
613	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
614	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
615	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
616	C₈	—	2	2	no	—
617	C₈	—	4	1	no	—
618	C₈	—	4	2	no	—
619	C₂²	—	2	1	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
625	C₂²	#548×C₆₇	1	1	yes	characteristic
626	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
627	C₄	—	2	2	no	—
628	C₄	—	2	1	no	—
629	C₄	—	2	2	no	—
630	C₄	—	2	2	no	—
631	C₄	—	2	1	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	2	no	—
635	C₄	—	4	2	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	1	no	—
641	C₄	—	8	2	no	—
642	C₄	#421×C₆₇	1	2	yes	—
643	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
644	C₄	#519×C₆₇	1	1	yes	characteristic
645	C₄	#548×C₆₇	1	1	yes	characteristic
646	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
647	C₂	—	2	1	no	—
648	C₂	—	2	1	no	—
649	C₂	—	4	2	no	—
650	C₂	Q₈×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
651	C₂	C₂×#421×C₆₇	1	1	yes	characteristic
652	C₂	C₆₇×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂×#41×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2295	C₂	semidirect
4. C₆₇×#1995	C₂	semidirect
5. C₆₇×#1994	C₂	semidirect
6. C₆₇×#1986	C₂	semidirect
7. C₆₇×#1986	C₂	semidirect
12. Q₈²×C₆₇	C₄	semidirect
13. #557×C₆₇	C₄	semidirect
14. #557×C₆₇	C₄	semidirect
17. #554×C₆₇	C₄	semidirect
18. #554×C₆₇	C₄	semidirect
20. D₄×Q₈×C₆₇	C₄	semidirect
134. #104×C₆₇	Q₈	semidirect
177. M₁₆×C₆₇	C₂×Q₈	semidirect
235. C₄²×C₆₇	C₂×Q₈	semidirect
320. #22529	C₆₇	direct (N × H)
328. Q₈×M₁₆	C₂×C₆₇	semidirect
329. #2295	C₂×C₆₇	semidirect
330. #1995	C₂×C₆₇	semidirect
331. #1994	C₂×C₆₇	semidirect
332. #1986	C₂×C₆₇	semidirect
333. #1986	C₂×C₆₇	semidirect
338. C₆₇	#22529	direct (N × H)
339. Q₈²	C₄×C₆₇	semidirect
340. #557	C₄×C₆₇	semidirect
341. #557	C₄×C₆₇	semidirect
344. #554	C₄×C₆₇	semidirect
345. #554	C₄×C₆₇	semidirect
347. D₄×Q₈	C₄×C₆₇	semidirect
461. #104	Q₈×C₆₇	semidirect
504. M₁₆	C₂×Q₈×C₆₇	semidirect
562. C₄²	C₂×Q₈×C₆₇	semidirect

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
Q₈×M₁₆×C₆₇	2
C₆₇×#2295	2
C₆₇×#1995	2
C₆₇×#1994	2
C₆₇×#1986	2
C₆₇×#1806	2
C₆₇×#1759	2
C₄×#104×C₆₇	2
#22529	67

Subgroups — C67×#22529 #30701780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22529 #30701780