Subgroups — C₆₇×#61370 #30740621

Order 17152.

1818 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3501	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3389	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3384	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3380	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3380	C₂	1	4	yes	maximal, complemented
6	Q₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3164	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3081	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3077	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2948	C₂	1	2	yes	maximal
13	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	#554×C₆₇	—	2	2	no	—
16	#554×C₆₇	—	2	4	no	—
17	#554×C₆₇	C₂²	1	4	yes	complemented
18	#554×C₆₇	C₂²	1	8	yes	complemented
19	D₄×Q₈×C₆₇	—	2	2	no	—
20	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
21	#544×C₆₇	C₂²	1	4	yes	—
22	#543×C₆₇	—	2	4	no	—
23	#533×C₆₇	C₂²	1	2	yes	—
24	#533×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₆₇	—	2	4	no	—
26	#518×C₆₇	C₂²	1	4	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	#501×C₆₇	C₂²	1	8	yes	complemented
29	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	#500×C₆₇	C₂²	1	4	yes	complemented
31	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	#499×C₆₇	C₂²	1	2	yes	complemented
33	#499×C₆₇	C₂²	1	4	yes	complemented
34	#499×C₆₇	C₂²	1	2	yes	complemented
35	#498×C₆₇	C₂²	1	2	yes	complemented
36	#478×C₆₇	C₂²	1	8	yes	complemented
37	#478×C₆₇	C₂²	1	8	yes	complemented
38	#478×C₆₇	C₂²	1	8	yes	complemented
39	#476×C₆₇	C₂²	1	4	yes	complemented
40	#476×C₆₇	C₂²	1	8	yes	complemented
41	#474×C₆₇	C₂²	1	4	yes	complemented
42	#474×C₆₇	C₂²	1	8	yes	complemented
43	#446×C₆₇	C₂²	1	2	yes	complemented
44	#446×C₆₇	C₂²	1	2	yes	complemented
45	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
46	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
47	#442×C₆₇	C₂²	1	8	yes	—
48	C₄×D₈×C₆₇	C₂²	1	2	yes	—
49	C₄×D₈×C₆₇	C₂²	1	4	yes	—
50	C₄×D₈×C₆₇	C₂²	1	2	yes	—
51	C₄×D₈×C₆₇	C₂²	1	4	yes	—
52	C₄×D₈×C₆₇	C₂²	1	2	yes	—
53	C₄×D₈×C₆₇	C₂²	1	2	yes	—
54	#428×C₆₇	C₂²	1	4	yes	—
55	#427×C₆₇	C₂²	1	2	yes	—
56	#427×C₆₇	C₂²	1	2	yes	—
57	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
58	#424×C₆₇	C₂²	1	2	yes	—
59	#423×C₆₇	C₂²	1	1	yes	characteristic
60	#418×C₆₇	C₂²	1	4	yes	—
61	C₂×#102×C₆₇	C₂²	1	2	yes	—
62	#405×C₆₇	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
64	C₂²×Q₈×C₆₇	—	2	4	no	—
65	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₆₇	—	2	2	no	—
67	C₂×D₈×C₆₇	Q₈	1	8	yes	complemented
68	C₂×D₈×C₆₇	C₂³	1	2	yes	—
69	C₂×D₈×C₆₇	C₂³	1	2	yes	—
70	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈×C₆₇	—	2	4	no	—
74	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
78	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
79	#125×C₆₇	—	2	4	no	—
80	#125×C₆₇	D₄	1	2	yes	—
81	#125×C₆₇	C₂³	1	4	yes	—
82	#125×C₆₇	C₂³	1	8	yes	—
83	C₂²⋊Q₈×C₆₇	—	2	4	no	—
84	C₂²⋊Q₈×C₆₇	—	2	8	no	—
85	C₂²⋊Q₈×C₆₇	—	2	2	no	—
86	C₂²⋊Q₈×C₆₇	—	2	2	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	—	2	8	no	—
89	C₂²⋊Q₈×C₆₇	—	2	8	no	—
90	C₂²⋊Q₈×C₆₇	—	2	8	no	—
91	C₂²⋊Q₈×C₆₇	—	2	8	no	—
92	C₂²⋊Q₈×C₆₇	—	2	8	no	—
93	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
94	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
95	C₄×Q₈×C₆₇	—	2	2	no	—
96	C₄×Q₈×C₆₇	—	2	2	no	—
97	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
98	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
99	C₄×D₄×C₆₇	—	2	4	no	—
100	C₄×D₄×C₆₇	—	2	4	no	—
101	C₄×D₄×C₆₇	—	2	2	no	—
102	C₄×D₄×C₆₇	—	2	2	no	—
103	C₄×D₄×C₆₇	C₂³	1	4	yes	—
104	C₄×D₄×C₆₇	C₂³	1	8	yes	—
105	C₄×D₄×C₆₇	C₂³	1	8	yes	—
106	C₄×D₄×C₆₇	C₂³	1	4	yes	—
107	#117×C₆₇	C₂³	1	4	yes	—
108	#117×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
110	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	#107×C₆₇	C₂³	1	2	yes	—
116	#107×C₆₇	C₂³	1	8	yes	—
117	#107×C₆₇	C₂³	1	2	yes	—
118	#107×C₆₇	C₂³	1	2	yes	—
119	#107×C₆₇	C₂³	1	2	yes	—
120	#107×C₆₇	C₂³	1	2	yes	—
121	#107×C₆₇	C₂³	1	2	yes	—
122	#106×C₆₇	C₂³	1	4	yes	—
123	#106×C₆₇	C₂³	1	8	yes	—
124	#106×C₆₇	C₂³	1	4	yes	—
125	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
126	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
127	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
128	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
129	#102×C₆₇	C₂³	1	8	yes	—
130	#102×C₆₇	C₂³	1	4	yes	—
131	#102×C₆₇	C₂³	1	4	yes	—
132	#102×C₆₇	C₂³	1	8	yes	—
133	#97×C₆₇	C₂³	1	4	yes	—
134	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
135	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
136	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₆₇	C₂³	1	4	yes	—
138	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
139	C₂⁴×C₆₇	—	2	2	no	—
140	C₂×Q₈×C₆₇	—	2	2	no	—
141	C₂×Q₈×C₆₇	—	2	2	no	—
142	C₂×Q₈×C₆₇	—	2	8	no	—
143	C₂×Q₈×C₆₇	—	8	8	no	—
144	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
145	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	2	4	no	—
148	C₂×D₄×C₆₇	C₂×Q₈	1	8	yes	complemented
149	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	8	no	—
153	C₂²×C₄×C₆₇	—	2	8	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	8	4	no	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
159	D₈×C₆₇	C₂×Q₈	1	8	yes	complemented
160	D₈×C₆₇	C₂×Q₈	1	8	yes	complemented
161	M₁₆×C₆₇	—	2	4	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
164	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
166	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	8	no	—
172	C₄⋊C₄×C₆₇	—	2	8	no	—
173	C₄⋊C₄×C₆₇	—	2	8	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	4	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
184	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
186	C₂²⋊C₄×C₆₇	—	2	4	no	—
187	C₂²⋊C₄×C₆₇	—	2	8	no	—
188	C₂²⋊C₄×C₆₇	—	2	8	no	—
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	2	4	no	—
191	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
192	C₄²×C₆₇	—	2	4	no	—
193	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₄²×C₆₇	C₂⁴	1	4	yes	—
198	C₂³×C₆₇	—	2	4	no	—
199	C₂³×C₆₇	—	2	8	no	—
200	C₂³×C₆₇	—	8	2	no	—
201	C₂³×C₆₇	—	8	2	no	—
202	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₆₇	—	2	4	no	—
204	Q₈×C₆₇	—	4	2	no	—
205	Q₈×C₆₇	—	4	4	no	—
206	Q₈×C₆₇	—	8	2	no	—
207	Q₈×C₆₇	—	8	8	no	—
208	D₄×C₆₇	—	2	8	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	C₂²×Q₈	1	8	yes	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	4	8	no	—
217	C₂×C₄×C₆₇	—	8	8	no	—
218	C₂×C₄×C₆₇	—	8	4	no	—
219	C₂×C₄×C₆₇	—	8	8	no	—
220	C₂×C₄×C₆₇	—	8	2	no	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
225	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
229	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
230	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
231	C₈×C₆₇	—	2	1	no	—
232	C₈×C₆₇	—	2	1	no	—
233	C₈×C₆₇	—	2	4	no	—
234	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
235	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
236	C₂²×C₆₇	—	2	1	no	—
237	C₂²×C₆₇	—	2	1	no	—
238	C₂²×C₆₇	—	2	8	no	—
239	C₂²×C₆₇	—	4	4	no	—
240	C₂²×C₆₇	—	4	4	no	—
241	C₂²×C₆₇	—	8	8	no	—
242	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
243	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
244	C₄×C₆₇	—	2	1	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	—	2	4	no	—
247	C₄×C₆₇	—	2	1	no	—
248	C₄×C₆₇	—	2	4	no	—
249	C₄×C₆₇	—	4	4	no	—
250	C₄×C₆₇	—	4	4	no	—
251	C₄×C₆₇	—	4	4	no	—
252	C₄×C₆₇	D₄×Q₈	1	2	yes	—
253	C₄×C₆₇	#576	1	2	yes	—
254	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
255	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
256	#61370	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₆₇	—	2	2	no	—
258	C₂×C₆₇	—	4	8	no	—
259	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
260	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
261	C₂×C₆₇	#3584	1	1	yes	characteristic
262	#3501	C₂×C₆₇	1	2	yes	complemented
263	#3389	C₂×C₆₇	1	4	yes	complemented
264	#3384	C₂×C₆₇	1	4	yes	complemented
265	#3380	C₂×C₆₇	1	2	yes	complemented
266	#3380	C₂×C₆₇	1	4	yes	complemented
267	Q₈×D₈	C₂×C₆₇	1	2	yes	complemented
268	#3164	C₂×C₆₇	1	2	yes	complemented
269	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
270	#3081	C₂×C₆₇	1	4	yes	complemented
271	#3077	C₂×C₆₇	1	2	yes	complemented
272	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
273	#2948	C₂×C₆₇	1	2	yes	—
274	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
275	C₆₇	#61370	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
277	#554	—	2	2	no	—
278	#554	—	2	4	no	—
279	#554	C₂²×C₆₇	1	4	yes	complemented
280	#554	C₂²×C₆₇	1	8	yes	complemented
281	D₄×Q₈	—	2	2	no	—
282	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
283	#544	C₂²×C₆₇	1	4	yes	—
284	#543	—	2	4	no	—
285	#533	C₂²×C₆₇	1	2	yes	—
286	#533	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×C₂²⋊Q₈	—	2	4	no	—
288	#518	C₂²×C₆₇	1	4	yes	—
289	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
290	#501	C₂²×C₆₇	1	8	yes	complemented
291	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
292	#500	C₂²×C₆₇	1	4	yes	complemented
293	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
294	#499	C₂²×C₆₇	1	2	yes	complemented
295	#499	C₂²×C₆₇	1	4	yes	complemented
296	#499	C₂²×C₆₇	1	2	yes	complemented
297	#498	C₂²×C₆₇	1	2	yes	complemented
298	#478	C₂²×C₆₇	1	8	yes	complemented
299	#478	C₂²×C₆₇	1	8	yes	complemented
300	#478	C₂²×C₆₇	1	8	yes	complemented
301	#476	C₂²×C₆₇	1	4	yes	complemented
302	#476	C₂²×C₆₇	1	8	yes	complemented
303	#474	C₂²×C₆₇	1	4	yes	complemented
304	#474	C₂²×C₆₇	1	8	yes	complemented
305	#446	C₂²×C₆₇	1	2	yes	complemented
306	#446	C₂²×C₆₇	1	2	yes	complemented
307	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
308	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
309	#442	C₂²×C₆₇	1	8	yes	—
310	C₄×D₈	C₂²×C₆₇	1	2	yes	—
311	C₄×D₈	C₂²×C₆₇	1	4	yes	—
312	C₄×D₈	C₂²×C₆₇	1	2	yes	—
313	C₄×D₈	C₂²×C₆₇	1	4	yes	—
314	C₄×D₈	C₂²×C₆₇	1	2	yes	—
315	C₄×D₈	C₂²×C₆₇	1	2	yes	—
316	#428	C₂²×C₆₇	1	4	yes	—
317	#427	C₂²×C₆₇	1	2	yes	—
318	#427	C₂²×C₆₇	1	2	yes	—
319	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
320	#424	C₂²×C₆₇	1	2	yes	—
321	#423	C₂²×C₆₇	1	1	yes	characteristic
322	#418	C₂²×C₆₇	1	4	yes	—
323	C₂×#102	C₂²×C₆₇	1	2	yes	—
324	#405	C₂²×C₆₇	1	2	yes	—
325	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
326	C₂²×Q₈	—	2	4	no	—
327	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
328	C₂³×C₄	—	2	2	no	—
329	C₂×D₈	Q₈×C₆₇	1	8	yes	complemented
330	C₂×D₈	C₂³×C₆₇	1	2	yes	—
331	C₂×D₈	C₂³×C₆₇	1	2	yes	—
332	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
333	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
335	C₄⋊Q₈	—	2	4	no	—
336	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
337	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
339	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
340	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
341	#125	—	2	4	no	—
342	#125	D₄×C₆₇	1	2	yes	—
343	#125	C₂³×C₆₇	1	4	yes	—
344	#125	C₂³×C₆₇	1	8	yes	—
345	C₂²⋊Q₈	—	2	4	no	—
346	C₂²⋊Q₈	—	2	8	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	2	no	—
349	C₂²⋊Q₈	—	2	4	no	—
350	C₂²⋊Q₈	—	2	8	no	—
351	C₂²⋊Q₈	—	2	8	no	—
352	C₂²⋊Q₈	—	2	8	no	—
353	C₂²⋊Q₈	—	2	8	no	—
354	C₂²⋊Q₈	—	2	8	no	—
355	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
356	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
360	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
361	C₄×D₄	—	2	4	no	—
362	C₄×D₄	—	2	4	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	C₂³×C₆₇	1	4	yes	—
366	C₄×D₄	C₂³×C₆₇	1	4	yes	—
367	C₄×D₄	C₂³×C₆₇	1	8	yes	—
368	C₄×D₄	C₂³×C₆₇	1	8	yes	—
369	#117	C₂³×C₆₇	1	4	yes	—
370	#117	C₂³×C₆₇	1	2	yes	—
371	C₂×C₄⋊C₄	—	2	8	no	—
372	C₂×C₄⋊C₄	—	2	4	no	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	8	no	—
376	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
377	#107	C₂³×C₆₇	1	2	yes	—
378	#107	C₂³×C₆₇	1	2	yes	—
379	#107	C₂³×C₆₇	1	2	yes	—
380	#107	C₂³×C₆₇	1	2	yes	—
381	#107	C₂³×C₆₇	1	2	yes	—
382	#107	C₂³×C₆₇	1	2	yes	—
383	#107	C₂³×C₆₇	1	8	yes	—
384	#106	C₂³×C₆₇	1	4	yes	—
385	#106	C₂³×C₆₇	1	4	yes	—
386	#106	C₂³×C₆₇	1	8	yes	—
387	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
388	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
389	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
390	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
391	#102	C₂³×C₆₇	1	4	yes	—
392	#102	C₂³×C₆₇	1	4	yes	—
393	#102	C₂³×C₆₇	1	8	yes	—
394	#102	C₂³×C₆₇	1	8	yes	—
395	#97	C₂³×C₆₇	1	4	yes	—
396	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
398	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
399	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₆₇	1	4	yes	—
401	C₂⁴	—	2	2	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	8	no	—
405	C₂×Q₈	—	8	8	no	—
406	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
407	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	C₂×Q₈×C₆₇	1	8	yes	complemented
411	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	8	4	no	—
418	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
421	D₈	C₂×Q₈×C₆₇	1	8	yes	complemented
422	D₈	C₂×Q₈×C₆₇	1	8	yes	complemented
423	M₁₆	—	2	4	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	complemented
426	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
427	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
428	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
429	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
430	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
440	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
441	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
442	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
443	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
444	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
445	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
446	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
454	C₄²	—	2	4	no	—
455	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
457	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
458	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
459	C₄²	C₂⁴×C₆₇	1	4	yes	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	8	no	—
462	C₂³	—	8	2	no	—
463	C₂³	—	8	2	no	—
464	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
465	Q₈	—	2	4	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	4	no	—
468	Q₈	—	8	8	no	—
469	Q₈	—	8	2	no	—
470	D₄	—	2	8	no	—
471	D₄	—	2	4	no	—
472	D₄	C₂²×Q₈×C₆₇	1	8	yes	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	—	8	8	no	—
481	C₂×C₄	—	8	8	no	—
482	C₂×C₄	—	8	2	no	—
483	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
487	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
488	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
489	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
490	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
491	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
492	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
493	C₈	—	2	1	no	—
494	C₈	—	2	1	no	—
495	C₈	—	2	4	no	—
496	C₈	C₂²×Q₈×C₆₇	1	2	yes	—
497	C₈	C₂²×Q₈×C₆₇	1	2	yes	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	8	no	—
504	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
505	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
506	C₄	—	2	1	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	D₄×Q₈×C₆₇	1	2	yes	—
515	C₄	#576×C₆₇	1	2	yes	—
516	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
517	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
518	C₂	—	2	2	no	—
519	C₂	—	4	8	no	—
520	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
521	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
522	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #533×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3501	C₂	semidirect
2. C₆₇×#3389	C₂	semidirect
3. C₆₇×#3384	C₂	semidirect
4. C₆₇×#3380	C₂	semidirect
5. C₆₇×#3380	C₂	semidirect
6. Q₈×D₈×C₆₇	C₂	semidirect
7. C₆₇×#3164	C₂	semidirect
8. C₆₇×#3160	C₂	semidirect
9. C₆₇×#3081	C₂	semidirect
10. C₆₇×#3077	C₂	semidirect
11. C₆₇×#2960	C₂	semidirect
17. #554×C₆₇	C₂²	semidirect
18. #554×C₆₇	C₂²	semidirect
20. D₄×Q₈×C₆₇	C₂²	semidirect
28. #501×C₆₇	C₂²	semidirect
29. #500×C₆₇	C₂²	semidirect
30. #500×C₆₇	C₂²	semidirect
31. #500×C₆₇	C₂²	semidirect
32. #499×C₆₇	C₂²	semidirect
33. #499×C₆₇	C₂²	semidirect
34. #499×C₆₇	C₂²	semidirect
35. #498×C₆₇	C₂²	semidirect
36. #478×C₆₇	C₂²	semidirect
37. #478×C₆₇	C₂²	semidirect
38. #478×C₆₇	C₂²	semidirect
39. #476×C₆₇	C₂²	semidirect
40. #476×C₆₇	C₂²	semidirect
41. #474×C₆₇	C₂²	semidirect
42. #474×C₆₇	C₂²	semidirect
43. #446×C₆₇	C₂²	semidirect
44. #446×C₆₇	C₂²	semidirect
45. C₈×Q₈×C₆₇	C₂²	semidirect
46. C₈×Q₈×C₆₇	C₂²	semidirect
67. C₂×D₈×C₆₇	Q₈	semidirect
148. C₂×D₄×C₆₇	C₂×Q₈	semidirect
159. D₈×C₆₇	C₂×Q₈	semidirect
160. D₈×C₆₇	C₂×Q₈	semidirect
163. C₂×C₈×C₆₇	C₂×Q₈	semidirect
256. #61370	C₆₇	direct (N × H)
262. #3501	C₂×C₆₇	semidirect
263. #3389	C₂×C₆₇	semidirect
264. #3384	C₂×C₆₇	semidirect
265. #3380	C₂×C₆₇	semidirect
266. #3380	C₂×C₆₇	semidirect
267. Q₈×D₈	C₂×C₆₇	semidirect
268. #3164	C₂×C₆₇	semidirect
269. #3160	C₂×C₆₇	semidirect
270. #3081	C₂×C₆₇	semidirect
271. #3077	C₂×C₆₇	semidirect
272. #2960	C₂×C₆₇	semidirect
275. C₆₇	#61370	direct (N × H)
279. #554	C₂²×C₆₇	semidirect
280. #554	C₂²×C₆₇	semidirect
282. D₄×Q₈	C₂²×C₆₇	semidirect
290. #501	C₂²×C₆₇	semidirect
291. #500	C₂²×C₆₇	semidirect
292. #500	C₂²×C₆₇	semidirect
293. #500	C₂²×C₆₇	semidirect
294. #499	C₂²×C₆₇	semidirect
295. #499	C₂²×C₆₇	semidirect
296. #499	C₂²×C₆₇	semidirect
297. #498	C₂²×C₆₇	semidirect
298. #478	C₂²×C₆₇	semidirect
299. #478	C₂²×C₆₇	semidirect
300. #478	C₂²×C₆₇	semidirect
301. #476	C₂²×C₆₇	semidirect
302. #476	C₂²×C₆₇	semidirect
303. #474	C₂²×C₆₇	semidirect
304. #474	C₂²×C₆₇	semidirect
305. #446	C₂²×C₆₇	semidirect
306. #446	C₂²×C₆₇	semidirect
307. C₈×Q₈	C₂²×C₆₇	semidirect
308. C₈×Q₈	C₂²×C₆₇	semidirect
329. C₂×D₈	Q₈×C₆₇	semidirect
410. C₂×D₄	C₂×Q₈×C₆₇	semidirect
421. D₈	C₂×Q₈×C₆₇	semidirect
422. D₈	C₂×Q₈×C₆₇	semidirect
425. C₂×C₈	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
C₆₇×#3501	2
C₆₇×#3389	2
C₆₇×#3384	2
C₆₇×#3380	2
Q₈×D₈×C₆₇	2
C₆₇×#3164	2
C₆₇×#3160	2
C₆₇×#3081	2
C₆₇×#3077	2
C₆₇×#2960	2
C₆₇×#2948	2
C₂×C₄×D₈×C₆₇	2
#61370	67

Subgroups — C67×#61370 #30740621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61370 #30740621