Subgroups — C₆₇×#61373 #30740624

Order 17152.

1778 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3503	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3386	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3385	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3383	C₂	1	2	yes	maximal, complemented
5	Q₈×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3380	C₂	1	2	yes	maximal, complemented
7	Q₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3164	C₂	1	2	yes	maximal, complemented
9	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3083	C₂	1	4	yes	maximal, complemented
11	C₆₇×#3079	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2951	C₂	1	2	yes	maximal
14	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
15	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
16	Q₈²×C₆₇	C₂²	1	4	yes	—
17	#557×C₆₇	C₂²	1	4	yes	—
18	#554×C₆₇	—	2	4	no	—
19	#554×C₆₇	C₂²	1	4	yes	—
20	D₄×Q₈×C₆₇	—	2	4	no	—
21	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
22	#544×C₆₇	—	2	4	no	—
23	#544×C₆₇	C₂²	1	4	yes	—
24	#533×C₆₇	—	2	2	no	—
25	#533×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
27	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
28	#520×C₆₇	C₂²	1	4	yes	—
29	C₄×#41×C₆₇	C₂²	1	2	yes	—
30	#501×C₆₇	C₂²	1	8	yes	—
31	#500×C₆₇	C₂²	1	4	yes	—
32	#500×C₆₇	C₂²	1	2	yes	—
33	#500×C₆₇	C₂²	1	2	yes	—
34	#500×C₆₇	C₂²	1	2	yes	—
35	#500×C₆₇	C₂²	1	2	yes	—
36	#499×C₆₇	C₂²	1	4	yes	—
37	#479×C₆₇	C₂²	1	8	yes	—
38	#478×C₆₇	C₂²	1	8	yes	—
39	#477×C₆₇	C₂²	1	4	yes	—
40	#477×C₆₇	C₂²	1	8	yes	—
41	#476×C₆₇	C₂²	1	4	yes	—
42	#475×C₆₇	C₂²	1	4	yes	—
43	#474×C₆₇	C₂²	1	4	yes	—
44	#474×C₆₇	C₂²	1	8	yes	—
45	#446×C₆₇	C₂²	1	2	yes	—
46	#446×C₆₇	C₂²	1	2	yes	—
47	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
48	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
49	#440×C₆₇	C₂²	1	8	yes	—
50	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
51	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
52	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
53	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
54	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
55	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₄×D₈×C₆₇	C₂²	1	4	yes	—
57	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
58	#428×C₆₇	C₂²	1	4	yes	—
59	#427×C₆₇	C₂²	1	1	yes	characteristic
60	C₂×#107×C₆₇	C₂²	1	2	yes	—
61	C₂×#107×C₆₇	C₂²	1	2	yes	—
62	#424×C₆₇	C₂²	1	2	yes	—
63	#423×C₆₇	C₂²	1	1	yes	characteristic
64	#417×C₆₇	C₂²	1	4	yes	—
65	#416×C₆₇	C₂²	1	2	yes	—
66	#405×C₆₇	C₂²	1	2	yes	—
67	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
68	C₂×#41×C₆₇	C₂³	1	2	yes	—
69	C₂²×Q₈×C₆₇	—	2	4	no	—
70	C₂²×Q₈×C₆₇	—	2	2	no	—
71	#135×C₆₇	Q₈	1	8	yes	complemented
72	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
74	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₆₇	—	2	4	no	—
79	C₄⋊Q₈×C₆₇	—	2	4	no	—
80	C₄⋊Q₈×C₆₇	—	2	4	no	—
81	C₄⋊Q₈×C₆₇	—	2	8	no	—
82	C₄⋊Q₈×C₆₇	—	2	2	no	—
83	C₄⋊Q₈×C₆₇	—	2	2	no	—
84	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
86	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
87	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
88	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
89	#125×C₆₇	—	2	4	no	—
90	#125×C₆₇	—	2	8	no	—
91	#125×C₆₇	D₄	1	2	yes	—
92	#125×C₆₇	C₂³	1	8	yes	—
93	C₂²⋊Q₈×C₆₇	—	2	4	no	—
94	C₂²⋊Q₈×C₆₇	—	2	4	no	—
95	C₂²⋊Q₈×C₆₇	—	2	8	no	—
96	C₂²⋊Q₈×C₆₇	—	2	8	no	—
97	C₂²⋊Q₈×C₆₇	—	2	8	no	—
98	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
99	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
100	C₄×Q₈×C₆₇	—	2	4	no	—
101	C₄×Q₈×C₆₇	—	2	4	no	—
102	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
103	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
104	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
105	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
106	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
107	C₄×D₄×C₆₇	—	2	2	no	—
108	C₄×D₄×C₆₇	—	2	2	no	—
109	C₄×D₄×C₆₇	—	2	8	no	—
110	C₄×D₄×C₆₇	C₂³	1	2	yes	—
111	C₄×D₄×C₆₇	C₂³	1	8	yes	—
112	C₄×D₄×C₆₇	C₂³	1	2	yes	—
113	#117×C₆₇	—	2	4	no	—
114	#117×C₆₇	—	2	2	no	—
115	#117×C₆₇	C₂³	1	2	yes	—
116	#117×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄²×C₆₇	—	2	2	no	—
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	#107×C₆₇	C₂³	1	8	yes	—
124	#107×C₆₇	C₂³	1	4	yes	—
125	#107×C₆₇	C₂³	1	4	yes	—
126	#107×C₆₇	C₂³	1	1	yes	characteristic
127	#107×C₆₇	C₂³	1	4	yes	—
128	#107×C₆₇	C₂³	1	1	yes	characteristic
129	#107×C₆₇	C₂³	1	4	yes	—
130	#106×C₆₇	C₂³	1	8	yes	—
131	#106×C₆₇	C₂³	1	2	yes	—
132	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
133	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
134	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
135	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
136	#103×C₆₇	C₂³	1	8	yes	—
137	#103×C₆₇	C₂³	1	2	yes	—
138	#103×C₆₇	C₂³	1	2	yes	—
139	#102×C₆₇	C₂³	1	2	yes	—
140	#102×C₆₇	C₂³	1	8	yes	—
141	#102×C₆₇	C₂³	1	2	yes	—
142	#97×C₆₇	C₂³	1	4	yes	—
143	C₄×C₈×C₆₇	C₂³	1	4	yes	—
144	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
148	#41×C₆₇	—	2	4	no	—
149	#41×C₆₇	C₂×Q₈	1	8	yes	complemented
150	C₂×Q₈×C₆₇	—	2	4	no	—
151	C₂×Q₈×C₆₇	—	2	8	no	—
152	C₂×Q₈×C₆₇	—	2	4	no	—
153	C₂×Q₈×C₆₇	—	8	4	no	—
154	C₂×Q₈×C₆₇	—	8	8	no	—
155	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
156	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
157	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
158	C₂×D₄×C₆₇	—	2	2	no	—
159	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	2	8	no	—
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
169	SD₁₆×C₆₇	C₂×Q₈	1	8	yes	complemented
170	D₈×C₆₇	C₂×Q₈	1	4	yes	complemented
171	M₁₆×C₆₇	—	2	4	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
174	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
175	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	8	no	—
182	C₄⋊C₄×C₆₇	—	2	4	no	—
183	C₄⋊C₄×C₆₇	—	2	8	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	8	no	—
187	C₄⋊C₄×C₆₇	—	2	8	no	—
188	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
190	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
194	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
197	C₂²⋊C₄×C₆₇	—	2	2	no	—
198	C₂²⋊C₄×C₆₇	—	2	2	no	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
202	C₄²×C₆₇	—	2	2	no	—
203	C₄²×C₆₇	—	2	2	no	—
204	C₄²×C₆₇	—	2	2	no	—
205	C₄²×C₆₇	—	2	8	no	—
206	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
209	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₄²×C₆₇	C₂⁴	1	4	yes	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₆₇	—	2	4	no	—
214	Q₈×C₆₇	—	2	4	no	—
215	Q₈×C₆₇	—	2	2	no	—
216	Q₈×C₆₇	—	4	8	no	—
217	Q₈×C₆₇	—	8	4	no	—
218	Q₈×C₆₇	—	8	8	no	—
219	Q₈×C₆₇	C₂²×Q₈	1	4	yes	—
220	D₄×C₆₇	—	2	8	no	—
221	D₄×C₆₇	—	2	2	no	—
222	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	8	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	8	no	—
230	C₂×C₄×C₆₇	—	4	4	no	—
231	C₂×C₄×C₆₇	—	4	4	no	—
232	C₂×C₄×C₆₇	—	8	8	no	—
233	C₂×C₄×C₆₇	—	8	4	no	—
234	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
242	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
243	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
244	C₈×C₆₇	—	2	1	no	—
245	C₈×C₆₇	—	2	1	no	—
246	C₈×C₆₇	—	2	4	no	—
247	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
248	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
255	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
256	C₄×C₆₇	—	2	4	no	—
257	C₄×C₆₇	—	2	1	no	—
258	C₄×C₆₇	—	2	4	no	—
259	C₄×C₆₇	—	2	1	no	—
260	C₄×C₆₇	—	2	4	no	—
261	C₄×C₆₇	—	4	2	no	—
262	C₄×C₆₇	—	4	8	no	—
263	C₄×C₆₇	—	4	2	no	—
264	C₄×C₆₇	D₄×Q₈	1	2	yes	—
265	C₄×C₆₇	#576	1	2	yes	—
266	C₄×C₆₇	#578	1	2	yes	—
267	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
268	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
269	#61373	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂×C₆₇	—	2	2	no	—
271	C₂×C₆₇	—	4	4	no	—
272	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
273	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
274	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
275	#3503	C₂×C₆₇	1	2	yes	complemented
276	#3386	C₂×C₆₇	1	4	yes	complemented
277	#3385	C₂×C₆₇	1	4	yes	complemented
278	#3383	C₂×C₆₇	1	2	yes	complemented
279	Q₈×Q₁₆	C₂×C₆₇	1	2	yes	complemented
280	#3380	C₂×C₆₇	1	2	yes	complemented
281	Q₈×D₈	C₂×C₆₇	1	2	yes	complemented
282	#3164	C₂×C₆₇	1	2	yes	complemented
283	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
284	#3083	C₂×C₆₇	1	4	yes	complemented
285	#3079	C₂×C₆₇	1	2	yes	complemented
286	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#2951	C₂×C₆₇	1	2	yes	—
288	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
289	C₆₇	#61373	1	1	yes	characteristic, direct product factor, Sylow
290	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
291	Q₈²	C₂²×C₆₇	1	4	yes	—
292	#557	C₂²×C₆₇	1	4	yes	—
293	#554	—	2	4	no	—
294	#554	C₂²×C₆₇	1	4	yes	—
295	D₄×Q₈	—	2	4	no	—
296	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
297	#544	—	2	4	no	—
298	#544	C₂²×C₆₇	1	4	yes	—
299	#533	—	2	2	no	—
300	#533	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊Q₈	—	2	2	no	—
302	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
303	#520	C₂²×C₆₇	1	4	yes	—
304	C₄×#41	C₂²×C₆₇	1	2	yes	—
305	#501	C₂²×C₆₇	1	8	yes	—
306	#500	C₂²×C₆₇	1	4	yes	—
307	#500	C₂²×C₆₇	1	2	yes	—
308	#500	C₂²×C₆₇	1	2	yes	—
309	#500	C₂²×C₆₇	1	2	yes	—
310	#500	C₂²×C₆₇	1	2	yes	—
311	#499	C₂²×C₆₇	1	4	yes	—
312	#479	C₂²×C₆₇	1	8	yes	—
313	#478	C₂²×C₆₇	1	8	yes	—
314	#477	C₂²×C₆₇	1	4	yes	—
315	#477	C₂²×C₆₇	1	8	yes	—
316	#476	C₂²×C₆₇	1	4	yes	—
317	#475	C₂²×C₆₇	1	4	yes	—
318	#474	C₂²×C₆₇	1	4	yes	—
319	#474	C₂²×C₆₇	1	8	yes	—
320	#446	C₂²×C₆₇	1	2	yes	—
321	#446	C₂²×C₆₇	1	2	yes	—
322	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
323	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
324	#440	C₂²×C₆₇	1	8	yes	—
325	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
326	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
327	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
328	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
329	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
330	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₄×D₈	C₂²×C₆₇	1	4	yes	—
332	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
333	#428	C₂²×C₆₇	1	4	yes	—
334	#427	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×#107	C₂²×C₆₇	1	2	yes	—
336	C₂×#107	C₂²×C₆₇	1	2	yes	—
337	#424	C₂²×C₆₇	1	2	yes	—
338	#423	C₂²×C₆₇	1	1	yes	characteristic
339	#417	C₂²×C₆₇	1	4	yes	—
340	#416	C₂²×C₆₇	1	2	yes	—
341	#405	C₂²×C₆₇	1	2	yes	—
342	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×#41	C₂³×C₆₇	1	2	yes	—
344	C₂²×Q₈	—	2	4	no	—
345	C₂²×Q₈	—	2	2	no	—
346	#135	Q₈×C₆₇	1	8	yes	complemented
347	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
348	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
349	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
350	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
351	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
353	C₄⋊Q₈	—	2	4	no	—
354	C₄⋊Q₈	—	2	4	no	—
355	C₄⋊Q₈	—	2	4	no	—
356	C₄⋊Q₈	—	2	8	no	—
357	C₄⋊Q₈	—	2	2	no	—
358	C₄⋊Q₈	—	2	2	no	—
359	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
360	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
361	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
362	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
363	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
364	#125	—	2	4	no	—
365	#125	—	2	8	no	—
366	#125	D₄×C₆₇	1	2	yes	—
367	#125	C₂³×C₆₇	1	8	yes	—
368	C₂²⋊Q₈	—	2	4	no	—
369	C₂²⋊Q₈	—	2	4	no	—
370	C₂²⋊Q₈	—	2	8	no	—
371	C₂²⋊Q₈	—	2	8	no	—
372	C₂²⋊Q₈	—	2	8	no	—
373	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
374	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
375	C₄×Q₈	—	2	4	no	—
376	C₄×Q₈	—	2	4	no	—
377	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
378	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
379	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
380	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
381	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
382	C₄×D₄	—	2	2	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	2	8	no	—
385	C₄×D₄	C₂³×C₆₇	1	2	yes	—
386	C₄×D₄	C₂³×C₆₇	1	2	yes	—
387	C₄×D₄	C₂³×C₆₇	1	8	yes	—
388	#117	—	2	4	no	—
389	#117	—	2	2	no	—
390	#117	C₂³×C₆₇	1	2	yes	—
391	#117	C₂³×C₆₇	1	2	yes	—
392	C₂×C₄⋊C₄	—	2	2	no	—
393	C₂×C₄⋊C₄	—	2	4	no	—
394	C₂×C₄⋊C₄	—	2	4	no	—
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
396	C₂×C₄²	—	2	2	no	—
397	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
398	#107	C₂³×C₆₇	1	4	yes	—
399	#107	C₂³×C₆₇	1	4	yes	—
400	#107	C₂³×C₆₇	1	4	yes	—
401	#107	C₂³×C₆₇	1	4	yes	—
402	#107	C₂³×C₆₇	1	1	yes	characteristic
403	#107	C₂³×C₆₇	1	1	yes	characteristic
404	#107	C₂³×C₆₇	1	8	yes	—
405	#106	C₂³×C₆₇	1	2	yes	—
406	#106	C₂³×C₆₇	1	8	yes	—
407	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
408	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
409	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
410	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
411	#103	C₂³×C₆₇	1	2	yes	—
412	#103	C₂³×C₆₇	1	2	yes	—
413	#103	C₂³×C₆₇	1	8	yes	—
414	#102	C₂³×C₆₇	1	2	yes	—
415	#102	C₂³×C₆₇	1	2	yes	—
416	#102	C₂³×C₆₇	1	8	yes	—
417	#97	C₂³×C₆₇	1	4	yes	—
418	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
419	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
420	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
421	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₆₇	1	4	yes	—
423	#41	—	2	4	no	—
424	#41	C₂×Q₈×C₆₇	1	8	yes	complemented
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	—	2	8	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	8	4	no	—
429	C₂×Q₈	—	8	8	no	—
430	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
432	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	8	no	—
440	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
443	Q₁₆	C₂×Q₈×C₆₇	1	4	yes	complemented
444	SD₁₆	C₂×Q₈×C₆₇	1	8	yes	complemented
445	D₈	C₂×Q₈×C₆₇	1	4	yes	complemented
446	M₁₆	—	2	4	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	complemented
449	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
450	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
451	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
452	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
453	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
464	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
465	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
466	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
467	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
468	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
469	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
470	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
471	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	8	no	—
481	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
482	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
483	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
484	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
485	C₄²	C₂⁴×C₆₇	1	4	yes	—
486	C₂³	—	2	2	no	—
487	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
488	Q₈	—	2	4	no	—
489	Q₈	—	2	4	no	—
490	Q₈	—	2	2	no	—
491	Q₈	—	4	8	no	—
492	Q₈	—	8	4	no	—
493	Q₈	—	8	8	no	—
494	Q₈	C₂²×Q₈×C₆₇	1	4	yes	—
495	D₄	—	2	8	no	—
496	D₄	—	2	2	no	—
497	D₄	C₂²×Q₈×C₆₇	1	4	yes	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	4	4	no	—
507	C₂×C₄	—	8	8	no	—
508	C₂×C₄	—	8	4	no	—
509	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
510	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
513	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
517	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
518	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
519	C₈	—	2	1	no	—
520	C₈	—	2	1	no	—
521	C₈	—	2	4	no	—
522	C₈	C₂²×Q₈×C₆₇	1	2	yes	—
523	C₈	C₂²×Q₈×C₆₇	1	2	yes	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
530	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
531	C₄	—	2	1	no	—
532	C₄	—	2	4	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	4	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	8	no	—
538	C₄	—	4	2	no	—
539	C₄	D₄×Q₈×C₆₇	1	2	yes	—
540	C₄	#576×C₆₇	1	2	yes	—
541	C₄	#578×C₆₇	1	2	yes	—
542	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
543	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
544	C₂	—	2	2	no	—
545	C₂	—	4	4	no	—
546	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
547	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
548	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#135×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×#3503	C₂	semidirect
2. C₆₇×#3386	C₂	semidirect
3. C₆₇×#3385	C₂	semidirect
4. C₆₇×#3383	C₂	semidirect
5. Q₈×Q₁₆×C₆₇	C₂	semidirect
6. C₆₇×#3380	C₂	semidirect
7. Q₈×D₈×C₆₇	C₂	semidirect
8. C₆₇×#3164	C₂	semidirect
9. C₂×#500×C₆₇	C₂	semidirect
10. C₆₇×#3083	C₂	semidirect
11. C₆₇×#3079	C₂	semidirect
12. C₆₇×#2960	C₂	semidirect
71. #135×C₆₇	Q₈	semidirect
149. #41×C₆₇	C₂×Q₈	semidirect
168. Q₁₆×C₆₇	C₂×Q₈	semidirect
169. SD₁₆×C₆₇	C₂×Q₈	semidirect
170. D₈×C₆₇	C₂×Q₈	semidirect
173. C₂×C₈×C₆₇	C₂×Q₈	semidirect
269. #61373	C₆₇	direct (N × H)
275. #3503	C₂×C₆₇	semidirect
276. #3386	C₂×C₆₇	semidirect
277. #3385	C₂×C₆₇	semidirect
278. #3383	C₂×C₆₇	semidirect
279. Q₈×Q₁₆	C₂×C₆₇	semidirect
280. #3380	C₂×C₆₇	semidirect
281. Q₈×D₈	C₂×C₆₇	semidirect
282. #3164	C₂×C₆₇	semidirect
283. C₂×#500	C₂×C₆₇	semidirect
284. #3083	C₂×C₆₇	semidirect
285. #3079	C₂×C₆₇	semidirect
286. #2960	C₂×C₆₇	semidirect
289. C₆₇	#61373	direct (N × H)
346. #135	Q₈×C₆₇	semidirect
424. #41	C₂×Q₈×C₆₇	semidirect
443. Q₁₆	C₂×Q₈×C₆₇	semidirect
444. SD₁₆	C₂×Q₈×C₆₇	semidirect
445. D₈	C₂×Q₈×C₆₇	semidirect
448. 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×#3503	2
C₆₇×#3386	2
C₆₇×#3385	2
C₆₇×#3383	2
Q₈×Q₁₆×C₆₇	2
C₆₇×#3380	2
Q₈×D₈×C₆₇	2
C₆₇×#3164	2
C₂×#500×C₆₇	2
C₆₇×#3083	2
C₆₇×#3079	2
C₆₇×#2960	2
C₆₇×#2951	2
C₄×#135×C₆₇	2
#61373	67

Subgroups — C67×#61373 #30740624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61373 #30740624