Subgroups — C₆₇×#34952 #30714203

Order 17152.

2634 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2582	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2581	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2580	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2579	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2538	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2487	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2409	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2396	C₂	1	4	yes	maximal, complemented
11	C₄×C₂²⋊Q₈×C₆₇	C₂	1	4	yes	maximal, complemented
12	C₆₇×#2297	C₂	1	2	yes	maximal
13	C₆₇×#2296	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
18	C₄×#41×C₆₇	—	2	8	no	—
19	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×#117×C₆₇	C₂²	1	2	yes	—
31	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#117×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
36	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
37	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
38	#400×C₆₇	C₂²	1	4	yes	—
39	#399×C₆₇	C₂²	1	4	yes	—
40	#398×C₆₇	C₂²	1	8	yes	—
41	#396×C₆₇	C₂²	1	8	yes	—
42	#395×C₆₇	C₂²	1	4	yes	—
43	#393×C₆₇	C₂²	1	4	yes	—
44	#391×C₆₇	C₂²	1	4	yes	—
45	#391×C₆₇	C₂²	1	4	yes	—
46	#390×C₆₇	C₂²	1	4	yes	—
47	#389×C₆₇	C₂²	1	4	yes	—
48	#389×C₆₇	C₂²	1	4	yes	—
49	#389×C₆₇	C₂²	1	4	yes	—
50	#387×C₆₇	C₂²	1	8	yes	—
51	#386×C₆₇	C₂²	1	4	yes	—
52	#386×C₆₇	C₂²	1	4	yes	—
53	#385×C₆₇	C₂²	1	4	yes	—
54	#385×C₆₇	C₂²	1	4	yes	—
55	#385×C₆₇	C₂²	1	8	yes	—
56	#384×C₆₇	C₂²	1	2	yes	—
57	#382×C₆₇	C₂²	1	2	yes	—
58	#380×C₆₇	C₂²	1	4	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	#380×C₆₇	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
65	C₂×#41×C₆₇	—	2	4	no	—
66	C₂×#41×C₆₇	—	2	4	no	—
67	C₂×#41×C₆₇	Q₈	1	8	yes	complemented
68	C₂²×Q₈×C₆₇	—	2	8	no	—
69	C₂²×Q₈×C₆₇	—	2	4	no	—
70	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂²⋊Q₈×C₆₇	—	2	16	no	—
84	C₄×Q₈×C₆₇	—	2	8	no	—
85	C₄×Q₈×C₆₇	—	2	8	no	—
86	C₄×Q₈×C₆₇	—	2	8	no	—
87	C₄×Q₈×C₆₇	—	2	8	no	—
88	C₄×Q₈×C₆₇	—	2	4	no	—
89	C₄×Q₈×C₆₇	—	2	4	no	—
90	C₄×D₄×C₆₇	—	2	8	no	—
91	C₄×D₄×C₆₇	—	2	8	no	—
92	C₄×D₄×C₆₇	—	2	4	no	—
93	C₄×D₄×C₆₇	—	2	4	no	—
94	#117×C₆₇	—	2	4	no	—
95	#117×C₆₇	—	2	8	no	—
96	#117×C₆₇	—	2	8	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
118	C₂×C₄²×C₆₇	—	2	8	no	—
119	C₂×C₄²×C₆₇	—	2	8	no	—
120	C₂×C₄²×C₆₇	D₄	1	8	yes	—
121	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
123	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
131	#95×C₆₇	C₂³	1	8	yes	—
132	#95×C₆₇	C₂³	1	8	yes	—
133	#95×C₆₇	C₂³	1	4	yes	—
134	#95×C₆₇	C₂³	1	4	yes	—
135	#95×C₆₇	C₂³	1	8	yes	—
136	#95×C₆₇	C₂³	1	8	yes	—
137	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
138	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
139	#41×C₆₇	—	2	32	no	—
140	C₂×Q₈×C₆₇	—	2	32	no	—
141	C₂×Q₈×C₆₇	—	2	4	no	—
142	C₂×Q₈×C₆₇	—	2	4	no	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	2	8	no	—
145	C₂×Q₈×C₆₇	—	2	4	no	—
146	C₂×Q₈×C₆₇	—	2	2	no	—
147	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	complemented
148	C₂×Q₈×C₆₇	#41	1	8	yes	complemented
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	4	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	C₂×Q₈	1	8	yes	—
154	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	complemented
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	8	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	8	no	—
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	—	4	4	no	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	complemented
172	C₂²×C₄×C₆₇	#41	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₄⋊C₄×C₆₇	—	2	8	no	—
186	C₄⋊C₄×C₆₇	—	2	8	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	8	no	—
194	C₄⋊C₄×C₆₇	—	2	8	no	—
195	C₄⋊C₄×C₆₇	—	2	16	no	—
196	C₄⋊C₄×C₆₇	—	4	4	no	—
197	C₄⋊C₄×C₆₇	—	4	8	no	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	16	no	—
203	C₂²⋊C₄×C₆₇	—	4	8	no	—
204	C₄²×C₆₇	—	2	4	no	—
205	C₄²×C₆₇	—	2	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	8	no	—
208	C₄²×C₆₇	—	2	8	no	—
209	C₄²×C₆₇	—	2	8	no	—
210	C₄²×C₆₇	C₂×D₄	1	4	yes	—
211	C₄²×C₆₇	C₂×D₄	1	4	yes	—
212	C₄²×C₆₇	C₂×D₄	1	4	yes	—
213	C₄²×C₆₇	C₂×D₄	1	4	yes	—
214	C₂³×C₆₇	—	2	2	no	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	2	2	no	—
217	C₂³×C₆₇	—	2	4	no	—
218	C₂³×C₆₇	—	2	2	no	—
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
221	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
222	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
223	C₂³×C₆₇	C₂×#41	1	2	yes	—
224	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
225	Q₈×C₆₇	—	2	16	no	—
226	Q₈×C₆₇	—	2	8	no	—
227	Q₈×C₆₇	—	2	16	no	—
228	D₄×C₆₇	—	2	16	no	—
229	D₄×C₆₇	—	2	8	no	—
230	C₂×C₄×C₆₇	—	2	16	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	16	no	—
239	C₂×C₄×C₆₇	—	2	8	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	8	no	—
244	C₂×C₄×C₆₇	—	2	8	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	8	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	4	8	no	—
252	C₂×C₄×C₆₇	—	4	8	no	—
253	C₂×C₄×C₆₇	—	4	8	no	—
254	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	complemented
255	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
262	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
263	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
264	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
265	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
266	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
267	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	2	2	no	—
270	C₂²×C₆₇	—	2	4	no	—
271	C₂²×C₆₇	—	2	4	no	—
272	C₂²×C₆₇	—	2	4	no	—
273	C₂²×C₆₇	—	2	8	no	—
274	C₂²×C₆₇	—	4	2	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
278	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
279	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
280	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
281	C₂²×C₆₇	C₂²×#41	1	2	yes	—
282	C₂²×C₆₇	#585	1	2	yes	—
283	C₄×C₆₇	—	2	4	no	—
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	8	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	4	2	no	—
289	C₄×C₆₇	—	4	2	no	—
290	C₄×C₆₇	—	4	4	no	—
291	C₄×C₆₇	—	4	8	no	—
292	C₄×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
293	C₄×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
294	C₄×C₆₇	#525	1	4	yes	—
295	#34952	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	2	4	no	—
297	C₂×C₆₇	—	4	4	no	—
298	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
300	C₂×C₆₇	#3449	1	1	yes	characteristic
301	C₂×C₆₇	Q₈×#41	1	2	yes	—
302	C₂×C₆₇	#3480	1	2	yes	—
303	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
304	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#2582	C₂×C₆₇	1	2	yes	complemented
306	#2581	C₂×C₆₇	1	2	yes	complemented
307	#2580	C₂×C₆₇	1	2	yes	complemented
308	#2579	C₂×C₆₇	1	4	yes	complemented
309	#2538	C₂×C₆₇	1	2	yes	complemented
310	#2487	C₂×C₆₇	1	2	yes	complemented
311	#2409	C₂×C₆₇	1	2	yes	complemented
312	#2396	C₂×C₆₇	1	4	yes	complemented
313	C₄×C₂²⋊Q₈	C₂×C₆₇	1	4	yes	complemented
314	#2297	C₂×C₆₇	1	2	yes	—
315	#2296	C₂×C₆₇	1	2	yes	complemented
316	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
317	C₆₇	#34952	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
319	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
320	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
321	C₄×#41	—	2	8	no	—
322	C₂×C₄×Q₈	—	2	8	no	—
323	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
324	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
327	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
328	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
329	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
330	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
332	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×#117	C₂²×C₆₇	1	2	yes	—
334	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×#117	C₂²×C₆₇	1	2	yes	—
336	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
337	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
338	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
339	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
340	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
341	#400	C₂²×C₆₇	1	4	yes	—
342	#399	C₂²×C₆₇	1	4	yes	—
343	#398	C₂²×C₆₇	1	8	yes	—
344	#396	C₂²×C₆₇	1	8	yes	—
345	#395	C₂²×C₆₇	1	4	yes	—
346	#393	C₂²×C₆₇	1	4	yes	—
347	#391	C₂²×C₆₇	1	4	yes	—
348	#391	C₂²×C₆₇	1	4	yes	—
349	#390	C₂²×C₆₇	1	4	yes	—
350	#389	C₂²×C₆₇	1	4	yes	—
351	#389	C₂²×C₆₇	1	4	yes	—
352	#389	C₂²×C₆₇	1	4	yes	—
353	#387	C₂²×C₆₇	1	8	yes	—
354	#386	C₂²×C₆₇	1	4	yes	—
355	#386	C₂²×C₆₇	1	4	yes	—
356	#385	C₂²×C₆₇	1	8	yes	—
357	#385	C₂²×C₆₇	1	4	yes	—
358	#385	C₂²×C₆₇	1	4	yes	—
359	#384	C₂²×C₆₇	1	2	yes	—
360	#382	C₂²×C₆₇	1	2	yes	—
361	#380	C₂²×C₆₇	1	2	yes	—
362	#380	C₂²×C₆₇	1	2	yes	—
363	#380	C₂²×C₆₇	1	4	yes	—
364	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
365	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
366	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
367	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
368	C₂×#41	—	2	4	no	—
369	C₂×#41	—	2	4	no	—
370	C₂×#41	Q₈×C₆₇	1	8	yes	complemented
371	C₂²×Q₈	—	2	8	no	—
372	C₂²×Q₈	—	2	4	no	—
373	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
374	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
375	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
376	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
377	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
379	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
380	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
382	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
383	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
384	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
386	C₂²⋊Q₈	—	2	16	no	—
387	C₄×Q₈	—	2	8	no	—
388	C₄×Q₈	—	2	8	no	—
389	C₄×Q₈	—	2	8	no	—
390	C₄×Q₈	—	2	8	no	—
391	C₄×Q₈	—	2	4	no	—
392	C₄×Q₈	—	2	4	no	—
393	C₄×D₄	—	2	8	no	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	#117	—	2	4	no	—
398	#117	—	2	8	no	—
399	#117	—	2	8	no	—
400	C₂×C₄⋊C₄	—	2	8	no	—
401	C₂×C₄⋊C₄	—	2	4	no	—
402	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
421	C₂×C₄²	—	2	8	no	—
422	C₂×C₄²	—	2	8	no	—
423	C₂×C₄²	D₄×C₆₇	1	8	yes	—
424	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
425	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
426	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
427	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
429	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
430	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
434	#95	C₂³×C₆₇	1	8	yes	—
435	#95	C₂³×C₆₇	1	8	yes	—
436	#95	C₂³×C₆₇	1	4	yes	—
437	#95	C₂³×C₆₇	1	4	yes	—
438	#95	C₂³×C₆₇	1	8	yes	—
439	#95	C₂³×C₆₇	1	8	yes	—
440	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
441	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
442	#41	—	2	32	no	—
443	C₂×Q₈	—	2	32	no	—
444	C₂×Q₈	—	2	4	no	—
445	C₂×Q₈	—	2	4	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	8	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	complemented
451	C₂×Q₈	#41×C₆₇	1	8	yes	complemented
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	C₂×Q₈×C₆₇	1	8	yes	—
457	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	complemented
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	8	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	4	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
471	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
472	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
473	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
474	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	complemented
475	C₂²×C₄	#41×C₆₇	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
477	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
488	C₄⋊C₄	—	2	8	no	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	8	no	—
497	C₄⋊C₄	—	2	8	no	—
498	C₄⋊C₄	—	2	16	no	—
499	C₄⋊C₄	—	4	4	no	—
500	C₄⋊C₄	—	4	8	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	16	no	—
506	C₂²⋊C₄	—	4	8	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	4	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	2	8	no	—
511	C₄²	—	2	8	no	—
512	C₄²	—	2	8	no	—
513	C₄²	C₂×D₄×C₆₇	1	4	yes	—
514	C₄²	C₂×D₄×C₆₇	1	4	yes	—
515	C₄²	C₂×D₄×C₆₇	1	4	yes	—
516	C₄²	C₂×D₄×C₆₇	1	4	yes	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	4	no	—
523	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
524	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
525	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
526	C₂³	C₂×#41×C₆₇	1	2	yes	—
527	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
528	Q₈	—	2	16	no	—
529	Q₈	—	2	8	no	—
530	Q₈	—	2	16	no	—
531	D₄	—	2	16	no	—
532	D₄	—	2	8	no	—
533	C₂×C₄	—	2	16	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	16	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	8	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	—	4	8	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	complemented
558	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
559	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
561	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
562	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
564	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
565	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
566	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
567	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
568	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
569	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
570	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	4	no	—
580	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
581	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
582	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
583	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
584	C₂²	C₂²×#41×C₆₇	1	2	yes	—
585	C₂²	#585×C₆₇	1	2	yes	—
586	C₄	—	2	4	no	—
587	C₄	—	2	2	no	—
588	C₄	—	2	8	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	2	no	—
593	C₄	—	4	4	no	—
594	C₄	—	4	8	no	—
595	C₄	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
596	C₄	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
597	C₄	#525×C₆₇	1	4	yes	—
598	C₂	—	2	4	no	—
599	C₂	—	4	4	no	—
600	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
601	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
602	C₂	C₆₇×#3449	1	1	yes	characteristic
603	C₂	Q₈×#41×C₆₇	1	2	yes	—
604	C₂	C₆₇×#3480	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂²×#41×C₆₇	C₂²
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
2. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2582	C₂	semidirect
4. C₆₇×#2581	C₂	semidirect
5. C₆₇×#2580	C₂	semidirect
6. C₆₇×#2579	C₂	semidirect
7. C₆₇×#2538	C₂	semidirect
8. C₆₇×#2487	C₂	semidirect
9. C₆₇×#2409	C₂	semidirect
10. C₆₇×#2396	C₂	semidirect
11. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
13. C₆₇×#2296	C₂	semidirect
14. C₆₇×#2285	C₂	semidirect
67. C₂×#41×C₆₇	Q₈	semidirect
147. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
148. C₂×Q₈×C₆₇	#41	semidirect
154. C₂×D₄×C₆₇	C₂×Q₈	semidirect
171. C₂²×C₄×C₆₇	C₂×Q₈	semidirect
254. C₂×C₄×C₆₇	C₂²⋊Q₈	semidirect
295. #34952	C₆₇	direct (N × H)
304. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
305. #2582	C₂×C₆₇	semidirect
306. #2581	C₂×C₆₇	semidirect
307. #2580	C₂×C₆₇	semidirect
308. #2579	C₂×C₆₇	semidirect
309. #2538	C₂×C₆₇	semidirect
310. #2487	C₂×C₆₇	semidirect
311. #2409	C₂×C₆₇	semidirect
312. #2396	C₂×C₆₇	semidirect
313. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
315. #2296	C₂×C₆₇	semidirect
316. #2285	C₂×C₆₇	semidirect
317. C₆₇	#34952	direct (N × H)
370. C₂×#41	Q₈×C₆₇	semidirect
450. C₂×Q₈	C₂×Q₈×C₆₇	semidirect
451. C₂×Q₈	#41×C₆₇	semidirect
457. C₂×D₄	C₂×Q₈×C₆₇	semidirect
474. C₂²×C₄	C₂×Q₈×C₆₇	semidirect
557. C₂×C₄	C₂²⋊Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2582	2
C₆₇×#2581	2
C₆₇×#2580	2
C₆₇×#2579	2
C₆₇×#2538	2
C₆₇×#2487	2
C₆₇×#2409	2
C₆₇×#2396	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#2297	2
C₆₇×#2296	2
C₆₇×#2285	2
#34952	67

Subgroups — C67×#34952 #30714203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34952 #30714203