Subgroups — C₆₇×#35092 #30714343

Order 17152.

2278 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#539×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2855	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2854	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2853	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2627	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2618	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#2614	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2469	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2465	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2462	C₂	1	4	yes	maximal, complemented
12	C₆₇×#2458	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2322	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2313	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#123×C₆₇	C₂	1	2	yes	maximal, complemented
16	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	#539×C₆₇	—	2	8	no	—
18	C₂×#126×C₆₇	C₂²	1	2	yes	—
19	C₂×#125×C₆₇	C₂²	1	2	yes	—
20	C₂×#123×C₆₇	C₂²	1	2	yes	—
21	C₂×#123×C₆₇	C₂²	1	2	yes	—
22	C₂×#123×C₆₇	C₂²	1	2	yes	—
23	C₂×#123×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
37	#400×C₆₇	C₂²	1	2	yes	—
38	#398×C₆₇	C₂²	1	2	yes	—
39	#398×C₆₇	C₂²	1	4	yes	—
40	#396×C₆₇	C₂²	1	4	yes	—
41	#395×C₆₇	C₂²	1	4	yes	—
42	#395×C₆₇	C₂²	1	4	yes	—
43	#395×C₆₇	C₂²	1	2	yes	—
44	#394×C₆₇	C₂²	1	2	yes	—
45	#393×C₆₇	C₂²	1	4	yes	—
46	#393×C₆₇	C₂²	1	4	yes	—
47	#391×C₆₇	C₂²	1	2	yes	—
48	#390×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	4	yes	—
50	#389×C₆₇	C₂²	1	2	yes	—
51	#389×C₆₇	C₂²	1	2	yes	—
52	#388×C₆₇	C₂²	1	4	yes	—
53	#388×C₆₇	C₂²	1	4	yes	—
54	#388×C₆₇	C₂²	1	4	yes	—
55	#388×C₆₇	C₂²	1	4	yes	—
56	#388×C₆₇	C₂²	1	4	yes	—
57	#387×C₆₇	C₂²	1	4	yes	—
58	#387×C₆₇	C₂²	1	4	yes	—
59	#387×C₆₇	C₂²	1	4	yes	—
60	#387×C₆₇	C₂²	1	4	yes	—
61	#387×C₆₇	C₂²	1	4	yes	—
62	#387×C₆₇	C₂²	1	4	yes	—
63	#387×C₆₇	C₂²	1	4	yes	—
64	#386×C₆₇	C₂²	1	2	yes	—
65	#386×C₆₇	C₂²	1	2	yes	—
66	#385×C₆₇	C₂²	1	2	yes	—
67	#385×C₆₇	C₂²	1	2	yes	—
68	#385×C₆₇	C₂²	1	4	yes	—
69	#381×C₆₇	C₂²	1	2	yes	—
70	#381×C₆₇	C₂²	1	2	yes	—
71	#381×C₆₇	C₂²	1	2	yes	—
72	#380×C₆₇	C₂²	1	1	yes	characteristic
73	#380×C₆₇	C₂²	1	1	yes	characteristic
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
75	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
76	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
77	C₂²×Q₈×C₆₇	—	2	4	no	—
78	C₂²×Q₈×C₆₇	—	2	4	no	—
79	C₂²×Q₈×C₆₇	—	2	4	no	—
80	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
81	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
83	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
90	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
92	#126×C₆₇	—	2	8	no	—
93	#125×C₆₇	—	2	8	no	—
94	#123×C₆₇	—	2	8	no	—
95	#123×C₆₇	—	2	8	no	—
96	#123×C₆₇	—	2	8	no	—
97	#123×C₆₇	—	2	8	no	—
98	C₂²⋊Q₈×C₆₇	—	2	4	no	—
99	C₂²⋊Q₈×C₆₇	—	2	8	no	—
100	#121×C₆₇	—	2	4	no	—
101	C₄×Q₈×C₆₇	—	2	16	no	—
102	C₄×Q₈×C₆₇	—	2	16	no	—
103	C₄×D₄×C₆₇	—	2	8	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	4	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
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×C₄²×C₆₇	—	2	4	no	—
136	C₂×C₄²×C₆₇	—	2	4	no	—
137	C₂×C₄²×C₆₇	—	2	4	no	—
138	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
139	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
140	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
141	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
143	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	4	yes	—
146	#95×C₆₇	C₂³	1	4	yes	—
147	#95×C₆₇	C₂³	1	4	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	2	yes	—
150	#95×C₆₇	C₂³	1	2	yes	—
151	#95×C₆₇	C₂³	1	4	yes	—
152	#95×C₆₇	C₂³	1	4	yes	—
153	#95×C₆₇	C₂³	1	4	yes	—
154	#95×C₆₇	C₂³	1	4	yes	—
155	#95×C₆₇	C₂³	1	4	yes	—
156	#95×C₆₇	C₂³	1	4	yes	—
157	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
158	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₂×Q₈×C₆₇	—	2	32	no	—
160	C₂×Q₈×C₆₇	—	2	8	no	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	2	8	no	—
163	C₂×Q₈×C₆₇	—	2	8	no	—
164	C₂×Q₈×C₆₇	—	4	2	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	2	4	no	—
167	C₂×D₄×C₆₇	—	2	4	no	—
168	C₂×D₄×C₆₇	—	2	2	no	—
169	C₂×D₄×C₆₇	—	4	2	no	—
170	C₂²×C₄×C₆₇	—	2	2	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	2	no	—
173	C₂²×C₄×C₆₇	—	2	2	no	—
174	C₂²×C₄×C₆₇	—	2	2	no	—
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	2	4	no	—
180	C₂²×C₄×C₆₇	—	4	2	no	—
181	C₂²×C₄×C₆₇	—	4	2	no	—
182	C₂²×C₄×C₆₇	—	4	2	no	—
183	C₂²×C₄×C₆₇	—	4	4	no	—
184	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₆₇	#41	1	4	yes	—
188	C₂²×C₄×C₆₇	#41	1	4	yes	—
189	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₆₇	—	2	8	no	—
203	C₄⋊C₄×C₆₇	—	2	4	no	—
204	C₄⋊C₄×C₆₇	—	2	4	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	2	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	—	2	4	no	—
209	C₄⋊C₄×C₆₇	—	2	2	no	—
210	C₄⋊C₄×C₆₇	—	2	4	no	—
211	C₄⋊C₄×C₆₇	—	2	8	no	—
212	C₄⋊C₄×C₆₇	—	2	8	no	—
213	C₄⋊C₄×C₆₇	—	2	8	no	—
214	C₄⋊C₄×C₆₇	—	4	2	no	—
215	C₄⋊C₄×C₆₇	—	4	2	no	—
216	C₄⋊C₄×C₆₇	—	4	4	no	—
217	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
218	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	2	4	no	—
221	C₂²⋊C₄×C₆₇	—	2	4	no	—
222	C₂²⋊C₄×C₆₇	—	2	4	no	—
223	C₂²⋊C₄×C₆₇	—	2	4	no	—
224	C₂²⋊C₄×C₆₇	—	2	4	no	—
225	C₂²⋊C₄×C₆₇	—	2	4	no	—
226	C₂²⋊C₄×C₆₇	—	2	8	no	—
227	C₂²⋊C₄×C₆₇	—	2	8	no	—
228	C₂²⋊C₄×C₆₇	—	2	8	no	—
229	C₂²⋊C₄×C₆₇	—	4	4	no	—
230	C₄²×C₆₇	—	2	8	no	—
231	C₄²×C₆₇	—	2	4	no	—
232	C₄²×C₆₇	—	2	4	no	—
233	C₄²×C₆₇	—	2	8	no	—
234	C₄²×C₆₇	—	2	4	no	—
235	C₄²×C₆₇	—	4	4	no	—
236	C₂³×C₆₇	—	2	2	no	—
237	C₂³×C₆₇	—	2	2	no	—
238	C₂³×C₆₇	—	2	2	no	—
239	C₂³×C₆₇	—	2	2	no	—
240	C₂³×C₆₇	—	2	2	no	—
241	C₂³×C₆₇	—	2	4	no	—
242	C₂³×C₆₇	—	2	4	no	—
243	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
244	C₂³×C₆₇	C₂×#41	1	2	yes	—
245	C₂³×C₆₇	C₂×#41	1	2	yes	—
246	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
247	Q₈×C₆₇	—	2	32	no	—
248	Q₈×C₆₇	—	8	2	no	—
249	D₄×C₆₇	—	4	8	no	—
250	D₄×C₆₇	—	8	2	no	—
251	C₂×C₄×C₆₇	—	2	8	no	—
252	C₂×C₄×C₆₇	—	2	8	no	—
253	C₂×C₄×C₆₇	—	2	8	no	—
254	C₂×C₄×C₆₇	—	2	8	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	4	no	—
276	C₂×C₄×C₆₇	—	2	4	no	—
277	C₂×C₄×C₆₇	—	2	4	no	—
278	C₂×C₄×C₆₇	—	2	4	no	—
279	C₂×C₄×C₆₇	—	2	4	no	—
280	C₂×C₄×C₆₇	—	2	4	no	—
281	C₂×C₄×C₆₇	—	4	4	no	—
282	C₂×C₄×C₆₇	—	4	4	no	—
283	C₂×C₄×C₆₇	—	4	4	no	—
284	C₂×C₄×C₆₇	—	8	4	no	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
290	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
291	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
292	C₂²×C₆₇	—	2	4	no	—
293	C₂²×C₆₇	—	2	2	no	—
294	C₂²×C₆₇	—	2	2	no	—
295	C₂²×C₆₇	—	2	4	no	—
296	C₂²×C₆₇	—	4	2	no	—
297	C₂²×C₆₇	—	4	4	no	—
298	C₂²×C₆₇	—	4	2	no	—
299	C₂²×C₆₇	—	4	2	no	—
300	C₂²×C₆₇	—	4	4	no	—
301	C₂²×C₆₇	#525	1	4	yes	—
302	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
303	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
304	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
305	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
306	C₂²×C₆₇	#585	1	2	yes	—
307	C₄×C₆₇	—	2	4	no	—
308	C₄×C₆₇	—	2	4	no	—
309	C₄×C₆₇	—	2	4	no	—
310	C₄×C₆₇	—	2	4	no	—
311	C₄×C₆₇	—	2	2	no	—
312	C₄×C₆₇	—	2	2	no	—
313	C₄×C₆₇	—	2	4	no	—
314	C₄×C₆₇	—	4	2	no	—
315	C₄×C₆₇	—	4	4	no	—
316	C₄×C₆₇	—	4	2	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	—	4	4	no	—
320	#35092	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
321	C₂×C₆₇	—	2	4	no	—
322	C₂×C₆₇	—	8	2	no	—
323	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
324	C₂×C₆₇	#3461	1	2	yes	—
325	C₂×C₆₇	#3485	1	1	yes	characteristic
326	C₂×C₆₇	#3486	1	1	yes	characteristic
327	C₂×C₆₇	#3507	1	2	yes	—
328	C₂×#539	C₂×C₆₇	1	1	yes	characteristic
329	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
330	#2855	C₂×C₆₇	1	1	yes	characteristic, complemented
331	#2854	C₂×C₆₇	1	1	yes	characteristic
332	#2853	C₂×C₆₇	1	2	yes	complemented
333	#2627	C₂×C₆₇	1	1	yes	characteristic, complemented
334	#2618	C₂×C₆₇	1	1	yes	characteristic
335	#2614	C₂×C₆₇	1	2	yes	complemented
336	#2469	C₂×C₆₇	1	4	yes	complemented
337	#2465	C₂×C₆₇	1	4	yes	complemented
338	#2462	C₂×C₆₇	1	4	yes	complemented
339	#2458	C₂×C₆₇	1	4	yes	complemented
340	#2322	C₂×C₆₇	1	2	yes	complemented
341	#2313	C₂×C₆₇	1	1	yes	characteristic, complemented
342	C₄×#123	C₂×C₆₇	1	2	yes	complemented
343	C₆₇	#35092	1	1	yes	characteristic, direct product factor, Sylow
344	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
345	#539	—	2	8	no	—
346	C₂×#126	C₂²×C₆₇	1	2	yes	—
347	C₂×#125	C₂²×C₆₇	1	2	yes	—
348	C₂×#123	C₂²×C₆₇	1	2	yes	—
349	C₂×#123	C₂²×C₆₇	1	2	yes	—
350	C₂×#123	C₂²×C₆₇	1	2	yes	—
351	C₂×#123	C₂²×C₆₇	1	2	yes	—
352	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
353	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
354	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
355	C₂×C₄×Q₈	—	2	8	no	—
356	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
357	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
358	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
359	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
360	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
361	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
362	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
363	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
364	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
365	#400	C₂²×C₆₇	1	2	yes	—
366	#398	C₂²×C₆₇	1	2	yes	—
367	#398	C₂²×C₆₇	1	4	yes	—
368	#396	C₂²×C₆₇	1	4	yes	—
369	#395	C₂²×C₆₇	1	4	yes	—
370	#395	C₂²×C₆₇	1	4	yes	—
371	#395	C₂²×C₆₇	1	2	yes	—
372	#394	C₂²×C₆₇	1	2	yes	—
373	#393	C₂²×C₆₇	1	4	yes	—
374	#393	C₂²×C₆₇	1	4	yes	—
375	#391	C₂²×C₆₇	1	2	yes	—
376	#390	C₂²×C₆₇	1	2	yes	—
377	#389	C₂²×C₆₇	1	4	yes	—
378	#389	C₂²×C₆₇	1	2	yes	—
379	#389	C₂²×C₆₇	1	2	yes	—
380	#388	C₂²×C₆₇	1	4	yes	—
381	#388	C₂²×C₆₇	1	4	yes	—
382	#388	C₂²×C₆₇	1	4	yes	—
383	#388	C₂²×C₆₇	1	4	yes	—
384	#388	C₂²×C₆₇	1	4	yes	—
385	#387	C₂²×C₆₇	1	4	yes	—
386	#387	C₂²×C₆₇	1	4	yes	—
387	#387	C₂²×C₆₇	1	4	yes	—
388	#387	C₂²×C₆₇	1	4	yes	—
389	#387	C₂²×C₆₇	1	4	yes	—
390	#387	C₂²×C₆₇	1	4	yes	—
391	#387	C₂²×C₆₇	1	4	yes	—
392	#386	C₂²×C₆₇	1	2	yes	—
393	#386	C₂²×C₆₇	1	2	yes	—
394	#385	C₂²×C₆₇	1	2	yes	—
395	#385	C₂²×C₆₇	1	4	yes	—
396	#385	C₂²×C₆₇	1	2	yes	—
397	#381	C₂²×C₆₇	1	2	yes	—
398	#381	C₂²×C₆₇	1	2	yes	—
399	#381	C₂²×C₆₇	1	2	yes	—
400	#380	C₂²×C₆₇	1	1	yes	characteristic
401	#380	C₂²×C₆₇	1	1	yes	characteristic
402	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
403	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
404	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
405	C₂²×Q₈	—	2	4	no	—
406	C₂²×Q₈	—	2	4	no	—
407	C₂²×Q₈	—	2	4	no	—
408	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
409	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
410	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
411	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
412	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
413	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
416	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
417	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
418	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
420	#126	—	2	8	no	—
421	#125	—	2	8	no	—
422	#123	—	2	8	no	—
423	#123	—	2	8	no	—
424	#123	—	2	8	no	—
425	#123	—	2	8	no	—
426	C₂²⋊Q₈	—	2	4	no	—
427	C₂²⋊Q₈	—	2	8	no	—
428	#121	—	2	4	no	—
429	C₄×Q₈	—	2	16	no	—
430	C₄×Q₈	—	2	16	no	—
431	C₄×D₄	—	2	8	no	—
432	C₂×C₄⋊C₄	—	2	4	no	—
433	C₂×C₄⋊C₄	—	2	4	no	—
434	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	—
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
436	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
441	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
453	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
454	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
455	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
457	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
461	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
463	C₂×C₄²	—	2	4	no	—
464	C₂×C₄²	—	2	4	no	—
465	C₂×C₄²	—	2	4	no	—
466	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
467	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
468	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
469	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
470	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
471	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
472	#95	C₂³×C₆₇	1	4	yes	—
473	#95	C₂³×C₆₇	1	4	yes	—
474	#95	C₂³×C₆₇	1	4	yes	—
475	#95	C₂³×C₆₇	1	4	yes	—
476	#95	C₂³×C₆₇	1	4	yes	—
477	#95	C₂³×C₆₇	1	2	yes	—
478	#95	C₂³×C₆₇	1	2	yes	—
479	#95	C₂³×C₆₇	1	4	yes	—
480	#95	C₂³×C₆₇	1	4	yes	—
481	#95	C₂³×C₆₇	1	4	yes	—
482	#95	C₂³×C₆₇	1	4	yes	—
483	#95	C₂³×C₆₇	1	4	yes	—
484	#95	C₂³×C₆₇	1	4	yes	—
485	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
486	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
487	C₂×Q₈	—	2	32	no	—
488	C₂×Q₈	—	2	8	no	—
489	C₂×Q₈	—	2	2	no	—
490	C₂×Q₈	—	2	8	no	—
491	C₂×Q₈	—	2	8	no	—
492	C₂×Q₈	—	4	2	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	4	2	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	2	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	4	2	no	—
509	C₂²×C₄	—	4	2	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	—	4	4	no	—
512	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
513	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
514	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
515	C₂²×C₄	#41×C₆₇	1	4	yes	—
516	C₂²×C₄	#41×C₆₇	1	4	yes	—
517	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
518	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
519	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
525	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
527	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
530	C₄⋊C₄	—	2	8	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	2	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	8	no	—
540	C₄⋊C₄	—	2	8	no	—
541	C₄⋊C₄	—	2	8	no	—
542	C₄⋊C₄	—	4	2	no	—
543	C₄⋊C₄	—	4	2	no	—
544	C₄⋊C₄	—	4	4	no	—
545	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
546	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	8	no	—
555	C₂²⋊C₄	—	2	8	no	—
556	C₂²⋊C₄	—	2	8	no	—
557	C₂²⋊C₄	—	4	4	no	—
558	C₄²	—	2	8	no	—
559	C₄²	—	2	4	no	—
560	C₄²	—	2	4	no	—
561	C₄²	—	2	8	no	—
562	C₄²	—	2	4	no	—
563	C₄²	—	4	4	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	2	no	—
566	C₂³	—	2	2	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	4	no	—
570	C₂³	—	2	4	no	—
571	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
572	C₂³	C₂×#41×C₆₇	1	2	yes	—
573	C₂³	C₂×#41×C₆₇	1	2	yes	—
574	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
575	Q₈	—	2	32	no	—
576	Q₈	—	8	2	no	—
577	D₄	—	4	8	no	—
578	D₄	—	8	2	no	—
579	C₂×C₄	—	2	8	no	—
580	C₂×C₄	—	2	8	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	8	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	4	4	no	—
611	C₂×C₄	—	4	4	no	—
612	C₂×C₄	—	8	4	no	—
613	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
614	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
616	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
617	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
618	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
619	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
620	C₂²	—	2	4	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	4	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	4	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	4	no	—
629	C₂²	#525×C₆₇	1	4	yes	—
630	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
631	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
632	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
633	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
634	C₂²	#585×C₆₇	1	2	yes	—
635	C₄	—	2	4	no	—
636	C₄	—	2	4	no	—
637	C₄	—	2	4	no	—
638	C₄	—	2	4	no	—
639	C₄	—	2	2	no	—
640	C₄	—	2	2	no	—
641	C₄	—	2	4	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	4	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	4	no	—
648	C₂	—	2	4	no	—
649	C₂	—	8	2	no	—
650	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
651	C₂	C₆₇×#3461	1	2	yes	—
652	C₂	C₆₇×#3485	1	1	yes	characteristic
653	C₂	C₆₇×#3486	1	1	yes	characteristic
654	C₂	C₆₇×#3507	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×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×#2855	C₂	semidirect
5. C₆₇×#2853	C₂	semidirect
6. C₆₇×#2627	C₂	semidirect
8. C₆₇×#2614	C₂	semidirect
9. C₆₇×#2469	C₂	semidirect
10. C₆₇×#2465	C₂	semidirect
11. C₆₇×#2462	C₂	semidirect
12. C₆₇×#2458	C₂	semidirect
13. C₆₇×#2322	C₂	semidirect
14. C₆₇×#2313	C₂	semidirect
15. C₄×#123×C₆₇	C₂	semidirect
320. #35092	C₆₇	direct (N × H)
329. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
330. #2855	C₂×C₆₇	semidirect
332. #2853	C₂×C₆₇	semidirect
333. #2627	C₂×C₆₇	semidirect
335. #2614	C₂×C₆₇	semidirect
336. #2469	C₂×C₆₇	semidirect
337. #2465	C₂×C₆₇	semidirect
338. #2462	C₂×C₆₇	semidirect
339. #2458	C₂×C₆₇	semidirect
340. #2322	C₂×C₆₇	semidirect
341. #2313	C₂×C₆₇	semidirect
342. C₄×#123	C₂×C₆₇	semidirect
343. C₆₇	#35092	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#539×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2855	2
C₆₇×#2854	2
C₆₇×#2853	2
C₆₇×#2627	2
C₆₇×#2618	2
C₆₇×#2614	2
C₆₇×#2469	2
C₆₇×#2465	2
C₆₇×#2462	2
C₆₇×#2458	2
C₆₇×#2322	2
C₆₇×#2313	2
C₄×#123×C₆₇	2
#35092	67

Subgroups — C67×#35092 #30714343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35092 #30714343