Subgroups — C₆₇×#35131 #30714382

Order 17152.

2670 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2861	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2858	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2662	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2610	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2545	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2544	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2435	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2434	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2433	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#2411	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2321	C₂	1	4	yes	maximal, complemented
15	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
19	C₂×#127×C₆₇	C₂²	1	2	yes	—
20	#529×C₆₇	—	2	8	no	—
21	C₂×#126×C₆₇	C₂²	1	2	yes	complemented
22	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×#124×C₆₇	—	2	8	no	—
24	C₂×#124×C₆₇	C₂²	1	2	yes	—
25	C₂×#124×C₆₇	C₂²	1	2	yes	—
26	C₂×#124×C₆₇	C₂²	1	2	yes	—
27	C₂×#124×C₆₇	C₂²	1	2	yes	—
28	C₂×#124×C₆₇	C₂²	1	2	yes	—
29	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#123×C₆₇	C₂²	1	2	yes	—
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#120×C₆₇	C₂²	1	4	yes	—
35	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	C₂×#117×C₆₇	C₂²	1	2	yes	—
39	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
40	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
41	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
42	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
43	#396×C₆₇	C₂²	1	8	yes	complemented
44	#395×C₆₇	C₂²	1	4	yes	complemented
45	#394×C₆₇	C₂²	1	4	yes	—
46	#394×C₆₇	C₂²	1	8	yes	complemented
47	#393×C₆₇	C₂²	1	4	yes	complemented
48	#392×C₆₇	C₂²	1	4	yes	—
49	#391×C₆₇	C₂²	1	2	yes	complemented
50	#391×C₆₇	C₂²	1	2	yes	complemented
51	#390×C₆₇	C₂²	1	2	yes	—
52	#390×C₆₇	C₂²	1	2	yes	—
53	#390×C₆₇	C₂²	1	8	yes	complemented
54	#390×C₆₇	C₂²	1	8	yes	complemented
55	#388×C₆₇	C₂²	1	8	yes	complemented
56	#388×C₆₇	C₂²	1	8	yes	complemented
57	#388×C₆₇	C₂²	1	8	yes	complemented
58	#386×C₆₇	C₂²	1	4	yes	—
59	#386×C₆₇	C₂²	1	8	yes	complemented
60	#385×C₆₇	C₂²	1	4	yes	—
61	#379×C₆₇	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
67	#376×C₆₇	C₂²	1	2	yes	complemented
68	C₄³×C₆₇	C₂²	1	2	yes	complemented
69	C₂⁵×C₆₇	C₂³	1	2	yes	—
70	C₂²×Q₈×C₆₇	—	2	4	no	—
71	C₂²×Q₈×C₆₇	—	4	4	no	—
72	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	—	2	4	no	—
76	C₂²×D₄×C₆₇	—	4	4	no	—
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
80	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
81	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
85	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
86	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₇	—	4	4	no	—
88	#127×C₆₇	—	4	4	no	—
89	#126×C₆₇	—	2	8	no	—
90	#125×C₆₇	—	2	4	no	—
91	#124×C₆₇	—	2	8	no	—
92	#124×C₆₇	—	2	8	no	—
93	#124×C₆₇	—	2	8	no	—
94	#124×C₆₇	—	2	8	no	—
95	#124×C₆₇	—	2	4	no	—
96	#124×C₆₇	—	4	4	no	—
97	#124×C₆₇	—	4	4	no	—
98	#123×C₆₇	—	2	8	no	—
99	C₂²⋊Q₈×C₆₇	—	2	4	no	—
100	#121×C₆₇	—	2	4	no	—
101	#120×C₆₇	—	2	16	no	—
102	C₄×Q₈×C₆₇	—	2	4	no	—
103	C₄×D₄×C₆₇	—	2	8	no	—
104	C₄×D₄×C₆₇	—	2	4	no	—
105	#117×C₆₇	—	2	8	no	—
106	#117×C₆₇	—	2	8	no	—
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	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₂²⋊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	8	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
133	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
138	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
142	#95×C₆₇	C₂³	1	4	yes	—
143	#95×C₆₇	C₂³	1	2	yes	—
144	#95×C₆₇	C₂³	1	2	yes	—
145	#95×C₆₇	C₂³	1	8	yes	—
146	#95×C₆₇	C₂³	1	8	yes	—
147	C₂⁴×C₆₇	—	2	4	no	—
148	C₂⁴×C₆₇	—	2	8	no	—
149	C₂⁴×C₆₇	—	4	4	no	—
150	C₂⁴×C₆₇	—	4	4	no	—
151	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
152	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
153	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₂×Q₈×C₆₇	—	2	2	no	—
155	C₂×Q₈×C₆₇	—	2	4	no	—
156	C₂×Q₈×C₆₇	—	2	4	no	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	4	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	—	4	4	no	—
161	C₂×Q₈×C₆₇	—	4	16	no	—
162	C₂×D₄×C₆₇	—	2	4	no	—
163	C₂×D₄×C₆₇	—	2	8	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	2	4	no	—
167	C₂×D₄×C₆₇	—	2	16	no	—
168	C₂×D₄×C₆₇	—	2	4	no	—
169	C₂×D₄×C₆₇	—	2	4	no	—
170	C₂×D₄×C₆₇	—	2	4	no	—
171	C₂×D₄×C₆₇	—	4	4	no	—
172	C₂×D₄×C₆₇	—	4	16	no	—
173	C₂×D₄×C₆₇	—	4	8	no	—
174	C₂×D₄×C₆₇	—	4	8	no	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	4	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	4	no	—
179	C₂²×C₄×C₆₇	—	4	2	no	—
180	C₂²×C₄×C₆₇	—	4	4	no	—
181	C₂²×C₄×C₆₇	—	4	4	no	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	4	no	—
199	C₄⋊C₄×C₆₇	—	2	2	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	2	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	8	no	—
204	C₄⋊C₄×C₆₇	—	4	8	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	—	2	8	no	—
208	C₂²⋊C₄×C₆₇	—	2	8	no	—
209	C₂²⋊C₄×C₆₇	—	2	8	no	—
210	C₂²⋊C₄×C₆₇	—	2	8	no	—
211	C₂²⋊C₄×C₆₇	—	2	4	no	—
212	C₂²⋊C₄×C₆₇	—	2	4	no	—
213	C₂²⋊C₄×C₆₇	—	2	4	no	—
214	C₂²⋊C₄×C₆₇	—	2	4	no	—
215	C₂²⋊C₄×C₆₇	—	2	16	no	—
216	C₂²⋊C₄×C₆₇	—	4	8	no	—
217	C₂²⋊C₄×C₆₇	—	4	8	no	—
218	C₂²⋊C₄×C₆₇	—	4	8	no	—
219	C₂²⋊C₄×C₆₇	#41	1	8	yes	—
220	C₄²×C₆₇	—	2	4	no	—
221	C₄²×C₆₇	—	2	8	no	—
222	C₄²×C₆₇	—	2	4	no	—
223	C₄²×C₆₇	—	2	4	no	—
224	C₄²×C₆₇	—	4	4	no	—
225	C₄²×C₆₇	C₂×D₄	1	4	yes	—
226	C₄²×C₆₇	C₂×D₄	1	4	yes	—
227	C₄²×C₆₇	C₂×D₄	1	4	yes	—
228	C₄²×C₆₇	C₂×D₄	1	4	yes	—
229	C₂³×C₆₇	—	2	2	no	—
230	C₂³×C₆₇	—	2	2	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	2	4	no	—
233	C₂³×C₆₇	—	2	8	no	—
234	C₂³×C₆₇	—	2	4	no	—
235	C₂³×C₆₇	—	2	4	no	—
236	C₂³×C₆₇	—	2	4	no	—
237	C₂³×C₆₇	—	2	4	no	—
238	C₂³×C₆₇	—	2	8	no	—
239	C₂³×C₆₇	—	2	4	no	—
240	C₂³×C₆₇	—	4	8	no	—
241	C₂³×C₆₇	—	4	8	no	—
242	C₂³×C₆₇	—	4	8	no	—
243	C₂³×C₆₇	—	8	16	no	—
244	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
245	C₂³×C₆₇	C₂×#41	1	4	yes	—
246	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
247	Q₈×C₆₇	—	4	8	no	—
248	Q₈×C₆₇	—	4	4	no	—
249	Q₈×C₆₇	—	4	8	no	—
250	D₄×C₆₇	—	4	8	no	—
251	D₄×C₆₇	—	4	8	no	—
252	D₄×C₆₇	—	4	4	no	—
253	D₄×C₆₇	—	4	8	no	—
254	D₄×C₆₇	—	8	16	no	—
255	C₂×C₄×C₆₇	—	2	8	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	8	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	4	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	4	no	—
276	C₂×C₄×C₆₇	—	4	4	no	—
277	C₂×C₄×C₆₇	—	4	8	no	—
278	C₂×C₄×C₆₇	—	4	8	no	—
279	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
286	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
287	C₂²×C₆₇	—	2	2	no	—
288	C₂²×C₆₇	—	2	2	no	—
289	C₂²×C₆₇	—	2	8	no	—
290	C₂²×C₆₇	—	4	2	no	—
291	C₂²×C₆₇	—	4	4	no	—
292	C₂²×C₆₇	—	4	4	no	—
293	C₂²×C₆₇	—	4	2	no	—
294	C₂²×C₆₇	—	4	2	no	—
295	C₂²×C₆₇	—	4	4	no	—
296	C₂²×C₆₇	—	4	8	no	—
297	C₂²×C₆₇	—	4	8	no	—
298	C₂²×C₆₇	—	4	8	no	—
299	C₂²×C₆₇	—	8	16	no	—
300	C₂²×C₆₇	#532	1	4	yes	—
301	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
302	C₂²×C₆₇	C₂²×#41	1	2	yes	—
303	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
304	C₂²×C₆₇	#585	1	2	yes	—
305	C₄×C₆₇	—	2	2	no	—
306	C₄×C₆₇	—	2	2	no	—
307	C₄×C₆₇	—	2	2	no	—
308	C₄×C₆₇	—	2	2	no	—
309	C₄×C₆₇	—	2	2	no	—
310	C₄×C₆₇	—	2	8	no	—
311	C₄×C₆₇	—	4	2	no	—
312	C₄×C₆₇	—	4	4	no	—
313	C₄×C₆₇	—	4	2	no	—
314	C₄×C₆₇	—	4	2	no	—
315	C₄×C₆₇	—	4	4	no	—
316	C₄×C₆₇	#535	1	4	yes	—
317	#35131	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
318	C₂×C₆₇	—	2	4	no	—
319	C₂×C₆₇	—	8	2	no	—
320	C₂×C₆₇	—	8	4	no	—
321	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
322	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
323	C₂×C₆₇	#3449	1	1	yes	characteristic
324	C₂×C₆₇	#3514	1	4	yes	—
325	C₂×#529	C₂×C₆₇	1	1	yes	characteristic, complemented
326	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#2861	C₂×C₆₇	1	2	yes	complemented
328	#2858	C₂×C₆₇	1	2	yes	complemented
329	#2662	C₂×C₆₇	1	4	yes	complemented
330	#2610	C₂×C₆₇	1	4	yes	complemented
331	#2545	C₂×C₆₇	1	2	yes	complemented
332	#2544	C₂×C₆₇	1	2	yes	complemented
333	#2436	C₂×C₆₇	1	1	yes	characteristic, complemented
334	#2435	C₂×C₆₇	1	1	yes	characteristic, complemented
335	#2434	C₂×C₆₇	1	1	yes	characteristic, complemented
336	#2433	C₂×C₆₇	1	1	yes	characteristic
337	#2411	C₂×C₆₇	1	4	yes	complemented
338	#2321	C₂×C₆₇	1	4	yes	complemented
339	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
340	C₆₇	#35131	1	1	yes	characteristic, direct product factor, Sylow
341	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
342	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
344	C₂×#127	C₂²×C₆₇	1	2	yes	—
345	#529	—	2	8	no	—
346	C₂×#126	C₂²×C₆₇	1	2	yes	complemented
347	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
348	C₂×#124	—	2	8	no	—
349	C₂×#124	C₂²×C₆₇	1	2	yes	—
350	C₂×#124	C₂²×C₆₇	1	2	yes	—
351	C₂×#124	C₂²×C₆₇	1	2	yes	—
352	C₂×#124	C₂²×C₆₇	1	2	yes	—
353	C₂×#124	C₂²×C₆₇	1	2	yes	—
354	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic, complemented
355	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
356	C₂×#123	C₂²×C₆₇	1	2	yes	—
357	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
358	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×#120	C₂²×C₆₇	1	4	yes	—
360	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
361	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
362	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
363	C₂×#117	C₂²×C₆₇	1	2	yes	—
364	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
365	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
366	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
367	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
368	#396	C₂²×C₆₇	1	8	yes	complemented
369	#395	C₂²×C₆₇	1	4	yes	complemented
370	#394	C₂²×C₆₇	1	4	yes	—
371	#394	C₂²×C₆₇	1	8	yes	complemented
372	#393	C₂²×C₆₇	1	4	yes	complemented
373	#392	C₂²×C₆₇	1	4	yes	—
374	#391	C₂²×C₆₇	1	2	yes	complemented
375	#391	C₂²×C₆₇	1	2	yes	complemented
376	#390	C₂²×C₆₇	1	2	yes	—
377	#390	C₂²×C₆₇	1	2	yes	—
378	#390	C₂²×C₆₇	1	8	yes	complemented
379	#390	C₂²×C₆₇	1	8	yes	complemented
380	#388	C₂²×C₆₇	1	8	yes	complemented
381	#388	C₂²×C₆₇	1	8	yes	complemented
382	#388	C₂²×C₆₇	1	8	yes	complemented
383	#386	C₂²×C₆₇	1	4	yes	—
384	#386	C₂²×C₆₇	1	8	yes	complemented
385	#385	C₂²×C₆₇	1	4	yes	—
386	#379	C₂²×C₆₇	1	4	yes	—
387	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
388	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
389	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
390	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
391	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
392	#376	C₂²×C₆₇	1	2	yes	complemented
393	C₄³	C₂²×C₆₇	1	2	yes	complemented
394	C₂⁵	C₂³×C₆₇	1	2	yes	—
395	C₂²×Q₈	—	2	4	no	—
396	C₂²×Q₈	—	4	4	no	—
397	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
398	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
399	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
400	C₂²×D₄	—	2	4	no	—
401	C₂²×D₄	—	4	4	no	—
402	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
403	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
404	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
405	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
406	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
407	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
409	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
410	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
411	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
412	C₄⋊Q₈	—	4	4	no	—
413	#127	—	4	4	no	—
414	#126	—	2	8	no	—
415	#125	—	2	4	no	—
416	#124	—	2	8	no	—
417	#124	—	2	8	no	—
418	#124	—	2	8	no	—
419	#124	—	2	8	no	—
420	#124	—	2	4	no	—
421	#124	—	4	4	no	—
422	#124	—	4	4	no	—
423	#123	—	2	8	no	—
424	C₂²⋊Q₈	—	2	4	no	—
425	#121	—	2	4	no	—
426	#120	—	2	16	no	—
427	C₄×Q₈	—	2	4	no	—
428	C₄×D₄	—	2	8	no	—
429	C₄×D₄	—	2	4	no	—
430	#117	—	2	8	no	—
431	#117	—	2	8	no	—
432	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
433	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
440	C₂×C₂²⋊C₄	—	2	8	no	—
441	C₂×C₂²⋊C₄	—	2	8	no	—
442	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
453	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
454	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
455	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
457	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
458	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
459	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
460	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
461	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
462	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
463	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
464	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
465	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
466	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
467	#95	C₂³×C₆₇	1	4	yes	—
468	#95	C₂³×C₆₇	1	2	yes	—
469	#95	C₂³×C₆₇	1	2	yes	—
470	#95	C₂³×C₆₇	1	8	yes	—
471	#95	C₂³×C₆₇	1	8	yes	—
472	C₂⁴	—	2	4	no	—
473	C₂⁴	—	2	8	no	—
474	C₂⁴	—	4	4	no	—
475	C₂⁴	—	4	4	no	—
476	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
477	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
478	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
479	C₂×Q₈	—	2	2	no	—
480	C₂×Q₈	—	2	4	no	—
481	C₂×Q₈	—	2	4	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	4	no	—
485	C₂×Q₈	—	4	4	no	—
486	C₂×Q₈	—	4	16	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	2	8	no	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	2	4	no	—
492	C₂×D₄	—	2	16	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	4	4	no	—
497	C₂×D₄	—	4	16	no	—
498	C₂×D₄	—	4	8	no	—
499	C₂×D₄	—	4	8	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	4	4	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	4	no	—
507	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
509	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
510	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
511	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
516	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
518	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
520	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	8	no	—
529	C₄⋊C₄	—	4	8	no	—
530	C₂²⋊C₄	—	2	4	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	8	no	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	8	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	16	no	—
541	C₂²⋊C₄	—	4	8	no	—
542	C₂²⋊C₄	—	4	8	no	—
543	C₂²⋊C₄	—	4	8	no	—
544	C₂²⋊C₄	#41×C₆₇	1	8	yes	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	8	no	—
547	C₄²	—	2	4	no	—
548	C₄²	—	2	4	no	—
549	C₄²	—	4	4	no	—
550	C₄²	C₂×D₄×C₆₇	1	4	yes	—
551	C₄²	C₂×D₄×C₆₇	1	4	yes	—
552	C₄²	C₂×D₄×C₆₇	1	4	yes	—
553	C₄²	C₂×D₄×C₆₇	1	4	yes	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	8	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	2	4	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	2	4	no	—
563	C₂³	—	2	8	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	4	8	no	—
566	C₂³	—	4	8	no	—
567	C₂³	—	4	8	no	—
568	C₂³	—	8	16	no	—
569	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
570	C₂³	C₂×#41×C₆₇	1	4	yes	—
571	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
572	Q₈	—	4	8	no	—
573	Q₈	—	4	8	no	—
574	Q₈	—	4	4	no	—
575	D₄	—	4	8	no	—
576	D₄	—	4	8	no	—
577	D₄	—	4	8	no	—
578	D₄	—	4	4	no	—
579	D₄	—	8	16	no	—
580	C₂×C₄	—	2	8	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	8	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	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	4	4	no	—
602	C₂×C₄	—	4	8	no	—
603	C₂×C₄	—	4	8	no	—
604	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
606	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
607	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
608	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
609	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
610	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
611	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	2	no	—
614	C₂²	—	2	8	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	4	no	—
617	C₂²	—	4	4	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	4	no	—
621	C₂²	—	4	8	no	—
622	C₂²	—	4	8	no	—
623	C₂²	—	4	8	no	—
624	C₂²	—	8	16	no	—
625	C₂²	#532×C₆₇	1	4	yes	—
626	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
627	C₂²	C₂²×#41×C₆₇	1	2	yes	—
628	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
629	C₂²	#585×C₆₇	1	2	yes	—
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	8	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	4	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	4	no	—
641	C₄	#535×C₆₇	1	4	yes	—
642	C₂	—	2	4	no	—
643	C₂	—	8	4	no	—
644	C₂	—	8	2	no	—
645	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
646	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
647	C₂	C₆₇×#3449	1	1	yes	characteristic
648	C₂	C₆₇×#3514	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₆₇×#2433	C₂
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
1. C₂×#529×C₆₇	C₂	semidirect
2. C₂²×#124×C₆₇	C₂	semidirect
3. C₆₇×#2861	C₂	semidirect
4. C₆₇×#2858	C₂	semidirect
5. C₆₇×#2662	C₂	semidirect
6. C₆₇×#2610	C₂	semidirect
7. C₆₇×#2545	C₂	semidirect
8. C₆₇×#2544	C₂	semidirect
9. C₆₇×#2436	C₂	semidirect
10. C₆₇×#2435	C₂	semidirect
11. C₆₇×#2434	C₂	semidirect
13. C₆₇×#2411	C₂	semidirect
14. C₆₇×#2321	C₂	semidirect
15. C₄×#117×C₆₇	C₂	semidirect
18. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
21. C₂×#126×C₆₇	C₂²	semidirect
22. C₂×#125×C₆₇	C₂²	semidirect
29. C₂×#124×C₆₇	C₂²	semidirect
30. C₂×#124×C₆₇	C₂²	semidirect
35. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
37. C₂×C₄×D₄×C₆₇	C₂²	semidirect
39. C₂×#117×C₆₇	C₂²	semidirect
43. #396×C₆₇	C₂²	semidirect
44. #395×C₆₇	C₂²	semidirect
46. #394×C₆₇	C₂²	semidirect
47. #393×C₆₇	C₂²	semidirect
49. #391×C₆₇	C₂²	semidirect
50. #391×C₆₇	C₂²	semidirect
53. #390×C₆₇	C₂²	semidirect
54. #390×C₆₇	C₂²	semidirect
55. #388×C₆₇	C₂²	semidirect
56. #388×C₆₇	C₂²	semidirect
57. #388×C₆₇	C₂²	semidirect
59. #386×C₆₇	C₂²	semidirect
62. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
63. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
67. #376×C₆₇	C₂²	semidirect
68. C₄³×C₆₇	C₂²	semidirect
132. C₂×C₄²×C₆₇	D₄	semidirect
317. #35131	C₆₇	direct (N × H)
325. C₂×#529	C₂×C₆₇	semidirect
326. C₂²×#124	C₂×C₆₇	semidirect
327. #2861	C₂×C₆₇	semidirect
328. #2858	C₂×C₆₇	semidirect
329. #2662	C₂×C₆₇	semidirect
330. #2610	C₂×C₆₇	semidirect
331. #2545	C₂×C₆₇	semidirect
332. #2544	C₂×C₆₇	semidirect
333. #2436	C₂×C₆₇	semidirect
334. #2435	C₂×C₆₇	semidirect
335. #2434	C₂×C₆₇	semidirect
337. #2411	C₂×C₆₇	semidirect
338. #2321	C₂×C₆₇	semidirect
339. C₄×#117	C₂×C₆₇	semidirect
340. C₆₇	#35131	direct (N × H)
343. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
346. C₂×#126	C₂²×C₆₇	semidirect
347. C₂×#125	C₂²×C₆₇	semidirect
354. C₂×#124	C₂²×C₆₇	semidirect
355. C₂×#124	C₂²×C₆₇	semidirect
360. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
362. C₂×C₄×D₄	C₂²×C₆₇	semidirect
364. C₂×#117	C₂²×C₆₇	semidirect
368. #396	C₂²×C₆₇	semidirect
369. #395	C₂²×C₆₇	semidirect
371. #394	C₂²×C₆₇	semidirect
372. #393	C₂²×C₆₇	semidirect
374. #391	C₂²×C₆₇	semidirect
375. #391	C₂²×C₆₇	semidirect
378. #390	C₂²×C₆₇	semidirect
379. #390	C₂²×C₆₇	semidirect
380. #388	C₂²×C₆₇	semidirect
381. #388	C₂²×C₆₇	semidirect
382. #388	C₂²×C₆₇	semidirect
384. #386	C₂²×C₆₇	semidirect
387. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
388. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
392. #376	C₂²×C₆₇	semidirect
393. C₄³	C₂²×C₆₇	semidirect
457. C₂×C₄²	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₂²×#124×C₆₇	2
C₆₇×#2861	2
C₆₇×#2858	2
C₆₇×#2662	2
C₆₇×#2610	2
C₆₇×#2545	2
C₆₇×#2544	2
C₆₇×#2436	2
C₆₇×#2435	2
C₆₇×#2434	2
C₆₇×#2433	2
C₆₇×#2411	2
C₆₇×#2321	2
C₄×#117×C₆₇	2
#35131	67

Subgroups — C67×#35131 #30714382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35131 #30714382