Subgroups — C₂×C₆₇×#3491 #30742026

Order 17152.

7390 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3491	C₂	1	32	yes	maximal, direct product factor
2	C₂×#552×C₆₇	C₂	1	8	yes	maximal, complemented
3	C₂×#551×C₆₇	C₂	1	8	yes	maximal, complemented
4	C₂×#543×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#539×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#534×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#529×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂²×#123×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#518×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	#552×C₆₇	C₂²	1	128	yes	complemented
14	#551×C₆₇	C₂²	1	128	yes	complemented
15	#543×C₆₇	C₂²	1	16	yes	complemented
16	#539×C₆₇	C₂²	1	64	yes	complemented
17	#534×C₆₇	C₂²	1	16	yes	—
18	#529×C₆₇	C₂²	1	64	yes	complemented
19	C₂×#126×C₆₇	C₂²	1	8	yes	complemented
20	C₂×#126×C₆₇	C₂²	1	8	yes	complemented
21	C₂×#125×C₆₇	C₂²	1	4	yes	complemented
22	C₂×#125×C₆₇	C₂²	1	8	yes	complemented
23	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#123×C₆₇	C₂²	1	8	yes	—
25	C₂×#123×C₆₇	C₂²	1	4	yes	—
26	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
27	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
28	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
29	C₂×#123×C₆₇	C₂²	1	32	yes	complemented
30	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
31	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#121×C₆₇	C₂²	1	4	yes	—
34	C₂×#120×C₆₇	C₂²	1	4	yes	—
35	#518×C₆₇	C₂²	1	32	yes	complemented
36	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
37	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
38	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
39	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
40	C₂×#117×C₆₇	C₂²	1	8	yes	complemented
41	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
42	C₂×#117×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#117×C₆₇	C₂²	1	16	yes	complemented
44	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
45	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
46	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
47	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
48	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
49	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
50	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
51	C₂²×C₄²×C₆₇	C₂²	1	2	yes	complemented
52	C₂⁵×C₆₇	C₂³	1	2	yes	—
53	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
55	C₂²×D₄×C₆₇	—	2	16	no	—
56	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
60	C₂³×C₄×C₆₇	—	2	8	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
68	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
69	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
70	#126×C₆₇	C₂³	1	64	yes	complemented
71	#126×C₆₇	C₂³	1	64	yes	complemented
72	#125×C₆₇	C₂³	1	32	yes	complemented
73	#125×C₆₇	C₂³	1	64	yes	complemented
74	#124×C₆₇	C₂³	1	32	yes	complemented
75	#123×C₆₇	C₂³	1	64	yes	—
76	#123×C₆₇	C₂³	1	32	yes	—
77	#123×C₆₇	C₂³	1	32	yes	complemented
78	#123×C₆₇	C₂³	1	32	yes	complemented
79	#123×C₆₇	C₂³	1	64	yes	complemented
80	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
81	C₂²⋊Q₈×C₆₇	C₂³	1	64	yes	complemented
82	C₂²⋊Q₈×C₆₇	C₂³	1	64	yes	complemented
83	#121×C₆₇	C₂³	1	32	yes	—
84	#120×C₆₇	C₂³	1	32	yes	—
85	C₄×Q₈×C₆₇	C₂³	1	32	yes	complemented
86	C₄×D₄×C₆₇	C₂³	1	64	yes	—
87	C₄×D₄×C₆₇	C₂³	1	64	yes	complemented
88	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
89	#117×C₆₇	C₂³	1	64	yes	complemented
90	#117×C₆₇	C₂³	1	32	yes	complemented
91	#117×C₆₇	C₂³	1	64	yes	complemented
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	complemented
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	complemented
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	complemented
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	complemented
114	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
115	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
116	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	16	yes	complemented
118	C₂⁴×C₆₇	—	2	8	no	—
119	C₂⁴×C₆₇	—	4	4	no	—
120	C₂⁴×C₆₇	—	4	8	no	—
121	C₂⁴×C₆₇	#41	1	8	yes	—
122	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
123	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
124	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
126	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
127	C₂×Q₈×C₆₇	—	4	8	no	—
128	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
129	C₂×D₄×C₆₇	—	2	16	no	—
130	C₂×D₄×C₆₇	—	2	16	no	—
131	C₂×D₄×C₆₇	—	2	64	no	—
132	C₂×D₄×C₆₇	—	4	8	no	—
133	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
134	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
135	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
136	C₂²×C₄×C₆₇	—	2	16	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	32	no	—
140	C₂²×C₄×C₆₇	—	2	8	no	—
141	C₂²×C₄×C₆₇	—	4	4	no	—
142	C₂²×C₄×C₆₇	—	4	16	no	—
143	C₂²×C₄×C₆₇	—	4	8	no	—
144	C₂²×C₄×C₆₇	—	4	4	no	—
145	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	32	yes	—
159	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
160	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
161	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
162	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
163	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
164	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
165	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
166	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
167	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
168	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
169	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
170	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
171	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
172	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
173	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
174	C₄²×C₆₇	C₂⁴	1	16	yes	—
175	C₄²×C₆₇	C₂⁴	1	16	yes	—
176	C₄²×C₆₇	C₂⁴	1	32	yes	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	8	no	—
179	C₂³×C₆₇	—	2	8	no	—
180	C₂³×C₆₇	—	2	16	no	—
181	C₂³×C₆₇	—	2	16	no	—
182	C₂³×C₆₇	—	4	16	no	—
183	C₂³×C₆₇	—	4	8	no	—
184	C₂³×C₆₇	—	4	16	no	—
185	C₂³×C₆₇	—	8	16	no	—
186	C₂³×C₆₇	C₂×#41	1	8	yes	—
187	C₂³×C₆₇	C₂×#41	1	4	yes	—
188	C₂³×C₆₇	C₂×#41	1	16	yes	—
189	C₂³×C₆₇	C₂⁵	1	8	yes	—
190	C₂³×C₆₇	C₂⁵	1	4	yes	—
191	C₂³×C₆₇	C₂⁵	1	2	yes	—
192	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
193	Q₈×C₆₇	—	4	16	no	—
194	D₄×C₆₇	—	2	32	no	—
195	D₄×C₆₇	—	2	32	no	—
196	D₄×C₆₇	—	4	16	no	—
197	C₂×C₄×C₆₇	—	2	32	no	—
198	C₂×C₄×C₆₇	—	2	8	no	—
199	C₂×C₄×C₆₇	—	2	8	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	16	no	—
203	C₂×C₄×C₆₇	—	2	8	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	2	16	no	—
206	C₂×C₄×C₆₇	—	2	8	no	—
207	C₂×C₄×C₆₇	—	4	8	no	—
208	C₂×C₄×C₆₇	—	4	32	no	—
209	C₂×C₄×C₆₇	—	4	16	no	—
210	C₂×C₄×C₆₇	—	4	8	no	—
211	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
212	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
213	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
214	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
215	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
216	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	2	8	no	—
219	C₂²×C₆₇	—	2	16	no	—
220	C₂²×C₆₇	—	2	8	no	—
221	C₂²×C₆₇	—	2	8	no	—
222	C₂²×C₆₇	—	2	16	no	—
223	C₂²×C₆₇	—	4	16	no	—
224	C₂²×C₆₇	—	4	16	no	—
225	C₂²×C₆₇	—	8	16	no	—
226	C₂²×C₆₇	C₂²×#41	1	8	yes	—
227	C₂²×C₆₇	C₂²×#41	1	4	yes	—
228	C₂²×C₆₇	C₂²×#41	1	2	yes	—
229	C₂²×C₆₇	#585	1	4	yes	—
230	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
231	C₄×C₆₇	—	2	4	no	—
232	C₄×C₆₇	—	2	4	no	—
233	C₄×C₆₇	—	2	8	no	—
234	C₄×C₆₇	—	2	8	no	—
235	C₄×C₆₇	—	2	16	no	—
236	C₄×C₆₇	—	2	8	no	—
237	C₂×#3491	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂×C₆₇	—	2	8	no	—
239	C₂×C₆₇	—	2	4	no	—
240	C₂×C₆₇	—	4	8	no	—
241	C₂×C₆₇	#3491	1	4	yes	direct product factor
242	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#585	1	2	yes	—
244	#3491	C₂×C₆₇	1	32	yes	direct product factor
245	C₂×#552	C₂×C₆₇	1	8	yes	complemented
246	C₂×#551	C₂×C₆₇	1	8	yes	complemented
247	C₂×#543	C₂×C₆₇	1	1	yes	characteristic, complemented
248	C₂×#539	C₂×C₆₇	1	4	yes	complemented
249	C₂×#534	C₂×C₆₇	1	1	yes	characteristic
250	C₂×#529	C₂×C₆₇	1	4	yes	complemented
251	C₂²×#123	C₂×C₆₇	1	2	yes	complemented
252	C₂×#518	C₂×C₆₇	1	2	yes	complemented
253	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
254	C₆₇	C₂×#3491	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
256	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
257	#552	C₂²×C₆₇	1	128	yes	complemented
258	#551	C₂²×C₆₇	1	128	yes	complemented
259	#543	C₂²×C₆₇	1	16	yes	complemented
260	#539	C₂²×C₆₇	1	64	yes	complemented
261	#534	C₂²×C₆₇	1	16	yes	—
262	#529	C₂²×C₆₇	1	64	yes	complemented
263	C₂×#126	C₂²×C₆₇	1	8	yes	complemented
264	C₂×#126	C₂²×C₆₇	1	8	yes	complemented
265	C₂×#125	C₂²×C₆₇	1	4	yes	complemented
266	C₂×#125	C₂²×C₆₇	1	8	yes	complemented
267	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
268	C₂×#123	C₂²×C₆₇	1	8	yes	—
269	C₂×#123	C₂²×C₆₇	1	4	yes	—
270	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
271	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
272	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
273	C₂×#123	C₂²×C₆₇	1	32	yes	complemented
274	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
275	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
276	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
277	C₂×#121	C₂²×C₆₇	1	4	yes	—
278	C₂×#120	C₂²×C₆₇	1	4	yes	—
279	#518	C₂²×C₆₇	1	32	yes	complemented
280	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
281	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
282	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
283	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
284	C₂×#117	C₂²×C₆₇	1	8	yes	complemented
285	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
286	C₂×#117	C₂²×C₆₇	1	8	yes	complemented
287	C₂×#117	C₂²×C₆₇	1	16	yes	complemented
288	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
289	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
290	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
291	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
292	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
293	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
294	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
295	C₂²×C₄²	C₂²×C₆₇	1	2	yes	complemented
296	C₂⁵	C₂³×C₆₇	1	2	yes	—
297	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
299	C₂²×D₄	—	2	16	no	—
300	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
301	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
302	C₂²×D₄	C₂³×C₆₇	1	16	yes	—
303	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
304	C₂³×C₄	—	2	8	no	—
305	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
306	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
307	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
311	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
312	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
313	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
314	#126	C₂³×C₆₇	1	64	yes	complemented
315	#126	C₂³×C₆₇	1	64	yes	complemented
316	#125	C₂³×C₆₇	1	32	yes	complemented
317	#125	C₂³×C₆₇	1	64	yes	complemented
318	#124	C₂³×C₆₇	1	32	yes	complemented
319	#123	C₂³×C₆₇	1	64	yes	—
320	#123	C₂³×C₆₇	1	32	yes	—
321	#123	C₂³×C₆₇	1	32	yes	complemented
322	#123	C₂³×C₆₇	1	32	yes	complemented
323	#123	C₂³×C₆₇	1	64	yes	complemented
324	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
325	C₂²⋊Q₈	C₂³×C₆₇	1	64	yes	complemented
326	C₂²⋊Q₈	C₂³×C₆₇	1	64	yes	complemented
327	#121	C₂³×C₆₇	1	32	yes	—
328	#120	C₂³×C₆₇	1	32	yes	—
329	C₄×Q₈	C₂³×C₆₇	1	32	yes	complemented
330	C₄×D₄	C₂³×C₆₇	1	64	yes	—
331	C₄×D₄	C₂³×C₆₇	1	64	yes	complemented
332	C₄×D₄	C₂³×C₆₇	1	32	yes	complemented
333	#117	C₂³×C₆₇	1	64	yes	complemented
334	#117	C₂³×C₆₇	1	32	yes	complemented
335	#117	C₂³×C₆₇	1	64	yes	complemented
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
339	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	complemented
344	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	complemented
345	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	complemented
357	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	complemented
358	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
359	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
360	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
361	C₂×C₄²	C₂³×C₆₇	1	16	yes	complemented
362	C₂⁴	—	2	8	no	—
363	C₂⁴	—	4	4	no	—
364	C₂⁴	—	4	8	no	—
365	C₂⁴	#41×C₆₇	1	8	yes	—
366	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
367	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
368	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
369	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
370	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
371	C₂×Q₈	—	4	8	no	—
372	C₂×Q₈	C₂⁴×C₆₇	1	16	yes	—
373	C₂×D₄	—	2	16	no	—
374	C₂×D₄	—	2	16	no	—
375	C₂×D₄	—	2	64	no	—
376	C₂×D₄	—	4	8	no	—
377	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
378	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
379	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
380	C₂²×C₄	—	2	16	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	32	no	—
384	C₂²×C₄	—	2	8	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	16	no	—
387	C₂²×C₄	—	4	8	no	—
388	C₂²×C₄	—	4	4	no	—
389	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
391	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
393	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
394	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
397	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
398	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
400	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
401	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
402	C₂²×C₄	C₂⁴×C₆₇	1	32	yes	—
403	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
404	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
405	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
406	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
407	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
408	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
409	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
410	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
411	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
412	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
413	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
414	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
415	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
416	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
417	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
418	C₄²	C₂⁴×C₆₇	1	16	yes	—
419	C₄²	C₂⁴×C₆₇	1	16	yes	—
420	C₄²	C₂⁴×C₆₇	1	32	yes	—
421	C₂³	—	2	4	no	—
422	C₂³	—	2	8	no	—
423	C₂³	—	2	8	no	—
424	C₂³	—	2	16	no	—
425	C₂³	—	2	16	no	—
426	C₂³	—	4	16	no	—
427	C₂³	—	4	8	no	—
428	C₂³	—	4	16	no	—
429	C₂³	—	8	16	no	—
430	C₂³	C₂×#41×C₆₇	1	8	yes	—
431	C₂³	C₂×#41×C₆₇	1	4	yes	—
432	C₂³	C₂×#41×C₆₇	1	16	yes	—
433	C₂³	C₂⁵×C₆₇	1	8	yes	—
434	C₂³	C₂⁵×C₆₇	1	4	yes	—
435	C₂³	C₂⁵×C₆₇	1	2	yes	—
436	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
437	Q₈	—	4	16	no	—
438	D₄	—	2	32	no	—
439	D₄	—	2	32	no	—
440	D₄	—	4	16	no	—
441	C₂×C₄	—	2	32	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	16	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	16	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	4	8	no	—
452	C₂×C₄	—	4	32	no	—
453	C₂×C₄	—	4	16	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
456	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
457	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
458	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
459	C₂×C₄	C₂⁵×C₆₇	1	16	yes	—
460	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
461	C₂²	—	2	4	no	—
462	C₂²	—	2	8	no	—
463	C₂²	—	2	16	no	—
464	C₂²	—	2	8	no	—
465	C₂²	—	2	8	no	—
466	C₂²	—	2	16	no	—
467	C₂²	—	4	16	no	—
468	C₂²	—	4	16	no	—
469	C₂²	—	8	16	no	—
470	C₂²	C₂²×#41×C₆₇	1	8	yes	—
471	C₂²	C₂²×#41×C₆₇	1	4	yes	—
472	C₂²	C₂²×#41×C₆₇	1	2	yes	—
473	C₂²	#585×C₆₇	1	4	yes	—
474	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
475	C₄	—	2	4	no	—
476	C₄	—	2	4	no	—
477	C₄	—	2	8	no	—
478	C₄	—	2	8	no	—
479	C₄	—	2	16	no	—
480	C₄	—	2	8	no	—
481	C₂	—	2	8	no	—
482	C₂	—	2	4	no	—
483	C₂	—	4	8	no	—
484	C₂	C₆₇×#3491	1	4	yes	direct product factor
485	C₂	C₂³×#41×C₆₇	1	1	yes	characteristic
486	C₂	C₂×#585×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
prime radical	C₂×#534×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3491	C₂	direct (N × H)
2. C₂×#552×C₆₇	C₂	semidirect
3. C₂×#551×C₆₇	C₂	semidirect
4. C₂×#543×C₆₇	C₂	semidirect
5. C₂×#539×C₆₇	C₂	semidirect
7. C₂×#529×C₆₇	C₂	semidirect
8. C₂²×#123×C₆₇	C₂	semidirect
9. C₂×#518×C₆₇	C₂	semidirect
10. C₂²×#117×C₆₇	C₂	semidirect
13. #552×C₆₇	C₂²	semidirect
14. #551×C₆₇	C₂²	semidirect
15. #543×C₆₇	C₂²	semidirect
16. #539×C₆₇	C₂²	semidirect
18. #529×C₆₇	C₂²	semidirect
19. C₂×#126×C₆₇	C₂²	semidirect
20. C₂×#126×C₆₇	C₂²	semidirect
21. C₂×#125×C₆₇	C₂²	semidirect
22. C₂×#125×C₆₇	C₂²	semidirect
23. C₂×#124×C₆₇	C₂²	semidirect
26. C₂×#123×C₆₇	C₂²	semidirect
27. C₂×#123×C₆₇	C₂²	semidirect
28. C₂×#123×C₆₇	C₂²	semidirect
29. C₂×#123×C₆₇	C₂²	semidirect
30. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
31. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
32. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
35. #518×C₆₇	C₂²	semidirect
36. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
38. C₂×C₄×D₄×C₆₇	C₂²	semidirect
39. C₂×C₄×D₄×C₆₇	C₂²	semidirect
40. C₂×#117×C₆₇	C₂²	semidirect
41. C₂×#117×C₆₇	C₂²	semidirect
42. C₂×#117×C₆₇	C₂²	semidirect
43. C₂×#117×C₆₇	C₂²	semidirect
44. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
45. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
49. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
50. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
51. C₂²×C₄²×C₆₇	C₂²	semidirect
70. #126×C₆₇	C₂³	semidirect
71. #126×C₆₇	C₂³	semidirect
72. #125×C₆₇	C₂³	semidirect
73. #125×C₆₇	C₂³	semidirect
74. #124×C₆₇	C₂³	semidirect
77. #123×C₆₇	C₂³	semidirect
78. #123×C₆₇	C₂³	semidirect
79. #123×C₆₇	C₂³	semidirect
80. C₂²⋊Q₈×C₆₇	C₂³	semidirect
81. C₂²⋊Q₈×C₆₇	C₂³	semidirect
82. C₂²⋊Q₈×C₆₇	C₂³	semidirect
85. C₄×Q₈×C₆₇	C₂³	semidirect
87. C₄×D₄×C₆₇	C₂³	semidirect
88. C₄×D₄×C₆₇	C₂³	semidirect
89. #117×C₆₇	C₂³	semidirect
90. #117×C₆₇	C₂³	semidirect
91. #117×C₆₇	C₂³	semidirect
99. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
100. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
112. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
113. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
117. C₂×C₄²×C₆₇	C₂³	semidirect
237. C₂×#3491	C₆₇	direct (N × H)
241. C₂×C₆₇	#3491	direct (N × H)
244. #3491	C₂×C₆₇	direct (N × H)
245. C₂×#552	C₂×C₆₇	semidirect
246. C₂×#551	C₂×C₆₇	semidirect
247. C₂×#543	C₂×C₆₇	semidirect
248. C₂×#539	C₂×C₆₇	semidirect
250. C₂×#529	C₂×C₆₇	semidirect
251. C₂²×#123	C₂×C₆₇	semidirect
252. C₂×#518	C₂×C₆₇	semidirect
253. C₂²×#117	C₂×C₆₇	semidirect
254. C₆₇	C₂×#3491	direct (N × H)
257. #552	C₂²×C₆₇	semidirect
258. #551	C₂²×C₆₇	semidirect
259. #543	C₂²×C₆₇	semidirect
260. #539	C₂²×C₆₇	semidirect
262. #529	C₂²×C₆₇	semidirect
263. C₂×#126	C₂²×C₆₇	semidirect
264. C₂×#126	C₂²×C₆₇	semidirect
265. C₂×#125	C₂²×C₆₇	semidirect
266. C₂×#125	C₂²×C₆₇	semidirect
267. C₂×#124	C₂²×C₆₇	semidirect
270. C₂×#123	C₂²×C₆₇	semidirect
271. C₂×#123	C₂²×C₆₇	semidirect
272. C₂×#123	C₂²×C₆₇	semidirect
273. C₂×#123	C₂²×C₆₇	semidirect
274. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
275. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
276. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
279. #518	C₂²×C₆₇	semidirect
280. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
282. C₂×C₄×D₄	C₂²×C₆₇	semidirect
283. C₂×C₄×D₄	C₂²×C₆₇	semidirect
284. C₂×#117	C₂²×C₆₇	semidirect
285. C₂×#117	C₂²×C₆₇	semidirect
286. C₂×#117	C₂²×C₆₇	semidirect
287. C₂×#117	C₂²×C₆₇	semidirect
288. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
289. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
293. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
294. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
295. C₂²×C₄²	C₂²×C₆₇	semidirect
314. #126	C₂³×C₆₇	semidirect
315. #126	C₂³×C₆₇	semidirect
316. #125	C₂³×C₆₇	semidirect
317. #125	C₂³×C₆₇	semidirect
318. #124	C₂³×C₆₇	semidirect
321. #123	C₂³×C₆₇	semidirect
322. #123	C₂³×C₆₇	semidirect
323. #123	C₂³×C₆₇	semidirect
324. C₂²⋊Q₈	C₂³×C₆₇	semidirect
325. C₂²⋊Q₈	C₂³×C₆₇	semidirect
326. C₂²⋊Q₈	C₂³×C₆₇	semidirect
329. C₄×Q₈	C₂³×C₆₇	semidirect
331. C₄×D₄	C₂³×C₆₇	semidirect
332. C₄×D₄	C₂³×C₆₇	semidirect
333. #117	C₂³×C₆₇	semidirect
334. #117	C₂³×C₆₇	semidirect
335. #117	C₂³×C₆₇	semidirect
343. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
344. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
356. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
357. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
361. C₂×C₄²	C₂³×C₆₇	semidirect
484. C₂	C₆₇×#3491	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3491	2
C₂×#552×C₆₇	2
C₂×#551×C₆₇	2
C₂×#543×C₆₇	2
C₂×#539×C₆₇	2
C₂×#534×C₆₇	2
C₂×#529×C₆₇	2
C₂²×#123×C₆₇	2
C₂×#518×C₆₇	2
C₂²×#117×C₆₇	2
C₂×#3491	67

Subgroups — C2×C67×#3491 #30742026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3491 #30742026