Subgroups — C₂²×C₇×#466 #13237651

Order 1792.

3694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#121	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#466	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×#102	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×D₄	—	2	2	no	—
8	C₂³×C₇×D₄	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇×D₈	—	2	8	no	—
11	C₂²×C₇×D₈	C₂²	1	12	yes	complemented
12	C₂³×C₇×C₈	—	2	8	no	—
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂³×C₇×C₈	C₂²	1	12	yes	complemented
16	C₂×C₇×#121	—	2	12	no	—
17	C₂×C₇×#121	C₂²	1	24	yes	complemented
18	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
20	C₇×#466	C₂²	1	64	yes	direct product factor
21	C₂×C₇×#107	C₂²	1	12	yes	complemented
22	C₂×C₇×#102	C₂²	1	24	yes	complemented
23	C₂²×#466	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₇	—	2	2	no	—
25	C₂⁵×C₇	D₄	1	1	yes	characteristic
26	C₂²×C₇×D₄	—	2	24	no	—
27	C₂²×C₇×D₄	—	2	8	no	—
28	C₂²×C₇×D₄	—	2	6	no	—
29	C₂²×C₇×D₄	—	2	8	no	—
30	C₂²×C₇×D₄	—	4	16	no	—
31	C₂²×C₇×D₄	C₂³	1	12	yes	—
32	C₂³×C₄×C₇	—	2	4	no	—
33	C₂³×C₄×C₇	—	2	4	no	—
34	C₂³×C₄×C₇	—	2	2	no	—
35	C₂³×C₄×C₇	D₄	1	6	yes	—
36	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₇	C₂³	1	6	yes	—
38	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₇×D₈	—	2	48	no	—
40	C₂×C₇×D₈	C₂³	1	16	yes	complemented
41	C₂²×C₇×C₈	—	2	4	no	—
42	C₂²×C₇×C₈	—	2	48	no	—
43	C₂²×C₇×C₈	D₄	1	8	yes	complemented
44	C₂²×C₇×C₈	C₂³	1	6	yes	—
45	C₂²×C₇×C₈	C₂³	1	6	yes	—
46	C₂²×C₇×C₈	C₂³	1	16	yes	complemented
47	C₇×#121	—	2	48	no	—
48	C₇×#121	C₂³	1	32	yes	complemented
49	C₂×C₇×C₄⋊C₄	—	2	6	no	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
51	C₂×C₇×C₂²⋊C₄	—	2	24	no	—
52	C₇×#107	C₂³	1	16	yes	complemented
53	C₇×#102	C₂³	1	32	yes	complemented
54	C₂³×D₈	C₂×C₇	1	1	yes	characteristic, complemented
55	C₂⁴×C₈	C₂×C₇	1	1	yes	characteristic, complemented
56	C₂²×#121	C₂×C₇	1	2	yes	complemented
57	C₂×#466	C₂×C₇	1	24	yes	direct product factor
58	C₂²×#107	C₂×C₇	1	1	yes	characteristic, complemented
59	C₂²×#102	C₂×C₇	1	2	yes	complemented
60	C₂⁴×C₇	—	2	4	no	—
61	C₂⁴×C₇	—	2	4	no	—
62	C₂⁴×C₇	—	2	12	no	—
63	C₂⁴×C₇	—	4	8	no	—
64	C₂⁴×C₇	—	4	8	no	—
65	C₂⁴×C₇	—	4	8	no	—
66	C₂⁴×C₇	D₈	1	8	yes	complemented
67	C₂⁴×C₇	C₂×D₄	1	6	yes	—
68	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₇×D₄	—	2	32	no	—
70	C₂×C₇×D₄	—	2	24	no	—
71	C₂×C₇×D₄	—	2	12	no	—
72	C₂×C₇×D₄	—	2	24	no	—
73	C₂×C₇×D₄	—	4	96	no	—
74	C₂×C₇×D₄	—	4	24	no	—
75	C₂×C₇×D₄	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₇	—	2	4	no	—
77	C₂²×C₄×C₇	—	2	24	no	—
78	C₂²×C₄×C₇	—	2	12	no	—
79	C₂²×C₄×C₇	—	2	12	no	—
80	C₂²×C₄×C₇	—	2	12	no	—
81	C₂²×C₄×C₇	—	4	8	no	—
82	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₇	#41	1	4	yes	—
87	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
89	C₇×D₈	—	2	32	no	—
90	C₂×C₇×C₈	—	2	12	no	—
91	C₂×C₇×C₈	—	2	32	no	—
92	C₂×C₇×C₈	C₂×D₄	1	24	yes	complemented
93	C₂×C₇×C₈	C₂⁴	1	4	yes	—
94	C₂×C₇×C₈	C₂⁴	1	4	yes	—
95	C₇×C₄⋊C₄	—	2	12	no	—
96	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
97	C₇×C₂²⋊C₄	—	2	32	no	—
98	C₂³×D₄	—	2	2	no	—
99	C₂³×D₄	C₂²×C₇	1	2	yes	—
100	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
101	C₂²×D₈	—	2	8	no	—
102	C₂²×D₈	C₂²×C₇	1	12	yes	complemented
103	C₂³×C₈	—	2	8	no	—
104	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
105	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
106	C₂³×C₈	C₂²×C₇	1	12	yes	complemented
107	C₂×#121	—	2	12	no	—
108	C₂×#121	C₂²×C₇	1	24	yes	complemented
109	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂²×C₂²⋊C₄	—	2	2	no	—
111	#466	C₂²×C₇	1	64	yes	direct product factor
112	C₂×#107	C₂²×C₇	1	12	yes	complemented
113	C₂×#102	C₂²×C₇	1	24	yes	complemented
114	C₂³×C₇	—	2	12	no	—
115	C₂³×C₇	—	2	12	no	—
116	C₂³×C₇	—	2	32	no	—
117	C₂³×C₇	—	2	8	no	—
118	C₂³×C₇	—	4	24	no	—
119	C₂³×C₇	—	4	24	no	—
120	C₂³×C₇	—	4	24	no	—
121	C₂³×C₇	—	8	32	no	—
122	C₂³×C₇	#121	1	4	yes	—
123	C₂³×C₇	C₂×D₈	1	4	yes	—
124	C₂³×C₇	C₂×D₈	1	24	yes	complemented
125	C₂³×C₇	#135	1	4	yes	—
126	C₂³×C₇	C₂²×D₄	1	4	yes	—
127	C₂³×C₇	C₂²×D₄	1	3	yes	—
128	C₇×D₄	—	2	8	no	—
129	C₇×D₄	—	2	8	no	—
130	C₇×D₄	—	4	64	no	—
131	C₇×D₄	—	4	24	no	—
132	C₂×C₄×C₇	—	2	12	no	—
133	C₂×C₄×C₇	—	2	24	no	—
134	C₂×C₄×C₇	—	2	4	no	—
135	C₂×C₄×C₇	—	2	4	no	—
136	C₂×C₄×C₇	—	2	8	no	—
137	C₂×C₄×C₇	—	4	24	no	—
138	C₂×C₄×C₇	#121	1	12	yes	—
139	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
140	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
141	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
142	C₂×C₄×C₇	C₂×#41	1	6	yes	—
143	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
144	C₇×C₈	—	2	4	no	—
145	C₇×C₈	C₂²×D₄	1	8	yes	complemented
146	C₂⁵	—	2	2	no	—
147	C₂⁵	C₇×D₄	1	1	yes	characteristic
148	C₂²×D₄	—	2	24	no	—
149	C₂²×D₄	—	2	8	no	—
150	C₂²×D₄	—	2	6	no	—
151	C₂²×D₄	—	2	8	no	—
152	C₂²×D₄	—	4	16	no	—
153	C₂²×D₄	C₂³×C₇	1	12	yes	—
154	C₂³×C₄	—	2	4	no	—
155	C₂³×C₄	—	2	4	no	—
156	C₂³×C₄	—	2	2	no	—
157	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
158	C₂³×C₄	C₇×D₄	1	6	yes	—
159	C₂³×C₄	C₂³×C₇	1	6	yes	—
160	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
161	C₂×D₈	—	2	48	no	—
162	C₂×D₈	C₂³×C₇	1	16	yes	complemented
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	—	2	48	no	—
165	C₂²×C₈	C₇×D₄	1	8	yes	complemented
166	C₂²×C₈	C₂³×C₇	1	6	yes	—
167	C₂²×C₈	C₂³×C₇	1	6	yes	—
168	C₂²×C₈	C₂³×C₇	1	16	yes	complemented
169	#121	—	2	48	no	—
170	#121	C₂³×C₇	1	32	yes	complemented
171	C₂×C₄⋊C₄	—	2	6	no	—
172	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
173	C₂×C₂²⋊C₄	—	2	24	no	—
174	#107	C₂³×C₇	1	16	yes	complemented
175	#102	C₂³×C₇	1	32	yes	complemented
176	C₂²×C₇	—	2	4	no	—
177	C₂²×C₇	—	2	48	no	—
178	C₂²×C₇	—	2	4	no	—
179	C₂²×C₇	—	4	8	no	—
180	C₂²×C₇	—	4	8	no	—
181	C₂²×C₇	—	4	8	no	—
182	C₂²×C₇	—	8	48	no	—
183	C₂²×C₇	#466	1	16	yes	direct product factor
184	C₂²×C₇	C₂×#121	1	6	yes	—
185	C₂²×C₇	C₂²×D₈	1	6	yes	—
186	C₂²×C₇	C₂²×D₈	1	8	yes	complemented
187	C₂²×C₇	C₂×#135	1	6	yes	—
188	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
189	C₄×C₇	—	2	4	no	—
190	C₄×C₇	—	4	8	no	—
191	C₄×C₇	C₂×#121	1	6	yes	—
192	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
193	C₄×C₇	C₂²×#41	1	1	yes	characteristic
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	2	12	no	—
197	C₂⁴	—	4	8	no	—
198	C₂⁴	—	4	8	no	—
199	C₂⁴	—	4	8	no	—
200	C₂⁴	C₇×D₈	1	8	yes	complemented
201	C₂⁴	C₂×C₇×D₄	1	6	yes	—
202	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
203	C₂×D₄	—	2	32	no	—
204	C₂×D₄	—	2	24	no	—
205	C₂×D₄	—	2	12	no	—
206	C₂×D₄	—	2	24	no	—
207	C₂×D₄	—	4	96	no	—
208	C₂×D₄	—	4	24	no	—
209	C₂×D₄	C₂⁴×C₇	1	8	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	24	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	12	no	—
214	C₂²×C₄	—	2	12	no	—
215	C₂²×C₄	—	4	8	no	—
216	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
217	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
218	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
219	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
220	C₂²×C₄	C₇×#41	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
222	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
223	D₈	—	2	32	no	—
224	C₂×C₈	—	2	12	no	—
225	C₂×C₈	—	2	32	no	—
226	C₂×C₈	C₂×C₇×D₄	1	24	yes	complemented
227	C₂×C₈	C₂⁴×C₇	1	4	yes	—
228	C₂×C₈	C₂⁴×C₇	1	4	yes	—
229	C₄⋊C₄	—	2	12	no	—
230	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
231	C₂²⋊C₄	—	2	32	no	—
232	C₂×C₇	—	2	8	no	—
233	C₂×C₇	—	8	8	no	—
234	C₂×C₇	C₂×#466	1	12	yes	direct product factor
235	C₂×C₇	C₂²×#121	1	1	yes	characteristic
236	C₂×C₇	C₂³×D₈	1	1	yes	characteristic
237	C₂×C₇	C₂²×#135	1	1	yes	characteristic
238	C₂³	—	2	32	no	—
239	C₂³	—	2	12	no	—
240	C₂³	—	2	12	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	4	24	no	—
243	C₂³	—	4	24	no	—
244	C₂³	—	4	24	no	—
245	C₂³	—	8	32	no	—
246	C₂³	C₇×#121	1	4	yes	—
247	C₂³	C₂×C₇×D₈	1	4	yes	—
248	C₂³	C₂×C₇×D₈	1	24	yes	complemented
249	C₂³	C₇×#135	1	4	yes	—
250	C₂³	C₂²×C₇×D₄	1	4	yes	—
251	C₂³	C₂²×C₇×D₄	1	3	yes	—
252	D₄	—	2	8	no	—
253	D₄	—	2	8	no	—
254	D₄	—	4	64	no	—
255	D₄	—	4	24	no	—
256	C₂×C₄	—	2	12	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	24	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	24	no	—
262	C₂×C₄	C₇×#121	1	12	yes	—
263	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
264	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
265	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
266	C₂×C₄	C₂×C₇×#41	1	6	yes	—
267	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	C₂²×C₇×D₄	1	8	yes	complemented
270	C₇	C₂²×#466	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	48	no	—
274	C₂²	—	4	8	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	4	8	no	—
277	C₂²	—	8	48	no	—
278	C₂²	C₇×#466	1	16	yes	direct product factor
279	C₂²	C₂×C₇×#121	1	6	yes	—
280	C₂²	C₂²×C₇×D₈	1	6	yes	—
281	C₂²	C₂²×C₇×D₈	1	8	yes	complemented
282	C₂²	C₂×C₇×#135	1	6	yes	—
283	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	4	8	no	—
286	C₄	C₂×C₇×#121	1	6	yes	—
287	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
288	C₄	C₂²×C₇×#41	1	1	yes	characteristic
289	C₂	—	2	8	no	—
290	C₂	—	8	8	no	—
291	C₂	C₂×C₇×#466	1	12	yes	direct product factor
292	C₂	C₂²×C₇×#121	1	1	yes	characteristic
293	C₂	C₂³×C₇×D₈	1	1	yes	characteristic
294	C₂	C₂²×C₇×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×D₈	C₂	semidirect
2. C₂⁴×C₇×C₈	C₂	semidirect
3. C₂²×C₇×#121	C₂	semidirect
4. C₂×C₇×#466	C₂	direct (N × H)
5. C₂²×C₇×#107	C₂	semidirect
6. C₂²×C₇×#102	C₂	semidirect
11. C₂²×C₇×D₈	C₂²	semidirect
15. C₂³×C₇×C₈	C₂²	semidirect
17. C₂×C₇×#121	C₂²	semidirect
20. C₇×#466	C₂²	direct (N × H)
21. C₂×C₇×#107	C₂²	semidirect
22. C₂×C₇×#102	C₂²	semidirect
23. C₂²×#466	C₇	direct (N × H)
40. C₂×C₇×D₈	C₂³	semidirect
43. C₂²×C₇×C₈	D₄	semidirect
46. C₂²×C₇×C₈	C₂³	semidirect
48. C₇×#121	C₂³	semidirect
52. C₇×#107	C₂³	semidirect
53. C₇×#102	C₂³	semidirect
54. C₂³×D₈	C₂×C₇	semidirect
55. C₂⁴×C₈	C₂×C₇	semidirect
56. C₂²×#121	C₂×C₇	semidirect
57. C₂×#466	C₂×C₇	direct (N × H)
58. C₂²×#107	C₂×C₇	semidirect
59. C₂²×#102	C₂×C₇	semidirect
66. C₂⁴×C₇	D₈	semidirect
92. C₂×C₇×C₈	C₂×D₄	semidirect
102. C₂²×D₈	C₂²×C₇	semidirect
106. C₂³×C₈	C₂²×C₇	semidirect
108. C₂×#121	C₂²×C₇	semidirect
111. #466	C₂²×C₇	direct (N × H)
112. C₂×#107	C₂²×C₇	semidirect
113. C₂×#102	C₂²×C₇	semidirect
124. C₂³×C₇	C₂×D₈	semidirect
145. C₇×C₈	C₂²×D₄	semidirect
162. C₂×D₈	C₂³×C₇	semidirect
165. C₂²×C₈	C₇×D₄	semidirect
168. C₂²×C₈	C₂³×C₇	semidirect
170. #121	C₂³×C₇	semidirect
174. #107	C₂³×C₇	semidirect
175. #102	C₂³×C₇	semidirect
183. C₂²×C₇	#466	direct (N × H)
186. C₂²×C₇	C₂²×D₈	semidirect
200. C₂⁴	C₇×D₈	semidirect
226. C₂×C₈	C₂×C₇×D₄	semidirect
234. C₂×C₇	C₂×#466	direct (N × H)
248. C₂³	C₂×C₇×D₈	semidirect
269. C₈	C₂²×C₇×D₄	semidirect
270. C₇	C₂²×#466	direct (N × H)
278. C₂²	C₇×#466	direct (N × H)
281. C₂²	C₂²×C₇×D₈	semidirect
291. C₂	C₂×C₇×#466	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×D₈	2
C₂⁴×C₇×C₈	2
C₂²×C₇×#121	2
C₂×C₇×#466	2
C₂²×C₇×#107	2
C₂²×C₇×#102	2
C₂²×#466	7

Subgroups — C22×C7×#466 #13237651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#466 #13237651