Subgroups — C₇×#2091 #11730560

Order 896.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#430	C₂	1	3	yes	maximal, complemented
2	C₇×#407	C₂	1	3	yes	maximal
3	C₇×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₇×M₁₆	C₂²	1	3	yes	—
6	C₂×C₇×M₁₆	C₂²	1	3	yes	—
7	C₇×#108	—	2	6	no	—
8	C₇×C₂²⋊C₈	—	2	6	no	—
9	#2091	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁴×C₇	Q₈	1	1	yes	characteristic
11	C₂²×C₄×C₇	—	2	3	no	—
12	C₂²×C₄×C₇	—	2	3	no	—
13	C₂²×C₄×C₇	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
15	C₇×M₁₆	—	2	6	no	—
16	C₇×M₁₆	—	2	6	no	—
17	C₂×C₇×C₈	—	2	3	no	—
18	C₂×C₇×C₈	D₄	1	6	yes	—
19	#430	C₂×C₇	1	3	yes	complemented
20	#407	C₂×C₇	1	3	yes	—
21	#338	C₂×C₇	1	1	yes	characteristic, complemented
22	C₂³×C₇	—	2	3	no	—
23	C₂³×C₇	—	4	2	no	—
24	C₂³×C₇	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₇	—	2	3	no	—
26	C₂×C₄×C₇	—	4	3	no	—
27	C₂×C₄×C₇	—	4	1	no	—
28	C₂×C₄×C₇	C₂×D₄	1	3	yes	—
29	C₂×C₄×C₇	#41	1	3	yes	—
30	C₇×C₈	—	2	6	no	—
31	C₇×C₈	—	4	3	no	—
32	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
33	C₂×M₁₆	C₂²×C₇	1	3	yes	—
34	C₂×M₁₆	C₂²×C₇	1	3	yes	—
35	#108	—	2	6	no	—
36	C₂²⋊C₈	—	2	6	no	—
37	C₂²×C₇	—	2	6	no	—
38	C₂²×C₇	—	4	1	no	—
39	C₂²×C₇	—	4	3	no	—
40	C₂²×C₇	—	4	1	no	—
41	C₂²×C₇	C₂²⋊Q₈	1	3	yes	—
42	C₄×C₇	—	4	1	no	—
43	C₄×C₇	#123	1	3	yes	—
44	C₄×C₇	#127	1	1	yes	characteristic
45	C₂⁴	C₇×Q₈	1	1	yes	characteristic
46	C₂²×C₄	—	2	3	no	—
47	C₂²×C₄	—	2	3	no	—
48	C₂²×C₄	C₇×Q₈	1	1	yes	characteristic
49	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
50	M₁₆	—	2	6	no	—
51	M₁₆	—	2	6	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	C₇×D₄	1	6	yes	—
54	C₂×C₇	—	2	3	no	—
55	C₂×C₇	—	4	2	no	—
56	C₂×C₇	#399	1	1	yes	characteristic
57	C₂³	—	2	3	no	—
58	C₂³	—	4	2	no	—
59	C₂³	C₂×C₇×Q₈	1	1	yes	characteristic
60	C₂×C₄	—	2	3	no	—
61	C₂×C₄	—	4	3	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	C₂×C₇×D₄	1	3	yes	—
64	C₂×C₄	C₇×#41	1	3	yes	—
65	C₈	—	2	6	no	—
66	C₈	—	4	3	no	—
67	C₇	#2091	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	—	2	6	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	3	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₇×C₂²⋊Q₈	1	3	yes	—
73	C₄	—	4	1	no	—
74	C₄	C₇×#123	1	3	yes	—
75	C₄	C₇×#127	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	4	2	no	—
78	C₂	C₇×#399	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle	C₂×C₇	#399
prime radical	C₂⁴×C₇	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#430	C₂	semidirect
3. C₇×#338	C₂	semidirect
9. #2091	C₇	direct (N × H)
19. #430	C₂×C₇	semidirect
21. #338	C₂×C₇	semidirect
67. C₇	#2091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#430	2
C₇×#407	2
C₇×#338	2
#2091	7

Subgroups — C7×#2091 #11730560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2091 #11730560