Subgroups — #11729051

Order 896.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	#89860	C₂	1	4	yes	maximal, complemented
3	#89467	C₂	1	1	yes	maximal, characteristic
4	C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×Q₁₆	—	2	2	no	—
6	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×D₅₆	—	2	2	no	—
8	#6437	C₂²	1	2	yes	—
9	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	#6383	—	2	8	no	—
11	C₄×C₇⋊C₈	C₂²	1	2	yes	—
12	#1709	—	7	1	no	maximal, Sylow
13	C₂×C₇×Q₈	—	2	2	no	—
14	C₂×D₂₈	—	2	2	no	—
15	C₂×D₂₈	—	2	4	no	—
16	C₇×Q₁₆	—	4	4	no	—
17	C₂×C₇×C₈	D₄	1	2	yes	—
18	C₇×C₄⋊C₄	—	2	4	no	—
19	C₄²×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₇⋊C₈	D₄	1	4	yes	—
21	D₅₆	—	4	4	no	—
22	#494	D₇	1	1	yes	characteristic, complemented
23	#493	—	7	1	no	—
24	#486	—	7	4	no	—
25	C₈²	—	7	1	no	—
26	C₂²×D₇	—	4	2	no	—
27	C₇×Q₈	—	4	4	no	—
28	C₂×C₄×C₇	—	4	2	no	—
29	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
30	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
31	D₂₈	—	4	4	no	—
32	D₂₈	—	4	8	no	—
33	C₇×C₈	D₈	1	4	yes	—
34	C₇⋊C₈	—	2	4	no	—
35	C₂×Q₁₆	—	2	2	no	—
36	C₂×D₈	—	14	2	no	—
37	C₄⋊Q₈	C₂×D₇	1	2	yes	—
38	#127	—	7	2	no	—
39	#102	—	14	8	no	—
40	C₄×C₈	—	7	2	no	—
41	C₄×C₈	C₂×D₇	1	1	yes	characteristic
42	C₂²×C₇	#127	1	1	yes	characteristic
43	C₂×D₇	—	8	2	no	—
44	C₂×D₇	—	8	4	no	—
45	C₄×C₇	—	8	2	no	—
46	C₄×C₇	C₂×D₈	1	2	yes	—
47	C₄×C₇	#135	1	4	yes	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×D₄	—	14	4	no	—
50	C₂×D₄	—	14	2	no	—
51	Q₁₆	—	4	4	no	—
52	D₈	—	28	4	no	—
53	C₂×C₈	—	7	4	no	—
54	C₂×C₈	C₂²⋊D₇	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄²	C₂²×D₇	1	1	yes	characteristic
57	C₂×C₇	#493	1	1	yes	characteristic
58	C₂×C₇	#495	1	2	yes	—
59	D₇	—	16	2	no	—
60	C₂³	—	28	2	no	—
61	Q₈	—	4	4	no	—
62	D₄	—	28	8	no	—
63	D₄	—	28	4	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	D₄×D₇	1	2	yes	—
66	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
67	C₈	—	14	4	no	—
68	C₈	#1141	1	4	yes	—
69	C₇	#1709	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	56	4	no	—
71	C₂²	—	56	2	no	—
72	C₂²	#6503	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	#6482	1	4	yes	—
75	C₄	C₂×#1141	1	2	yes	—
76	C₂	—	112	2	no	—
77	C₂	#89845	1	1	yes	characteristic
78	C₂	#89963	1	2	yes	—

	subgroup	G/H
center	C₂²	#6503
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#494	C₂
socle	C₂²×C₇	#127
prime radical	#89467	C₂
hypercenter	#494	D₇
hypocenter	C₇	#1709
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₇×#494	C₂	semidirect
2. #89860	C₂	semidirect
4. C₈×C₇⋊C₈	C₂	semidirect
22. #494	D₇	semidirect
69. C₇	#1709	semidirect

Maximal subgroups

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

subgroup	index
C₇×#494	2
#89860	2
#89467	2
C₈×C₇⋊C₈	2
#1709	7