Subgroups — #11729052

Order 896.

192 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
2	#89850	C₂	1	4	yes	maximal
3	#89471	C₂	1	1	yes	maximal, characteristic
4	C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×Q₁₆	—	2	2	no	—
6	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×Dic₂₈	—	2	2	no	—
8	#6432	C₂²	1	2	yes	—
9	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	#6381	—	2	8	no	—
11	C₄×C₇⋊C₈	C₂²	1	2	yes	—
12	#1713	—	7	1	no	maximal, Sylow
13	C₂×C₇×Q₈	—	2	2	no	—
14	C₂×Dic₁₄	—	2	2	no	—
15	C₇×Q₁₆	—	4	4	no	—
16	C₂×C₇×C₈	D₄	1	2	yes	—
17	C₇×C₄⋊C₄	—	2	4	no	—
18	C₄²×C₇	C₂³	1	1	yes	characteristic
19	C₄⋊Dic₇	—	2	4	no	—
20	C₂×C₇⋊C₈	D₄	1	4	yes	—
21	Dic₂₈	—	4	4	no	—
22	#500	—	7	4	no	—
23	#494	—	7	1	no	—
24	#494	D₇	1	1	yes	characteristic
25	C₈²	—	7	1	no	—
26	C₇×Q₈	—	4	4	no	—
27	C₂×C₄×C₇	—	4	2	no	—
28	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₇	—	4	2	no	—
31	Dic₁₄	—	4	4	no	—
32	C₇×C₈	D₈	1	4	yes	—
33	C₇⋊C₈	Q₁₆	1	8	yes	complemented
34	C₂×Q₁₆	—	2	2	no	—
35	C₂×Q₁₆	—	14	2	no	—
36	C₄⋊Q₈	—	7	2	no	—
37	C₄⋊Q₈	C₂×D₇	1	2	yes	—
38	#107	—	14	8	no	—
39	C₄×C₈	—	7	2	no	—
40	C₄×C₈	C₂×D₇	1	1	yes	characteristic
41	C₂²×C₇	#127	1	1	yes	characteristic
42	C₄×C₇	—	8	2	no	—
43	C₄×C₇	C₂×D₈	1	2	yes	—
44	C₄×C₇	C₂×Q₁₆	1	4	yes	—
45	Dic₇	—	8	2	no	—
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	—	14	2	no	—
48	Q₁₆	—	4	4	no	—
49	Q₁₆	—	28	4	no	—
50	C₂×C₈	—	7	4	no	—
51	C₂×C₈	C₂²⋊D₇	1	2	yes	—
52	C₄⋊C₄	—	2	4	no	—
53	C₄⋊C₄	—	14	4	no	—
54	C₄²	C₂²×D₇	1	1	yes	characteristic
55	C₂×C₇	#493	1	1	yes	characteristic
56	C₂×C₇	#494	1	2	yes	—
57	Q₈	—	4	4	no	—
58	Q₈	—	28	4	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	—	28	2	no	—
61	C₂×C₄	D₄×D₇	1	2	yes	—
62	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
63	C₈	—	7	8	no	—
64	C₈	#1141	1	4	yes	—
65	C₇	#1713	1	1	yes	characteristic, complemented, Sylow
66	C₂²	#6503	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	56	2	no	—
69	C₄	D₇×Q₁₆	1	4	yes	—
70	C₄	C₂×#1141	1	2	yes	—
71	C₂	#89845	1	1	yes	characteristic
72	C₂	#89953	1	2	yes	—

	subgroup	G/H
center	C₂²	#6503
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#494	C₂
socle, prime radical	C₂²×C₇	#127
hypercenter	#494	D₇
hypocenter	C₇	#1713
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
33. C₇⋊C₈	Q₁₆	semidirect
65. C₇	#1713	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
#89850	2
#89471	2
C₈×C₇⋊C₈	2
#1713	7