Subgroups — #11727075

Order 896.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89650	C₂	1	2	yes	maximal
2	C₇×#366	C₂	1	1	yes	maximal, characteristic
3	#89299	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
5	#89285	C₂	1	2	yes	maximal
6	#6453	—	2	2	no	—
7	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₇×#107	C₂²	1	2	yes	—
9	#6392	C₂²	1	2	yes	—
10	C₈×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#6382	—	2	2	no	—
13	#2253	—	7	1	no	maximal, Sylow
14	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
15	C₇×C₄⋊C₄	—	2	2	no	—
16	C₄⋊Dic₇	—	2	2	no	—
17	C₄⋊Dic₇	—	4	2	no	—
18	Dic₇⋊C₄	—	4	2	no	—
19	C₄×Dic₇	D₄	1	1	yes	characteristic
20	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
21	C₇×C₁₆	Q₈	1	2	yes	—
22	C₇⋊C₁₆	Q₈	1	2	yes	—
23	#499	—	7	2	no	—
24	#367	—	7	2	no	—
25	#366	—	7	1	no	—
26	#366	D₇	1	1	yes	characteristic
27	C₄×C₁₆	—	7	1	no	—
28	C₂×C₄×C₇	—	4	2	no	—
29	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₇	—	4	2	no	—
31	C₂×Dic₇	D₈	1	1	yes	characteristic
32	C₂×Dic₇	D₈	1	1	yes	characteristic
33	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₇⋊C₈	Q₁₆	1	2	yes	—
36	#125	—	14	2	no	—
37	C₂×C₁₆	—	7	1	no	—
38	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
39	#107	—	7	2	no	—
40	#107	C₂×D₇	1	2	yes	—
41	#106	—	14	2	no	—
42	C₄×C₈	—	7	1	no	—
43	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
44	C₄×C₇	—	8	2	no	—
45	C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
46	C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
47	Dic₇	—	2	1	no	—
48	Dic₇	—	2	1	no	—
49	Dic₇	—	8	2	no	—
50	C₂×C₈	—	7	1	no	—
51	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	14	2	no	—
54	C₄⋊C₄	—	28	2	no	—
55	C₄⋊C₄	—	28	2	no	—
56	C₄²	—	7	1	no	—
57	C₁₆	—	7	2	no	—
58	C₁₆	Dic₁₄	1	2	yes	—
59	C₂×C₇	#500	1	1	yes	characteristic
60	C₂×C₇	#508	1	1	yes	characteristic
61	C₂×C₇	#508	1	1	yes	characteristic
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	7	1	no	—
64	C₂×C₄	—	7	1	no	—
65	C₂×C₄	—	28	2	no	—
66	C₂×C₄	D₄×D₇	1	1	yes	characteristic
67	C₈	—	7	2	no	—
68	C₈	C₂×Dic₁₄	1	1	yes	characteristic
69	C₈	Q₈×D₇	1	1	yes	characteristic
70	C₇	#2253	1	1	yes	characteristic, complemented, Sylow
71	C₂²	D₇×D₈	1	1	yes	characteristic
72	C₄	—	8	2	no	—
73	C₄	—	14	1	no	—
74	C₄	—	14	1	no	—
75	C₄	—	56	2	no	—
76	C₄	#6451	1	1	yes	characteristic
77	C₄	D₇×Q₁₆	1	1	yes	characteristic
78	C₂	#89646	1	1	yes	characteristic
79	C₂	#89684	1	1	yes	characteristic
80	C₂	#89691	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#366	C₂
socle, prime radical	C₂²×C₇	C₂×D₈
hypercenter	#366	D₇
hypocenter	C₇	#2253
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
70. C₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#89650	2
C₇×#366	2
#89299	2
C₁₆×Dic₇	2
#89285	2
#2253	7