Subgroups — #11727074

Order 896.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89646	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	#89284	C₂	1	2	yes	maximal
6	#6451	—	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	#6381	—	2	2	no	—
13	#2252	—	7	1	no	maximal, Sylow
14	C₂×Dic₁₄	—	4	2	no	—
15	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
16	C₇×C₄⋊C₄	—	2	2	no	—
17	C₄⋊Dic₇	—	2	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	complemented
22	C₇⋊C₁₆	Q₈	1	2	yes	complemented
23	#500	—	7	2	no	—
24	#366	—	7	1	no	—
25	#366	—	7	2	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	Dic₁₄	—	8	4	no	—
34	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
36	C₇⋊C₈	Q₁₆	1	2	yes	—
37	C₄⋊Q₈	—	14	2	no	—
38	C₂×C₁₆	—	7	1	no	—
39	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
40	#107	—	7	2	no	—
41	#107	—	14	2	no	—
42	#107	C₂×D₇	1	2	yes	—
43	C₄×C₈	—	7	1	no	—
44	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
45	C₄×C₇	—	8	2	no	—
46	C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₇	—	8	2	no	—
49	Dic₇	D₁₆	1	2	yes	—
50	Dic₇	Q₃₂	1	2	yes	—
51	C₂×Q₈	—	28	2	no	—
52	C₂×C₈	—	7	1	no	—
53	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	14	2	no	—
56	C₄⋊C₄	—	28	2	no	—
57	C₄²	—	7	1	no	—
58	C₁₆	—	7	2	no	—
59	C₁₆	Dic₁₄	1	2	yes	complemented
60	C₂×C₇	#500	1	1	yes	characteristic
61	C₂×C₇	C₂×D₁₆	1	1	yes	characteristic
62	C₂×C₇	C₂×Q₃₂	1	1	yes	characteristic
63	Q₈	—	56	4	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	7	1	no	—
66	C₂×C₄	—	7	1	no	—
67	C₂×C₄	—	28	2	no	—
68	C₂×C₄	D₄×D₇	1	1	yes	characteristic
69	C₈	—	7	2	no	—
70	C₈	C₂×Dic₁₄	1	1	yes	characteristic
71	C₈	Q₈×D₇	1	1	yes	characteristic
72	C₇	#2252	1	1	yes	characteristic, complemented, Sylow
73	C₂²	D₇×D₈	1	1	yes	characteristic
74	C₄	—	7	2	no	—
75	C₄	—	7	2	no	—
76	C₄	—	8	2	no	—
77	C₄	—	56	2	no	—
78	C₄	#6451	1	1	yes	characteristic
79	C₄	D₇×Q₁₆	1	1	yes	characteristic
80	C₂	#89646	1	1	yes	characteristic
81	C₂	D₇×D₁₆	1	1	yes	characteristic
82	C₂	D₇×Q₃₂	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	#2252
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
21. C₇×C₁₆	Q₈	semidirect
22. C₇⋊C₁₆	Q₈	semidirect
59. C₁₆	Dic₁₄	semidirect
72. C₇	#2252	semidirect

Maximal subgroups

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

subgroup	index
#89646	2
C₇×#366	2
#89299	2
C₁₆×Dic₇	2
#89284	2
#2252	7