Subgroups — #11727035

Order 896.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#89297	C₂	1	2	yes	maximal
4	Dic₇⋊C₁₆	C₂	1	2	yes	maximal
5	C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₄	C₄	1	1	yes	characteristic
7	C₂×C₇×C₁₆	C₂²	1	2	yes	—
8	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	Dic₇⋊C₈	C₄	1	2	yes	—
10	C₈×Dic₇	C₂²	1	2	yes	—
11	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
14	#2183	—	7	1	no	maximal, Sylow
15	C₂×Dic₁₄	C₈	1	1	yes	characteristic
16	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
19	C₄⋊Dic₇	C₈	1	1	yes	characteristic
20	Dic₇⋊C₄	C₈	1	2	yes	—
21	C₄×Dic₇	C₂×C₄	1	2	yes	—
22	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₇×C₁₆	—	2	2	no	—
24	C₇⋊C₁₆	—	2	1	no	—
25	C₇⋊C₁₆	Q₈	1	2	yes	complemented
26	C₈×Q₈	—	7	1	no	—
27	C₄⋊C₁₆	—	7	2	no	—
28	C₄⋊C₁₆	D₇	1	1	yes	characteristic
29	#346	—	7	2	no	—
30	C₄×C₁₆	—	7	1	no	—
31	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₇	C₂×C₈	1	2	yes	—
35	C₂×Dic₇	C₂×C₈	1	2	yes	—
36	Dic₁₄	—	2	2	no	—
37	C₇×C₈	—	2	1	no	—
38	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₇×C₈	#41	1	1	yes	characteristic
40	C₇⋊C₈	—	2	2	no	—
41	C₄×Q₈	—	7	1	no	—
42	C₂×C₁₆	—	7	1	no	—
43	C₂×C₁₆	—	7	1	no	—
44	C₂×C₁₆	C₂×D₇	1	2	yes	—
45	C₄⋊C₈	—	7	2	no	—
46	C₄⋊C₈	—	7	1	no	—
47	C₄×C₈	—	7	2	no	—
48	C₄×C₈	C₂×D₇	1	1	yes	characteristic
49	C₂²×C₇	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₇	—	2	1	no	—
51	C₄×C₇	M₃₂	1	2	yes	—
52	C₄×C₇	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₇	#131	1	1	yes	characteristic
54	Dic₇	—	2	2	no	—
55	Dic₇	—	2	2	no	—
56	C₂×Q₈	—	7	1	no	—
57	C₂×C₈	—	7	2	no	—
58	C₂×C₈	C₄×D₇	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
60	C₄⋊C₄	—	7	2	no	—
61	C₄⋊C₄	—	7	1	no	—
62	C₄²	—	7	2	no	—
63	C₄²	C₄×D₇	1	1	yes	characteristic
64	C₁₆	—	2	2	no	—
65	C₁₆	—	7	2	no	—
66	C₁₆	—	14	1	no	—
67	C₂×C₇	C₈×Q₈	1	1	yes	characteristic
68	C₂×C₇	C₂×M₃₂	1	1	yes	characteristic
69	C₂×C₇	#504	1	1	yes	characteristic
70	Q₈	—	14	2	no	—
71	C₂×C₄	—	7	2	no	—
72	C₂×C₄	—	7	2	no	—
73	C₂×C₄	C₈×D₇	1	1	yes	characteristic
74	C₂×C₄	C₈×D₇	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	14	2	no	—
78	C₈	#1159	1	1	yes	characteristic
79	C₈	Q₈×D₇	1	1	yes	characteristic
80	C₇	#2183	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂×C₈×D₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	14	2	no	—
84	C₄	—	14	2	no	—
85	C₄	#6372	1	2	yes	—
86	C₄	#6450	1	1	yes	characteristic
87	C₄	#6470	1	1	yes	characteristic
88	C₂	#89602	1	1	yes	characteristic
89	C₂	C₂×#6372	1	1	yes	characteristic
90	C₂	#89679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇
commutator, socle, prime radical	C₂²×C₇	C₂²×C₈
Fitting	C₇×C₄⋊C₁₆	C₂
cosocle	C₂×C₇×C₈	C₂³
hypercenter	C₄⋊C₁₆	D₇
hypocenter	C₇	#2183
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. C₇⋊C₁₆	Q₈	semidirect
80. C₇	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₁₄	2
C₇×C₄⋊C₁₆	2
#89297	2
Dic₇⋊C₁₆	2
C₄×C₇⋊C₁₆	2
#2183	7