Subgroups — #11729175

Order 896.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#89873	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
6	#6489	C₄	1	1	yes	characteristic
7	C₂×#1136	C₄	1	1	yes	characteristic
8	C₇×M₃₂	—	2	2	no	—
9	C₂×C₇×C₁₆	C₂²	1	2	yes	—
10	#6389	C₄	1	2	yes	—
11	C₈×Dic₇	C₂²	1	2	yes	—
12	C₂×C₇⋊C₁₆	—	2	2	no	—
13	C₂×C₇⋊C₁₆	C₂²	1	2	yes	—
14	#2151	—	7	1	no	maximal, Sylow
15	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇×C₈	D₄	1	2	yes	—
18	C₂×C₇×C₈	Q₈	1	2	yes	—
19	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
20	C₂²⋊Dic₇	C₈	1	2	yes	—
21	C₄⋊Dic₇	C₈	1	2	yes	—
22	C₄×Dic₇	C₂×C₄	1	2	yes	—
23	#1136	—	2	2	no	—
24	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₇×C₁₆	—	2	2	no	—
26	C₇⋊C₁₆	—	2	2	no	—
27	C₂×M₃₂	D₇	1	1	yes	characteristic
28	C₂²×C₁₆	—	7	1	no	—
29	#405	—	7	1	no	—
30	C₄⋊C₁₆	—	7	4	no	—
31	C₂³×C₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₇	M₁₆	1	2	yes	—
35	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₇	C₂×C₈	1	4	yes	—
37	C₇×C₈	C₄⋊C₄	1	2	yes	—
38	C₇×C₈	C₂×D₄	1	1	yes	characteristic
39	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₇⋊C₈	—	2	2	no	—
41	C₂×M₁₆	—	7	1	no	—
42	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
43	#117	—	7	1	no	—
44	M₃₂	—	2	2	no	—
45	C₂×C₁₆	—	7	2	no	—
46	C₂×C₁₆	—	14	2	no	—
47	C₂×C₁₆	C₂×D₇	1	2	yes	—
48	#97	—	7	2	no	—
49	C₄×C₈	—	7	2	no	—
50	C₂²×C₇	—	2	2	no	—
51	C₂²×C₇	C₄⋊C₈	1	2	yes	—
52	C₂²×C₇	C₂²×C₈	1	1	yes	characteristic
53	C₄×C₇	C₄⋊C₈	1	2	yes	—
54	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
56	Dic₇	—	2	4	no	—
57	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
58	M₁₆	—	14	2	no	—
59	C₂×C₈	—	7	2	no	—
60	C₂×C₈	Dic₁₄	1	2	yes	—
61	C₂×C₈	C₄×D₇	1	1	yes	characteristic
62	C₂×C₈	C₂²⋊D₇	1	2	yes	—
63	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
64	C₄⋊C₄	—	7	2	no	—
65	C₂²⋊C₄	—	7	2	no	—
66	C₄²	—	7	2	no	—
67	C₁₆	—	2	2	no	—
68	C₁₆	—	14	2	no	—
69	C₂×C₇	—	2	2	no	—
70	C₂×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₇	#504	1	2	yes	—
72	C₂³	C₈×D₇	1	1	yes	characteristic
73	C₂×C₄	—	7	4	no	—
74	C₂×C₄	C₈×D₇	1	1	yes	characteristic
75	C₂×C₄	#1131	1	2	yes	—
76	C₂×C₄	Dic₇⋊C₄	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
78	C₈	—	14	2	no	—
79	C₈	Dic₇⋊C₄	1	2	yes	—
80	C₈	C₂×Dic₁₄	1	1	yes	characteristic
81	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
82	C₇	#2151	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	2	2	no	—
84	C₂²	Dic₇⋊C₈	1	2	yes	—
85	C₂²	C₂×C₈×D₇	1	1	yes	characteristic
86	C₄	—	14	4	no	—
87	C₄	Dic₇⋊C₈	1	2	yes	—
88	C₄	C₂×#1131	1	1	yes	characteristic
89	C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	#89679	1	2	yes	—
92	C₂	C₂×Dic₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₈
Fitting	C₂×C₇×M₃₂	C₂
prime radical	C₂³×C₇	C₂×C₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	C₂×M₃₂	D₇
hypocenter	C₇	#2151
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
4. Dic₇⋊C₁₆	C₂	semidirect
82. C₇	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₃₂	2
#89873	2
C₂²×C₇⋊C₁₆	2
Dic₇⋊C₁₆	2
#2151	7