Subgroups — #11727497

Order 896.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6386	C₂	1	2	yes	maximal
3	C₄⋊(C₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₇	C₄	1	1	yes	characteristic
7	C₄×C₇×C₈	C₄	1	2	yes	—
8	C₄×C₇×C₈	C₂²	1	2	yes	—
9	#6386	—	2	4	no	—
10	C₂×C₇⋊C₁₆	C₂²	1	4	yes	—
11	#2150	—	7	1	no	maximal, Sylow
12	C₂²×C₄×C₇	C₈	1	2	yes	—
13	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₇×C₈	—	2	2	no	—
15	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
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	C₄²×C₇	C₈	1	1	yes	characteristic
21	C₄²×C₇	C₈	1	1	yes	characteristic
22	C₄²×C₇	C₂×C₄	1	2	yes	—
23	C₇⋊C₁₆	—	2	4	no	—
24	C₂×M₃₂	—	7	2	no	—
25	C₂×C₄×C₈	D₇	1	1	yes	characteristic
26	C₄⋊C₁₆	—	7	4	no	—
27	C₂³×C₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₇	—	2	4	no	—
29	C₂×C₄×C₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₇	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₇	M₁₆	1	2	yes	—
34	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₇×C₈	—	2	2	no	—
36	C₇×C₈	C₄⋊C₄	1	2	yes	—
37	C₇×C₈	C₂×D₄	1	1	yes	characteristic
38	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₈	Dic₇	1	1	yes	characteristic
40	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
41	C₂×C₄²	Dic₇	1	1	yes	characteristic
42	M₃₂	—	14	4	no	—
43	C₂×C₁₆	—	7	4	no	—
44	C₄×C₈	Dic₇	1	2	yes	—
45	C₄×C₈	C₂×D₇	1	2	yes	—
46	C₂²×C₇	—	2	2	no	—
47	C₂²×C₇	C₄⋊C₈	1	2	yes	—
48	C₂²×C₇	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₇	—	2	2	no	—
50	C₄×C₇	C₄⋊C₈	1	2	yes	—
51	C₄×C₇	M₃₂	1	4	yes	complemented
52	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
54	C₂²×C₄	C₇⋊C₈	1	2	yes	—
55	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	Dic₁₄	1	2	yes	—
58	C₂×C₈	D₂₈	1	2	yes	—
59	C₂×C₈	C₂×Dic₇	1	2	yes	—
60	C₂×C₈	C₂×Dic₇	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
62	C₄²	C₇⋊C₈	1	1	yes	characteristic
63	C₄²	C₇⋊C₈	1	1	yes	characteristic
64	C₄²	C₂×Dic₇	1	2	yes	—
65	C₁₆	—	14	4	no	—
66	C₂×C₇	—	2	2	no	—
67	C₂×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₇	C₂×M₃₂	1	2	yes	—
69	C₂³	C₂×C₇⋊C₈	1	1	yes	characteristic
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	C₂×C₇⋊C₈	1	2	yes	—
72	C₂×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
73	C₂×C₄	C₂×C₇⋊C₈	1	2	yes	—
74	C₂×C₄	#1136	1	2	yes	—
75	C₂×C₄	C₄⋊Dic₇	1	2	yes	—
76	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	C₄⋊Dic₇	1	2	yes	—
79	C₈	C₂×Dic₁₄	1	1	yes	characteristic
80	C₈	C₂×D₂₈	1	1	yes	characteristic
81	C₇	#2150	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊(C₇⋊C₈)	1	2	yes	—
84	C₂²	C₂²×C₇⋊C₈	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
87	C₄	#6386	1	4	yes	complemented
88	C₄	C₂×#1136	1	1	yes	characteristic
89	C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
92	C₂	C₂×#6386	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₈
Fitting	C₂×C₄×C₇×C₈	C₂
prime radical	C₂³×C₇	C₂×C₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	C₂×C₄×C₈	D₇
hypocenter	C₇	#2150
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₇⋊C₁₆)	C₂	semidirect
51. C₄×C₇	M₃₂	semidirect
81. C₇	#2150	semidirect
87. C₄	#6386	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×C₈	2
C₂×#6386	2
C₄⋊(C₇⋊C₁₆)	2
#2150	7