Subgroups — #11727639

Order 896.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#6386	C₂	1	2	yes	maximal
3	#89258	C₂	1	4	yes	maximal, complemented
4	C₂×C₇×M₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₇×#117	C₄	1	1	yes	characteristic
7	C₇×#97	C₄	1	2	yes	—
8	C₄×C₇×C₈	C₂²	1	2	yes	—
9	#6386	—	2	2	no	—
10	#6386	C₂²	1	4	yes	—
11	C₂×C₇⋊C₁₆	—	2	2	no	—
12	#2153	—	7	1	no	maximal, Sylow
13	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₇×M₁₆	C₈	1	4	yes	—
15	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
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	C₇×C₄⋊C₄	—	2	1	no	—
21	C₇×C₂²⋊C₄	—	2	1	no	—
22	C₄²×C₇	C₂×C₄	1	2	yes	—
23	C₇⋊C₁₆	—	2	4	no	—
24	C₂×M₃₂	—	7	2	no	—
25	#405	D₇	1	1	yes	characteristic
26	#364	—	7	4	no	—
27	C₂³×C₇	M₁₆	1	1	yes	characteristic
28	C₂×C₄×C₇	—	2	2	no	—
29	C₂×C₄×C₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₇	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₇	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₇×C₈	C₄⋊C₄	1	2	yes	—
34	C₇×C₈	C₂×C₈	1	4	yes	—
35	C₇×C₈	C₂×D₄	1	1	yes	characteristic
36	C₇×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₂×M₁₆	Dic₇	1	1	yes	characteristic
38	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
39	#117	Dic₇	1	1	yes	characteristic
40	M₃₂	—	7	4	no	—
41	M₃₂	—	14	2	no	—
42	C₂×C₁₆	—	14	2	no	—
43	#97	Dic₇	1	2	yes	—
44	C₄×C₈	C₂×D₇	1	2	yes	—
45	C₂²×C₇	—	4	1	no	—
46	C₂²×C₇	C₄⋊C₈	1	2	yes	—
47	C₂²×C₇	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₇	—	4	2	no	—
49	C₄×C₇	C₄⋊C₈	1	2	yes	—
50	C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₇	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
53	M₁₆	C₇⋊C₈	1	4	yes	—
54	C₂×C₈	Dic₁₄	1	2	yes	—
55	C₂×C₈	D₂₈	1	2	yes	—
56	C₂×C₈	C₂×Dic₇	1	2	yes	—
57	C₂×C₈	C₂×Dic₇	1	1	yes	characteristic
58	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
59	C₄⋊C₄	—	2	1	no	—
60	C₂²⋊C₄	—	2	1	no	—
61	C₄²	C₂×Dic₇	1	2	yes	—
62	C₁₆	—	14	4	no	—
63	C₂×C₇	—	2	1	no	—
64	C₂×C₇	—	2	2	no	—
65	C₂×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
66	C₂³	#1136	1	1	yes	characteristic
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₂×C₇⋊C₈	1	2	yes	—
69	C₂×C₄	#1136	1	1	yes	characteristic
70	C₂×C₄	C₄⋊Dic₇	1	2	yes	—
71	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
72	C₈	C₂×C₇⋊C₈	1	4	yes	—
73	C₈	C₄⋊Dic₇	1	2	yes	—
74	C₈	C₂×Dic₁₄	1	1	yes	characteristic
75	C₈	C₂×D₂₈	1	1	yes	characteristic
76	C₇	#2153	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	4	1	no	—
78	C₂²	C₄⋊(C₇⋊C₈)	1	2	yes	—
79	C₂²	C₂×#1136	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
82	C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
83	C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#405	C₂
socle	C₂×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	M₁₆
cosocle	C₂×C₇×C₈	C₂³
hypercenter	#405	D₇
hypocenter	C₇	#2153
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
3. #89258	C₂	semidirect
76. C₇	#2153	semidirect

Maximal subgroups

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

subgroup	index
C₇×#405	2
C₂×#6386	2
#89258	2
#2153	7