Subgroups — #11725626

Order 896.

124 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	C₂×#6393	C₂	1	2	yes	maximal
3	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
4	C₂×#1136	—	2	2	no	—
5	C₇×M₃₂	—	2	2	no	—
6	C₂×C₇×C₁₆	C₄	1	2	yes	—
7	#6393	—	2	2	no	—
8	#6393	C₄	1	4	yes	—
9	#1385	—	7	1	no	maximal, Sylow
10	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂×C₇×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₇×C₈	D₄	1	2	yes	—
14	C₂×C₇×C₈	Q₈	1	1	yes	characteristic
15	#1136	—	2	4	no	—
16	#1136	—	4	2	no	—
17	C₂×C₇⋊C₈	—	4	2	no	—
18	C₇×C₁₆	—	2	2	no	—
19	C₂×M₃₂	D₇	1	1	yes	characteristic
20	C₂×#108	—	7	2	no	—
21	C₂³×C₇	Q₁₆	1	1	yes	characteristic
22	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₇	D₈	1	1	yes	characteristic
25	C₂×C₄×C₇	SD₁₆	1	2	yes	—
26	C₇×C₈	C₄²	1	1	yes	characteristic
27	C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
28	C₇×C₈	C₄⋊C₄	1	2	yes	—
29	C₇⋊C₈	—	4	4	no	—
30	C₂×M₁₆	—	14	2	no	—
31	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
32	M₃₂	—	2	2	no	—
33	C₂×C₁₆	Dic₇	1	2	yes	—
34	#108	—	7	4	no	—
35	#108	—	14	2	no	—
36	C₂²×C₇	—	4	1	no	—
37	C₂²×C₇	#103	1	2	yes	—
38	C₂²×C₇	#107	1	1	yes	characteristic
39	C₄×C₇	#95	1	1	yes	characteristic
40	C₄×C₇	#102	1	2	yes	—
41	C₄×C₇	#106	1	1	yes	characteristic
42	C₂²×C₄	D₂₈	1	1	yes	characteristic
43	M₁₆	—	14	4	no	—
44	M₁₆	—	28	2	no	—
45	C₂×C₈	—	28	2	no	—
46	C₂×C₈	Dic₁₄	1	1	yes	characteristic
47	C₂×C₈	C₄×D₇	1	2	yes	—
48	C₂×C₈	C₂×Dic₇	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊D₇	1	2	yes	—
50	C₁₆	—	2	2	no	—
51	C₂×C₇	—	2	2	no	—
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	#340	1	1	yes	characteristic
54	C₂³	Dic₂₈	1	1	yes	characteristic
55	C₂×C₄	#1132	1	2	yes	—
56	C₂×C₄	D₅₆	1	1	yes	characteristic
57	C₂×C₄	C₄⋊Dic₇	1	1	yes	characteristic
58	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
59	C₈	—	28	4	no	—
60	C₈	C₄×Dic₇	1	1	yes	characteristic
61	C₈	Dic₇⋊C₄	1	2	yes	—
62	C₈	C₂²⋊Dic₇	1	1	yes	characteristic
63	C₇	#1385	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	4	1	no	—
65	C₂²	#6390	1	2	yes	—
66	C₂²	#6392	1	1	yes	characteristic
67	C₄	#6391	1	1	yes	characteristic
68	C₄	#6395	1	2	yes	—
69	C₄	C₇⋊#95	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	#89346	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#6391
commutator	C₇×C₈	C₄²
Frattini	C₂²×C₈	C₂×D₇
Fitting	C₂×C₇×M₃₂	C₂
socle	C₂×C₇	#340
prime radical	C₂³×C₇	Q₁₆
cosocle	C₂²×C₇×C₈	C₂²
hypercenter	C₂×M₃₂	D₇
hypocenter	C₇	#1385
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
63. C₇	#1385	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
C₂×#6393	2
#1385	7