Subgroups — #89772

Order 448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
3	C₄⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#6377	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₇×C₄⋊C₄	C₄	1	2	yes	—
8	C₇×C₂²⋊C₄	C₄	1	2	yes	—
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₇⋊C₈	C₂²	1	2	yes	—
13	#433	—	7	1	no	maximal, Sylow
14	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	—	2	2	no	—
16	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₇⋊C₈	—	2	2	no	—
21	C₇⋊C₈	—	2	2	no	—
22	#117	D₇	1	1	yes	characteristic
23	C₄⋊C₈	—	7	2	no	—
24	C₂²⋊C₈	—	7	2	no	—
25	#97	—	7	1	no	—
26	C₄×C₈	—	7	1	no	—
27	C₂²×C₇	—	2	1	no	—
28	C₂²×C₇	—	2	2	no	—
29	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₇	—	2	1	no	—
31	C₄×C₇	—	2	2	no	—
32	C₄×C₇	—	2	2	no	—
33	C₄×C₇	#41	1	2	yes	—
34	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
35	C₂×C₈	—	7	2	no	—
36	C₂×C₈	—	7	2	no	—
37	C₄⋊C₄	Dic₇	1	2	yes	—
38	C₂²⋊C₄	Dic₇	1	2	yes	—
39	C₄²	C₂×D₇	1	1	yes	characteristic
40	C₄²	C₂×D₇	1	1	yes	characteristic
41	C₂×C₇	—	4	1	no	—
42	C₂×C₇	#117	1	1	yes	characteristic
43	C₂×C₇	#131	1	2	yes	—
44	C₂³	C₂×Dic₇	1	1	yes	characteristic
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	C₂×Dic₇	1	2	yes	—
47	C₂×C₄	C₂×Dic₇	1	2	yes	—
48	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
49	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
50	C₈	—	14	2	no	—
51	C₈	—	14	2	no	—
52	C₇	#433	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	2	1	no	—
54	C₂²	—	2	2	no	—
55	C₂²	C₂²×Dic₇	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	2	no	—
58	C₄	—	2	2	no	—
59	C₄	#1157	1	2	yes	—
60	C₂	—	4	1	no	—
61	C₂	#6489	1	1	yes	characteristic
62	C₂	#6511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₇×#117	C₂
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#433
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₇⋊C₈)	C₂	semidirect
4. #6377	C₂	semidirect
5. C₄×C₇⋊C₈	C₂	semidirect
52. C₇	#433	semidirect

Maximal subgroups

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

subgroup	index
C₇×#117	2
C₂²⋊(C₇⋊C₈)	2
C₄⋊(C₇⋊C₈)	2
#6377	2
C₄×C₇⋊C₈	2
#433	7