Subgroups — #89462

Order 448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6435	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₇)⋊C₈	C₂	1	2	yes	maximal
4	Dic₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	#6377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₈	C₂²	1	2	yes	—
8	C₄²×C₇	C₂²	1	1	yes	characteristic
9	(C₂×D₇)⋊C₄	C₄	1	2	yes	—
10	Dic₇⋊C₄	C₄	1	2	yes	—
11	C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₈	C₂²	1	2	yes	—
13	#433	—	7	1	no	maximal, Sylow
14	C₂²×D₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₇	C₂×C₄	1	2	yes	—
18	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₇	—	2	2	no	—
20	C₇×C₈	—	2	2	no	—
21	C₇⋊C₈	—	2	2	no	—
22	#117	—	7	1	no	—
23	C₄⋊C₈	—	7	2	no	—
24	C₂²⋊C₈	—	7	2	no	—
25	#97	—	7	1	no	—
26	C₄×C₈	D₇	1	1	yes	characteristic, complemented
27	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₇	—	2	1	no	—
29	C₂×D₇	—	2	2	no	—
30	C₄×C₇	—	2	2	no	—
31	C₄×C₇	#41	1	2	yes	—
32	Dic₇	—	2	1	no	—
33	Dic₇	—	2	2	no	—
34	C₂²×C₄	—	7	1	no	—
35	C₂×C₈	—	7	2	no	—
36	C₂×C₈	C₂×D₇	1	2	yes	—
37	C₄⋊C₄	—	7	2	no	—
38	C₂²⋊C₄	—	7	2	no	—
39	C₄²	—	7	1	no	—
40	C₄²	C₂×D₇	1	1	yes	characteristic
41	C₂×C₇	#117	1	1	yes	characteristic
42	C₂×C₇	#131	1	2	yes	—
43	D₇	—	4	1	no	—
44	C₂³	—	7	1	no	—
45	C₂×C₄	—	7	2	no	—
46	C₂×C₄	—	7	1	no	—
47	C₂×C₄	—	14	2	no	—
48	C₂×C₄	C₄×D₇	1	2	yes	—
49	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	14	2	no	—
52	C₇	#433	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	14	1	no	—
54	C₂²	—	14	2	no	—
55	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	14	1	no	—
58	C₄	—	14	2	no	—
59	C₄	#1157	1	2	yes	—
60	C₂	—	28	1	no	—
61	C₂	#6435	1	1	yes	characteristic
62	C₂	#6464	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₄×C₇×C₈	C₂
prime radical	C₂²×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×C₈	D₇
hypocenter	C₇	#433
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×C₈	C₂	semidirect
4. Dic₇⋊C₈	C₂	semidirect
5. #6377	C₂	semidirect
26. C₄×C₈	D₇	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
#6435	2
C₄×C₇×C₈	2
(C₂×D₇)⋊C₈	2
Dic₇⋊C₈	2
#6377	2
#433	7