Subgroups — #89652

Order 448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₇	C₂	1	1	yes	maximal, characteristic
2	#6455	C₂	1	2	yes	maximal
3	C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	#6392	C₂	1	1	yes	maximal, characteristic, complemented
5	#6382	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	(C₂×D₇)⋊C₄	—	2	2	no	—
10	C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₄×Dic₇	—	2	2	no	—
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₈×D₇	C₄	1	4	yes	complemented
14	#427	—	7	1	no	maximal, Sylow
15	C₂²×D₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₇	—	2	2	no	—
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	—	2	2	no	—
19	C₂×Dic₇	D₄	1	1	yes	characteristic
20	C₄×D₇	C₂×C₄	1	2	yes	—
21	C₄×D₇	Q₈	1	2	yes	—
22	C₇×C₈	C₂×C₄	1	2	yes	—
23	C₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₈	—	7	1	no	—
25	#117	—	7	2	no	—
26	#107	—	7	1	no	—
27	#107	D₇	1	1	yes	characteristic, complemented
28	#106	—	7	2	no	—
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₇	—	2	1	no	—
31	C₂×D₇	—	2	1	no	—
32	C₂×D₇	C₄⋊C₄	1	2	yes	—
33	C₄×C₇	—	4	2	no	—
34	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
36	Dic₇	—	4	2	no	—
37	Dic₇	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄	—	7	1	no	—
39	C₂×C₈	—	7	1	no	—
40	C₂×C₈	—	7	4	no	—
41	C₂×C₈	C₂×D₇	1	1	yes	characteristic
42	C₄⋊C₄	—	7	2	no	—
43	C₄⋊C₄	C₂×D₇	1	2	yes	—
44	C₂²⋊C₄	—	14	2	no	—
45	C₄²	—	14	2	no	—
46	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₇	#135	1	1	yes	characteristic
48	C₂×C₇	#135	1	1	yes	characteristic
49	D₇	—	2	2	no	—
50	C₂³	—	7	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	7	1	no	—
53	C₂×C₄	—	7	2	no	—
54	C₂×C₄	—	7	2	no	—
55	C₂×C₄	—	14	2	no	—
56	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
57	C₈	—	7	2	no	—
58	C₈	C₄×D₇	1	2	yes	—
59	C₇	#427	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	7	2	no	—
61	C₂²	—	14	1	no	—
62	C₂²	—	14	1	no	—
63	C₂²	D₄×D₇	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	7	2	no	—
66	C₄	—	28	2	no	—
67	C₄	C₂×C₄×D₇	1	1	yes	characteristic
68	C₄	Q₈×D₇	1	1	yes	characteristic
69	C₂	—	14	2	no	—
70	C₂	D₇×C₄⋊C₄	1	1	yes	characteristic
71	C₂	#6475	1	1	yes	characteristic
72	C₂	#6482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂²×D₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#107	D₇
hypocenter	C₇	#427
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
3. C₇×#107	C₂	semidirect
4. #6392	C₂	semidirect
5. #6382	C₂	semidirect
13. C₈×D₇	C₄	semidirect
27. #107	D₇	semidirect
59. C₇	#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₇	2
#6455	2
C₇×#107	2
#6392	2
#6382	2
#427	7