Subgroups — #89646

Order 448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6451	C₂	1	2	yes	maximal
2	C₇×#107	C₂	1	1	yes	maximal, characteristic
3	#6392	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
5	#6381	C₂	1	2	yes	maximal
6	C₂×Dic₁₄	—	2	2	no	—
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄⋊Dic₇	C₂²	1	2	yes	—
10	Dic₇⋊C₄	—	2	2	no	—
11	C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	#500	—	7	1	no	maximal, Sylow
14	C₂×C₄×C₇	—	2	2	no	—
15	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
16	C₂×Dic₇	—	2	2	no	—
17	C₂×Dic₇	D₄	1	1	yes	characteristic
18	C₂×Dic₇	D₄	1	1	yes	characteristic
19	Dic₁₄	—	4	4	no	—
20	C₇×C₈	Q₈	1	2	yes	complemented
21	C₇⋊C₈	Q₈	1	2	yes	complemented
22	C₄⋊Q₈	—	7	2	no	—
23	#107	—	7	1	no	—
24	#107	—	7	2	no	—
25	#107	D₇	1	1	yes	characteristic
26	C₄×C₈	—	7	1	no	—
27	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
28	C₄×C₇	—	4	2	no	—
29	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
31	Dic₇	—	4	2	no	—
32	Dic₇	D₈	1	2	yes	—
33	Dic₇	Q₁₆	1	2	yes	—
34	C₂×Q₈	—	14	2	no	—
35	C₂×C₈	—	7	1	no	—
36	C₂×C₈	C₂×D₇	1	1	yes	characteristic
37	C₄⋊C₄	—	7	2	no	—
38	C₄⋊C₄	—	14	2	no	—
39	C₄⋊C₄	C₂×D₇	1	2	yes	—
40	C₄²	—	7	1	no	—
41	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₇	C₂×D₈	1	1	yes	characteristic
43	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
44	Q₈	—	28	4	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	7	1	no	—
47	C₂×C₄	—	7	1	no	—
48	C₂×C₄	—	14	2	no	—
49	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
50	C₈	—	7	2	no	—
51	C₈	Dic₁₄	1	2	yes	complemented
52	C₇	#500	1	1	yes	characteristic, complemented, Sylow
53	C₂²	D₄×D₇	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	7	2	no	—
56	C₄	—	7	2	no	—
57	C₄	—	28	2	no	—
58	C₄	C₂×Dic₁₄	1	1	yes	characteristic
59	C₄	Q₈×D₇	1	1	yes	characteristic
60	C₂	#6451	1	1	yes	characteristic
61	C₂	D₇×D₈	1	1	yes	characteristic
62	C₂	D₇×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#107	D₇
hypocenter	C₇	#500
LCS	G ▷ C₂×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
20. C₇×C₈	Q₈	semidirect
21. C₇⋊C₈	Q₈	semidirect
51. C₈	Dic₁₄	semidirect
52. C₇	#500	semidirect

Maximal subgroups

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

subgroup	index
#6451	2
C₇×#107	2
#6392	2
C₈×Dic₇	2
#6381	2
#500	7