Subgroups — #89630

Order 448.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6453	C₂	1	2	yes	maximal
2	C₇×#106	C₂	1	1	yes	maximal, characteristic
3	#6391	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
5	#6381	C₂	1	2	yes	maximal
6	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇×C₄⋊C₄	C₂²	1	2	yes	—
8	C₄⋊Dic₇	—	2	2	no	—
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	#499	—	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	2	yes	—
18	C₇×C₈	Q₈	1	2	yes	—
19	C₇⋊C₈	Q₈	1	2	yes	—
20	#125	—	7	2	no	—
21	#107	—	7	2	no	—
22	#106	—	7	1	no	—
23	#106	D₇	1	1	yes	characteristic
24	C₄×C₈	—	7	1	no	—
25	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₇	—	4	2	no	—
27	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
28	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
29	Dic₇	—	2	2	no	—
30	Dic₇	—	4	2	no	—
31	C₂×C₈	—	7	1	no	—
32	C₂×C₈	C₂×D₇	1	1	yes	characteristic
33	C₄⋊C₄	—	7	2	no	—
34	C₄⋊C₄	—	14	2	no	—
35	C₄⋊C₄	—	14	2	no	—
36	C₄⋊C₄	C₂×D₇	1	2	yes	—
37	C₄²	—	7	1	no	—
38	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₇	#135	1	2	yes	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	7	2	no	—
42	C₂×C₄	—	14	2	no	—
43	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
44	C₈	—	7	2	no	—
45	C₈	Dic₁₄	1	2	yes	—
46	C₇	#499	1	1	yes	characteristic, complemented, Sylow
47	C₂²	D₄×D₇	1	1	yes	characteristic
48	C₄	—	4	2	no	—
49	C₄	—	14	2	no	—
50	C₄	—	28	2	no	—
51	C₄	C₂×Dic₁₄	1	1	yes	characteristic
52	C₄	Q₈×D₇	1	1	yes	characteristic
53	C₂	#6451	1	1	yes	characteristic
54	C₂	#6479	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#106	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#106	D₇
hypocenter	C₇	#499
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
46. C₇	#499	semidirect

Maximal subgroups

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

subgroup	index
#6453	2
C₇×#106	2
#6391	2
C₈×Dic₇	2
#6381	2
#499	7