Subgroups — #89627

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×#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	#6382	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	#498	—	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	Dic₁₄	—	4	4	no	—
19	C₇×C₈	Q₈	1	2	yes	complemented
20	C₇⋊C₈	Q₈	1	2	yes	complemented
21	C₄⋊Q₈	—	7	2	no	—
22	#106	—	7	1	no	—
23	#106	—	7	2	no	—
24	#106	D₇	1	1	yes	characteristic
25	C₄×C₈	—	7	1	no	—
26	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
27	C₄×C₇	—	4	2	no	—
28	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
30	Dic₇	—	4	2	no	—
31	Dic₇	SD₁₆	1	4	yes	—
32	C₂×Q₈	—	14	2	no	—
33	C₂×C₈	—	7	1	no	—
34	C₂×C₈	C₂×D₇	1	1	yes	characteristic
35	C₄⋊C₄	—	7	2	no	—
36	C₄⋊C₄	—	14	2	no	—
37	C₄⋊C₄	C₂×D₇	1	2	yes	—
38	C₄²	—	7	1	no	—
39	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₇	C₂×SD₁₆	1	2	yes	—
41	Q₈	—	28	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	7	2	no	—
44	C₂×C₄	—	14	2	no	—
45	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
46	C₈	—	7	2	no	—
47	C₈	Dic₁₄	1	2	yes	complemented
48	C₇	#498	1	1	yes	characteristic, complemented, Sylow
49	C₂²	D₄×D₇	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	7	4	no	—
52	C₄	—	28	2	no	—
53	C₄	C₂×Dic₁₄	1	1	yes	characteristic
54	C₄	Q₈×D₇	1	1	yes	characteristic
55	C₂	#6451	1	1	yes	characteristic
56	C₂	D₇×SD₁₆	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	#498
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
19. C₇×C₈	Q₈	semidirect
20. C₇⋊C₈	Q₈	semidirect
47. C₈	Dic₁₄	semidirect
48. C₇	#498	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×#106	2
#6391	2
C₈×Dic₇	2
#6382	2
#498	7