Subgroups — C₇×#501 #90147

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#125	C₂	1	1	yes	maximal, characteristic
3	C₇×#107	C₂	1	2	yes	maximal
4	C₇×#106	C₂	1	2	yes	maximal
5	C₇×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	—	2	1	no	—
7	C₂×C₇×C₈	C₂²	1	2	yes	—
8	C₇×C₄⋊C₄	—	2	1	no	—
9	C₇×C₄⋊C₄	—	2	1	no	—
10	C₇×C₄⋊C₄	—	2	1	no	—
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	#501	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₇×Q₈	—	4	2	no	—
17	C₂×C₄×C₇	—	2	1	no	—
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₇×C₈	Q₈	1	4	yes	complemented
24	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
25	#125	C₂×C₇	1	1	yes	characteristic
26	#107	C₂×C₇	1	2	yes	—
27	#106	C₂×C₇	1	2	yes	—
28	#97	C₂×C₇	1	1	yes	characteristic
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₇	—	2	1	no	—
31	C₄×C₇	—	2	1	no	—
32	C₄×C₇	—	4	1	no	—
33	C₄×C₇	—	4	2	no	—
34	C₄×C₇	—	4	1	no	—
35	C₄×C₇	C₂×Q₈	1	2	yes	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×C₈	C₂²×C₇	1	2	yes	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
42	C₄⋊C₄	C₂²×C₇	1	2	yes	—
43	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
44	C₄²	C₂²×C₇	1	1	yes	characteristic
45	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₂×C₇	#137	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	C₇×D₄	1	1	yes	characteristic
53	C₂×C₄	C₇×D₄	1	1	yes	characteristic
54	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
55	C₈	C₇×Q₈	1	4	yes	complemented
56	C₇	#501	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	4	1	no	—
62	C₄	—	4	1	no	—
63	C₄	C₂×C₇×Q₈	1	2	yes	—
64	C₂	C₇×C₄⋊Q₈	1	1	yes	characteristic
65	C₂	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂	C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. #501	C₇	direct (N × H)
23. C₇×C₈	Q₈	semidirect
55. C₈	C₇×Q₈	semidirect
56. C₇	#501	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×C₄⋊Q₈	2
C₇×#125	2
C₇×#107	2
C₇×#106	2
C₇×#97	2
#501	7

Subgroups — C7×#501 #90147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#501 #90147