Subgroups — C₇×#477 #90123

Order 448.

96 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	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#107	C₂	1	2	yes	maximal
4	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇×#103	C₂	1	2	yes	maximal
6	C₂×C₇×Q₈	—	2	1	no	—
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	2	yes	—
9	C₇×C₄⋊C₄	—	2	1	no	—
10	C₇×C₄⋊C₄	—	2	1	no	—
11	C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₄²×C₇	—	2	1	no	—
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	#477	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₇×Q₈	—	2	1	no	—
17	C₇×Q₈	—	4	2	no	—
18	C₇×Q₈	Q₈	1	2	yes	complemented
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×C₈	—	2	2	no	—
26	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
27	C₄×Q₈	C₂×C₇	1	1	yes	characteristic
28	#107	C₂×C₇	1	2	yes	—
29	C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
30	#103	C₂×C₇	1	2	yes	—
31	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₇	—	2	2	no	—
33	C₄×C₇	—	2	1	no	—
34	C₄×C₇	—	4	1	no	—
35	C₄×C₇	—	4	2	no	—
36	C₄×C₇	Q₁₆	1	2	yes	—
37	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₇	#41	1	1	yes	characteristic
39	C₂×Q₈	—	2	1	no	—
40	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
41	C₂×C₈	C₂²×C₇	1	2	yes	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	C₂²×C₇	1	2	yes	—
45	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
46	C₄²	—	2	1	no	—
47	C₄²	C₂²×C₇	1	1	yes	characteristic
48	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
49	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	Q₈	—	2	1	no	—
52	Q₈	—	4	2	no	—
53	Q₈	C₇×Q₈	1	2	yes	complemented
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₇×D₄	1	1	yes	characteristic
58	C₂×C₄	C₇×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₇	#477	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	—	4	1	no	—
67	C₄	C₇×Q₁₆	1	2	yes	—
68	C₄	C₂×C₇×Q₈	1	1	yes	characteristic
69	C₄	C₇×#41	1	1	yes	characteristic
70	C₂	C₇×C₂²⋊Q₈	1	1	yes	characteristic
71	C₂	C₂×C₇×Q₁₆	1	1	yes	characteristic
72	C₂	C₇×AGL(1,ℤ/8)	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. #477	C₇	direct (N × H)
18. C₇×Q₈	Q₈	semidirect
53. Q₈	C₇×Q₈	semidirect
61. C₇	#477	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
C₄×C₇×Q₈	2
C₇×#107	2
C₇×C₄⋊C₈	2
C₇×#103	2
#477	7

Subgroups — C7×#477 #90123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#477 #90123