Subgroups — C₇×#474 #90120

Order 448.

108 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, complemented
2	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#107	C₂	1	2	yes	maximal, complemented
4	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#102	C₂	1	2	yes	maximal
6	C₂×C₇×Q₈	—	2	1	no	—
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₇	—	2	1	no	—
9	C₂×C₇×C₈	C₂²	1	2	yes	—
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	C₇×C₂²⋊C₄	—	2	1	no	—
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	#474	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	—	2	1	no	—
17	C₇×Q₈	—	4	2	no	—
18	C₇×D₄	—	2	1	no	—
19	C₇×D₄	Q₈	1	2	yes	complemented
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	—	4	2	no	—
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₇×C₈	—	2	2	no	—
27	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
28	C₄×D₄	C₂×C₇	1	1	yes	characteristic
29	#107	C₂×C₇	1	2	yes	complemented
30	C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
31	#102	C₂×C₇	1	2	yes	—
32	C₂²×C₇	—	2	2	no	—
33	C₂²×C₇	—	4	1	no	—
34	C₂²×C₇	—	4	1	no	—
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	—	4	1	no	—
38	C₄×C₇	—	4	2	no	—
39	C₄×C₇	D₈	1	2	yes	—
40	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
41	C₄×C₇	#41	1	1	yes	characteristic
42	C₂×Q₈	—	2	1	no	—
43	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
44	C₂²×C₄	—	2	1	no	—
45	C₂×C₈	C₂²×C₇	1	2	yes	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂²×C₇	1	2	yes	—
48	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
49	C₂²⋊C₄	—	2	1	no	—
50	C₄²	C₂²×C₇	1	1	yes	characteristic
51	C₂×C₇	—	4	2	no	—
52	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂×C₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₇	#137	1	1	yes	characteristic
55	C₂³	—	2	1	no	—
56	Q₈	—	4	2	no	—
57	D₄	—	2	1	no	—
58	D₄	C₇×Q₈	1	2	yes	complemented
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	C₇×D₄	1	1	yes	characteristic
63	C₂×C₄	C₇×D₄	1	1	yes	characteristic
64	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₇	#474	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	2	no	—
74	C₄	C₇×D₈	1	2	yes	—
75	C₄	C₂×C₇×Q₈	1	1	yes	characteristic
76	C₄	C₇×#41	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	C₇×C₂²⋊Q₈	1	1	yes	characteristic
79	C₂	C₂×C₇×D₈	1	1	yes	characteristic
80	C₂	C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂×C₇×D₄	C₂²
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
1. C₇×C₄⋊Q₈	C₂	semidirect
3. C₇×#107	C₂	semidirect
4. C₇×C₄⋊C₈	C₂	semidirect
15. #474	C₇	direct (N × H)
19. C₇×D₄	Q₈	semidirect
27. C₄⋊Q₈	C₂×C₇	semidirect
29. #107	C₂×C₇	semidirect
30. C₄⋊C₈	C₂×C₇	semidirect
58. D₄	C₇×Q₈	semidirect
66. C₇	#474	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×D₄	2
C₇×#107	2
C₇×C₄⋊C₈	2
C₇×#102	2
#474	7

Subgroups — C7×#474 #90120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#474 #90120