Subgroups — C₇×#562 #90583

Order 448.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#124	C₂	1	2	yes	maximal, complemented
3	C₇×#123	C₂	1	4	yes	maximal, complemented
4	C₇×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
5	C₇×#121	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₇×D₄	C₂²	1	4	yes	—
8	C₂×C₇×D₄	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₂²	1	4	yes	—
10	C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₇×C₄⋊C₄	C₂²	1	4	yes	—
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₄²×C₇	C₂²	1	1	yes	characteristic
17	#562	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₇	C₂³	1	4	yes	—
19	C₇×Q₈	—	4	2	no	—
20	C₇×D₄	—	4	4	no	—
21	C₇×D₄	—	4	2	no	—
22	C₂×C₄×C₇	—	4	4	no	—
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₇	C₂³	1	2	yes	—
25	C₂×C₄×C₇	C₂³	1	4	yes	—
26	C₂×C₄×C₇	C₂³	1	4	yes	—
27	#125	C₂×C₇	1	1	yes	characteristic, complemented
28	#124	C₂×C₇	1	2	yes	complemented
29	#123	C₂×C₇	1	4	yes	complemented
30	C₂²⋊Q₈	C₂×C₇	1	4	yes	complemented
31	#121	C₂×C₇	1	4	yes	complemented
32	C₂²×C₇	—	2	4	no	—
33	C₂²×C₇	—	2	4	no	—
34	C₂²×C₇	—	2	4	no	—
35	C₂²×C₇	C₂⁴	1	1	yes	characteristic
36	C₄×C₇	—	2	4	no	—
37	C₄×C₇	—	2	4	no	—
38	C₄×C₇	—	2	2	no	—
39	C₄×C₇	—	2	1	no	—
40	C₂×Q₈	C₂²×C₇	1	2	yes	—
41	C₂×D₄	C₂²×C₇	1	4	yes	—
42	C₂×D₄	C₂²×C₇	1	2	yes	—
43	C₂²×C₄	C₂²×C₇	1	4	yes	—
44	C₄⋊C₄	C₂²×C₇	1	4	yes	—
45	C₄⋊C₄	C₂²×C₇	1	4	yes	—
46	C₄⋊C₄	C₂²×C₇	1	2	yes	—
47	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
48	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
49	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
50	C₄²	C₂²×C₇	1	1	yes	characteristic
51	C₂×C₇	—	4	4	no	—
52	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
53	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
54	C₂³	C₂³×C₇	1	4	yes	—
55	Q₈	—	4	2	no	—
56	D₄	—	4	4	no	—
57	D₄	—	4	2	no	—
58	C₂×C₄	—	4	4	no	—
59	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
60	C₂×C₄	C₂³×C₇	1	2	yes	—
61	C₂×C₄	C₂³×C₇	1	4	yes	—
62	C₂×C₄	C₂³×C₇	1	4	yes	—
63	C₇	#562	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²	—	2	4	no	—
65	C₂²	—	2	4	no	—
66	C₂²	—	2	4	no	—
67	C₂²	C₂⁴×C₇	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	4	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₂	—	4	4	no	—
73	C₂	C₇×H(5,𝔽₂)	1	2	yes	—
74	C₂	C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#125	C₂	semidirect
2. C₇×#124	C₂	semidirect
3. C₇×#123	C₂	semidirect
4. C₇×C₂²⋊Q₈	C₂	semidirect
5. C₇×#121	C₂	semidirect
17. #562	C₇	direct (N × H)
27. #125	C₂×C₇	semidirect
28. #124	C₂×C₇	semidirect
29. #123	C₂×C₇	semidirect
30. C₂²⋊Q₈	C₂×C₇	semidirect
31. #121	C₂×C₇	semidirect
63. C₇	#562	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×#125	2
C₇×#124	2
C₇×#123	2
C₇×C₂²⋊Q₈	2
C₇×#121	2
#562	7

Subgroups — C7×#562 #90583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#562 #90583