Subgroups — C₇×#560 #90581

Order 448.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#126	C₂	1	2	yes	maximal, complemented
3	C₇×#123	C₂	1	3	yes	maximal, complemented
4	C₇×#121	C₂	1	6	yes	maximal, complemented
5	C₇×#120	C₂	1	3	yes	maximal, complemented
6	C₂⁴×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×D₄	C₂²	1	6	yes	—
8	C₂×C₇×D₄	C₂²	1	3	yes	—
9	C₂×C₇×D₄	C₂²	1	3	yes	complemented
10	C₂²×C₄×C₇	C₂²	1	3	yes	complemented
11	C₇×C₄⋊C₄	C₂²	1	6	yes	complemented
12	C₇×C₂²⋊C₄	C₂²	1	6	yes	—
13	C₇×C₂²⋊C₄	C₂²	1	3	yes	—
14	C₇×C₂²⋊C₄	C₂²	1	3	yes	complemented
15	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
16	#560	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	—	4	3	no	—
18	C₂³×C₇	C₂³	1	2	yes	—
19	C₂³×C₇	C₂³	1	1	yes	characteristic
20	C₂³×C₇	C₂³	1	3	yes	—
21	C₇×D₄	—	4	6	no	—
22	C₇×D₄	—	4	3	no	—
23	C₇×D₄	—	4	3	no	—
24	C₂×C₄×C₇	—	4	3	no	—
25	C₂×C₄×C₇	C₂³	1	3	yes	—
26	C₂×C₄×C₇	C₂³	1	6	yes	—
27	#127	C₂×C₇	1	1	yes	characteristic, complemented
28	#126	C₂×C₇	1	2	yes	complemented
29	#123	C₂×C₇	1	3	yes	complemented
30	#121	C₂×C₇	1	6	yes	complemented
31	#120	C₂×C₇	1	3	yes	complemented
32	C₂²×C₇	—	2	3	no	—
33	C₂²×C₇	—	2	6	no	—
34	C₂²×C₇	—	2	6	no	—
35	C₂²×C₇	—	2	3	no	—
36	C₂²×C₇	—	4	4	no	—
37	C₂²×C₇	C₂⁴	1	1	yes	characteristic
38	C₄×C₇	—	2	6	no	—
39	C₄×C₇	—	2	3	no	—
40	C₂⁴	C₂²×C₇	1	1	yes	characteristic
41	C₂×D₄	C₂²×C₇	1	6	yes	—
42	C₂×D₄	C₂²×C₇	1	3	yes	—
43	C₂×D₄	C₂²×C₇	1	3	yes	complemented
44	C₂²×C₄	C₂²×C₇	1	3	yes	complemented
45	C₄⋊C₄	C₂²×C₇	1	6	yes	complemented
46	C₂²⋊C₄	C₂²×C₇	1	6	yes	—
47	C₂²⋊C₄	C₂²×C₇	1	3	yes	—
48	C₂²⋊C₄	C₂²×C₇	1	3	yes	complemented
49	C₄²	C₂²×C₇	1	1	yes	characteristic, complemented
50	C₂×C₇	—	4	3	no	—
51	C₂×C₇	—	4	2	no	—
52	C₂×C₇	—	4	1	no	—
53	C₂×C₇	H(5,𝔽₂)	1	3	yes	—
54	C₂³	—	4	3	no	—
55	C₂³	C₂³×C₇	1	2	yes	—
56	C₂³	C₂³×C₇	1	1	yes	characteristic
57	C₂³	C₂³×C₇	1	3	yes	—
58	D₄	—	4	6	no	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	C₂×C₄	—	4	3	no	—
62	C₂×C₄	C₂³×C₇	1	3	yes	—
63	C₂×C₄	C₂³×C₇	1	6	yes	—
64	C₇	#560	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	6	no	—
68	C₂²	—	2	3	no	—
69	C₂²	—	4	4	no	—
70	C₂²	C₂⁴×C₇	1	1	yes	characteristic
71	C₄	—	2	6	no	—
72	C₄	—	2	3	no	—
73	C₂	—	4	2	no	—
74	C₂	—	4	1	no	—
75	C₂	—	4	3	no	—
76	C₂	C₇×H(5,𝔽₂)	1	3	yes	—

	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×#127	C₂	semidirect
2. C₇×#126	C₂	semidirect
3. C₇×#123	C₂	semidirect
4. C₇×#121	C₂	semidirect
5. C₇×#120	C₂	semidirect
9. C₂×C₇×D₄	C₂²	semidirect
10. C₂²×C₄×C₇	C₂²	semidirect
11. C₇×C₄⋊C₄	C₂²	semidirect
14. C₇×C₂²⋊C₄	C₂²	semidirect
15. C₄²×C₇	C₂²	semidirect
16. #560	C₇	direct (N × H)
27. #127	C₂×C₇	semidirect
28. #126	C₂×C₇	semidirect
29. #123	C₂×C₇	semidirect
30. #121	C₂×C₇	semidirect
31. #120	C₂×C₇	semidirect
43. C₂×D₄	C₂²×C₇	semidirect
44. C₂²×C₄	C₂²×C₇	semidirect
45. C₄⋊C₄	C₂²×C₇	semidirect
48. C₂²⋊C₄	C₂²×C₇	semidirect
49. C₄²	C₂²×C₇	semidirect
64. C₇	#560	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×#127	2
C₇×#126	2
C₇×#123	2
C₇×#121	2
C₇×#120	2
#560	7

Subgroups — C7×#560 #90581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#560 #90581