Subgroups — C₇×#509 #90155

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×SD₃₂	C₂	1	2	yes	maximal, complemented
4	C₇×D₁₆	C₂	1	2	yes	maximal, complemented
5	C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	1	no	—
7	C₂×C₇×D₄	—	2	1	no	—
8	C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
9	C₇×SD₁₆	—	2	1	no	—
10	C₇×D₈	—	2	1	no	—
11	C₇×D₈	C₂²	1	2	yes	—
12	C₇×D₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×C₁₆	C₂²	1	2	yes	complemented
15	#509	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	—	4	1	no	—
17	C₇×Q₈	—	2	1	no	—
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	—	2	1	no	—
20	C₇×D₄	—	4	1	no	—
21	C₇×D₄	—	4	1	no	—
22	C₂×C₄×C₇	—	4	1	no	—
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₇×C₈	D₄	1	1	yes	characteristic
25	C₇×C₈	C₂³	1	1	yes	characteristic
26	#135	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂×D₈	C₂×C₇	1	1	yes	characteristic, complemented
28	SD₃₂	C₂×C₇	1	2	yes	complemented
29	D₁₆	C₂×C₇	1	2	yes	complemented
30	M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
31	C₂²×C₇	—	4	1	no	—
32	C₂²×C₇	—	4	2	no	—
33	C₂²×C₇	—	8	2	no	—
34	C₂²×C₇	D₈	1	1	yes	characteristic
35	C₄×C₇	—	4	1	no	—
36	C₄×C₇	D₈	1	1	yes	characteristic
37	C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	#41	—	2	1	no	—
39	C₂×D₄	—	2	1	no	—
40	Q₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
41	SD₁₆	—	2	1	no	—
42	D₈	—	2	1	no	—
43	D₈	C₂²×C₇	1	2	yes	—
44	D₈	C₂²×C₇	1	1	yes	characteristic, complemented
45	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
46	C₁₆	C₂²×C₇	1	2	yes	complemented
47	C₂×C₇	—	2	1	no	—
48	C₂×C₇	—	8	1	no	—
49	C₂×C₇	—	8	2	no	—
50	C₂×C₇	C₂×D₈	1	1	yes	characteristic
51	C₂³	—	4	1	no	—
52	Q₈	—	2	1	no	—
53	D₄	—	2	2	no	—
54	D₄	—	2	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	4	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	C₇×D₄	1	1	yes	characteristic
59	C₈	C₇×D₄	1	1	yes	characteristic
60	C₈	C₂³×C₇	1	1	yes	characteristic
61	C₇	#509	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²	—	4	1	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	8	2	no	—
65	C₂²	C₇×D₈	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	C₇×D₈	1	1	yes	characteristic
68	C₄	C₂×C₇×D₄	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	2	no	—
71	C₂	—	8	1	no	—
72	C₂	C₂×C₇×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³×C₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#135	C₂	semidirect
2. C₂×C₇×D₈	C₂	semidirect
3. C₇×SD₃₂	C₂	semidirect
4. C₇×D₁₆	C₂	semidirect
5. C₇×M₃₂	C₂	semidirect
8. C₇×Q₁₆	C₂²	semidirect
12. C₇×D₈	C₂²	semidirect
14. C₇×C₁₆	C₂²	semidirect
15. #509	C₇	direct (N × H)
26. #135	C₂×C₇	semidirect
27. C₂×D₈	C₂×C₇	semidirect
28. SD₃₂	C₂×C₇	semidirect
29. D₁₆	C₂×C₇	semidirect
30. M₃₂	C₂×C₇	semidirect
40. Q₁₆	C₂²×C₇	semidirect
44. D₈	C₂²×C₇	semidirect
46. C₁₆	C₂²×C₇	semidirect
61. C₇	#509	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×#135	2
C₂×C₇×D₈	2
C₇×SD₃₂	2
C₇×D₁₆	2
C₇×M₃₂	2
#509	7

Subgroups — C7×#509 #90155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#509 #90155