Subgroups — C₇×#576 #90597

Order 448.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₇×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
3	C₇×#135	C₂	1	3	yes	maximal, complemented
4	C₂×C₇×D₈	C₂	1	3	yes	maximal, complemented
5	C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	2	no	—
7	C₇×#41	C₂²	1	1	yes	characteristic
8	C₇×#41	C₂²	1	6	yes	complemented
9	C₂×C₇×D₄	—	2	6	no	—
10	C₂×C₇×D₄	C₂²	1	6	yes	complemented
11	C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
12	C₇×SD₁₆	C₂²	1	6	yes	complemented
13	C₇×D₈	C₂²	1	3	yes	complemented
14	C₇×D₈	C₂²	1	6	yes	complemented
15	C₇×M₁₆	C₂²	1	3	yes	complemented
16	C₂×C₇×C₈	C₂²	1	3	yes	complemented
17	#576	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₇	—	2	6	no	—
19	C₇×Q₈	D₄	1	1	yes	characteristic
20	C₇×Q₈	C₂³	1	2	yes	—
21	C₇×D₄	—	2	6	no	—
22	C₇×D₄	—	2	6	no	—
23	C₇×D₄	D₄	1	3	yes	—
24	C₇×D₄	C₂³	1	6	yes	—
25	C₂×C₄×C₇	—	2	6	no	—
26	C₂×C₄×C₇	C₂³	1	3	yes	—
27	C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×C₈	C₂³	1	3	yes	—
29	H(5,𝔽₂)	C₂×C₇	1	2	yes	complemented
30	AGL(1,ℤ/8)	C₂×C₇	1	6	yes	complemented
31	#135	C₂×C₇	1	3	yes	complemented
32	C₂×D₈	C₂×C₇	1	3	yes	complemented
33	#131	C₂×C₇	1	1	yes	characteristic, complemented
34	C₂²×C₇	—	2	6	no	—
35	C₂²×C₇	—	8	6	no	—
36	C₂²×C₇	C₂×D₄	1	3	yes	—
37	C₄×C₇	—	2	2	no	—
38	C₄×C₇	C₂×D₄	1	3	yes	—
39	C₄×C₇	C₂⁴	1	1	yes	characteristic
40	#41	—	2	2	no	—
41	#41	C₂²×C₇	1	1	yes	characteristic
42	#41	C₂²×C₇	1	6	yes	complemented
43	C₂×D₄	—	2	6	no	—
44	C₂×D₄	C₂²×C₇	1	6	yes	complemented
45	Q₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
46	SD₁₆	C₂²×C₇	1	6	yes	complemented
47	D₈	C₂²×C₇	1	3	yes	complemented
48	D₈	C₂²×C₇	1	6	yes	complemented
49	M₁₆	C₂²×C₇	1	3	yes	complemented
50	C₂×C₈	C₂²×C₇	1	3	yes	complemented
51	C₂×C₇	—	2	3	no	—
52	C₂×C₇	—	4	6	no	—
53	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
54	C₂³	—	2	6	no	—
55	Q₈	C₇×D₄	1	1	yes	characteristic
56	Q₈	C₂³×C₇	1	2	yes	—
57	D₄	—	2	6	no	—
58	D₄	—	2	6	no	—
59	D₄	C₇×D₄	1	3	yes	—
60	D₄	C₂³×C₇	1	6	yes	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	C₂³×C₇	1	3	yes	—
63	C₈	C₂³×C₇	1	1	yes	characteristic
64	C₈	C₂³×C₇	1	3	yes	—
65	C₇	#576	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	—	2	6	no	—
67	C₂²	—	8	6	no	—
68	C₂²	C₂×C₇×D₄	1	3	yes	—
69	C₄	—	2	2	no	—
70	C₄	C₂×C₇×D₄	1	3	yes	—
71	C₄	C₂⁴×C₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	4	6	no	—
74	C₂	C₂²×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×H(5,𝔽₂)	C₂	semidirect
2. C₇×AGL(1,ℤ/8)	C₂	semidirect
3. C₇×#135	C₂	semidirect
4. C₂×C₇×D₈	C₂	semidirect
5. C₇×#131	C₂	semidirect
8. C₇×#41	C₂²	semidirect
10. C₂×C₇×D₄	C₂²	semidirect
11. C₇×Q₁₆	C₂²	semidirect
12. C₇×SD₁₆	C₂²	semidirect
13. C₇×D₈	C₂²	semidirect
14. C₇×D₈	C₂²	semidirect
15. C₇×M₁₆	C₂²	semidirect
16. C₂×C₇×C₈	C₂²	semidirect
17. #576	C₇	direct (N × H)
29. H(5,𝔽₂)	C₂×C₇	semidirect
30. AGL(1,ℤ/8)	C₂×C₇	semidirect
31. #135	C₂×C₇	semidirect
32. C₂×D₈	C₂×C₇	semidirect
33. #131	C₂×C₇	semidirect
42. #41	C₂²×C₇	semidirect
44. C₂×D₄	C₂²×C₇	semidirect
45. Q₁₆	C₂²×C₇	semidirect
46. SD₁₆	C₂²×C₇	semidirect
47. D₈	C₂²×C₇	semidirect
48. D₈	C₂²×C₇	semidirect
49. M₁₆	C₂²×C₇	semidirect
50. C₂×C₈	C₂²×C₇	semidirect
65. C₇	#576	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×H(5,𝔽₂)	2
C₇×AGL(1,ℤ/8)	2
C₇×#135	2
C₂×C₇×D₈	2
C₇×#131	2
#576	7

Subgroups — C7×#576 #90597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#576 #90597