Subgroups — C₇×#518 #90539

Order 448.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×D₄	C₂	1	8	yes	maximal, complemented
3	C₇×#117	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₇	C₂²	1	2	yes	—
6	C₂×C₇×D₄	C₄	1	8	yes	complemented
7	C₂×C₇×D₄	C₂²	1	4	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₇	C₂²	1	8	yes	complemented
10	C₇×C₄⋊C₄	C₂²	1	4	yes	complemented
11	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
12	C₇×C₂²⋊C₄	C₂²	1	8	yes	complemented
13	C₄²×C₇	C₂²	1	4	yes	complemented
14	#518	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₇	—	4	4	no	—
16	C₂³×C₇	C₂×C₄	1	8	yes	complemented
17	C₂³×C₇	C₂³	1	1	yes	characteristic
18	C₂³×C₇	C₂³	1	4	yes	—
19	C₇×D₄	C₂×C₄	1	16	yes	complemented
20	C₂×C₄×C₇	—	4	8	no	—
21	C₂×C₄×C₇	C₂×C₄	1	4	yes	complemented
22	C₂×C₄×C₇	C₂³	1	8	yes	—
23	C₂×C₄×C₇	C₂³	1	2	yes	—
24	C₂²×D₄	C₂×C₇	1	1	yes	characteristic
25	C₄×D₄	C₂×C₇	1	8	yes	complemented
26	#117	C₂×C₇	1	2	yes	complemented
27	C₂×C₂²⋊C₄	C₂×C₇	1	4	yes	complemented
28	C₂²×C₇	—	2	8	no	—
29	C₂²×C₇	—	2	2	no	—
30	C₂²×C₇	—	4	8	no	—
31	C₂²×C₇	C₂²×C₄	1	2	yes	—
32	C₂²×C₇	C₂²×C₄	1	8	yes	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	—	2	8	no	—
35	C₄×C₇	C₂²×C₄	1	4	yes	—
36	C₂⁴	C₂²×C₇	1	2	yes	—
37	C₂×D₄	C₄×C₇	1	8	yes	complemented
38	C₂×D₄	C₂²×C₇	1	4	yes	—
39	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
40	C₂²×C₄	C₂²×C₇	1	8	yes	complemented
41	C₄⋊C₄	C₂²×C₇	1	4	yes	complemented
42	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
43	C₂²⋊C₄	C₂²×C₇	1	8	yes	complemented
44	C₄²	C₂²×C₇	1	4	yes	complemented
45	C₂×C₇	—	2	2	no	—
46	C₂×C₇	—	2	8	no	—
47	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
49	C₂³	—	4	4	no	—
50	C₂³	C₂×C₄×C₇	1	8	yes	complemented
51	C₂³	C₂³×C₇	1	1	yes	characteristic
52	C₂³	C₂³×C₇	1	4	yes	—
53	D₄	C₂×C₄×C₇	1	16	yes	complemented
54	C₂×C₄	—	4	8	no	—
55	C₂×C₄	C₂×C₄×C₇	1	4	yes	complemented
56	C₂×C₄	C₂³×C₇	1	8	yes	—
57	C₂×C₄	C₂³×C₇	1	2	yes	—
58	C₇	#518	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	2	8	no	—
60	C₂²	—	2	2	no	—
61	C₂²	—	4	8	no	—
62	C₂²	C₂²×C₄×C₇	1	2	yes	—
63	C₂²	C₂²×C₄×C₇	1	8	yes	—
64	C₂²	C₂⁴×C₇	1	1	yes	characteristic
65	C₄	—	2	8	no	—
66	C₄	C₂²×C₄×C₇	1	4	yes	—
67	C₂	—	2	2	no	—
68	C₂	—	2	8	no	—
69	C₂	C₂³×C₄×C₇	1	1	yes	characteristic
70	C₂	C₇×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator	C₂	C₂³×C₄×C₇
Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×D₄	C₂	semidirect
3. C₇×#117	C₂	semidirect
4. C₂×C₇×C₂²⋊C₄	C₂	semidirect
6. C₂×C₇×D₄	C₄	semidirect
9. C₂²×C₄×C₇	C₂²	semidirect
10. C₇×C₄⋊C₄	C₂²	semidirect
12. C₇×C₂²⋊C₄	C₂²	semidirect
13. C₄²×C₇	C₂²	semidirect
14. #518	C₇	direct (N × H)
16. C₂³×C₇	C₂×C₄	semidirect
19. C₇×D₄	C₂×C₄	semidirect
21. C₂×C₄×C₇	C₂×C₄	semidirect
25. C₄×D₄	C₂×C₇	semidirect
26. #117	C₂×C₇	semidirect
27. C₂×C₂²⋊C₄	C₂×C₇	semidirect
37. C₂×D₄	C₄×C₇	semidirect
40. C₂²×C₄	C₂²×C₇	semidirect
41. C₄⋊C₄	C₂²×C₇	semidirect
43. C₂²⋊C₄	C₂²×C₇	semidirect
44. C₄²	C₂²×C₇	semidirect
50. C₂³	C₂×C₄×C₇	semidirect
53. D₄	C₂×C₄×C₇	semidirect
55. C₂×C₄	C₂×C₄×C₇	semidirect
58. C₇	#518	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×D₄	2
C₄×C₇×D₄	2
C₇×#117	2
C₂×C₇×C₂²⋊C₄	2
#518	7

Subgroups — C7×#518 #90539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#518 #90539