Subgroups — C₇×#520 #90541

Order 448.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×Q₈	C₂	1	2	yes	maximal, complemented
3	C₄×C₇×D₄	C₂	1	6	yes	maximal, complemented
4	C₇×#117	C₂	1	3	yes	maximal, complemented
5	C₂×C₇×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₇×#41	C₄	1	8	yes	complemented
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₄	C₂²	1	3	yes	—
9	C₂²×C₄×C₇	C₂²	1	6	yes	—
10	C₂²×C₄×C₇	C₂²	1	3	yes	—
11	C₇×C₄⋊C₄	C₂²	1	3	yes	—
12	C₇×C₄⋊C₄	C₂²	1	6	yes	—
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₇×C₂²⋊C₄	C₂²	1	6	yes	—
15	C₄²×C₇	C₂²	1	6	yes	—
16	#520	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	C₂³	1	3	yes	—
18	C₇×Q₈	C₂×C₄	1	4	yes	complemented
19	C₇×D₄	C₂×C₄	1	12	yes	complemented
20	C₂×C₄×C₇	—	4	6	no	—
21	C₂×C₄×C₇	C₂×C₄	1	12	yes	complemented
22	C₂×C₄×C₇	C₂³	1	2	yes	—
23	C₂×C₄×C₇	C₂³	1	6	yes	—
24	C₂×C₄×C₇	C₂³	1	3	yes	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂×#41	C₂×C₇	1	1	yes	characteristic
27	C₄×Q₈	C₂×C₇	1	2	yes	complemented
28	C₄×D₄	C₂×C₇	1	6	yes	complemented
29	#117	C₂×C₇	1	3	yes	complemented
30	C₂×C₄⋊C₄	C₂×C₇	1	3	yes	complemented
31	C₂²×C₇	—	2	3	no	—
32	C₂²×C₇	—	2	3	no	—
33	C₂²×C₇	C₂²×C₄	1	6	yes	—
34	C₂²×C₇	C₂⁴	1	1	yes	characteristic
35	C₄×C₇	—	2	6	no	—
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	C₂²×C₄	1	2	yes	—
38	C₄×C₇	C₂²×C₄	1	6	yes	—
39	#41	C₄×C₇	1	8	yes	complemented
40	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
41	C₂×D₄	C₂²×C₇	1	3	yes	—
42	C₂²×C₄	C₂²×C₇	1	6	yes	—
43	C₂²×C₄	C₂²×C₇	1	3	yes	—
44	C₄⋊C₄	C₂²×C₇	1	3	yes	—
45	C₄⋊C₄	C₂²×C₇	1	6	yes	—
46	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
47	C₂²⋊C₄	C₂²×C₇	1	6	yes	—
48	C₄²	C₂²×C₇	1	6	yes	—
49	C₂×C₇	—	2	6	no	—
50	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
51	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
53	C₂³	C₂³×C₇	1	3	yes	—
54	Q₈	C₂×C₄×C₇	1	4	yes	complemented
55	D₄	C₂×C₄×C₇	1	12	yes	complemented
56	C₂×C₄	—	4	6	no	—
57	C₂×C₄	C₂×C₄×C₇	1	12	yes	complemented
58	C₂×C₄	C₂³×C₇	1	2	yes	—
59	C₂×C₄	C₂³×C₇	1	6	yes	—
60	C₂×C₄	C₂³×C₇	1	3	yes	—
61	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
62	C₇	#520	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²	—	2	3	no	—
64	C₂²	—	2	3	no	—
65	C₂²	C₂²×C₄×C₇	1	6	yes	—
66	C₂²	C₂⁴×C₇	1	1	yes	characteristic
67	C₄	—	2	6	no	—
68	C₄	—	2	2	no	—
69	C₄	C₂²×C₄×C₇	1	2	yes	—
70	C₄	C₂²×C₄×C₇	1	6	yes	—
71	C₂	—	2	6	no	—
72	C₂	C₂³×C₄×C₇	1	1	yes	characteristic
73	C₂	C₇×H(5,𝔽₂)	1	1	yes	characteristic
74	C₂	C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator	C₂	C₂³×C₄×C₇
Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂×C₇×#41	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×Q₈	C₂	semidirect
3. C₄×C₇×D₄	C₂	semidirect
4. C₇×#117	C₂	semidirect
5. C₂×C₇×C₄⋊C₄	C₂	semidirect
6. C₇×#41	C₄	semidirect
16. #520	C₇	direct (N × H)
18. C₇×Q₈	C₂×C₄	semidirect
19. C₇×D₄	C₂×C₄	semidirect
21. C₂×C₄×C₇	C₂×C₄	semidirect
27. C₄×Q₈	C₂×C₇	semidirect
28. C₄×D₄	C₂×C₇	semidirect
29. #117	C₂×C₇	semidirect
30. C₂×C₄⋊C₄	C₂×C₇	semidirect
39. #41	C₄×C₇	semidirect
54. Q₈	C₂×C₄×C₇	semidirect
55. D₄	C₂×C₄×C₇	semidirect
57. C₂×C₄	C₂×C₄×C₇	semidirect
62. C₇	#520	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#41	2
C₄×C₇×Q₈	2
C₄×C₇×D₄	2
C₇×#117	2
C₂×C₇×C₄⋊C₄	2
#520	7

Subgroups — C7×#520 #90541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#520 #90541