Subgroups — C₇×#544 #90565

Order 448.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	4	yes	maximal, complemented
2	C₇×#125	C₂	1	4	yes	maximal, complemented
3	C₇×C₂²⋊Q₈	C₂	1	4	yes	maximal
4	C₇×#117	C₂	1	2	yes	maximal
5	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	C₂²	1	4	yes	—
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₇×C₄⋊C₄	C₂²	1	8	yes	—
14	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₄²×C₇	C₂²	1	4	yes	—
16	#544	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	C₂³	1	1	yes	characteristic
18	C₇×Q₈	—	4	4	no	—
19	C₂×C₄×C₇	—	4	2	no	—
20	C₂×C₄×C₇	Q₈	1	4	yes	complemented
21	C₂×C₄×C₇	C₂³	1	4	yes	—
22	C₂×C₄×C₇	C₂³	1	4	yes	—
23	C₂×C₄×C₇	C₂³	1	4	yes	—
24	C₂×C₄×C₇	C₂³	1	2	yes	—
25	C₄⋊Q₈	C₂×C₇	1	4	yes	complemented
26	#125	C₂×C₇	1	4	yes	complemented
27	C₂²⋊Q₈	C₂×C₇	1	4	yes	—
28	#117	C₂×C₇	1	2	yes	—
29	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
30	C₂²×C₇	—	2	1	no	—
31	C₂²×C₇	—	2	1	no	—
32	C₂²×C₇	C₂×Q₈	1	2	yes	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	—	2	4	no	—
35	C₄×C₇	—	2	4	no	—
36	C₄×C₇	—	2	4	no	—
37	C₄×C₇	C₂×Q₈	1	4	yes	—
38	C₂×Q₈	C₂²×C₇	1	4	yes	—
39	C₂²×C₄	C₂²×C₇	1	2	yes	—
40	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
41	C₄⋊C₄	C₂²×C₇	1	4	yes	—
42	C₄⋊C₄	C₂²×C₇	1	2	yes	—
43	C₄⋊C₄	C₂²×C₇	1	2	yes	—
44	C₄⋊C₄	C₂²×C₇	1	4	yes	—
45	C₄⋊C₄	C₂²×C₇	1	8	yes	—
46	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
47	C₄²	C₂²×C₇	1	4	yes	—
48	C₂×C₇	—	2	2	no	—
49	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
52	C₂³	C₂³×C₇	1	1	yes	characteristic
53	Q₈	—	4	4	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	C₇×Q₈	1	4	yes	complemented
56	C₂×C₄	C₂³×C₇	1	4	yes	—
57	C₂×C₄	C₂³×C₇	1	4	yes	—
58	C₂×C₄	C₂³×C₇	1	4	yes	—
59	C₂×C₄	C₂³×C₇	1	2	yes	—
60	C₇	#544	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×C₇×Q₈	1	2	yes	—
64	C₂²	C₂⁴×C₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	2	4	no	—
67	C₄	—	2	4	no	—
68	C₄	C₂×C₇×Q₈	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×C₇×Q₈	1	1	yes	characteristic
71	C₂	C₇×H(5,𝔽₂)	1	1	yes	characteristic
72	C₂	C₇×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂³×C₇	C₂³
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×C₄⋊Q₈	C₂	semidirect
2. C₇×#125	C₂	semidirect
16. #544	C₇	direct (N × H)
20. C₂×C₄×C₇	Q₈	semidirect
25. C₄⋊Q₈	C₂×C₇	semidirect
26. #125	C₂×C₇	semidirect
55. C₂×C₄	C₇×Q₈	semidirect
60. C₇	#544	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×C₄⋊Q₈	2
C₇×#125	2
C₇×C₂²⋊Q₈	2
C₇×#117	2
C₂×C₇×C₄⋊C₄	2
#544	7

Subgroups — C7×#544 #90565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#544 #90565