Subgroups — C₇×#563 #90584

Order 448.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₇×#126	C₂	1	4	yes	maximal, complemented
3	C₇×#125	C₂	1	2	yes	maximal, complemented
4	C₇×#124	C₂	1	1	yes	maximal, characteristic
5	C₇×#123	C₂	1	2	yes	maximal
6	C₇×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	C₂×C₇×Q₈	C₂²	1	2	yes	—
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₇	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	4	yes	—
14	C₇×C₄⋊C₄	C₂²	1	4	yes	—
15	C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
18	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₄²×C₇	C₂²	1	2	yes	—
21	#563	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₇	C₂³	1	2	yes	—
23	C₇×Q₈	—	4	2	no	—
24	C₇×Q₈	—	4	1	no	—
25	C₇×D₄	—	4	1	no	—
26	C₂×C₄×C₇	—	4	2	no	—
27	C₂×C₄×C₇	C₂³	1	2	yes	—
28	C₂×C₄×C₇	C₂³	1	4	yes	—
29	C₂×C₄×C₇	C₂³	1	2	yes	—
30	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₇	C₂³	1	4	yes	—
32	C₄⋊Q₈	C₂×C₇	1	2	yes	complemented
33	#126	C₂×C₇	1	4	yes	complemented
34	#125	C₂×C₇	1	2	yes	complemented
35	#124	C₂×C₇	1	1	yes	characteristic
36	#123	C₂×C₇	1	2	yes	—
37	C₂²⋊Q₈	C₂×C₇	1	4	yes	complemented
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	—	2	4	no	—
40	C₂²×C₇	C₂⁴	1	1	yes	characteristic
41	C₄×C₇	—	2	4	no	—
42	C₄×C₇	—	2	1	no	—
43	C₄×C₇	—	2	2	no	—
44	C₄×C₇	—	2	2	no	—
45	C₄×C₇	—	2	4	no	—
46	C₂×Q₈	C₂²×C₇	1	2	yes	—
47	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
48	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
49	C₂²×C₄	C₂²×C₇	1	2	yes	—
50	C₄⋊C₄	C₂²×C₇	1	2	yes	—
51	C₄⋊C₄	C₂²×C₇	1	4	yes	—
52	C₄⋊C₄	C₂²×C₇	1	4	yes	—
53	C₄⋊C₄	C₂²×C₇	1	4	yes	—
54	C₄⋊C₄	C₂²×C₇	1	2	yes	—
55	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
56	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
57	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
58	C₄²	C₂²×C₇	1	1	yes	characteristic
59	C₄²	C₂²×C₇	1	2	yes	—
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂×C₇	2₋¹⁺⁴	1	2	yes	—
63	C₂³	C₂³×C₇	1	2	yes	—
64	Q₈	—	4	2	no	—
65	Q₈	—	4	1	no	—
66	D₄	—	4	1	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	C₂³×C₇	1	2	yes	—
69	C₂×C₄	C₂³×C₇	1	4	yes	—
70	C₂×C₄	C₂³×C₇	1	2	yes	—
71	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
72	C₂×C₄	C₂³×C₇	1	4	yes	—
73	C₇	#563	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂⁴×C₇	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	2	4	no	—
82	C₂	—	4	2	no	—
83	C₂	C₇×H(5,𝔽₂)	1	1	yes	characteristic
84	C₂	C₇×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂×C₇×D₄	C₂²
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×#126	C₂	semidirect
3. C₇×#125	C₂	semidirect
6. C₇×C₂²⋊Q₈	C₂	semidirect
21. #563	C₇	direct (N × H)
32. C₄⋊Q₈	C₂×C₇	semidirect
33. #126	C₂×C₇	semidirect
34. #125	C₂×C₇	semidirect
37. C₂²⋊Q₈	C₂×C₇	semidirect
73. C₇	#563	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×#126	2
C₇×#125	2
C₇×#124	2
C₇×#123	2
C₇×C₂²⋊Q₈	2
#563	7

Subgroups — C7×#563 #90584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#563 #90584