Subgroups — C₇×#578 #90599

Order 448.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	C₇×#137	C₂	1	6	yes	maximal, complemented
3	C₇×#135	C₂	1	3	yes	maximal, complemented
4	C₂×C₇×Q₁₆	C₂	1	3	yes	maximal, complemented
5	C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	6	no	—
7	C₇×#41	C₂²	1	6	yes	—
8	C₇×#41	C₂²	1	1	yes	characteristic
9	C₂×C₇×Q₈	—	2	2	no	—
10	C₂×C₇×Q₈	C₂²	1	6	yes	—
11	C₇×Q₁₆	C₂²	1	3	yes	—
12	C₇×Q₁₆	C₂²	1	6	yes	—
13	C₇×SD₁₆	C₂²	1	6	yes	—
14	C₇×D₈	C₂²	1	1	yes	characteristic
15	C₇×M₁₆	C₂²	1	3	yes	—
16	C₂×C₇×C₈	C₂²	1	3	yes	—
17	#578	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₇×Q₈	—	2	6	no	—
19	C₇×Q₈	D₄	1	1	yes	characteristic
20	C₇×Q₈	C₂³	1	6	yes	—
21	C₇×D₄	—	2	6	no	—
22	C₇×D₄	D₄	1	3	yes	—
23	C₇×D₄	C₂³	1	2	yes	—
24	C₂×C₄×C₇	—	2	6	no	—
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	2₋¹⁺⁴	C₂×C₇	1	2	yes	complemented
30	#137	C₂×C₇	1	6	yes	complemented
31	#135	C₂×C₇	1	3	yes	complemented
32	C₂×Q₁₆	C₂×C₇	1	3	yes	complemented
33	#131	C₂×C₇	1	1	yes	characteristic, complemented
34	C₂²×C₇	—	2	2	no	—
35	C₂²×C₇	C₂×D₄	1	3	yes	—
36	C₄×C₇	—	2	6	no	—
37	C₄×C₇	C₂×D₄	1	3	yes	—
38	C₄×C₇	C₂⁴	1	1	yes	characteristic
39	#41	—	2	6	no	—
40	#41	C₂²×C₇	1	6	yes	—
41	#41	C₂²×C₇	1	1	yes	characteristic
42	C₂×Q₈	—	2	2	no	—
43	C₂×Q₈	C₂²×C₇	1	6	yes	—
44	Q₁₆	C₂²×C₇	1	3	yes	—
45	Q₁₆	C₂²×C₇	1	6	yes	—
46	SD₁₆	C₂²×C₇	1	6	yes	—
47	D₈	C₂²×C₇	1	1	yes	characteristic
48	M₁₆	C₂²×C₇	1	3	yes	—
49	C₂×C₈	C₂²×C₇	1	3	yes	—
50	C₂×C₇	—	2	3	no	—
51	C₂×C₇	—	4	2	no	—
52	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
53	Q₈	—	2	6	no	—
54	Q₈	C₇×D₄	1	1	yes	characteristic
55	Q₈	C₂³×C₇	1	6	yes	—
56	D₄	—	2	6	no	—
57	D₄	C₇×D₄	1	3	yes	—
58	D₄	C₂³×C₇	1	2	yes	—
59	C₂×C₄	—	2	6	no	—
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	C₂³×C₇	1	3	yes	—
62	C₈	C₂³×C₇	1	1	yes	characteristic
63	C₈	C₂³×C₇	1	3	yes	—
64	C₇	#578	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	2	2	no	—
66	C₂²	C₂×C₇×D₄	1	3	yes	—
67	C₄	—	2	6	no	—
68	C₄	C₂×C₇×D₄	1	3	yes	—
69	C₄	C₂⁴×C₇	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	4	2	no	—
72	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. C₇×2₋¹⁺⁴	C₂	semidirect
2. C₇×#137	C₂	semidirect
3. C₇×#135	C₂	semidirect
4. C₂×C₇×Q₁₆	C₂	semidirect
5. C₇×#131	C₂	semidirect
17. #578	C₇	direct (N × H)
29. 2₋¹⁺⁴	C₂×C₇	semidirect
30. #137	C₂×C₇	semidirect
31. #135	C₂×C₇	semidirect
32. C₂×Q₁₆	C₂×C₇	semidirect
33. #131	C₂×C₇	semidirect
64. C₇	#578	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×2₋¹⁺⁴	2
C₇×#137	2
C₇×#135	2
C₂×C₇×Q₁₆	2
C₇×#131	2
#578	7

Subgroups — C7×#578 #90599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#578 #90599