Subgroups — C₇×#486 #90132

Order 448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#127	C₂	1	1	yes	maximal, characteristic
3	C₇×#102	C₂	1	4	yes	maximal, complemented
4	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×Q₈	—	2	1	no	—
6	C₂×C₇×D₄	—	2	1	no	—
7	C₂×C₇×D₄	—	2	1	no	—
8	C₂×C₇×D₄	C₂²	1	2	yes	—
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₇×C₄⋊C₄	—	2	1	no	—
11	C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₄²×C₇	C₂²	1	1	yes	characteristic
13	#486	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₇	—	2	2	no	—
15	C₇×Q₈	—	4	2	no	—
16	C₇×D₄	—	2	4	no	—
17	C₇×D₄	—	4	2	no	—
18	C₇×D₄	—	4	2	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₇×C₈	—	2	2	no	—
24	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
25	#127	C₂×C₇	1	1	yes	characteristic
26	#102	C₂×C₇	1	4	yes	complemented
27	C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
28	C₂²×C₇	—	4	2	no	—
29	C₂²×C₇	—	4	2	no	—
30	C₂²×C₇	—	4	2	no	—
31	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₇	—	4	2	no	—
33	C₄×C₇	D₈	1	2	yes	—
34	C₄×C₇	SD₁₆	1	2	yes	—
35	C₄×C₇	#41	1	2	yes	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×D₄	—	2	1	no	—
38	C₂×D₄	—	2	1	no	—
39	C₂×D₄	C₂²×C₇	1	2	yes	—
40	C₂×C₈	C₂²×C₇	1	2	yes	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	C₂²×C₇	1	2	yes	—
43	C₄²	C₂²×C₇	1	1	yes	characteristic
44	C₂×C₇	—	8	2	no	—
45	C₂×C₇	#124	1	1	yes	characteristic
46	C₂×C₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
48	C₂³	—	2	2	no	—
49	Q₈	—	4	2	no	—
50	D₄	—	2	4	no	—
51	D₄	—	4	2	no	—
52	D₄	—	4	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₇×D₄	1	1	yes	characteristic
55	C₂×C₄	C₇×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₇	#486	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	C₇×D₈	1	2	yes	—
65	C₄	C₇×SD₁₆	1	2	yes	—
66	C₄	C₇×#41	1	2	yes	—
67	C₂	—	8	2	no	—
68	C₂	C₇×#124	1	1	yes	characteristic
69	C₂	C₂×C₇×D₈	1	1	yes	characteristic
70	C₂	C₂×C₇×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₇×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×C₄⋊Q₈	C₂	semidirect
3. C₇×#102	C₂	semidirect
4. C₄×C₇×C₈	C₂	semidirect
13. #486	C₇	direct (N × H)
24. C₄⋊Q₈	C₂×C₇	semidirect
26. #102	C₂×C₇	semidirect
27. C₄×C₈	C₂×C₇	semidirect
58. C₇	#486	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×#127	2
C₇×#102	2
C₄×C₇×C₈	2
#486	7

Subgroups — C7×#486 #90132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#486 #90132