Subgroups — C₇×#427 #90073

Order 448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	2	yes	maximal
3	C₇×#107	C₂	1	2	yes	maximal, complemented
4	C₇×#106	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇×C₈	C₄	1	4	yes	complemented
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₇×C₂²⋊C₄	—	2	2	no	—
11	C₄²×C₇	—	2	2	no	—
12	#427	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₇	—	2	4	no	—
15	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₇	Q₈	1	2	yes	—
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₇×C₈	C₂×C₄	1	2	yes	—
20	C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
22	#117	C₂×C₇	1	2	yes	—
23	#107	C₂×C₇	1	2	yes	complemented
24	#106	C₂×C₇	1	2	yes	complemented
25	C₂²×C₇	—	2	2	no	—
26	C₂²×C₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
28	C₄×C₇	—	4	4	no	—
29	C₄×C₇	C₄⋊C₄	1	2	yes	—
30	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
33	C₂×C₈	C₄×C₇	1	4	yes	complemented
34	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
35	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
36	C₄⋊C₄	C₂²×C₇	1	4	yes	—
37	C₂²⋊C₄	—	2	2	no	—
38	C₄²	—	2	2	no	—
39	C₂×C₇	—	2	2	no	—
40	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₇	#135	1	2	yes	—
42	C₂³	C₇×D₄	1	1	yes	characteristic
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
45	C₂×C₄	C₇×D₄	1	1	yes	characteristic
46	C₂×C₄	C₇×Q₈	1	2	yes	—
47	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
48	C₈	C₂×C₄×C₇	1	2	yes	—
49	C₈	C₂×C₄×C₇	1	2	yes	—
50	C₇	#427	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²	—	2	2	no	—
52	C₂²	C₇×C₄⋊C₄	1	2	yes	—
53	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
54	C₄	—	4	4	no	—
55	C₄	C₇×C₄⋊C₄	1	2	yes	—
56	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
57	C₄	C₂×C₇×Q₈	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
60	C₂	C₇×#135	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₇×#107	C₂	semidirect
4. C₇×#106	C₂	semidirect
6. C₂×C₇×C₈	C₄	semidirect
12. #427	C₇	direct (N × H)
23. #107	C₂×C₇	semidirect
24. #106	C₂×C₇	semidirect
33. C₂×C₈	C₄×C₇	semidirect
50. C₇	#427	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
C₇×#117	2
C₇×#107	2
C₇×#106	2
#427	7

Subgroups — C7×#427 #90073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#427 #90073