Subgroups — C₇×#442 #90088

Order 448.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
3	C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#102	C₂	1	2	yes	maximal, complemented
5	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₂²	1	2	yes	—
7	C₂²×C₄×C₇	—	2	2	no	—
8	C₇×D₈	C₄	1	4	yes	complemented
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₇×C₂²⋊C₄	—	2	2	no	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic
14	#442	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₇	—	2	2	no	—
16	C₇×D₄	—	2	2	no	—
17	C₇×D₄	C₂×C₄	1	4	yes	complemented
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	—	4	4	no	—
20	C₂×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₇×C₈	—	2	1	no	—
23	C₇×C₈	C₂×C₄	1	2	yes	complemented
24	C₂×D₈	C₂×C₇	1	1	yes	characteristic
25	C₄×D₄	C₂×C₇	1	2	yes	complemented
26	#106	C₂×C₇	1	1	yes	characteristic, complemented
27	#102	C₂×C₇	1	2	yes	complemented
28	#97	C₂×C₇	1	1	yes	characteristic, complemented
29	C₂²×C₇	—	2	4	no	—
30	C₂²×C₇	—	4	2	no	—
31	C₂²×C₇	—	4	2	no	—
32	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₇	—	2	2	no	—
34	C₄×C₇	—	4	2	no	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	#41	1	1	yes	characteristic
37	C₂×D₄	C₂²×C₇	1	2	yes	—
38	C₂²×C₄	—	2	2	no	—
39	D₈	C₄×C₇	1	4	yes	complemented
40	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
41	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
42	C₄⋊C₄	C₂²×C₇	1	2	yes	—
43	C₂²⋊C₄	—	2	2	no	—
44	C₄²	C₂²×C₇	1	1	yes	characteristic
45	C₂×C₇	—	4	4	no	—
46	C₂×C₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
48	C₂³	—	2	2	no	—
49	D₄	—	2	2	no	—
50	D₄	C₂×C₄×C₇	1	4	yes	complemented
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	4	4	no	—
53	C₂×C₄	C₇×D₄	1	2	yes	—
54	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₂×C₄×C₇	1	2	yes	complemented
57	C₇	#442	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	4	2	no	—
64	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
65	C₄	C₇×#41	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	C₄×C₇×D₄	1	1	yes	characteristic
68	C₂	C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator	C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂×C₇×D₈	C₂
LCS	G ▷ 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
2. C₄×C₇×D₄	C₂	semidirect
3. C₇×#106	C₂	semidirect
4. C₇×#102	C₂	semidirect
5. C₇×#97	C₂	semidirect
8. C₇×D₈	C₄	semidirect
14. #442	C₇	direct (N × H)
17. C₇×D₄	C₂×C₄	semidirect
23. C₇×C₈	C₂×C₄	semidirect
25. C₄×D₄	C₂×C₇	semidirect
26. #106	C₂×C₇	semidirect
27. #102	C₂×C₇	semidirect
28. #97	C₂×C₇	semidirect
39. D₈	C₄×C₇	semidirect
50. D₄	C₂×C₄×C₇	semidirect
56. C₈	C₂×C₄×C₇	semidirect
57. C₇	#442	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×D₈	2
C₄×C₇×D₄	2
C₇×#106	2
C₇×#102	2
C₇×#97	2
#442	7

Subgroups — C7×#442 #90088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#442 #90088