Subgroups — C₇×#481 #90127

Order 448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#106	C₂	1	2	yes	maximal, complemented
4	C₇×#102	C₂	1	2	yes	maximal, complemented
5	C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	—	2	1	no	—
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₇	—	2	1	no	—
9	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	—	2	1	no	—
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₇×C₂²⋊C₄	—	2	1	no	—
15	#481	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	—	2	1	no	—
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	—	4	2	no	—
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	—	2	2	no	—
23	C₂×C₄×C₇	—	4	2	no	—
24	C₂×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₇×C₈	—	2	2	no	—
27	#121	C₂×C₇	1	1	yes	characteristic
28	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
29	#106	C₂×C₇	1	2	yes	complemented
30	#102	C₂×C₇	1	2	yes	complemented
31	C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	—	4	1	no	—
35	C₂²×C₇	—	4	1	no	—
36	C₂²×C₇	—	4	1	no	—
37	C₂²×C₇	SD₁₆	1	2	yes	—
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	—	4	1	no	—
41	C₄×C₇	—	4	2	no	—
42	C₄×C₇	#41	1	2	yes	—
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
47	C₂×C₈	C₂²×C₇	1	2	yes	—
48	C₄⋊C₄	—	2	1	no	—
49	C₄⋊C₄	C₂²×C₇	1	2	yes	—
50	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
51	C₂²⋊C₄	—	2	1	no	—
52	C₂×C₇	—	2	2	no	—
53	C₂×C₇	—	8	1	no	—
54	C₂×C₇	#123	1	1	yes	characteristic
55	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂³	—	2	1	no	—
58	C₂³	C₇×D₄	1	1	yes	characteristic
59	D₄	—	2	2	no	—
60	D₄	—	4	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	C₇×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₇	#481	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₇×SD₁₆	1	2	yes	—
75	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	4	2	no	—
79	C₄	C₇×#41	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	8	1	no	—
82	C₂	C₇×#123	1	1	yes	characteristic
83	C₂	C₂×C₇×SD₁₆	1	1	yes	characteristic
84	C₂	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₇×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₄⋊C₄	C₂	semidirect
3. C₇×#106	C₂	semidirect
4. C₇×#102	C₂	semidirect
5. C₇×C₂²⋊C₈	C₂	semidirect
15. #481	C₇	direct (N × H)
28. C₂×C₄⋊C₄	C₂×C₇	semidirect
29. #106	C₂×C₇	semidirect
30. #102	C₂×C₇	semidirect
31. C₂²⋊C₈	C₂×C₇	semidirect
68. C₇	#481	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×#121	2
C₂×C₇×C₄⋊C₄	2
C₇×#106	2
C₇×#102	2
C₇×C₂²⋊C₈	2
#481	7

Subgroups — C7×#481 #90127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#481 #90127