Subgroups — C₇×#453 #90099

Order 448.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#104	C₂	1	2	yes	maximal, complemented
5	C₇×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	2	no	—
7	C₇×#41	C₂²	1	2	yes	—
8	C₂×C₇×D₄	—	2	2	no	—
9	C₂×C₇×D₄	—	2	2	no	—
10	C₂×C₇×D₄	—	2	2	no	—
11	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₇×SD₁₆	—	2	2	no	—
14	C₇×D₈	—	2	2	no	—
15	C₇×M₁₆	C₂²	1	2	yes	complemented
16	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
17	#453	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₇	—	2	2	no	—
19	C₂³×C₇	—	2	1	no	—
20	C₂³×C₇	D₄	1	2	yes	—
21	C₇×Q₈	D₄	1	2	yes	complemented
22	C₇×D₄	—	2	4	no	—
23	C₇×D₄	—	2	2	no	—
24	C₇×D₄	—	2	2	no	—
25	C₇×D₄	—	2	2	no	—
26	C₇×D₄	—	4	4	no	—
27	C₇×D₄	D₄	1	2	yes	complemented
28	C₂×C₄×C₇	—	2	2	no	—
29	C₂×C₄×C₇	—	2	1	no	—
30	C₂×C₄×C₇	—	2	2	no	—
31	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₇×C₈	—	2	2	no	—
33	H(5,𝔽₂)	C₂×C₇	1	1	yes	characteristic, complemented
34	AGL(1,ℤ/8)	C₂×C₇	1	2	yes	complemented
35	#127	C₂×C₇	1	1	yes	characteristic, complemented
36	#104	C₂×C₇	1	2	yes	complemented
37	#100	C₂×C₇	1	1	yes	characteristic, complemented
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	—	2	2	no	—
40	C₂²×C₇	—	4	2	no	—
41	C₂²×C₇	—	4	2	no	—
42	C₂²×C₇	—	4	1	no	—
43	C₂²×C₇	—	8	2	no	—
44	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
45	C₄×C₇	—	2	2	no	—
46	C₄×C₇	—	2	2	no	—
47	C₄×C₇	C₂×D₄	1	2	yes	—
48	#41	—	2	2	no	—
49	#41	C₂²×C₇	1	2	yes	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	2	2	no	—
52	C₂×D₄	—	2	2	no	—
53	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
54	C₂×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
55	SD₁₆	—	2	2	no	—
56	D₈	—	2	2	no	—
57	M₁₆	C₂²×C₇	1	2	yes	complemented
58	C₄²	C₂²×C₇	1	1	yes	characteristic, complemented
59	C₂×C₇	—	2	1	no	—
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	—	4	2	no	—
62	C₂×C₇	—	8	1	no	—
63	C₂×C₇	#120	1	1	yes	characteristic
64	C₂³	—	2	2	no	—
65	C₂³	—	2	1	no	—
66	C₂³	C₇×D₄	1	2	yes	—
67	Q₈	C₇×D₄	1	2	yes	complemented
68	D₄	—	2	4	no	—
69	D₄	—	2	2	no	—
70	D₄	—	2	2	no	—
71	D₄	—	2	2	no	—
72	D₄	—	4	4	no	—
73	D₄	C₇×D₄	1	2	yes	complemented
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₇	#453	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	2	no	—
86	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	2	2	no	—
89	C₄	C₂×C₇×D₄	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	8	1	no	—
94	C₂	C₇×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×H(5,𝔽₂)	C₂	semidirect
2. C₇×AGL(1,ℤ/8)	C₂	semidirect
3. C₇×#127	C₂	semidirect
4. C₇×#104	C₂	semidirect
5. C₇×#100	C₂	semidirect
12. C₂×C₇×D₄	C₂²	semidirect
15. C₇×M₁₆	C₂²	semidirect
16. C₄²×C₇	C₂²	semidirect
17. #453	C₇	direct (N × H)
21. C₇×Q₈	D₄	semidirect
27. C₇×D₄	D₄	semidirect
33. H(5,𝔽₂)	C₂×C₇	semidirect
34. AGL(1,ℤ/8)	C₂×C₇	semidirect
35. #127	C₂×C₇	semidirect
36. #104	C₂×C₇	semidirect
37. #100	C₂×C₇	semidirect
54. C₂×D₄	C₂²×C₇	semidirect
57. M₁₆	C₂²×C₇	semidirect
58. C₄²	C₂²×C₇	semidirect
67. Q₈	C₇×D₄	semidirect
73. D₄	C₇×D₄	semidirect
79. C₇	#453	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×H(5,𝔽₂)	2
C₇×AGL(1,ℤ/8)	2
C₇×#127	2
C₇×#104	2
C₇×#100	2
#453	7

Subgroups — C7×#453 #90099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#453 #90099