Subgroups — C₇×#901 #10621298

Order 672.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#235	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
5	D₃×C₇×C₈	C₂	1	2	yes	maximal, complemented
6	C₇×#131	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×#67	C₄	1	2	yes	—
9	C₇×D₁₂	C₄	1	1	yes	characteristic
10	C₄×D₃×C₇	C₂²	1	2	yes	—
11	C₇×Dic₆	C₄	1	1	yes	characteristic
12	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×C₃⋊C₈	C₂²	1	2	yes	—
15	C₇×#41	—	3	1	no	—
16	C₇×M₁₆	—	3	1	no	—
17	C₇×M₁₆	—	3	2	no	—
18	C₂×C₇×C₈	—	3	2	no	—
19	C₂×C₇×C₈	D₃	1	1	yes	characteristic, complemented
20	#901	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×D₃×C₇	C₂×C₄	1	2	yes	—
23	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×Dic₃	C₂×C₄	1	2	yes	—
26	C₇×Q₈	—	3	1	no	—
27	C₇×D₄	—	3	2	no	—
28	C₇×D₄	—	3	1	no	—
29	C₂×C₄×C₇	—	3	2	no	—
30	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
31	C₇×C₈	—	3	2	no	—
32	C₇×C₈	C₂×D₃	1	1	yes	characteristic
33	C₇×C₈	C₂×D₃	1	1	yes	characteristic
34	#235	C₂×C₇	1	1	yes	characteristic
35	C₂×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
36	#208	C₂×C₇	1	1	yes	characteristic, complemented
37	ΓL(1,𝔽₂₅)	C₂×C₇	1	2	yes	complemented
38	D₃×C₈	C₂×C₇	1	2	yes	complemented
39	C₂×C₃×C₇	—	2	1	no	—
40	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
41	D₃×C₇	—	2	2	no	—
42	#131	—	3	1	no	Sylow
43	C₂²×C₇	—	3	2	no	—
44	C₂²×C₇	C₄×D₃	1	1	yes	characteristic
45	C₄×C₇	—	3	2	no	—
46	C₄×C₇	C₄×D₃	1	1	yes	characteristic
47	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
49	#67	C₄×C₇	1	2	yes	—
50	D₁₂	C₄×C₇	1	1	yes	characteristic
51	C₄×D₃	C₂²×C₇	1	2	yes	—
52	Dic₆	C₄×C₇	1	1	yes	characteristic
53	C₃×C₈	C₂²×C₇	1	1	yes	characteristic
54	C₃×C₈	C₂²×C₇	1	1	yes	characteristic
55	C₃⋊C₈	C₂²×C₇	1	2	yes	—
56	C₃×C₇	#131	1	1	yes	characteristic, complemented, Hall
57	#41	—	3	1	no	—
58	M₁₆	—	3	1	no	—
59	M₁₆	—	3	2	no	—
60	C₂×C₈	—	3	2	no	—
61	C₂×C₈	D₃×C₇	1	1	yes	characteristic, complemented
62	C₂×C₇	—	2	1	no	—
63	C₂×C₇	—	6	2	no	—
64	C₂×C₇	C₂×C₄×D₃	1	1	yes	characteristic
65	C₂²×C₃	C₂×C₄×C₇	1	1	yes	characteristic
66	C₂×D₃	C₂×C₄×C₇	1	2	yes	—
67	C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
68	C₃×C₄	C₂³×C₇	1	1	yes	characteristic
69	Dic₃	C₂×C₄×C₇	1	2	yes	—
70	Q₈	—	3	1	no	—
71	D₄	—	3	2	no	—
72	D₄	—	3	1	no	—
73	C₂×C₄	—	3	2	no	—
74	C₂×C₄	C₂×D₃×C₇	1	1	yes	characteristic
75	C₈	—	3	2	no	—
76	C₈	C₂×D₃×C₇	1	1	yes	characteristic
77	C₈	C₂×D₃×C₇	1	1	yes	characteristic
78	C₇	#901	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
81	D₃	—	2	2	no	—
82	C₂²	—	3	2	no	—
83	C₂²	C₄×D₃×C₇	1	1	yes	characteristic
84	C₄	—	3	2	no	—
85	C₄	C₄×D₃×C₇	1	1	yes	characteristic
86	C₄	C₂²×D₃×C₇	1	1	yes	characteristic
87	C₃	C₇×#131	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	6	2	no	—
90	C₂	C₂×C₄×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₇
Frattini	C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×C₈	C₂
socle	C₂×C₃×C₇	C₂²×C₄
prime radical	C₇×#235	C₂
cosocle	C₃×C₄	C₂³×C₇
hypercenter	C₂×C₇×C₈	D₃
hypocenter	C₃	C₇×#131
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×C₈	C₂	semidirect
3. C₇×#208	C₂	semidirect
4. C₇×ΓL(1,𝔽₂₅)	C₂	semidirect
5. D₃×C₇×C₈	C₂	semidirect
19. C₂×C₇×C₈	D₃	semidirect
20. #901	C₇	direct (N × H)
35. C₂×C₃×C₈	C₂×C₇	semidirect
36. #208	C₂×C₇	semidirect
37. ΓL(1,𝔽₂₅)	C₂×C₇	semidirect
38. D₃×C₈	C₂×C₇	semidirect
56. C₃×C₇	#131	semidirect
61. C₂×C₈	D₃×C₇	semidirect
78. C₇	#901	direct (N × H)
87. C₃	C₇×#131	semidirect

Maximal subgroups

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

subgroup	index
C₇×#235	2
C₂×C₃×C₇×C₈	2
C₇×#208	2
C₇×ΓL(1,𝔽₂₅)	2
D₃×C₇×C₈	2
C₇×#131	3
#901	7

Subgroups — C7×#901 #10621298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#901 #10621298