Subgroups — C₇×#4512 #13051294

Order 1344.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₇×#815	C₂	1	1	yes	maximal, characteristic
3	C₇×#812	C₂	1	1	yes	maximal, characteristic
4	C₇×#347	—	3	1	no	maximal, Hall
5	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₄×C₇×Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₇×C₃⋊C₈	C₄	1	1	yes	characteristic
8	C₃×C₇×C₁₆	C₄	1	2	yes	complemented
9	C₇×C₃⋊C₁₆	C₄	1	2	yes	complemented
10	C₇×M₃₂	—	3	1	no	—
11	C₇×M₃₂	D₃	1	1	yes	characteristic
12	C₇×#97	—	3	1	no	—
13	#4512	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₇×Dic₃	—	2	1	no	—
16	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₇×C₃⋊C₈	—	2	1	no	—
19	C₂×C₇×C₈	—	3	1	no	—
20	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
21	C₄²×C₇	—	3	1	no	—
22	C₇×C₁₆	—	3	2	no	—
23	C₇×C₁₆	Dic₃	1	2	yes	complemented
24	C₃×M₃₂	C₂×C₇	1	1	yes	characteristic
25	#815	C₂×C₇	1	1	yes	characteristic
26	#812	C₂×C₇	1	1	yes	characteristic
27	C₂²×C₃×C₇	M₁₆	1	1	yes	characteristic
28	C₃×C₄×C₇	C₄²	1	1	yes	characteristic
29	C₃×C₄×C₇	M₁₆	1	1	yes	characteristic
30	C₇×Dic₃	—	4	1	no	—
31	#347	—	3	1	no	Sylow
32	C₂×C₄×C₇	—	6	1	no	—
33	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
34	C₇×C₈	—	6	1	no	—
35	C₇×C₈	C₄×D₃	1	1	yes	characteristic
36	C₇×C₈	C₂×Dic₃	1	1	yes	characteristic
37	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
38	C₄×Dic₃	C₄×C₇	1	1	yes	characteristic
39	C₂×C₃⋊C₈	C₄×C₇	1	1	yes	characteristic
40	C₃×C₁₆	C₄×C₇	1	2	yes	complemented
41	C₃⋊C₁₆	C₄×C₇	1	2	yes	complemented
42	C₂×C₃×C₇	—	2	1	no	—
43	C₂×C₃×C₇	#97	1	1	yes	characteristic
44	M₃₂	—	3	1	no	—
45	M₃₂	D₃×C₇	1	1	yes	characteristic
46	#97	—	3	1	no	—
47	C₂²×C₇	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
48	C₄×C₇	—	12	1	no	—
49	C₄×C₇	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
50	C₄×C₇	C₄×Dic₃	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
52	C₂×Dic₃	—	2	1	no	—
53	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
54	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
55	C₃⋊C₈	—	2	1	no	—
56	C₃×C₇	#347	1	1	yes	characteristic, complemented, Hall
57	C₂×C₈	—	3	1	no	—
58	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
59	C₄²	—	3	1	no	—
60	C₁₆	—	3	2	no	—
61	C₁₆	C₇×Dic₃	1	2	yes	complemented
62	C₂×C₇	—	2	1	no	—
63	C₂×C₇	#815	1	1	yes	characteristic
64	C₂²×C₃	C₇×M₁₆	1	1	yes	characteristic
65	C₃×C₄	C₄²×C₇	1	1	yes	characteristic
66	C₃×C₄	C₇×M₁₆	1	1	yes	characteristic
67	Dic₃	—	4	1	no	—
68	C₂×C₄	—	6	1	no	—
69	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
70	C₈	—	6	1	no	—
71	C₈	C₄×D₃×C₇	1	1	yes	characteristic
72	C₈	C₂×C₇×Dic₃	1	1	yes	characteristic
73	C₇	#4512	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	C₇×#97	1	1	yes	characteristic
76	C₂²	C₇×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
77	C₄	—	12	1	no	—
78	C₄	C₇×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
79	C₄	C₄×C₇×Dic₃	1	1	yes	characteristic
80	C₃	C₇×#347	1	1	yes	characteristic, complemented, Sylow
81	C₂	—	2	1	no	—
82	C₂	C₇×#815	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₄×Dic₃
commutator	C₃×C₄	C₄²×C₇
Frattini	C₂×C₈	C₂×D₃×C₇
Fitting	C₃×C₇×M₃₂	C₂
socle	C₂×C₃×C₇	#97
prime radical	C₂²×C₃×C₇	M₁₆
cosocle	C₂×C₃×C₈	C₂²×C₇
hypercenter	C₇×M₃₂	D₃
hypocenter	C₃	C₇×#347
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₇×M₃₂

Complements

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

N	H ≅ G/N	type
8. C₃×C₇×C₁₆	C₄	semidirect
9. C₇×C₃⋊C₁₆	C₄	semidirect
13. #4512	C₇	direct (N × H)
23. C₇×C₁₆	Dic₃	semidirect
40. C₃×C₁₆	C₄×C₇	semidirect
41. C₃⋊C₁₆	C₄×C₇	semidirect
56. C₃×C₇	#347	semidirect
61. C₁₆	C₇×Dic₃	semidirect
73. C₇	#4512	direct (N × H)
80. C₃	C₇×#347	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×M₃₂	2
C₇×#815	2
C₇×#812	2
C₇×#347	3
#4512	7

Subgroups — C7×#4512 #13051294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4512 #13051294