Subgroups — #37524463

Order 22932.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#13109897	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇⋊(C₇⋊#261)	C₃	1	1	yes	maximal, characteristic
3	C₇×#11717942	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇²×D₁₃	C₂×C₃	1	1	yes	characteristic
5	#15692591	—	7	1	no	maximal
6	#15692586	—	7	1	no	maximal
7	C₇⋊(C₇⋊#261)	C₉	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic
9	(C₇⋊C₉)⋊Dic₇	—	13	1	no	maximal, Hall
10	#13109897	D₇	1	1	yes	characteristic
11	C₇×D₁₃⋊C₉	—	7	1	no	—
12	C₇²×D₁₃	C₂×C₉	1	1	yes	characteristic
13	C₃×C₇⋊#261	—	7	1	no	—
14	C₃×C₇⋊#261	—	21	2	no	—
15	C₃×C₇⋊#261	—	21	1	no	—
16	C₂×C₇×C₇⋊C₉	—	13	1	no	—
17	#11717942	Dic₇	1	1	yes	characteristic, complemented
18	C₇×C₁₃⋊C₉	—	7	1	no	—
19	C₇²×C₁₃	C₄×C₉	1	1	yes	characteristic, complemented, Hall
20	C₃×C₇⋊Dic₇	—	39	1	no	—
21	C₃×C₇×D₁₃	—	3	2	no	—
22	C₃×C₇×D₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
23	C₃×C₇×D₁₃	C₃×D₇	1	1	yes	characteristic
24	#261⋊C₉	—	49	1	no	Hall
25	C₇×C₇⋊C₉	—	13	1	no	Hall
26	C₇⋊#261	—	7	1	no	—
27	C₇⋊#261	—	21	2	no	—
28	C₇⋊#261	—	21	1	no	—
29	C₂×C₃×C₇²	—	39	1	no	—
30	C₃×C₇×C₁₃	—	3	2	no	—
31	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
32	C₃×C₇×C₁₃	C₃×Dic₇	1	1	yes	characteristic
33	C₉×Dic₇	—	91	1	no	—
34	Dic₇⋊C₉	—	91	1	no	—
35	D₁₃⋊C₉	—	7	1	no	—
36	C₇⋊Dic₇	—	39	1	no	Hall
37	C₇×D₁₃	—	3	2	no	—
38	C₇×D₁₃	D₇⋊C₉	1	1	yes	characteristic
39	C₇×D₁₃	C₉×D₇	1	1	yes	characteristic
40	C₃×#261	—	147	1	no	—
41	C₃×C₇²	AGL(1,𝔽₁₃)	1	1	yes	characteristic
42	C₂×C₇×C₉	—	91	1	no	—
43	C₂×C₇⋊C₉	—	13	1	no	—
44	C₁₃⋊C₉	—	7	1	no	Hall
45	C₂×C₇²	—	39	1	no	—
46	C₇×C₁₃	—	3	2	no	—
47	C₇×C₁₃	Dic₇⋊C₉	1	1	yes	characteristic, complemented
48	C₇×C₁₃	C₉×Dic₇	1	1	yes	characteristic, complemented
49	C₃×Dic₇	—	273	1	no	—
50	C₃×Dic₇	—	273	2	no	—
51	C₃×Dic₇	—	273	1	no	—
52	C₃×D₁₃	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic
53	C₇×C₉	—	91	1	no	—
54	C₇⋊C₉	—	13	1	no	—
55	#261	—	147	1	no	Hall
56	C₇²	#261⋊C₉	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃×C₇	—	39	2	no	—
58	C₂×C₃×C₇	—	39	1	no	—
59	C₂×C₃×C₇	—	39	1	no	—
60	C₃×C₁₃	ΓL(1,𝔽₈)⋊Dic₇	1	1	yes	characteristic
61	C₄×C₉	—	637	1	no	Hall
62	Dic₇	—	273	1	no	—
63	Dic₇	—	273	2	no	—
64	Dic₇	—	273	1	no	—
65	D₁₃	(C₇⋊C₉)⋊D₇	1	1	yes	characteristic
66	C₃×C₇	—	3	2	no	—
67	C₃×C₇	#11763979	1	1	yes	characteristic
68	C₃×C₇	#11763984	1	1	yes	characteristic
69	C₂×C₉	—	91	1	no	—
70	C₂×C₇	—	39	2	no	—
71	C₂×C₇	—	39	1	no	—
72	C₂×C₇	—	39	1	no	—
73	C₁₃	(C₇⋊C₉)⋊Dic₇	1	1	yes	characteristic, complemented, Sylow
74	C₃×C₄	—	1911	1	no	—
75	C₉	—	91	1	no	Sylow
76	C₇	—	3	2	no	—
77	C₇	#15692586	1	1	yes	characteristic, complemented
78	C₇	#15692591	1	1	yes	characteristic, complemented
79	C₂×C₃	—	39	1	no	—
80	C₄	—	1911	1	no	Sylow
81	C₃	#21439648	1	1	yes	characteristic
82	C₂	—	39	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#21439648
commutator, hypocenter	C₇²×C₁₃	C₄×C₉
Fitting, socle	C₃×C₇²×C₁₃	C₃×C₄
prime radical, cosocle	C₃×C₇²×D₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
3. C₇×#11717942	C₄	semidirect
7. C₇⋊(C₇⋊#261)	C₉	semidirect
17. #11717942	Dic₇	semidirect
19. C₇²×C₁₃	C₄×C₉	semidirect
47. C₇×C₁₃	Dic₇⋊C₉	semidirect
48. C₇×C₁₃	C₉×Dic₇	semidirect
56. C₇²	#261⋊C₉	semidirect
73. C₁₃	(C₇⋊C₉)⋊Dic₇	semidirect
77. C₇	#15692586	semidirect
78. C₇	#15692591	semidirect

Maximal subgroups

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

subgroup	index
C₇×#13109897	2
C₃×C₇⋊(C₇⋊#261)	3
#15692591	7
#15692586	7
(C₇⋊C₉)⋊Dic₇	13