Subgroups — #37524445

Order 22932.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#89205	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₇⋊#261	C₃	1	1	yes	maximal, characteristic
3	C₁₃⋊#89205	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇²×D₁₃	C₂×C₃	1	1	yes	characteristic
5	#15692591	—	7	1	no	maximal
6	#15692589	—	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	#13183531	—	13	1	no	maximal, Hall
10	#13109897	—	7	1	no	—
11	#13109896	—	7	1	no	—
12	C₇²×D₁₃	C₂×C₉	1	1	yes	characteristic
13	C₃×C₇⋊#261	ΓL(1,𝔽₈)	1	1	yes	characteristic
14	C₃×C₇×#261	—	21	1	no	—
15	C₂×#89205	—	13	1	no	—
16	#11717942	—	7	1	no	—
17	#11717941	—	7	1	no	—
18	C₇²×C₁₃	C₄×C₉	1	1	yes	characteristic, complemented, Hall
19	C₃×C₇×Dic₇	—	39	1	no	—
20	C₃×C₇×D₁₃	—	6	1	no	—
21	C₃×C₇×D₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
22	C₃×C₇×D₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
23	#261⋊C₉	—	49	1	no	Hall
24	#89205	—	13	1	no	Hall
25	C₇⋊#261	C₇⋊C₉	1	1	yes	characteristic, complemented
26	C₇×#261	—	21	1	no	—
27	C₂×C₃×C₇²	—	39	1	no	—
28	C₃×C₇×C₁₃	—	6	1	no	—
29	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
30	C₃×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	C₄×C₇⋊C₉	—	91	1	no	—
32	Dic₇⋊C₉	—	91	1	no	—
33	D₁₃⋊C₉	—	49	1	no	—
34	C₇×Dic₇	—	39	1	no	Hall
35	C₇×D₁₃	—	6	1	no	—
36	C₇×D₁₃	D₇⋊C₉	1	1	yes	characteristic
37	C₇×D₁₃	C₂×C₇⋊C₉	1	1	yes	characteristic
38	C₃×#261	—	21	1	no	—
39	C₃×C₇²	AGL(1,𝔽₁₃)	1	1	yes	characteristic
40	C₂×C₇⋊C₉	—	91	1	no	—
41	C₂×C₇⋊C₉	—	91	1	no	—
42	C₁₃⋊C₉	—	49	1	no	Hall
43	C₂×C₇²	—	39	1	no	—
44	C₇×C₁₃	—	6	1	no	—
45	C₇×C₁₃	Dic₇⋊C₉	1	1	yes	characteristic, complemented
46	C₇×C₁₃	C₄×C₇⋊C₉	1	1	yes	characteristic, complemented
47	C₃×C₄×C₇	—	273	1	no	—
48	C₃×Dic₇	—	39	1	no	—
49	C₃×D₁₃	#64422	1	1	yes	characteristic
50	C₇⋊C₉	—	91	1	no	—
51	C₇⋊C₉	—	91	1	no	—
52	#261	—	21	1	no	Hall
53	C₇²	#261⋊C₉	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃×C₇	—	39	1	no	—
55	C₂×C₃×C₇	—	39	1	no	—
56	C₂×C₃×C₇	—	78	1	no	—
57	C₃×C₁₃	#10596872	1	1	yes	characteristic
58	C₄×C₉	—	637	1	no	Hall
59	C₄×C₇	—	273	1	no	—
60	Dic₇	—	39	1	no	—
61	D₁₃	#11725075	1	1	yes	characteristic
62	C₃×C₇	—	6	1	no	—
63	C₃×C₇	#11763982	1	1	yes	characteristic
64	C₃×C₇	#11763984	1	1	yes	characteristic
65	C₂×C₉	—	637	1	no	—
66	C₂×C₇	—	39	1	no	—
67	C₂×C₇	—	39	1	no	—
68	C₂×C₇	—	78	1	no	—
69	C₁₃	#13183531	1	1	yes	characteristic, complemented, Sylow
70	C₃×C₄	—	273	1	no	—
71	C₉	—	637	1	no	Sylow
72	C₇	—	6	1	no	—
73	C₇	#15692589	1	1	yes	characteristic, complemented
74	C₇	#15692591	1	1	yes	characteristic, complemented
75	C₂×C₃	—	39	1	no	—
76	C₄	—	273	1	no	Sylow
77	C₃	#21439630	1	1	yes	characteristic
78	C₂	—	39	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#21439630
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. C₁₃⋊#89205	C₄	semidirect
7. C₇×C₇⋊#261	C₉	semidirect
18. C₇²×C₁₃	C₄×C₉	semidirect
25. C₇⋊#261	C₇⋊C₉	semidirect
45. C₇×C₁₃	Dic₇⋊C₉	semidirect
46. C₇×C₁₃	C₄×C₇⋊C₉	semidirect
53. C₇²	#261⋊C₉	semidirect
69. C₁₃	#13183531	semidirect
73. C₇	#15692589	semidirect
74. C₇	#15692591	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#89205	2
C₃×C₇×C₇⋊#261	3
#15692591	7
#15692589	7
#13183531	13