Subgroups — #37524413

Order 22932.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26039472	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇⋊(C₇⋊#261)	C₃	1	1	yes	maximal, characteristic
3	#19153199	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇²×D₁₃	C₂×C₃	1	1	yes	characteristic
5	#15692591	—	7	2	no	maximal
6	#15692591	—	7	1	no	maximal
7	#15692591	—	7	5	no	maximal
8	C₇⋊(C₇⋊#261)	C₉	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic
10	#13183533	—	13	1	no	maximal, Hall
11	#13109897	—	7	1	no	—
12	#13109897	—	7	2	no	—
13	#13109897	—	7	5	no	—
14	C₇²×D₁₃	C₂×C₉	1	1	yes	characteristic
15	C₃×C₇⋊#261	—	7	2	no	—
16	C₃×C₇⋊#261	—	21	1	no	—
17	C₃×C₇⋊#261	—	21	5	no	—
18	C₂×#89204	—	13	1	no	—
19	#11717942	—	7	1	no	—
20	#11717942	—	7	2	no	—
21	#11717942	—	7	5	no	—
22	C₇²×C₁₃	C₄×C₉	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇⋊Dic₇	—	39	1	no	—
24	C₃×C₇×D₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
25	C₃×C₇×D₁₃	AGL(1,𝔽₇)	1	2	yes	—
26	C₃×C₇×D₁₃	AGL(1,𝔽₇)	1	5	yes	—
27	#261⋊C₉	—	49	1	no	Hall
28	#89204	—	13	1	no	Hall
29	C₇⋊#261	—	7	2	no	—
30	C₇⋊#261	—	21	1	no	—
31	C₇⋊#261	—	21	5	no	—
32	C₂×C₃×C₇²	—	39	1	no	—
33	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
34	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	2	yes	—
35	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	5	yes	—
36	Dic₇⋊C₉	—	91	1	no	—
37	Dic₇⋊C₉	—	91	2	no	—
38	Dic₇⋊C₉	—	91	5	no	—
39	D₁₃⋊C₉	—	49	1	no	—
40	C₇⋊Dic₇	—	39	1	no	Hall
41	C₇×D₁₃	D₇⋊C₉	1	1	yes	characteristic
42	C₇×D₁₃	D₇⋊C₉	1	2	yes	—
43	C₇×D₁₃	D₇⋊C₉	1	5	yes	—
44	C₃×#261	—	147	1	no	—
45	C₃×C₇²	AGL(1,𝔽₁₃)	1	1	yes	characteristic
46	C₂×C₇⋊C₉	—	91	1	no	—
47	C₂×C₇⋊C₉	—	91	2	no	—
48	C₂×C₇⋊C₉	—	91	5	no	—
49	C₁₃⋊C₉	—	49	1	no	Hall
50	C₂×C₇²	—	39	1	no	—
51	C₇×C₁₃	Dic₇⋊C₉	1	1	yes	characteristic, complemented
52	C₇×C₁₃	Dic₇⋊C₉	1	2	yes	complemented
53	C₇×C₁₃	Dic₇⋊C₉	1	5	yes	complemented
54	C₃×Dic₇	—	273	1	no	—
55	C₃×Dic₇	—	273	2	no	—
56	C₃×Dic₇	—	273	5	no	—
57	C₃×D₁₃	#64424	1	1	yes	characteristic
58	C₇⋊C₉	—	91	1	no	—
59	C₇⋊C₉	—	91	2	no	—
60	C₇⋊C₉	—	91	5	no	—
61	#261	—	147	1	no	Hall
62	C₇²	#261⋊C₉	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₃×C₇	—	13	1	no	—
64	C₂×C₃×C₇	—	39	2	no	—
65	C₂×C₃×C₇	—	39	5	no	—
66	C₃×C₁₃	#10596874	1	1	yes	characteristic
67	C₄×C₉	—	637	1	no	Hall
68	Dic₇	—	273	1	no	—
69	Dic₇	—	273	2	no	—
70	Dic₇	—	273	5	no	—
71	D₁₃	#11725077	1	1	yes	characteristic
72	C₃×C₇	#11763984	1	1	yes	characteristic
73	C₃×C₇	#11763984	1	2	yes	—
74	C₃×C₇	#11763984	1	5	yes	—
75	C₂×C₉	—	637	1	no	—
76	C₂×C₇	—	13	1	no	—
77	C₂×C₇	—	39	2	no	—
78	C₂×C₇	—	39	5	no	—
79	C₁₃	#13183533	1	1	yes	characteristic, complemented, Sylow
80	C₃×C₄	—	1911	1	no	—
81	C₉	—	637	1	no	Sylow
82	C₇	#15692591	1	1	yes	characteristic, complemented
83	C₇	#15692591	1	2	yes	complemented
84	C₇	#15692591	1	5	yes	complemented
85	C₂×C₃	—	39	1	no	—
86	C₄	—	1911	1	no	Sylow
87	C₃	#21439598	1	1	yes	characteristic
88	C₂	—	39	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#21439598
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. #19153199	C₄	semidirect
8. C₇⋊(C₇⋊#261)	C₉	semidirect
22. C₇²×C₁₃	C₄×C₉	semidirect
51. C₇×C₁₃	Dic₇⋊C₉	semidirect
52. C₇×C₁₃	Dic₇⋊C₉	semidirect
53. C₇×C₁₃	Dic₇⋊C₉	semidirect
62. C₇²	#261⋊C₉	semidirect
79. C₁₃	#13183533	semidirect
82. C₇	#15692591	semidirect
83. C₇	#15692591	semidirect
84. C₇	#15692591	semidirect

Maximal subgroups

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

subgroup	index
#26039472	2
C₃×C₇⋊(C₇⋊#261)	3
#15692591	7
#13183533	13