Subgroups — #37524340

Order 22932.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19153199	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₃×C₇⋊Dic₇	C₃	1	1	yes	maximal, characteristic
3	#19153199	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
5	#15692570	—	7	2	no	maximal
6	#15692570	—	7	1	no	maximal
7	#15692570	—	7	5	no	maximal
8	C₁₃×C₇⋊Dic₇	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	C₂×#11717942	—	7	2	no	—
12	C₂×#11717942	—	7	1	no	—
13	C₂×#11717942	—	7	5	no	—
14	C₂×C₇²×C₁₃	C₂×C₉	1	1	yes	characteristic
15	C₃×C₁₃×Dic₇	—	7	2	no	—
16	C₃×C₁₃×Dic₇	—	21	1	no	—
17	C₃×C₁₃×Dic₇	—	21	5	no	—
18	C₂×#89204	—	13	1	no	—
19	#11717942	—	7	2	no	—
20	#11717942	—	7	1	no	—
21	#11717942	—	7	5	no	—
22	C₇²×C₁₃	C₄×C₉	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇⋊Dic₇	C₁₃⋊C₃	1	1	yes	characteristic
24	C₂×C₃×C₇×C₁₃	AGL(1,𝔽₇)	1	2	yes	—
25	C₂×C₃×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
26	C₂×C₃×C₇×C₁₃	AGL(1,𝔽₇)	1	5	yes	—
27	C₄×C₁₃⋊C₉	—	49	1	no	Hall
28	#89204	—	13	1	no	Hall
29	C₁₃×Dic₇	—	7	2	no	—
30	C₁₃×Dic₇	—	21	1	no	—
31	C₁₃×Dic₇	—	21	5	no	—
32	C₂×C₃×C₇²	C₂×C₁₃⋊C₃	1	1	yes	characteristic
33	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	2	yes	—
34	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
35	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	5	yes	—
36	Dic₇⋊C₉	—	91	2	no	—
37	Dic₇⋊C₉	—	91	1	no	—
38	Dic₇⋊C₉	—	91	5	no	—
39	C₂×C₁₃⋊C₉	—	49	1	no	—
40	C₇⋊Dic₇	C₁₃⋊C₉	1	1	yes	characteristic, complemented, Hall
41	C₂×C₇×C₁₃	D₇⋊C₉	1	2	yes	—
42	C₂×C₇×C₁₃	D₇⋊C₉	1	1	yes	characteristic
43	C₂×C₇×C₁₃	D₇⋊C₉	1	5	yes	—
44	C₃×C₄×C₁₃	—	147	1	no	—
45	C₃×C₇²	C₄×C₁₃⋊C₃	1	1	yes	characteristic
46	C₂×C₇⋊C₉	—	91	2	no	—
47	C₂×C₇⋊C₉	—	91	1	no	—
48	C₂×C₇⋊C₉	—	91	5	no	—
49	C₁₃⋊C₉	—	49	1	no	Hall
50	C₂×C₇²	C₂×C₁₃⋊C₉	1	1	yes	characteristic
51	C₇×C₁₃	Dic₇⋊C₉	1	2	yes	complemented
52	C₇×C₁₃	Dic₇⋊C₉	1	1	yes	characteristic, complemented
53	C₇×C₁₃	Dic₇⋊C₉	1	5	yes	complemented
54	C₃×Dic₇	—	7	2	no	—
55	C₃×Dic₇	—	7	1	no	—
56	C₃×Dic₇	—	21	5	no	—
57	C₂×C₃×C₁₃	#64424	1	1	yes	characteristic
58	C₇⋊C₉	—	91	2	no	—
59	C₇⋊C₉	—	91	1	no	—
60	C₇⋊C₉	—	91	5	no	—
61	C₄×C₁₃	—	147	1	no	Hall
62	C₇²	C₄×C₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₃×C₇	#10587721	1	2	yes	—
64	C₂×C₃×C₇	#10587721	1	1	yes	characteristic
65	C₂×C₃×C₇	#10587721	1	5	yes	—
66	C₃×C₁₃	#10596874	1	1	yes	characteristic
67	C₄×C₉	—	637	1	no	Hall
68	Dic₇	—	7	2	no	—
69	Dic₇	—	7	1	no	—
70	Dic₇	—	21	5	no	—
71	C₂×C₁₃	#11725077	1	1	yes	characteristic
72	C₃×C₇	#11763962	1	2	yes	—
73	C₃×C₇	#11763962	1	1	yes	characteristic
74	C₃×C₇	#11763962	1	5	yes	—
75	C₂×C₉	—	637	1	no	—
76	C₂×C₇	#13109901	1	2	yes	—
77	C₂×C₇	#13109901	1	1	yes	characteristic
78	C₂×C₇	#13109901	1	5	yes	—
79	C₁₃	#13183533	1	1	yes	characteristic, complemented, Sylow
80	C₃×C₄	—	147	1	no	—
81	C₉	—	637	1	no	Sylow
82	C₇	#15692570	1	2	yes	complemented
83	C₇	#15692570	1	1	yes	characteristic, complemented
84	C₇	#15692570	1	5	yes	complemented
85	C₂×C₃	#16838449	1	1	yes	characteristic
86	C₄	—	147	1	no	Sylow
87	C₃	#21439525	1	1	yes	characteristic
88	C₂	#26039447	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	#16838449
commutator, hypocenter	C₇²×C₁₃	C₄×C₉
Fitting, socle, prime radical, cosocle	C₂×C₃×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
3. #19153199	C₄	semidirect
8. C₁₃×C₇⋊Dic₇	C₉	semidirect
22. C₇²×C₁₃	C₄×C₉	semidirect
40. C₇⋊Dic₇	C₁₃⋊C₉	semidirect
51. C₇×C₁₃	Dic₇⋊C₉	semidirect
52. C₇×C₁₃	Dic₇⋊C₉	semidirect
53. C₇×C₁₃	Dic₇⋊C₉	semidirect
62. C₇²	C₄×C₁₃⋊C₉	semidirect
79. C₁₃	#13183533	semidirect
82. C₇	#15692570	semidirect
83. C₇	#15692570	semidirect
84. C₇	#15692570	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19153199	2
C₃×C₁₃×C₇⋊Dic₇	3
#15692570	7
#13183533	13