Subgroups — #37524356

Order 22932.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃⋊#89205	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₁₃×Dic₇	C₃	1	1	yes	maximal, characteristic
3	C₁₃⋊#89205	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
5	C₄×#11717941	—	7	1	no	maximal
6	#15692570	—	7	1	no	maximal
7	C₇×C₁₃×Dic₇	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	C₂×#11717942	—	7	1	no	—
11	C₂×#11717941	—	7	1	no	—
12	C₂×C₇²×C₁₃	C₂×C₉	1	1	yes	characteristic
13	C₃×C₄×C₇×C₁₃	—	21	1	no	—
14	C₃×C₁₃×Dic₇	ΓL(1,𝔽₈)	1	1	yes	characteristic
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	C₁₃⋊C₃	1	1	yes	characteristic
20	C₂×C₃×C₇×C₁₃	—	6	1	no	—
21	C₂×C₃×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
22	C₂×C₃×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
23	C₄×C₁₃⋊C₉	—	49	1	no	Hall
24	#89205	—	13	1	no	Hall
25	C₄×C₇×C₁₃	—	21	1	no	—
26	C₁₃×Dic₇	C₇⋊C₉	1	1	yes	characteristic, complemented
27	C₂×C₃×C₇²	C₂×C₁₃⋊C₃	1	1	yes	characteristic
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	C₂×C₁₃⋊C₉	—	49	1	no	—
34	C₇×Dic₇	C₁₃⋊C₉	1	1	yes	characteristic, complemented, Hall
35	C₂×C₇×C₁₃	—	6	1	no	—
36	C₂×C₇×C₁₃	D₇⋊C₉	1	1	yes	characteristic
37	C₂×C₇×C₁₃	C₂×C₇⋊C₉	1	1	yes	characteristic
38	C₃×C₄×C₁₃	—	21	1	no	—
39	C₃×C₇²	C₄×C₁₃⋊C₃	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	C₂×C₁₃⋊C₉	1	1	yes	characteristic
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	—	21	1	no	—
48	C₃×Dic₇	#63270	1	1	yes	characteristic
49	C₂×C₃×C₁₃	#64422	1	1	yes	characteristic
50	C₇⋊C₉	—	91	1	no	—
51	C₇⋊C₉	—	91	1	no	—
52	C₄×C₁₃	—	21	1	no	Hall
53	C₇²	C₄×C₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃×C₇	—	6	1	no	—
55	C₂×C₃×C₇	#10587721	1	1	yes	characteristic
56	C₂×C₃×C₇	C₂×#63270	1	1	yes	characteristic
57	C₃×C₁₃	#10596872	1	1	yes	characteristic
58	C₄×C₉	—	637	1	no	Hall
59	C₄×C₇	—	21	1	no	—
60	Dic₇	#11717941	1	1	yes	characteristic, complemented
61	C₂×C₁₃	#11725075	1	1	yes	characteristic
62	C₃×C₇	—	6	1	no	—
63	C₃×C₇	#11763962	1	1	yes	characteristic
64	C₃×C₇	C₄×#63270	1	1	yes	characteristic
65	C₂×C₉	—	637	1	no	—
66	C₂×C₇	—	6	1	no	—
67	C₂×C₇	#13109901	1	1	yes	characteristic
68	C₂×C₇	C₂×#11717941	1	1	yes	characteristic
69	C₁₃	#13183531	1	1	yes	characteristic, complemented, Sylow
70	C₃×C₄	—	21	1	no	—
71	C₉	—	637	1	no	Sylow
72	C₇	—	6	1	no	—
73	C₇	#15692570	1	1	yes	characteristic, complemented
74	C₇	C₄×#11717941	1	1	yes	characteristic, complemented
75	C₂×C₃	#16838465	1	1	yes	characteristic
76	C₄	—	21	1	no	Sylow
77	C₃	#21439541	1	1	yes	characteristic
78	C₂	#26039463	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	#16838465
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. C₁₃⋊#89205	C₄	semidirect
7. C₇×C₁₃×Dic₇	C₉	semidirect
18. C₇²×C₁₃	C₄×C₉	semidirect
26. C₁₃×Dic₇	C₇⋊C₉	semidirect
34. C₇×Dic₇	C₁₃⋊C₉	semidirect
45. C₇×C₁₃	Dic₇⋊C₉	semidirect
46. C₇×C₁₃	C₄×C₇⋊C₉	semidirect
53. C₇²	C₄×C₁₃⋊C₉	semidirect
60. Dic₇	#11717941	semidirect
69. C₁₃	#13183531	semidirect
73. C₇	#15692570	semidirect
74. C₇	C₄×#11717941	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃⋊#89205	2
C₃×C₇×C₁₃×Dic₇	3
C₄×#11717941	7
#15692570	7
#13183531	13