Subgroups — #37590648

Order 24684.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×C₁₁⋊D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₁×D₅₆₁	C₂	1	2	yes	maximal, complemented
3	(C₁₁⋊D₁₁)⋊D₁₇	—	3	1	no	maximal, Hall
4	C₃×C₁₁²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₁₇×C₁₁⋊D₁₁	D₃	1	1	yes	characteristic, complemented
6	C₁₁×D₁₈₇	—	3	2	no	—
7	D₁₁×D₅₁	—	11	2	no	maximal
8	C₁₁²×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
9	D₃₃⋊D₁₁	—	17	1	no	maximal, Hall
10	D₅₆₁	D₁₁	1	2	yes	complemented
11	C₁₁×D₅₁	—	11	2	no	—
12	C₃×C₁₇×D₁₁	—	11	2	no	—
13	C₃×C₁₇×D₁₁	—	22	5	no	—
14	D₁₁×D₁₇	—	33	2	no	—
15	C₁₁×D₃₃	—	17	2	no	—
16	C₃×C₁₁⋊D₁₁	D₁₇	1	1	yes	characteristic, complemented
17	C₃×C₁₁×C₁₇	—	2	5	no	—
18	C₃×C₁₁×C₁₇	C₂×D₁₁	1	2	yes	complemented
19	D₁₁²	—	51	1	no	Hall
20	D₁₈₇	—	3	2	no	—
21	C₁₁×D₁₇	—	33	2	no	—
22	C₁₇×D₁₁	—	11	2	no	—
23	C₁₇×D₁₁	—	22	5	no	—
24	C₃×C₁₁²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
25	C₁₁⋊D₁₁	D₅₁	1	1	yes	characteristic, complemented
26	C₁₁×D₁₁	—	51	2	no	—
27	C₂×D₅₁	—	121	1	no	Hall
28	C₁₁×C₁₇	—	2	5	no	—
29	C₁₁×C₁₇	D₃×D₁₁	1	2	yes	complemented
30	D₃×D₁₁	—	187	2	no	—
31	C₁₁²	C₂×D₅₁	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃×C₁₇	—	121	1	no	—
33	D₅₁	—	11	2	no	—
34	C₂×D₁₇	—	363	1	no	Hall
35	D₃₃	—	17	2	no	—
36	C₃×D₁₁	—	11	2	no	—
37	C₃×D₁₁	—	22	5	no	—
38	D₃×C₁₁	—	187	2	no	—
39	C₃×C₁₇	D₁₁²	1	1	yes	characteristic, complemented, Hall
40	C₂×D₁₁	—	561	2	no	—
41	C₂×C₁₇	—	121	1	no	—
42	D₁₇	—	33	2	no	—
43	C₃×C₁₁	—	2	5	no	—
44	C₃×C₁₁	D₁₁×D₁₇	1	2	yes	complemented
45	C₂×C₁₁	—	561	2	no	—
46	D₁₁	—	11	2	no	—
47	D₁₁	—	22	5	no	—
48	D₁₁	—	51	2	no	—
49	C₁₇	D₃₃⋊D₁₁	1	1	yes	characteristic, complemented, Sylow
50	C₂×D₃	—	2057	1	no	Hall
51	C₁₁	—	2	5	no	—
52	C₁₁	D₁₁×D₅₁	1	2	yes	complemented
53	C₂×C₃	—	121	1	no	—
54	D₃	—	187	2	no	—
55	C₂²	—	6171	1	no	Sylow
56	C₃	(C₁₁⋊D₁₁)⋊D₁₇	1	1	yes	characteristic, complemented, Sylow
57	C₂	—	561	2	no	—
58	C₂	—	121	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₁²×C₁₇	C₂²
LCS	G ▷ C₃×C₁₁²×C₁₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₇×C₁₁⋊D₁₁	C₂	semidirect
2. C₁₁×D₅₆₁	C₂	semidirect
4. C₃×C₁₁²×C₁₇	C₂²	semidirect
5. C₁₇×C₁₁⋊D₁₁	D₃	semidirect
8. C₁₁²×C₁₇	C₂×D₃	semidirect
10. D₅₆₁	D₁₁	semidirect
16. C₃×C₁₁⋊D₁₁	D₁₇	semidirect
18. C₃×C₁₁×C₁₇	C₂×D₁₁	semidirect
24. C₃×C₁₁²	C₂×D₁₇	semidirect
25. C₁₁⋊D₁₁	D₅₁	semidirect
29. C₁₁×C₁₇	D₃×D₁₁	semidirect
31. C₁₁²	C₂×D₅₁	semidirect
39. C₃×C₁₇	D₁₁²	semidirect
44. C₃×C₁₁	D₁₁×D₁₇	semidirect
49. C₁₇	D₃₃⋊D₁₁	semidirect
52. C₁₁	D₁₁×D₅₁	semidirect
56. C₃	(C₁₁⋊D₁₁)⋊D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₇×C₁₁⋊D₁₁	2
C₁₁×D₅₆₁	2
(C₁₁⋊D₁₁)⋊D₁₇	3
D₁₁×D₅₁	11
D₃₃⋊D₁₁	17