Subgroups — #44540752

Order 31740.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₂₃⋊D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×D₃₄₅	C₂	1	2	yes	maximal, complemented
3	D₁₁₅⋊D₂₃	—	3	1	no	maximal, Hall
4	C₃×C₅×C₂₃²	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₆₉⋊D₂₃	—	5	1	no	maximal, Hall
6	C₂₃×D₁₁₅	—	3	2	no	—
7	C₅×C₂₃⋊D₂₃	D₃	1	1	yes	characteristic, complemented
8	C₂₃×D₆₉	—	5	2	no	—
9	C₃×C₂₃⋊D₂₃	D₅	1	1	yes	characteristic, complemented
10	C₅×C₂₃²	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	D₂₃²	—	15	1	no	Hall
12	C₃×C₂₃²	C₂×D₅	1	1	yes	characteristic, complemented, Hall
13	D₁₅×D₂₃	—	23	2	no	maximal
14	C₂₃⋊D₂₃	D₁₅	1	1	yes	characteristic, complemented
15	C₂₃×D₂₃	—	15	2	no	—
16	D₃₄₅	D₂₃	1	2	yes	complemented
17	C₂₃×D₁₅	—	23	2	no	—
18	C₃×C₅×D₂₃	—	23	2	no	—
19	C₃×C₅×D₂₃	—	46	11	no	—
20	C₂₃²	C₂×D₁₅	1	1	yes	characteristic, complemented, Sylow
21	D₅×D₂₃	—	69	2	no	—
22	C₃×C₅×C₂₃	—	2	11	no	—
23	C₃×C₅×C₂₃	C₂×D₂₃	1	2	yes	complemented
24	D₃×D₂₃	—	115	2	no	—
25	D₁₁₅	—	3	2	no	—
26	C₅×D₂₃	—	23	2	no	—
27	C₅×D₂₃	—	46	11	no	—
28	D₅×C₂₃	—	69	2	no	—
29	D₆₉	—	5	2	no	—
30	C₃×D₂₃	—	23	2	no	—
31	C₃×D₂₃	—	46	11	no	—
32	D₃×C₂₃	—	115	2	no	—
33	C₅×C₂₃	—	2	11	no	—
34	C₅×C₂₃	D₃×D₂₃	1	2	yes	complemented
35	C₂×D₂₃	—	345	2	no	—
36	C₃×C₂₃	—	2	11	no	—
37	C₃×C₂₃	D₅×D₂₃	1	2	yes	complemented
38	C₂×D₁₅	—	529	1	no	Hall
39	C₂×C₂₃	—	345	2	no	—
40	D₂₃	—	15	2	no	—
41	D₂₃	—	23	2	no	—
42	D₂₃	—	46	11	no	—
43	C₂×C₃×C₅	—	529	1	no	—
44	D₁₅	—	23	2	no	—
45	C₂₃	—	2	11	no	—
46	C₂₃	D₁₅×D₂₃	1	2	yes	complemented
47	C₂×D₅	—	1587	1	no	Hall
48	C₃×C₅	D₂₃²	1	1	yes	characteristic, complemented, Hall
49	C₂×D₃	—	2645	1	no	Hall
50	C₂×C₅	—	529	1	no	—
51	D₅	—	69	2	no	—
52	C₂×C₃	—	529	1	no	—
53	D₃	—	115	2	no	—
54	C₅	D₆₉⋊D₂₃	1	1	yes	characteristic, complemented, Sylow
55	C₂²	—	7935	1	no	Sylow
56	C₃	D₁₁₅⋊D₂₃	1	1	yes	characteristic, complemented, Sylow
57	C₂	—	529	1	no	—
58	C₂	—	345	2	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₂₃²	C₂²
LCS	G ▷ C₃×C₅×C₂₃²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₂₃⋊D₂₃	C₂	semidirect
2. C₂₃×D₃₄₅	C₂	semidirect
4. C₃×C₅×C₂₃²	C₂²	semidirect
7. C₅×C₂₃⋊D₂₃	D₃	semidirect
9. C₃×C₂₃⋊D₂₃	D₅	semidirect
10. C₅×C₂₃²	C₂×D₃	semidirect
12. C₃×C₂₃²	C₂×D₅	semidirect
14. C₂₃⋊D₂₃	D₁₅	semidirect
16. D₃₄₅	D₂₃	semidirect
20. C₂₃²	C₂×D₁₅	semidirect
23. C₃×C₅×C₂₃	C₂×D₂₃	semidirect
34. C₅×C₂₃	D₃×D₂₃	semidirect
37. C₃×C₂₃	D₅×D₂₃	semidirect
46. C₂₃	D₁₅×D₂₃	semidirect
48. C₃×C₅	D₂₃²	semidirect
54. C₅	D₆₉⋊D₂₃	semidirect
56. C₃	D₁₁₅⋊D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₂₃⋊D₂₃	2
C₂₃×D₃₄₅	2
D₁₁₅⋊D₂₃	3
D₆₉⋊D₂₃	5
D₁₅×D₂₃	23