Subgroups — D₃×C₂₃×D₁₁₁ #44499708

Order 30636.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×C₂₃×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂₃×D₁₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×C₃⋊D₁₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×D₁₁₁	—	3	1	no	maximal
5	D₃×C₂₃×D₃₇	—	3	1	no	maximal
6	C₃²×C₂₃×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₂₃×D₁₁₁	—	3	1	no	—
8	C₂₃×D₁₁₁	—	3	1	no	—
9	C₂₃×D₁₁₁	—	6	1	no	—
10	C₂₃×D₁₁₁	D₃	1	1	yes	characteristic, direct product factor
11	D₃×C₂₃×C₃₇	D₃	1	1	yes	characteristic, complemented
12	C₃×C₂₃×D₃₇	—	3	1	no	—
13	C₂×C₃×C₂₃×C₃₇	—	3	1	no	—
14	C₂×C₂₃×D₃₇	—	9	1	no	Hall
15	C₃×C₂₃×C₃₇	—	2	1	no	—
16	C₃×C₂₃×C₃₇	C₂×D₃	1	1	yes	characteristic, complemented
17	C₃×C₂₃×C₃₇	C₂×D₃	1	1	yes	characteristic, complemented
18	C₂×C₂₃×C₃₇	—	3	1	no	—
19	C₂₃×D₃₇	—	3	1	no	—
20	C₂₃×D₃₇	—	9	1	no	—
21	D₃×D₁₁₁	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂₃×C₃₇	D₃²	1	1	yes	characteristic, complemented, Hall
23	D₃²×C₂₃	—	37	1	no	maximal, Hall
24	C₃⋊D₁₁₁	C₂×C₂₃	1	1	yes	characteristic, complemented
25	C₃×D₁₁₁	C₂×C₂₃	1	1	yes	characteristic, complemented
26	C₃×D₃×C₃₇	C₂×C₂₃	1	1	yes	characteristic, complemented
27	C₂×D₁₁₁	—	3	1	no	—
28	D₃×D₃₇	—	3	1	no	—
29	C₃⋊D₃×C₂₃	—	37	1	no	—
30	C₃×D₃×C₂₃	—	37	1	no	—
31	C₃×D₃×C₂₃	D₃₇	1	1	yes	characteristic, complemented
32	C₃²×C₃₇	C₂²×C₂₃	1	1	yes	characteristic, complemented, Hall
33	C₂×D₃×C₂₃	—	111	1	no	—
34	C₂×D₃×C₂₃	—	111	1	no	—
35	C₂×C₃×C₃₇	—	3	1	no	—
36	D₁₁₁	—	3	1	no	—
37	D₁₁₁	—	3	1	no	—
38	D₁₁₁	—	6	1	no	—
39	D₁₁₁	D₃×C₂₃	1	1	yes	characteristic, direct product factor
40	C₃×D₃₇	—	3	1	no	—
41	D₃×C₃₇	D₃×C₂₃	1	1	yes	characteristic, complemented
42	C₃²×C₂₃	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
43	C₂×D₃₇	—	9	1	no	Hall
44	C₂×C₃×C₂₃	—	3	1	no	—
45	C₂×C₃×C₂₃	—	111	1	no	—
46	D₃×C₂₃	—	37	1	no	—
47	D₃×C₂₃	—	111	1	no	—
48	D₃×C₂₃	—	111	1	no	—
49	D₃×C₂₃	—	222	1	no	—
50	D₃×C₂₃	D₁₁₁	1	1	yes	characteristic, direct product factor
51	C₃×C₃₇	—	2	1	no	—
52	C₃×C₃₇	C₂×D₃×C₂₃	1	1	yes	characteristic, complemented
53	C₃×C₃₇	C₂×D₃×C₂₃	1	1	yes	characteristic, complemented
54	C₂²×C₂₃	—	333	1	no	Hall
55	C₂×C₃₇	—	3	1	no	—
56	D₃₇	—	3	1	no	—
57	D₃₇	—	9	1	no	—
58	C₃×C₂₃	—	2	1	no	—
59	C₃×C₂₃	D₃×D₃₇	1	1	yes	characteristic, complemented
60	C₃×C₂₃	C₂×D₁₁₁	1	1	yes	characteristic, complemented
61	C₂×C₂₃	—	3	1	no	—
62	C₂×C₂₃	—	333	1	no	—
63	C₂×C₂₃	—	111	1	no	—
64	C₃₇	D₃²×C₂₃	1	1	yes	characteristic, complemented, Sylow
65	D₃²	—	37	1	no	Hall
66	C₂₃	D₃×D₁₁₁	1	1	yes	characteristic, direct product factor, Sylow
67	C₃⋊D₃	—	37	1	no	—
68	C₃×D₃	—	37	1	no	—
69	C₃×D₃	C₂₃×D₃₇	1	1	yes	characteristic, complemented
70	C₂×D₃	—	111	1	no	—
71	C₂×D₃	—	111	1	no	—
72	C₃²	C₂×C₂₃×D₃₇	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	3	1	no	—
74	C₂×C₃	—	111	1	no	—
75	D₃	—	37	1	no	—
76	D₃	—	111	1	no	—
77	D₃	—	111	1	no	—
78	D₃	—	222	1	no	—
79	D₃	C₂₃×D₁₁₁	1	1	yes	characteristic, direct product factor
80	C₂²	—	333	1	no	Sylow
81	C₃	—	2	1	no	—
82	C₃	D₃×C₂₃×D₃₇	1	1	yes	characteristic, complemented
83	C₃	C₂×C₂₃×D₁₁₁	1	1	yes	characteristic, complemented
84	C₂	—	3	1	no	—
85	C₂	—	333	1	no	—
86	C₂	—	111	1	no	—

	subgroup	G/H
center, hypercenter	C₂₃	D₃×D₁₁₁
commutator, cosocle, hypocenter	C₃²×C₃₇	C₂²×C₂₃
Frattini	C₁	G
Fitting, socle	C₃²×C₂₃×C₃₇	C₂²
LCS	G ▷ C₃²×C₃₇
UCS	C₁ ◁ C₂₃

Complements

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

N	H ≅ G/N	type
1. C₃×D₃×C₂₃×C₃₇	C₂	semidirect
2. C₃×C₂₃×D₁₁₁	C₂	semidirect
3. C₂₃×C₃⋊D₁₁₁	C₂	semidirect
6. C₃²×C₂₃×C₃₇	C₂²	semidirect
10. C₂₃×D₁₁₁	D₃	direct (N × H)
11. D₃×C₂₃×C₃₇	D₃	semidirect
16. C₃×C₂₃×C₃₇	C₂×D₃	semidirect
17. C₃×C₂₃×C₃₇	C₂×D₃	semidirect
21. D₃×D₁₁₁	C₂₃	direct (N × H)
22. C₂₃×C₃₇	D₃²	semidirect
24. C₃⋊D₁₁₁	C₂×C₂₃	semidirect
25. C₃×D₁₁₁	C₂×C₂₃	semidirect
26. C₃×D₃×C₃₇	C₂×C₂₃	semidirect
31. C₃×D₃×C₂₃	D₃₇	semidirect
32. C₃²×C₃₇	C₂²×C₂₃	semidirect
39. D₁₁₁	D₃×C₂₃	direct (N × H)
41. D₃×C₃₇	D₃×C₂₃	semidirect
42. C₃²×C₂₃	C₂×D₃₇	semidirect
50. D₃×C₂₃	D₁₁₁	direct (N × H)
52. C₃×C₃₇	C₂×D₃×C₂₃	semidirect
53. C₃×C₃₇	C₂×D₃×C₂₃	semidirect
59. C₃×C₂₃	D₃×D₃₇	semidirect
60. C₃×C₂₃	C₂×D₁₁₁	semidirect
64. C₃₇	D₃²×C₂₃	semidirect
66. C₂₃	D₃×D₁₁₁	direct (N × H)
69. C₃×D₃	C₂₃×D₃₇	semidirect
72. C₃²	C₂×C₂₃×D₃₇	semidirect
79. D₃	C₂₃×D₁₁₁	direct (N × H)
82. C₃	D₃×C₂₃×D₃₇	semidirect
83. C₃	C₂×C₂₃×D₁₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×C₂₃×C₃₇	2
C₃×C₂₃×D₁₁₁	2
C₂₃×C₃⋊D₁₁₁	2
C₂×C₂₃×D₁₁₁	3
D₃×C₂₃×D₃₇	3
D₃×D₁₁₁	23
D₃²×C₂₃	37

Subgroups — D3×C23×D111 #44499708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₂₃×D₁₁₁ #44499708