Subgroups — C₂×D₃×C₅²⋊(C₃₁⋊C₃) #42124855

Order 27900.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²⋊(C₃₁⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×C₅²⋊(C₃₁⋊C₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅²×D₃×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₅²⋊(C₃₁⋊C₃)	—	3	1	no	maximal
5	C₃×C₅²⋊(C₃₁⋊C₃)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅²×D₃×C₃₁	C₂×C₃	1	2	yes	complemented
7	C₂×C₅²⋊(C₃₁⋊C₃)	—	2	1	no	—
8	C₂×C₅²⋊(C₃₁⋊C₃)	—	3	2	no	—
9	C₂×C₅²⋊(C₃₁⋊C₃)	D₃	1	1	yes	characteristic, direct product factor
10	C₂×C₃×C₅²×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂²×C₅²×C₃₁	—	3	1	no	Hall
12	C₅²⋊(C₃₁⋊C₃)	—	2	1	no	—
13	C₅²⋊(C₃₁⋊C₃)	C₂×D₃	1	1	yes	characteristic, direct product factor
14	C₃×C₅²×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented
15	C₂×C₅×D₃×C₃₁	—	3	2	no	—
16	C₂×C₅²×C₃₁	—	3	2	no	—
17	C₂×C₅²×C₃₁	C₃×D₃	1	1	yes	characteristic, complemented
18	C₂×D₃×C₃₁⋊C₃	—	25	1	no	maximal, Hall
19	C₂×C₃×C₅×C₃₁	—	3	2	no	—
20	C₅×D₃×C₃₁	—	3	4	no	—
21	C₂×D₃×C₅²⋊C₃	—	31	1	no	maximal, Hall
22	C₅²×C₃₁	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
23	C₂²×C₅×C₃₁	—	9	2	no	—
24	C₂×C₃×C₃₁⋊C₃	—	25	1	no	—
25	D₃×C₃₁⋊C₃	—	25	2	no	—
26	C₃×C₅×C₃₁	—	3	2	no	—
27	C₂×C₃×C₅²⋊C₃	—	31	1	no	—
28	D₃×C₅²⋊C₃	—	31	2	no	—
29	C₂×D₃×C₃₁	C₅²⋊C₃	1	1	yes	characteristic, complemented
30	C₂²×C₃₁⋊C₃	—	75	1	no	—
31	C₂×C₅×C₃₁	—	3	2	no	—
32	C₂×C₅×C₃₁	—	9	4	no	—
33	C₂×C₅²×D₃	C₃₁⋊C₃	1	1	yes	characteristic, complemented
34	C₂²×C₅²⋊C₃	—	93	1	no	—
35	C₃×C₃₁⋊C₃	—	25	1	no	Hall
36	C₃×C₅²⋊C₃	—	31	1	no	Hall
37	C₂×C₃×C₃₁	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
38	D₃×C₃₁	C₂×C₅²⋊C₃	1	2	yes	complemented
39	C₂×C₃₁⋊C₃	—	25	1	no	—
40	C₂×C₃₁⋊C₃	—	50	1	no	—
41	C₂×C₃₁⋊C₃	—	75	2	no	—
42	C₅×C₃₁	—	3	2	no	—
43	C₂×C₃×C₅²	C₂×C₃₁⋊C₃	1	1	yes	characteristic, complemented
44	C₅²×D₃	C₂×C₃₁⋊C₃	1	2	yes	complemented
45	C₂×C₅²⋊C₃	—	31	1	no	—
46	C₂×C₅²⋊C₃	—	62	1	no	—
47	C₂×C₅²⋊C₃	—	93	2	no	—
48	C₂²×C₃₁	—	3	1	no	Hall
49	C₂²×C₅²	—	3	1	no	Hall
50	C₃×C₃₁	C₂²×C₅²⋊C₃	1	1	yes	characteristic, complemented
51	C₃₁⋊C₃	—	25	1	no	—
52	C₃₁⋊C₃	—	50	1	no	—
53	C₃×C₅²	C₂²×C₃₁⋊C₃	1	1	yes	characteristic, complemented
54	C₅²⋊C₃	—	31	1	no	—
55	C₅²⋊C₃	—	62	1	no	—
56	C₂×C₃₁	—	3	2	no	—
57	C₂×C₃₁	D₃×C₅²⋊C₃	1	1	yes	characteristic, complemented
58	C₂×C₅×D₃	—	3	2	no	—
59	C₂×C₅²	—	3	2	no	—
60	C₂×C₅²	D₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented
61	C₂×C₃×D₃	—	775	1	no	Hall
62	C₃₁	C₂×D₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₃×C₅	—	3	2	no	—
64	C₅×D₃	—	3	4	no	—
65	C₅²	C₂×D₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₅	—	9	2	no	—
67	C₂×C₃²	—	775	1	no	—
68	C₃×D₃	—	775	2	no	—
69	C₃×C₅	—	3	2	no	—
70	C₂²×C₃	—	2325	1	no	—
71	C₂×D₃	C₅²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor
72	C₂×C₅	—	3	2	no	—
73	C₂×C₅	—	9	4	no	—
74	C₃²	—	775	1	no	Sylow
75	C₂×C₃	—	775	1	no	—
76	C₂×C₃	—	1550	1	no	—
77	C₂×C₃	—	2325	2	no	—
78	C₂×C₃	C₂×C₅²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, complemented
79	D₃	C₂×C₅²⋊(C₃₁⋊C₃)	1	2	yes	direct product factor
80	C₅	—	3	2	no	—
81	C₂²	—	3	1	no	Sylow
82	C₃	—	775	1	no	—
83	C₃	—	1550	1	no	—
84	C₃	C₂²×C₅²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, complemented
85	C₂	—	3	2	no	—
86	C₂	D₃×C₅²⋊(C₃₁⋊C₃)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₃×C₅²⋊(C₃₁⋊C₃)
commutator, cosocle, hypocenter	C₃×C₅²×C₃₁	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₃₁	C₂×C₃
LCS	G ▷ C₃×C₅²×C₃₁
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²⋊(C₃₁⋊C₃)	C₂	semidirect
2. D₃×C₅²⋊(C₃₁⋊C₃)	C₂	direct (N × H)
3. C₂×C₅²×D₃×C₃₁	C₃	semidirect
5. C₃×C₅²⋊(C₃₁⋊C₃)	C₂²	semidirect
6. C₅²×D₃×C₃₁	C₂×C₃	semidirect
9. C₂×C₅²⋊(C₃₁⋊C₃)	D₃	direct (N × H)
10. C₂×C₃×C₅²×C₃₁	C₂×C₃	semidirect
13. C₅²⋊(C₃₁⋊C₃)	C₂×D₃	direct (N × H)
14. C₃×C₅²×C₃₁	C₂²×C₃	semidirect
17. C₂×C₅²×C₃₁	C₃×D₃	semidirect
22. C₅²×C₃₁	C₂×C₃×D₃	semidirect
29. C₂×D₃×C₃₁	C₅²⋊C₃	semidirect
33. C₂×C₅²×D₃	C₃₁⋊C₃	semidirect
37. C₂×C₃×C₃₁	C₂×C₅²⋊C₃	semidirect
38. D₃×C₃₁	C₂×C₅²⋊C₃	semidirect
43. C₂×C₃×C₅²	C₂×C₃₁⋊C₃	semidirect
44. C₅²×D₃	C₂×C₃₁⋊C₃	semidirect
50. C₃×C₃₁	C₂²×C₅²⋊C₃	semidirect
53. C₃×C₅²	C₂²×C₃₁⋊C₃	semidirect
57. C₂×C₃₁	D₃×C₅²⋊C₃	semidirect
60. C₂×C₅²	D₃×C₃₁⋊C₃	semidirect
62. C₃₁	C₂×D₃×C₅²⋊C₃	semidirect
65. C₅²	C₂×D₃×C₃₁⋊C₃	semidirect
71. C₂×D₃	C₅²⋊(C₃₁⋊C₃)	direct (N × H)
78. C₂×C₃	C₂×C₅²⋊(C₃₁⋊C₃)	semidirect
79. D₃	C₂×C₅²⋊(C₃₁⋊C₃)	direct (N × H)
84. C₃	C₂²×C₅²⋊(C₃₁⋊C₃)	semidirect
86. C₂	D₃×C₅²⋊(C₃₁⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅²⋊(C₃₁⋊C₃)	2
D₃×C₅²⋊(C₃₁⋊C₃)	2
C₂×C₅²×D₃×C₃₁	3
C₂²×C₅²⋊(C₃₁⋊C₃)	3
C₂×D₃×C₃₁⋊C₃	25
C₂×D₃×C₅²⋊C₃	31

Subgroups — C2×D3×C52⋊(C31⋊C3) #42124855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₅²⋊(C₃₁⋊C₃) #42124855