Subgroups — D₃₁⋊((C₅⋊D₅)⋊C₉) #42124513

Order 27900.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₁⋊(C₅²⋊C₉)	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₅⋊D₅)⋊(C₃₁⋊C₉)	C₂	1	1	yes	maximal, characteristic, complemented
3	#28358426	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅⋊D₅×D₃₁	C₃	1	1	yes	maximal, characteristic
5	C₅²⋊(C₃₁⋊C₉)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₅⋊D₁₅₅	C₂×C₃	1	1	yes	characteristic
7	C₃×C₃₁×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
8	C₃×C₅²×D₃₁	C₂×C₃	1	1	yes	characteristic
9	C₅⋊D₅×D₃₁	C₉	1	1	yes	characteristic, complemented, Hall
10	C₃×C₅²×C₃₁	C₂²×C₃	1	1	yes	characteristic
11	C₃×D₅×D₃₁	—	15	2	no	—
12	C₅⋊D₁₅₅	C₂×C₉	1	1	yes	characteristic, complemented
13	C₃₁×C₅⋊D₅	C₂×C₉	1	1	yes	characteristic, complemented
14	C₅²×D₃₁	C₂×C₉	1	1	yes	characteristic, complemented
15	C₂×D₃₁⋊C₉	—	25	1	no	maximal, Hall
16	C₃×D₁₅₅	—	15	2	no	—
17	C₃×D₅×C₃₁	—	15	2	no	—
18	C₃×C₅×D₃₁	—	3	2	no	—
19	C₂×(C₅⋊D₅)⋊C₉	—	31	1	no	maximal, Hall
20	C₅²×C₃₁	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
21	D₅×D₃₁	—	15	2	no	—
22	C₂×C₃₁⋊C₉	—	25	1	no	—
23	D₃₁⋊C₉	—	25	1	no	—
24	D₃₁⋊C₉	—	25	1	no	—
25	C₃×C₅×C₃₁	—	3	2	no	—
26	C₂×C₅²⋊C₉	—	31	1	no	—
27	(C₅⋊D₅)⋊C₉	—	31	1	no	—
28	(C₅⋊D₅)⋊C₉	—	31	1	no	—
29	C₂×C₃×D₃₁	—	25	1	no	—
30	D₁₅₅	—	15	2	no	—
31	C₅×D₃₁	—	3	2	no	—
32	D₅×C₃₁	—	15	2	no	—
33	C₂×C₃×C₅⋊D₅	—	31	1	no	—
34	C₃₁⋊C₉	—	25	1	no	Hall
35	C₅²⋊C₉	—	31	1	no	Hall
36	C₂×C₃×C₃₁	—	25	1	no	—
37	C₃×D₃₁	—	25	1	no	—
38	C₃×D₃₁	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic
39	C₅×C₃₁	—	3	2	no	—
40	C₂×C₃×C₅²	—	31	1	no	—
41	C₃×C₅⋊D₅	—	31	1	no	—
42	C₃×C₅⋊D₅	D₃₁⋊C₃	1	1	yes	characteristic
43	C₂×D₃₁	—	25	1	no	Hall
44	C₂×C₅⋊D₅	—	31	1	no	Hall
45	C₃×C₃₁	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic
46	C₃×C₅²	C₂×D₃₁⋊C₃	1	1	yes	characteristic
47	C₂×C₃₁	—	25	1	no	—
48	D₃₁	—	25	1	no	—
49	D₃₁	(C₅⋊D₅)⋊C₉	1	1	yes	characteristic, complemented
50	C₂×C₃×D₅	—	465	2	no	—
51	C₂×C₅²	—	31	1	no	—
52	C₅⋊D₅	—	31	1	no	—
53	C₅⋊D₅	D₃₁⋊C₉	1	1	yes	characteristic, complemented
54	C₂²×C₉	—	775	1	no	Hall
55	C₃₁	C₂×(C₅⋊D₅)⋊C₉	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃×C₅	—	93	2	no	—
57	C₃×D₅	—	15	2	no	—
58	C₃×D₅	—	465	2	no	—
59	C₅²	C₂×D₃₁⋊C₉	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₅	—	465	2	no	—
61	C₂×C₉	—	775	1	no	—
62	C₂×C₉	—	775	1	no	—
63	C₂×C₉	—	775	1	no	—
64	C₃×C₅	—	3	2	no	—
65	C₂²×C₃	—	775	1	no	—
66	C₂×C₅	—	93	2	no	—
67	D₅	—	15	2	no	—
68	D₅	—	465	2	no	—
69	C₉	—	775	1	no	Sylow
70	C₂×C₃	—	775	1	no	—
71	C₂×C₃	—	25	1	no	—
72	C₂×C₃	—	31	1	no	—
73	C₅	—	3	2	no	—
74	C₂²	—	775	1	no	Sylow
75	C₃	D₃₁⋊((C₅⋊D₅)⋊C₃)	1	1	yes	characteristic
76	C₂	—	775	1	no	—
77	C₂	—	25	1	no	—
78	C₂	—	31	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₃₁⋊((C₅⋊D₅)⋊C₃)
commutator, hypocenter	C₅²×C₃₁	C₂²×C₉
Fitting, socle, cosocle	C₃×C₅²×C₃₁	C₂²×C₃
prime radical	C₃×C₅⋊D₅×D₃₁	C₃
LCS	G ▷ C₅²×C₃₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. D₃₁⋊(C₅²⋊C₉)	C₂	semidirect
2. (C₅⋊D₅)⋊(C₃₁⋊C₉)	C₂	semidirect
3. #28358426	C₂	semidirect
5. C₅²⋊(C₃₁⋊C₉)	C₂²	semidirect
9. C₅⋊D₅×D₃₁	C₉	semidirect
12. C₅⋊D₁₅₅	C₂×C₉	semidirect
13. C₃₁×C₅⋊D₅	C₂×C₉	semidirect
14. C₅²×D₃₁	C₂×C₉	semidirect
20. C₅²×C₃₁	C₂²×C₉	semidirect
49. D₃₁	(C₅⋊D₅)⋊C₉	semidirect
53. C₅⋊D₅	D₃₁⋊C₉	semidirect
55. C₃₁	C₂×(C₅⋊D₅)⋊C₉	semidirect
59. C₅²	C₂×D₃₁⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
D₃₁⋊(C₅²⋊C₉)	2
(C₅⋊D₅)⋊(C₃₁⋊C₉)	2
#28358426	2
C₃×C₅⋊D₅×D₃₁	3
C₂×D₃₁⋊C₉	25
C₂×(C₅⋊D₅)⋊C₉	31

Subgroups — D31⋊((C5⋊D5)⋊C9) #42124513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁⋊((C₅⋊D₅)⋊C₉) #42124513