Subgroups — C₃×(C₃₁⋊C₃)⋊#1044 #42124764

Order 27900.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅⋊D₅×C₃₁⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃₁⋊#1044	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃₁⋊C₃)⋊#1044	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅²×C₃₁⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×(C₃₁⋊C₃)⋊AGL(1,𝔽₅)	—	5	2	no	maximal
6	C₃×C₃₁×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
7	C₅⋊D₅×C₃₁⋊C₃	C₂×C₃	1	3	yes	—
8	C₃₁⋊#1044	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×D₅×C₃₁⋊C₃	—	5	2	no	—
10	C₃×D₅×C₃₁⋊C₃	—	10	2	no	—
11	C₅²×C₃₁⋊C₃	C₃×C₄	1	3	yes	complemented
12	C₃×C₅²×C₃₁	C₃×C₄	1	1	yes	characteristic, complemented
13	C₃×C₃₁⋊AGL(1,𝔽₅)	—	5	2	no	—
14	(C₃₁⋊C₃)⋊AGL(1,𝔽₅)	—	5	6	no	—
15	C₃₁×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic
16	C₃×C₅×C₃₁⋊C₃	—	2	2	no	—
17	C₃×C₅×C₃₁⋊C₃	AGL(1,𝔽₅)	1	2	yes	complemented
18	C₃×(C₃₁⋊C₃)⋊C₄	—	25	1	no	Hall
19	C₃×D₅×C₃₁	—	5	2	no	—
20	C₃×D₅×C₃₁	—	10	2	no	—
21	D₅×C₃₁⋊C₃	—	5	6	no	—
22	D₅×C₃₁⋊C₃	—	10	6	no	—
23	C₃²×#1044	—	31	1	no	maximal, Hall
24	C₅²×C₃₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
25	C₃₁⋊AGL(1,𝔽₅)	—	5	2	no	—
26	C₂×C₃×C₃₁⋊C₃	—	25	1	no	—
27	C₃×C₅×C₃₁	—	2	2	no	—
28	C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	1	2	yes	complemented
29	C₅×C₃₁⋊C₃	—	2	6	no	—
30	C₅×C₃₁⋊C₃	C₃×AGL(1,𝔽₅)	1	6	yes	complemented
31	C₃²×C₅⋊D₅	—	31	1	no	—
32	C₃×Dic₃₁	—	25	1	no	—
33	(C₃₁⋊C₃)⋊C₄	—	25	3	no	—
34	D₅×C₃₁	—	5	2	no	—
35	D₅×C₃₁	—	10	2	no	—
36	C₃×#1044	—	31	3	no	—
37	C₃×#1044	—	31	1	no	—
38	C₃×C₃₁⋊C₃	#1044	1	1	yes	characteristic, complemented, Hall
39	C₃²×C₅²	—	31	1	no	Hall
40	C₂×C₃×C₃₁	—	25	1	no	—
41	C₂×C₃₁⋊C₃	—	25	3	no	—
42	C₃²×AGL(1,𝔽₅)	—	155	2	no	—
43	C₅×C₃₁	—	2	2	no	—
44	C₅×C₃₁	C₃²×AGL(1,𝔽₅)	1	2	yes	complemented
45	C₃×C₅⋊D₅	—	31	3	no	—
46	C₃×C₅⋊D₅	D₃₁⋊C₃	1	1	yes	characteristic
47	Dic₃₁	—	25	1	no	Hall
48	#1044	—	31	1	no	Hall
49	C₃×C₃₁	C₃×#1044	1	1	yes	characteristic, complemented
50	C₃₁⋊C₃	C₃×#1044	1	3	yes	complemented
51	C₃²×D₅	—	310	2	no	—
52	C₃²×D₅	—	155	2	no	—
53	C₃×C₅²	—	31	3	no	—
54	C₃×C₅²	(C₃₁⋊C₃)⋊C₄	1	1	yes	characteristic, complemented
55	C₂×C₃₁	—	25	1	no	—
56	C₃×AGL(1,𝔽₅)	—	155	2	no	—
57	C₃×AGL(1,𝔽₅)	—	155	6	no	—
58	C₅⋊D₅	C₃×D₃₁⋊C₃	1	1	yes	characteristic
59	C₃²×C₅	—	31	2	no	—
60	C₃²×C₅	—	62	2	no	—
61	C₃²×C₄	—	775	1	no	Hall
62	C₃₁	C₃²×#1044	1	1	yes	characteristic, complemented, Sylow
63	C₃×D₅	—	5	2	no	—
64	C₃×D₅	—	10	2	no	—
65	C₃×D₅	—	310	6	no	—
66	C₃×D₅	—	155	6	no	—
67	C₅²	C₃×(C₃₁⋊C₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
68	AGL(1,𝔽₅)	—	155	2	no	—
69	C₂×C₃²	—	775	1	no	—
70	C₃×C₅	—	2	2	no	—
71	C₃×C₅	—	31	6	no	—
72	C₃×C₅	—	62	6	no	—
73	C₃×C₅	(C₃₁⋊C₃)⋊AGL(1,𝔽₅)	1	2	yes	complemented
74	C₃×C₄	—	775	3	no	—
75	C₃×C₄	—	775	1	no	—
76	D₅	—	5	2	no	—
77	D₅	—	10	2	no	—
78	C₃²	—	31	1	no	Sylow
79	C₂×C₃	—	775	3	no	—
80	C₂×C₃	—	25	1	no	—
81	C₅	—	2	2	no	—
82	C₅	C₃×(C₃₁⋊C₃)⋊AGL(1,𝔽₅)	1	2	yes	complemented
83	C₄	—	775	1	no	Sylow
84	C₃	—	31	3	no	—
85	C₃	(C₃₁⋊C₃)⋊#1044	1	1	yes	characteristic, direct product factor
86	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃	(C₃₁⋊C₃)⋊#1044
commutator, hypocenter	C₅²×C₃₁	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₃₁	C₃×C₄
prime radical	C₃×C₅⋊D₅×C₃₁⋊C₃	C₂
cosocle	C₃₁×C₅⋊D₅	C₂×C₃²
LCS	G ▷ C₅²×C₃₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₃₁⋊#1044	C₃	semidirect
3. (C₃₁⋊C₃)⋊#1044	C₃	direct (N × H)
4. C₃×C₅²×C₃₁⋊C₃	C₄	semidirect
8. C₃₁⋊#1044	C₃²	semidirect
11. C₅²×C₃₁⋊C₃	C₃×C₄	semidirect
12. C₃×C₅²×C₃₁	C₃×C₄	semidirect
17. C₃×C₅×C₃₁⋊C₃	AGL(1,𝔽₅)	semidirect
24. C₅²×C₃₁	C₃²×C₄	semidirect
28. C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	semidirect
30. C₅×C₃₁⋊C₃	C₃×AGL(1,𝔽₅)	semidirect
38. C₃×C₃₁⋊C₃	#1044	semidirect
44. C₅×C₃₁	C₃²×AGL(1,𝔽₅)	semidirect
49. C₃×C₃₁	C₃×#1044	semidirect
50. C₃₁⋊C₃	C₃×#1044	semidirect
54. C₃×C₅²	(C₃₁⋊C₃)⋊C₄	semidirect
62. C₃₁	C₃²×#1044	semidirect
67. C₅²	C₃×(C₃₁⋊C₃)⋊C₄	semidirect
73. C₃×C₅	(C₃₁⋊C₃)⋊AGL(1,𝔽₅)	semidirect
82. C₅	C₃×(C₃₁⋊C₃)⋊AGL(1,𝔽₅)	semidirect
85. C₃	(C₃₁⋊C₃)⋊#1044	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅⋊D₅×C₃₁⋊C₃	2
C₃×C₃₁⋊#1044	3
(C₃₁⋊C₃)⋊#1044	3
C₃×(C₃₁⋊C₃)⋊AGL(1,𝔽₅)	5
C₃²×#1044	31

Subgroups — C3×(C31⋊C3)⋊#1044 #42124764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₃₁⋊C₃)⋊#1044 #42124764