Subgroups — C₃₁⋊C₃×C₃⋊#1043 #42124819

Order 27900.

224 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	C₃₁×C₃⋊#1043	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₁⋊C₃×#1043	—	3	1	no	maximal
4	C₃×C₅²×C₃₁⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	ΓL(1,𝔽₁₆)×C₃₁⋊C₃	—	5	6	no	maximal
6	C₃×C₃₁×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
7	C₅⋊D₅×C₃₁⋊C₃	—	2	1	no	—
8	C₅⋊D₅×C₃₁⋊C₃	D₃	1	1	yes	characteristic
9	C₃₁×#1043	—	3	1	no	Hall
10	C₃×D₅×C₃₁⋊C₃	—	5	6	no	—
11	C₅²×C₃₁⋊C₃	—	2	1	no	—
12	C₅²×C₃₁⋊C₃	Dic₃	1	1	yes	characteristic, complemented
13	C₃×C₅²×C₃₁	C₃×C₄	1	1	yes	characteristic, complemented
14	C₃₁×ΓL(1,𝔽₁₆)	—	5	6	no	—
15	AGL(1,𝔽₅)×C₃₁⋊C₃	—	15	6	no	—
16	C₃₁×C₅⋊D₅	C₃×D₃	1	1	yes	characteristic
17	C₃×C₅×C₃₁⋊C₃	AGL(1,𝔽₅)	1	6	yes	complemented
18	Dic₃×C₃₁⋊C₃	—	25	1	no	Hall
19	C₃×D₅×C₃₁	—	5	6	no	—
20	D₅×C₃₁⋊C₃	—	5	6	no	—
21	D₅×C₃₁⋊C₃	—	10	6	no	—
22	C₃×C₃⋊#1043	—	31	1	no	maximal, Hall
23	C₅²×C₃₁	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
24	AGL(1,𝔽₅)×C₃₁	—	15	6	no	—
25	C₂×C₃×C₃₁⋊C₃	—	25	1	no	—
26	C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	1	6	yes	complemented
27	C₅×C₃₁⋊C₃	—	2	6	no	—
28	C₅×C₃₁⋊C₃	ΓL(1,𝔽₁₆)	1	6	yes	complemented
29	C₃²×C₅⋊D₅	—	31	1	no	—
30	Dic₃×C₃₁	—	25	1	no	—
31	C₄×C₃₁⋊C₃	—	75	1	no	—
32	D₅×C₃₁	—	5	6	no	—
33	C₃⋊#1043	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
34	C₃×#1043	—	93	1	no	—
35	C₃×C₃₁⋊C₃	#1043	1	1	yes	characteristic, complemented, Hall
36	C₃²×C₅²	—	31	1	no	Hall
37	C₂×C₃×C₃₁	—	25	1	no	—
38	C₂×C₃₁⋊C₃	—	25	1	no	—
39	C₂×C₃₁⋊C₃	—	50	1	no	—
40	C₃×ΓL(1,𝔽₁₆)	—	155	6	no	—
41	C₅×C₃₁	C₃×ΓL(1,𝔽₁₆)	1	6	yes	complemented
42	C₃×C₅⋊D₅	—	31	1	no	—
43	C₃×C₅⋊D₅	—	62	1	no	—
44	C₃×C₅⋊D₅	C₂×C₃₁⋊C₃	1	1	yes	characteristic
45	C₄×C₃₁	—	75	1	no	Hall
46	#1043	—	3	1	no	Hall
47	C₃×C₃₁	C₃×#1043	1	1	yes	characteristic, complemented
48	C₃₁⋊C₃	—	2	1	no	—
49	C₃₁⋊C₃	C₃⋊#1043	1	1	yes	characteristic, direct product factor
50	C₃²×D₅	—	155	6	no	—
51	C₃×C₅²	—	31	1	no	—
52	C₃×C₅²	—	62	1	no	—
53	C₃×C₅²	C₄×C₃₁⋊C₃	1	1	yes	characteristic, complemented
54	C₂×C₃₁	—	25	1	no	—
55	ΓL(1,𝔽₁₆)	—	5	6	no	—
56	C₃×AGL(1,𝔽₅)	—	465	6	no	—
57	C₅⋊D₅	D₃×C₃₁⋊C₃	1	1	yes	characteristic
58	C₃²×C₅	—	31	6	no	—
59	C₃×Dic₃	—	775	1	no	Hall
60	C₃₁	C₃×C₃⋊#1043	1	1	yes	characteristic, complemented, Sylow
61	C₃×D₅	—	5	6	no	—
62	C₃×D₅	—	310	6	no	—
63	C₃×D₅	—	155	6	no	—
64	C₅²	Dic₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
65	AGL(1,𝔽₅)	—	15	6	no	—
66	C₂×C₃²	—	775	1	no	—
67	C₃×C₅	—	31	6	no	—
68	C₃×C₅	—	62	6	no	—
69	C₃×C₅	AGL(1,𝔽₅)×C₃₁⋊C₃	1	6	yes	complemented
70	C₃×C₄	—	2325	1	no	—
71	Dic₃	—	25	1	no	—
72	D₅	—	5	6	no	—
73	C₃²	—	31	1	no	Sylow
74	C₂×C₃	—	775	1	no	—
75	C₂×C₃	—	1550	1	no	—
76	C₂×C₃	—	25	1	no	—
77	C₅	ΓL(1,𝔽₁₆)×C₃₁⋊C₃	1	6	yes	complemented
78	C₄	—	75	1	no	Sylow
79	C₃	—	31	1	no	—
80	C₃	—	62	1	no	—
81	C₃	C₃₁⋊C₃×#1043	1	1	yes	characteristic, complemented
82	C₂	—	25	1	no	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×C₃⋊#1043	C₃	semidirect
4. C₃×C₅²×C₃₁⋊C₃	C₄	semidirect
12. C₅²×C₃₁⋊C₃	Dic₃	semidirect
13. C₃×C₅²×C₃₁	C₃×C₄	semidirect
17. C₃×C₅×C₃₁⋊C₃	AGL(1,𝔽₅)	semidirect
23. C₅²×C₃₁	C₃×Dic₃	semidirect
26. C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	semidirect
28. C₅×C₃₁⋊C₃	ΓL(1,𝔽₁₆)	semidirect
33. C₃⋊#1043	C₃₁⋊C₃	direct (N × H)
35. C₃×C₃₁⋊C₃	#1043	semidirect
41. C₅×C₃₁	C₃×ΓL(1,𝔽₁₆)	semidirect
47. C₃×C₃₁	C₃×#1043	semidirect
49. C₃₁⋊C₃	C₃⋊#1043	direct (N × H)
53. C₃×C₅²	C₄×C₃₁⋊C₃	semidirect
60. C₃₁	C₃×C₃⋊#1043	semidirect
64. C₅²	Dic₃×C₃₁⋊C₃	semidirect
69. C₃×C₅	AGL(1,𝔽₅)×C₃₁⋊C₃	semidirect
77. C₅	ΓL(1,𝔽₁₆)×C₃₁⋊C₃	semidirect
81. C₃	C₃₁⋊C₃×#1043	semidirect

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
C₃₁×C₃⋊#1043	3
C₃₁⋊C₃×#1043	3
ΓL(1,𝔽₁₆)×C₃₁⋊C₃	5
C₃×C₃⋊#1043	31

Subgroups — C31⋊C3×C3⋊#1043 #42124819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁⋊C₃×C₃⋊#1043 #42124819