Subgroups — C₃×A₅×ΓL(1,𝔽₃₂) #42124999

Order 27900.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₅×ΓL(1,𝔽₃₂)	C₃	1	1	yes	maximal, characteristic, direct product factor
2	C₃×C₃₁×A₅	C₅	1	1	yes	maximal, characteristic, complemented
3	C₃×A₄×ΓL(1,𝔽₃₂)	—	5	1	no	maximal
4	C₃×D₅×ΓL(1,𝔽₃₂)	—	6	1	no	maximal
5	C₃×D₃×ΓL(1,𝔽₃₂)	—	10	1	no	maximal
6	C₃×C₅×ΓL(1,𝔽₃₂)	—	6	1	no	—
7	C₂²×C₃×ΓL(1,𝔽₃₂)	—	5	1	no	—
8	A₄×ΓL(1,𝔽₃₂)	—	5	1	no	—
9	A₄×ΓL(1,𝔽₃₂)	—	5	2	no	—
10	C₃₁×A₅	C₃×C₅	1	1	yes	characteristic, complemented
11	D₅×ΓL(1,𝔽₃₂)	—	6	1	no	—
12	C₃²×ΓL(1,𝔽₃₂)	—	10	1	no	—
13	C₃×A₄×C₃₁	—	5	1	no	Hall
14	C₃×D₅×C₃₁	—	6	1	no	—
15	C₂×C₃×ΓL(1,𝔽₃₂)	—	15	1	no	—
16	D₃×ΓL(1,𝔽₃₂)	—	10	1	no	—
17	C₃×C₅×A₅	—	31	1	no	maximal, Hall
18	C₅×ΓL(1,𝔽₃₂)	—	6	1	no	Hall
19	C₂²×ΓL(1,𝔽₃₂)	—	5	1	no	—
20	C₃×D₃×C₃₁	—	10	1	no	—
21	C₃×C₅×C₃₁	—	6	1	no	—
22	C₃×ΓL(1,𝔽₃₂)	—	10	1	no	—
23	C₃×ΓL(1,𝔽₃₂)	—	12	2	no	—
24	C₃×ΓL(1,𝔽₃₂)	—	20	1	no	—
25	C₃×ΓL(1,𝔽₃₂)	A₅	1	1	yes	maximal normal, characteristic, direct product factor
26	C₂²×C₃×C₃₁	—	5	1	no	—
27	A₄×C₃₁	—	5	1	no	—
28	A₄×C₃₁	—	5	2	no	—
29	D₅×C₃₁	—	6	1	no	—
30	C₂×ΓL(1,𝔽₃₂)	—	15	1	no	—
31	C₅×A₅	—	31	1	no	—
32	C₃²×C₃₁	—	10	1	no	Hall
33	C₂×C₃×C₃₁	—	15	1	no	—
34	D₃×C₃₁	—	10	1	no	—
35	C₃×C₅×A₄	—	155	1	no	—
36	C₃×A₅	ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
37	C₅×C₃₁	—	6	1	no	—
38	ΓL(1,𝔽₃₂)	—	12	2	no	—
39	ΓL(1,𝔽₃₂)	C₃×A₅	1	1	yes	characteristic, direct product factor
40	C₃×C₅×D₅	—	186	1	no	—
41	C₂²×C₃₁	—	5	1	no	Hall
42	C₃×C₃₁	—	10	1	no	—
43	C₃×C₃₁	—	20	1	no	—
44	C₃×C₃₁	C₅×A₅	1	1	yes	characteristic, complemented
45	C₃×C₅×D₃	—	310	1	no	—
46	C₃×C₅²	—	186	1	no	—
47	C₂×C₃₁	—	15	1	no	—
48	C₂²×C₃×C₅	—	155	1	no	—
49	C₅×A₄	—	155	1	no	—
50	C₅×A₄	—	155	2	no	—
51	A₅	C₃×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
52	C₅×D₅	—	186	1	no	—
53	C₃²×C₅	—	310	1	no	—
54	C₃×A₄	—	5	1	no	Hall
55	C₃₁	C₃×C₅×A₅	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃×C₅	—	465	1	no	—
57	C₃×D₅	—	6	1	no	—
58	C₅×D₃	—	310	1	no	—
59	C₅²	—	186	1	no	Sylow
60	C₂²×C₅	—	155	1	no	—
61	C₃×D₃	—	10	1	no	—
62	C₃×C₅	—	6	1	no	—
63	C₃×C₅	—	31	1	no	—
64	C₃×C₅	—	310	1	no	—
65	C₃×C₅	—	620	1	no	—
66	C₃×C₅	—	372	2	no	—
67	C₂²×C₃	—	5	1	no	—
68	A₄	—	5	1	no	—
69	A₄	—	5	2	no	—
70	C₂×C₅	—	465	1	no	—
71	D₅	—	6	1	no	—
72	C₃²	—	10	1	no	Sylow
73	C₂×C₃	—	15	1	no	—
74	D₃	—	10	1	no	—
75	C₅	—	6	1	no	—
76	C₅	—	31	1	no	—
77	C₅	—	372	2	no	—
78	C₂²	—	5	1	no	Sylow
79	C₃	—	10	1	no	—
80	C₃	—	20	1	no	—
81	C₃	A₅×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
82	C₂	—	15	1	no	—

	subgroup	G/H
center, hypercenter	C₃	A₅×ΓL(1,𝔽₃₂)
commutator, hypocenter	C₃₁×A₅	C₃×C₅
Frattini	C₁	G
Fitting	C₃×C₃₁	C₅×A₅
socle	C₃×C₃₁×A₅	C₅
cosocle	C₃₁	C₃×C₅×A₅
LCS	G ▷ C₃₁×A₅
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. A₅×ΓL(1,𝔽₃₂)	C₃	direct (N × H)
2. C₃×C₃₁×A₅	C₅	semidirect
10. C₃₁×A₅	C₃×C₅	semidirect
25. C₃×ΓL(1,𝔽₃₂)	A₅	direct (N × H)
36. C₃×A₅	ΓL(1,𝔽₃₂)	direct (N × H)
39. ΓL(1,𝔽₃₂)	C₃×A₅	direct (N × H)
44. C₃×C₃₁	C₅×A₅	semidirect
51. A₅	C₃×ΓL(1,𝔽₃₂)	direct (N × H)
55. C₃₁	C₃×C₅×A₅	semidirect
81. C₃	A₅×ΓL(1,𝔽₃₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
A₅×ΓL(1,𝔽₃₂)	3
C₃×C₃₁×A₅	5
C₃×A₄×ΓL(1,𝔽₃₂)	5
C₃×D₅×ΓL(1,𝔽₃₂)	6
C₃×D₃×ΓL(1,𝔽₃₂)	10
C₃×C₅×A₅	31

Subgroups — C3×A5×ΓL(1,𝔽32) #42124999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×A₅×ΓL(1,𝔽₃₂) #42124999