Subgroups — #36382955

Order 22599.

1062 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁⋊H(5,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	#21419406	C₃	1	24	yes	maximal, complemented
3	C₃×#14578900	C₃	1	12	yes	maximal, complemented
4	C₃×C₃₁×#737	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	36	yes	complemented
6	H(3,𝔽₃)×C₃₁⋊C₃	—	3	27	no	—
7	#14578900	C₃²	1	9	yes	complemented
8	#14578900	C₃²	1	72	yes	complemented
9	C₃₁×#737	—	3	9	no	—
10	C₃×3₋¹⁺²×C₃₁	C₃²	1	8	yes	complemented
11	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₃₁	C₃²	1	4	yes	complemented
13	C₃³×C₃₁	C₃³	1	4	yes	—
14	C₃²×C₃₁⋊C₃	—	3	36	no	—
15	3₋¹⁺²×C₃₁	—	3	24	no	—
16	H(3,𝔽₃)×C₃₁	H(3,𝔽₃)	1	9	yes	—
17	C₃×C₉×C₃₁	—	3	12	no	—
18	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
19	C₃×C₉×C₃₁	C₃³	1	8	yes	—
20	C₃²⋊(C₃₁⋊C₃)	—	3	72	no	—
21	C₃²⋊(C₃₁⋊C₃)	C₃³	1	27	yes	—
22	#10624932	—	31	1	no	maximal, Sylow
23	C₃²×C₃₁	—	9	4	no	—
24	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	12	yes	—
25	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
26	C₃×C₃₁⋊C₃	—	3	27	no	—
27	C₃×C₃₁⋊C₃	—	27	36	no	—
28	C₉×C₃₁	—	3	1	no	—
29	C₉×C₃₁	—	3	8	no	—
30	H(5,𝔽₃)	—	31	3	no	—
31	C₃×#737	C₃₁⋊C₃	1	1	yes	characteristic, complemented
32	#6911	—	31	24	no	—
33	C₃×#732	—	31	12	no	—
34	C₃×C₃₁	—	3	12	no	—
35	C₃×C₃₁	—	3	1	no	—
36	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃₁⋊C₃	—	9	27	no	—
38	#737	—	3	9	no	—
39	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	8	yes	complemented
40	C₃×H(3,𝔽₃)	—	31	36	no	—
41	C₃×H(3,𝔽₃)	—	93	27	no	—
42	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
43	C₃²×C₉	C₃×C₃₁⋊C₃	1	4	yes	complemented
44	#732	—	31	9	no	—
45	#732	—	31	72	no	—
46	C₃₁	#10624932	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	93	36	no	—
48	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
49	3₋¹⁺²	—	3	24	no	—
50	H(3,𝔽₃)	—	31	27	no	—
51	H(3,𝔽₃)	—	93	72	no	—
52	H(3,𝔽₃)	C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
53	C₃×C₉	—	3	12	no	—
54	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
55	C₃×C₉	C₃²×C₃₁⋊C₃	1	8	yes	—
56	C₃²	—	9	4	no	—
57	C₃²	—	837	36	no	—
58	C₃²	—	93	27	no	—
59	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	12	yes	—
60	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
61	C₉	—	3	1	no	—
62	C₉	—	3	8	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	12	no	—
65	C₃	—	279	27	no	—
66	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃×C₃₁×#737	C₃
socle	C₃×C₃₁	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₃₁⋊C₃
hypocenter	C₃₁	#10624932
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. C₃₁⋊H(5,𝔽₃)	C₃	semidirect
2. #21419406	C₃	semidirect
3. C₃×#14578900	C₃	semidirect
4. C₃×C₃₁×#737	C₃	semidirect
5. C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	semidirect
7. #14578900	C₃²	semidirect
8. #14578900	C₃²	semidirect
10. C₃×3₋¹⁺²×C₃₁	C₃²	semidirect
12. C₃²×C₉×C₃₁	C₃²	semidirect
31. C₃×#737	C₃₁⋊C₃	semidirect
39. C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	semidirect
43. C₃²×C₉	C₃×C₃₁⋊C₃	semidirect
46. C₃₁	#10624932	semidirect

Maximal subgroups

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

subgroup	index
C₃₁⋊H(5,𝔽₃)	3
#21419406	3
C₃×#14578900	3
C₃×C₃₁×#737	3
#10624932	31