Subgroups — #36379699

Order 22599.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419258	C₃	1	2	yes	maximal, complemented
2	C₃×C₃₁⋊C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#6903	C₃	1	1	yes	maximal, characteristic
4	C₃₁⋊C₈₁	—	3	1	no	—
5	C₃₁⋊C₈₁	—	3	2	no	—
6	C₃₁×#737	C₉	1	1	yes	characteristic
7	C₃₁×C₂₇⋊C₃	—	3	2	no	—
8	C₃₁×C₂₇⋊C₃	C₉	1	2	yes	—
9	C₃×C₂₇×C₃₁	—	3	1	no	—
10	C₃×C₂₇×C₃₁	C₃²	1	1	yes	characteristic
11	3₋¹⁺²×C₃₁	—	3	2	no	—
12	3₋¹⁺²×C₃₁	C₂₇	1	2	yes	—
13	H(3,𝔽₃)×C₃₁	C₂₇	1	1	yes	characteristic
14	C₃×C₉×C₃₁	—	3	1	no	—
15	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
16	C₂₇×C₃₁	—	3	2	no	—
17	C₂₇×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
18	C₂₇×C₃₁	3₋¹⁺²	1	2	yes	—
19	#10624503	—	31	1	no	maximal, Sylow
20	C₃²×C₃₁	—	3	1	no	—
21	C₃²×C₃₁	C₃×C₂₇	1	1	yes	characteristic
22	C₉×C₃₁	—	3	2	no	—
23	C₉×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
24	C₉×C₃₁	C₂₇⋊C₃	1	2	yes	—
25	#6903	C₃₁⋊C₃	1	1	yes	characteristic
26	C₈₁⋊C₃	—	31	2	no	—
27	C₃×C₈₁	—	31	1	no	—
28	C₃×C₃₁	—	3	1	no	—
29	C₃×C₃₁	—	9	1	no	—
30	C₃×C₃₁	C₃²⋊C₂₇	1	1	yes	characteristic
31	#737	C₃₁⋊C₉	1	1	yes	characteristic
32	C₂₇⋊C₃	—	3	2	no	—
33	C₂₇⋊C₃	C₃₁⋊C₉	1	2	yes	—
34	C₃×C₂₇	—	3	1	no	—
35	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
36	C₈₁	—	93	1	no	—
37	C₈₁	—	93	2	no	—
38	C₃₁	#10624503	1	1	yes	characteristic, complemented, Sylow
39	3₋¹⁺²	—	3	2	no	—
40	3₋¹⁺²	C₃₁⋊C₂₇	1	2	yes	—
41	H(3,𝔽₃)	C₃₁⋊C₂₇	1	1	yes	characteristic
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	C₃×C₃₁⋊C₉	1	1	yes	characteristic
44	C₂₇	—	3	2	no	—
45	C₂₇	#11719689	1	2	yes	—
46	C₂₇	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
47	C₃²	—	3	1	no	—
48	C₃²	C₃×C₃₁⋊C₂₇	1	1	yes	characteristic
49	C₉	—	3	2	no	—
50	C₉	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
51	C₉	#14578892	1	2	yes	—
52	C₃	—	3	1	no	—
53	C₃	—	9	1	no	—
54	C₃	C₃²⋊(C₃₁⋊C₂₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂₇	C₃²⋊(C₃₁⋊C₃)
commutator	C₃²×C₃₁	C₃×C₂₇
Frattini	C₃×C₂₇	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6903	C₃
socle	C₃×C₃₁	C₃²⋊C₂₇
prime radical	H(3,𝔽₃)×C₃₁	C₂₇
cosocle	C₃×C₂₇×C₃₁	C₃²
hypercenter	#6903	C₃₁⋊C₃
hypocenter	C₃₁	#10624503
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇ ◁ #6903

Complements

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

N	H ≅ G/N	type
1. #21419258	C₃	semidirect
2. C₃×C₃₁⋊C₈₁	C₃	semidirect
38. C₃₁	#10624503	semidirect

Maximal subgroups

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

subgroup	index
#21419258	3
C₃×C₃₁⋊C₈₁	3
C₃₁×#6903	3
#10624503	31