Subgroups — #36382067

Order 22599.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419258	C₃	1	8	yes	maximal, complemented
2	C₃×C₃₁⋊C₈₁	C₃	1	4	yes	maximal, complemented
3	C₃₁×#6903	C₃	1	1	yes	maximal, characteristic
4	C₃₁⋊C₈₁	C₃²	1	1	yes	characteristic
5	C₃₁⋊C₈₁	C₃²	1	8	yes	—
6	C₃₁×#737	C₉	1	1	yes	characteristic
7	C₃₁×C₂₇⋊C₃	C₉	1	8	yes	—
8	C₃×C₂₇×C₃₁	C₃²	1	4	yes	—
9	3₋¹⁺²×C₃₁	C₂₇	1	8	yes	—
10	H(3,𝔽₃)×C₃₁	C₂₇	1	1	yes	characteristic
11	C₃×C₉×C₃₁	C₃×C₉	1	4	yes	—
12	C₂₇×C₃₁	C₃×C₉	1	8	yes	—
13	C₂₇×C₃₁	C₃³	1	1	yes	characteristic
14	#10624833	—	31	1	no	maximal, Sylow
15	C₃²×C₃₁	C₃×C₂₇	1	4	yes	—
16	C₉×C₃₁	C₃×C₂₇	1	8	yes	—
17	C₉×C₃₁	C₃²×C₉	1	1	yes	characteristic
18	#6903	C₃₁⋊C₃	1	1	yes	characteristic
19	C₈₁⋊C₃	—	31	8	no	—
20	C₃×C₈₁	—	31	4	no	—
21	C₃×C₃₁	—	3	4	no	—
22	C₃×C₃₁	C₃²×C₂₇	1	1	yes	characteristic
23	#737	C₃₁⋊C₉	1	1	yes	characteristic
24	C₂₇⋊C₃	C₃₁⋊C₉	1	8	yes	—
25	C₃×C₂₇	C₃×C₃₁⋊C₃	1	4	yes	—
26	C₈₁	—	31	1	no	—
27	C₈₁	—	31	8	no	—
28	C₃₁	#10624833	1	1	yes	characteristic, complemented, Sylow
29	3₋¹⁺²	C₃₁⋊C₂₇	1	8	yes	—
30	H(3,𝔽₃)	C₃₁⋊C₂₇	1	1	yes	characteristic
31	C₃×C₉	C₃×C₃₁⋊C₉	1	4	yes	—
32	C₂₇	C₃×C₃₁⋊C₉	1	8	yes	—
33	C₂₇	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
34	C₃²	C₃×C₃₁⋊C₂₇	1	4	yes	—
35	C₉	C₃×C₃₁⋊C₂₇	1	8	yes	—
36	C₉	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
37	C₃	—	3	4	no	—
38	C₃	C₃²×C₃₁⋊C₂₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂₇	C₃²×C₃₁⋊C₃
commutator, socle	C₃×C₃₁	C₃²×C₂₇
Fitting	C₃₁×#6903	C₃
prime radical	H(3,𝔽₃)×C₃₁	C₂₇
cosocle	C₂₇×C₃₁	C₃³
hypercenter	#6903	C₃₁⋊C₃
hypocenter	C₃₁	#10624833
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ 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
28. C₃₁	#10624833	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
#10624833	31