Subgroups — #36379529

Order 22599.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	3	yes	maximal
2	C₃₁×#6887	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₁⋊C₂₇	—	3	3	no	—
4	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
5	C₃₁×C₉⋊C₉	—	3	2	no	—
6	C₃₁×C₃²⋊C₉	C₉	1	3	yes	—
7	C₉²×C₃₁	—	3	1	no	—
8	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
9	C₃×C₉×C₃₁	—	3	1	no	—
10	C₃×C₉×C₃₁	—	3	2	no	—
11	C₃×C₉×C₃₁	—	3	3	no	—
12	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃×C₉×C₃₁	3₋¹⁺²	1	2	yes	—
14	C₃₁⋊C₂₇	—	9	3	no	—
15	#10624458	—	31	1	no	maximal, Sylow
16	C₃²×C₃₁	—	3	1	no	—
17	C₃²×C₃₁	—	3	3	no	—
18	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
19	C₉×C₃₁	—	3	2	no	—
20	C₉×C₃₁	—	9	3	no	—
21	C₉×C₃₁	#732	1	1	yes	characteristic
22	C₉×C₃₁	#733	1	2	yes	—
23	#6887	C₃₁⋊C₃	1	1	yes	characteristic
24	C₃²⋊C₂₇	—	31	3	no	—
25	C₃×C₃₁	—	9	1	no	—
26	C₃×C₃₁	#6868	1	1	yes	characteristic
27	C₃×C₃₁	#6872	1	3	yes	—
28	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
29	C₃×C₂₇	—	93	3	no	—
30	C₉⋊C₉	—	3	2	no	—
31	C₃²⋊C₉	C₃₁⋊C₉	1	3	yes	—
32	C₉²	—	3	1	no	—
33	C₃₁	#10624458	1	1	yes	characteristic, complemented, Sylow
34	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
35	C₃×C₉	—	3	1	no	—
36	C₃×C₉	—	3	2	no	—
37	C₃×C₉	—	3	3	no	—
38	C₃×C₉	#11719689	1	2	yes	—
39	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
40	C₂₇	—	279	3	no	—
41	C₃²	—	3	1	no	—
42	C₃²	—	3	3	no	—
43	C₃²	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
44	C₉	—	3	2	no	—
45	C₉	—	9	3	no	—
46	C₉	#14578900	1	1	yes	characteristic
47	C₉	#14578902	1	2	yes	—
48	C₃	—	9	1	no	—
49	C₃	#21419230	1	1	yes	characteristic
50	C₃	#21419241	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₁⋊C₃)
commutator, prime radical	C₃³×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6887	C₃
socle	C₃²×C₃₁	C₃²⋊C₉
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	#6887	C₃₁⋊C₃
hypocenter	C₃₁	#10624458
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
33. C₃₁	#10624458	semidirect

Maximal subgroups

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

subgroup	index
C₃²⋊(C₃₁⋊C₂₇)	3
C₃₁×#6887	3
#10624458	31