Subgroups — #36379621

Order 22599.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578900	C₃	1	1	yes	maximal, characteristic
2	#21419336	C₃	1	2	yes	maximal, complemented
3	C₃₁×#6867	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
5	#14578900	—	3	3	no	—
6	(C₃₁⋊C₉)⋊C₉	—	3	6	no	—
7	C₃²⋊(C₃₁⋊C₉)	—	3	2	no	—
8	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
9	C₃₁×C₃²⋊C₉	—	3	1	no	—
10	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
11	C₃²×C₃₁⋊C₃	—	9	1	no	—
12	C₃×C₉×C₃₁	—	3	2	no	—
13	C₃×C₉×C₃₁	—	3	1	no	—
14	C₃×C₉×C₃₁	—	3	1	no	—
15	C₃×C₉×C₃₁	—	9	1	no	—
16	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
17	C₃²⋊(C₃₁⋊C₃)	—	9	2	no	—
18	C₃×C₃₁⋊C₉	—	3	6	no	—
19	#10624484	—	31	1	no	maximal, Sylow
20	C₃²×C₃₁	—	9	1	no	—
21	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
22	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	2	yes	—
23	C₃²×C₃₁	#731	1	1	yes	characteristic
24	C₃×C₃₁⋊C₃	—	9	3	no	—
25	C₃×C₃₁⋊C₃	—	27	3	no	—
26	C₉×C₃₁	—	9	1	no	—
27	C₉×C₃₁	—	27	1	no	—
28	C₃₁⋊C₉	—	9	6	no	—
29	C₃×#732	—	31	1	no	—
30	#6895	—	31	2	no	—
31	#6867	C₃₁⋊C₃	1	1	yes	characteristic, complemented
32	C₃×C₃₁	—	3	2	no	—
33	C₃×C₃₁	—	3	1	no	—
34	C₃×C₃₁	—	3	1	no	—
35	C₃×C₃₁	#6857	1	1	yes	characteristic
36	C₃₁⋊C₃	—	27	3	no	—
37	C₃×H(3,𝔽₃)	—	93	1	no	—
38	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
39	#732	—	93	3	no	—
40	C₉⋊C₉	—	93	6	no	—
41	C₃²⋊C₉	—	3	1	no	—
42	C₃²⋊C₉	—	93	2	no	—
43	C₃₁	#10624484	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	279	1	no	—
45	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
46	H(3,𝔽₃)	—	279	2	no	—
47	H(3,𝔽₃)	—	93	3	no	—
48	C₃×C₉	—	3	1	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	9	1	no	—
52	C₃×C₉	—	93	6	no	—
53	C₃²	—	9	1	no	—
54	C₃²	—	279	3	no	—
55	C₃²	—	837	3	no	—
56	C₃²	#14578893	1	1	yes	characteristic
57	C₃²	#14578895	1	2	yes	—
58	C₃²	#14578897	1	1	yes	characteristic
59	C₉	—	9	1	no	—
60	C₉	—	279	6	no	—
61	C₉	—	27	1	no	—
62	C₃	—	3	2	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	1	no	—
65	C₃	—	837	3	no	—
66	C₃	#21419204	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419204
commutator, cosocle	C₃²×C₉×C₃₁	C₃²
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6867	C₃
socle	C₃×C₃₁	#6857
prime radical	C₃×#14578900	C₃
hypercenter	#6867	C₃₁⋊C₃
hypocenter	C₃₁	#10624484
LCS	G ▷ C₃²×C₉×C₃₁ ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6867

Complements

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

N	H ≅ G/N	type
2. #21419336	C₃	semidirect
3. C₃₁×#6867	C₃	semidirect
31. #6867	C₃₁⋊C₃	semidirect
43. C₃₁	#10624484	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578900	3
#21419336	3
C₃₁×#6867	3
#10624484	31