Subgroups — #36379622

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	1	yes	maximal, characteristic, complemented
3	#21419228	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#6895	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
6	#14578900	—	3	3	no	—
7	(C₃₁⋊C₉)⋊C₉	—	3	3	no	—
8	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
9	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
10	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
11	C₃₁×C₉⋊C₉	—	3	3	no	—
12	C₃₁×C₃²⋊C₉	—	3	1	no	—
13	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃²×C₃₁⋊C₃	—	9	1	no	—
15	C₃×C₉×C₃₁	—	3	3	no	—
16	C₃×C₉×C₃₁	—	3	1	no	—
17	C₃×C₉×C₃₁	—	3	1	no	—
18	C₃×C₉×C₃₁	—	3	1	no	—
19	C₃×C₉×C₃₁	—	3	1	no	—
20	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
21	C₃²⋊(C₃₁⋊C₃)	—	9	1	no	—
22	C₃²⋊(C₃₁⋊C₃)	—	9	1	no	—
23	C₃×C₃₁⋊C₉	—	3	3	no	—
24	C₃×C₃₁⋊C₉	—	9	1	no	—
25	#10624485	—	31	1	no	maximal, Sylow
26	C₃²×C₃₁	—	9	1	no	—
27	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
28	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
29	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃²×C₃₁	#731	1	1	yes	characteristic
31	C₃×C₃₁⋊C₃	—	9	3	no	—
32	C₃×C₃₁⋊C₃	—	27	3	no	—
33	C₉×C₃₁	—	9	3	no	—
34	C₉×C₃₁	—	9	1	no	—
35	C₃₁⋊C₉	—	9	3	no	—
36	C₃₁⋊C₉	—	27	1	no	—
37	C₃×#732	—	31	1	no	—
38	#6895	—	31	1	no	—
39	#6895	C₃₁⋊C₃	1	1	yes	characteristic, complemented
40	#6867	—	31	1	no	—
41	C₃×C₃₁	—	3	1	no	—
42	C₃×C₃₁	—	3	1	no	—
43	C₃×C₃₁	—	3	1	no	—
44	C₃×C₃₁	—	3	1	no	—
45	C₃×C₃₁	#6857	1	1	yes	characteristic
46	C₃₁⋊C₃	—	27	3	no	—
47	C₃×H(3,𝔽₃)	—	93	1	no	—
48	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
49	#732	—	93	3	no	—
50	C₉⋊C₉	—	3	3	no	—
51	C₉⋊C₉	—	93	3	no	—
52	C₃²⋊C₉	—	3	1	no	—
53	C₃²⋊C₉	—	93	1	no	—
54	C₃²⋊C₉	—	93	1	no	—
55	C₃₁	#10624485	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	279	1	no	—
57	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
58	H(3,𝔽₃)	—	279	1	no	—
59	H(3,𝔽₃)	—	279	1	no	—
60	H(3,𝔽₃)	—	93	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	3	1	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	279	1	no	—
67	C₃×C₉	—	93	3	no	—
68	C₃²	—	9	1	no	—
69	C₃²	—	279	3	no	—
70	C₃²	—	837	3	no	—
71	C₃²	#14578894	1	1	yes	characteristic
72	C₃²	#14578895	1	1	yes	characteristic
73	C₃²	#14578895	1	1	yes	characteristic
74	C₃²	#14578898	1	1	yes	characteristic
75	C₉	—	9	3	no	—
76	C₉	—	9	1	no	—
77	C₉	—	279	3	no	—
78	C₉	—	837	1	no	—
79	C₃	—	3	1	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	1	no	—
82	C₃	—	3	1	no	—
83	C₃	—	837	3	no	—
84	C₃	#21419202	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419202
commutator, cosocle	C₃²×C₉×C₃₁	C₃²
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6895	C₃
socle	C₃×C₃₁	#6857
prime radical	C₃×#14578900	C₃
hypercenter	#6895	C₃₁⋊C₃
hypocenter	C₃₁	#10624485
LCS	G ▷ C₃²×C₉×C₃₁ ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6895

Complements

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

N	H ≅ G/N	type
2. #21419336	C₃	semidirect
3. #21419228	C₃	semidirect
4. C₃₁×#6895	C₃	semidirect
39. #6895	C₃₁⋊C₃	semidirect
55. C₃₁	#10624485	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
#21419228	3
C₃₁×#6895	3
#10624485	31