Subgroups — #36381009

Order 22599.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578892	C₃	1	3	yes	maximal, complemented
2	#21419232	C₃	1	9	yes	maximal, complemented
3	C₃×C₃₁×#737	C₃	1	1	yes	maximal, characteristic
4	#14578892	—	3	6	no	—
5	#14578892	C₃²	1	9	yes	—
6	C₃×C₃₁⋊C₂₇	—	3	3	no	—
7	C₃₁×#737	—	3	3	no	—
8	C₃₁×#737	—	3	6	no	—
9	C₃×3₋¹⁺²×C₃₁	C₉	1	2	yes	—
10	C₃×3₋¹⁺²×C₃₁	C₉	1	6	yes	—
11	C₃×H(3,𝔽₃)×C₃₁	C₉	1	1	yes	characteristic
12	C₃²×C₉×C₃₁	C₃²	1	3	yes	—
13	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
14	C₃³×C₃₁	C₃×C₉	1	3	yes	—
15	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
16	3₋¹⁺²×C₃₁	—	3	6	no	—
17	3₋¹⁺²×C₃₁	—	3	18	no	—
18	H(3,𝔽₃)×C₃₁	—	3	3	no	—
19	C₃×C₉×C₃₁	—	3	3	no	—
20	C₃×C₉×C₃₁	—	3	6	no	—
21	C₃×C₉×C₃₁	C₃×C₉	1	6	yes	—
22	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
23	C₃×C₉×C₃₁	H(3,𝔽₃)	1	3	yes	—
24	C₃×C₉×C₃₁	3₋¹⁺²	1	6	yes	—
25	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
26	C₃₁⋊C₂₇	—	3	9	no	—
27	#10624694	—	31	1	no	maximal, Sylow
28	C₃²×C₃₁	—	3	3	no	—
29	C₃²×C₃₁	—	9	1	no	—
30	C₃²×C₃₁	—	9	3	no	—
31	C₃²×C₃₁	C₃²⋊C₉	1	3	yes	—
32	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
33	C₉×C₃₁	—	3	6	no	—
34	C₉×C₃₁	C₃²⋊C₉	1	6	yes	—
35	C₉×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₉×C₃₁	C₃×3₋¹⁺²	1	2	yes	—
37	C₃×#737	C₃₁⋊C₃	1	1	yes	characteristic
38	C₃×C₂₇⋊C₃	—	31	3	no	—
39	#6869	—	31	9	no	—
40	C₃×C₃₁	—	3	1	no	—
41	C₃×C₃₁	—	3	3	no	—
42	C₃×C₃₁	—	9	3	no	—
43	C₃×C₃₁	C₃×C₃²⋊C₉	1	1	yes	characteristic
44	#737	—	3	6	no	—
45	#737	—	3	3	no	—
46	C₃×3₋¹⁺²	C₃₁⋊C₉	1	2	yes	—
47	C₃×3₋¹⁺²	C₃₁⋊C₉	1	6	yes	—
48	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	1	yes	characteristic
49	C₃²×C₉	C₃×C₃₁⋊C₃	1	3	yes	—
50	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
51	C₂₇⋊C₃	—	31	9	no	—
52	C₂₇⋊C₃	—	93	6	no	—
53	C₃×C₂₇	—	93	3	no	—
54	C₃₁	#10624694	1	1	yes	characteristic, complemented, Sylow
55	C₃³	C₃×C₃₁⋊C₉	1	3	yes	—
56	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
57	3₋¹⁺²	—	3	6	no	—
58	3₋¹⁺²	—	3	18	no	—
59	H(3,𝔽₃)	—	3	3	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	6	no	—
62	C₃×C₉	C₃×C₃₁⋊C₉	1	6	yes	—
63	C₃×C₉	C₃×C₃₁⋊C₉	1	2	yes	—
64	C₃×C₉	#11719689	1	6	yes	—
65	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
66	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
67	C₂₇	—	93	9	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	3	no	—
71	C₃²	C₃²⋊(C₃₁⋊C₉)	1	3	yes	—
72	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
73	C₉	—	3	6	no	—
74	C₉	C₃²⋊(C₃₁⋊C₉)	1	6	yes	—
75	C₉	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
76	C₉	C₃×#11719689	1	2	yes	—
77	C₃	—	3	1	no	—
78	C₃	—	3	3	no	—
79	C₃	—	9	3	no	—
80	C₃	C₃×C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃×C₃²⋊(C₃₁⋊C₃)
commutator	C₃²×C₃₁	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃×C₃₁×#737	C₃
socle	C₃×C₃₁	C₃×C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)×C₃₁	C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₃×#737	C₃₁⋊C₃
hypocenter	C₃₁	#10624694
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. C₃×#14578892	C₃	semidirect
2. #21419232	C₃	semidirect
54. C₃₁	#10624694	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578892	3
#21419232	3
C₃×C₃₁×#737	3
#10624694	31