Subgroups — #36380966

Order 22599.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578892	C₃	1	2	yes	maximal, complemented
2	C₃²×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₁×#737	C₃	1	1	yes	maximal, characteristic
5	#14578892	—	3	6	no	—
6	C₃×C₃₁⋊C₂₇	—	3	3	no	—
7	C₃×C₃₁⋊C₂₇	C₃²	1	3	yes	—
8	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
9	C₃₁×#737	—	3	3	no	—
10	C₃₁×#737	—	3	6	no	—
11	C₃×3₋¹⁺²×C₃₁	C₉	1	2	yes	—
12	C₃×3₋¹⁺²×C₃₁	C₉	1	6	yes	—
13	C₃×H(3,𝔽₃)×C₃₁	C₉	1	1	yes	characteristic
14	C₃²×C₉×C₃₁	C₃²	1	3	yes	—
15	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
16	C₃³×C₃₁	C₃×C₉	1	3	yes	—
17	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
18	3₋¹⁺²×C₃₁	—	3	6	no	—
19	3₋¹⁺²×C₃₁	—	3	18	no	—
20	H(3,𝔽₃)×C₃₁	—	3	3	no	—
21	C₃×C₉×C₃₁	—	3	3	no	—
22	C₃×C₉×C₃₁	—	3	6	no	—
23	C₃×C₉×C₃₁	C₃×C₉	1	6	yes	—
24	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
25	C₃×C₉×C₃₁	H(3,𝔽₃)	1	3	yes	—
26	C₃×C₉×C₃₁	3₋¹⁺²	1	6	yes	—
27	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
28	C₃₁⋊C₂₇	—	3	3	no	—
29	C₃₁⋊C₂₇	—	3	6	no	—
30	#10624684	—	31	1	no	maximal, Sylow
31	C₃²×C₃₁	—	3	3	no	—
32	C₃²×C₃₁	—	3	3	no	—
33	C₃²×C₃₁	—	3	9	no	—
34	C₃²×C₃₁	C₃²⋊C₉	1	3	yes	—
35	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
36	C₉×C₃₁	—	3	6	no	—
37	C₉×C₃₁	C₃²⋊C₉	1	6	yes	—
38	C₉×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₉×C₃₁	C₃×3₋¹⁺²	1	2	yes	—
40	C₃×#737	C₃₁⋊C₃	1	1	yes	characteristic
41	C₃×C₂₇⋊C₃	—	31	2	no	—
42	C₃²×C₂₇	—	31	1	no	—
43	C₃²⋊C₂₇	—	31	9	no	—
44	C₃×C₃₁	—	3	3	no	—
45	C₃×C₃₁	—	9	3	no	—
46	C₃×C₃₁	C₃×C₃²⋊C₉	1	1	yes	characteristic
47	C₃×C₃₁	#6903	1	3	yes	—
48	#737	—	3	6	no	—
49	#737	—	3	3	no	—
50	C₃×3₋¹⁺²	C₃₁⋊C₉	1	2	yes	—
51	C₃×3₋¹⁺²	C₃₁⋊C₉	1	6	yes	—
52	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	1	yes	characteristic
53	C₃²×C₉	C₃×C₃₁⋊C₃	1	3	yes	—
54	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
55	C₂₇⋊C₃	—	93	6	no	—
56	C₃×C₂₇	—	31	3	no	—
57	C₃×C₂₇	—	31	6	no	—
58	C₃×C₂₇	—	93	3	no	—
59	C₃₁	#10624684	1	1	yes	characteristic, complemented, Sylow
60	C₃³	C₃×C₃₁⋊C₉	1	3	yes	—
61	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
62	3₋¹⁺²	—	3	6	no	—
63	3₋¹⁺²	—	3	18	no	—
64	H(3,𝔽₃)	—	3	3	no	—
65	C₃×C₉	—	3	3	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	C₃×C₃₁⋊C₉	1	6	yes	—
68	C₃×C₉	C₃×C₃₁⋊C₉	1	2	yes	—
69	C₃×C₉	#11719689	1	6	yes	—
70	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
71	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
72	C₂₇	—	93	3	no	—
73	C₂₇	—	93	6	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	9	no	—
77	C₃²	C₃²⋊(C₃₁⋊C₉)	1	3	yes	—
78	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
79	C₉	—	3	6	no	—
80	C₉	C₃²⋊(C₃₁⋊C₉)	1	6	yes	—
81	C₉	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
82	C₉	C₃×#11719689	1	2	yes	—
83	C₃	—	3	3	no	—
84	C₃	—	9	3	no	—
85	C₃	C₃×C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
86	C₃	#21419371	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃²×C₉
Fitting	C₃×C₃₁×#737	C₃
prime radical	C₃×H(3,𝔽₃)×C₃₁	C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₃×#737	C₃₁⋊C₃
hypocenter	C₃₁	#10624684
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×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. C₃²×C₃₁⋊C₂₇	C₃	semidirect
3. C₃²⋊(C₃₁⋊C₂₇)	C₃	semidirect
59. C₃₁	#10624684	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
C₃²×C₃₁⋊C₂₇	3
C₃²⋊(C₃₁⋊C₂₇)	3
C₃×C₃₁×#737	3
#10624684	31