Subgroups — #36381116

Order 22599.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578891	C₃	1	3	yes	maximal, complemented
2	C₉×C₃²⋊(C₃₁⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	2	yes	maximal, complemented
4	(C₃×C₃₁⋊C₃)⋊C₂₇	C₃	1	6	yes	maximal, complemented
5	C₃²×C₂₇×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×C₃²⋊(C₃₁⋊C₃)	C₉	1	1	yes	characteristic
7	C₃×C₉×C₃₁⋊C₃	C₃²	1	3	yes	—
8	#14578891	—	3	9	no	—
9	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
10	C₃²⋊(C₃₁⋊C₉)	C₉	1	2	yes	—
11	(C₃×C₃₁⋊C₃)⋊C₉	C₉	1	6	yes	—
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃×C₂₇×C₃₁	—	3	3	no	—
14	C₃×C₂₇×C₃₁	C₃²	1	2	yes	—
15	C₃×C₂₇×C₃₁	C₃²	1	1	yes	characteristic
16	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
17	C₃²×C₃₁⋊C₃	C₃×C₉	1	3	yes	—
18	C₃×C₉×C₃₁	—	3	1	no	—
19	C₃×C₉×C₃₁	—	3	2	no	—
20	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
21	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
22	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
23	C₃×C₃₁⋊C₉	C₃×C₉	1	6	yes	—
24	C₉×C₃₁⋊C₃	—	3	3	no	—
25	C₉×C₃₁⋊C₃	—	3	6	no	—
26	C₂₇×C₃₁	—	3	2	no	—
27	C₂₇×C₃₁	H(3,𝔽₃)	1	3	yes	complemented
28	C₃₁⋊C₂₇	—	3	6	no	—
29	C₂₇⋊H(3,𝔽₃)	—	31	1	no	maximal, Sylow
30	C₃²×C₃₁	—	3	1	no	—
31	C₃²×C₃₁	—	3	2	no	—
32	C₃²×C₃₁	C₂₇⋊C₃	1	3	yes	complemented
33	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
34	C₃×C₃₁⋊C₃	—	3	3	no	—
35	C₃×C₃₁⋊C₃	—	3	3	no	—
36	C₃×C₃₁⋊C₃	—	3	6	no	—
37	C₉×C₃₁	—	3	2	no	—
38	C₉×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₉×C₃₁	#737	1	2	yes	—
40	C₃₁⋊C₉	—	3	6	no	—
41	C₃×C₂₇⋊C₃	—	31	3	no	—
42	C₃²×C₂₇	C₃₁⋊C₃	1	1	yes	characteristic, complemented
43	C₉×H(3,𝔽₃)	—	31	1	no	—
44	C₃²⋊C₂₇	—	31	2	no	—
45	C₃²⋊C₂₇	—	31	6	no	—
46	C₃×C₃₁	—	3	3	no	—
47	C₃×C₃₁	C₉×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₃₁	C₃×C₂₇⋊C₃	1	1	yes	characteristic
49	C₃×C₃₁	#6903	1	2	yes	—
50	C₃₁⋊C₃	—	9	3	no	—
51	C₃×H(3,𝔽₃)	—	31	1	no	—
52	C₃²×C₉	—	31	3	no	—
53	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
54	C₂₇⋊C₃	—	93	9	no	—
55	C₃×C₂₇	—	3	3	no	—
56	C₃×C₂₇	—	31	6	no	—
57	C₃×C₂₇	C₃×C₃₁⋊C₃	1	2	yes	—
58	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
59	C₃²⋊C₉	—	31	2	no	—
60	C₃²⋊C₉	—	31	6	no	—
61	C₃₁	C₂₇⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	31	3	no	—
63	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
64	H(3,𝔽₃)	—	93	3	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	31	6	no	—
68	C₃×C₉	—	93	3	no	—
69	C₃×C₉	—	93	6	no	—
70	C₃×C₉	C₉×C₃₁⋊C₃	1	2	yes	—
71	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
72	C₂₇	—	3	2	no	—
73	C₂₇	—	93	6	no	—
74	C₂₇	C₃²⋊(C₃₁⋊C₃)	1	3	yes	complemented
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	93	3	no	—
78	C₃²	—	93	3	no	—
79	C₃²	—	93	6	no	—
80	C₃²	#14578891	1	3	yes	complemented
81	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
82	C₉	—	3	2	no	—
83	C₉	—	93	6	no	—
84	C₉	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
85	C₉	#14578912	1	2	yes	—
86	C₃	—	3	3	no	—
87	C₃	—	279	3	no	—
88	C₃	C₉×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
89	C₃	C₃×#14578891	1	1	yes	characteristic
90	C₃	#21419370	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×#14578891	C₃	semidirect
3. C₃²⋊(C₃₁⋊C₂₇)	C₃	semidirect
4. (C₃×C₃₁⋊C₃)⋊C₂₇	C₃	semidirect
5. C₃²×C₂₇×C₃₁	C₃	semidirect
27. C₂₇×C₃₁	H(3,𝔽₃)	semidirect
32. C₃²×C₃₁	C₂₇⋊C₃	semidirect
42. C₃²×C₂₇	C₃₁⋊C₃	semidirect
61. C₃₁	C₂₇⋊H(3,𝔽₃)	semidirect
74. C₂₇	C₃²⋊(C₃₁⋊C₃)	semidirect
80. C₃²	#14578891	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578891	3
C₉×C₃²⋊(C₃₁⋊C₃)	3
C₃²⋊(C₃₁⋊C₂₇)	3
(C₃×C₃₁⋊C₃)⋊C₂₇	3
C₃²×C₂₇×C₃₁	3
C₂₇⋊H(3,𝔽₃)	31