Subgroups — #36381118

Order 22599.

256 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	C₃²×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	2	yes	maximal, complemented
4	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	6	yes	maximal, complemented
5	C₉×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic
6	#14578892	—	3	9	no	—
7	C₃×C₃₁⋊C₂₇	—	3	3	no	—
8	C₃×C₃₁⋊C₂₇	C₃²	1	2	yes	—
9	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
10	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
11	C₃×H(3,𝔽₃)×C₃₁	C₉	1	1	yes	characteristic
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₃₁	C₃²	1	3	yes	—
14	C₃₁×C₃²⋊C₉	C₉	1	6	yes	—
15	C₃₁×C₃²⋊C₉	C₉	1	2	yes	—
16	C₃³×C₃₁	C₃×C₉	1	3	yes	—
17	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
18	H(3,𝔽₃)×C₃₁	—	3	3	no	—
19	C₃×C₉×C₃₁	—	3	1	no	—
20	C₃×C₉×C₃₁	—	3	3	no	—
21	C₃×C₉×C₃₁	—	3	2	no	—
22	C₃×C₉×C₃₁	—	3	6	no	—
23	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
24	C₃×C₉×C₃₁	C₃×C₉	1	6	yes	—
25	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
26	C₃₁⋊C₂₇	—	3	2	no	—
27	C₃₁⋊C₂₇	—	3	6	no	—
28	C₃₁⋊C₂₇	H(3,𝔽₃)	1	3	yes	complemented
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	—	3	3	no	—
33	C₃²×C₃₁	—	3	3	no	—
34	C₃²×C₃₁	—	3	6	no	—
35	C₃²×C₃₁	C₂₇⋊C₃	1	3	yes	complemented
36	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
37	C₉×C₃₁	—	3	2	no	—
38	C₉×C₃₁	—	3	6	no	—
39	C₉×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₉×C₃₁	#737	1	2	yes	—
41	C₃×C₂₇⋊C₃	—	31	3	no	—
42	C₃²×C₂₇	—	31	1	no	—
43	C₉×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
44	C₃²⋊C₂₇	—	31	6	no	—
45	C₃²⋊C₂₇	—	31	2	no	—
46	C₃×C₃₁	—	3	3	no	—
47	C₃×C₃₁	—	9	3	no	—
48	C₃×C₃₁	C₉×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₃₁	C₃×C₂₇⋊C₃	1	1	yes	characteristic
50	C₃×C₃₁	#6903	1	2	yes	—
51	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	1	yes	characteristic
52	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
53	C₃²×C₉	C₃×C₃₁⋊C₃	1	3	yes	—
54	C₂₇⋊C₃	—	93	9	no	—
55	C₃×C₂₇	—	31	1	no	—
56	C₃×C₂₇	—	31	2	no	—
57	C₃×C₂₇	—	31	6	no	—
58	C₃×C₂₇	—	93	3	no	—
59	C₃²⋊C₉	C₃₁⋊C₉	1	6	yes	—
60	C₃²⋊C₉	C₃₁⋊C₉	1	2	yes	—
61	C₃₁	C₂₇⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
62	C₃³	C₃×C₃₁⋊C₉	1	3	yes	—
63	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
64	H(3,𝔽₃)	—	3	3	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	3	no	—
67	C₃×C₉	—	3	2	no	—
68	C₃×C₉	—	3	6	no	—
69	C₃×C₉	C₃×C₃₁⋊C₉	1	2	yes	—
70	C₃×C₉	C₃×C₃₁⋊C₉	1	6	yes	—
71	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
72	C₂₇	—	31	3	no	—
73	C₂₇	—	93	2	no	—
74	C₂₇	—	93	6	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	6	no	—
80	C₃²	#14578892	1	3	yes	complemented
81	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
82	C₉	—	3	2	no	—
83	C₉	—	3	6	no	—
84	C₉	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
85	C₉	#14578911	1	2	yes	—
86	C₃	—	3	3	no	—
87	C₃	—	9	3	no	—
88	C₃	H(3,𝔽₃)×C₃₁⋊C₉	1	1	yes	characteristic
89	C₃	C₃×#14578892	1	1	yes	characteristic
90	C₃	#21419371	1	2	yes	—

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

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
4. C₃²⋊(C₃₁⋊C₂₇)	C₃	semidirect
28. C₃₁⋊C₂₇	H(3,𝔽₃)	semidirect
35. C₃²×C₃₁	C₂₇⋊C₃	semidirect
61. C₃₁	C₂₇⋊H(3,𝔽₃)	semidirect
80. C₃²	#14578892	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
C₉×H(3,𝔽₃)×C₃₁	3
C₂₇⋊H(3,𝔽₃)	31