Subgroups — #36382154

Order 22599.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419391	C₃	1	3	yes	maximal, complemented
2	#21419281	C₃	1	3	yes	maximal, complemented
3	#21419276	C₃	1	3	yes	maximal, complemented
4	#21419271	C₃	1	3	yes	maximal, complemented
5	C₃₁×#6887	C₃	1	1	yes	maximal, characteristic, complemented
6	#14578912	—	3	3	no	—
7	#14578902	C₃²	1	3	yes	—
8	#14578902	C₃²	1	3	yes	—
9	#14578900	C₃²	1	3	yes	—
10	#14578897	—	3	3	no	—
11	#14578894	—	3	3	no	—
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃₁×C₉⋊C₉	—	3	1	no	—
14	C₃₁×C₉⋊C₉	C₃²	1	3	yes	—
15	C₃₁×C₃²⋊C₉	—	3	1	no	—
16	C₉²×C₃₁	—	3	1	no	—
17	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×C₉×C₃₁	—	3	3	no	—
19	C₃×C₉×C₃₁	—	3	3	no	—
20	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
23	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
24	#11719689	—	3	3	no	—
25	#11719689	—	3	3	no	—
26	#11719687	—	9	3	no	—
27	#11719686	—	9	3	no	—
28	C₉×C₃₁⋊C₃	—	9	3	no	—
29	#10624849	—	31	1	no	maximal, Sylow
30	C₃²×C₃₁	—	3	1	no	—
31	C₃²×C₃₁	—	9	1	no	—
32	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₃₁⋊C₃	—	9	3	no	—
34	C₉×C₃₁	—	3	1	no	—
35	C₉×C₃₁	—	3	1	no	—
36	C₉×C₃₁	—	9	3	no	—
37	C₉×C₃₁	#732	1	3	yes	—
38	C₃₁⋊C₉	—	9	3	no	—
39	C₃₁⋊C₉	—	9	3	no	—
40	#6908	—	31	3	no	—
41	#6887	C₃₁⋊C₃	1	1	yes	characteristic, complemented
42	#6883	—	31	3	no	—
43	#6882	—	31	3	no	—
44	#6881	—	31	3	no	—
45	C₃×C₃₁	—	3	1	no	—
46	C₃×C₃₁	—	9	1	no	—
47	C₃×C₃₁	C₃×#732	1	1	yes	characteristic
48	C₃₁⋊C₃	—	27	3	no	—
49	#737	—	93	3	no	—
50	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
51	#733	—	31	3	no	—
52	#733	—	31	3	no	—
53	#732	—	31	3	no	—
54	#731	—	93	3	no	—
55	AΣL(1,𝔽₂₇)	—	93	3	no	—
56	C₉⋊C₉	—	3	1	no	—
57	C₉⋊C₉	C₃×C₃₁⋊C₃	1	3	yes	—
58	C₃²⋊C₉	—	3	1	no	—
59	C₉²	—	3	1	no	—
60	C₃₁	#10624849	1	1	yes	characteristic, complemented, Sylow
61	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
62	3₋¹⁺²	—	279	3	no	—
63	3₋¹⁺²	—	279	3	no	—
64	3₋¹⁺²	—	93	3	no	—
65	3₋¹⁺²	—	93	3	no	—
66	H(3,𝔽₃)	—	93	3	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	279	3	no	—
70	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
71	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
72	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
73	C₃²	—	3	1	no	—
74	C₃²	—	9	1	no	—
75	C₃²	—	279	3	no	—
76	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
77	C₉	—	3	1	no	—
78	C₉	—	3	1	no	—
79	C₉	—	9	3	no	—
80	C₉	—	279	3	no	—
81	C₉	—	279	3	no	—
82	C₉	#14578900	1	3	yes	—
83	C₃	—	3	1	no	—
84	C₃	—	9	1	no	—
85	C₃	—	837	3	no	—
86	C₃	C₃×#14578900	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×#14578900
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6887	C₃
socle	C₃×C₃₁	C₃×#732
hypercenter	#6887	C₃₁⋊C₃
hypocenter	C₃₁	#10624849
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. #21419391	C₃	semidirect
2. #21419281	C₃	semidirect
3. #21419276	C₃	semidirect
4. #21419271	C₃	semidirect
5. C₃₁×#6887	C₃	semidirect
41. #6887	C₃₁⋊C₃	semidirect
60. C₃₁	#10624849	semidirect

Maximal subgroups

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

subgroup	index
#21419391	3
#21419281	3
#21419276	3
#21419271	3
C₃₁×#6887	3
#10624849	31