Subgroups — #36382951

Order 22599.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419438	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃₁⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#14578911	C₃	1	1	yes	maximal, characteristic, complemented
4	#21419402	C₃	1	24	yes	maximal, complemented
5	C₃×#14578898	C₃	1	12	yes	maximal, complemented
6	C₃₁×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
7	#14578911	—	3	9	no	—
8	#14578911	—	3	9	no	—
9	C₃×#11719689	C₃²	1	12	yes	complemented
10	C₃×#11719689	C₃²	1	8	yes	complemented
11	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	12	yes	—
12	H(3,𝔽₃)×C₃₁⋊C₃	—	3	9	no	—
13	C₃²×C₃₁⋊C₉	C₃²	1	4	yes	complemented
14	#14578898	C₃²	1	9	yes	complemented
15	#14578898	C₃²	1	72	yes	complemented
16	C₃₁×#737	—	3	9	no	—
17	C₃×3₋¹⁺²×C₃₁	C₃²	1	12	yes	complemented
18	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
19	C₃³×C₃₁	C₃³	1	4	yes	—
20	C₃²×C₃₁⋊C₃	—	3	12	no	—
21	3₋¹⁺²×C₃₁	—	3	24	no	—
22	3₋¹⁺²×C₃₁	C₃³	1	9	yes	—
23	H(3,𝔽₃)×C₃₁	H(3,𝔽₃)	1	9	yes	—
24	C₃×C₉×C₃₁	—	3	12	no	—
25	C₃²⋊(C₃₁⋊C₃)	—	3	24	no	—
26	C₃²⋊(C₃₁⋊C₃)	C₃³	1	9	yes	—
27	#11719689	—	3	24	no	—
28	#11719689	—	3	24	no	—
29	#11719689	C₃³	1	9	yes	—
30	C₃×C₃₁⋊C₉	—	3	12	no	—
31	C₃×C₃₁⋊C₉	—	3	12	no	—
32	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
33	C₃×C₃₁⋊C₉	C₃³	1	8	yes	—
34	#10624931	—	31	1	no	maximal, Sylow
35	C₃²×C₃₁	—	9	4	no	—
36	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	12	yes	—
37	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
38	C₃×C₃₁⋊C₃	—	3	9	no	—
39	C₃×C₃₁⋊C₃	—	27	12	no	—
40	C₉×C₃₁	—	3	9	no	—
41	C₃₁⋊C₉	—	3	1	no	—
42	C₃₁⋊C₉	—	3	9	no	—
43	C₃₁⋊C₉	—	3	8	no	—
44	3₋¹⁺⁴	—	31	1	no	—
45	3₋¹⁺⁴	C₃₁⋊C₃	1	1	yes	characteristic, complemented
46	H(5,𝔽₃)	—	31	1	no	—
47	C₃×#737	—	31	1	no	—
48	#6910	—	31	24	no	—
49	C₃×#731	—	31	12	no	—
50	C₃×C₃₁	—	3	12	no	—
51	C₃×C₃₁	—	3	1	no	—
52	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃₁⋊C₃	—	9	9	no	—
54	#737	—	3	9	no	—
55	#737	—	93	9	no	—
56	#737	—	93	9	no	—
57	C₃×3₋¹⁺²	—	31	8	no	—
58	C₃×3₋¹⁺²	—	31	12	no	—
59	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	12	yes	complemented
60	C₃×H(3,𝔽₃)	—	31	12	no	—
61	C₃×H(3,𝔽₃)	—	93	9	no	—
62	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
63	C₃²×C₉	—	31	4	no	—
64	#731	—	31	9	no	—
65	#731	—	31	72	no	—
66	C₃₁	#10624931	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	93	12	no	—
68	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
69	3₋¹⁺²	—	3	24	no	—
70	3₋¹⁺²	—	31	9	no	—
71	3₋¹⁺²	—	93	24	no	—
72	3₋¹⁺²	—	93	24	no	—
73	3₋¹⁺²	C₃²×C₃₁⋊C₃	1	9	yes	—
74	H(3,𝔽₃)	—	31	9	no	—
75	H(3,𝔽₃)	—	93	24	no	—
76	H(3,𝔽₃)	C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
77	C₃×C₉	—	3	12	no	—
78	C₃×C₉	—	31	1	no	—
79	C₃×C₉	—	31	8	no	—
80	C₃×C₉	—	93	12	no	—
81	C₃×C₉	—	93	12	no	—
82	C₃²	—	9	4	no	—
83	C₃²	—	837	12	no	—
84	C₃²	—	93	9	no	—
85	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	12	yes	—
86	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
87	C₉	—	3	9	no	—
88	C₉	—	93	1	no	—
89	C₉	—	93	8	no	—
90	C₉	—	93	9	no	—
91	C₃	—	3	1	no	—
92	C₃	—	3	12	no	—
93	C₃	—	279	9	no	—
94	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃₁×3₋¹⁺⁴	C₃
socle	C₃×C₃₁	C₃²×H(3,𝔽₃)
prime radical	C₃₁⋊H(5,𝔽₃)	C₃
hypercenter	3₋¹⁺⁴	C₃₁⋊C₃
hypocenter	C₃₁	#10624931
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ 3₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #21419438	C₃	semidirect
3. C₃×#14578911	C₃	semidirect
4. #21419402	C₃	semidirect
5. C₃×#14578898	C₃	semidirect
6. C₃₁×3₋¹⁺⁴	C₃	semidirect
9. C₃×#11719689	C₃²	semidirect
10. C₃×#11719689	C₃²	semidirect
13. C₃²×C₃₁⋊C₉	C₃²	semidirect
14. #14578898	C₃²	semidirect
15. #14578898	C₃²	semidirect
17. C₃×3₋¹⁺²×C₃₁	C₃²	semidirect
45. 3₋¹⁺⁴	C₃₁⋊C₃	semidirect
59. C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	semidirect
66. C₃₁	#10624931	semidirect

Maximal subgroups

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

subgroup	index
#21419438	3
C₃₁⋊H(5,𝔽₃)	3
C₃×#14578911	3
#21419402	3
C₃×#14578898	3
C₃₁×3₋¹⁺⁴	3
#10624931	31