Subgroups — #36382948

Order 22599.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419438	C₃	1	2	yes	maximal, complemented
2	C₃×#14578911	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419400	C₃	1	24	yes	maximal, complemented
4	C₃×#14578896	C₃	1	12	yes	maximal, complemented
5	C₃₁×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	#14578911	—	3	9	no	—
7	#14578911	—	3	18	no	—
8	C₃×#11719689	C₃²	1	24	yes	complemented
9	C₃×#11719689	C₃²	1	8	yes	complemented
10	C₃²×C₃₁⋊C₉	C₃²	1	4	yes	complemented
11	#14578896	C₃²	1	9	yes	complemented
12	#14578896	C₃²	1	72	yes	complemented
13	C₃×H(3,𝔽₃)×C₃₁	—	3	9	no	—
14	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	12	yes	—
15	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
16	C₃³×C₃₁	—	3	12	no	—
17	C₃³×C₃₁	C₃³	1	4	yes	—
18	H(3,𝔽₃)×C₃₁	—	3	24	no	—
19	H(3,𝔽₃)×C₃₁	H(3,𝔽₃)	1	9	yes	—
20	H(3,𝔽₃)×C₃₁	C₃³	1	9	yes	—
21	#11719689	—	3	48	no	—
22	#11719689	—	3	24	no	—
23	#11719689	C₃³	1	18	yes	—
24	C₃×C₃₁⋊C₉	—	3	12	no	—
25	C₃×C₃₁⋊C₉	—	3	24	no	—
26	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
27	C₃×C₃₁⋊C₉	C₃³	1	8	yes	—
28	#10624931	—	31	1	no	maximal, Sylow
29	C₃²×C₃₁	—	3	9	no	—
30	C₃²×C₃₁	—	9	4	no	—
31	C₃²×C₃₁	—	27	12	no	—
32	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	12	yes	—
33	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
34	C₃₁⋊C₉	—	3	1	no	—
35	C₃₁⋊C₉	—	3	18	no	—
36	C₃₁⋊C₉	—	3	8	no	—
37	3₋¹⁺⁴	—	31	2	no	—
38	H(5,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
39	C₃×#737	—	31	1	no	—
40	#6910	—	31	24	no	—
41	C₃×#731	—	31	12	no	—
42	C₃×C₃₁	—	3	1	no	—
43	C₃×C₃₁	—	3	12	no	—
44	C₃×C₃₁	—	9	9	no	—
45	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
46	#737	—	93	9	no	—
47	#737	—	93	18	no	—
48	C₃×3₋¹⁺²	—	31	8	no	—
49	C₃×3₋¹⁺²	—	31	24	no	—
50	C₃×H(3,𝔽₃)	—	3	9	no	—
51	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	12	yes	—
52	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
53	C₃²×C₉	—	31	4	no	—
54	#731	—	31	9	no	—
55	#731	—	31	72	no	—
56	C₃₁	#10624931	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	3	12	no	—
58	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
59	3₋¹⁺²	—	31	18	no	—
60	3₋¹⁺²	—	93	24	no	—
61	3₋¹⁺²	—	93	48	no	—
62	H(3,𝔽₃)	—	3	24	no	—
63	H(3,𝔽₃)	C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
64	H(3,𝔽₃)	C₃²×C₃₁⋊C₃	1	9	yes	—
65	C₃×C₉	—	31	1	no	—
66	C₃×C₉	—	31	8	no	—
67	C₃×C₉	—	93	12	no	—
68	C₃×C₉	—	93	24	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	9	4	no	—
71	C₃²	—	27	12	no	—
72	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	12	yes	—
73	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
74	C₉	—	93	1	no	—
75	C₉	—	93	8	no	—
76	C₉	—	93	18	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	12	no	—
79	C₃	—	9	9	no	—
80	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, prime radical	C₃₁×H(5,𝔽₃)	C₃
socle	C₃×C₃₁	C₃²×H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	#10624931
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #21419438	C₃	semidirect
2. C₃×#14578911	C₃	semidirect
3. #21419400	C₃	semidirect
4. C₃×#14578896	C₃	semidirect
8. C₃×#11719689	C₃²	semidirect
9. C₃×#11719689	C₃²	semidirect
10. C₃²×C₃₁⋊C₉	C₃²	semidirect
11. #14578896	C₃²	semidirect
12. #14578896	C₃²	semidirect
56. 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
C₃×#14578911	3
#21419400	3
C₃×#14578896	3
C₃₁×H(5,𝔽₃)	3
#10624931	31