Subgroups — #36382945

Order 22599.

1118 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	C₃₁⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419396	C₃	1	24	yes	maximal, complemented
4	C₃×#14578894	C₃	1	12	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
6	#14578911	—	3	18	no	—
7	C₃×#11719689	C₃²	1	24	yes	complemented
8	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	12	yes	complemented
9	H(3,𝔽₃)×C₃₁⋊C₃	—	3	9	no	—
10	#14578894	C₃²	1	9	yes	complemented
11	#14578894	C₃²	1	72	yes	complemented
12	C₃⁴×C₃₁	C₃²	1	4	yes	complemented
13	C₃×H(3,𝔽₃)×C₃₁	—	3	9	no	—
14	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
15	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	8	yes	complemented
16	C₃³×C₃₁	—	3	12	no	—
17	C₃³×C₃₁	—	9	12	no	—
18	C₃³×C₃₁	C₃³	1	1	yes	characteristic
19	C₃³×C₃₁	C₃³	1	8	yes	—
20	C₃³×C₃₁	C₃³	1	4	yes	—
21	C₃²×C₃₁⋊C₃	—	3	12	no	—
22	H(3,𝔽₃)×C₃₁	—	3	24	no	—
23	H(3,𝔽₃)×C₃₁	H(3,𝔽₃)	1	9	yes	—
24	C₃²⋊(C₃₁⋊C₃)	—	3	24	no	—
25	C₃²⋊(C₃₁⋊C₃)	C₃³	1	9	yes	—
26	#11719689	—	3	48	no	—
27	#11719689	C₃³	1	18	yes	—
28	C₃×C₃₁⋊C₉	—	3	24	no	—
29	#10624930	—	31	1	no	maximal, Sylow
30	C₃²×C₃₁	—	3	3	no	—
31	C₃²×C₃₁	—	3	1	no	—
32	C₃²×C₃₁	—	3	8	no	—
33	C₃²×C₃₁	—	9	4	no	—
34	C₃²×C₃₁	—	9	36	no	—
35	C₃²×C₃₁	—	9	8	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	C₃₁⋊C₉	—	3	18	no	—
41	3₋¹⁺⁴	—	31	2	no	—
42	H(5,𝔽₃)	—	31	1	no	—
43	C₃²×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic, complemented
44	#6909	—	31	24	no	—
45	C₃×AΣL(1,𝔽₂₇)	—	31	12	no	—
46	C₃×C₃₁	—	3	12	no	—
47	C₃×C₃₁	—	3	3	no	—
48	C₃×C₃₁	—	3	1	no	—
49	C₃×C₃₁	—	9	8	no	—
50	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃₁⋊C₃	—	9	9	no	—
52	C₃⁴	C₃×C₃₁⋊C₃	1	4	yes	complemented
53	#737	—	93	18	no	—
54	C₃×3₋¹⁺²	—	31	24	no	—
55	C₃×H(3,𝔽₃)	—	3	9	no	—
56	C₃×H(3,𝔽₃)	—	31	12	no	—
57	C₃×H(3,𝔽₃)	—	93	9	no	—
58	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
59	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	8	yes	complemented
60	AΣL(1,𝔽₂₇)	—	31	9	no	—
61	AΣL(1,𝔽₂₇)	—	31	72	no	—
62	C₃₁	#10624930	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	3	12	no	—
64	C₃³	—	9	12	no	—
65	C₃³	—	93	12	no	—
66	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
67	C₃³	C₃²×C₃₁⋊C₃	1	8	yes	—
68	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
69	3₋¹⁺²	—	31	18	no	—
70	3₋¹⁺²	—	93	48	no	—
71	H(3,𝔽₃)	—	3	24	no	—
72	H(3,𝔽₃)	—	31	9	no	—
73	H(3,𝔽₃)	—	93	24	no	—
74	H(3,𝔽₃)	C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
75	C₃×C₉	—	93	24	no	—
76	C₃²	—	3	3	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	8	no	—
79	C₃²	—	9	4	no	—
80	C₃²	—	9	36	no	—
81	C₃²	—	9	8	no	—
82	C₃²	—	837	12	no	—
83	C₃²	—	93	9	no	—
84	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	12	yes	—
85	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
86	C₉	—	93	18	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	3	no	—
89	C₃	—	3	12	no	—
90	C₃	—	9	8	no	—
91	C₃	—	279	9	no	—
92	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	C₃²×H(3,𝔽₃)×C₃₁	C₃
socle	C₃×C₃₁	C₃²×H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	#10624930
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #21419438	C₃	semidirect
2. C₃₁⋊H(5,𝔽₃)	C₃	semidirect
3. #21419396	C₃	semidirect
4. C₃×#14578894	C₃	semidirect
5. C₃²×H(3,𝔽₃)×C₃₁	C₃	semidirect
7. C₃×#11719689	C₃²	semidirect
8. C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	semidirect
10. #14578894	C₃²	semidirect
11. #14578894	C₃²	semidirect
12. C₃⁴×C₃₁	C₃²	semidirect
15. C₃×H(3,𝔽₃)×C₃₁	C₃²	semidirect
43. C₃²×H(3,𝔽₃)	C₃₁⋊C₃	semidirect
52. C₃⁴	C₃×C₃₁⋊C₃	semidirect
59. C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	semidirect
62. C₃₁	#10624930	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
#21419396	3
C₃×#14578894	3
C₃²×H(3,𝔽₃)×C₃₁	3
#10624930	31