Subgroups — C₃²×#14578896 #36382882

Order 22599.

1160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11719689	C₃	1	2	yes	maximal, complemented
2	C₃³×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#14578896	C₃	1	36	yes	maximal, direct product factor
4	C₃²×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic
5	C₃×#11719689	—	3	16	no	—
6	C₃×#11719689	C₃²	1	24	yes	complemented
7	C₃²×C₃₁⋊C₉	—	3	8	no	—
8	C₃²×C₃₁⋊C₉	—	3	1	no	—
9	C₃²×C₃₁⋊C₉	—	3	2	no	—
10	C₃²×C₃₁⋊C₉	C₃²	1	12	yes	complemented
11	#14578896	C₃²	1	81	yes	direct product factor
12	C₃⁴×C₃₁	—	3	1	no	—
13	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
14	C₃×H(3,𝔽₃)×C₃₁	—	3	8	no	—
15	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	12	yes	—
16	C₃³×C₃₁	—	3	9	no	—
17	C₃³×C₃₁	—	3	12	no	—
18	C₃³×C₃₁	—	9	9	no	—
19	C₃³×C₃₁	H(3,𝔽₃)	1	8	yes	—
20	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃³×C₃₁	C₃³	1	4	yes	—
22	H(3,𝔽₃)×C₃₁	—	3	24	no	—
23	H(3,𝔽₃)×C₃₁	C₃³	1	9	yes	—
24	#11719689	—	3	48	no	—
25	#11719689	C₃³	1	18	yes	complemented
26	C₃×C₃₁⋊C₉	—	3	24	no	—
27	C₃×C₃₁⋊C₉	—	3	12	no	—
28	C₃×C₃₁⋊C₉	—	3	24	no	—
29	C₃×C₃₁⋊C₉	C₃³	1	9	yes	complemented
30	C₃²×#731	—	31	1	no	maximal, Sylow
31	C₃²×C₃₁	—	3	24	no	—
32	C₃²×C₃₁	—	3	12	no	—
33	C₃²×C₃₁	—	3	9	no	—
34	C₃²×C₃₁	—	9	36	no	—
35	C₃²×C₃₁	#731	1	9	yes	complemented
36	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	8	yes	—
37	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	4	yes	—
38	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
39	C₃₁⋊C₉	—	3	9	no	—
40	C₃₁⋊C₉	—	3	18	no	—
41	C₃²×3₋¹⁺²	—	31	2	no	—
42	C₃²×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
43	C₃³×C₉	—	31	1	no	—
44	C₃×#731	—	31	36	no	—
45	C₃×C₃₁	—	3	8	no	—
46	C₃×C₃₁	—	3	1	no	—
47	C₃×C₃₁	—	9	9	no	—
48	C₃×C₃₁	C₃×#731	1	12	yes	complemented
49	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃⁴	—	3	1	no	—
51	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	31	24	no	—
53	C₃×3₋¹⁺²	—	93	16	no	—
54	C₃×H(3,𝔽₃)	—	3	8	no	—
55	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	12	yes	—
56	C₃²×C₉	—	31	12	no	—
57	C₃²×C₉	—	93	8	no	—
58	C₃²×C₉	—	93	1	no	—
59	C₃²×C₉	—	93	2	no	—
60	#731	—	31	81	no	—
61	C₃₁	C₃²×#731	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	3	9	no	—
63	C₃³	—	3	12	no	—
64	C₃³	—	9	9	no	—
65	C₃³	C₃²⋊(C₃₁⋊C₃)	1	8	yes	—
66	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
67	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
68	3₋¹⁺²	—	31	18	no	—
69	3₋¹⁺²	—	93	48	no	—
70	H(3,𝔽₃)	—	3	24	no	—
71	H(3,𝔽₃)	C₃²×C₃₁⋊C₃	1	9	yes	—
72	C₃×C₉	—	31	9	no	—
73	C₃×C₉	—	93	24	no	—
74	C₃×C₉	—	93	12	no	—
75	C₃×C₉	—	93	24	no	—
76	C₃²	—	3	24	no	—
77	C₃²	—	3	12	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	9	36	no	—
80	C₃²	#14578896	1	9	yes	direct product factor
81	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	8	yes	—
82	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	4	yes	—
83	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
84	C₉	—	93	9	no	—
85	C₉	—	93	18	no	—
86	C₃	—	3	8	no	—
87	C₃	—	3	1	no	—
88	C₃	—	9	9	no	—
89	C₃	C₃×#14578896	1	12	yes	direct product factor
90	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃²⋊(C₃₁⋊C₃)
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₁	C₃
socle	C₃³×C₃₁	H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	C₃²×#731
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×#11719689	C₃	semidirect
2. C₃³×C₃₁⋊C₉	C₃	semidirect
3. C₃×#14578896	C₃	direct (N × H)
6. C₃×#11719689	C₃²	semidirect
10. C₃²×C₃₁⋊C₉	C₃²	semidirect
11. #14578896	C₃²	direct (N × H)
25. #11719689	C₃³	semidirect
29. C₃×C₃₁⋊C₉	C₃³	semidirect
35. C₃²×C₃₁	#731	semidirect
48. C₃×C₃₁	C₃×#731	semidirect
61. C₃₁	C₃²×#731	semidirect
80. C₃²	#14578896	direct (N × H)
89. C₃	C₃×#14578896	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#11719689	3
C₃³×C₃₁⋊C₉	3
C₃×#14578896	3
C₃²×H(3,𝔽₃)×C₃₁	3
C₃²×#731	31

Subgroups — C32×#14578896 #36382882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#14578896 #36382882