Subgroups — C₃×#21419320 #36382297

Order 22599.

1232 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	#21419320	C₃	1	27	yes	maximal, direct product factor
4	C₃×C₃²⋊(C₃₁⋊C₉)	C₃	1	9	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic
6	C₃×#11719689	—	3	18	no	—
7	C₃×#11719689	C₃²	1	18	yes	complemented
8	C₃²×C₃₁⋊C₉	—	3	9	no	—
9	C₃²×C₃₁⋊C₉	C₃²	1	3	yes	—
10	C₃²×C₃₁⋊C₉	C₃²	1	6	yes	—
11	C₃²×C₃₁⋊C₉	C₃²	1	9	yes	complemented
12	C₃²⋊(C₃₁⋊C₉)	C₃²	1	81	yes	complemented
13	C₃⁴×C₃₁	C₃²	1	3	yes	—
14	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
15	C₃×H(3,𝔽₃)×C₃₁	—	3	9	no	—
16	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	9	yes	—
17	C₃³×C₃₁	—	3	9	no	—
18	C₃³×C₃₁	—	3	9	no	—
19	C₃³×C₃₁	—	3	27	no	—
20	C₃³×C₃₁	H(3,𝔽₃)	1	9	yes	—
21	C₃³×C₃₁	C₃³	1	9	yes	—
22	C₃³×C₃₁	C₃³	1	3	yes	—
23	C₃³×C₃₁	C₃³	1	1	yes	characteristic
24	H(3,𝔽₃)×C₃₁	—	3	27	no	—
25	#11719689	—	3	54	no	—
26	C₃×C₃₁⋊C₉	—	3	9	no	—
27	C₃×C₃₁⋊C₉	—	3	27	no	—
28	C₃×C₃₁⋊C₉	—	3	18	no	—
29	C₃×C₃₁⋊C₉	C₃³	1	9	yes	—
30	C₃×C₃₁⋊C₉	C₃³	1	18	yes	—
31	C₃×#6893	—	31	1	no	maximal, Sylow
32	C₃²×C₃₁	—	3	9	no	—
33	C₃²×C₃₁	—	3	27	no	—
34	C₃²×C₃₁	—	3	9	no	—
35	C₃²×C₃₁	—	3	27	no	—
36	C₃²×C₃₁	—	9	27	no	—
37	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	9	yes	—
38	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	3	yes	—
39	C₃²×C₃₁	#737	1	9	yes	—
40	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
41	C₃₁⋊C₉	—	3	9	no	—
42	C₃₁⋊C₉	—	3	18	no	—
43	C₃²×3₋¹⁺²	—	31	2	no	—
44	C₃²×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
45	C₃³×C₉	—	31	1	no	—
46	#6893	—	31	27	no	—
47	C₃×C₃²⋊C₉	—	31	9	no	—
48	C₃×C₃₁	—	3	9	no	—
49	C₃×C₃₁	—	9	9	no	—
50	C₃×C₃₁	#6893	1	9	yes	complemented
51	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₃₁	C₃×#737	1	3	yes	—
53	C₃⁴	C₃×C₃₁⋊C₃	1	3	yes	—
54	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	31	18	no	—
56	C₃×3₋¹⁺²	—	93	18	no	—
57	C₃×H(3,𝔽₃)	—	3	9	no	—
58	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	9	yes	—
59	C₃²×C₉	—	31	9	no	—
60	C₃²×C₉	—	31	3	no	—
61	C₃²×C₉	—	31	6	no	—
62	C₃²×C₉	—	93	9	no	—
63	C₃²⋊C₉	—	31	81	no	—
64	C₃₁	C₃×#6893	1	1	yes	characteristic, complemented, Sylow
65	C₃³	—	3	9	no	—
66	C₃³	—	3	9	no	—
67	C₃³	—	3	27	no	—
68	C₃³	C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
69	C₃³	C₃²×C₃₁⋊C₃	1	9	yes	—
70	C₃³	C₃²×C₃₁⋊C₃	1	3	yes	—
71	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	93	54	no	—
73	H(3,𝔽₃)	—	3	27	no	—
74	C₃×C₉	—	31	9	no	—
75	C₃×C₉	—	31	18	no	—
76	C₃×C₉	—	93	27	no	—
77	C₃×C₉	—	93	9	no	—
78	C₃×C₉	—	93	18	no	—
79	C₃²	—	3	27	no	—
80	C₃²	—	3	9	no	—
81	C₃²	—	3	9	no	—
82	C₃²	—	3	27	no	—
83	C₃²	—	9	27	no	—
84	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
85	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
86	C₃²	#14578911	1	9	yes	—
87	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
88	C₉	—	93	9	no	—
89	C₉	—	93	18	no	—
90	C₃	—	3	9	no	—
91	C₃	—	9	9	no	—
92	C₃	#21419320	1	9	yes	direct product factor
93	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
94	C₃	C₃×#14578911	1	3	yes	—

	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	C₃³
hypercenter	C₃²×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6893
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ 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. #21419320	C₃	direct (N × H)
4. C₃×C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
7. C₃×#11719689	C₃²	semidirect
11. C₃²×C₃₁⋊C₉	C₃²	semidirect
12. C₃²⋊(C₃₁⋊C₉)	C₃²	semidirect
50. C₃×C₃₁	#6893	semidirect
64. C₃₁	C₃×#6893	semidirect
92. C₃	#21419320	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
#21419320	3
C₃×C₃²⋊(C₃₁⋊C₉)	3
C₃²×H(3,𝔽₃)×C₃₁	3
C₃×#6893	31

Subgroups — C3×#21419320 #36382297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419320 #36382297