Group #36379875 — subgroups

Subgroups — #36379875

Order 22599.

842 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊(C₃₁⋊C₉)	C₃	1	9	yes	maximal, complemented
3	C₃×C₃²⋊(C₃₁⋊C₉)	C₃	1	2	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic
5	C₃²×C₃₁⋊C₉	—	3	3	no	—
6	C₃²×C₃₁⋊C₉	—	3	9	no	—
7	C₃²×C₃₁⋊C₉	C₃²	1	6	yes	—
8	C₃²×C₃₁⋊C₉	C₃²	1	3	yes	—
9	C₃²⋊(C₃₁⋊C₉)	—	3	6	no	—
10	C₃²⋊(C₃₁⋊C₉)	—	3	27	no	—
11	C₃⁴×C₃₁	C₃²	1	3	yes	—
12	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
13	C₃×H(3,𝔽₃)×C₃₁	—	3	3	no	—
14	C₃×H(3,𝔽₃)×C₃₁	—	3	6	no	—
15	C₃×H(3,𝔽₃)×C₃₁	C₉	1	9	yes	complemented
16	C₃³×C₃₁	—	3	3	no	—
17	C₃³×C₃₁	—	3	6	no	—
18	C₃³×C₃₁	—	3	3	no	—
19	C₃³×C₃₁	—	3	9	no	—
20	C₃³×C₃₁	—	3	6	no	—
21	C₃³×C₃₁	—	3	18	no	—
22	C₃³×C₃₁	C₃×C₉	1	3	yes	—
23	C₃³×C₃₁	C₃×C₉	1	9	yes	complemented
24	C₃³×C₃₁	H(3,𝔽₃)	1	3	yes	—
25	C₃³×C₃₁	3₋¹⁺²	1	6	yes	—
26	C₃³×C₃₁	C₃³	1	1	yes	characteristic
27	H(3,𝔽₃)×C₃₁	—	3	27	no	—
28	C₃×C₃₁⋊C₉	—	3	27	no	—
29	C₃×C₃₁⋊C₉	—	3	6	no	—
30	C₃×C₃₁⋊C₉	—	3	3	no	—
31	C₃×C₃₁⋊C₉	—	3	18	no	—
32	C₃×C₃₁⋊C₉	—	3	6	no	—
33	C₃×C₃₁⋊C₉	H(3,𝔽₃)	1	9	yes	complemented
34	#10624555	—	31	1	no	maximal, Sylow
35	C₃²×C₃₁	—	3	9	no	—
36	C₃²×C₃₁	—	3	3	no	—
37	C₃²×C₃₁	—	3	18	no	—
38	C₃²×C₃₁	—	3	6	no	—
39	C₃²×C₃₁	—	3	9	no	—
40	C₃²×C₃₁	—	3	27	no	—
41	C₃²×C₃₁	—	9	3	no	—
42	C₃²×C₃₁	—	9	18	no	—
43	C₃²×C₃₁	—	9	6	no	—
44	C₃²×C₃₁	C₃²⋊C₉	1	9	yes	complemented
45	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
46	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	3	yes	—
47	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₃₁	C₃×3₋¹⁺²	1	2	yes	—
49	C₃²×C₃₁	#737	1	6	yes	—
50	C₃₁⋊C₉	—	3	9	no	—
51	C₃₁⋊C₉	—	9	6	no	—
52	C₃²×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
53	C₃³×C₉	—	31	1	no	—
54	C₃×C₃²⋊C₉	—	31	9	no	—
55	C₃×C₃²⋊C₉	—	31	2	no	—
56	C₃×C₃₁	—	3	9	no	—
57	C₃×C₃₁	—	9	9	no	—
58	C₃×C₃₁	C₃×C₃²⋊C₉	1	1	yes	characteristic
59	C₃×C₃₁	C₉×H(3,𝔽₃)	1	3	yes	—
60	C₃×C₃₁	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
61	C₃×C₃₁	C₉⋊H(3,𝔽₃)	1	6	yes	—
62	C₃×C₃₁	#6893	1	2	yes	—
63	C₃⁴	C₃×C₃₁⋊C₃	1	3	yes	—
64	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
65	C₃×H(3,𝔽₃)	—	3	6	no	—
66	C₃×H(3,𝔽₃)	—	3	3	no	—
67	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	9	yes	complemented
68	C₃²×C₉	—	31	3	no	—
69	C₃²×C₉	—	31	6	no	—
70	C₃²×C₉	—	93	9	no	—
71	C₃²×C₉	—	93	3	no	—
72	C₃²⋊C₉	—	93	27	no	—
73	C₃²⋊C₉	—	93	6	no	—
74	C₃₁	#10624555	1	1	yes	characteristic, complemented, Sylow
75	C₃³	—	3	3	no	—
76	C₃³	—	3	6	no	—
77	C₃³	—	3	9	no	—
78	C₃³	—	3	3	no	—
79	C₃³	—	3	6	no	—
80	C₃³	—	3	18	no	—
81	C₃³	C₃×C₃₁⋊C₉	1	3	yes	—
82	C₃³	C₃×C₃₁⋊C₉	1	9	yes	complemented
83	C₃³	#11719689	1	6	yes	—
84	C₃³	C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
85	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
86	H(3,𝔽₃)	—	3	27	no	—
87	C₃×C₉	—	31	9	no	—
88	C₃×C₉	—	93	18	no	—
89	C₃×C₉	—	93	6	no	—
90	C₃×C₉	—	93	27	no	—
91	C₃×C₉	—	93	6	no	—
92	C₃×C₉	—	93	3	no	—
93	C₃²	—	3	9	no	—
94	C₃²	—	3	3	no	—
95	C₃²	—	3	18	no	—
96	C₃²	—	3	6	no	—
97	C₃²	—	3	9	no	—
98	C₃²	—	3	27	no	—
99	C₃²	—	9	3	no	—
100	C₃²	—	9	18	no	—
101	C₃²	—	9	6	no	—
102	C₃²	C₃²⋊(C₃₁⋊C₉)	1	9	yes	complemented
103	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
104	C₃²	H(3,𝔽₃)×C₃₁⋊C₃	1	3	yes	—
105	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
106	C₃²	C₃×#11719689	1	2	yes	—
107	C₃²	#14578911	1	6	yes	—
108	C₉	—	279	6	no	—
109	C₉	—	93	9	no	—
110	C₃	—	3	9	no	—
111	C₃	—	9	9	no	—
112	C₃	C₃×C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
113	C₃	H(3,𝔽₃)×C₃₁⋊C₉	1	3	yes	—
114	C₃	(C₃×C₃₁⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
115	C₃	#21419314	1	6	yes	—
116	C₃	#21419320	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₃₁⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊(C₃₁⋊C₉)

C₃

yes

maximal, complemented

C₃×C₃²⋊(C₃₁⋊C₉)

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)×C₃₁

C₃

yes

maximal, characteristic

C₃²×C₃₁⋊C₉

—

C₃²×C₃₁⋊C₉

—

C₃²×C₃₁⋊C₉

C₃²

yes

—

C₃²×C₃₁⋊C₉

C₃²

yes

—

C₃²⋊(C₃₁⋊C₉)

—

C₃²⋊(C₃₁⋊C₉)

—

C₃⁴×C₃₁

C₃²

yes

—

C₃⁴×C₃₁

C₃²

yes

characteristic

C₃×H(3,𝔽₃)×C₃₁

—

C₃×H(3,𝔽₃)×C₃₁

—

C₃×H(3,𝔽₃)×C₃₁

C₉

yes

complemented

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

C₃×C₉

yes

—

C₃³×C₃₁

C₃×C₉

yes

complemented

C₃³×C₃₁

H(3,𝔽₃)

yes

—

C₃³×C₃₁

3₋¹⁺²

yes

—

C₃³×C₃₁

C₃³

yes

characteristic

H(3,𝔽₃)×C₃₁

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

yes

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

C₃²⋊C₉

yes

complemented

C₃²×C₃₁

C₃²×C₉

yes

characteristic

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₁

C₃×3₋¹⁺²

yes

—

C₃²×C₃₁

#737

yes

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃²×H(3,𝔽₃)

C₃₁⋊C₃

yes

characteristic

C₃³×C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₁

C₉×H(3,𝔽₃)

yes

—

C₃×C₃₁

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₁

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₃₁

#6893

yes

—

C₃⁴

C₃×C₃₁⋊C₃

yes

—

C₃⁴

C₃×C₃₁⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃₁⋊C₉

yes

complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃₁

#10624555

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₃₁⋊C₉

yes

—

C₃³

C₃×C₃₁⋊C₉

yes

complemented

C₃³

#11719689

yes

—

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

—

C₃³

C₃²×C₃₁⋊C₃

yes

characteristic

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

C₃²⋊(C₃₁⋊C₉)

yes

complemented

103

C₃²

C₃²×C₃₁⋊C₉

yes

characteristic

104

C₃²

H(3,𝔽₃)×C₃₁⋊C₃

yes

—

105

C₃²

C₃×C₃²⋊(C₃₁⋊C₃)

yes

characteristic

106

C₃²

C₃×#11719689

yes

—

107

C₃²

#14578911

yes

—

108

C₉

—

279

—

109

C₉

—

110

C₃

—

111

C₃

—

112

C₃

C₃×C₃²⋊(C₃₁⋊C₉)

yes

characteristic

113

C₃

H(3,𝔽₃)×C₃₁⋊C₉

yes

—

114

C₃

(C₃×C₃₁⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

115

C₃

#21419314

yes

—

116

C₃

#21419320

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₁⋊C₃
commutator	C₃²×C₃₁	C₃²×C₉
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₁	C₃
socle, cosocle	C₃³×C₃₁	C₃³
hypercenter	C₃²×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	#10624555
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₁⋊C₃

commutator

C₃²×C₃₁

C₃²×C₉

Fitting, prime radical

C₃²×H(3,𝔽₃)×C₃₁

C₃

socle, cosocle

C₃³×C₃₁

C₃³

hypercenter

C₃²×H(3,𝔽₃)

C₃₁⋊C₃

C₃₁

G ▷ C₃²×C₃₁ ▷ C₃₁

C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃³×C₃₁⋊C₉	C₃	semidirect
2. C₃×C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
3. C₃×C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
15. C₃×H(3,𝔽₃)×C₃₁	C₉	semidirect
23. C₃³×C₃₁	C₃×C₉	semidirect
33. C₃×C₃₁⋊C₉	H(3,𝔽₃)	semidirect
44. C₃²×C₃₁	C₃²⋊C₉	semidirect
67. C₃×H(3,𝔽₃)	C₃₁⋊C₉	semidirect
74. C₃₁	#10624555	semidirect
82. C₃³	C₃×C₃₁⋊C₉	semidirect
102. C₃²	C₃²⋊(C₃₁⋊C₉)	semidirect

H ≅ G/N

type

1. C₃³×C₃₁⋊C₉

C₃

semidirect

2. C₃×C₃²⋊(C₃₁⋊C₉)

C₃

semidirect

3. C₃×C₃²⋊(C₃₁⋊C₉)

C₃

semidirect

15. C₃×H(3,𝔽₃)×C₃₁

C₉

semidirect

23. C₃³×C₃₁

C₃×C₉

semidirect

33. C₃×C₃₁⋊C₉

H(3,𝔽₃)

semidirect

44. C₃²×C₃₁

C₃²⋊C₉

semidirect

67. C₃×H(3,𝔽₃)

C₃₁⋊C₉

semidirect

74. C₃₁

#10624555

semidirect

82. C₃³

C₃×C₃₁⋊C₉

semidirect

102. C₃²

C₃²⋊(C₃₁⋊C₉)

semidirect

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

subgroup	index
C₃³×C₃₁⋊C₉	3
C₃×C₃²⋊(C₃₁⋊C₉)	3
C₃²×H(3,𝔽₃)×C₃₁	3
#10624555	31

subgroup

index

C₃³×C₃₁⋊C₉

C₃×C₃²⋊(C₃₁⋊C₉)

C₃²×H(3,𝔽₃)×C₃₁

#10624555

Subgroups — #36379875

Conjugacy classes of subgroups

Complements

Maximal subgroups