Group #36380191 — subgroups

Subgroups — #36380191

Order 22599.

452 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₃²×#11719689	C₃	1	1	yes	maximal, characteristic
2	#21419312	C₃	1	1	yes	maximal, characteristic
3	#21419312	C₃	1	1	yes	maximal, characteristic
4	#21419212	C₃	1	9	yes	maximal, complemented
5	C₃²×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic
6	C₃×#11719689	—	3	1	no	—
7	C₃×#11719689	—	3	1	no	—
8	C₃×#11719689	—	3	9	no	—
9	C₃×#11719689	—	3	1	no	—
10	C₃×#11719689	—	3	1	no	—
11	C₃×#11719689	C₃²	1	3	yes	—
12	C₃²×C₃₁⋊C₉	—	3	1	no	—
13	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
14	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
15	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
16	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
17	C₃²⋊(C₃₁⋊C₉)	C₃²	1	3	yes	—
18	C₃²⋊(C₃₁⋊C₉)	C₃²	1	3	yes	—
19	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
20	C₃×3₋¹⁺²×C₃₁	—	3	1	no	—
21	C₃×3₋¹⁺²×C₃₁	—	3	1	no	—
22	C₃×3₋¹⁺²×C₃₁	—	3	9	no	—
23	C₃×3₋¹⁺²×C₃₁	C₃²	1	3	yes	—
24	C₃²×C₉×C₃₁	—	3	1	no	—
25	C₃³×C₃₁	—	3	3	no	—
26	C₃³×C₃₁	—	3	3	no	—
27	C₃³×C₃₁	—	9	2	no	—
28	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₃₁	C₃³	1	1	yes	characteristic
32	3₋¹⁺²×C₃₁	—	3	9	no	—
33	3₋¹⁺²×C₃₁	—	3	9	no	—
34	3₋¹⁺²×C₃₁	—	3	9	no	—
35	C₃×C₉×C₃₁	—	3	9	no	—
36	C₃×C₉×C₃₁	—	3	3	no	—
37	#11719689	—	3	9	no	—
38	#11719689	—	3	9	no	—
39	#11719689	—	3	9	no	—
40	#11719689	—	9	3	no	—
41	#11719689	—	9	3	no	—
42	C₃×C₃₁⋊C₉	—	3	9	no	—
43	C₃×C₃₁⋊C₉	—	3	3	no	—
44	C₃×C₃₁⋊C₉	—	3	3	no	—
45	C₃×C₃₁⋊C₉	—	3	3	no	—
46	#10624607	—	31	1	no	maximal, Sylow
47	C₃²×C₃₁	—	3	2	no	—
48	C₃²×C₃₁	—	3	2	no	—
49	C₃²×C₃₁	—	3	1	no	—
50	C₃²×C₃₁	—	3	1	no	—
51	C₃²×C₃₁	—	3	2	no	—
52	C₃²×C₃₁	—	3	2	no	—
53	C₃²×C₃₁	—	3	1	no	—
54	C₃²×C₃₁	—	3	1	no	—
55	C₃²×C₃₁	—	3	1	no	—
56	C₃²×C₃₁	—	3	1	no	—
57	C₃²×C₃₁	—	9	9	no	—
58	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
59	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
60	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₉×C₃₁	—	3	9	no	—
62	C₃₁⋊C₉	—	3	9	no	—
63	C₃₁⋊C₉	—	9	3	no	—
64	C₃₁⋊C₉	—	9	3	no	—
65	C₃²×3₋¹⁺²	—	31	1	no	—
66	C₃²×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic
67	#6891	—	31	1	no	—
68	#6891	—	31	1	no	—
69	#6860	—	31	9	no	—
70	C₃×C₃₁	—	3	3	no	—
71	C₃×C₃₁	—	3	3	no	—
72	C₃×C₃₁	—	9	1	no	—
73	C₃×C₃₁	—	9	1	no	—
74	C₃×C₃₁	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
75	C₃×C₃₁	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
76	C₃×C₃₁	#6912	1	2	yes	—
77	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
78	C₃×3₋¹⁺²	—	3	1	no	—
79	C₃×3₋¹⁺²	—	3	1	no	—
80	C₃×3₋¹⁺²	—	3	9	no	—
81	C₃×3₋¹⁺²	—	31	3	no	—
82	C₃×3₋¹⁺²	—	93	1	no	—
83	C₃×3₋¹⁺²	—	93	1	no	—
84	C₃×3₋¹⁺²	—	93	1	no	—
85	C₃×3₋¹⁺²	—	93	1	no	—
86	C₃×3₋¹⁺²	—	93	9	no	—
87	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	3	yes	—
88	C₃²×C₉	—	3	1	no	—
89	C₃²×C₉	—	93	1	no	—
90	C₃²⋊C₉	—	31	3	no	—
91	C₃²⋊C₉	—	31	3	no	—
92	C₃²⋊C₉	—	93	1	no	—
93	C₃²⋊C₉	—	93	1	no	—
94	C₃²⋊C₉	—	93	1	no	—
95	C₃²⋊C₉	—	93	1	no	—
96	C₃₁	#10624607	1	1	yes	characteristic, complemented, Sylow
97	C₃³	—	3	3	no	—
98	C₃³	—	3	3	no	—
99	C₃³	—	9	2	no	—
100	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
101	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
102	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
103	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
104	3₋¹⁺²	—	3	9	no	—
105	3₋¹⁺²	—	3	9	no	—
106	3₋¹⁺²	—	3	9	no	—
107	3₋¹⁺²	—	279	3	no	—
108	3₋¹⁺²	—	279	3	no	—
109	3₋¹⁺²	—	93	9	no	—
110	3₋¹⁺²	—	93	9	no	—
111	3₋¹⁺²	—	93	9	no	—
112	C₃×C₉	—	3	9	no	—
113	C₃×C₉	—	3	3	no	—
114	C₃×C₉	—	93	3	no	—
115	C₃×C₉	—	93	3	no	—
116	C₃×C₉	—	93	9	no	—
117	C₃×C₉	—	93	3	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	3	2	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	1	no	—
126	C₃²	—	3	2	no	—
127	C₃²	—	3	2	no	—
128	C₃²	—	9	9	no	—
129	C₃²	#14578894	1	3	yes	—
130	C₃²	#14578895	1	3	yes	—
131	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
132	C₉	—	3	9	no	—
133	C₉	—	279	3	no	—
134	C₉	—	279	3	no	—
135	C₉	—	93	9	no	—
136	C₃	—	3	3	no	—
137	C₃	—	3	3	no	—
138	C₃	—	9	1	no	—
139	C₃	—	9	1	no	—
140	C₃	C₃×#14578894	1	1	yes	characteristic
141	C₃	C₃×#14578895	1	1	yes	characteristic
142	C₃	#21419409	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11719689

C₃

yes

maximal, characteristic

#21419312

C₃

yes

maximal, characteristic

#21419312

C₃

yes

maximal, characteristic

#21419212

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₁

C₃

yes

maximal, characteristic

—

—

—

—

—

C₃²

yes

—

C₃²×C₃₁⋊C₉

—

C₃²⋊(C₃₁⋊C₉)

—

C₃²⋊(C₃₁⋊C₉)

—

C₃²⋊(C₃₁⋊C₉)

—

C₃²⋊(C₃₁⋊C₉)

—

C₃²⋊(C₃₁⋊C₉)

C₃²

yes

—

C₃²⋊(C₃₁⋊C₉)

C₃²

yes

—

C₃⁴×C₃₁

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₁

—

C₃×3₋¹⁺²×C₃₁

—

C₃×3₋¹⁺²×C₃₁

—

C₃×3₋¹⁺²×C₃₁

C₃²

yes

—

C₃²×C₉×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

yes

C₃³×C₃₁

yes

C₃³×C₃₁

yes

C₃³×C₃₁

C₃³

yes

characteristic

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

—

—

—

—

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

#10624607

—

maximal, Sylow

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₁

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₁

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₁⋊C₃

yes

—

—

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₁

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₁

#6912

yes

—

C₃⁴

C₃×C₃₁⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₁⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃₁

#10624607

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

100

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

101

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

102

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

103

C₃³

C₃²×C₃₁⋊C₃

yes

characteristic

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

279

—

108

3₋¹⁺²

—

279

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

3₋¹⁺²

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

#14578894

yes

—

130

C₃²

#14578895

yes

—

131

C₃²

C₃×C₃²⋊(C₃₁⋊C₃)

yes

characteristic

132

C₉

—

133

C₉

—

279

—

134

C₉

—

279

—

135

C₉

—

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

C₃×#14578894

yes

characteristic

141

C₃

C₃×#14578895

yes

characteristic

142

C₃

#21419409

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)
commutator, cosocle	C₃³×C₃₁	C₃³
Frattini	C₃³	C₃²×C₃₁⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₁	C₃
socle	C₃²×C₃₁	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₁	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	#10624607
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₁⋊C₃)

commutator, cosocle

C₃³×C₃₁

C₃³

Frattini

C₃³

C₃²×C₃₁⋊C₃

Fitting

C₃²×3₋¹⁺²×C₃₁

C₃

socle

C₃²×C₃₁

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₁

C₃²

hypercenter

C₃²×3₋¹⁺²

C₃₁⋊C₃

hypocenter

C₃₁

#10624607

LCS

G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
4. #21419212	C₃	semidirect
96. C₃₁	#10624607	semidirect

H ≅ G/N

type

4. #21419212

C₃

semidirect

96. C₃₁

#10624607

semidirect

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

subgroup	index
C₃²×#11719689	3
#21419312	3
#21419212	3
C₃²×3₋¹⁺²×C₃₁	3
#10624607	31

subgroup

index

C₃²×3₋¹⁺²×C₃₁

Subgroups — #36380191

Conjugacy classes of subgroups

Complements

Maximal subgroups