Group #36382930 — subgroups

Subgroups — C₃×#21419420 #36382930

Order 22599.

992 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 274 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	C₃×#14578913	C₃	1	3	yes	maximal, complemented
2	C₃²×#11719686	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419420	C₃	1	27	yes	maximal, direct product factor
4	C₃×#14578902	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#14578900	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#14578898	C₃	1	3	yes	maximal, complemented
7	C₃×#14578895	C₃	1	3	yes	maximal, complemented
8	C₃×C₃₁×#733	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃³×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
10	#14578913	—	3	18	no	—
11	#14578913	C₃²	1	27	yes	complemented
12	C₃×#11719689	C₃²	1	3	yes	—
13	C₃×#11719686	—	3	3	no	—
14	C₃×#11719686	—	3	6	no	—
15	C₃×#11719686	—	3	3	no	—
16	C₃×#11719686	C₃²	1	9	yes	complemented
17	C₃×#11719687	—	3	3	no	—
18	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	3	yes	—
19	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
20	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₃₁⋊C₃	—	3	3	no	—
22	#14578902	C₃²	1	9	yes	complemented
23	#14578900	C₃²	1	9	yes	complemented
24	#14578898	C₃²	1	27	yes	complemented
25	#14578895	C₃²	1	27	yes	complemented
26	C₃×3₋¹⁺²×C₃₁	C₃²	1	3	yes	—
27	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
28	C₃₁×#733	C₃²	1	9	yes	complemented
29	C₃³×C₃₁	C₃³	1	1	yes	characteristic
30	C₃²×C₃₁⋊C₃	—	3	3	no	—
31	C₃²×C₃₁⋊C₃	—	3	2	no	—
32	C₃²×C₃₁⋊C₃	—	3	1	no	—
33	C₃²×C₃₁⋊C₃	—	9	3	no	—
34	C₃²×C₃₁⋊C₃	C₃³	1	3	yes	—
35	3₋¹⁺²×C₃₁	—	3	6	no	—
36	3₋¹⁺²×C₃₁	C₃³	1	9	yes	—
37	C₃×C₉×C₃₁	—	3	2	no	—
38	C₃×C₉×C₃₁	—	3	1	no	—
39	C₃×C₉×C₃₁	—	3	3	no	—
40	C₃×C₉×C₃₁	C₃³	1	3	yes	—
41	C₃²⋊(C₃₁⋊C₃)	—	3	6	no	—
42	C₃²⋊(C₃₁⋊C₃)	C₃³	1	9	yes	—
43	#11719689	—	3	6	no	—
44	#11719689	C₃³	1	9	yes	—
45	C₃×C₃₁⋊C₉	—	3	3	no	—
46	C₃×C₃₁⋊C₉	—	3	2	no	—
47	C₃×C₃₁⋊C₉	—	3	1	no	—
48	C₃×C₃₁⋊C₉	H(3,𝔽₃)	1	3	yes	—
49	C₃×C₃₁⋊C₉	H(3,𝔽₃)	1	6	yes	—
50	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
51	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
52	#11719687	—	3	27	no	—
53	#11719686	—	3	27	no	—
54	#11719686	—	3	27	no	—
55	C₉×C₃₁⋊C₃	—	3	27	no	—
56	C₃×#6913	—	31	1	no	maximal, Sylow
57	C₃²×C₃₁	—	3	3	no	—
58	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	2	yes	—
59	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
61	C₃×C₃₁⋊C₃	—	3	9	no	—
62	C₃×C₃₁⋊C₃	—	3	3	no	—
63	C₃×C₃₁⋊C₃	—	9	6	no	—
64	C₃×C₃₁⋊C₃	—	9	9	no	—
65	C₃×C₃₁⋊C₃	—	9	3	no	—
66	C₃×C₃₁⋊C₃	—	9	3	no	—
67	C₉×C₃₁	—	3	9	no	—
68	C₉×C₃₁	—	3	3	no	—
69	C₃₁⋊C₉	—	3	9	no	—
70	C₃₁⋊C₉	—	3	3	no	—
71	C₃₁⋊C₉	C₃×H(3,𝔽₃)	1	9	yes	—
72	C₃×#737	—	31	3	no	—
73	C₃²×3₋¹⁺²	—	31	1	no	—
74	#6913	—	31	27	no	—
75	C₃×#733	—	31	1	no	—
76	C₃×#733	C₃₁⋊C₃	1	1	yes	characteristic, complemented
77	C₃×#732	—	31	1	no	—
78	C₃×#731	—	31	3	no	—
79	C₃×AΣL(1,𝔽₂₇)	—	31	3	no	—
80	C₃×C₃₁	—	3	2	no	—
81	C₃×C₃₁	—	3	1	no	—
82	C₃×C₃₁	#6913	1	3	yes	complemented
83	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
84	C₃₁⋊C₃	—	9	9	no	—
85	C₃₁⋊C₃	—	9	3	no	—
86	C₃⁴	—	31	1	no	—
87	#737	—	31	27	no	—
88	#737	—	93	18	no	—
89	C₃×3₋¹⁺²	—	31	9	no	—
90	C₃×3₋¹⁺²	—	31	3	no	—
91	C₃×3₋¹⁺²	—	93	6	no	—
92	C₃×3₋¹⁺²	—	93	3	no	—
93	C₃×3₋¹⁺²	—	93	3	no	—
94	C₃×3₋¹⁺²	—	93	3	no	—
95	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	3	yes	—
96	C₃×H(3,𝔽₃)	—	31	3	no	—
97	C₃²×C₉	—	31	1	no	—
98	C₃²×C₉	—	31	1	no	—
99	C₃²×C₉	—	93	3	no	—
100	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
101	#733	—	31	9	no	—
102	#733	C₃×C₃₁⋊C₃	1	9	yes	complemented
103	#732	—	31	9	no	—
104	#731	—	31	27	no	—
105	AΣL(1,𝔽₂₇)	—	31	27	no	—
106	C₃₁	C₃×#6913	1	1	yes	characteristic, complemented, Sylow
107	C₃³	—	279	3	no	—
108	C₃³	—	31	3	no	—
109	C₃³	—	93	2	no	—
110	C₃³	—	93	3	no	—
111	C₃³	—	93	1	no	—
112	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
113	3₋¹⁺²	—	3	6	no	—
114	3₋¹⁺²	—	31	9	no	—
115	3₋¹⁺²	—	93	27	no	—
116	3₋¹⁺²	—	93	27	no	—
117	3₋¹⁺²	—	93	27	no	—
118	3₋¹⁺²	—	93	6	no	—
119	3₋¹⁺²	C₃²×C₃₁⋊C₃	1	9	yes	—
120	H(3,𝔽₃)	—	31	9	no	—
121	H(3,𝔽₃)	—	93	6	no	—
122	C₃×C₉	—	3	2	no	—
123	C₃×C₉	—	3	3	no	—
124	C₃×C₉	—	3	1	no	—
125	C₃×C₉	—	31	6	no	—
126	C₃×C₉	—	31	3	no	—
127	C₃×C₉	—	31	3	no	—
128	C₃×C₉	—	31	3	no	—
129	C₃×C₉	—	93	27	no	—
130	C₃×C₉	—	93	2	no	—
131	C₃×C₉	—	93	1	no	—
132	C₃×C₉	—	93	3	no	—
133	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
134	C₃²	—	3	3	no	—
135	C₃²	—	279	3	no	—
136	C₃²	—	279	6	no	—
137	C₃²	—	279	3	no	—
138	C₃²	—	279	9	no	—
139	C₃²	—	93	9	no	—
140	C₃²	—	93	3	no	—
141	C₃²	H(3,𝔽₃)×C₃₁⋊C₃	1	2	yes	—
142	C₃²	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
143	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
144	C₉	—	3	3	no	—
145	C₉	—	3	9	no	—
146	C₉	—	31	9	no	—
147	C₉	—	93	3	no	—
148	C₉	—	93	9	no	—
149	C₃	—	3	2	no	—
150	C₃	—	3	1	no	—
151	C₃	—	279	3	no	—
152	C₃	—	279	9	no	—
153	C₃	#21419420	1	3	yes	direct product factor
154	C₃	C₃×H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#14578913

C₃

yes

maximal, complemented

C₃²×#11719686

C₃

yes

maximal, characteristic, complemented

#21419420

C₃

yes

maximal, direct product factor

C₃×#14578902

C₃

yes

maximal, characteristic, complemented

C₃×#14578900

C₃

yes

maximal, characteristic, complemented

C₃×#14578898

C₃

yes

maximal, complemented

C₃×#14578895

C₃

yes

maximal, complemented

C₃×C₃₁×#733

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₁⋊C₃

C₃²

yes

characteristic

#14578913

—

#14578913

C₃²

yes

complemented

C₃×#11719689

C₃²

yes

—

—

—

—

C₃²

yes

complemented

C₃×#11719687

—

C₃×C₃²⋊(C₃₁⋊C₃)

C₃²

yes

—

C₃²×C₃₁⋊C₉

C₃²

yes

characteristic

C₃²×C₃₁⋊C₉

C₃²

yes

characteristic

C₃×C₉×C₃₁⋊C₃

—

#14578902

C₃²

yes

complemented

#14578900

C₃²

yes

complemented

#14578898

C₃²

yes

complemented

#14578895

C₃²

yes

complemented

C₃×3₋¹⁺²×C₃₁

C₃²

yes

—

C₃²×C₉×C₃₁

C₃²

yes

characteristic

C₃₁×#733

C₃²

yes

complemented

C₃³×C₃₁

C₃³

yes

characteristic

C₃²×C₃₁⋊C₃

—

C₃²×C₃₁⋊C₃

—

C₃²×C₃₁⋊C₃

—

C₃²×C₃₁⋊C₃

—

C₃²×C₃₁⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

C₃³

yes

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

C₃³

yes

—

C₃²⋊(C₃₁⋊C₃)

—

C₃²⋊(C₃₁⋊C₃)

C₃³

yes

—

#11719689

—

#11719689

C₃³

yes

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

H(3,𝔽₃)

yes

—

C₃×C₃₁⋊C₉

H(3,𝔽₃)

yes

—

C₃×C₃₁⋊C₉

C₃³

yes

—

C₃×C₃₁⋊C₉

C₃³

yes

—

#11719687

—

#11719686

—

#11719686

—

C₉×C₃₁⋊C₃

—

C₃×#6913

—

maximal, Sylow

C₃²×C₃₁

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₁

C₃⁴

yes

characteristic

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₉×C₃₁

—

C₉×C₃₁

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

C₃×H(3,𝔽₃)

yes

—

C₃×#737

—

C₃²×3₋¹⁺²

—

#6913

—

C₃×#733

—

C₃×#733

C₃₁⋊C₃

yes

characteristic, complemented

C₃×#732

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

#6913

yes

complemented

C₃×C₃₁

C₃²×H(3,𝔽₃)

yes

characteristic

C₃₁⋊C₃

—

C₃₁⋊C₃

—

C₃⁴

—

#737

—

#737

—

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

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

100

C₃²×C₉

C₃×C₃₁⋊C₃

yes

characteristic

101

#733

—

102

#733

C₃×C₃₁⋊C₃

yes

complemented

103

#732

—

104

#731

—

105

AΣL(1,𝔽₂₇)

—

106

C₃₁

C₃×#6913

yes

characteristic, complemented, Sylow

107

C₃³

—

279

—

108

C₃³

—

109

C₃³

—

110

C₃³

—

111

C₃³

—

112

C₃³

C₃²×C₃₁⋊C₃

yes

characteristic

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

—

119

3₋¹⁺²

C₃²×C₃₁⋊C₃

yes

—

120

H(3,𝔽₃)

—

121

H(3,𝔽₃)

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

C₃²×C₃₁⋊C₃

yes

—

134

C₃²

—

135

C₃²

—

279

—

136

C₃²

—

279

—

137

C₃²

—

279

—

138

C₃²

—

279

—

139

C₃²

—

140

C₃²

—

141

C₃²

H(3,𝔽₃)×C₃₁⋊C₃

yes

—

142

C₃²

H(3,𝔽₃)×C₃₁⋊C₃

yes

characteristic

143

C₃²

C₃³×C₃₁⋊C₃

yes

characteristic

144

C₉

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₉

—

149

C₃

—

150

C₃

—

151

C₃

—

279

—

152

C₃

—

279

—

153

C₃

#21419420

yes

direct product factor

154

C₃

C₃×H(3,𝔽₃)×C₃₁⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₁⋊C₃
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃×C₃₁×#733	C₃
socle	C₃²×C₃₁	C₃×H(3,𝔽₃)
hypercenter	C₃×#733	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6913
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₁⋊C₃

commutator, cosocle

C₃²×C₃₁

C₃⁴

Frattini

C₃²

C₃³×C₃₁⋊C₃

Fitting

C₃×C₃₁×#733

C₃

socle

C₃²×C₃₁

C₃×H(3,𝔽₃)

C₃₁⋊C₃

C₃₁

G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
1. C₃×#14578913	C₃	semidirect
2. C₃²×#11719686	C₃	semidirect
3. #21419420	C₃	direct (N × H)
4. C₃×#14578902	C₃	semidirect
5. C₃×#14578900	C₃	semidirect
6. C₃×#14578898	C₃	semidirect
7. C₃×#14578895	C₃	semidirect
8. C₃×C₃₁×#733	C₃	semidirect
11. #14578913	C₃²	semidirect
16. C₃×#11719686	C₃²	semidirect
22. #14578902	C₃²	semidirect
23. #14578900	C₃²	semidirect
24. #14578898	C₃²	semidirect
25. #14578895	C₃²	semidirect
28. C₃₁×#733	C₃²	semidirect
76. C₃×#733	C₃₁⋊C₃	semidirect
82. C₃×C₃₁	#6913	semidirect
102. #733	C₃×C₃₁⋊C₃	semidirect
106. C₃₁	C₃×#6913	semidirect
153. C₃	#21419420	direct (N × H)

H ≅ G/N

type

1. C₃×#14578913

C₃

semidirect

2. C₃²×#11719686

C₃

semidirect

3. #21419420

C₃

direct (N × H)

4. C₃×#14578902

C₃

semidirect

5. C₃×#14578900

C₃

semidirect

6. C₃×#14578898

C₃

semidirect

7. C₃×#14578895

C₃

semidirect

8. C₃×C₃₁×#733

C₃

semidirect

11. #14578913

C₃²

semidirect

16. C₃×#11719686

C₃²

semidirect

22. #14578902

C₃²

semidirect

23. #14578900

C₃²

semidirect

24. #14578898

C₃²

semidirect

25. #14578895

C₃²

semidirect

28. C₃₁×#733

C₃²

semidirect

76. C₃×#733

C₃₁⋊C₃

semidirect

82. C₃×C₃₁

#6913

semidirect

102. #733

C₃×C₃₁⋊C₃

semidirect

106. C₃₁

C₃×#6913

semidirect

153. C₃

#21419420

direct (N × H)

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

subgroup	index
C₃×#14578913	3
C₃²×#11719686	3
#21419420	3
C₃×#14578902	3
C₃×#14578900	3
C₃×#14578898	3
C₃×#14578895	3
C₃×C₃₁×#733	3
C₃×#6913	31

subgroup

index

C₃×C₃₁×#733

Subgroups — C₃×#21419420 #36382930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#21419420 #36382930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419420 #36382930