Group #13135711 — subgroups

Subgroups — #10588242⋊C₃ #13135711

Order 1728.

192 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 66 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	C₂×Q₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	#63372⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#63371⋊C₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#4565	—	3	1	no	maximal
5	#10588242	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×3₋¹⁺²	—	2	1	no	—
7	Q₁₆×3₋¹⁺²	—	2	1	no	—
8	Q₁₆×3₋¹⁺²	C₄	1	2	yes	—
9	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	#3667⋊C₃	—	2	1	no	—
11	3₋¹⁺²⋊C₁₆	—	2	1	no	—
12	C₂×C₃²×Q₁₆	D₃	1	1	yes	characteristic
13	C₃×#819	—	3	1	no	—
14	C₃×#812	—	3	1	no	—
15	C₂×C₉×Q₁₆	—	2	1	no	—
16	C₂×C₉×Q₁₆	C₂×C₃	1	1	yes	characteristic
17	#63372	C₂×C₃	1	1	yes	characteristic
18	#63371	C₂×C₃	1	1	yes	characteristic
19	Q₈×3₋¹⁺²	—	2	2	no	—
20	Q₈×3₋¹⁺²	—	4	1	no	—
21	C₂×C₄×3₋¹⁺²	—	4	1	no	—
22	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
23	C₈×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
24	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
25	3₋¹⁺²⋊C₈	—	4	1	no	—
26	C₃×#362	—	9	1	no	—
27	#4565	—	3	1	no	—
28	C₂×C₃²×Q₈	—	2	1	no	—
29	C₃²×Q₁₆	—	2	1	no	—
30	C₃²×Q₁₆	Dic₃	1	2	yes	—
31	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
32	C₃×#208	—	6	1	no	—
33	C₂×Q₈×C₉	—	2	2	no	—
34	C₂×Q₈×C₉	—	2	1	no	—
35	C₃×C₃⋊C₁₆	—	6	1	no	—
36	C₉×Q₁₆	—	2	2	no	—
37	C₉×Q₁₆	—	2	2	no	—
38	C₉×Q₁₆	—	2	1	no	—
39	C₉×Q₁₆	C₃×C₄	1	2	yes	—
40	C₂×C₈×C₉	—	2	1	no	—
41	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
42	#3667	—	2	1	no	—
43	C₉⋊C₁₆	—	2	1	no	—
44	C₂²×3₋¹⁺²	SD₁₆	1	1	yes	characteristic
45	C₄×3₋¹⁺²	—	4	2	no	—
46	C₄×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×3₋¹⁺²	D₈	1	1	yes	characteristic
48	C₂×C₃×Q₁₆	—	3	1	no	—
49	C₂×C₃×Q₁₆	C₃×D₃	1	1	yes	characteristic
50	C₃×M₃₂	—	9	1	no	—
51	C₃×#108	—	9	1	no	—
52	#819	—	3	1	no	—
53	#812	—	3	1	no	—
54	C₃²×Q₈	—	2	2	no	—
55	C₃²×Q₈	—	4	1	no	—
56	C₂×C₃²×C₄	—	4	1	no	—
57	C₂×C₃²×C₄	#67	1	1	yes	characteristic
58	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
59	C₃²×C₈	#67	1	1	yes	characteristic
60	C₃×C₃⋊C₈	—	12	1	no	—
61	Q₈×C₉	—	2	4	no	—
62	Q₈×C₉	—	2	2	no	—
63	Q₈×C₉	—	4	2	no	—
64	Q₈×C₉	—	4	1	no	—
65	C₂×C₄×C₉	—	2	1	no	—
66	C₂×C₄×C₉	—	4	2	no	—
67	C₂×C₄×C₉	—	4	1	no	—
68	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
69	C₈×C₉	—	2	1	no	—
70	C₈×C₉	—	2	1	no	—
71	C₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
72	C₈×C₉	C₃×D₄	1	1	yes	characteristic
73	C₉⋊C₈	—	4	1	no	—
74	#362	—	9	1	no	Sylow
75	C₂×3₋¹⁺²	—	2	1	no	—
76	C₂×3₋¹⁺²	#102	1	1	yes	characteristic
77	C₂×C₃×Q₈	—	2	1	no	—
78	C₂×C₃×Q₈	—	6	1	no	—
79	C₃×Q₁₆	—	2	1	no	—
80	C₃×Q₁₆	—	3	2	no	—
81	C₃×Q₁₆	—	6	1	no	—
82	C₃×Q₁₆	C₃×Dic₃	1	2	yes	—
83	C₃×M₁₆	—	18	1	no	—
84	C₂×C₃×C₈	—	3	1	no	—
85	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
86	#208	—	6	1	no	—
87	C₃×C₁₆	—	18	1	no	—
88	C₃⋊C₁₆	—	6	1	no	—
89	C₂²×C₃²	#214	1	1	yes	characteristic
90	C₃²×C₄	—	4	2	no	—
91	C₃²×C₄	#213	1	1	yes	characteristic
92	C₃²×C₄	#217	1	1	yes	characteristic
93	C₂²×C₉	—	2	1	no	—
94	C₂²×C₉	C₃×SD₁₆	1	1	yes	characteristic
95	C₄×C₉	—	2	1	no	—
96	C₄×C₉	—	2	1	no	—
97	C₄×C₉	—	4	4	no	—
98	C₄×C₉	—	4	2	no	—
99	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₉	C₃×D₈	1	1	yes	characteristic
101	C₂×Q₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic
102	M₃₂	—	9	1	no	—
103	#108	—	9	1	no	—
104	3₋¹⁺²	#362	1	1	yes	characteristic, complemented, Sylow
105	C₃×Q₈	—	2	2	no	—
106	C₃×Q₈	—	4	1	no	—
107	C₃×Q₈	—	6	2	no	—
108	C₃×Q₈	—	12	1	no	—
109	C₂×C₃×C₄	—	3	1	no	—
110	C₂×C₃×C₄	—	4	1	no	—
111	C₂×C₃×C₄	—	12	1	no	—
112	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
113	C₃×C₈	—	3	1	no	—
114	C₃×C₈	—	3	1	no	—
115	C₃×C₈	—	36	1	no	—
116	C₃×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
117	C₃×C₈	C₃×#67	1	1	yes	characteristic
118	C₃⋊C₈	—	12	1	no	—
119	C₂×C₃²	—	2	1	no	—
120	C₂×C₃²	#832	1	1	yes	characteristic
121	C₂×C₉	—	2	2	no	—
122	C₂×C₉	—	2	1	no	—
123	C₂×C₉	—	2	1	no	—
124	C₂×C₉	C₃×#102	1	1	yes	characteristic
125	C₂×Q₈	—	2	1	no	—
126	Q₁₆	—	2	1	no	—
127	Q₁₆	3₋¹⁺²⋊C₄	1	2	yes	—
128	M₁₆	—	18	1	no	—
129	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
130	C₁₆	—	18	1	no	—
131	C₂²×C₃	—	3	1	no	—
132	C₂²×C₃	C₃×#214	1	1	yes	characteristic
133	C₃×C₄	—	3	1	no	—
134	C₃×C₄	—	3	1	no	—
135	C₃×C₄	—	4	2	no	—
136	C₃×C₄	—	12	2	no	—
137	C₃×C₄	C₃×#213	1	1	yes	characteristic
138	C₃×C₄	C₃×#217	1	1	yes	characteristic
139	C₃²	#4565	1	1	yes	characteristic
140	C₉	—	2	1	no	—
141	C₉	C₃×#362	1	1	yes	characteristic, complemented
142	Q₈	—	2	2	no	—
143	Q₈	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
146	C₈	—	36	1	no	—
147	C₈	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
148	C₈	#611⋊C₃	1	1	yes	characteristic
149	C₂×C₃	—	2	1	no	—
150	C₂×C₃	—	3	1	no	—
151	C₂×C₃	—	6	1	no	—
152	C₂×C₃	C₃×#832	1	1	yes	characteristic
153	C₂²	#3672⋊C₃	1	1	yes	characteristic
154	C₄	—	4	2	no	—
155	C₄	#3673⋊C₃	1	1	yes	characteristic
156	C₄	#3676⋊C₃	1	1	yes	characteristic
157	C₃	—	3	1	no	—
158	C₃	C₃×#4565	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	#63392⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic

#63372⋊C₃

C₂

yes

maximal, characteristic

#63371⋊C₃

C₂

yes

maximal, characteristic

C₃×#4565

—

maximal

#10588242

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

—

Q₁₆×3₋¹⁺²

—

Q₁₆×3₋¹⁺²

C₄

yes

—

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

#3667⋊C₃

—

3₋¹⁺²⋊C₁₆

—

C₂×C₃²×Q₁₆

D₃

yes

characteristic

C₃×#819

—

C₃×#812

—

C₂×C₉×Q₁₆

—

C₂×C₉×Q₁₆

C₂×C₃

yes

characteristic

#63372

C₂×C₃

yes

characteristic

#63371

C₂×C₃

yes

characteristic

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₈×3₋¹⁺²

C₂×C₄

yes

characteristic

C₈×3₋¹⁺²

D₄

yes

characteristic

3₋¹⁺²⋊C₈

—

C₃×#362

—

#4565

—

C₂×C₃²×Q₈

—

C₃²×Q₁₆

—

C₃²×Q₁₆

Dic₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

characteristic

C₃×#208

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₃×C₃⋊C₁₆

—

C₉×Q₁₆

—

C₉×Q₁₆

—

C₉×Q₁₆

—

C₉×Q₁₆

C₃×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

characteristic

#3667

—

C₉⋊C₁₆

—

C₂²×3₋¹⁺²

SD₁₆

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

D₈

yes

characteristic

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

C₃×D₃

yes

characteristic

C₃×M₃₂

—

C₃×#108

—

#819

—

#812

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

characteristic

C₃²×C₈

C₂×Dic₃

yes

characteristic

C₃²×C₈

#67

yes

characteristic

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₂×C₃×C₄

yes

characteristic

C₈×C₉

C₃×D₄

yes

characteristic

C₉⋊C₈

—

#362

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#102

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

C₃×Dic₃

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

characteristic

#208

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

#214

yes

characteristic

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₂²×C₉

—

C₂²×C₉

C₃×SD₁₆

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×C₂²⋊C₄

yes

characteristic

100

C₄×C₉

C₃×D₈

yes

characteristic

101

C₂×Q₁₆

AGL(1,ℤ/9)

yes

characteristic

102

M₃₂

—

103

#108

—

104

3₋¹⁺²

#362

yes

characteristic, complemented, Sylow

105

C₃×Q₈

—

106

C₃×Q₈

—

107

C₃×Q₈

—

108

C₃×Q₈

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

C₃×#67

yes

characteristic

113

C₃×C₈

—

114

C₃×C₈

—

115

C₃×C₈

—

116

C₃×C₈

C₂×C₃×Dic₃

yes

characteristic

117

C₃×C₈

C₃×#67

yes

characteristic

118

C₃⋊C₈

—

119

C₂×C₃²

—

120

C₂×C₃²

#832

yes

characteristic

121

C₂×C₉

—

122

C₂×C₉

—

123

C₂×C₉

—

124

C₂×C₉

C₃×#102

yes

characteristic

125

C₂×Q₈

—

126

Q₁₆

—

127

Q₁₆

3₋¹⁺²⋊C₄

yes

—

128

M₁₆

—

129

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

characteristic

130

C₁₆

—

131

C₂²×C₃

—

132

C₂²×C₃

C₃×#214

yes

characteristic

133

C₃×C₄

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

C₃×#213

yes

characteristic

138

C₃×C₄

C₃×#217

yes

characteristic

139

C₃²

#4565

yes

characteristic

140

C₉

—

141

C₉

C₃×#362

yes

characteristic, complemented

142

Q₈

—

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

#611⋊C₃

yes

characteristic

146

C₈

—

147

C₈

C₂×3₋¹⁺²⋊C₄

yes

characteristic

148

C₈

#611⋊C₃

yes

characteristic

149

C₂×C₃

—

150

C₂×C₃

—

151

C₂×C₃

—

152

C₂×C₃

C₃×#832

yes

characteristic

153

C₂²

#3672⋊C₃

yes

characteristic

154

C₄

—

155

C₄

#3673⋊C₃

yes

characteristic

156

C₄

#3676⋊C₃

yes

characteristic

157

C₃

—

158

C₃

C₃×#4565

yes

characteristic

159

C₂

—

160

C₂

#63392⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂	#63392⋊C₃
commutator	C₈×C₉	C₂×C₃×C₄
Frattini	C₂×C₃×C₈	C₂×C₃×D₃
Fitting	C₂×Q₁₆×3₋¹⁺²	C₂
socle	C₂×C₃	C₃×#832
prime radical	C₂²×C₃²	#214
cosocle	C₂×C₈×C₉	C₂²×C₃
hypercenter	C₂×Q₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#362
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂

#63392⋊C₃

commutator

C₈×C₉

C₂×C₃×C₄

Frattini

C₂×C₃×C₈

C₂×C₃×D₃

Fitting

C₂×Q₁₆×3₋¹⁺²

C₂

C₂×C₃

C₂²×C₃²

C₂×C₈×C₉

C₂²×C₃

C₂×Q₁₆

C₉

G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

N	H ≅ G/N	type
5. #10588242	C₃	semidirect
104. 3₋¹⁺²	#362	semidirect
141. C₉	C₃×#362	semidirect

H ≅ G/N

type

5. #10588242

C₃

semidirect

104. 3₋¹⁺²

#362

semidirect

141. C₉

C₃×#362

semidirect

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

subgroup	index
C₂×Q₁₆×3₋¹⁺²	2
#63372⋊C₃	2
#63371⋊C₃	2
C₃×#4565	3
#10588242	3

subgroup

index

C₂×Q₁₆×3₋¹⁺²

Subgroups — #10588242⋊C₃ #13135711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588242⋊C3 #13135711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588242⋊C₃ #13135711