Group #13144504 — subgroups

Subgroups — #13144504

Order 1728.

682 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 113 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	H(3,𝔽₃)×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#88550	C₂	1	2	yes	maximal, complemented
3	C₂×#88546	C₂	1	2	yes	maximal
4	#11721715	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720520	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5060	—	3	4	no	maximal
7	C₂×Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₂×D₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	C₂×#6229	C₂²	1	2	yes	—
10	C₂²⋊C₄×H(3,𝔽₃)	—	2	2	no	—
11	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	#88550	—	2	4	no	—
13	#88546	—	2	4	no	—
14	C₄⋊#1084	—	2	2	no	—
15	C₂×#6180	C₂²	1	2	yes	—
16	C₃²×#124	D₃	1	4	yes	—
17	C₂×C₃×#216	—	3	8	no	—
18	C₂×C₃×#214	—	3	8	no	—
19	C₃×#869	—	3	4	no	—
20	C₃×#803	—	3	4	no	—
21	C₂³×H(3,𝔽₃)	—	2	1	no	—
22	Q₈×H(3,𝔽₃)	—	2	2	no	—
23	D₄×H(3,𝔽₃)	—	2	2	no	—
24	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
25	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
26	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
27	C₂×#1084	—	2	2	no	—
28	#6229	—	2	2	no	—
29	#6229	—	2	2	no	—
30	#6180	D₄	1	4	yes	—
31	C₃×#497	—	9	1	no	—
32	#5060	—	3	4	no	—
33	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
34	C₂×C₃²×D₄	C₂×D₃	1	4	yes	—
35	C₂×C₃×Dic₆	—	3	8	no	—
36	C₃²×C₂²⋊C₄	—	2	8	no	—
37	C₃²×C₄²	C₂×D₃	1	4	yes	—
38	C₃×#216	—	6	16	no	—
39	C₃×#214	—	6	16	no	—
40	C₃×C₄⋊Dic₃	—	6	8	no	—
41	C₂×C₃×C₃⋊C₈	—	3	8	no	—
42	C₂²×H(3,𝔽₃)	—	4	1	no	—
43	C₂²×H(3,𝔽₃)	—	4	2	no	—
44	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
45	C₄×H(3,𝔽₃)	—	2	2	no	—
46	C₄×H(3,𝔽₃)	—	4	1	no	—
47	C₄×H(3,𝔽₃)	C₂×D₄	1	2	yes	—
48	#1084	—	4	2	no	—
49	C₂×C₃×Q₁₆	—	9	2	no	—
50	C₂×C₃×SD₁₆	—	9	2	no	—
51	C₃×C₄⋊Q₈	—	9	1	no	—
52	C₃×#124	—	3	4	no	—
53	C₃×#124	C₃⋊D₃	1	1	yes	characteristic
54	C₂×#216	—	3	8	no	—
55	C₂×#214	—	3	8	no	—
56	#869	—	3	4	no	—
57	C₃×#97	—	9	1	no	—
58	#803	—	3	4	no	—
59	C₂³×C₃²	—	2	4	no	—
60	C₃²×Q₈	—	2	8	no	—
61	C₃²×D₄	—	2	8	no	—
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	#67	1	8	yes	—
64	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
65	C₂×C₃×Dic₃	—	6	8	no	—
66	C₃×Dic₆	—	6	8	no	—
67	C₃×Dic₆	—	6	8	no	—
68	C₃×C₃⋊C₈	—	3	16	no	—
69	#497	—	9	1	no	Sylow
70	C₂×H(3,𝔽₃)	—	8	1	no	—
71	C₂×H(3,𝔽₃)	#127	1	1	yes	characteristic
72	C₂×H(3,𝔽₃)	#137	1	2	yes	—
73	C₂×C₃×Q₈	—	3	4	no	—
74	C₂×C₃×Q₈	—	9	2	no	—
75	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
76	C₂×C₃×D₄	—	3	4	no	—
77	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
78	C₂×Dic₆	—	3	8	no	—
79	C₃×Q₁₆	—	18	4	no	—
80	C₃×SD₁₆	—	18	4	no	—
81	C₂×C₃×C₈	—	9	2	no	—
82	C₃×C₄⋊C₄	—	18	2	no	—
83	C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₂²⋊C₄	—	6	8	no	—
85	C₃×C₄²	—	3	4	no	—
86	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
87	#216	—	6	16	no	—
88	#214	—	6	16	no	—
89	C₄⋊Dic₃	—	6	8	no	—
90	C₂×C₃⋊C₈	—	3	8	no	—
91	C₂²×C₃²	—	4	4	no	—
92	C₂²×C₃²	—	4	8	no	—
93	C₂²×C₃²	C₂×#67	1	4	yes	—
94	C₃²×C₄	—	2	8	no	—
95	C₃²×C₄	—	4	4	no	—
96	C₃²×C₄	D₃×D₄	1	8	yes	—
97	C₃×Dic₃	—	12	8	no	—
98	C₂×Q₁₆	—	9	2	no	—
99	C₂×SD₁₆	—	9	2	no	—
100	C₄⋊Q₈	—	9	1	no	—
101	#124	#272	1	1	yes	characteristic
102	#97	—	9	1	no	—
103	H(3,𝔽₃)	#497	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	—	6	4	no	—
106	C₃×Q₈	—	2	2	no	—
107	C₃×Q₈	—	6	8	no	—
108	C₃×Q₈	—	18	2	no	—
109	C₃×Q₈	—	18	2	no	—
110	C₃×D₄	—	2	2	no	—
111	C₃×D₄	—	6	8	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	3	8	no	—
114	C₂×C₃×C₄	—	3	4	no	—
115	C₂×C₃×C₄	—	6	4	no	—
116	C₂×C₃×C₄	—	18	2	no	—
117	C₂×C₃×C₄	#638	1	2	yes	—
118	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
119	C₂×Dic₃	—	6	8	no	—
120	Dic₆	—	6	8	no	—
121	Dic₆	—	6	8	no	—
122	C₃×C₈	—	9	4	no	—
123	C₃⋊C₈	—	3	16	no	—
124	C₂×C₃²	—	8	4	no	—
125	C₂×C₃²	#940	1	4	yes	—
126	C₂×C₃²	#951	1	8	yes	—
127	C₂×Q₈	—	9	2	no	—
128	C₂×Q₈	C₂×#272	1	1	yes	characteristic
129	C₂×D₄	C₂×#272	1	1	yes	characteristic
130	Q₁₆	—	18	4	no	—
131	SD₁₆	—	18	4	no	—
132	C₂×C₈	—	9	2	no	—
133	C₄⋊C₄	—	18	2	no	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₄²	C₂×#272	1	1	yes	characteristic
136	C₂²×C₃	—	3	4	no	—
137	C₂²×C₃	—	4	1	no	—
138	C₂²×C₃	—	4	2	no	—
139	C₂²×C₃	—	12	4	no	—
140	C₂²×C₃	—	12	8	no	—
141	C₂²×C₃	C₂×#638	1	1	yes	characteristic
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	3	8	no	—
144	C₃×C₄	—	4	1	no	—
145	C₃×C₄	—	6	8	no	—
146	C₃×C₄	—	12	4	no	—
147	C₃×C₄	—	36	2	no	—
148	C₃×C₄	D₄×C₃⋊D₃	1	2	yes	—
149	Dic₃	—	12	8	no	—
150	C₃²	#5060	1	4	yes	complemented
151	C₂³	—	2	1	no	—
152	Q₈	—	2	2	no	—
153	Q₈	—	18	2	no	—
154	Q₈	—	18	2	no	—
155	D₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	18	2	no	—
158	C₂×C₄	#6235	1	2	yes	—
159	C₂×C₄	C₂²×#272	1	1	yes	characteristic
160	C₈	—	9	4	no	—
161	C₂×C₃	—	3	4	no	—
162	C₂×C₃	—	3	8	no	—
163	C₂×C₃	—	8	1	no	—
164	C₂×C₃	—	24	4	no	—
165	C₂×C₃	#64149	1	1	yes	characteristic
166	C₂×C₃	#64160	1	2	yes	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	2	no	—
169	C₂²	C₂×#6235	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	1	no	—
172	C₄	—	36	2	no	—
173	C₄	D₄×#272	1	2	yes	—
174	C₃	—	3	4	no	—
175	C₃	#10592759	1	1	yes	characteristic
176	C₂	—	8	1	no	—
177	C₂	#11721850	1	1	yes	characteristic
178	C₂	#11721872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×#124

C₂

yes

maximal, characteristic

C₂×#88550

C₂

yes

maximal, complemented

C₂×#88546

C₂

yes

maximal

#11721715

C₂

yes

maximal, characteristic, complemented

#11720520

C₂

yes

maximal, characteristic, complemented

C₃×#5060

—

maximal

C₂×Q₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×D₄×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×#6229

C₂²

yes

—

C₂²⋊C₄×H(3,𝔽₃)

—

C₄²×H(3,𝔽₃)

C₂²

yes

—

—

—

C₂²

yes

—

C₃²×#124

D₃

yes

—

C₂×C₃×#216

—

C₂×C₃×#214

—

C₃×#869

—

C₃×#803

—

C₂³×H(3,𝔽₃)

—

Q₈×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

—

—

—

D₄

yes

—

C₃×#497

—

#5060

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×C₃²×D₄

C₂×D₃

yes

—

C₂×C₃×Dic₆

—

C₃²×C₂²⋊C₄

—

C₃²×C₄²

C₂×D₃

yes

—

C₃×#216

—

C₃×#214

—

C₃×C₄⋊Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₄

yes

—

#1084

—

C₂×C₃×Q₁₆

—

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

—

C₃×#124

—

C₃×#124

C₃⋊D₃

yes

—

—

—

C₃×#97

—

#803

—

C₂³×C₃²

—

C₃²×Q₈

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃×C₃⋊C₈

—

#497

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#127

yes

characteristic

C₂×H(3,𝔽₃)

#137

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×C₃⋊D₃

yes

characteristic

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×SD₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

#216

—

#214

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×#67

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

D₃×D₄

yes

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₂×SD₁₆

—

100

C₄⋊Q₈

—

101

yes

102

—

103

H(3,𝔽₃)

#497

yes

characteristic, complemented, Sylow

104

C₂³×C₃

—

105

C₂³×C₃

—

106

C₃×Q₈

—

107

C₃×Q₈

—

108

C₃×Q₈

—

109

C₃×Q₈

—

110

C₃×D₄

—

111

C₃×D₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

#638

yes

—

118

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

119

C₂×Dic₃

—

120

Dic₆

—

121

Dic₆

—

122

C₃×C₈

—

123

C₃⋊C₈

—

124

C₂×C₃²

—

125

C₂×C₃²

#940

yes

—

126

C₂×C₃²

#951

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

C₂×#272

yes

characteristic

129

C₂×D₄

C₂×#272

yes

characteristic

130

Q₁₆

—

131

SD₁₆

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

C₂×#272

yes

characteristic

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

—

139

C₂²×C₃

—

140

C₂²×C₃

—

141

C₂²×C₃

C₂×#638

yes

characteristic

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

D₄×C₃⋊D₃

yes

—

149

Dic₃

—

150

C₃²

#5060

yes

complemented

151

C₂³

—

152

Q₈

—

153

Q₈

—

154

Q₈

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#6235

yes

—

159

C₂×C₄

C₂²×#272

yes

characteristic

160

C₈

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

—

164

C₂×C₃

—

165

C₂×C₃

#64149

yes

characteristic

166

C₂×C₃

#64160

yes

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂×#6235

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

D₄×#272

yes

—

174

C₃

—

175

C₃

#10592759

yes

characteristic

176

C₂

—

177

C₂

#11721850

yes

characteristic

178

C₂

#11721872

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#638
commutator, cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	H(3,𝔽₃)×#124	C₂
prime radical	C₂×D₄×H(3,𝔽₃)	C₂²
hypercenter	C₃×#124	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#497
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#124

subgroup

G/H

center, socle

C₂²×C₃

C₂×#638

commutator, cosocle

C₂×C₄×H(3,𝔽₃)

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

H(3,𝔽₃)×#124

C₂

prime radical

C₂×D₄×H(3,𝔽₃)

C₂²

C₃⋊D₃

G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#124

N	H ≅ G/N	type
2. C₂×#88550	C₂	semidirect
4. #11721715	C₂	semidirect
5. #11720520	C₂	semidirect
103. H(3,𝔽₃)	#497	semidirect
150. C₃²	#5060	semidirect

H ≅ G/N

type

2. C₂×#88550

C₂

semidirect

4. #11721715

C₂

semidirect

5. #11720520

C₂

semidirect

103. H(3,𝔽₃)

#497

semidirect

150. C₃²

#5060

semidirect

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

subgroup	index
H(3,𝔽₃)×#124	2
C₂×#88550	2
C₂×#88546	2
#11721715	2
#11720520	2
C₃×#5060	3

subgroup

index

Subgroups — #13144504

Conjugacy classes of subgroups

Complements

Maximal subgroups