Group #11782490 — subgroups

Subgroups — #11782490

Order 1152.

397 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 104 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×#4681	C₂	1	2	yes	maximal
2	#10590387	C₂	1	1	yes	maximal, characteristic
3	#10590312	C₂	1	1	yes	maximal, characteristic
4	#10590309	C₂	1	2	yes	maximal
5	C₈×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#71212	—	3	2	no	maximal
7	C₃×#869	C₂²	1	2	yes	—
8	C₃×#869	C₂²	1	2	yes	—
9	C₂×#3718	—	2	2	no	—
10	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃×#816	—	2	4	no	—
14	#63576	—	2	2	no	—
15	#63574	—	2	4	no	—
16	C₃×#487	—	3	2	no	—
17	#5092	—	3	2	no	—
18	#5080	—	3	2	no	—
19	#5080	—	3	2	no	—
20	#5078	—	3	2	no	—
21	#4681	D₃	1	2	yes	—
22	C₈×C₃⋊C₈	—	3	2	no	—
23	C₈×C₃⋊C₈	—	6	1	no	—
24	C₂×C₃×Dic₆	—	2	2	no	—
25	C₂×C₃×Dic₆	—	2	2	no	—
26	C₂×C₃²×C₈	D₄	1	2	yes	—
27	C₃²×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
29	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
30	C₃×C₄⋊Dic₃	—	2	2	no	—
31	C₃×C₄⋊Dic₃	—	2	2	no	—
32	C₃×C₄⋊Dic₃	—	2	2	no	—
33	C₃×C₄⋊Dic₃	—	2	2	no	—
34	#3718	—	4	4	no	—
35	#1707	—	9	1	no	Sylow
36	C₃×C₄⋊Q₈	—	3	2	no	—
37	C₃×C₄⋊Q₈	—	3	2	no	—
38	C₂×#216	—	6	4	no	—
39	#869	C₂×D₃	1	2	yes	—
40	#869	C₂×D₃	1	2	yes	—
41	C₃×#103	—	6	4	no	—
42	C₃×C₄×C₈	—	2	1	no	—
43	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
44	#816	—	2	4	no	—
45	#808	—	6	4	no	—
46	#808	—	6	4	no	—
47	#807	—	6	4	no	—
48	C₄×C₃⋊C₈	—	3	2	no	—
49	C₄×C₃⋊C₈	—	3	2	no	—
50	C₄×C₃⋊C₈	—	6	1	no	—
51	C₄×C₃⋊C₈	—	6	1	no	—
52	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×Dic₃	—	4	2	no	—
56	C₂×C₃×Dic₃	—	4	2	no	—
57	C₃×Dic₆	—	4	4	no	—
58	C₃×Dic₆	—	4	4	no	—
59	C₃²×C₈	—	2	2	no	—
60	C₃⋊(C₃⋊C₈)	SD₁₆	1	4	yes	—
61	C₃⋊(C₃⋊C₈)	Q₁₆	1	4	yes	complemented
62	#500	—	9	1	no	—
63	#498	—	9	2	no	—
64	#494	—	9	1	no	—
65	#487	—	3	2	no	—
66	C₈²	—	9	1	no	—
67	C₂×C₃×Q₈	—	6	2	no	—
68	C₂×C₃×Q₈	—	6	2	no	—
69	C₂×Dic₆	—	2	2	no	—
70	C₂×Dic₆	—	2	2	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	#67	1	4	yes	—
73	C₃×C₄⋊C₄	—	6	2	no	—
74	C₃×C₄⋊C₄	—	6	2	no	—
75	C₃×C₄⋊C₄	—	6	2	no	—
76	C₃×C₄⋊C₄	—	6	2	no	—
77	C₃×C₄²	—	2	1	no	—
78	C₃×C₄²	C₂²×D₃	1	2	yes	—
79	#216	—	12	8	no	—
80	C₄⋊Dic₃	—	2	2	no	—
81	C₄⋊Dic₃	—	2	2	no	—
82	C₄⋊Dic₃	—	2	2	no	—
83	C₄⋊Dic₃	—	2	2	no	—
84	C₂×C₃⋊C₈	—	3	4	no	—
85	C₂×C₃⋊C₈	—	3	4	no	—
86	C₂×C₃⋊C₈	—	6	2	no	—
87	C₂×C₃⋊C₈	—	6	2	no	—
88	C₂²×C₃²	#127	1	1	yes	characteristic
89	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
90	C₃²×C₄	C₂×Q₁₆	1	2	yes	—
91	C₃²×C₄	#135	1	2	yes	—
92	C₃×Dic₃	—	8	2	no	—
93	C₃×Dic₃	—	8	2	no	—
94	C₂×Q₁₆	—	18	2	no	—
95	C₄⋊Q₈	—	3	2	no	—
96	C₄⋊Q₈	—	3	2	no	—
97	#107	—	18	2	no	—
98	#106	—	18	4	no	—
99	#103	—	6	4	no	—
100	C₄×C₈	—	9	1	no	—
101	C₄×C₈	—	9	1	no	—
102	C₄×C₈	D₃²	1	1	yes	characteristic
103	C₃×Q₈	—	12	4	no	—
104	C₃×Q₈	—	12	4	no	—
105	C₂×C₃×C₄	—	2	1	no	—
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	12	2	no	—
109	C₂×C₃×C₄	—	12	2	no	—
110	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
111	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
112	C₂×C₃×C₄	C₂×#67	1	2	yes	—
113	C₂×Dic₃	—	4	2	no	—
114	C₂×Dic₃	—	4	2	no	—
115	Dic₆	—	4	4	no	—
116	Dic₆	—	4	4	no	—
117	C₃×C₈	—	2	4	no	—
118	C₃×C₈	—	2	4	no	—
119	C₃⋊C₈	—	3	8	no	—
120	C₃⋊C₈	—	3	8	no	—
121	C₃⋊C₈	—	6	4	no	—
122	C₃⋊C₈	—	6	4	no	—
123	C₂×C₃²	#492	1	1	yes	characteristic
124	C₂×C₃²	#494	1	1	yes	characteristic
125	C₂×C₃²	#495	1	1	yes	characteristic
126	C₂×Q₈	—	6	2	no	—
127	C₂×Q₈	—	6	2	no	—
128	Q₁₆	—	36	4	no	—
129	C₂×C₈	—	9	2	no	—
130	C₂×C₈	—	9	2	no	—
131	C₂×C₈	#625	1	2	yes	—
132	C₄⋊C₄	—	6	2	no	—
133	C₄⋊C₄	—	6	2	no	—
134	C₄⋊C₄	—	6	2	no	—
135	C₄⋊C₄	—	6	2	no	—
136	C₄²	C₂×D₃²	1	1	yes	characteristic
137	C₂²×C₃	—	2	1	no	—
138	C₂²×C₃	#940	1	2	yes	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	—	2	2	no	—
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	24	2	no	—
143	C₃×C₄	—	24	2	no	—
144	C₃×C₄	D₃×SD₁₆	1	4	yes	—
145	C₃×C₄	D₃×Q₁₆	1	4	yes	—
146	C₃×C₄	#950	1	4	yes	—
147	Dic₃	—	8	2	no	—
148	Dic₃	—	8	2	no	—
149	C₃²	#1707	1	1	yes	characteristic, complemented, Sylow
150	Q₈	—	12	4	no	—
151	Q₈	—	12	4	no	—
152	C₂×C₄	—	12	2	no	—
153	C₂×C₄	—	12	2	no	—
154	C₂×C₄	#3802	1	1	yes	characteristic
155	C₂×C₄	#3802	1	1	yes	characteristic
156	C₂×C₄	C₂×#625	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	9	4	no	—
159	C₈	—	9	4	no	—
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	#5059	1	2	yes	—
164	C₂×C₃	#5175	1	2	yes	—
165	C₂×C₃	#5183	1	2	yes	—
166	C₂²	#63914	1	1	yes	characteristic
167	C₄	—	24	2	no	—
168	C₄	—	24	2	no	—
169	C₄	#63796	1	2	yes	—
170	C₄	#63802	1	2	yes	—
171	C₄	#63822	1	2	yes	—
172	C₃	—	2	1	no	—
173	C₃	#71212	1	2	yes	complemented
174	C₂	#10590353	1	1	yes	characteristic
175	C₂	#10590884	1	1	yes	characteristic
176	C₂	#10590905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#4681

C₂

yes

maximal

#10590387

C₂

yes

maximal, characteristic

#10590312

C₂

yes

maximal, characteristic

#10590309

C₂

yes

maximal

C₈×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#71212

—

maximal

C₃×#869

C₂²

yes

—

C₃×#869

C₂²

yes

—

C₂×#3718

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

—

—

—

—

D₃

yes

—

C₈×C₃⋊C₈

—

C₈×C₃⋊C₈

—

C₂×C₃×Dic₆

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄²

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

#3718

—

#1707

—

Sylow

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₂×#216

—

#869

C₂×D₃

yes

—

#869

C₂×D₃

yes

—

C₃×#103

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

—

—

—

—

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

SD₁₆

yes

—

C₃⋊(C₃⋊C₈)

Q₁₆

yes

—

—

—

—

C₈²

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂²×D₃

yes

—

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

C₂×SD₁₆

yes

—

C₃²×C₄

C₂×Q₁₆

yes

—

C₃²×C₄

#135

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#107

—

#106

—

#103

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

D₃²

yes

characteristic

103

C₃×Q₈

—

104

C₃×Q₈

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

D₃×D₄

yes

—

111

C₂×C₃×C₄

D₃×D₄

yes

—

112

C₂×C₃×C₄

C₂×#67

yes

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

Dic₆

—

116

Dic₆

—

117

C₃×C₈

—

118

C₃×C₈

—

119

C₃⋊C₈

—

120

C₃⋊C₈

—

121

C₃⋊C₈

—

122

C₃⋊C₈

—

123

C₂×C₃²

#492

yes

characteristic

124

C₂×C₃²

#494

yes

characteristic

125

C₂×C₃²

#495

yes

characteristic

126

C₂×Q₈

—

127

C₂×Q₈

—

128

Q₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

#625

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂×D₃²

yes

characteristic

137

C₂²×C₃

—

138

C₂²×C₃

#940

yes

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

D₃×SD₁₆

yes

—

145

C₃×C₄

D₃×Q₁₆

yes

—

146

C₃×C₄

#950

yes

—

147

Dic₃

—

148

Dic₃

—

149

C₃²

#1707

yes

characteristic, complemented, Sylow

150

Q₈

—

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#3802

yes

characteristic

155

C₂×C₄

#3802

yes

characteristic

156

C₂×C₄

C₂×#625

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₂×C₃

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

#5059

yes

—

164

C₂×C₃

#5175

yes

—

165

C₂×C₃

#5183

yes

—

166

C₂²

#63914

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

#63796

yes

—

170

C₄

#63802

yes

—

171

C₄

#63822

yes

—

172

C₃

—

173

C₃

#71212

yes

complemented

174

C₂

#10590353

yes

characteristic

175

C₂

#10590884

yes

characteristic

176

C₂

#10590905

yes

characteristic

	subgroup	G/H
center	C₂²	#63914
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle, prime radical	C₂²×C₃²	#127
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1707
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#63914

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂×D₃²

Fitting

C₃²×C₄×C₈

C₂²

socle, prime radical

C₂²×C₃²

#127

hypercenter

C₄×C₈

D₃²

hypocenter

C₃²

#1707

LCS

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
61. C₃⋊(C₃⋊C₈)	Q₁₆	semidirect
149. C₃²	#1707	semidirect
173. C₃	#71212	semidirect

H ≅ G/N

type

61. C₃⋊(C₃⋊C₈)

Q₁₆

semidirect

149. C₃²

#1707

semidirect

173. C₃

#71212

semidirect

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

subgroup	index
C₃×#4681	2
#10590387	2
#10590312	2
#10590309	2
C₈×C₃⋊(C₃⋊C₈)	2
#71212	3

subgroup

index

C₈×C₃⋊(C₃⋊C₈)

Subgroups — #11782490

Conjugacy classes of subgroups

Complements

Maximal subgroups