Group #10694583 — subgroups

Subgroups — #10694583

Order 768.

478 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 115 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	#78467	C₂	1	1	yes	maximal, characteristic
2	C₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
3	#70790	C₂	1	4	yes	maximal, complemented
4	#69763	C₂	1	1	yes	maximal, characteristic, complemented
5	#9760	—	3	1	no	maximal, Sylow
6	C₂×#904	—	2	2	no	—
7	#5477	C₂²	1	2	yes	—
8	C₃×#431	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#4936	C₂²	1	2	yes	—
11	#4699	C₄	1	1	yes	characteristic, complemented
12	#4696	—	2	2	no	—
13	#4695	C₄	1	1	yes	characteristic, complemented
14	#4693	C₄	1	2	yes	complemented
15	C₃×#322	—	2	2	no	—
16	#4551	—	2	2	no	—
17	#4470	—	2	8	no	—
18	#3147	—	3	1	no	—
19	#1835	—	3	1	no	—
20	#1517	—	3	4	no	—
21	#1452	D₃	1	1	yes	characteristic, complemented
22	C₂×#235	—	2	2	no	—
23	C₂²×C₃×C₈	D₄	1	2	yes	—
24	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
25	#930	—	2	4	no	—
26	C₂×C₄⋊Dic₃	—	2	2	no	—
27	C₂×Dic₁₂	—	2	2	no	—
28	#904	—	4	8	no	—
29	C₂×D₂₄	—	2	2	no	—
30	C₂×#204	—	2	4	no	—
31	#875	—	2	2	no	—
32	#874	C₂×C₄	1	2	yes	—
33	C₄×D₁₂	—	2	4	no	—
34	#869	C₂×C₄	1	2	yes	—
35	C₃×C₄⋊C₈	—	2	2	no	—
36	C₃×C₂²⋊C₈	D₄	1	4	yes	—
37	C₃×C₄×C₈	—	2	2	no	—
38	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄×C₈	D₄	1	2	yes	—
41	C₃⋊#95	—	2	4	no	—
42	C₂×#135	—	6	2	no	—
43	#532	—	3	2	no	—
44	#495	—	6	2	no	—
45	#494	—	3	1	no	—
46	#493	—	3	1	no	—
47	#492	—	3	2	no	—
48	#431	C₂×D₃	1	2	yes	—
49	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
50	#382	—	3	2	no	—
51	#340	—	6	2	no	—
52	#327	—	6	8	no	—
53	#322	—	2	2	no	—
54	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
55	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×#67	—	4	2	no	—
57	C₂²×Dic₃	—	4	2	no	—
58	#235	—	4	8	no	—
59	C₂×D₁₂	—	2	2	no	—
60	C₂×D₁₂	—	2	4	no	—
61	C₂×C₄×D₃	—	4	2	no	—
62	C₂×Dic₆	—	2	2	no	—
63	C₂×C₃×C₈	—	2	4	no	—
64	C₂×C₃×C₈	—	2	4	no	—
65	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
67	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₃×C₄²	#41	1	2	yes	—
70	(C₂×D₃)⋊C₄	—	4	4	no	—
71	C₄⋊Dic₃	—	2	4	no	—
72	C₄⋊Dic₃	—	4	4	no	—
73	Dic₁₂	—	4	4	no	—
74	D₂₄	—	4	4	no	—
75	#204	—	4	8	no	—
76	C₂×#41	—	6	2	no	—
77	#135	—	12	8	no	—
78	C₂×Q₁₆	—	6	2	no	—
79	C₂×SD₁₆	—	6	4	no	—
80	C₂×D₈	—	6	2	no	—
81	C₂²×C₈	D₁₂	1	2	yes	—
82	C₄⋊Q₈	—	3	2	no	—
83	#127	—	3	2	no	—
84	#124	—	6	2	no	—
85	#121	—	6	4	no	—
86	C₄×D₄	—	6	4	no	—
87	C₂×C₄⋊C₄	—	6	2	no	—
88	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	C₂²⋊C₈	D₁₂	1	4	yes	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	C₄×D₃	1	1	yes	characteristic
93	C₄×C₈	C₄×D₃	1	1	yes	characteristic
94	C₄×C₈	#67	1	2	yes	—
95	#95	—	6	4	no	—
96	C₂³×C₃	#127	1	1	yes	characteristic
97	C₂²×D₃	—	4	2	no	—
98	C₂×C₃×C₄	—	4	2	no	—
99	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
101	C₂×C₃×C₄	#121	1	2	yes	—
102	C₂×C₃×C₄	#124	1	1	yes	characteristic
103	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
104	#67	—	8	4	no	—
105	C₂×Dic₃	—	4	2	no	—
106	C₂×Dic₃	—	8	4	no	—
107	C₂×Dic₃	—	8	2	no	—
108	D₁₂	—	4	4	no	—
109	D₁₂	—	8	4	no	—
110	C₄×D₃	—	8	4	no	—
111	Dic₆	—	4	4	no	—
112	C₃×C₈	—	2	2	no	—
113	C₃×C₈	—	2	2	no	—
114	C₃×C₈	—	4	4	no	—
115	#41	—	12	8	no	—
116	C₂×Q₈	—	6	2	no	—
117	C₂×D₄	—	6	4	no	—
118	C₂×D₄	—	6	2	no	—
119	C₂×D₄	—	12	2	no	—
120	C₂²×C₄	—	12	2	no	—
121	C₂²×C₄	—	12	2	no	—
122	C₂²×C₄	C₂×D₁₂	1	2	yes	—
123	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
124	Q₁₆	—	12	4	no	—
125	SD₁₆	—	12	8	no	—
126	D₈	—	12	4	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
130	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
131	C₄⋊C₄	—	6	4	no	—
132	C₄⋊C₄	—	12	4	no	—
133	C₂²⋊C₄	—	12	4	no	—
134	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
135	C₄²	#235	1	2	yes	—
136	C₄²	C₂×#67	1	1	yes	characteristic
137	C₂²×C₃	—	2	2	no	—
138	C₂²×C₃	#390	1	1	yes	characteristic
139	C₂²×C₃	#496	1	2	yes	—
140	C₂×D₃	—	8	2	no	—
141	C₂×D₃	—	8	4	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	#418	1	2	yes	—
145	C₃×C₄	#490	1	2	yes	—
146	Dic₃	—	8	2	no	—
147	Dic₃	—	16	2	no	—
148	C₂³	—	12	2	no	—
149	C₂³	#874	1	1	yes	characteristic
150	Q₈	—	12	4	no	—
151	D₄	—	12	4	no	—
152	D₄	—	24	4	no	—
153	D₄	—	24	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	12	2	no	—
156	C₂×C₄	—	24	4	no	—
157	C₂×C₄	—	24	4	no	—
158	C₂×C₄	—	24	2	no	—
159	C₂×C₄	C₄×D₁₂	1	2	yes	—
160	C₂×C₄	#875	1	1	yes	characteristic
161	C₂×C₄	#908	1	4	yes	—
162	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#930	1	2	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	4	4	no	—
167	C₂×C₃	—	2	2	no	—
168	C₂×C₃	#1961	1	1	yes	characteristic
169	C₂×C₃	#1976	1	2	yes	—
170	D₃	—	16	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	24	4	no	—
173	C₂²	—	24	2	no	—
174	C₂²	#4712	1	2	yes	—
175	C₂²	#4939	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	24	2	no	—
179	C₄	—	48	2	no	—
180	C₄	#4713	1	2	yes	—
181	C₄	#5131	1	2	yes	—
182	C₃	#9760	1	1	yes	characteristic, complemented, Sylow
183	C₂	—	2	2	no	—
184	C₂	—	48	2	no	—
185	C₂	#69823	1	1	yes	characteristic
186	C₂	#69897	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#78467

C₂

yes

maximal, characteristic

C₃×#1452

C₂

yes

maximal, characteristic, complemented

#70790

C₂

yes

maximal, complemented

#69763

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

C₃×#431

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4936

C₂²

yes

—

#4699

C₄

yes

characteristic, complemented

#4696

—

#4695

C₄

yes

characteristic, complemented

#4693

C₄

yes

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂×#235

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

#930

—

C₂×C₄⋊Dic₃

—

C₂×Dic₁₂

—

#904

—

C₂×D₂₄

—

C₂×#204

—

#875

—

#874

C₂×C₄

yes

—

C₄×D₁₂

—

#869

C₂×C₄

yes

—

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

D₄

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

—

—

—

—

—

—

—

C₂×D₃

yes

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂×#67

—

C₂²×Dic₃

—

#235

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

#41

yes

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

D₂₄

—

#204

—

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

D₁₂

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

D₁₂

yes

—

C₄×C₈

—

C₄×C₈

C₄×D₃

yes

characteristic

C₄×C₈

C₄×D₃

yes

characteristic

C₄×C₈

#67

yes

—

#95

—

C₂³×C₃

#127

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₃×C₄

C₄×D₄

yes

—

101

C₂×C₃×C₄

#121

yes

—

102

C₂×C₃×C₄

#124

yes

characteristic

103

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

104

#67

—

105

C₂×Dic₃

—

106

C₂×Dic₃

—

107

C₂×Dic₃

—

108

D₁₂

—

109

D₁₂

—

110

C₄×D₃

—

111

Dic₆

—

112

C₃×C₈

—

113

C₃×C₈

—

114

C₃×C₈

—

115

#41

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×D₁₂

yes

—

123

C₂²×C₄

C₂×D₁₂

yes

characteristic

124

Q₁₆

—

125

SD₁₆

—

126

D₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

130

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

C₂×C₄×D₃

yes

characteristic

135

C₄²

#235

yes

—

136

C₄²

C₂×#67

yes

characteristic

137

C₂²×C₃

—

138

C₂²×C₃

#390

yes

characteristic

139

C₂²×C₃

#496

yes

—

140

C₂×D₃

—

141

C₂×D₃

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

#418

yes

—

145

C₃×C₄

#490

yes

—

146

Dic₃

—

147

Dic₃

—

148

C₂³

—

149

C₂³

#874

yes

characteristic

150

Q₈

—

151

D₄

—

152

D₄

—

153

D₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

C₄×D₁₂

yes

—

160

C₂×C₄

#875

yes

characteristic

161

C₂×C₄

#908

yes

—

162

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

163

C₂×C₄

#930

yes

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₂×C₃

—

168

C₂×C₃

#1961

yes

characteristic

169

C₂×C₃

#1976

yes

—

170

D₃

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#4712

yes

—

175

C₂²

#4939

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#4713

yes

—

181

C₄

#5131

yes

—

182

C₃

#9760

yes

characteristic, complemented, Sylow

183

C₂

—

184

C₂

—

185

C₂

#69823

yes

characteristic

186

C₂

#69897

yes

—

	subgroup	G/H
center	C₂²	#4939
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1452	C₂
socle	C₂²×C₃	#390
prime radical	#78467	C₂
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1452	D₃
hypocenter	C₃	#9760
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#4939

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
2. C₃×#1452	C₂	semidirect
3. #70790	C₂	semidirect
4. #69763	C₂	semidirect
11. #4699	C₄	semidirect
13. #4695	C₄	semidirect
14. #4693	C₄	semidirect
21. #1452	D₃	semidirect
182. C₃	#9760	semidirect

H ≅ G/N

type

2. C₃×#1452

C₂

semidirect

3. #70790

C₂

semidirect

4. #69763

C₂

semidirect

11. #4699

C₄

semidirect

13. #4695

C₄

semidirect

14. #4693

C₄

semidirect

21. #1452

D₃

semidirect

182. C₃

#9760

semidirect

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

subgroup	index
#78467	2
C₃×#1452	2
#70790	2
#69763	2
#9760	3

subgroup

index

Subgroups — #10694583

Conjugacy classes of subgroups

Complements

Maximal subgroups