Group #10694621 — subgroups

Subgroups — #10694621

Order 768.

414 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 155 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×#4680	C₂	1	1	yes	maximal, characteristic
2	C₃×#1450	C₂	1	1	yes	maximal, characteristic
3	#70780	C₂	1	2	yes	maximal
4	#70779	C₂	1	2	yes	maximal
5	#69792	C₂	1	1	yes	maximal, characteristic
6	#9767	—	3	1	no	maximal, Sylow
7	C₂×#869	C₂²	1	2	yes	—
8	C₃×#431	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#819	—	2	2	no	—
11	C₂×#817	—	2	2	no	—
12	#4924	C₂²	1	1	yes	characteristic
13	#4924	C₂²	1	1	yes	characteristic
14	#4680	—	2	1	no	—
15	#4680	—	2	1	no	—
16	#4680	C₄	1	4	yes	—
17	C₃×#322	—	2	1	no	—
18	C₃×C₈²	—	2	1	no	—
19	#4482	—	2	4	no	—
20	#4481	—	2	4	no	—
21	C₂×#498	—	3	1	no	—
22	#1848	—	3	1	no	—
23	#1549	—	3	2	no	—
24	#1548	—	3	2	no	—
25	#1450	D₃	1	1	yes	characteristic
26	C₂²×Dic₆	—	2	2	no	—
27	C₂²×C₃×C₈	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×#208	—	2	1	no	—
32	C₂×#208	—	2	1	no	—
33	#869	—	2	2	no	—
34	#869	—	2	2	no	—
35	#869	—	2	2	no	—
36	#869	C₂×C₄	1	4	yes	—
37	C₃×C₄⋊C₈	D₄	1	4	yes	—
38	C₃×C₂²⋊C₈	—	2	2	no	—
39	C₃×C₄×C₈	—	2	2	no	—
40	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
41	C₃×C₄×C₈	D₄	1	1	yes	characteristic
42	C₃×C₄×C₈	D₄	1	1	yes	characteristic
43	#819	—	4	4	no	—
44	#817	—	2	4	no	—
45	#817	—	2	4	no	—
46	C₄⋊(C₃⋊C₈)	—	2	2	no	—
47	C₄⋊(C₃⋊C₈)	—	2	2	no	—
48	C₂×C₄⋊Q₈	—	3	2	no	—
49	#498	—	3	4	no	—
50	#498	—	6	1	no	—
51	#498	—	6	1	no	—
52	#431	C₂×D₃	1	2	yes	—
53	C₂×#108	—	6	2	no	—
54	C₂×#106	—	6	2	no	—
55	#423	—	3	1	no	—
56	#423	—	3	1	no	—
57	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	#333	—	6	4	no	—
59	#332	—	6	4	no	—
60	#322	—	2	1	no	—
61	C₈²	—	2	1	no	—
62	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	#41	1	2	yes	—
64	C₂²×Dic₃	—	4	2	no	—
65	C₂×Dic₆	—	2	4	no	—
66	C₂×Dic₆	—	4	2	no	—
67	C₂×Dic₆	—	8	4	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₃×C₈	SD₁₆	1	4	yes	—
72	C₂×C₃×C₈	Q₁₆	1	4	yes	—
73	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₄²	C₂×D₄	1	2	yes	—
75	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₄⋊Dic₃	—	2	4	no	—
77	C₄⋊Dic₃	—	2	4	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₄⋊Dic₃	—	4	2	no	—
80	#208	—	4	2	no	—
81	#208	—	4	2	no	—
82	C₂×C₃⋊C₈	—	4	1	no	—
83	C₂×C₃⋊C₈	—	4	1	no	—
84	C₂²×Q₈	—	6	2	no	—
85	C₂×M₁₆	—	6	1	no	—
86	C₂×M₁₆	—	6	1	no	—
87	C₂²×C₈	D₁₂	1	2	yes	—
88	C₄⋊Q₈	—	3	4	no	—
89	C₄⋊Q₈	—	6	2	no	—
90	C₄⋊Q₈	—	6	2	no	—
91	C₄⋊Q₈	—	6	2	no	—
92	C₂×C₄⋊C₄	—	6	2	no	—
93	C₂×C₄⋊C₄	—	6	2	no	—
94	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
95	#108	—	12	4	no	—
96	#106	—	6	4	no	—
97	#106	—	6	4	no	—
98	C₄⋊C₈	—	6	2	no	—
99	C₄⋊C₈	—	6	2	no	—
100	C₄⋊C₈	D₁₂	1	4	yes	—
101	C₂²⋊C₈	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	C₄×D₃	1	2	yes	—
104	C₄×C₈	#67	1	1	yes	characteristic
105	C₄×C₈	#67	1	1	yes	characteristic
106	C₂³×C₃	#124	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
110	C₂×C₃×C₄	#121	1	2	yes	—
111	C₂×C₃×C₄	#127	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
113	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
114	C₂×Dic₃	—	4	4	no	—
115	C₂×Dic₃	—	8	4	no	—
116	Dic₆	—	8	4	no	—
117	Dic₆	—	8	4	no	—
118	C₃×C₈	—	4	4	no	—
119	C₃×C₈	#103	1	4	yes	—
120	C₃×C₈	#103	1	4	yes	—
121	C₃⋊C₈	—	8	1	no	—
122	C₃⋊C₈	—	8	1	no	—
123	C₂×Q₈	—	6	4	no	—
124	C₂×Q₈	—	12	2	no	—
125	C₂×Q₈	—	24	4	no	—
126	C₂²×C₄	—	12	2	no	—
127	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
128	C₂²×C₄	#235	1	2	yes	—
129	M₁₆	—	12	2	no	—
130	M₁₆	—	12	2	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	12	1	no	—
133	C₂×C₈	—	12	1	no	—
134	C₂×C₈	#204	1	4	yes	—
135	C₂×C₈	Dic₁₂	1	4	yes	—
136	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
137	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
138	C₄⋊C₄	—	6	4	no	—
139	C₄⋊C₄	—	6	4	no	—
140	C₄⋊C₄	—	12	2	no	—
141	C₄⋊C₄	—	12	2	no	—
142	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
143	C₄²	C₂×D₁₂	1	2	yes	—
144	C₄²	C₂×#67	1	1	yes	characteristic
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	#390	1	1	yes	characteristic
147	C₂²×C₃	#487	1	2	yes	—
148	C₃×C₄	—	2	2	no	—
149	C₃×C₄	C₂×#103	1	2	yes	—
150	C₃×C₄	#471	1	4	yes	—
151	C₃×C₄	#492	1	1	yes	characteristic
152	C₃×C₄	#494	1	1	yes	characteristic
153	Dic₃	—	8	4	no	—
154	C₂³	#875	1	1	yes	characteristic
155	Q₈	—	24	4	no	—
156	Q₈	—	24	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	12	4	no	—
159	C₂×C₄	—	24	4	no	—
160	C₂×C₄	C₄×D₁₂	1	2	yes	—
161	C₂×C₄	#874	1	1	yes	characteristic
162	C₂×C₄	C₂×#204	1	2	yes	—
163	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
164	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
165	C₂×C₄	#930	1	2	yes	—
166	C₈	—	4	4	no	—
167	C₈	—	24	1	no	—
168	C₈	—	24	1	no	—
169	C₈	#816	1	4	yes	—
170	C₈	#816	1	4	yes	—
171	C₂×C₃	—	2	2	no	—
172	C₂×C₃	#1956	1	1	yes	characteristic
173	C₂×C₃	#1977	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#4681	1	2	yes	—
176	C₂²	#4939	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	24	4	no	—
179	C₄	#4693	1	1	yes	characteristic
180	C₄	#4699	1	1	yes	characteristic
181	C₄	C₂×#816	1	2	yes	—
182	C₄	#5141	1	4	yes	—
183	C₃	#9767	1	1	yes	characteristic, complemented, Sylow
184	C₂	—	2	2	no	—
185	C₂	#69771	1	1	yes	characteristic
186	C₂	#69902	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4680

C₂

yes

maximal, characteristic

C₃×#1450

C₂

yes

maximal, characteristic

#70780

C₂

yes

maximal

#70779

C₂

yes

maximal

#69792

C₂

yes

maximal, characteristic

#9767

—

maximal, Sylow

C₂×#869

C₂²

yes

—

C₃×#431

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

—

C₂²

yes

characteristic

#4924

C₂²

yes

—

—

C₄

yes

—

C₃×#322

—

C₃×C₈²

—

—

—

—

—

—

—

D₃

yes

characteristic

C₂²×Dic₆

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

—

—

—

—

—

C₂×C₄

yes

—

C₃×C₄⋊C₈

D₄

yes

—

C₃×C₂²⋊C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

—

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊Q₈

—

—

—

—

C₂×D₃

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

C₈²

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₂×C₃×C₈

Q₁₆

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₁₂

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

D₁₂

yes

—

101

C₂²⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₄×D₃

yes

—

104

C₄×C₈

#67

yes

characteristic

105

C₄×C₈

#67

yes

characteristic

106

C₂³×C₃

#124

yes

characteristic

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

109

C₂×C₃×C₄

C₄×D₄

yes

—

110

C₂×C₃×C₄

#121

yes

—

111

C₂×C₃×C₄

#127

yes

characteristic

112

C₂×C₃×C₄

C₂×SD₁₆

yes

—

113

C₂×C₃×C₄

C₂×Q₁₆

yes

—

114

C₂×Dic₃

—

115

C₂×Dic₃

—

116

Dic₆

—

117

Dic₆

—

118

C₃×C₈

—

119

C₃×C₈

#103

yes

—

120

C₃×C₈

#103

yes

—

121

C₃⋊C₈

—

122

C₃⋊C₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×D₁₂

yes

characteristic

128

C₂²×C₄

#235

yes

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

#204

yes

—

135

C₂×C₈

Dic₁₂

yes

—

136

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

137

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

C₂×C₄×D₃

yes

characteristic

143

C₄²

C₂×D₁₂

yes

—

144

C₄²

C₂×#67

yes

characteristic

145

C₂²×C₃

—

146

C₂²×C₃

#390

yes

characteristic

147

C₂²×C₃

#487

yes

—

148

C₃×C₄

—

149

C₃×C₄

C₂×#103

yes

—

150

C₃×C₄

#471

yes

—

151

C₃×C₄

#492

yes

characteristic

152

C₃×C₄

#494

yes

characteristic

153

Dic₃

—

154

C₂³

#875

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

C₄×D₁₂

yes

—

161

C₂×C₄

#874

yes

characteristic

162

C₂×C₄

C₂×#204

yes

—

163

C₂×C₄

C₂×Dic₁₂

yes

—

164

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

165

C₂×C₄

#930

yes

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

#816

yes

—

170

C₈

#816

yes

—

171

C₂×C₃

—

172

C₂×C₃

#1956

yes

characteristic

173

C₂×C₃

#1977

yes

—

174

C₂²

—

175

C₂²

#4681

yes

—

176

C₂²

#4939

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

#4693

yes

characteristic

180

C₄

#4699

yes

characteristic

181

C₄

C₂×#816

yes

—

182

C₄

#5141

yes

—

183

C₃

#9767

yes

characteristic, complemented, Sylow

184

C₂

—

185

C₂

#69771

yes

characteristic

186

C₂

#69902

yes

—

	subgroup	G/H
center	C₂²	#4939
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1450	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1450	D₃
hypocenter	C₃	#9767
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#4939

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
183. C₃	#9767	semidirect

H ≅ G/N

type

183. C₃

#9767

semidirect

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

subgroup	index
C₂×#4680	2
C₃×#1450	2
#70780	2
#70779	2
#69792	2
#9767	3

subgroup

index

Subgroups — #10694621

Conjugacy classes of subgroups

Complements

Maximal subgroups