Group #10694650 — subgroups

Subgroups — #10694650

Order 768.

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

normal subgroups only

This group has 478 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×#4699	C₂	1	1	yes	maximal, characteristic
2	C₃×#1450	C₂	1	1	yes	maximal, characteristic
3	#70778	C₂	1	4	yes	maximal
4	#69792	C₂	1	1	yes	maximal, characteristic
5	#9772	—	3	1	no	maximal, Sylow
6	C₂²×Dic₁₂	—	2	2	no	—
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	#4924	C₂²	1	2	yes	—
12	#4699	—	2	1	no	—
13	#4699	—	2	1	no	—
14	#4699	C₄	1	4	yes	—
15	C₃×#322	—	2	1	no	—
16	C₃×C₈²	—	2	1	no	—
17	#4486	—	2	8	no	—
18	C₂×#494	—	3	1	no	—
19	#1848	—	3	1	no	—
20	#1546	—	3	4	no	—
21	#1450	D₃	1	1	yes	characteristic
22	C₂²×Dic₆	—	2	2	no	—
23	C₂²×C₃×C₈	D₄	1	2	yes	—
24	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃	—	2	4	no	—
26	C₂×#208	—	2	2	no	—
27	C₂×Dic₁₂	—	2	4	no	—
28	C₂×Dic₁₂	—	2	4	no	—
29	C₂×Dic₁₂	—	4	4	no	—
30	C₂×Dic₁₂	—	4	4	no	—
31	#869	—	2	4	no	—
32	#869	—	2	2	no	—
33	#869	C₂×C₄	1	4	yes	—
34	C₃×C₄⋊C₈	D₄	1	4	yes	—
35	C₃×C₂²⋊C₈	—	2	2	no	—
36	C₃×C₄×C₈	—	2	2	no	—
37	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄×C₈	D₄	1	1	yes	characteristic
39	C₃×C₄×C₈	D₄	1	1	yes	characteristic
40	#819	—	4	4	no	—
41	C₄⋊(C₃⋊C₈)	—	2	4	no	—
42	C₂²×Q₁₆	—	6	2	no	—
43	C₂×C₄⋊Q₈	—	3	2	no	—
44	#494	—	3	4	no	—
45	#494	—	6	1	no	—
46	#494	—	6	1	no	—
47	#431	C₂×D₃	1	2	yes	—
48	C₂×#108	—	6	2	no	—
49	#423	—	3	2	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	#332	—	6	8	no	—
52	#322	—	2	1	no	—
53	C₈²	—	2	1	no	—
54	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	#41	1	2	yes	—
56	C₂²×Dic₃	—	4	2	no	—
57	C₂×Dic₆	—	2	4	no	—
58	C₂×Dic₆	—	4	4	no	—
59	C₂×Dic₆	—	4	4	no	—
60	C₂×Dic₆	—	4	2	no	—
61	C₂×C₃×C₈	—	2	4	no	—
62	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
63	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₈	D₈	1	4	yes	—
65	C₂×C₃×C₈	SD₁₆	1	4	yes	—
66	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
67	C₃×C₄²	C₂×D₄	1	2	yes	—
68	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₄⋊Dic₃	—	2	8	no	—
70	C₄⋊Dic₃	—	4	4	no	—
71	#208	—	4	4	no	—
72	C₂×C₃⋊C₈	—	4	2	no	—
73	Dic₁₂	—	4	8	no	—
74	Dic₁₂	—	4	8	no	—
75	C₂²×Q₈	—	6	2	no	—
76	C₂×Q₁₆	—	6	4	no	—
77	C₂×Q₁₆	—	6	4	no	—
78	C₂×Q₁₆	—	12	4	no	—
79	C₂×Q₁₆	—	12	4	no	—
80	C₂×M₁₆	—	6	2	no	—
81	C₂²×C₈	D₁₂	1	2	yes	—
82	C₄⋊Q₈	—	3	4	no	—
83	C₄⋊Q₈	—	6	4	no	—
84	C₄⋊Q₈	—	6	2	no	—
85	C₂×C₄⋊C₄	—	6	4	no	—
86	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
87	#108	—	12	4	no	—
88	C₄⋊C₈	—	6	4	no	—
89	C₄⋊C₈	D₁₂	1	4	yes	—
90	C₂²⋊C₈	—	2	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	C₄×D₃	1	2	yes	—
93	C₄×C₈	#67	1	1	yes	characteristic
94	C₄×C₈	#67	1	1	yes	characteristic
95	C₂³×C₃	#124	1	1	yes	characteristic
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
99	C₂×C₃×C₄	#121	1	2	yes	—
100	C₂×C₃×C₄	#127	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
102	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
103	C₂×Dic₃	—	4	4	no	—
104	C₂×Dic₃	—	8	4	no	—
105	Dic₆	—	4	8	no	—
106	Dic₆	—	8	2	no	—
107	Dic₆	—	8	2	no	—
108	C₃×C₈	—	4	4	no	—
109	C₃×C₈	#102	1	4	yes	—
110	C₃×C₈	#102	1	4	yes	—
111	C₃⋊C₈	—	8	2	no	—
112	C₂×Q₈	—	6	4	no	—
113	C₂×Q₈	—	12	2	no	—
114	C₂×Q₈	—	12	4	no	—
115	C₂×Q₈	—	12	4	no	—
116	C₂²×C₄	—	12	2	no	—
117	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
118	C₂²×C₄	#235	1	2	yes	—
119	Q₁₆	—	12	8	no	—
120	Q₁₆	—	12	8	no	—
121	M₁₆	—	12	4	no	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	—	12	2	no	—
124	C₂×C₈	#204	1	4	yes	—
125	C₂×C₈	D₂₄	1	4	yes	—
126	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
127	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
128	C₄⋊C₄	—	6	8	no	—
129	C₄⋊C₄	—	12	4	no	—
130	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
131	C₄²	C₂×D₁₂	1	2	yes	—
132	C₄²	C₂×#67	1	1	yes	characteristic
133	C₂²×C₃	—	2	2	no	—
134	C₂²×C₃	#390	1	1	yes	characteristic
135	C₂²×C₃	#486	1	2	yes	—
136	C₃×C₄	—	2	2	no	—
137	C₃×C₄	C₂×#102	1	2	yes	—
138	C₃×C₄	#473	1	4	yes	—
139	C₃×C₄	#492	1	1	yes	characteristic
140	C₃×C₄	#493	1	1	yes	characteristic
141	Dic₃	—	8	4	no	—
142	C₂³	#875	1	1	yes	characteristic
143	Q₈	—	12	8	no	—
144	Q₈	—	24	2	no	—
145	Q₈	—	24	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	12	4	no	—
148	C₂×C₄	—	24	4	no	—
149	C₂×C₄	C₄×D₁₂	1	2	yes	—
150	C₂×C₄	#874	1	1	yes	characteristic
151	C₂×C₄	C₂×#204	1	2	yes	—
152	C₂×C₄	C₂×D₂₄	1	2	yes	—
153	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#930	1	2	yes	—
155	C₈	—	4	4	no	—
156	C₈	—	24	2	no	—
157	C₈	#821	1	4	yes	—
158	C₈	#821	1	4	yes	—
159	C₂×C₃	—	2	2	no	—
160	C₂×C₃	#1957	1	1	yes	characteristic
161	C₂×C₃	#1979	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	#4694	1	2	yes	—
164	C₂²	#4939	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	24	4	no	—
167	C₄	#4693	1	1	yes	characteristic
168	C₄	#4695	1	1	yes	characteristic
169	C₄	C₂×#821	1	2	yes	—
170	C₄	#5143	1	4	yes	—
171	C₃	#9772	1	1	yes	characteristic, complemented, Sylow
172	C₂	—	2	2	no	—
173	C₂	#69772	1	1	yes	characteristic
174	C₂	#69904	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4699

C₂

yes

maximal, characteristic

C₃×#1450

C₂

yes

maximal, characteristic

#70778

C₂

yes

maximal

#69792

C₂

yes

maximal, characteristic

#9772

—

maximal, Sylow

C₂²×Dic₁₂

—

C₂×#869

C₂²

yes

—

C₃×#431

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#819

—

#4924

C₂²

yes

—

#4699

—

#4699

—

#4699

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×#208

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

#869

—

#869

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

characteristic

#819

—

C₄⋊(C₃⋊C₈)

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

C₂×D₃

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#332

—

#322

—

C₈²

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

#208

—

C₂×C₃⋊C₈

—

Dic₁₂

—

Dic₁₂

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₁₂

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

D₁₂

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₃

yes

—

C₄×C₈

yes

C₄×C₈

yes

C₂³×C₃

yes

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄×D₄

yes

—

C₂×C₃×C₄

#121

yes

—

100

C₂×C₃×C₄

#127

yes

characteristic

101

C₂×C₃×C₄

C₂×D₈

yes

—

102

C₂×C₃×C₄

C₂×SD₁₆

yes

—

103

C₂×Dic₃

—

104

C₂×Dic₃

—

105

Dic₆

—

106

Dic₆

—

107

Dic₆

—

108

C₃×C₈

—

109

C₃×C₈

#102

yes

—

110

C₃×C₈

#102

yes

—

111

C₃⋊C₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×D₁₂

yes

characteristic

118

C₂²×C₄

#235

yes

—

119

Q₁₆

—

120

Q₁₆

—

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#204

yes

—

125

C₂×C₈

D₂₄

yes

—

126

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

127

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

C₂×C₄×D₃

yes

characteristic

131

C₄²

C₂×D₁₂

yes

—

132

C₄²

C₂×#67

yes

characteristic

133

C₂²×C₃

—

134

C₂²×C₃

#390

yes

characteristic

135

C₂²×C₃

#486

yes

—

136

C₃×C₄

—

137

C₃×C₄

C₂×#102

yes

—

138

C₃×C₄

#473

yes

—

139

C₃×C₄

#492

yes

characteristic

140

C₃×C₄

#493

yes

characteristic

141

Dic₃

—

142

C₂³

#875

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₄×D₁₂

yes

—

150

C₂×C₄

#874

yes

characteristic

151

C₂×C₄

C₂×#204

yes

—

152

C₂×C₄

C₂×D₂₄

yes

—

153

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

154

C₂×C₄

#930

yes

—

155

C₈

—

156

C₈

—

157

C₈

#821

yes

—

158

C₈

#821

yes

—

159

C₂×C₃

—

160

C₂×C₃

#1957

yes

characteristic

161

C₂×C₃

#1979

yes

—

162

C₂²

—

163

C₂²

#4694

yes

—

164

C₂²

#4939

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

#4693

yes

characteristic

168

C₄

#4695

yes

characteristic

169

C₄

C₂×#821

yes

—

170

C₄

#5143

yes

—

171

C₃

#9772

yes

characteristic, complemented, Sylow

172

C₂

—

173

C₂

#69772

yes

characteristic

174

C₂

#69904

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	#9772
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
171. C₃	#9772	semidirect

H ≅ G/N

type

171. C₃

#9772

semidirect

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

subgroup	index
C₂×#4699	2
C₃×#1450	2
#70778	2
#69792	2
#9772	3

subgroup

index

Subgroups — #10694650

Conjugacy classes of subgroups

Complements

Maximal subgroups