Group #10695037 — subgroups

Subgroups — #10695037

Order 768.

542 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 199 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×C₃×#494	C₂	1	1	yes	maximal, characteristic
2	#71787	C₂	1	4	yes	maximal
3	#69764	C₂	1	1	yes	maximal, characteristic
4	C₂×#4455	C₂	1	1	yes	maximal, characteristic
5	#9839	—	3	1	no	maximal, Sylow
6	C₂²×C₃×Q₁₆	—	2	2	no	—
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#494	—	2	2	no	—
9	C₃×#494	C₄	1	4	yes	complemented
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#817	—	2	2	no	—
12	#4935	C₂²	1	2	yes	—
13	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
14	#4529	—	2	8	no	—
15	#4455	—	2	2	no	—
16	C₂×#494	D₃	1	1	yes	characteristic
17	#1990	—	3	4	no	—
18	#1844	—	3	1	no	—
19	C₂×#322	—	3	1	no	—
20	C₂²×C₃×Q₈	—	2	2	no	—
21	C₂×C₃×Q₁₆	—	2	8	no	—
22	C₂×C₃×Q₁₆	—	4	4	no	—
23	C₂×C₃×Q₁₆	—	4	4	no	—
24	C₂²×C₃×C₈	D₄	1	2	yes	—
25	C₃×C₄⋊Q₈	—	2	2	no	—
26	C₃×C₄⋊Q₈	—	2	4	no	—
27	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄⋊C₄	—	2	4	no	—
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₃	—	2	4	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂²×C₃⋊C₈	D₄	1	4	yes	—
33	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄×C₈	D₄	1	2	yes	—
35	#817	—	4	4	no	—
36	C₄×C₃⋊C₈	—	2	2	no	—
37	C₄×C₃⋊C₈	—	2	2	no	—
38	C₂²×Q₁₆	—	2	2	no	—
39	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
40	#494	—	2	2	no	—
41	#494	Dic₃	1	4	yes	complemented
42	C₂×#106	—	6	2	no	—
43	C₂×C₄×C₈	—	3	2	no	—
44	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	#384	—	3	2	no	—
46	#340	—	6	8	no	—
47	#322	—	6	2	no	—
48	C₂×C₃×Q₈	—	2	4	no	—
49	C₂×C₃×Q₈	—	4	2	no	—
50	C₂×C₃×Q₈	—	4	4	no	—
51	C₂×C₃×Q₈	—	4	4	no	—
52	C₂²×C₃×C₄	—	4	2	no	—
53	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
54	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₃	—	4	2	no	—
56	C₃×Q₁₆	—	4	16	no	—
57	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
58	C₂×C₃×C₈	D₈	1	4	yes	—
59	C₂×C₃×C₈	SD₁₆	1	4	yes	—
60	C₃×C₄⋊C₄	—	2	8	no	—
61	C₃×C₄⋊C₄	—	4	4	no	—
62	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₃×C₄²	#41	1	2	yes	—
65	C₄⋊Dic₃	—	4	4	no	—
66	C₄⋊Dic₃	—	4	8	no	—
67	C₂×C₃⋊C₈	—	2	4	no	—
68	C₂×C₃⋊C₈	—	2	8	no	—
69	C₂²×Q₈	—	2	2	no	—
70	C₂×Q₁₆	—	2	8	no	—
71	C₂×Q₁₆	—	4	4	no	—
72	C₂×Q₁₆	—	4	4	no	—
73	C₂²×C₈	—	3	4	no	—
74	C₂²×C₈	#67	1	2	yes	—
75	C₄⋊Q₈	—	2	4	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
78	C₂×C₄⋊C₄	—	2	4	no	—
79	C₂×C₄⋊C₄	—	6	4	no	—
80	C₂×C₄⋊C₄	—	6	2	no	—
81	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
82	#106	—	12	4	no	—
83	C₄×C₈	—	6	2	no	—
84	C₄×C₈	—	6	2	no	—
85	C₄×C₈	C₂×Dic₃	1	2	yes	—
86	C₄×C₈	#67	1	2	yes	—
87	C₂³×C₃	#127	1	1	yes	characteristic
88	C₃×Q₈	—	4	8	no	—
89	C₃×Q₈	—	8	4	no	—
90	C₂×C₃×C₄	—	4	4	no	—
91	C₂×C₃×C₄	—	8	2	no	—
92	C₂×C₃×C₄	—	8	2	no	—
93	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
95	C₂×C₃×C₄	#121	1	2	yes	—
96	C₂×C₃×C₄	#124	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
98	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
99	C₂×C₃×C₄	#137	1	8	yes	—
100	C₂×Dic₃	—	8	2	no	—
101	C₂×Dic₃	—	8	4	no	—
102	C₃×C₈	#102	1	8	yes	—
103	C₃⋊C₈	—	4	4	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	4	2	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂×Q₈	—	4	4	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	—	12	2	no	—
110	C₂²×C₄	D₃×D₄	1	2	yes	—
111	C₂²×C₄	C₂×#67	1	1	yes	characteristic
112	Q₁₆	—	4	16	no	—
113	C₂×C₈	—	6	8	no	—
114	C₂×C₈	—	6	4	no	—
115	C₂×C₈	#213	1	4	yes	—
116	C₂×C₈	#214	1	4	yes	—
117	C₂×C₈	#217	1	4	yes	—
118	C₄⋊C₄	—	2	8	no	—
119	C₄⋊C₄	—	4	4	no	—
120	C₄⋊C₄	—	12	8	no	—
121	C₄⋊C₄	—	12	4	no	—
122	C₄²	#237	1	2	yes	—
123	C₄²	C₂²×Dic₃	1	1	yes	characteristic
124	C₄²	C₂×#67	1	1	yes	characteristic
125	C₂²×C₃	#390	1	1	yes	characteristic
126	C₂²×C₃	#492	1	1	yes	characteristic
127	C₂²×C₃	#493	1	1	yes	characteristic
128	C₂²×C₃	#497	1	4	yes	—
129	C₃×C₄	—	8	4	no	—
130	C₃×C₄	C₂×#102	1	2	yes	—
131	C₃×C₄	#441	1	4	yes	—
132	C₃×C₄	#470	1	4	yes	—
133	C₃×C₄	#486	1	2	yes	—
134	Dic₃	—	16	2	no	—
135	C₂³	#940	1	1	yes	characteristic
136	Q₈	—	4	8	no	—
137	Q₈	—	8	4	no	—
138	C₂×C₄	—	4	4	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	24	4	no	—
142	C₂×C₄	—	24	2	no	—
143	C₂×C₄	#918	1	8	yes	—
144	C₂×C₄	C₂×#213	1	2	yes	—
145	C₂×C₄	C₂×#214	1	2	yes	—
146	C₂×C₄	D₄×Dic₃	1	2	yes	—
147	C₂×C₄	#936	1	1	yes	characteristic
148	C₂×C₄	#938	1	2	yes	—
149	C₂×C₄	C₂×#217	1	1	yes	characteristic
150	C₈	—	12	4	no	—
151	C₈	#832	1	8	yes	—
152	C₂×C₃	#1716	1	2	yes	—
153	C₂×C₃	#1722	1	2	yes	—
154	C₂×C₃	#1957	1	1	yes	characteristic
155	C₂×C₃	#1971	1	2	yes	—
156	C₂²	#5073	1	1	yes	characteristic
157	C₂²	#5076	1	1	yes	characteristic
158	C₂²	#5190	1	4	yes	—
159	C₂²	#5221	1	1	yes	characteristic
160	C₄	—	8	4	no	—
161	C₄	—	48	2	no	—
162	C₄	#5070	1	2	yes	—
163	C₄	#5184	1	4	yes	—
164	C₄	#5189	1	4	yes	—
165	C₄	C₂×#832	1	2	yes	—
166	C₃	#9839	1	1	yes	characteristic, complemented, Sylow
167	C₂	#71299	1	2	yes	—
168	C₂	#71301	1	2	yes	—
169	C₂	#71764	1	1	yes	characteristic
170	C₂	#72067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#494

C₂

yes

maximal, characteristic

#71787

C₂

yes

maximal

#69764

C₂

yes

maximal, characteristic

C₂×#4455

C₂

yes

maximal, characteristic

#9839

—

maximal, Sylow

C₂²×C₃×Q₁₆

—

C₂×C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#494

—

C₃×#494

C₄

yes

complemented

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#817

—

#4935

C₂²

yes

—

C₂×C₄×C₃⋊C₈

C₂²

yes

—

#4529

—

#4455

—

C₂×#494

D₃

yes

—

—

—

C₂²×C₃×Q₈

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂²×C₃⋊C₈

D₄

yes

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#817

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₃

yes

—

#494

—

#494

Dic₃

yes

complemented

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₃

yes

—

C₄×C₈

#67

yes

—

C₂³×C₃

#127

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄×D₄

yes

—

C₂×C₃×C₄

#121

yes

—

C₂×C₃×C₄

#124

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

C₂×C₃×C₄

C₂×SD₁₆

yes

—

C₂×C₃×C₄

#137

yes

—

100

C₂×Dic₃

—

101

C₂×Dic₃

—

102

C₃×C₈

#102

yes

—

103

C₃⋊C₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

D₃×D₄

yes

—

111

C₂²×C₄

C₂×#67

yes

characteristic

112

Q₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

#213

yes

—

116

C₂×C₈

#214

yes

—

117

C₂×C₈

#217

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

#237

yes

—

123

C₄²

C₂²×Dic₃

yes

characteristic

124

C₄²

C₂×#67

yes

characteristic

125

C₂²×C₃

#390

yes

characteristic

126

C₂²×C₃

#492

yes

characteristic

127

C₂²×C₃

#493

yes

characteristic

128

C₂²×C₃

#497

yes

—

129

C₃×C₄

—

130

C₃×C₄

C₂×#102

yes

—

131

C₃×C₄

#441

yes

—

132

C₃×C₄

#470

yes

—

133

C₃×C₄

#486

yes

—

134

Dic₃

—

135

C₂³

#940

yes

characteristic

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#918

yes

—

144

C₂×C₄

C₂×#213

yes

—

145

C₂×C₄

C₂×#214

yes

—

146

C₂×C₄

D₄×Dic₃

yes

—

147

C₂×C₄

#936

yes

characteristic

148

C₂×C₄

#938

yes

—

149

C₂×C₄

C₂×#217

yes

characteristic

150

C₈

—

151

C₈

#832

yes

—

152

C₂×C₃

#1716

yes

—

153

C₂×C₃

#1722

yes

—

154

C₂×C₃

#1957

yes

characteristic

155

C₂×C₃

#1971

yes

—

156

C₂²

#5073

yes

characteristic

157

C₂²

#5076

yes

characteristic

158

C₂²

#5190

yes

—

159

C₂²

#5221

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

#5070

yes

—

163

C₄

#5184

yes

—

164

C₄

#5189

yes

—

165

C₄

C₂×#832

yes

—

166

C₃

#9839

yes

characteristic, complemented, Sylow

167

C₂

#71299

yes

—

168

C₂

#71301

yes

—

169

C₂

#71764

yes

characteristic

170

C₂

#72067

yes

—

	subgroup	G/H
center	C₂³	#940
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#494	C₂
socle, prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#494	D₃
hypocenter	C₃	#9839
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

#940

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₂×C₃×#494

C₂

socle, prime radical

C₂³×C₃

#127

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
9. C₃×#494	C₄	semidirect
41. #494	Dic₃	semidirect
166. C₃	#9839	semidirect

H ≅ G/N

type

9. C₃×#494

C₄

semidirect

41. #494

Dic₃

semidirect

166. C₃

#9839

semidirect

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

subgroup	index
C₂×C₃×#494	2
#71787	2
#69764	2
C₂×#4455	2
#9839	3

subgroup

index

C₂×C₃×#494

Subgroups — #10695037

Conjugacy classes of subgroups

Complements

Maximal subgroups