Group #72226171 — subgroups

Subgroups — #72226171

Order 1536.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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	#10709931	C₂	1	1	yes	maximal, characteristic, complemented
2	#10709905	C₂	1	2	yes	maximal, complemented
3	#10709899	C₂	1	1	yes	maximal, characteristic
4	#10709873	C₂	1	2	yes	maximal, complemented
5	C₃×#7052	C₂	1	1	yes	maximal, characteristic, complemented
6	#123419	—	3	1	no	maximal, Sylow
7	#69061	—	2	4	no	—
8	#69060	—	2	2	no	—
9	#69058	—	2	4	no	—
10	#69046	—	2	2	no	—
11	#68065	C₂²	1	2	yes	—
12	#68062	C₂²	1	1	yes	characteristic
13	#68062	C₂²	1	1	yes	characteristic
14	C₃×C₈×C₁₆	C₂²	1	2	yes	—
15	C₃×C₈×C₁₆	C₂²	1	1	yes	characteristic
16	#12523	—	3	1	no	—
17	#12517	—	3	2	no	—
18	#12515	—	3	1	no	—
19	#12509	—	3	2	no	—
20	#7052	D₃	1	1	yes	characteristic, complemented
21	C₂×#800	—	4	8	no	—
22	C₂×D₄₈	—	4	4	no	—
23	#4699	—	2	2	no	—
24	#4695	—	2	2	no	—
25	#4695	—	2	1	no	—
26	#4695	—	2	1	no	—
27	#4695	—	2	2	no	—
28	#4682	—	2	2	no	—
29	#4682	—	2	2	no	—
30	C₃×C₄×C₁₆	D₄	1	2	yes	—
31	C₃×C₄×C₁₆	D₄	1	4	yes	—
32	C₃×C₈²	C₂³	1	1	yes	characteristic
33	#4509	—	4	8	no	—
34	#4505	—	4	4	no	—
35	#2254	—	6	2	no	—
36	#2248	—	6	4	no	—
37	#2246	—	6	2	no	—
38	#2240	—	6	4	no	—
39	#1713	—	3	2	no	—
40	#1712	—	3	1	no	—
41	#1712	—	3	1	no	—
42	C₈×C₁₆	C₂×D₃	1	2	yes	—
43	C₈×C₁₆	C₂×D₃	1	1	yes	characteristic
44	C₂×Dic₁₂	—	4	4	no	—
45	C₂×D₂₄	—	4	2	no	—
46	C₂×D₂₄	—	4	2	no	—
47	C₂×D₂₄	—	4	4	no	—
48	C₂×D₂₄	—	4	4	no	—
49	#874	—	4	1	no	—
50	#874	—	4	1	no	—
51	#869	—	4	2	no	—
52	C₂×C₃×C₁₆	D₈	1	8	yes	—
53	C₂×C₃×C₁₆	D₈	1	4	yes	—
54	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
56	#818	—	4	4	no	—
57	#818	—	4	4	no	—
58	#800	—	8	16	no	—
59	D₄₈	—	8	8	no	—
60	C₂×SD₃₂	—	12	8	no	—
61	C₂×D₁₆	—	12	4	no	—
62	#500	—	6	2	no	—
63	#500	—	6	2	no	—
64	#494	—	6	2	no	—
65	#493	—	6	2	no	—
66	#493	—	6	1	no	—
67	#493	—	6	1	no	—
68	#493	—	6	2	no	—
69	#367	—	12	4	no	—
70	#357	—	12	8	no	—
71	C₄×C₁₆	D₁₂	1	4	yes	—
72	C₄×C₁₆	D₁₂	1	2	yes	—
73	C₈²	C₂²×D₃	1	1	yes	characteristic
74	C₂×D₁₂	—	8	1	no	—
75	C₂×D₁₂	—	8	1	no	—
76	C₂×D₁₂	—	8	2	no	—
77	C₂×D₁₂	—	8	2	no	—
78	C₂×Dic₆	—	8	2	no	—
79	C₂×C₃×C₈	C₂×D₈	1	2	yes	—
80	C₂×C₃×C₈	C₂×D₈	1	4	yes	—
81	C₃×C₄²	#127	1	1	yes	characteristic
82	C₄⋊Dic₃	—	8	2	no	—
83	C₄⋊Dic₃	—	8	2	no	—
84	Dic₁₂	—	8	8	no	—
85	D₂₄	—	8	4	no	—
86	D₂₄	—	8	4	no	—
87	D₂₄	—	8	8	no	—
88	D₂₄	—	8	8	no	—
89	C₃×C₁₆	—	2	8	no	—
90	C₃×C₁₆	SD₃₂	1	8	yes	complemented
91	C₂×Q₁₆	—	12	4	no	—
92	C₂×D₈	—	12	4	no	—
93	C₂×D₈	—	12	2	no	—
94	C₂×D₈	—	12	2	no	—
95	C₂×D₈	—	12	4	no	—
96	C₄⋊Q₈	—	12	2	no	—
97	#127	—	12	1	no	—
98	#127	—	12	1	no	—
99	SD₃₂	—	24	16	no	—
100	D₁₆	—	24	8	no	—
101	C₂×C₁₆	D₂₄	1	8	yes	—
102	C₂×C₁₆	D₂₄	1	4	yes	—
103	#107	—	12	4	no	—
104	#107	—	12	4	no	—
105	C₄×C₈	C₂×D₁₂	1	2	yes	—
106	C₄×C₈	C₂×D₁₂	1	1	yes	characteristic
107	C₂²×D₃	—	16	1	no	—
108	C₂²×D₃	—	16	1	no	—
109	C₂×C₃×C₄	#493	1	1	yes	characteristic
110	C₂×C₃×C₄	#493	1	2	yes	—
111	C₂×Dic₃	—	16	2	no	—
112	D₁₂	—	16	2	no	—
113	D₁₂	—	16	2	no	—
114	D₁₂	—	16	4	no	—
115	D₁₂	—	16	4	no	—
116	Dic₆	—	16	4	no	—
117	C₃×C₈	C₂×SD₃₂	1	4	yes	—
118	C₃×C₈	#509	1	8	yes	—
119	C₂×Q₈	—	24	2	no	—
120	C₂×D₄	—	24	1	no	—
121	C₂×D₄	—	24	1	no	—
122	C₂×D₄	—	24	2	no	—
123	C₂×D₄	—	24	2	no	—
124	Q₁₆	—	24	8	no	—
125	D₈	—	24	4	no	—
126	D₈	—	24	4	no	—
127	D₈	—	24	8	no	—
128	D₈	—	24	8	no	—
129	C₂×C₈	C₂×D₂₄	1	2	yes	—
130	C₂×C₈	C₂×D₂₄	1	4	yes	—
131	C₄⋊C₄	—	24	2	no	—
132	C₄⋊C₄	—	24	2	no	—
133	C₄²	#874	1	1	yes	characteristic
134	C₁₆	—	2	8	no	—
135	C₁₆	#800	1	8	yes	complemented
136	C₂²×C₃	#1712	1	1	yes	characteristic
137	C₂×D₃	—	32	1	no	—
138	C₂×D₃	—	32	1	no	—
139	C₂×D₃	—	32	2	no	—
140	C₂×D₃	—	32	2	no	—
141	C₃×C₄	#2248	1	2	yes	—
142	C₃×C₄	#2250	1	4	yes	—
143	Dic₃	—	32	2	no	—
144	C₂³	—	48	1	no	—
145	C₂³	—	48	1	no	—
146	Q₈	—	48	4	no	—
147	D₄	—	48	2	no	—
148	D₄	—	48	2	no	—
149	D₄	—	48	4	no	—
150	D₄	—	48	4	no	—
151	C₂×C₄	—	48	2	no	—
152	C₂×C₄	#4695	1	1	yes	characteristic
153	C₂×C₄	#4695	1	2	yes	—
154	C₈	C₂×#800	1	4	yes	—
155	C₈	#4908	1	8	yes	—
156	C₂×C₃	#12517	1	1	yes	characteristic
157	C₂×C₃	#12519	1	2	yes	—
158	D₃	—	64	1	no	—
159	D₃	—	64	1	no	—
160	C₂²	—	96	1	no	—
161	C₂²	—	96	1	no	—
162	C₂²	—	96	2	no	—
163	C₂²	—	96	2	no	—
164	C₂²	#68062	1	1	yes	characteristic
165	C₄	—	96	2	no	—
166	C₄	#69061	1	2	yes	—
167	C₄	#69081	1	4	yes	—
168	C₃	#123419	1	1	yes	characteristic, complemented, Sylow
169	C₂	—	192	1	no	—
170	C₂	—	192	1	no	—
171	C₂	#10709905	1	1	yes	characteristic
172	C₂	#10709915	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10709931

C₂

yes

maximal, characteristic, complemented

#10709905

C₂

yes

maximal, complemented

#10709899

C₂

yes

maximal, characteristic

#10709873

C₂

yes

maximal, complemented

C₃×#7052

C₂

yes

maximal, characteristic, complemented

—

—

—

—

—

C₂²

yes

—

#68062

C₂²

yes

characteristic

#68062

C₂²

yes

characteristic

C₃×C₈×C₁₆

C₂²

yes

—

C₃×C₈×C₁₆

C₂²

yes

—

—

—

—

D₃

yes

characteristic, complemented

C₂×#800

—

C₂×D₄₈

—

—

—

—

—

—

—

—

C₃×C₄×C₁₆

D₄

yes

—

C₃×C₄×C₁₆

D₄

yes

—

C₃×C₈²

C₂³

yes

—

—

—

—

—

—

—

—

—

C₈×C₁₆

C₂×D₃

yes

—

C₈×C₁₆

C₂×D₃

yes

characteristic

C₂×Dic₁₂

—

C₂×D₂₄

—

C₂×D₂₄

—

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

#874

—

#869

—

C₂×C₃×C₁₆

D₈

yes

—

C₂×C₃×C₁₆

D₈

yes

—

C₃×C₄×C₈

C₂×D₄

yes

characteristic

C₃×C₄×C₈

C₂×D₄

yes

—

#818

—

#818

—

#800

—

D₄₈

—

C₂×SD₃₂

—

C₂×D₁₆

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

D₁₂

yes

—

C₄×C₁₆

D₁₂

yes

—

C₈²

C₂²×D₃

yes

characteristic

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

C₂×D₈

yes

—

C₂×C₃×C₈

C₂×D₈

yes

—

C₃×C₄²

#127

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

D₂₄

—

D₂₄

—

D₂₄

—

D₂₄

—

C₃×C₁₆

—

C₃×C₁₆

SD₃₂

yes

complemented

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

100

D₁₆

—

101

C₂×C₁₆

D₂₄

yes

—

102

C₂×C₁₆

D₂₄

yes

—

103

#107

—

104

#107

—

105

C₄×C₈

C₂×D₁₂

yes

—

106

C₄×C₈

C₂×D₁₂

yes

characteristic

107

C₂²×D₃

—

108

C₂²×D₃

—

109

C₂×C₃×C₄

#493

yes

characteristic

110

C₂×C₃×C₄

#493

yes

—

111

C₂×Dic₃

—

112

D₁₂

—

113

D₁₂

—

114

D₁₂

—

115

D₁₂

—

116

Dic₆

—

117

C₃×C₈

C₂×SD₃₂

yes

—

118

C₃×C₈

#509

yes

—

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

Q₁₆

—

125

D₈

—

126

D₈

—

127

D₈

—

128

D₈

—

129

C₂×C₈

C₂×D₂₄

yes

—

130

C₂×C₈

C₂×D₂₄

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄²

#874

yes

characteristic

134

C₁₆

—

135

C₁₆

#800

yes

complemented

136

C₂²×C₃

#1712

yes

characteristic

137

C₂×D₃

—

138

C₂×D₃

—

139

C₂×D₃

—

140

C₂×D₃

—

141

C₃×C₄

#2248

yes

—

142

C₃×C₄

#2250

yes

—

143

Dic₃

—

144

C₂³

—

145

C₂³

—

146

Q₈

—

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

#4695

yes

characteristic

153

C₂×C₄

#4695

yes

—

154

C₈

C₂×#800

yes

—

155

C₈

#4908

yes

—

156

C₂×C₃

#12517

yes

characteristic

157

C₂×C₃

#12519

yes

—

158

D₃

—

159

D₃

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

#68062

yes

characteristic

165

C₄

—

166

C₄

#69061

yes

—

167

C₄

#69081

yes

—

168

C₃

#123419

yes

characteristic, complemented, Sylow

169

C₂

—

192

—

170

C₂

—

192

—

171

C₂

#10709905

yes

characteristic

172

C₂

#10709915

yes

—

	subgroup	G/H
center	C₂²	#68062
commutator, cosocle	C₃×C₈²	C₂³
Frattini	C₈²	C₂²×D₃
Fitting	C₃×#7052	C₂
socle	C₂²×C₃	#1712
prime radical	#10709899	C₂
hypercenter	#7052	D₃
hypocenter	C₃	#123419
LCS	G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #7052

subgroup

G/H

center

C₂²

#68062

commutator, cosocle

C₃×C₈²

C₂³

Frattini

C₈²

C₂²×D₃

C₂

C₂²×C₃

C₂

D₃

C₃

G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #7052

N	H ≅ G/N	type
1. #10709931	C₂	semidirect
2. #10709905	C₂	semidirect
4. #10709873	C₂	semidirect
5. C₃×#7052	C₂	semidirect
20. #7052	D₃	semidirect
90. C₃×C₁₆	SD₃₂	semidirect
135. C₁₆	#800	semidirect
168. C₃	#123419	semidirect

H ≅ G/N

type

1. #10709931

C₂

semidirect

2. #10709905

C₂

semidirect

4. #10709873

C₂

semidirect

5. C₃×#7052

C₂

semidirect

20. #7052

D₃

semidirect

90. C₃×C₁₆

SD₃₂

semidirect

135. C₁₆

#800

semidirect

168. C₃

#123419

semidirect

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

subgroup	index
#10709931	2
#10709905	2
#10709899	2
#10709873	2
C₃×#7052	2
#123419	3

subgroup

index

Subgroups — #72226171

Conjugacy classes of subgroups

Complements

Maximal subgroups