Group #72226155 — subgroups

Subgroups — #72226155

Order 1536.

566 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 133 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	#10709899	C₂	1	1	yes	maximal, characteristic, complemented
2	#10709898	C₂	1	2	yes	maximal, complemented
3	#10709874	C₂	1	2	yes	maximal, complemented
4	#10681801	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#12517	C₂	1	1	yes	maximal, characteristic, complemented
6	#123412	—	3	1	no	maximal, Sylow
7	C₃×#2248	—	2	2	no	—
8	C₃×#1713	C₂²	1	1	yes	characteristic
9	C₃×#1712	C₂²	1	1	yes	characteristic
10	#71381	—	2	4	no	—
11	#71290	—	2	4	no	—
12	#69060	—	2	2	no	—
13	#68062	C₂²	1	2	yes	—
14	C₃×C₈×C₁₆	C₂²	1	1	yes	characteristic
15	C₈×C₃⋊C₁₆	C₂²	1	2	yes	—
16	#12517	D₃	1	1	yes	characteristic, complemented
17	#12515	—	3	2	no	—
18	#12515	—	3	1	no	—
19	#12509	—	3	2	no	—
20	#7052	—	3	1	no	—
21	C₂×C₃×SD₃₂	—	4	4	no	—
22	C₃×#500	—	2	2	no	—
23	C₃×#494	—	2	1	no	—
24	C₃×#493	—	2	1	no	—
25	C₃×#493	—	2	2	no	—
26	C₂×#826	—	4	8	no	—
27	C₂×D₄₈	—	4	4	no	—
28	#4695	—	2	2	no	—
29	#4695	—	2	2	no	—
30	#4695	—	2	2	no	—
31	C₃×C₄×C₁₆	D₄	1	2	yes	—
32	C₃×C₈²	C₂³	1	1	yes	characteristic
33	#4491	—	4	8	no	—
34	C₄×C₃⋊C₁₆	D₄	1	4	yes	—
35	#2248	—	2	2	no	—
36	#2246	—	6	4	no	—
37	#2246	—	6	2	no	—
38	#2240	—	6	4	no	—
39	#1713	C₂×D₃	1	1	yes	characteristic
40	#1712	—	3	2	no	—
41	#1712	C₂×D₃	1	1	yes	characteristic
42	C₈×C₁₆	—	3	2	no	—
43	C₈×C₁₆	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×Q₁₆	—	4	2	no	—
45	C₂×C₃×D₈	—	4	2	no	—
46	C₂×C₃×D₈	—	4	4	no	—
47	C₃×C₄⋊Q₈	—	4	1	no	—
48	C₃×#127	—	4	1	no	—
49	C₂×D₂₄	—	4	4	no	—
50	C₂×D₂₄	—	4	4	no	—
51	C₂×D₂₄	—	4	4	no	—
52	#874	—	4	2	no	—
53	C₃×SD₃₂	—	8	8	no	—
54	C₂×C₃×C₁₆	D₈	1	4	yes	—
55	C₃×#107	—	4	4	no	—
56	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
57	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
58	#826	—	8	16	no	—
59	C₂×C₃⋊C₁₆	D₈	1	8	yes	—
60	D₄₈	—	8	8	no	—
61	C₂×SD₃₂	—	4	4	no	—
62	C₂×D₁₆	—	12	8	no	—
63	C₂×D₁₆	—	12	4	no	—
64	#500	—	2	2	no	—
65	#494	—	2	1	no	—
66	#493	—	2	2	no	—
67	#493	—	2	1	no	—
68	#493	—	6	2	no	—
69	#493	—	6	2	no	—
70	#493	—	6	2	no	—
71	#357	—	12	8	no	—
72	C₄×C₁₆	—	3	4	no	—
73	C₄×C₁₆	#67	1	2	yes	—
74	C₈²	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃×Q₈	—	8	1	no	—
76	C₂×C₃×D₄	—	8	2	no	—
77	C₂×C₃×D₄	—	8	1	no	—
78	C₂×D₁₂	—	8	2	no	—
79	C₂×D₁₂	—	8	2	no	—
80	C₂×D₁₂	—	8	2	no	—
81	C₃×Q₁₆	—	8	4	no	—
82	C₃×D₈	—	8	4	no	—
83	C₃×D₈	—	8	8	no	—
84	C₂×C₃×C₈	C₂×D₈	1	4	yes	—
85	C₂×C₃×C₈	C₂×D₈	1	2	yes	—
86	C₃×C₄⋊C₄	—	8	2	no	—
87	C₃×C₄²	#127	1	1	yes	characteristic
88	D₂₄	—	8	8	no	—
89	D₂₄	—	8	8	no	—
90	D₂₄	—	8	8	no	—
91	C₃×C₁₆	D₁₆	1	8	yes	complemented
92	C₃⋊C₁₆	—	2	8	no	—
93	C₂×Q₁₆	—	4	2	no	—
94	C₂×D₈	—	4	4	no	—
95	C₂×D₈	—	4	2	no	—
96	C₂×D₈	—	12	4	no	—
97	C₂×D₈	—	12	4	no	—
98	C₂×D₈	—	12	4	no	—
99	C₄⋊Q₈	—	4	1	no	—
100	#127	—	4	1	no	—
101	#127	—	12	2	no	—
102	SD₃₂	—	8	8	no	—
103	D₁₆	—	24	16	no	—
104	D₁₆	—	24	8	no	—
105	C₂×C₁₆	—	3	8	no	—
106	C₂×C₁₆	#213	1	4	yes	—
107	#107	—	4	4	no	—
108	C₄×C₈	D₃×D₄	1	2	yes	—
109	C₄×C₈	C₂×#67	1	1	yes	characteristic
110	C₂³×C₃	—	16	1	no	—
111	C₂²×D₃	—	16	2	no	—
112	C₃×Q₈	—	16	2	no	—
113	C₃×D₄	—	16	2	no	—
114	C₃×D₄	—	16	4	no	—
115	C₂×C₃×C₄	—	16	1	no	—
116	C₂×C₃×C₄	#493	1	2	yes	—
117	C₂×C₃×C₄	#493	1	1	yes	characteristic
118	D₁₂	—	16	4	no	—
119	D₁₂	—	16	4	no	—
120	D₁₂	—	16	4	no	—
121	C₃×C₈	C₂×D₁₆	1	4	yes	—
122	C₃×C₈	#509	1	8	yes	—
123	C₂×Q₈	—	8	1	no	—
124	C₂×D₄	—	8	2	no	—
125	C₂×D₄	—	8	1	no	—
126	C₂×D₄	—	24	2	no	—
127	C₂×D₄	—	24	2	no	—
128	C₂×D₄	—	24	2	no	—
129	Q₁₆	—	8	4	no	—
130	D₈	—	8	4	no	—
131	D₈	—	8	8	no	—
132	D₈	—	24	8	no	—
133	D₈	—	24	8	no	—
134	D₈	—	24	8	no	—
135	C₂×C₈	D₃×D₈	1	4	yes	—
136	C₂×C₈	C₂×#213	1	2	yes	—
137	C₄⋊C₄	—	8	2	no	—
138	C₄²	#940	1	1	yes	characteristic
139	C₁₆	—	6	8	no	—
140	C₁₆	#826	1	8	yes	complemented
141	C₂²×C₃	—	32	1	no	—
142	C₂²×C₃	—	32	2	no	—
143	C₂²×C₃	#1712	1	1	yes	characteristic
144	C₂×D₃	—	32	2	no	—
145	C₂×D₃	—	32	2	no	—
146	C₂×D₃	—	32	2	no	—
147	C₃×C₄	—	32	1	no	—
148	C₃×C₄	#2246	1	2	yes	—
149	C₃×C₄	#2250	1	4	yes	—
150	C₂³	—	16	1	no	—
151	C₂³	—	48	2	no	—
152	Q₈	—	16	2	no	—
153	D₄	—	16	4	no	—
154	D₄	—	16	2	no	—
155	D₄	—	48	4	no	—
156	D₄	—	48	4	no	—
157	D₄	—	48	4	no	—
158	C₂×C₄	—	16	1	no	—
159	C₂×C₄	#5073	1	1	yes	characteristic
160	C₂×C₄	#5151	1	2	yes	—
161	C₈	#4914	1	8	yes	—
162	C₈	C₂×#826	1	4	yes	—
163	C₂×C₃	—	64	1	no	—
164	C₂×C₃	#12515	1	1	yes	characteristic
165	C₂×C₃	#12519	1	2	yes	—
166	D₃	—	64	2	no	—
167	C₂²	—	32	1	no	—
168	C₂²	—	32	2	no	—
169	C₂²	—	96	2	no	—
170	C₂²	—	96	2	no	—
171	C₂²	—	96	2	no	—
172	C₂²	#71276	1	1	yes	characteristic
173	C₄	—	32	1	no	—
174	C₄	#71290	1	2	yes	—
175	C₄	#71481	1	4	yes	—
176	C₃	#123412	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	64	1	no	—
178	C₂	—	192	2	no	—
179	C₂	#10709898	1	1	yes	characteristic
180	C₂	#10709914	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10709899

C₂

yes

maximal, characteristic, complemented

#10709898

C₂

yes

maximal, complemented

#10709874

C₂

yes

maximal, complemented

#10681801

C₂

yes

maximal, characteristic, complemented

C₃×#12517

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

characteristic

C₃×#1712

C₂²

yes

—

—

—

C₂²

yes

—

C₃×C₈×C₁₆

C₂²

yes

characteristic

C₈×C₃⋊C₁₆

C₂²

yes

—

#12517

D₃

yes

characteristic, complemented

—

—

—

—

C₂×C₃×SD₃₂

—

—

—

—

—

—

C₂×D₄₈

—

#4695

—

#4695

—

#4695

—

C₃×C₄×C₁₆

D₄

yes

—

C₃×C₈²

C₂³

yes

characteristic

#4491

—

C₄×C₃⋊C₁₆

D₄

yes

—

—

—

—

—

C₂×D₃

yes

characteristic

#1712

—

#1712

C₂×D₃

yes

characteristic

C₈×C₁₆

—

C₈×C₁₆

C₂×D₃

yes

characteristic

C₂×C₃×Q₁₆

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₃×C₄⋊Q₈

—

C₃×#127

—

C₂×D₂₄

—

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

C₃×SD₃₂

—

C₂×C₃×C₁₆

D₈

yes

—

C₃×#107

—

C₃×C₄×C₈

C₂×D₄

yes

—

C₃×C₄×C₈

C₂×D₄

yes

characteristic

#826

—

C₂×C₃⋊C₁₆

D₈

yes

—

D₄₈

—

C₂×SD₃₂

—

C₂×D₁₆

—

C₂×D₁₆

—

—

—

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

#67

yes

—

C₈²

C₂²×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₃×Q₁₆

—

C₃×D₈

—

C₃×D₈

—

C₂×C₃×C₈

C₂×D₈

yes

—

C₂×C₃×C₈

C₂×D₈

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

#127

yes

characteristic

D₂₄

—

D₂₄

—

D₂₄

—

C₃×C₁₆

D₁₆

yes

complemented

C₃⋊C₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

100

#127

—

101

#127

—

102

SD₃₂

—

103

D₁₆

—

104

D₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

#213

yes

—

107

#107

—

108

C₄×C₈

D₃×D₄

yes

—

109

C₄×C₈

C₂×#67

yes

characteristic

110

C₂³×C₃

—

111

C₂²×D₃

—

112

C₃×Q₈

—

113

C₃×D₄

—

114

C₃×D₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

#493

yes

—

117

C₂×C₃×C₄

#493

yes

characteristic

118

D₁₂

—

119

D₁₂

—

120

D₁₂

—

121

C₃×C₈

C₂×D₁₆

yes

—

122

C₃×C₈

#509

yes

—

123

C₂×Q₈

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

Q₁₆

—

130

D₈

—

131

D₈

—

132

D₈

—

133

D₈

—

134

D₈

—

135

C₂×C₈

D₃×D₈

yes

—

136

C₂×C₈

C₂×#213

yes

—

137

C₄⋊C₄

—

138

C₄²

#940

yes

characteristic

139

C₁₆

—

140

C₁₆

#826

yes

complemented

141

C₂²×C₃

—

142

C₂²×C₃

—

143

C₂²×C₃

#1712

yes

characteristic

144

C₂×D₃

—

145

C₂×D₃

—

146

C₂×D₃

—

147

C₃×C₄

—

148

C₃×C₄

#2246

yes

—

149

C₃×C₄

#2250

yes

—

150

C₂³

—

151

C₂³

—

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

C₂×C₄

—

159

C₂×C₄

#5073

yes

characteristic

160

C₂×C₄

#5151

yes

—

161

C₈

#4914

yes

—

162

C₈

C₂×#826

yes

—

163

C₂×C₃

—

164

C₂×C₃

#12515

yes

characteristic

165

C₂×C₃

#12519

yes

—

166

D₃

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#71276

yes

characteristic

173

C₄

—

174

C₄

#71290

yes

—

175

C₄

#71481

yes

—

176

C₃

#123412

yes

characteristic, complemented, Sylow

177

C₂

—

178

C₂

—

192

—

179

C₂

#10709898

yes

characteristic

180

C₂

#10709914

yes

—

	subgroup	G/H
center	C₂²	#71276
commutator, cosocle	C₃×C₈²	C₂³
Frattini	C₈²	C₂²×D₃
Fitting	C₃×#12517	C₂
socle	C₂²×C₃	#1712
hypercenter	#12517	D₃
hypocenter	C₃	#123412
LCS	G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #12517

subgroup

G/H

center

C₂²

#71276

commutator, cosocle

C₃×C₈²

C₂³

Frattini

C₈²

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #12517

N	H ≅ G/N	type
1. #10709899	C₂	semidirect
2. #10709898	C₂	semidirect
3. #10709874	C₂	semidirect
4. #10681801	C₂	semidirect
5. C₃×#12517	C₂	semidirect
16. #12517	D₃	semidirect
91. C₃×C₁₆	D₁₆	semidirect
140. C₁₆	#826	semidirect
176. C₃	#123412	semidirect

H ≅ G/N

type

1. #10709899

C₂

semidirect

2. #10709898

C₂

semidirect

3. #10709874

C₂

semidirect

4. #10681801

C₂

semidirect

5. C₃×#12517

C₂

semidirect

16. #12517

D₃

semidirect

91. C₃×C₁₆

D₁₆

semidirect

140. C₁₆

#826

semidirect

176. C₃

#123412

semidirect

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

subgroup	index
#10709899	2
#10709898	2
#10709874	2
#10681801	2
C₃×#12517	2
#123412	3

subgroup

index

Subgroups — #72226155

Conjugacy classes of subgroups

Complements

Maximal subgroups