Group #27172220 — subgroups

Subgroups — #27172220

Order 12928.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#13108062	C₂	1	2	yes	maximal, complemented
2	C₂×#15692176	C₂	1	2	yes	maximal, complemented
3	#20265083	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264932	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
7	C₂×#13108062	—	2	4	no	—
8	C₂×#13108062	—	2	4	no	—
9	C₂×#13108062	—	2	4	no	—
10	C₂²×#11717579	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
12	#15692176	—	2	4	no	—
13	#15692162	—	2	2	no	—
14	C₂²×D₄×C₁₀₁	—	2	2	no	—
15	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
16	#127×C₁₀₁	—	2	4	no	—
17	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₀₂	—	2	2	no	—
19	C₂×Dic₂₀₂	—	2	2	no	—
20	C₂×Dic₂₀₂	—	2	1	no	—
21	C₂×Dic₂₀₂	D₄	1	2	yes	complemented
22	C₂²×Dic₁₀₁	—	2	1	no	—
23	C₂²×Dic₁₀₁	—	2	1	no	—
24	#13108062	—	2	16	no	—
25	C₂×#11717579	—	2	2	no	—
26	C₂×#11717579	—	2	2	no	—
27	C₂×#11717579	D₄	1	4	yes	complemented
28	C₄⋊Dic₁₀₁	—	2	2	no	—
29	C₂⁴×C₁₀₁	—	2	2	no	—
30	C₂×D₄×C₁₀₁	—	2	8	no	—
31	C₂×D₄×C₁₀₁	—	2	2	no	—
32	C₂×D₄×C₁₀₁	—	2	2	no	—
33	C₂×D₄×C₁₀₁	—	2	2	no	—
34	C₂×D₄×C₁₀₁	—	2	2	no	—
35	C₂×D₄×C₁₀₁	—	2	2	no	—
36	C₂×D₄×C₁₀₁	—	4	4	no	—
37	C₂×D₄×C₁₀₁	—	4	4	no	—
38	C₂×D₄×C₁₀₁	D₄	1	4	yes	—
39	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₄²×C₁₀₁	—	2	2	no	—
42	C₂³×C₁₀₁	—	2	4	no	—
43	C₂³×C₁₀₁	—	4	2	no	—
44	C₂³×C₁₀₁	—	4	2	no	—
45	C₂³×C₁₀₁	—	4	4	no	—
46	C₂³×C₁₀₁	—	4	4	no	—
47	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
48	D₄×C₁₀₁	—	2	4	no	—
49	D₄×C₁₀₁	—	2	4	no	—
50	D₄×C₁₀₁	—	4	16	no	—
51	D₄×C₁₀₁	—	4	2	no	—
52	D₄×C₁₀₁	—	4	2	no	—
53	C₂×C₄×C₁₀₁	—	2	2	no	—
54	C₂×C₄×C₁₀₁	—	2	2	no	—
55	C₂×C₄×C₁₀₁	SD₁₆	1	4	yes	complemented
56	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
60	C₂×Dic₁₀₁	—	2	2	no	—
61	C₂×Dic₁₀₁	—	4	1	no	—
62	C₂×Dic₁₀₁	—	4	1	no	—
63	C₂×Dic₁₀₁	—	4	1	no	—
64	C₂×Dic₁₀₁	—	4	2	no	—
65	Dic₂₀₂	—	2	4	no	—
66	Dic₂₀₂	—	4	2	no	—
67	#11717579	—	2	4	no	—
68	C₂²×C₁₀₁	—	4	4	no	—
69	C₂²×C₁₀₁	—	4	8	no	—
70	C₂²×C₁₀₁	—	8	4	no	—
71	C₂²×C₁₀₁	—	8	2	no	—
72	C₂²×C₁₀₁	—	8	2	no	—
73	C₂²×C₁₀₁	#120	1	1	yes	characteristic
74	C₂²×C₁₀₁	#121	1	1	yes	characteristic
75	C₂²×C₁₀₁	#127	1	1	yes	characteristic
76	C₂²×C₁₀₁	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
78	C₄×C₁₀₁	—	2	4	no	—
79	C₄×C₁₀₁	#120	1	2	yes	—
80	C₄×C₁₀₁	#121	1	2	yes	—
81	Dic₁₀₁	—	4	2	no	—
82	Dic₁₀₁	—	8	1	no	—
83	C₂×C₁₀₁	—	8	4	no	—
84	C₂×C₁₀₁	#392	1	1	yes	characteristic
85	C₂×C₁₀₁	#450	1	2	yes	—
86	C₂×C₁₀₁	#460	1	2	yes	—
87	C₂×C₁₀₁	#492	1	1	yes	characteristic
88	C₂×C₁₀₁	#496	1	1	yes	characteristic
89	#2014	—	101	1	no	maximal, Sylow
90	C₁₀₁	#2014	1	1	yes	characteristic, complemented, Sylow
91	C₂²×SD₁₆	—	101	2	no	—
92	C₂×#127	D₁₀₁	1	1	yes	characteristic
93	C₂×#102	—	101	2	no	—
94	#391	—	101	1	no	—
95	#336	—	101	1	no	—
96	C₂²×Q₈	—	101	1	no	—
97	C₂²×D₄	—	2	2	no	—
98	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
99	C₂×SD₁₆	—	202	4	no	—
100	C₂×SD₁₆	—	202	4	no	—
101	C₂×SD₁₆	—	202	4	no	—
102	C₂²×C₈	—	101	2	no	—
103	#127	—	2	4	no	—
104	C₂×C₄⋊C₄	—	101	1	no	—
105	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
106	#102	—	202	4	no	—
107	#95	—	202	2	no	—
108	C₂⁴	—	2	2	no	—
109	C₂×Q₈	—	101	2	no	—
110	C₂×Q₈	—	202	2	no	—
111	C₂×Q₈	—	202	2	no	—
112	C₂×Q₈	—	202	1	no	—
113	C₂×D₄	—	2	8	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	—	4	4	no	—
120	C₂×D₄	—	4	4	no	—
121	C₂×D₄	C₂²⋊D₁₀₁	1	4	yes	—
122	C₂²×C₄	—	202	1	no	—
123	C₂²×C₄	—	202	1	no	—
124	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
126	SD₁₆	—	202	16	no	—
127	C₂×C₈	—	101	4	no	—
128	C₂×C₈	—	202	2	no	—
129	C₂×C₈	—	202	2	no	—
130	C₄⋊C₄	—	202	2	no	—
131	C₄²	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	4	4	no	—
136	C₂³	—	4	4	no	—
137	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
138	Q₈	—	404	2	no	—
139	Q₈	—	202	4	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	16	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	404	1	no	—
148	C₂×C₄	—	404	2	no	—
149	C₂×C₄	—	404	1	no	—
150	C₂×C₄	—	404	1	no	—
151	C₂×C₄	—	202	2	no	—
152	C₂×C₄	#13108062	1	4	yes	complemented
153	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
154	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
156	C₂×C₄	#13108073	1	1	yes	characteristic
157	C₈	—	202	4	no	—
158	C₂²	—	4	8	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	4	no	—
163	C₂²	#15692224	1	1	yes	characteristic
164	C₂²	#15692226	1	1	yes	characteristic
165	C₂²	#15692251	1	1	yes	characteristic
166	C₂²	C₂×#13108062	1	2	yes	—
167	C₂²	#15692281	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	808	1	no	—
170	C₄	—	404	2	no	—
171	C₄	#15692223	1	2	yes	—
172	C₄	#15692227	1	2	yes	—
173	C₂	—	8	4	no	—
174	C₂	#20265085	1	1	yes	characteristic
175	C₂	#20265328	1	2	yes	—
176	C₂	#20265374	1	2	yes	—
177	C₂	#20265549	1	1	yes	characteristic
178	C₂	#20265564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13108062

C₂

yes

maximal, complemented

C₂×#15692176

C₂

yes

maximal, complemented

#20265083

C₂

yes

maximal, characteristic, complemented

#20264932

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

#15692176

—

#15692162

—

C₂²×D₄×C₁₀₁

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

#127×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

D₄

yes

complemented

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

—

—

—

D₄

yes

complemented

C₄⋊Dic₁₀₁

—

C₂⁴×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

SD₁₆

yes

complemented

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

C₂×SD₁₆

yes

—

C₂²×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

—

C₄×C₁₀₁

#121

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#392

yes

characteristic

C₂×C₁₀₁

#450

yes

—

C₂×C₁₀₁

#460

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#2014

yes

characteristic, complemented, Sylow

C₂²×SD₁₆

—

101

—

C₂×#127

D₁₀₁

yes

characteristic

C₂×#102

—

101

—

#391

—

101

—

#336

—

101

—

C₂²×Q₈

—

101

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₂×SD₁₆

—

202

—

100

C₂×SD₁₆

—

202

—

101

C₂×SD₁₆

—

202

—

102

C₂²×C₈

—

101

—

103

#127

—

104

C₂×C₄⋊C₄

—

101

—

105

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

106

#102

—

202

—

107

#95

—

202

—

108

C₂⁴

—

109

C₂×Q₈

—

101

—

110

C₂×Q₈

—

202

—

111

C₂×Q₈

—

202

—

112

C₂×Q₈

—

202

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

C₂²⋊D₁₀₁

yes

—

122

C₂²×C₄

—

202

—

123

C₂²×C₄

—

202

—

124

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

125

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

126

SD₁₆

—

202

—

127

C₂×C₈

—

101

—

128

C₂×C₈

—

202

—

129

C₂×C₈

—

202

—

130

C₄⋊C₄

—

202

—

131

C₄²

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

138

Q₈

—

404

—

139

Q₈

—

202

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

404

—

148

C₂×C₄

—

404

—

149

C₂×C₄

—

404

—

150

C₂×C₄

—

404

—

151

C₂×C₄

—

202

—

152

C₂×C₄

#13108062

yes

complemented

153

C₂×C₄

D₄×D₁₀₁

yes

characteristic

154

C₂×C₄

D₄×D₁₀₁

yes

—

155

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

156

C₂×C₄

#13108073

yes

characteristic

157

C₈

—

202

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#15692224

yes

characteristic

164

C₂²

#15692226

yes

characteristic

165

C₂²

#15692251

yes

characteristic

166

C₂²

C₂×#13108062

yes

—

167

C₂²

#15692281

yes

—

168

C₄

—

169

C₄

—

808

—

170

C₄

—

404

—

171

C₄

#15692223

yes

—

172

C₄

#15692227

yes

—

173

C₂

—

174

C₂

#20265085

yes

characteristic

175

C₂

#20265328

yes

—

176

C₂

#20265374

yes

—

177

C₂

#20265549

yes

characteristic

178

C₂

#20265564

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting, prime radical	C₂×#127×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
hypercenter	C₂×#127	D₁₀₁
hypocenter	C₁₀₁	#2014
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting, prime radical

C₂×#127×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂²×#13108062	C₂	semidirect
2. C₂×#15692176	C₂	semidirect
3. #20265083	C₂	semidirect
4. #20264932	C₂	semidirect
21. C₂×Dic₂₀₂	D₄	semidirect
27. C₂×#11717579	D₄	semidirect
55. C₂×C₄×C₁₀₁	SD₁₆	semidirect
90. C₁₀₁	#2014	semidirect
152. C₂×C₄	#13108062	semidirect

H ≅ G/N

type

1. C₂²×#13108062

C₂

semidirect

2. C₂×#15692176

C₂

semidirect

3. #20265083

C₂

semidirect

4. #20264932

C₂

semidirect

21. C₂×Dic₂₀₂

D₄

semidirect

27. C₂×#11717579

D₄

semidirect

55. C₂×C₄×C₁₀₁

SD₁₆

semidirect

90. C₁₀₁

#2014

semidirect

152. C₂×C₄

#13108062

semidirect

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

subgroup	index
C₂²×#13108062	2
C₂×#15692176	2
#20265083	2
#20264932	2
C₂×#127×C₁₀₁	2
#2014	101

subgroup

index

C₂×#127×C₁₀₁

101

Subgroups — #27172220

Conjugacy classes of subgroups

Complements

Maximal subgroups