Group #27171122 — subgroups

Subgroups — #27171122

Order 12928.

336 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 95 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×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265200	C₂	1	2	yes	maximal, complemented
3	C₂×#15692172	C₂	1	1	yes	maximal, characteristic
4	#20264945	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₁	C₄	1	2	yes	—
7	C₂×#13108058	C₂²	1	2	yes	—
8	D₄×Dic₁₀₁	—	2	2	no	—
9	D₄×Dic₁₀₁	—	2	2	no	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₁₀₁	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	#15692184	—	2	2	no	—
14	#15692184	—	2	2	no	—
15	#15692172	—	2	2	no	—
16	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
18	#100×C₁₀₁	—	2	2	no	—
19	C₂²×Dic₁₀₁	—	2	2	no	—
20	C₂²×Dic₁₀₁	—	2	2	no	—
21	C₂²×Dic₁₀₁	—	4	4	no	—
22	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
25	#13108058	—	2	2	no	—
26	#13108058	—	2	2	no	—
27	C₂×#11717579	—	2	2	no	—
28	C₄⋊Dic₁₀₁	—	2	2	no	—
29	C₂²⋊Dic₁₀₁	—	2	4	no	—
30	C₄×Dic₁₀₁	D₄	1	2	yes	—
31	C₄×Dic₁₀₁	D₄	1	2	yes	—
32	C₂⁴×C₁₀₁	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₁₀₁	—	2	2	no	—
34	C₂×D₄×C₁₀₁	—	2	2	no	—
35	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
36	C₂×D₄×C₁₀₁	Q₈	1	2	yes	—
37	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₀₁	—	2	2	no	—
39	M₁₆×C₁₀₁	—	2	2	no	—
40	C₂×C₈×C₁₀₁	—	2	2	no	—
41	C₂³×C₁₀₁	—	2	2	no	—
42	C₂³×C₁₀₁	—	2	2	no	—
43	C₂³×C₁₀₁	—	4	2	no	—
44	C₂³×C₁₀₁	—	4	2	no	—
45	C₂³×C₁₀₁	C₄⋊C₄	1	4	yes	—
46	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
47	D₄×C₁₀₁	—	2	4	no	—
48	D₄×C₁₀₁	—	2	4	no	—
49	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₀₁	#41	1	2	yes	—
53	C₂×Dic₁₀₁	—	2	2	no	—
54	C₂×Dic₁₀₁	—	2	2	no	—
55	C₂×Dic₁₀₁	—	4	1	no	—
56	C₂×Dic₁₀₁	—	4	4	no	—
57	C₂×Dic₁₀₁	—	4	4	no	—
58	C₂×Dic₁₀₁	—	4	2	no	—
59	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
60	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₈×C₁₀₁	—	4	2	no	—
63	#11717579	—	4	2	no	—
64	C₂²×C₁₀₁	—	2	1	no	—
65	C₂²×C₁₀₁	—	2	1	no	—
66	C₂²×C₁₀₁	—	2	4	no	—
67	C₂²×C₁₀₁	—	4	4	no	—
68	C₂²×C₁₀₁	—	4	4	no	—
69	C₂²×C₁₀₁	—	4	2	no	—
70	C₂²×C₁₀₁	—	4	2	no	—
71	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₀₁	#117	1	1	yes	characteristic
74	C₄×C₁₀₁	#121	1	2	yes	—
75	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
76	Dic₁₀₁	—	2	1	no	—
77	Dic₁₀₁	—	2	2	no	—
78	Dic₁₀₁	—	4	2	no	—
79	Dic₁₀₁	#100	1	2	yes	complemented
80	C₂×C₁₀₁	—	2	1	no	—
81	C₂×C₁₀₁	—	2	1	no	—
82	C₂×C₁₀₁	—	4	4	no	—
83	C₂×C₁₀₁	#380	1	1	yes	characteristic
84	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
85	C₂×C₁₀₁	#413	1	1	yes	characteristic
86	#1800	—	101	1	no	maximal, Sylow
87	C₁₀₁	#1800	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄×D₄	—	101	1	no	—
89	#423	—	101	2	no	—
90	C₂×#100	—	101	1	no	—
91	C₂×#100	D₁₀₁	1	1	yes	characteristic
92	#343	—	101	2	no	—
93	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
94	C₂³×C₄	—	101	2	no	—
95	C₂×M₁₆	—	101	2	no	—
96	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
97	C₄×D₄	—	202	2	no	—
98	C₄×D₄	—	202	2	no	—
99	C₂×C₄⋊C₄	—	101	1	no	—
100	C₂×C₂²⋊C₄	—	101	2	no	—
101	C₂×C₄²	—	101	1	no	—
102	C₄⋊C₈	—	202	2	no	—
103	C₄⋊C₈	—	202	2	no	—
104	#100	—	2	2	no	—
105	#100	—	202	2	no	—
106	C₂⁴	C₄×D₁₀₁	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	Dic₂₀₂	1	2	yes	—
110	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
111	C₂²×C₄	—	101	1	no	—
112	C₂²×C₄	—	101	1	no	—
113	C₂²×C₄	—	101	2	no	—
114	C₂²×C₄	—	404	4	no	—
115	C₂²×C₄	—	202	2	no	—
116	C₂²×C₄	—	202	2	no	—
117	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
118	M₁₆	—	2	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	202	2	no	—
121	M₁₆	—	202	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	202	2	no	—
124	C₄⋊C₄	—	202	2	no	—
125	C₂²⋊C₄	—	202	4	no	—
126	C₄²	—	101	2	no	—
127	C₄²	—	101	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	2	no	—
132	C₂³	#13108053	1	4	yes	—
133	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
134	D₄	—	2	4	no	—
135	D₄	—	2	4	no	—
136	C₂×C₄	—	101	1	no	—
137	C₂×C₄	—	101	2	no	—
138	C₂×C₄	—	101	1	no	—
139	C₂×C₄	—	404	4	no	—
140	C₂×C₄	—	404	4	no	—
141	C₂×C₄	—	404	2	no	—
142	C₂×C₄	—	404	1	no	—
143	C₂×C₄	—	202	2	no	—
144	C₂×C₄	—	202	2	no	—
145	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
147	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
148	C₂×C₄	#13108073	1	2	yes	—
149	C₈	—	4	2	no	—
150	C₈	—	404	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
159	C₂²	C₂×#13108053	1	1	yes	characteristic
160	C₂²	#15692211	1	1	yes	characteristic
161	C₄	—	101	2	no	—
162	C₄	—	404	2	no	—
163	C₄	—	202	1	no	—
164	C₄	—	202	2	no	—
165	C₄	#15692227	1	2	yes	—
166	C₄	#15692232	1	2	yes	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	4	4	no	—
170	C₂	#20265035	1	1	yes	characteristic
171	C₂	#100×D₁₀₁	1	1	yes	characteristic
172	C₂	#20265158	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×Dic₁₀₁

C₂

yes

maximal, characteristic

#20265200

C₂

yes

maximal, complemented

C₂×#15692172

C₂

yes

maximal, characteristic

#20264945

C₂

yes

maximal, complemented

C₂×#100×C₁₀₁

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₁

C₄

yes

—

C₂×#13108058

C₂²

yes

—

D₄×Dic₁₀₁

—

D₄×Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₀₁

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

—

—

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

—

#100×C₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

—

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

D₄

yes

—

C₂⁴×C₁₀₁

C₂×C₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#100

yes

complemented

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1800

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

—

101

—

#423

—

101

—

C₂×#100

—

101

—

C₂×#100

D₁₀₁

yes

characteristic

#343

—

101

—

C₂²×D₄

C₂×D₁₀₁

yes

characteristic

C₂³×C₄

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₄×D₄

—

202

—

C₄×D₄

—

202

—

C₂×C₄⋊C₄

—

101

—

100

C₂×C₂²⋊C₄

—

101

—

101

C₂×C₄²

—

101

—

102

C₄⋊C₈

—

202

—

103

C₄⋊C₈

—

202

—

104

#100

—

105

#100

—

202

—

106

C₂⁴

C₄×D₁₀₁

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

Dic₂₀₂

yes

—

110

C₂×D₄

C₂²⋊D₁₀₁

yes

—

111

C₂²×C₄

—

101

—

112

C₂²×C₄

—

101

—

113

C₂²×C₄

—

101

—

114

C₂²×C₄

—

404

—

115

C₂²×C₄

—

202

—

116

C₂²×C₄

—

202

—

117

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

202

—

121

M₁₆

—

202

—

122

C₂×C₈

—

123

C₂×C₈

—

202

—

124

C₄⋊C₄

—

202

—

125

C₂²⋊C₄

—

202

—

126

C₄²

—

101

—

127

C₄²

—

101

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

#13108053

yes

—

133

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

134

D₄

—

135

D₄

—

136

C₂×C₄

—

101

—

137

C₂×C₄

—

101

—

138

C₂×C₄

—

101

—

139

C₂×C₄

—

404

—

140

C₂×C₄

—

404

—

141

C₂×C₄

—

404

—

142

C₂×C₄

—

404

—

143

C₂×C₄

—

202

—

144

C₂×C₄

—

202

—

145

C₂×C₄

D₄×D₁₀₁

yes

—

146

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

147

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

148

C₂×C₄

#13108073

yes

—

149

C₈

—

150

C₈

—

404

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂²⋊C₄×D₁₀₁

yes

characteristic

159

C₂²

C₂×#13108053

yes

characteristic

160

C₂²

#15692211

yes

characteristic

161

C₄

—

101

—

162

C₄

—

404

—

163

C₄

—

202

—

164

C₄

—

202

—

165

C₄

#15692227

yes

—

166

C₄

#15692232

yes

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#20265035

yes

characteristic

171

C₂

#100×D₁₀₁

yes

characteristic

172

C₂

#20265158

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₀₁
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#100×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#100	D₁₀₁
hypocenter	C₁₀₁	#1800
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₀₁

commutator

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#100×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #20265200	C₂	semidirect
4. #20264945	C₂	semidirect
79. Dic₁₀₁	#100	semidirect
87. C₁₀₁	#1800	semidirect

H ≅ G/N

type

2. #20265200

C₂

semidirect

4. #20264945

C₂

semidirect

79. Dic₁₀₁

#100

semidirect

87. C₁₀₁

#1800

semidirect

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

subgroup	index
C₂×D₄×Dic₁₀₁	2
#20265200	2
C₂×#15692172	2
#20264945	2
C₂×#100×C₁₀₁	2
#1800	101

subgroup

index

C₂×D₄×Dic₁₀₁

C₂×#100×C₁₀₁

101

Subgroups — #27171122

Conjugacy classes of subgroups

Complements

Maximal subgroups