Group #27179454 — subgroups

Subgroups — #27179454

Order 12928.

582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15692238	C₂	1	1	yes	maximal, characteristic
2	#20265068	C₂	1	4	yes	maximal, complemented
3	#20265061	C₂	1	4	yes	maximal, complemented
4	#20265056	C₂	1	2	yes	maximal
5	#20265050	C₂	1	2	yes	maximal, complemented
6	#20265040	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄²×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
9	#15692238	—	2	4	no	—
10	C₂×#13108053	C₂²	1	4	yes	—
11	C₂×#13108053	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×#13108051	C₂²	1	4	yes	—
15	C₂×#13108051	C₂²	1	2	yes	—
16	#15692162	C₂²	1	4	yes	—
17	#15692162	C₂²	1	4	yes	—
18	#15692162	C₂²	1	4	yes	—
19	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₀₁	—	2	4	no	—
22	C₂×C₄²×C₁₀₁	C₂²	1	4	yes	—
23	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
25	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
26	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
28	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
29	#13108053	—	2	4	no	—
30	#13108053	—	2	8	no	—
31	C₂²⋊Dic₁₀₁	—	2	2	no	—
32	C₂²⋊Dic₁₀₁	—	2	4	no	—
33	#13108051	—	2	4	no	—
34	#13108051	—	2	8	no	—
35	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₁	—	2	4	no	—
37	C₂²×C₄×C₁₀₁	—	2	4	no	—
38	C₂²×C₄×C₁₀₁	—	2	8	no	—
39	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
42	C₄²×C₁₀₁	—	2	8	no	—
43	C₂³×C₁₀₁	—	2	4	no	—
44	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
45	C₂³×C₁₀₁	#41	1	4	yes	—
46	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
47	C₂²×D₁₀₁	—	2	1	no	—
48	C₂²×D₁₀₁	—	2	4	no	—
49	C₂²×D₁₀₁	—	2	2	no	—
50	C₂×C₄×C₁₀₁	—	2	16	no	—
51	C₂×C₄×C₁₀₁	—	2	8	no	—
52	C₂×C₄×C₁₀₁	—	2	4	no	—
53	C₂×C₄×C₁₀₁	—	2	4	no	—
54	C₂×C₄×C₁₀₁	—	2	4	no	—
55	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
56	C₂×C₄×C₁₀₁	#41	1	8	yes	—
57	C₂²⋊D₁₀₁	—	4	4	no	—
58	C₂×Dic₁₀₁	—	2	2	no	—
59	C₂×Dic₁₀₁	—	2	1	no	—
60	C₂×Dic₁₀₁	—	2	4	no	—
61	C₂×Dic₁₀₁	—	2	4	no	—
62	C₂×Dic₁₀₁	—	2	4	no	—
63	C₂×Dic₁₀₁	—	2	4	no	—
64	C₂×Dic₁₀₁	—	2	2	no	—
65	C₂²×C₁₀₁	—	2	4	no	—
66	C₂²×C₁₀₁	—	2	8	no	—
67	C₂²×C₁₀₁	#123	1	4	yes	—
68	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
70	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
71	C₂×D₁₀₁	—	4	1	no	—
72	C₂×D₁₀₁	—	4	2	no	—
73	C₂×D₁₀₁	—	4	4	no	—
74	C₄×C₁₀₁	—	2	4	no	—
75	C₄×C₁₀₁	—	2	8	no	—
76	Dic₁₀₁	—	4	1	no	—
77	Dic₁₀₁	—	4	2	no	—
78	Dic₁₀₁	—	4	4	no	—
79	C₂×C₁₀₁	—	2	4	no	—
80	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
81	C₂×C₁₀₁	#525	1	2	yes	—
82	C₂×C₁₀₁	#529	1	4	yes	—
83	D₁₀₁	—	8	1	no	—
84	#2853	—	101	1	no	maximal, Sylow
85	C₁₀₁	#2853	1	1	yes	characteristic, complemented, Sylow
86	C₂×#123	—	101	1	no	—
87	C₂²×C₄²	D₁₀₁	1	1	yes	characteristic, complemented
88	#388	—	101	4	no	—
89	#387	—	101	4	no	—
90	#386	—	101	2	no	—
91	#385	—	101	2	no	—
92	#381	—	101	1	no	—
93	C₂²×D₄	—	101	1	no	—
94	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
95	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
96	#123	—	202	4	no	—
97	C₂×C₄⋊C₄	—	101	2	no	—
98	C₂×C₄⋊C₄	—	101	4	no	—
99	C₂×C₂²⋊C₄	—	101	1	no	—
100	C₂×C₂²⋊C₄	—	101	2	no	—
101	C₂×C₂²⋊C₄	—	101	2	no	—
102	C₂×C₂²⋊C₄	—	101	4	no	—
103	C₂×C₄²	—	2	4	no	—
104	C₂×C₄²	C₂×D₁₀₁	1	4	yes	—
105	#95	—	101	4	no	—
106	#95	—	101	4	no	—
107	#95	—	101	4	no	—
108	C₂⁴	—	101	1	no	—
109	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
110	C₂×D₄	—	202	2	no	—
111	C₂×D₄	—	202	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	101	4	no	—
116	C₂²×C₄	—	101	1	no	—
117	C₂²×C₄	—	101	2	no	—
118	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
119	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
121	C₄⋊C₄	—	202	4	no	—
122	C₄⋊C₄	—	202	8	no	—
123	C₂²⋊C₄	—	202	2	no	—
124	C₂²⋊C₄	—	202	4	no	—
125	C₂²⋊C₄	—	202	4	no	—
126	C₂²⋊C₄	—	202	8	no	—
127	C₄²	—	2	8	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	202	1	no	—
130	C₂³	—	202	4	no	—
131	C₂³	—	202	2	no	—
132	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
133	C₂³	#13108074	1	4	yes	—
134	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
135	D₄	—	404	4	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	202	2	no	—
142	C₂×C₄	—	202	1	no	—
143	C₂×C₄	—	202	4	no	—
144	C₂×C₄	—	202	4	no	—
145	C₂×C₄	—	202	4	no	—
146	C₂×C₄	—	202	4	no	—
147	C₂×C₄	—	202	2	no	—
148	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
149	C₂×C₄	#13108074	1	8	yes	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	8	no	—
152	C₂²	—	404	1	no	—
153	C₂²	—	404	2	no	—
154	C₂²	—	404	4	no	—
155	C₂²	#15692238	1	4	yes	—
156	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
157	C₂²	C₂×#13108074	1	2	yes	—
158	C₂²	C₂×#13108074	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	8	no	—
161	C₄	—	404	1	no	—
162	C₄	—	404	2	no	—
163	C₄	—	404	4	no	—
164	C₂	—	2	4	no	—
165	C₂	—	808	1	no	—
166	C₂	C₂×#15692238	1	1	yes	characteristic
167	C₂	#20265688	1	2	yes	—
168	C₂	#20265711	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692238

C₂

yes

maximal, characteristic

#20265068

C₂

yes

maximal, complemented

#20265061

C₂

yes

maximal, complemented

#20265056

C₂

yes

maximal

#20265050

C₂

yes

maximal, complemented

#20265040

C₂

yes

maximal, characteristic, complemented

C₂²×C₄²×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₁₀₁

C₂²

yes

characteristic

#15692238

—

C₂×#13108053

C₂²

yes

—

C₂×#13108053

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×#13108051

C₂²

yes

—

C₂×#13108051

C₂²

yes

—

#15692162

C₂²

yes

—

#15692162

C₂²

yes

—

#15692162

C₂²

yes

—

C₂³×C₄×C₁₀₁

C₂²

yes

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂×C₄²×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

C₂³×D₁₀₁

C₂³

yes

characteristic

C₂×C₂²⋊D₁₀₁

—

C₂×C₂²⋊D₁₀₁

—

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₂²×Dic₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

#13108053

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

#13108051

—

#13108051

—

C₂⁴×C₁₀₁

C₂³

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

—

C₂³×C₁₀₁

#41

yes

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#123

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₂²×C₁₀₁

C₂×#41

yes

—

C₂²×C₁₀₁

C₂×#41

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#123

yes

characteristic

C₂×C₁₀₁

#525

yes

—

C₂×C₁₀₁

#529

yes

—

D₁₀₁

—

#2853

—

101

maximal, Sylow

C₁₀₁

#2853

yes

characteristic, complemented, Sylow

C₂×#123

—

101

—

C₂²×C₄²

D₁₀₁

yes

characteristic, complemented

#388

—

101

—

#387

—

101

—

#386

—

101

—

#385

—

101

—

#381

—

101

—

C₂²×D₄

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

#123

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

100

C₂×C₂²⋊C₄

—

101

—

101

C₂×C₂²⋊C₄

—

101

—

102

C₂×C₂²⋊C₄

—

101

—

103

C₂×C₄²

—

104

C₂×C₄²

C₂×D₁₀₁

yes

—

105

#95

—

101

—

106

#95

—

101

—

107

#95

—

101

—

108

C₂⁴

—

101

—

109

C₂⁴

C₂²×D₁₀₁

yes

characteristic

110

C₂×D₄

—

202

—

111

C₂×D₄

—

202

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

101

—

116

C₂²×C₄

—

101

—

117

C₂²×C₄

—

101

—

118

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

119

C₂²×C₄

C₂²×D₁₀₁

yes

—

120

C₂²×C₄

C₂²×D₁₀₁

yes

—

121

C₄⋊C₄

—

202

—

122

C₄⋊C₄

—

202

—

123

C₂²⋊C₄

—

202

—

124

C₂²⋊C₄

—

202

—

125

C₂²⋊C₄

—

202

—

126

C₂²⋊C₄

—

202

—

127

C₄²

—

128

C₂³

—

129

C₂³

—

202

—

130

C₂³

—

202

—

131

C₂³

—

202

—

132

C₂³

C₂×C₂²⋊D₁₀₁

yes

—

133

C₂³

#13108074

yes

—

134

C₂³

C₂³×D₁₀₁

yes

characteristic

135

D₄

—

404

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

202

—

142

C₂×C₄

—

202

—

143

C₂×C₄

—

202

—

144

C₂×C₄

—

202

—

145

C₂×C₄

—

202

—

146

C₂×C₄

—

202

—

147

C₂×C₄

—

202

—

148

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

149

C₂×C₄

#13108074

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

404

—

153

C₂²

—

404

—

154

C₂²

—

404

—

155

C₂²

#15692238

yes

—

156

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

157

C₂²

C₂×#13108074

yes

—

158

C₂²

C₂×#13108074

yes

—

159

C₄

—

160

C₄

—

161

C₄

—

404

—

162

C₄

—

404

—

163

C₄

—

404

—

164

C₂

—

165

C₂

—

808

—

166

C₂

C₂×#15692238

yes

characteristic

167

C₂

#20265688

yes

—

168

C₂

#20265711

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	C₂²×C₄²×C₁₀₁	C₂
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	C₂²×C₄²	D₁₀₁
hypocenter	C₁₀₁	#2853
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator, socle, cosocle

C₂³×C₁₀₁

C₂⁴

Fitting

C₂²×C₄²×C₁₀₁

C₂

prime radical

C₂²×C₂²⋊D₁₀₁

C₂²

hypercenter

C₂²×C₄²

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #20265068	C₂	semidirect
3. #20265061	C₂	semidirect
5. #20265050	C₂	semidirect
6. #20265040	C₂	semidirect
7. C₂²×C₄²×C₁₀₁	C₂	semidirect
85. C₁₀₁	#2853	semidirect
87. C₂²×C₄²	D₁₀₁	semidirect

H ≅ G/N

type

2. #20265068

C₂

semidirect

3. #20265061

C₂

semidirect

5. #20265050

C₂

semidirect

6. #20265040

C₂

semidirect

7. C₂²×C₄²×C₁₀₁

C₂

semidirect

85. C₁₀₁

#2853

semidirect

87. C₂²×C₄²

D₁₀₁

semidirect

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

subgroup	index
C₂×#15692238	2
#20265068	2
#20265061	2
#20265056	2
#20265050	2
#20265040	2
C₂²×C₄²×C₁₀₁	2
#2853	101

subgroup

index

C₂²×C₄²×C₁₀₁

101

Subgroups — #27179454

Conjugacy classes of subgroups

Complements

Maximal subgroups