Group #27172239 — subgroups

Subgroups — #27172239

Order 12928.

382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15692281	C₂	1	2	yes	maximal, complemented
2	C₂×#15692172	C₂	1	2	yes	maximal, complemented
3	#20265143	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108074	C₂²	1	1	yes	characteristic
7	#15692281	—	2	4	no	—
8	#15692281	—	2	4	no	—
9	C₂×#13108062	—	2	2	no	—
10	C₂×#13108059	—	2	2	no	—
11	C₂×#13108058	C₂²	1	1	yes	characteristic
12	C₂×#13108058	C₂²	1	2	yes	—
13	#15692172	—	2	4	no	—
14	#15692167	—	2	2	no	—
15	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
16	C₂×D₈×C₁₀₁	—	2	4	no	—
17	C₂×D₈×C₁₀₁	—	2	2	no	—
18	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
19	#13108074	—	2	2	no	—
20	#13108074	—	2	2	no	—
21	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
22	C₂×D₄₀₄	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
24	C₂×C₄×D₁₀₁	—	2	1	no	—
25	#13108062	—	2	8	no	—
26	#13108059	—	2	8	no	—
27	#13108058	—	2	2	no	—
28	#13108058	—	2	2	no	—
29	#13108058	D₄	1	4	yes	—
30	C₂×#11717579	—	2	2	no	—
31	C₂×#11717579	—	2	1	no	—
32	C₂⁴×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	—	2	2	no	—
37	C₂×D₄×C₁₀₁	—	2	2	no	—
38	C₂×D₄×C₁₀₁	D₄	1	4	yes	—
39	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
40	D₈×C₁₀₁	—	2	8	no	—
41	C₂×C₈×C₁₀₁	—	2	2	no	—
42	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
43	C₂³×C₁₀₁	—	2	4	no	—
44	C₂³×C₁₀₁	—	4	2	no	—
45	C₂³×C₁₀₁	—	4	2	no	—
46	C₂³×C₁₀₁	—	8	4	no	—
47	C₂³×C₁₀₁	#41	1	1	yes	characteristic
48	C₂²×D₁₀₁	—	2	1	no	—
49	D₄×C₁₀₁	—	2	4	no	—
50	D₄×C₁₀₁	—	2	4	no	—
51	D₄×C₁₀₁	—	4	2	no	—
52	D₄×C₁₀₁	—	4	2	no	—
53	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₁₀₁	—	4	2	no	—
58	C₂×Dic₁₀₁	—	2	1	no	—
59	D₄₀₄	—	2	2	no	—
60	C₄×D₁₀₁	—	4	1	no	—
61	C₄×D₁₀₁	—	4	1	no	—
62	Dic₂₀₂	—	2	2	no	—
63	C₈×C₁₀₁	—	2	2	no	—
64	#11717579	—	2	4	no	—
65	#11717579	—	4	1	no	—
66	C₂²×C₁₀₁	—	2	2	no	—
67	C₂²×C₁₀₁	—	4	4	no	—
68	C₂²×C₁₀₁	—	8	4	no	—
69	C₂²×C₁₀₁	—	8	4	no	—
70	C₂²×C₁₀₁	—	8	4	no	—
71	C₂²×C₁₀₁	#121	1	1	yes	characteristic
72	C₂²×C₁₀₁	#121	1	2	yes	—
73	C₂×D₁₀₁	—	4	1	no	—
74	C₂×D₁₀₁	—	4	2	no	—
75	C₄×C₁₀₁	#120	1	2	yes	—
76	C₄×C₁₀₁	#120	1	1	yes	characteristic
77	C₄×C₁₀₁	#127	1	1	yes	characteristic
78	Dic₁₀₁	—	4	1	no	—
79	C₂×C₁₀₁	—	2	2	no	—
80	C₂×C₁₀₁	—	8	4	no	—
81	C₂×C₁₀₁	#392	1	1	yes	characteristic
82	C₂×C₁₀₁	#472	1	2	yes	—
83	D₁₀₁	—	8	1	no	—
84	#2018	—	101	1	no	maximal, Sylow
85	C₁₀₁	#2018	1	1	yes	characteristic, complemented, Sylow
86	C₂×AGL(1,ℤ/8)	—	101	2	no	—
87	C₂²×D₈	D₁₀₁	1	1	yes	characteristic, complemented
88	C₂×#100	—	101	2	no	—
89	#408	—	101	1	no	—
90	#341	—	101	1	no	—
91	C₂×#41	—	101	1	no	—
92	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
93	AGL(1,ℤ/8)	—	202	4	no	—
94	AGL(1,ℤ/8)	—	202	4	no	—
95	C₂×SD₁₆	—	202	2	no	—
96	C₂×D₈	—	2	4	no	—
97	C₂×D₈	—	2	2	no	—
98	C₂×D₈	—	202	2	no	—
99	C₂×M₁₆	—	101	2	no	—
100	C₂×M₁₆	—	101	1	no	—
101	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
102	#100	—	202	4	no	—
103	C₂²⋊C₈	—	202	2	no	—
104	C₂⁴	—	2	2	no	—
105	#41	—	202	2	no	—
106	#41	—	202	2	no	—
107	C₂×Q₈	—	101	1	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	101	1	no	—
114	C₂×D₄	—	202	1	no	—
115	C₂×D₄	C₂²⋊D₁₀₁	1	4	yes	—
116	C₂²×C₄	—	202	1	no	—
117	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
118	SD₁₆	—	202	8	no	—
119	D₈	—	2	8	no	—
120	D₈	—	202	8	no	—
121	M₁₆	—	101	4	no	—
122	M₁₆	—	202	2	no	—
123	M₁₆	—	202	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	202	1	no	—
126	C₂×C₈	—	202	2	no	—
127	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
128	C₂³	—	2	4	no	—
129	C₂³	—	4	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	8	4	no	—
132	C₂³	—	202	1	no	—
133	C₂³	#13108073	1	1	yes	characteristic
134	Q₈	—	202	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	4	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	404	2	no	—
140	D₄	—	202	2	no	—
141	C₂×C₄	—	404	1	no	—
142	C₂×C₄	—	404	1	no	—
143	C₂×C₄	—	202	1	no	—
144	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
145	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	404	1	no	—
150	C₈	—	202	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	4	no	—
153	C₂²	—	8	4	no	—
154	C₂²	—	8	4	no	—
155	C₂²	—	8	4	no	—
156	C₂²	—	404	1	no	—
157	C₂²	—	404	2	no	—
158	C₂²	#15692226	1	1	yes	characteristic
159	C₂²	#15692227	1	2	yes	—
160	C₄	—	404	1	no	—
161	C₄	#15692223	1	2	yes	—
162	C₄	#15692224	1	1	yes	characteristic
163	C₄	#15692251	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	4	no	—
166	C₂	—	808	1	no	—
167	C₂	#20265085	1	1	yes	characteristic
168	C₂	#20265444	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692281

C₂

yes

maximal, complemented

C₂×#15692172

C₂

yes

maximal, complemented

#20265143

C₂

yes

maximal, characteristic, complemented

#20264942

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108074

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#13108058

C₂²

yes

—

#15692172

—

#15692167

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

C₂×D₈×C₁₀₁

—

C₂×D₈×C₁₀₁

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

#13108074

—

#13108074

—

C₂×Dic₂₀₂

D₄

yes

characteristic

C₂×D₄₀₄

D₄

yes

characteristic

C₂×C₂²⋊D₁₀₁

—

C₂×C₄×D₁₀₁

—

—

—

—

—

D₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂⁴×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

D₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

#41

yes

characteristic

C₂²×D₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₈×C₁₀₁

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#121

yes

characteristic

C₂²×C₁₀₁

#121

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

#120

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#392

yes

characteristic

C₂×C₁₀₁

#472

yes

—

D₁₀₁

—

#2018

—

101

maximal, Sylow

C₁₀₁

#2018

yes

characteristic, complemented, Sylow

C₂×AGL(1,ℤ/8)

—

101

—

C₂²×D₈

D₁₀₁

yes

characteristic, complemented

C₂×#100

—

101

—

#408

—

101

—

#341

—

101

—

C₂×#41

—

101

—

C₂²×D₄

C₂×D₁₀₁

yes

—

AGL(1,ℤ/8)

—

202

—

AGL(1,ℤ/8)

—

202

—

C₂×SD₁₆

—

202

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

202

—

C₂×M₁₆

—

101

—

100

C₂×M₁₆

—

101

—

101

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

102

#100

—

202

—

103

C₂²⋊C₈

—

202

—

104

C₂⁴

—

105

#41

—

202

—

106

#41

—

202

—

107

C₂×Q₈

—

101

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

101

—

114

C₂×D₄

—

202

—

115

C₂×D₄

C₂²⋊D₁₀₁

yes

—

116

C₂²×C₄

—

202

—

117

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

118

SD₁₆

—

202

—

119

D₈

—

120

D₈

—

202

—

121

M₁₆

—

101

—

122

M₁₆

—

202

—

123

M₁₆

—

202

—

124

C₂×C₈

—

125

C₂×C₈

—

202

—

126

C₂×C₈

—

202

—

127

C₂×C₈

C₂²⋊D₁₀₁

yes

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

202

—

133

C₂³

#13108073

yes

characteristic

134

Q₈

—

202

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

404

—

140

D₄

—

202

—

141

C₂×C₄

—

404

—

142

C₂×C₄

—

404

—

143

C₂×C₄

—

202

—

144

C₂×C₄

D₄×D₁₀₁

yes

characteristic

145

C₂×C₄

D₄×D₁₀₁

yes

—

146

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

148

C₈

—

149

C₈

—

404

—

150

C₈

—

202

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

404

—

157

C₂²

—

404

—

158

C₂²

#15692226

yes

characteristic

159

C₂²

#15692227

yes

—

160

C₄

—

404

—

161

C₄

#15692223

yes

—

162

C₄

#15692224

yes

characteristic

163

C₄

#15692251

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

808

—

167

C₂

#20265085

yes

characteristic

168

C₂

#20265444

yes

—

	subgroup	G/H
center	C₂²	#15692226
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂²×D₈×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#121
hypercenter	C₂²×D₈	D₁₀₁
hypocenter	C₁₀₁	#2018
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

#15692226

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂²×D₈×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#121

hypercenter

C₂²×D₈

D₁₀₁

hypocenter

C₁₀₁

#2018

LCS

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#15692281	C₂	semidirect
2. C₂×#15692172	C₂	semidirect
3. #20265143	C₂	semidirect
4. #20264942	C₂	semidirect
5. C₂²×D₈×C₁₀₁	C₂	semidirect
85. C₁₀₁	#2018	semidirect
87. C₂²×D₈	D₁₀₁	semidirect

H ≅ G/N

type

1. C₂×#15692281

C₂

semidirect

2. C₂×#15692172

C₂

semidirect

3. #20265143

C₂

semidirect

4. #20264942

C₂

semidirect

5. C₂²×D₈×C₁₀₁

C₂

semidirect

85. C₁₀₁

#2018

semidirect

87. C₂²×D₈

D₁₀₁

semidirect

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

subgroup	index
C₂×#15692281	2
C₂×#15692172	2
#20265143	2
#20264942	2
C₂²×D₈×C₁₀₁	2
#2018	101

subgroup

index

C₂²×D₈×C₁₀₁

101

Subgroups — #27172239

Conjugacy classes of subgroups

Complements

Maximal subgroups