Group #27172251 — subgroups

Subgroups — #27172251

Order 12928.

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

normal subgroups only

This group has 342 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×#15692286	C₂	1	2	yes	maximal
2	C₂×#15692174	C₂	1	2	yes	maximal, complemented
3	#20265143	C₂	1	1	yes	maximal, characteristic
4	#20264942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108074	C₂²	1	1	yes	characteristic
7	#15692286	—	2	4	no	—
8	#15692286	—	2	4	no	—
9	C₂×#13108064	—	2	2	no	—
10	C₂×#13108061	—	2	2	no	—
11	C₂×#13108058	C₂²	1	1	yes	characteristic
12	C₂×#13108058	C₂²	1	2	yes	—
13	#15692174	—	2	4	no	—
14	#15692167	—	2	2	no	—
15	C₂²×Q₈×C₁₀₁	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₁₀₁	—	2	4	no	—
17	C₂×Q₁₆×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	#13108064	—	2	8	no	—
26	#13108061	—	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	C₂×Q₈×C₁₀₁	—	2	2	no	—
33	C₂×Q₈×C₁₀₁	—	2	2	no	—
34	C₂×Q₈×C₁₀₁	—	2	2	no	—
35	C₂×Q₈×C₁₀₁	—	2	2	no	—
36	C₂×Q₈×C₁₀₁	—	2	2	no	—
37	C₂×Q₈×C₁₀₁	D₄	1	4	yes	—
38	C₂²×C₄×C₁₀₁	—	2	2	no	—
39	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
40	Q₁₆×C₁₀₁	—	2	8	no	—
41	C₂×C₈×C₁₀₁	—	2	2	no	—
42	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
43	C₂³×C₁₀₁	#41	1	1	yes	characteristic
44	C₂²×D₁₀₁	—	2	1	no	—
45	Q₈×C₁₀₁	—	2	4	no	—
46	Q₈×C₁₀₁	—	2	4	no	—
47	Q₈×C₁₀₁	—	4	2	no	—
48	Q₈×C₁₀₁	—	4	2	no	—
49	C₂×C₄×C₁₀₁	—	2	4	no	—
50	C₂×C₄×C₁₀₁	—	4	2	no	—
51	C₂×C₄×C₁₀₁	—	4	2	no	—
52	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
56	C₂²⋊D₁₀₁	—	4	2	no	—
57	C₂×Dic₁₀₁	—	2	1	no	—
58	D₄₀₄	—	2	2	no	—
59	C₄×D₁₀₁	—	4	1	no	—
60	C₄×D₁₀₁	—	4	1	no	—
61	Dic₂₀₂	—	2	2	no	—
62	C₈×C₁₀₁	—	2	2	no	—
63	#11717579	—	2	4	no	—
64	#11717579	—	4	1	no	—
65	C₂²×C₁₀₁	—	2	2	no	—
66	C₂²×C₁₀₁	#121	1	1	yes	characteristic
67	C₂²×C₁₀₁	#121	1	2	yes	—
68	C₂×D₁₀₁	—	4	1	no	—
69	C₂×D₁₀₁	—	4	2	no	—
70	C₄×C₁₀₁	—	4	4	no	—
71	C₄×C₁₀₁	#120	1	2	yes	—
72	C₄×C₁₀₁	#120	1	1	yes	characteristic
73	C₄×C₁₀₁	#127	1	1	yes	characteristic
74	Dic₁₀₁	—	4	1	no	—
75	C₂×C₁₀₁	—	2	2	no	—
76	C₂×C₁₀₁	#392	1	1	yes	characteristic
77	C₂×C₁₀₁	#473	1	2	yes	—
78	D₁₀₁	—	8	1	no	—
79	#2020	—	101	1	no	maximal, Sylow
80	C₁₀₁	#2020	1	1	yes	characteristic, complemented, Sylow
81	C₂×#137	—	101	2	no	—
82	C₂²×Q₁₆	D₁₀₁	1	1	yes	characteristic, complemented
83	C₂×#101	—	101	2	no	—
84	#408	—	101	1	no	—
85	#341	—	101	1	no	—
86	C₂×#41	—	101	1	no	—
87	C₂²×Q₈	C₂×D₁₀₁	1	2	yes	—
88	#137	—	202	4	no	—
89	#137	—	202	4	no	—
90	C₂×Q₁₆	—	2	4	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	202	2	no	—
93	C₂×SD₁₆	—	202	2	no	—
94	C₂×M₁₆	—	101	2	no	—
95	C₂×M₁₆	—	101	1	no	—
96	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
97	#101	—	202	4	no	—
98	C₂²⋊C₈	—	202	2	no	—
99	#41	—	202	2	no	—
100	#41	—	202	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	101	1	no	—
107	C₂×Q₈	C₂²⋊D₁₀₁	1	4	yes	—
108	C₂×D₄	—	101	1	no	—
109	C₂×D₄	—	202	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	202	1	no	—
112	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	202	8	no	—
115	SD₁₆	—	202	8	no	—
116	M₁₆	—	101	4	no	—
117	M₁₆	—	202	2	no	—
118	M₁₆	—	202	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	202	1	no	—
121	C₂×C₈	—	202	2	no	—
122	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
123	C₂³	—	202	1	no	—
124	C₂³	#13108073	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	202	2	no	—
130	D₄	—	404	2	no	—
131	D₄	—	202	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	404	1	no	—
136	C₂×C₄	—	404	1	no	—
137	C₂×C₄	—	202	1	no	—
138	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
139	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	404	1	no	—
144	C₈	—	202	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	404	1	no	—
147	C₂²	—	404	2	no	—
148	C₂²	#15692226	1	1	yes	characteristic
149	C₂²	#15692227	1	2	yes	—
150	C₄	—	4	4	no	—
151	C₄	—	404	1	no	—
152	C₄	#15692223	1	2	yes	—
153	C₄	#15692224	1	1	yes	characteristic
154	C₄	#15692251	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	808	1	no	—
157	C₂	#20265085	1	1	yes	characteristic
158	C₂	#20265449	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692286

C₂

yes

maximal

C₂×#15692174

C₂

yes

maximal, complemented

#20265143

C₂

yes

maximal, characteristic

#20264942

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108074

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#13108058

C₂²

yes

—

#15692174

—

#15692167

—

C₂²×Q₈×C₁₀₁

C₂²

yes

—

C₂×Q₁₆×C₁₀₁

—

C₂×Q₁₆×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

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

Q₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

#41

yes

characteristic

C₂²×D₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×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

#121

yes

characteristic

C₂²×C₁₀₁

#121

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#392

yes

characteristic

C₂×C₁₀₁

#473

yes

—

D₁₀₁

—

#2020

—

101

maximal, Sylow

C₁₀₁

#2020

yes

characteristic, complemented, Sylow

C₂×#137

—

101

—

C₂²×Q₁₆

D₁₀₁

yes

characteristic, complemented

C₂×#101

—

101

—

#408

—

101

—

#341

—

101

—

C₂×#41

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

—

#137

—

202

—

#137

—

202

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

202

—

C₂×SD₁₆

—

202

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

#101

—

202

—

C₂²⋊C₈

—

202

—

#41

—

202

—

100

#41

—

202

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

101

—

107

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

108

C₂×D₄

—

101

—

109

C₂×D₄

—

202

—

110

C₂²×C₄

—

111

C₂²×C₄

—

202

—

112

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

202

—

115

SD₁₆

—

202

—

116

M₁₆

—

101

—

117

M₁₆

—

202

—

118

M₁₆

—

202

—

119

C₂×C₈

—

120

C₂×C₈

—

202

—

121

C₂×C₈

—

202

—

122

C₂×C₈

C₂²⋊D₁₀₁

yes

—

123

C₂³

—

202

—

124

C₂³

#13108073

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

202

—

130

D₄

—

404

—

131

D₄

—

202

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

404

—

136

C₂×C₄

—

404

—

137

C₂×C₄

—

202

—

138

C₂×C₄

D₄×D₁₀₁

yes

characteristic

139

C₂×C₄

D₄×D₁₀₁

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

142

C₈

—

143

C₈

—

404

—

144

C₈

—

202

—

145

C₂²

—

146

C₂²

—

404

—

147

C₂²

—

404

—

148

C₂²

#15692226

yes

characteristic

149

C₂²

#15692227

yes

—

150

C₄

—

151

C₄

—

404

—

152

C₄

#15692223

yes

—

153

C₄

#15692224

yes

characteristic

154

C₄

#15692251

yes

characteristic

155

C₂

—

156

C₂

—

808

—

157

C₂

#20265085

yes

characteristic

158

C₂

#20265449

yes

—

	subgroup	G/H
center	C₂²	#15692226
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂²×Q₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#121
prime radical	C₂×#13108074	C₂²
hypercenter	C₂²×Q₁₆	D₁₀₁
hypocenter	C₁₀₁	#2020
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#15692226

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂²×Q₁₆×C₁₀₁

C₂

C₂²×C₁₀₁

C₂²

C₂²×Q₁₆

D₁₀₁

hypocenter

C₁₀₁

#2020

LCS

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂×#15692174	C₂	semidirect
4. #20264942	C₂	semidirect
5. C₂²×Q₁₆×C₁₀₁	C₂	semidirect
80. C₁₀₁	#2020	semidirect
82. C₂²×Q₁₆	D₁₀₁	semidirect

H ≅ G/N

type

2. C₂×#15692174

C₂

semidirect

4. #20264942

C₂

semidirect

5. C₂²×Q₁₆×C₁₀₁

C₂

semidirect

80. C₁₀₁

#2020

semidirect

82. C₂²×Q₁₆

D₁₀₁

semidirect

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

subgroup	index
C₂×#15692286	2
C₂×#15692174	2
#20265143	2
#20264942	2
C₂²×Q₁₆×C₁₀₁	2
#2020	101

subgroup

index

C₂²×Q₁₆×C₁₀₁

101

Subgroups — #27172251

Conjugacy classes of subgroups

Complements

Maximal subgroups