Group #27172253 — subgroups

Subgroups — #27172253

Order 12928.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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×#15692285	C₂	1	2	yes	maximal
2	C₂²×Dic₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692173	C₂	1	2	yes	maximal, complemented
4	#20264942	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
7	#15692285	—	2	4	no	—
8	#15692285	—	2	4	no	—
9	C₂×Dic₄₀₄	—	2	4	no	—
10	C₂×Dic₄₀₄	—	2	2	no	—
11	C₂×#13108064	—	2	2	no	—
12	C₂×#13108062	—	2	2	no	—
13	C₂×#13108058	C₂²	1	2	yes	—
14	#15692173	—	2	4	no	—
15	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
19	C₂×Dic₂₀₂	—	2	2	no	—
20	C₂×Dic₂₀₂	—	2	2	no	—
21	C₂×Dic₂₀₂	—	2	2	no	—
22	C₂×Dic₂₀₂	—	2	2	no	—
23	C₂×Dic₂₀₂	—	2	2	no	—
24	C₂×Dic₂₀₂	D₄	1	4	yes	—
25	C₂²×Dic₁₀₁	—	2	2	no	—
26	Dic₄₀₄	—	2	8	no	—
27	#13108064	—	2	8	no	—
28	#13108062	—	2	8	no	—
29	#13108058	—	2	2	no	—
30	#13108058	D₄	1	4	yes	—
31	C₂×#11717579	—	2	2	no	—
32	#41×C₁₀₁	—	2	2	no	—
33	#41×C₁₀₁	—	2	2	no	—
34	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₀₁	—	2	1	no	—
36	C₂×D₄×C₁₀₁	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₁	—	2	1	no	—
38	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
39	M₁₆×C₁₀₁	—	2	2	no	—
40	C₂×C₈×C₁₀₁	—	2	2	no	—
41	C₂×C₈×C₁₀₁	—	2	1	no	—
42	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
43	C₂³×C₁₀₁	—	2	1	no	—
44	C₂³×C₁₀₁	#41	1	1	yes	characteristic
45	Q₈×C₁₀₁	—	2	2	no	—
46	D₄×C₁₀₁	—	2	2	no	—
47	D₄×C₁₀₁	—	4	2	no	—
48	C₂×C₄×C₁₀₁	—	2	1	no	—
49	C₂×C₄×C₁₀₁	—	4	1	no	—
50	C₂×C₄×C₁₀₁	—	4	1	no	—
51	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₁₀₁	—	2	4	no	—
56	C₂×Dic₁₀₁	—	4	2	no	—
57	C₂×Dic₁₀₁	—	4	2	no	—
58	Dic₂₀₂	—	2	4	no	—
59	Dic₂₀₂	—	2	4	no	—
60	Dic₂₀₂	—	4	2	no	—
61	Dic₂₀₂	—	4	2	no	—
62	C₈×C₁₀₁	—	2	2	no	—
63	C₈×C₁₀₁	—	4	1	no	—
64	#11717579	—	2	4	no	—
65	C₂²×C₁₀₁	—	2	2	no	—
66	C₂²×C₁₀₁	—	4	1	no	—
67	C₂²×C₁₀₁	—	4	2	no	—
68	C₂²×C₁₀₁	#121	1	2	yes	—
69	C₂²×C₁₀₁	#121	1	1	yes	characteristic
70	C₄×C₁₀₁	—	4	1	no	—
71	C₄×C₁₀₁	#120	1	1	yes	characteristic
72	C₄×C₁₀₁	#120	1	2	yes	—
73	C₄×C₁₀₁	#127	1	1	yes	characteristic
74	Dic₁₀₁	—	4	4	no	—
75	C₂×C₁₀₁	—	2	2	no	—
76	C₂×C₁₀₁	—	8	1	no	—
77	C₂×C₁₀₁	#392	1	1	yes	characteristic
78	C₂×C₁₀₁	#473	1	2	yes	—
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	—	101	1	no	—
83	C₂×#101	—	101	2	no	—
84	#408	D₁₀₁	1	1	yes	characteristic
85	#341	—	101	1	no	—
86	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
87	C₂²×Q₈	—	101	2	no	—
88	#137	—	202	4	no	—
89	#137	—	202	4	no	—
90	C₂×Q₁₆	—	202	4	no	—
91	C₂×Q₁₆	—	202	2	no	—
92	C₂×Q₁₆	—	202	2	no	—
93	C₂×SD₁₆	—	202	2	no	—
94	C₂×M₁₆	—	101	2	no	—
95	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
97	#101	—	202	4	no	—
98	C₂²⋊C₈	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	—	101	4	no	—
102	C₂×Q₈	—	202	2	no	—
103	C₂×Q₈	—	202	2	no	—
104	C₂×Q₈	—	202	2	no	—
105	C₂×Q₈	—	202	2	no	—
106	C₂×Q₈	—	202	2	no	—
107	C₂×Q₈	D₄₀₄	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	D₄₀₄	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	202	2	no	—
112	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
113	Q₁₆	—	202	8	no	—
114	Q₁₆	—	202	8	no	—
115	SD₁₆	—	202	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	101	4	no	—
118	M₁₆	—	202	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	202	2	no	—
122	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
123	C₂³	—	2	1	no	—
124	C₂³	#13108074	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	404	2	no	—
127	Q₈	—	404	2	no	—
128	Q₈	—	202	4	no	—
129	Q₈	—	202	4	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	404	2	no	—
136	C₂×C₄	—	404	2	no	—
137	C₂×C₄	—	202	4	no	—
138	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
139	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	202	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	#15692228	1	2	yes	—
149	C₂²	#15692229	1	1	yes	characteristic
150	C₄	—	4	1	no	—
151	C₄	—	404	4	no	—
152	C₄	#15692223	1	1	yes	characteristic
153	C₄	#15692225	1	2	yes	—
154	C₄	#15692251	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	8	1	no	—
157	C₂	#20265086	1	1	yes	characteristic
158	C₂	#20265452	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692285

C₂

yes

maximal

C₂²×Dic₄₀₄

C₂

yes

maximal, characteristic, complemented

C₂×#15692173

C₂

yes

maximal, complemented

#20264942

C₂

yes

maximal, characteristic, complemented

#408×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

—

#15692285

—

#15692285

—

C₂×Dic₄₀₄

—

C₂×Dic₄₀₄

—

C₂×#13108064

—

C₂×#13108062

—

C₂×#13108058

C₂²

yes

—

#15692173

—

C₂×#41×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

C₂²⋊C₈×C₁₀₁

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

D₄

yes

—

C₂²×Dic₁₀₁

—

Dic₄₀₄

—

—

—

—

D₄

yes

—

C₂×#11717579

—

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

characteristic

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

#41

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#121

yes

—

C₂²×C₁₀₁

#121

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

#120

yes

—

C₄×C₁₀₁

#127

yes

characteristic

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#392

yes

characteristic

C₂×C₁₀₁

#473

yes

—

#2020

—

101

maximal, Sylow

C₁₀₁

#2020

yes

characteristic, complemented, Sylow

C₂×#137

—

101

—

C₂²×Q₁₆

—

101

—

C₂×#101

—

101

—

#408

D₁₀₁

yes

characteristic

#341

—

101

—

C₂×#41

C₂×D₁₀₁

yes

characteristic

C₂²×Q₈

—

101

—

#137

—

202

—

#137

—

202

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

—

202

—

C₂×SD₁₆

—

202

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

#101

—

202

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

—

101

—

102

C₂×Q₈

—

202

—

103

C₂×Q₈

—

202

—

104

C₂×Q₈

—

202

—

105

C₂×Q₈

—

202

—

106

C₂×Q₈

—

202

—

107

C₂×Q₈

D₄₀₄

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

D₄₀₄

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

202

—

112

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

113

Q₁₆

—

202

—

114

Q₁₆

—

202

—

115

SD₁₆

—

202

—

116

M₁₆

—

117

M₁₆

—

101

—

118

M₁₆

—

202

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

202

—

122

C₂×C₈

C₂²⋊D₁₀₁

yes

—

123

C₂³

—

124

C₂³

#13108074

yes

characteristic

125

Q₈

—

126

Q₈

—

404

—

127

Q₈

—

404

—

128

Q₈

—

202

—

129

Q₈

—

202

—

130

D₄

—

131

D₄

—

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

—

139

C₂×C₄

D₄×D₁₀₁

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

141

C₂×C₄

C₂×D₄₀₄

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

202

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#15692228

yes

—

149

C₂²

#15692229

yes

characteristic

150

C₄

—

151

C₄

—

404

—

152

C₄

#15692223

yes

characteristic

153

C₄

#15692225

yes

—

154

C₄

#15692251

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#20265086

yes

characteristic

158

C₂

#20265452

yes

—

	subgroup	G/H
center	C₂²	#15692229
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#408×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#121
prime radical	C₂×#41×C₁₀₁	C₂²
hypercenter	#408	D₁₀₁
hypocenter	C₁₀₁	#2020
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#15692229

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#408×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#121

prime radical

C₂×#41×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. C₂²×Dic₄₀₄	C₂	semidirect
3. C₂×#15692173	C₂	semidirect
4. #20264942	C₂	semidirect
80. C₁₀₁	#2020	semidirect

H ≅ G/N

type

2. C₂²×Dic₄₀₄

C₂

semidirect

3. C₂×#15692173

C₂

semidirect

4. #20264942

C₂

semidirect

80. C₁₀₁

#2020

semidirect

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

subgroup	index
C₂×#15692285	2
C₂²×Dic₄₀₄	2
C₂×#15692173	2
#20264942	2
#408×C₁₀₁	2
#2020	101

subgroup

index

C₂²×Dic₄₀₄

#408×C₁₀₁

101

Subgroups — #27172253

Conjugacy classes of subgroups

Complements

Maximal subgroups