Group #27171342 — subgroups

Subgroups — #27171342

Order 12928.

324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20265220	C₂	1	2	yes	maximal, complemented
2	C₂×#15692172	C₂	1	1	yes	maximal, characteristic
3	#20265080	C₂	1	1	yes	maximal, characteristic
4	#20264936	C₂	1	2	yes	maximal, complemented
5	#518×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Dic₁₀₁	—	2	2	no	—
9	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
10	#15692188	—	2	2	no	—
11	#15692186	—	2	2	no	—
12	#15692172	C₄	1	4	yes	complemented
13	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₀₁	—	2	2	no	—
15	C₄×D₄×C₁₀₁	—	2	2	no	—
16	#117×C₁₀₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
18	C₂²×Dic₁₀₁	—	2	1	no	—
19	C₂²×Dic₁₀₁	—	2	1	no	—
20	#13108058	—	2	2	no	—
21	#13108058	C₂×C₄	1	4	yes	complemented
22	C₂×#11717579	—	2	2	no	—
23	C₄⋊Dic₁₀₁	—	2	2	no	—
24	C₂²⋊Dic₁₀₁	—	4	4	no	—
25	C₄×Dic₁₀₁	—	2	2	no	—
26	C₂⁴×C₁₀₁	D₄	1	2	yes	—
27	C₂×D₄×C₁₀₁	—	2	2	no	—
28	C₂×D₄×C₁₀₁	—	2	2	no	—
29	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₀₁	Q₈	1	2	yes	—
31	C₂²×C₄×C₁₀₁	—	2	2	no	—
32	C₂²×C₄×C₁₀₁	—	2	2	no	—
33	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
35	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
36	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
37	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
38	C₄²×C₁₀₁	—	2	2	no	—
39	C₂³×C₁₀₁	—	2	2	no	—
40	C₂³×C₁₀₁	—	2	2	no	—
41	C₂³×C₁₀₁	—	4	2	no	—
42	C₂³×C₁₀₁	—	4	2	no	—
43	C₂³×C₁₀₁	C₄⋊C₄	1	4	yes	—
44	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
45	D₄×C₁₀₁	—	2	4	no	—
46	D₄×C₁₀₁	—	2	4	no	—
47	C₂×C₄×C₁₀₁	—	2	2	no	—
48	C₂×C₄×C₁₀₁	—	2	2	no	—
49	C₂×C₄×C₁₀₁	—	8	2	no	—
50	C₂×C₄×C₁₀₁	—	8	2	no	—
51	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₀₁	#41	1	2	yes	—
55	C₂×Dic₁₀₁	—	2	2	no	—
56	C₂×Dic₁₀₁	—	4	1	no	—
57	C₂×Dic₁₀₁	—	4	1	no	—
58	C₂×Dic₁₀₁	—	4	2	no	—
59	C₂×Dic₁₀₁	—	4	1	no	—
60	#11717579	—	2	4	no	—
61	C₂²×C₁₀₁	—	2	1	no	—
62	C₂²×C₁₀₁	—	2	1	no	—
63	C₂²×C₁₀₁	—	2	4	no	—
64	C₂²×C₁₀₁	—	4	4	no	—
65	C₂²×C₁₀₁	—	4	2	no	—
66	C₂²×C₁₀₁	—	4	2	no	—
67	C₂²×C₁₀₁	—	8	2	no	—
68	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₀₁	#120	1	1	yes	characteristic
70	C₂²×C₁₀₁	#123	1	1	yes	characteristic
71	C₄×C₁₀₁	—	4	2	no	—
72	C₄×C₁₀₁	—	4	2	no	—
73	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
74	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
75	Dic₁₀₁	—	4	2	no	—
76	Dic₁₀₁	—	8	1	no	—
77	C₂×C₁₀₁	—	2	1	no	—
78	C₂×C₁₀₁	—	2	1	no	—
79	C₂×C₁₀₁	—	4	4	no	—
80	C₂×C₁₀₁	#385	1	1	yes	characteristic
81	C₂×C₁₀₁	#453	1	1	yes	characteristic
82	C₂×C₁₀₁	#455	1	1	yes	characteristic
83	#1856	—	101	1	no	maximal, Sylow
84	C₁₀₁	#1856	1	1	yes	characteristic, complemented, Sylow
85	#518	D₁₀₁	1	1	yes	characteristic
86	#428	—	101	2	no	—
87	C₂×#100	—	101	1	no	—
88	#390	—	101	1	no	—
89	#339	—	101	2	no	—
90	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
91	C₂×M₁₆	—	101	2	no	—
92	C₄×D₄	—	2	2	no	—
93	C₄×D₄	—	2	2	no	—
94	#117	C₂×D₁₀₁	1	2	yes	—
95	C₂×C₄⋊C₄	—	101	1	no	—
96	C₂×C₂²⋊C₄	—	2	2	no	—
97	C₂×C₂²⋊C₄	—	202	2	no	—
98	C₂×C₄²	—	101	1	no	—
99	#107	—	202	2	no	—
100	#106	—	202	2	no	—
101	#100	—	101	4	no	—
102	C₂⁴	C₂²⋊D₁₀₁	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	Dic₂₀₂	1	2	yes	—
106	C₂×D₄	C₄×D₁₀₁	1	2	yes	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	202	1	no	—
110	C₂²×C₄	—	202	1	no	—
111	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
112	M₁₆	—	101	4	no	—
113	M₁₆	—	202	2	no	—
114	C₂×C₈	—	202	2	no	—
115	C₄⋊C₄	—	202	2	no	—
116	C₄⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
117	C₄⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
118	C₂²⋊C₄	—	2	4	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	404	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	202	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	#13108053	1	4	yes	—
128	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	404	2	no	—
136	C₂×C₄	—	404	1	no	—
137	C₂×C₄	—	404	1	no	—
138	C₂×C₄	—	404	1	no	—
139	C₂×C₄	—	202	2	no	—
140	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
142	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
143	C₂×C₄	#13108074	1	2	yes	—
144	C₈	—	202	4	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	8	2	no	—
152	C₂²	C₂×#13108053	1	1	yes	characteristic
153	C₂²	#15692224	1	1	yes	characteristic
154	C₂²	#15692238	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	4	2	no	—
157	C₄	—	808	1	no	—
158	C₄	—	404	2	no	—
159	C₄	C₄×C₂²⋊D₁₀₁	1	2	yes	—
160	C₄	#15692234	1	2	yes	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	4	no	—
164	C₂	#20265050	1	1	yes	characteristic
165	C₂	#20265345	1	1	yes	characteristic
166	C₂	#20265355	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265220

C₂

yes

maximal, complemented

C₂×#15692172

C₂

yes

maximal, characteristic

#20265080

C₂

yes

maximal, characteristic

#20264936

C₂

yes

maximal, complemented

#518×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₀₁

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

—

C₄

yes

complemented

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

C₄×D₄×C₁₀₁

—

C₄×D₄×C₁₀₁

—

#117×C₁₀₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

#13108058

—

#13108058

C₂×C₄

yes

complemented

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂×D₄×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1856

yes

characteristic, complemented, Sylow

#518

D₁₀₁

yes

characteristic

#428

—

101

—

C₂×#100

—

101

—

#390

—

101

—

#339

—

101

—

C₂²×D₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₄×D₄

—

C₄×D₄

—

#117

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

202

—

C₂×C₄²

—

101

—

#107

—

202

—

100

#106

—

202

—

101

#100

—

101

—

102

C₂⁴

C₂²⋊D₁₀₁

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

Dic₂₀₂

yes

—

106

C₂×D₄

C₄×D₁₀₁

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

202

—

110

C₂²×C₄

—

202

—

111

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

112

M₁₆

—

101

—

113

M₁₆

—

202

—

114

C₂×C₈

—

202

—

115

C₄⋊C₄

—

202

—

116

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

117

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

404

—

121

C₄²

—

122

C₄²

—

202

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

#13108053

yes

—

128

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

404

—

136

C₂×C₄

—

404

—

137

C₂×C₄

—

404

—

138

C₂×C₄

—

404

—

139

C₂×C₄

—

202

—

140

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

142

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

143

C₂×C₄

#13108074

yes

—

144

C₈

—

202

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×#13108053

yes

characteristic

153

C₂²

#15692224

yes

characteristic

154

C₂²

#15692238

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

808

—

158

C₄

—

404

—

159

C₄

C₄×C₂²⋊D₁₀₁

yes

—

160

C₄

#15692234

yes

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#20265050

yes

characteristic

165

C₂

#20265345

yes

characteristic

166

C₂

#20265355

yes

characteristic

	subgroup	G/H
center	C₂²	#15692224
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#518×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	C₂²×D₄×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#518	D₁₀₁
hypocenter	C₁₀₁	#1856
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #518

subgroup

G/H

center

C₂²

#15692224

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#518×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#120

prime radical

C₂²×D₄×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #518

N	H ≅ G/N	type
1. #20265220	C₂	semidirect
4. #20264936	C₂	semidirect
12. #15692172	C₄	semidirect
21. #13108058	C₂×C₄	semidirect
84. C₁₀₁	#1856	semidirect

H ≅ G/N

type

1. #20265220

C₂

semidirect

4. #20264936

C₂

semidirect

12. #15692172

C₄

semidirect

21. #13108058

C₂×C₄

semidirect

84. C₁₀₁

#1856

semidirect

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

subgroup	index
#20265220	2
C₂×#15692172	2
#20265080	2
#20264936	2
#518×C₁₀₁	2
#1856	101

subgroup

index

#518×C₁₀₁

101

Subgroups — #27171342

Conjugacy classes of subgroups

Complements

Maximal subgroups