Group #27171536 — subgroups

Subgroups — #27171536

Order 12928.

344 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20265655	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265182	C₂	1	2	yes	maximal
3	C₂×#15692174	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264937	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	Q₈×Dic₁₀₁	—	2	4	no	—
8	#15692214	—	2	2	no	—
9	#15692214	C₂²	1	2	yes	—
10	#15692176	—	2	4	no	—
11	#15692174	—	2	2	no	—
12	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	#124×C₁₀₁	—	2	2	no	—
15	#124×C₁₀₁	C₄	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
17	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
18	#13108058	—	2	2	no	—
19	#13108058	—	2	2	no	—
20	C₂×#11717579	—	2	2	no	—
21	C₄⋊Dic₁₀₁	—	2	4	no	—
22	C₄⋊Dic₁₀₁	D₄	1	4	yes	—
23	C₂²⋊Dic₁₀₁	—	2	2	no	—
24	C₄×Dic₁₀₁	—	2	4	no	—
25	C₄×Dic₁₀₁	—	2	2	no	—
26	C₂⁴×C₁₀₁	—	2	1	no	—
27	C₂×Q₈×C₁₀₁	—	2	2	no	—
28	C₂×Q₈×C₁₀₁	—	2	2	no	—
29	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂×Q₈×C₁₀₁	D₄	1	2	yes	—
31	C₂×D₄×C₁₀₁	—	2	2	no	—
32	C₂×D₄×C₁₀₁	—	2	1	no	—
33	C₂×D₄×C₁₀₁	—	2	2	no	—
34	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₀₁	—	2	1	no	—
36	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
37	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
38	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
39	C₄²×C₁₀₁	—	2	1	no	—
40	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
41	C₂³×C₁₀₁	—	2	2	no	—
42	C₂³×C₁₀₁	—	4	4	no	—
43	C₂³×C₁₀₁	—	4	1	no	—
44	C₂³×C₁₀₁	—	4	1	no	—
45	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₀₁	—	2	4	no	—
47	Q₈×C₁₀₁	—	2	4	no	—
48	D₄×C₁₀₁	—	2	4	no	—
49	D₄×C₁₀₁	—	4	2	no	—
50	C₂×C₄×C₁₀₁	—	2	2	no	—
51	C₂×C₄×C₁₀₁	—	2	2	no	—
52	C₂×C₄×C₁₀₁	—	2	2	no	—
53	C₂×C₄×C₁₀₁	—	4	2	no	—
54	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₁	#41	1	2	yes	—
59	C₂×Dic₁₀₁	—	2	4	no	—
60	#11717579	—	4	2	no	—
61	C₂²×C₁₀₁	—	2	2	no	—
62	C₂²×C₁₀₁	—	4	4	no	—
63	C₂²×C₁₀₁	—	4	2	no	—
64	C₂²×C₁₀₁	—	8	2	no	—
65	C₂²×C₁₀₁	—	8	2	no	—
66	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₀₁	#120	1	1	yes	characteristic
68	C₂²×C₁₀₁	#123	1	1	yes	characteristic
69	C₄×C₁₀₁	—	2	4	no	—
70	C₄×C₁₀₁	—	4	2	no	—
71	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
72	C₄×C₁₀₁	#121	1	2	yes	—
73	Dic₁₀₁	—	4	4	no	—
74	C₂×C₁₀₁	—	2	1	no	—
75	C₂×C₁₀₁	—	2	1	no	—
76	C₂×C₁₀₁	—	8	2	no	—
77	C₂×C₁₀₁	#386	1	1	yes	characteristic
78	C₂×C₁₀₁	#454	1	2	yes	—
79	#1888	—	101	1	no	maximal, Sylow
80	C₁₀₁	#1888	1	1	yes	characteristic, complemented, Sylow
81	C₂×#124	D₁₀₁	1	1	yes	characteristic
82	#519	—	101	1	no	—
83	#418	—	101	2	no	—
84	C₂×#101	—	101	1	no	—
85	#339	—	101	2	no	—
86	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
88	C₂×M₁₆	—	101	2	no	—
89	#124	—	2	2	no	—
90	#124	Dic₁₀₁	1	4	yes	complemented
91	C₄×Q₈	—	202	4	no	—
92	#117	—	101	2	no	—
93	#117	—	202	2	no	—
94	C₂×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
96	#102	—	202	4	no	—
97	#101	—	202	2	no	—
98	C₂⁴	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₁₀₁	1	2	yes	—
102	C₂×Q₈	C₂²⋊D₁₀₁	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	C₂×Dic₁₀₁	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
110	M₁₆	—	202	2	no	—
111	M₁₆	—	202	2	no	—
112	C₂×C₈	—	202	2	no	—
113	C₄⋊C₄	—	101	4	no	—
114	C₄⋊C₄	—	202	4	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₂²⋊C₄	—	202	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	202	4	no	—
119	C₄²	—	202	2	no	—
120	C₄²	C₂×Dic₁₀₁	1	2	yes	—
121	C₂³	—	2	2	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	202	4	no	—
135	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
136	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
137	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
139	C₂×C₄	#13108073	1	2	yes	—
140	C₈	—	404	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
147	C₂²	#15692223	1	1	yes	characteristic
148	C₂²	#15692236	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	404	4	no	—
152	C₄	D₄×Dic₁₀₁	1	2	yes	—
153	C₄	#15692227	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	8	2	no	—
157	C₂	#20265057	1	1	yes	characteristic
158	C₂	#20265349	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265655

C₂

yes

maximal, characteristic, complemented

#20265182

C₂

yes

maximal

C₂×#15692174

C₂

yes

maximal, characteristic, complemented

#20264937

C₂

yes

maximal, complemented

C₂×#124×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

Q₈×Dic₁₀₁

—

#15692214

—

#15692214

C₂²

yes

—

#15692176

—

#15692174

—

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

#124×C₁₀₁

—

#124×C₁₀₁

C₄

yes

complemented

C₂×C₂²⋊C₄×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

—

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

D₄

yes

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

#121

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#386

yes

characteristic

C₂×C₁₀₁

#454

yes

—

#1888

—

101

maximal, Sylow

C₁₀₁

#1888

yes

characteristic, complemented, Sylow

C₂×#124

D₁₀₁

yes

characteristic

#519

—

101

—

#418

—

101

—

C₂×#101

—

101

—

#339

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

C₂²×D₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

#124

—

#124

Dic₁₀₁

yes

complemented

C₄×Q₈

—

202

—

#117

—

101

—

#117

—

202

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#102

—

202

—

#101

—

202

—

C₂⁴

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₁₀₁

yes

—

102

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×Dic₁₀₁

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

109

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

110

M₁₆

—

202

—

111

M₁₆

—

202

—

112

C₂×C₈

—

202

—

113

C₄⋊C₄

—

101

—

114

C₄⋊C₄

—

202

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

202

—

117

C₄²

—

118

C₄²

—

202

—

119

C₄²

—

202

—

120

C₄²

C₂×Dic₁₀₁

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

202

—

135

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

136

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

137

C₂×C₄

D₄×D₁₀₁

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

139

C₂×C₄

#13108073

yes

—

140

C₈

—

404

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

147

C₂²

#15692223

yes

characteristic

148

C₂²

#15692236

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

404

—

152

C₄

D₄×Dic₁₀₁

yes

—

153

C₄

#15692227

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

157

C₂

#20265057

yes

characteristic

158

C₂

#20265349

yes

—

	subgroup	G/H
center	C₂²	#15692223
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#124×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	C₂²×D₄×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	#1888
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

subgroup

G/H

center

C₂²

#15692223

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#124×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 ◁ C₂×#124

N	H ≅ G/N	type
1. #20265655	C₂	semidirect
3. C₂×#15692174	C₂	semidirect
4. #20264937	C₂	semidirect
15. #124×C₁₀₁	C₄	semidirect
80. C₁₀₁	#1888	semidirect
90. #124	Dic₁₀₁	semidirect

H ≅ G/N

type

1. #20265655

C₂

semidirect

3. C₂×#15692174

C₂

semidirect

4. #20264937

C₂

semidirect

15. #124×C₁₀₁

C₄

semidirect

80. C₁₀₁

#1888

semidirect

90. #124

Dic₁₀₁

semidirect

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

subgroup	index
#20265655	2
#20265182	2
C₂×#15692174	2
#20264937	2
C₂×#124×C₁₀₁	2
#1888	101

subgroup

index

C₂×#124×C₁₀₁

101

Subgroups — #27171536

Conjugacy classes of subgroups

Complements

Maximal subgroups