Group #27171685 — subgroups

Subgroups — #27171685

Order 12928.

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

normal subgroups only

This group has 308 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	#20265944	C₂	1	1	yes	maximal, characteristic
2	#20265161	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692174	C₂	1	2	yes	maximal, complemented
4	#20264947	C₂	1	2	yes	maximal, complemented
5	#536×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	#15692262	—	2	4	no	—
7	C₂×#13108058	—	2	2	no	—
8	C₂×#13108058	C₂²	1	2	yes	—
9	C₂×#13108058	C₂²	1	2	yes	—
10	#15692174	—	2	4	no	—
11	#15692174	—	2	1	no	—
12	#15692172	—	2	1	no	—
13	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₁	—	2	1	no	—
16	#124×C₁₀₁	—	2	1	no	—
17	#123×C₁₀₁	C₄	1	4	yes	—
18	C₂²⋊Q₈×C₁₀₁	—	2	2	no	—
19	#117×C₁₀₁	C₂²	1	1	yes	characteristic
20	#13108058	—	2	4	no	—
21	#13108058	—	2	2	no	—
22	#13108058	—	2	2	no	—
23	#13108058	—	2	2	no	—
24	#13108058	D₄	1	4	yes	—
25	C₂×#11717579	—	2	4	no	—
26	C₂×#11717579	—	2	2	no	—
27	C₂×#11717579	—	2	2	no	—
28	#41×C₁₀₁	—	2	2	no	—
29	#41×C₁₀₁	—	2	2	no	—
30	C₂×Q₈×C₁₀₁	—	2	1	no	—
31	C₂×Q₈×C₁₀₁	—	2	1	no	—
32	C₂×Q₈×C₁₀₁	—	2	2	no	—
33	C₂×Q₈×C₁₀₁	—	2	2	no	—
34	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₁₀₁	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
39	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₀₁	—	2	4	no	—
41	C₄⋊C₄×C₁₀₁	—	2	1	no	—
42	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
43	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
44	C₄²×C₁₀₁	—	2	1	no	—
45	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₁₀₁	—	2	2	no	—
48	Q₈×C₁₀₁	—	4	1	no	—
49	Q₈×C₁₀₁	—	4	2	no	—
50	Q₈×C₁₀₁	—	4	1	no	—
51	D₄×C₁₀₁	—	2	4	no	—
52	D₄×C₁₀₁	—	2	2	no	—
53	C₂×C₄×C₁₀₁	—	2	2	no	—
54	C₂×C₄×C₁₀₁	—	2	2	no	—
55	C₂×C₄×C₁₀₁	—	2	2	no	—
56	C₂×C₄×C₁₀₁	—	2	2	no	—
57	C₂×C₄×C₁₀₁	—	2	4	no	—
58	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₁	#41	1	2	yes	—
63	#11717579	—	2	4	no	—
64	#11717579	—	4	2	no	—
65	C₂²×C₁₀₁	—	2	2	no	—
66	C₂²×C₁₀₁	—	4	1	no	—
67	C₂²×C₁₀₁	—	4	2	no	—
68	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
70	C₄×C₁₀₁	—	2	2	no	—
71	C₄×C₁₀₁	—	4	2	no	—
72	C₄×C₁₀₁	—	4	2	no	—
73	C₄×C₁₀₁	#120	1	1	yes	characteristic
74	C₄×C₁₀₁	#121	1	2	yes	—
75	C₄×C₁₀₁	#123	1	1	yes	characteristic
76	C₂×C₁₀₁	—	2	1	no	—
77	C₂×C₁₀₁	—	2	2	no	—
78	C₂×C₁₀₁	—	4	2	no	—
79	C₂×C₁₀₁	#386	1	1	yes	characteristic
80	#1915	—	101	1	no	maximal, Sylow
81	C₁₀₁	#1915	1	1	yes	characteristic, complemented, Sylow
82	#568	—	101	1	no	—
83	#536	D₁₀₁	1	1	yes	characteristic
84	#413	—	101	1	no	—
85	C₂×#101	—	101	2	no	—
86	#344	—	101	2	no	—
87	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
88	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
89	#131	—	202	4	no	—
90	C₂×M₁₆	—	101	2	no	—
91	C₂×M₁₆	—	101	2	no	—
92	C₂×M₁₆	—	202	2	no	—
93	C₄⋊Q₈	—	2	1	no	—
94	#124	—	2	1	no	—
95	#123	Dic₁₀₁	1	4	yes	—
96	C₂²⋊Q₈	—	2	2	no	—
97	#117	C₂×D₁₀₁	1	1	yes	characteristic
98	#101	—	202	4	no	—
99	#101	—	202	1	no	—
100	#100	—	202	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
108	C₂×D₄	C₂×Dic₁₀₁	1	2	yes	—
109	C₂×D₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
111	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
113	M₁₆	—	101	4	no	—
114	M₁₆	—	202	2	no	—
115	M₁₆	—	202	2	no	—
116	M₁₆	—	202	4	no	—
117	M₁₆	—	202	2	no	—
118	C₂×C₈	—	202	2	no	—
119	C₂×C₈	—	202	4	no	—
120	C₂×C₈	—	202	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	C₂×Dic₁₀₁	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₂³	C₂²⋊Dic₁₀₁	1	2	yes	—
127	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
140	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
143	C₂×C₄	#13108073	1	2	yes	—
144	C₈	—	404	2	no	—
145	C₈	—	202	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
150	C₂²	D₄×Dic₁₀₁	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#15692223	1	1	yes	characteristic
155	C₄	#15692227	1	2	yes	—
156	C₄	#15692236	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#20265057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265944

C₂

yes

maximal, characteristic

#20265161

C₂

yes

maximal, characteristic

C₂×#15692174

C₂

yes

maximal, complemented

#20264947

C₂

yes

maximal, complemented

#536×C₁₀₁

C₂

yes

maximal, characteristic

#15692262

—

C₂×#13108058

—

C₂×#13108058

C₂²

yes

—

C₂×#13108058

C₂²

yes

—

#15692174

—

#15692174

—

#15692172

—

C₂×#41×C₁₀₁

C₂²

yes

characteristic

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

—

#124×C₁₀₁

—

#123×C₁₀₁

C₄

yes

—

C₂²⋊Q₈×C₁₀₁

—

#117×C₁₀₁

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

—

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

characteristic

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂×D₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

#123

yes

characteristic

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#386

yes

characteristic

#1915

—

101

maximal, Sylow

C₁₀₁

#1915

yes

characteristic, complemented, Sylow

#568

—

101

—

#536

D₁₀₁

yes

characteristic

#413

—

101

—

C₂×#101

—

101

—

#344

—

101

—

C₂×#41

C₂×D₁₀₁

yes

characteristic

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

#131

—

202

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

202

—

C₄⋊Q₈

—

#124

—

#123

Dic₁₀₁

yes

—

C₂²⋊Q₈

—

#117

C₂×D₁₀₁

yes

characteristic

#101

—

202

—

#101

—

202

—

100

#100

—

202

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₁₀₁

yes

characteristic

108

C₂×D₄

C₂×Dic₁₀₁

yes

—

109

C₂×D₄

C₂²⋊D₁₀₁

yes

characteristic

110

C₂²×C₄

C₂×Dic₁₀₁

yes

—

111

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

112

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

113

M₁₆

—

101

—

114

M₁₆

—

202

—

115

M₁₆

—

202

—

116

M₁₆

—

202

—

117

M₁₆

—

202

—

118

C₂×C₈

—

202

—

119

C₂×C₈

—

202

—

120

C₂×C₈

—

202

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

C₂×Dic₁₀₁

yes

—

125

C₄²

—

126

C₂³

C₂²⋊Dic₁₀₁

yes

—

127

C₂³

C₂²×Dic₁₀₁

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

140

C₂×C₄

D₄×D₁₀₁

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

143

C₂×C₄

#13108073

yes

—

144

C₈

—

404

—

145

C₈

—

202

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

150

C₂²

D₄×Dic₁₀₁

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#15692223

yes

characteristic

155

C₄

#15692227

yes

—

156

C₄

#15692236

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#20265057

yes

characteristic

	subgroup	G/H
center	C₂	#20265057
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#536×C₁₀₁	C₂
socle	C₂×C₁₀₁	#386
prime radical	C₂×#41×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#536	D₁₀₁
hypocenter	C₁₀₁	#1915
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#20265057

commutator

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#536×C₁₀₁

C₂

socle

C₂×C₁₀₁

#386

prime radical

C₂×#41×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
3. C₂×#15692174	C₂	semidirect
4. #20264947	C₂	semidirect
81. C₁₀₁	#1915	semidirect

H ≅ G/N

type

3. C₂×#15692174

C₂

semidirect

4. #20264947

C₂

semidirect

81. C₁₀₁

#1915

semidirect

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

subgroup	index
#20265944	2
#20265161	2
C₂×#15692174	2
#20264947	2
#536×C₁₀₁	2
#1915	101

subgroup

index

#536×C₁₀₁

101

Subgroups — #27171685

Conjugacy classes of subgroups

Complements

Maximal subgroups