Group #27170965 — subgroups

Subgroups — #27170965

Order 12928.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692178	C₂	1	2	yes	maximal
3	C₂×#15692166	C₂	1	1	yes	maximal, characteristic
4	#20264939	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
7	C₂²×#11717579	C₂²	1	2	yes	—
8	C₄×Dic₂₀₂	—	2	2	no	—
9	C₄×Dic₂₀₂	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×#13108053	—	2	1	no	—
12	C₂×C₄×Dic₁₀₁	—	2	1	no	—
13	#15692178	C₄	1	8	yes	complemented
14	#15692166	—	2	2	no	—
15	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₀₁	—	2	1	no	—
17	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
18	#95×C₁₀₁	—	2	1	no	—
19	C₂×Dic₂₀₂	—	2	1	no	—
20	C₂×Dic₂₀₂	—	2	2	no	—
21	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
22	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₀₁	—	2	1	no	—
24	C₂²×Dic₁₀₁	—	2	1	no	—
25	C₂×#11717579	—	2	2	no	—
26	C₂×#11717579	—	2	2	no	—
27	C₂×#11717579	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₁₀₁	—	2	1	no	—
29	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
30	#13108053	—	2	4	no	—
31	C₄×Dic₁₀₁	—	2	4	no	—
32	C₂²×C₄×C₁₀₁	—	2	2	no	—
33	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₁	—	2	2	no	—
37	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₄²×C₁₀₁	—	4	2	no	—
39	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
40	C₄²×C₁₀₁	D₄	1	2	yes	—
41	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₁	—	2	1	no	—
43	C₂×C₄×C₁₀₁	—	2	2	no	—
44	C₂×C₄×C₁₀₁	—	2	1	no	—
45	C₂×C₄×C₁₀₁	—	2	4	no	—
46	C₂×C₄×C₁₀₁	—	4	2	no	—
47	C₂×C₄×C₁₀₁	—	4	2	no	—
48	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₁	#41	1	2	yes	—
54	C₂×Dic₁₀₁	—	2	2	no	—
55	C₂×Dic₁₀₁	—	2	4	no	—
56	C₂×Dic₁₀₁	—	2	2	no	—
57	C₂×Dic₁₀₁	—	4	1	no	—
58	C₂×Dic₁₀₁	—	4	1	no	—
59	Dic₂₀₂	—	2	2	no	—
60	Dic₂₀₂	—	2	4	no	—
61	Dic₂₀₂	C₄²	1	4	yes	complemented
62	#11717579	—	2	4	no	—
63	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
65	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₁₀₁	#137	1	2	yes	—
67	C₂²×C₁₀₁	#137	1	1	yes	characteristic
68	C₄×C₁₀₁	—	2	2	no	—
69	C₄×C₁₀₁	—	2	2	no	—
70	C₄×C₁₀₁	—	4	4	no	—
71	C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₁₀₁	#117	1	1	yes	characteristic
73	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
74	Dic₁₀₁	—	2	4	no	—
75	Dic₁₀₁	—	4	2	no	—
76	C₂×C₁₀₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₁₀₁	#417	1	1	yes	characteristic
78	C₂×C₁₀₁	#419	1	1	yes	characteristic
79	C₂×C₁₀₁	#440	1	2	yes	—
80	C₂×C₁₀₁	#441	1	2	yes	—
81	#1763	—	101	1	no	maximal, Sylow
82	C₁₀₁	#1763	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄×Q₈	—	101	1	no	—
84	C₂×#103	—	101	2	no	—
85	C₂×#97	—	101	1	no	—
86	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
87	#340	—	101	2	no	—
88	C₂²×Q₈	—	101	1	no	—
89	C₂²×C₈	—	101	2	no	—
90	C₄×Q₈	—	101	4	no	—
91	C₄×Q₈	—	202	2	no	—
92	C₂×C₄⋊C₄	—	101	1	no	—
93	C₂×C₄⋊C₄	—	202	1	no	—
94	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	202	1	no	—
97	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
98	#103	—	101	8	no	—
99	#97	—	202	2	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	101	4	no	—
102	C₂×Q₈	—	101	2	no	—
103	C₂×Q₈	—	202	1	no	—
104	C₂×Q₈	—	202	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	202	1	no	—
107	C₂²×C₄	—	202	1	no	—
108	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
111	C₂×C₈	—	101	4	no	—
112	C₂×C₈	—	202	2	no	—
113	C₂×C₈	—	202	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	101	2	no	—
116	C₄⋊C₄	—	202	4	no	—
117	C₄⋊C₄	—	202	1	no	—
118	C₄⋊C₄	C₄×D₁₀₁	1	4	yes	—
119	C₄²	—	4	2	no	—
120	C₄²	—	202	4	no	—
121	C₄²	C₄×D₁₀₁	1	2	yes	—
122	C₄²	D₄₀₄	1	2	yes	—
123	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
124	Q₈	—	101	4	no	—
125	Q₈	—	202	2	no	—
126	Q₈	—	202	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	404	1	no	—
134	C₂×C₄	—	404	1	no	—
135	C₂×C₄	—	202	2	no	—
136	C₂×C₄	—	202	2	no	—
137	C₂×C₄	—	202	4	no	—
138	C₂×C₄	#13108051	1	2	yes	—
139	C₂×C₄	#13108051	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
142	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
143	C₂×C₄	#13108074	1	2	yes	—
144	C₈	—	202	4	no	—
145	C₂²	C₂×#13108051	1	1	yes	characteristic
146	C₂²	C₄×C₂²⋊D₁₀₁	1	2	yes	—
147	C₂²	#15692281	1	1	yes	characteristic
148	C₂²	#15692285	1	2	yes	—
149	C₂²	#15692286	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	404	2	no	—
154	C₄	—	202	4	no	—
155	C₄	C₄²×D₁₀₁	1	1	yes	characteristic
156	C₄	#15692212	1	1	yes	characteristic
157	C₄	C₄×D₄₀₄	1	2	yes	—
158	C₂	C₄×#13108051	1	1	yes	characteristic
159	C₂	#20265176	1	1	yes	characteristic
160	C₂	#20265185	1	1	yes	characteristic
161	C₂	#20265277	1	2	yes	—
162	C₂	#20265282	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₀₂

C₂

yes

maximal, characteristic

C₂×#15692178

C₂

yes

maximal

C₂×#15692166

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

—

C₄×Dic₂₀₂

—

C₄×Dic₂₀₂

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×#13108053

—

C₂×C₄×Dic₁₀₁

—

#15692178

C₄

yes

complemented

#15692166

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#95×C₁₀₁

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

#13108053

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

C₄²

yes

complemented

#11717579

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×D₄

yes

—

C₂²×C₁₀₁

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₁₀₁

#137

yes

—

C₂²×C₁₀₁

#137

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₄²

yes

characteristic

C₄×C₁₀₁

#117

yes

characteristic

C₄×C₁₀₁

C₄×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

C₄×C₂²⋊C₄

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#441

yes

—

#1763

—

101

maximal, Sylow

C₁₀₁

#1763

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

101

—

C₂×#103

—

101

—

C₂×#97

—

101

—

C₄×C₄⋊C₄

D₁₀₁

yes

characteristic

#340

—

101

—

C₂²×Q₈

—

101

—

C₂²×C₈

—

101

—

C₄×Q₈

—

101

—

C₄×Q₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#103

—

101

—

#97

—

202

—

100

#95

—

101

C₂×Q₈

—

101

—

102

C₂×Q₈

—

101

—

103

C₂×Q₈

—

202

—

104

C₂×Q₈

—

202

—

105

C₂²×C₄

—

106

C₂²×C₄

—

202

—

107

C₂²×C₄

—

202

—

108

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

109

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

110

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

111

C₂×C₈

—

101

—

112

C₂×C₈

—

202

—

113

C₂×C₈

—

202

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

101

—

116

C₄⋊C₄

—

202

—

117

C₄⋊C₄

—

202

—

118

C₄⋊C₄

C₄×D₁₀₁

yes

—

119

C₄²

—

120

C₄²

—

202

—

121

C₄²

C₄×D₁₀₁

yes

—

122

C₄²

D₄₀₄

yes

—

123

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

124

Q₈

—

101

—

125

Q₈

—

202

—

126

Q₈

—

202

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

404

—

134

C₂×C₄

—

404

—

135

C₂×C₄

—

202

—

136

C₂×C₄

—

202

—

137

C₂×C₄

—

202

—

138

C₂×C₄

#13108051

yes

—

139

C₂×C₄

#13108051

yes

—

140

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

142

C₂×C₄

C₂×D₄₀₄

yes

characteristic

143

C₂×C₄

#13108074

yes

—

144

C₈

—

202

—

145

C₂²

C₂×#13108051

yes

characteristic

146

C₂²

C₄×C₂²⋊D₁₀₁

yes

—

147

C₂²

#15692281

yes

characteristic

148

C₂²

#15692285

yes

—

149

C₂²

#15692286

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

404

—

154

C₄

—

202

—

155

C₄

C₄²×D₁₀₁

yes

characteristic

156

C₄

#15692212

yes

characteristic

157

C₄

C₄×D₄₀₄

yes

—

158

C₂

C₄×#13108051

yes

characteristic

159

C₂

#20265176

yes

characteristic

160

C₂

#20265185

yes

characteristic

161

C₂

#20265277

yes

—

162

C₂

#20265282

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₄×C₄⋊C₄×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#1763
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₄×C₁₀₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₄×C₄⋊C₄×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₀₁

hypocenter

C₁₀₁

#1763

LCS

G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄×Dic₂₀₂	C₄	semidirect
13. #15692178	C₄	semidirect
61. Dic₂₀₂	C₄²	semidirect
82. C₁₀₁	#1763	semidirect

H ≅ G/N

type

9. C₄×Dic₂₀₂

C₄

semidirect

13. #15692178

C₄

semidirect

61. Dic₂₀₂

C₄²

semidirect

82. C₁₀₁

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₂₀₂	2
C₂×#15692178	2
C₂×#15692166	2
#20264939	2
C₄×C₄⋊C₄×C₁₀₁	2
#1763	101

subgroup

index

C₂×C₄×Dic₂₀₂

C₄×C₄⋊C₄×C₁₀₁

101

Subgroups — #27170965

Conjugacy classes of subgroups

Complements

Maximal subgroups