Group #27172275 — subgroups

Subgroups — #27172275

Order 12928.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15692175	C₂	1	2	yes	maximal
2	#20265079	C₂	1	1	yes	maximal, characteristic
3	#20264939	C₂	1	2	yes	maximal, complemented
4	#20264932	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
7	C₂²×#11717579	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×#13108051	—	2	2	no	—
10	#15692175	—	2	4	no	—
11	#125×C₁₀₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂³×D₁₀₁	—	2	1	no	—
16	C₂×D₄₀₄	—	2	2	no	—
17	C₂×D₄₀₄	—	2	2	no	—
18	C₂×D₄₀₄	—	2	1	no	—
19	C₂×D₄₀₄	Q₈	1	2	yes	—
20	C₂²×Dic₁₀₁	—	2	1	no	—
21	C₂×#11717579	—	2	2	no	—
22	C₂×#11717579	—	2	2	no	—
23	C₂×#11717579	—	2	2	no	—
24	C₄⋊Dic₁₀₁	—	2	2	no	—
25	#13108051	—	4	4	no	—
26	C₂²×C₄×C₁₀₁	—	2	2	no	—
27	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
28	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₁	—	2	8	no	—
30	C₄⋊C₄×C₁₀₁	—	2	2	no	—
31	C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
32	C₄²×C₁₀₁	—	2	2	no	—
33	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₁₀₁	—	2	2	no	—
35	C₂²×D₁₀₁	—	4	1	no	—
36	C₂²×D₁₀₁	—	4	1	no	—
37	C₂²×D₁₀₁	—	4	2	no	—
38	C₂²×D₁₀₁	—	4	2	no	—
39	C₂×C₄×C₁₀₁	—	2	2	no	—
40	C₂×C₄×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	—	4	2	no	—
44	C₂×C₄×C₁₀₁	—	4	2	no	—
45	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	#41	1	2	yes	—
48	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
49	C₂×Dic₁₀₁	—	4	1	no	—
50	C₂×Dic₁₀₁	—	4	2	no	—
51	D₄₀₄	—	2	4	no	—
52	D₄₀₄	—	4	2	no	—
53	#11717579	—	4	2	no	—
54	C₂²×C₁₀₁	#120	1	1	yes	characteristic
55	C₂²×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₀₁	#124	1	1	yes	characteristic
57	C₂²×C₁₀₁	#135	1	2	yes	—
58	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₁₀₁	—	4	2	no	—
60	C₂×D₁₀₁	—	4	4	no	—
61	C₂×D₁₀₁	—	8	2	no	—
62	C₂×D₁₀₁	—	8	1	no	—
63	C₂×D₁₀₁	—	8	1	no	—
64	C₄×C₁₀₁	—	4	2	no	—
65	C₄×C₁₀₁	—	4	4	no	—
66	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₀₁	#124	1	2	yes	—
68	Dic₁₀₁	—	8	1	no	—
69	C₂×C₁₀₁	#393	1	1	yes	characteristic
70	C₂×C₁₀₁	#449	1	2	yes	—
71	C₂×C₁₀₁	#478	1	2	yes	—
72	C₂×C₁₀₁	#488	1	1	yes	characteristic
73	C₂×C₁₀₁	#490	1	1	yes	characteristic
74	D₁₀₁	—	8	2	no	—
75	#2025	—	101	1	no	maximal, Sylow
76	C₁₀₁	#2025	1	1	yes	characteristic, complemented, Sylow
77	C₂×#125	D₁₀₁	1	1	yes	characteristic, complemented
78	C₂×#102	—	101	2	no	—
79	#390	—	101	1	no	—
80	#340	—	101	2	no	—
81	#336	—	101	1	no	—
82	C₂²×D₄	—	101	1	no	—
83	C₂²×C₈	—	101	2	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	101	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
88	C₂×C₂²⋊C₄	—	202	2	no	—
89	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
90	#102	—	202	4	no	—
91	C₂⁴	—	202	1	no	—
92	C₂×D₄	—	101	2	no	—
93	C₂×D₄	—	202	2	no	—
94	C₂×D₄	—	202	1	no	—
95	C₂×D₄	—	202	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	202	1	no	—
98	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
100	C₂×C₈	—	202	2	no	—
101	C₂×C₈	—	202	2	no	—
102	C₂×C₈	—	202	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	202	2	no	—
106	C₄⋊C₄	C₂²⋊D₁₀₁	1	4	yes	—
107	C₂²⋊C₄	—	404	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	404	1	no	—
110	C₂³	—	404	1	no	—
111	C₂³	—	404	2	no	—
112	C₂³	—	404	2	no	—
113	C₂³	—	202	2	no	—
114	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
115	D₄	—	404	2	no	—
116	D₄	—	202	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	404	1	no	—
124	C₂×C₄	—	404	2	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
126	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
127	C₂×C₄	#13108075	1	2	yes	—
128	C₂×C₄	#13108075	1	1	yes	characteristic
129	C₈	—	404	2	no	—
130	C₂²	—	808	1	no	—
131	C₂²	—	808	1	no	—
132	C₂²	—	808	2	no	—
133	C₂²	—	404	4	no	—
134	C₂²	—	404	2	no	—
135	C₂²	#15692224	1	1	yes	characteristic
136	C₂²	#15692231	1	1	yes	characteristic
137	C₂²	#15692244	1	1	yes	characteristic
138	C₂²	#15692275	1	2	yes	—
139	C₂²	#15692280	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	808	1	no	—
143	C₄	#15692231	1	2	yes	—
144	C₄	#15692244	1	2	yes	—
145	C₂	—	808	2	no	—
146	C₂	#20265088	1	1	yes	characteristic
147	C₂	#20265322	1	2	yes	—
148	C₂	#20265474	1	2	yes	—
149	C₂	#20265534	1	1	yes	characteristic
150	C₂	#20265543	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692175

C₂

yes

maximal

#20265079

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal, complemented

#20264932

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₀₄

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×#13108051

—

#15692175

—

#125×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂³×D₁₀₁

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

Q₈

yes

—

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

#13108051

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

D₄₀₄

—

D₄₀₄

—

#11717579

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

#124

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#393

yes

characteristic

C₂×C₁₀₁

#449

yes

—

C₂×C₁₀₁

#478

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

D₁₀₁

—

#2025

—

101

maximal, Sylow

C₁₀₁

#2025

yes

characteristic, complemented, Sylow

C₂×#125

D₁₀₁

yes

characteristic, complemented

C₂×#102

—

101

—

#390

—

101

—

#340

—

101

—

#336

—

101

—

C₂²×D₄

—

101

—

C₂²×C₈

—

101

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₂²⋊C₄

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#102

—

202

—

C₂⁴

—

202

—

C₂×D₄

—

101

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

100

C₂×C₈

—

202

—

101

C₂×C₈

—

202

—

102

C₂×C₈

—

202

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

202

—

106

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

107

C₂²⋊C₄

—

404

—

108

C₄²

—

109

C₂³

—

404

—

110

C₂³

—

404

—

111

C₂³

—

404

—

112

C₂³

—

404

—

113

C₂³

—

202

—

114

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

115

D₄

—

404

—

116

D₄

—

202

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

404

—

124

C₂×C₄

—

404

—

125

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

126

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

127

C₂×C₄

#13108075

yes

—

128

C₂×C₄

#13108075

yes

characteristic

129

C₈

—

404

—

130

C₂²

—

808

—

131

C₂²

—

808

—

132

C₂²

—

808

—

133

C₂²

—

404

—

134

C₂²

—

404

—

135

C₂²

#15692224

yes

characteristic

136

C₂²

#15692231

yes

characteristic

137

C₂²

#15692244

yes

characteristic

138

C₂²

#15692275

yes

—

139

C₂²

#15692280

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

808

—

143

C₄

#15692231

yes

—

144

C₄

#15692244

yes

—

145

C₂

—

808

—

146

C₂

#20265088

yes

characteristic

147

C₂

#20265322

yes

—

148

C₂

#20265474

yes

—

149

C₂

#20265534

yes

characteristic

150

C₂

#20265543

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#125×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
prime radical	C₂²×D₄₀₄	C₂²
hypercenter	C₂×#125	D₁₀₁
hypocenter	C₁₀₁	#2025
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#125×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

prime radical

C₂²×D₄₀₄

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
3. #20264939	C₂	semidirect
4. #20264932	C₂	semidirect
5. C₂×#125×C₁₀₁	C₂	semidirect
76. C₁₀₁	#2025	semidirect
77. C₂×#125	D₁₀₁	semidirect

H ≅ G/N

type

3. #20264939

C₂

semidirect

4. #20264932

C₂

semidirect

5. C₂×#125×C₁₀₁

C₂

semidirect

76. C₁₀₁

#2025

semidirect

77. C₂×#125

D₁₀₁

semidirect

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

subgroup	index
C₂×#15692175	2
#20265079	2
#20264939	2
#20264932	2
C₂×#125×C₁₀₁	2
#2025	101

subgroup

index

C₂×#125×C₁₀₁

101

Subgroups — #27172275

Conjugacy classes of subgroups

Complements

Maximal subgroups