Group #27172416 — subgroups

Subgroups — #27172416

Order 12928.

266 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20265200	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692182	C₂	1	2	yes	maximal, complemented
3	#20265161	C₂	1	2	yes	maximal, complemented
4	#20264935	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	1	yes	characteristic
7	C₂×#13108058	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	#15692184	—	2	2	no	—
10	#15692182	—	2	2	no	—
11	#15692182	—	2	2	no	—
12	#15692174	—	2	2	no	—
13	#15692172	—	2	2	no	—
14	C₂×#41×C₁₀₁	C₂²	1	2	yes	—
15	#137×C₁₀₁	—	2	2	no	—
16	AGL(1,ℤ/8)×C₁₀₁	—	2	2	no	—
17	#135×C₁₀₁	—	2	2	no	—
18	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₀₁	—	2	1	no	—
20	#13108058	—	2	2	no	—
21	#13108058	—	2	2	no	—
22	#13108058	—	2	2	no	—
23	C₂×#11717579	—	2	1	no	—
24	C₂×#11717579	—	2	2	no	—
25	C₄×Dic₁₀₁	—	2	1	no	—
26	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
27	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
28	#41×C₁₀₁	—	2	2	no	—
29	#41×C₁₀₁	—	2	2	no	—
30	#41×C₁₀₁	—	2	2	no	—
31	#41×C₁₀₁	—	2	2	no	—
32	C₂×Q₈×C₁₀₁	D₄	1	2	yes	—
33	C₂×D₄×C₁₀₁	—	2	2	no	—
34	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
35	C₂²×C₄×C₁₀₁	—	2	2	no	—
36	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
37	Q₁₆×C₁₀₁	—	2	2	no	—
38	SD₁₆×C₁₀₁	—	2	4	no	—
39	D₈×C₁₀₁	—	2	2	no	—
40	M₁₆×C₁₀₁	—	2	2	no	—
41	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
42	C₂³×C₁₀₁	—	2	2	no	—
43	C₂³×C₁₀₁	#41	1	1	yes	characteristic
44	Q₈×C₁₀₁	—	2	2	no	—
45	Q₈×C₁₀₁	—	2	2	no	—
46	D₄×C₁₀₁	—	2	2	no	—
47	D₄×C₁₀₁	—	2	2	no	—
48	D₄×C₁₀₁	—	4	2	no	—
49	D₄×C₁₀₁	—	4	2	no	—
50	C₂×C₄×C₁₀₁	—	2	2	no	—
51	C₂×C₄×C₁₀₁	—	4	2	no	—
52	C₂×C₄×C₁₀₁	—	4	2	no	—
53	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₁	#41	1	2	yes	—
57	C₂×Dic₁₀₁	—	2	1	no	—
58	C₂×Dic₁₀₁	—	2	1	no	—
59	C₂×Dic₁₀₁	—	4	2	no	—
60	C₈×C₁₀₁	—	2	2	no	—
61	#11717579	—	4	1	no	—
62	#11717579	—	4	2	no	—
63	C₂²×C₁₀₁	—	4	1	no	—
64	C₂²×C₁₀₁	—	4	2	no	—
65	C₂²×C₁₀₁	—	8	2	no	—
66	C₂²×C₁₀₁	#121	1	1	yes	characteristic
67	C₂²×C₁₀₁	#123	1	2	yes	—
68	C₄×C₁₀₁	—	4	2	no	—
69	C₄×C₁₀₁	#120	1	1	yes	characteristic
70	C₄×C₁₀₁	#121	1	2	yes	—
71	C₄×C₁₀₁	#124	1	1	yes	characteristic
72	Dic₁₀₁	—	2	2	no	—
73	Dic₁₀₁	—	4	1	no	—
74	C₂×C₁₀₁	—	2	1	no	—
75	C₂×C₁₀₁	—	2	2	no	—
76	C₂×C₁₀₁	—	8	2	no	—
77	C₂×C₁₀₁	#394	1	1	yes	characteristic
78	#2050	—	101	1	no	maximal, Sylow
79	C₁₀₁	#2050	1	1	yes	characteristic, complemented, Sylow
80	#575	D₁₀₁	1	1	yes	characteristic
81	#423	—	101	1	no	—
82	C₂×#104	—	101	2	no	—
83	#413	—	101	2	no	—
84	#338	—	101	1	no	—
85	C₂×#41	C₂×D₁₀₁	1	2	yes	—
86	#137	—	2	2	no	—
87	AGL(1,ℤ/8)	—	2	2	no	—
88	#135	—	2	2	no	—
89	C₂×M₁₆	—	101	2	no	—
90	C₂×M₁₆	—	101	1	no	—
91	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
92	C₂×C₄²	—	101	1	no	—
93	C₄⋊C₈	—	202	2	no	—
94	#104	—	202	2	no	—
95	#104	—	202	2	no	—
96	#101	—	202	2	no	—
97	#100	—	202	2	no	—
98	#41	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	C₂²⋊D₁₀₁	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	202	1	no	—
107	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
108	Q₁₆	—	2	2	no	—
109	SD₁₆	—	2	4	no	—
110	D₈	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	202	2	no	—
113	M₁₆	—	202	2	no	—
114	M₁₆	—	202	2	no	—
115	C₂×C₈	—	202	1	no	—
116	C₂×C₈	—	202	2	no	—
117	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
118	C₄²	—	101	1	no	—
119	C₄²	—	101	1	no	—
120	C₄²	—	202	1	no	—
121	C₂³	—	2	2	no	—
122	C₂³	#13108073	1	1	yes	characteristic
123	Q₈	—	2	2	no	—
124	Q₈	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	404	2	no	—
133	C₂×C₄	—	202	1	no	—
134	C₂×C₄	—	202	1	no	—
135	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
138	C₂×C₄	#13108073	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	404	1	no	—
141	C₈	—	404	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	#15692226	1	1	yes	characteristic
146	C₂²	#15692236	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	404	1	no	—
149	C₄	—	202	2	no	—
150	C₄	#15692224	1	1	yes	characteristic
151	C₄	#15692227	1	2	yes	—
152	C₄	#15692242	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	8	2	no	—
156	C₂	#20265091	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265200

C₂

yes

maximal, characteristic, complemented

C₂×#15692182

C₂

yes

maximal, complemented

#20265161

C₂

yes

maximal, complemented

#20264935

C₂

yes

maximal, characteristic, complemented

#575×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

characteristic

C₂×#13108058

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

—

—

—

—

C₂×#41×C₁₀₁

C₂²

yes

—

#137×C₁₀₁

—

AGL(1,ℤ/8)×C₁₀₁

—

#135×C₁₀₁

—

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

—

—

—

—

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

characteristic

#41×C₁₀₁

—

#41×C₁₀₁

—

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

Q₁₆×C₁₀₁

—

SD₁₆×C₁₀₁

—

D₈×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

#41

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#121

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

#124

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#394

yes

characteristic

#2050

—

101

maximal, Sylow

C₁₀₁

#2050

yes

characteristic, complemented, Sylow

#575

D₁₀₁

yes

characteristic

#423

—

101

—

C₂×#104

—

101

—

#413

—

101

—

#338

—

101

—

C₂×#41

C₂×D₁₀₁

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

characteristic

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

#104

—

202

—

#104

—

202

—

#101

—

202

—

#100

—

202

—

#41

—

#41

—

100

#41

—

101

#41

—

102

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

103

C₂×D₄

—

104

C₂×D₄

C₂²⋊D₁₀₁

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

—

202

—

107

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

108

Q₁₆

—

109

SD₁₆

—

110

D₈

—

111

M₁₆

—

112

M₁₆

—

202

—

113

M₁₆

—

202

—

114

M₁₆

—

202

—

115

C₂×C₈

—

202

—

116

C₂×C₈

—

202

—

117

C₂×C₈

C₂²⋊D₁₀₁

yes

—

118

C₄²

—

101

—

119

C₄²

—

101

—

120

C₄²

—

202

—

121

C₂³

—

122

C₂³

#13108073

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

404

—

133

C₂×C₄

—

202

—

134

C₂×C₄

—

202

—

135

C₂×C₄

D₄×D₁₀₁

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

138

C₂×C₄

#13108073

yes

—

139

C₈

—

140

C₈

—

404

—

141

C₈

—

404

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#15692226

yes

characteristic

146

C₂²

#15692236

yes

—

147

C₄

—

148

C₄

—

404

—

149

C₄

—

202

—

150

C₄

#15692224

yes

characteristic

151

C₄

#15692227

yes

—

152

C₄

#15692242

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#20265091

yes

characteristic

	subgroup	G/H
center	C₄	#15692242
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting, prime radical	#575×C₁₀₁	C₂
socle	C₂×C₁₀₁	#394
hypercenter	#575	D₁₀₁
hypocenter	C₁₀₁	#2050
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#15692242

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting, prime radical

#575×C₁₀₁

C₂

C₂×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #20265200	C₂	semidirect
2. C₂×#15692182	C₂	semidirect
3. #20265161	C₂	semidirect
4. #20264935	C₂	semidirect
79. C₁₀₁	#2050	semidirect

H ≅ G/N

type

1. #20265200

C₂

semidirect

2. C₂×#15692182

C₂

semidirect

3. #20265161

C₂

semidirect

4. #20264935

C₂

semidirect

79. C₁₀₁

#2050

semidirect

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

subgroup	index
#20265200	2
C₂×#15692182	2
#20265161	2
#20264935	2
#575×C₁₀₁	2
#2050	101

subgroup

index

#575×C₁₀₁

101

Subgroups — #27172416

Conjugacy classes of subgroups

Complements

Maximal subgroups