Group #27171237 — subgroups

Subgroups — #27171237

Order 12928.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265151	C₂	1	1	yes	maximal, characteristic
3	#20264938	C₂	1	2	yes	maximal, complemented
4	#20264934	C₂	1	2	yes	maximal, complemented
5	#413×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₁	—	2	1	no	—
7	D₄×Dic₁₀₁	—	2	2	no	—
8	D₄×Dic₁₀₁	—	2	1	no	—
9	#15692214	C₄	1	2	yes	complemented
10	#15692214	C₂²	1	2	yes	—
11	#15692214	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₀₁	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	#15692170	—	2	2	no	—
15	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
17	#101×C₁₀₁	—	2	1	no	—
18	#100×C₁₀₁	—	2	1	no	—
19	C₂²×Dic₁₀₁	—	2	2	no	—
20	C₂²×Dic₁₀₁	—	2	1	no	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₄⋊Dic₁₀₁	—	2	1	no	—
23	C₄⋊Dic₁₀₁	—	2	2	no	—
24	C₂²⋊Dic₁₀₁	—	2	2	no	—
25	C₂²⋊Dic₁₀₁	—	2	1	no	—
26	C₂²⋊Dic₁₀₁	—	2	2	no	—
27	C₄×Dic₁₀₁	—	2	2	no	—
28	C₄×Dic₁₀₁	—	2	1	no	—
29	C₄×Dic₁₀₁	—	2	2	no	—
30	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
33	#41×C₁₀₁	—	2	2	no	—
34	#41×C₁₀₁	—	2	2	no	—
35	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
37	C₂×D₄×C₁₀₁	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₀₁	—	2	2	no	—
41	M₁₆×C₁₀₁	—	2	2	no	—
42	C₂×C₈×C₁₀₁	—	2	2	no	—
43	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₀₁	—	2	2	no	—
46	D₄×C₁₀₁	—	2	4	no	—
47	D₄×C₁₀₁	—	2	2	no	—
48	C₂×C₄×C₁₀₁	—	2	2	no	—
49	C₂×C₄×C₁₀₁	—	2	2	no	—
50	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₁	#41	1	2	yes	—
54	C₂×C₄×C₁₀₁	#41	1	2	yes	—
55	C₂×Dic₁₀₁	—	2	1	no	—
56	C₂×Dic₁₀₁	—	2	2	no	—
57	C₂×Dic₁₀₁	—	2	1	no	—
58	C₂×Dic₁₀₁	—	4	2	no	—
59	C₂×Dic₁₀₁	—	4	4	no	—
60	C₂×Dic₁₀₁	—	4	2	no	—
61	C₈×C₁₀₁	—	4	2	no	—
62	C₂²×C₁₀₁	—	2	2	no	—
63	C₂²×C₁₀₁	—	4	1	no	—
64	C₂²×C₁₀₁	—	4	2	no	—
65	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₀₁	#123	1	2	yes	—
67	C₄×C₁₀₁	—	2	2	no	—
68	C₄×C₁₀₁	#117	1	1	yes	characteristic
69	C₄×C₁₀₁	#117	1	1	yes	characteristic
70	C₄×C₁₀₁	#123	1	2	yes	—
71	Dic₁₀₁	—	2	2	no	—
72	Dic₁₀₁	—	4	2	no	—
73	Dic₁₀₁	—	4	1	no	—
74	Dic₁₀₁	—	8	2	no	—
75	C₂×C₁₀₁	—	2	1	no	—
76	C₂×C₁₀₁	—	2	2	no	—
77	C₂×C₁₀₁	—	4	2	no	—
78	C₂×C₁₀₁	#381	1	1	yes	characteristic
79	#1827	—	101	1	no	maximal, Sylow
80	C₁₀₁	#1827	1	1	yes	characteristic, complemented, Sylow
81	C₄×#41	—	101	1	no	—
82	#413	D₁₀₁	1	1	yes	characteristic
83	#410	—	101	1	no	—
84	#339	—	101	2	no	—
85	H(3,ℤ/4)	—	101	2	no	—
86	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
87	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
88	C₄×Q₈	—	202	1	no	—
89	C₄×D₄	—	202	2	no	—
90	C₄×D₄	—	202	1	no	—
91	#117	—	101	2	no	—
92	#117	—	101	1	no	—
93	#117	—	101	2	no	—
94	C₂×C₄²	—	101	2	no	—
95	C₂×C₄²	—	101	1	no	—
96	#101	—	2	1	no	—
97	#100	—	2	1	no	—
98	#99	—	202	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
102	C₂×D₄	D₄₀₄	1	2	yes	—
103	C₂×D₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
104	C₂²×C₄	—	202	2	no	—
105	C₂²×C₄	—	202	1	no	—
106	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₄⋊C₄	—	202	2	no	—
112	C₄⋊C₄	—	202	2	no	—
113	C₄⋊C₄	—	202	1	no	—
114	C₂²⋊C₄	—	202	2	no	—
115	C₂²⋊C₄	—	202	2	no	—
116	C₂²⋊C₄	—	202	1	no	—
117	C₄²	—	101	1	no	—
118	C₄²	—	101	2	no	—
119	C₄²	—	101	1	no	—
120	C₄²	—	202	2	no	—
121	C₄²	—	202	2	no	—
122	C₄²	—	202	1	no	—
123	C₂³	#13108051	1	2	yes	—
124	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	404	2	no	—
131	C₂×C₄	—	404	2	no	—
132	C₂×C₄	—	404	4	no	—
133	C₂×C₄	—	202	2	no	—
134	C₂×C₄	—	202	1	no	—
135	C₂×C₄	—	202	1	no	—
136	C₂×C₄	#13108051	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
139	C₂×C₄	#13108073	1	2	yes	—
140	C₂×C₄	#13108073	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×#13108051	1	1	yes	characteristic
146	C₂²	#15692239	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	808	2	no	—
149	C₄	—	404	1	no	—
150	C₄	—	404	2	no	—
151	C₄	—	202	2	no	—
152	C₄	#15692211	1	1	yes	characteristic
153	C₄	#15692211	1	1	yes	characteristic
154	C₄	#15692236	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	—	2	1	no	—
157	C₂	—	4	2	no	—
158	C₂	#20265038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₁₀₁

C₂

yes

maximal, characteristic

#20265151

C₂

yes

maximal, characteristic

#20264938

C₂

yes

maximal, complemented

#20264934

C₂

yes

maximal, complemented

#413×C₁₀₁

C₂

yes

maximal, characteristic

Q₈×Dic₁₀₁

—

D₄×Dic₁₀₁

—

D₄×Dic₁₀₁

—

#15692214

C₄

yes

complemented

#15692214

C₂²

yes

—

#15692214

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₁

C₄

yes

complemented

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

#15692170

—

C₂×#41×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

—

#101×C₁₀₁

—

#100×C₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

C₂×C₄

yes

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

characteristic

C₂×D₄×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×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

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#381

yes

characteristic

#1827

—

101

maximal, Sylow

C₁₀₁

#1827

yes

characteristic, complemented, Sylow

C₄×#41

—

101

—

#413

D₁₀₁

yes

characteristic

#410

—

101

—

#339

—

101

—

H(3,ℤ/4)

—

101

—

C₂×#41

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₄×Q₈

—

202

—

C₄×D₄

—

202

—

C₄×D₄

—

202

—

#117

—

101

—

#117

—

101

—

#117

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

#101

—

#100

—

#99

—

202

—

#41

—

100

#41

—

101

C₂×Q₈

C₂²⋊D₁₀₁

yes

characteristic

102

C₂×D₄

D₄₀₄

yes

—

103

C₂×D₄

C₂²⋊D₁₀₁

yes

characteristic

104

C₂²×C₄

—

202

—

105

C₂²×C₄

—

202

—

106

C₂²×C₄

C₄×D₁₀₁

yes

—

107

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

202

—

112

C₄⋊C₄

—

202

—

113

C₄⋊C₄

—

202

—

114

C₂²⋊C₄

—

202

—

115

C₂²⋊C₄

—

202

—

116

C₂²⋊C₄

—

202

—

117

C₄²

—

101

—

118

C₄²

—

101

—

119

C₄²

—

101

—

120

C₄²

—

202

—

121

C₄²

—

202

—

122

C₄²

—

202

—

123

C₂³

#13108051

yes

—

124

C₂³

C₂×D₄₀₄

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

404

—

131

C₂×C₄

—

404

—

132

C₂×C₄

—

404

—

133

C₂×C₄

—

202

—

134

C₂×C₄

—

202

—

135

C₂×C₄

—

202

—

136

C₂×C₄

#13108051

yes

—

137

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

139

C₂×C₄

#13108073

yes

—

140

C₂×C₄

#13108073

yes

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#13108051

yes

characteristic

146

C₂²

#15692239

yes

—

147

C₄

—

148

C₄

—

808

—

149

C₄

—

404

—

150

C₄

—

404

—

151

C₄

—

202

—

152

C₄

#15692211

yes

characteristic

153

C₄

#15692211

yes

characteristic

154

C₄

#15692236

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#20265038

yes

characteristic

	subgroup	G/H
center	C₄	#15692211
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#413×C₁₀₁	C₂
socle	C₂×C₁₀₁	#381
prime radical	C₂×#41×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#413	D₁₀₁
hypocenter	C₁₀₁	#1827
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#15692211

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#413×C₁₀₁

C₂

socle

C₂×C₁₀₁

#381

prime radical

C₂×#41×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #20264938	C₂	semidirect
4. #20264934	C₂	semidirect
9. #15692214	C₄	semidirect
12. C₂×C₄×Dic₁₀₁	C₄	semidirect
80. C₁₀₁	#1827	semidirect

H ≅ G/N

type

3. #20264938

C₂

semidirect

4. #20264934

C₂

semidirect

9. #15692214

C₄

semidirect

12. C₂×C₄×Dic₁₀₁

C₄

semidirect

80. C₁₀₁

#1827

semidirect

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

subgroup	index
#41×Dic₁₀₁	2
#20265151	2
#20264938	2
#20264934	2
#413×C₁₀₁	2
#1827	101

subgroup

index

#41×Dic₁₀₁

#413×C₁₀₁

101

Subgroups — #27171237

Conjugacy classes of subgroups

Complements

Maximal subgroups