Group #27172234 — subgroups

Subgroups — #27172234

Order 12928.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×Dic₄₀₄	C₂	1	2	yes	maximal
2	C₂×#15692255	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692180	C₂	1	2	yes	maximal
4	#20265083	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
8	C₂×Dic₄₀₄	—	2	4	no	—
9	C₂×Dic₄₀₄	—	2	4	no	—
10	C₂×Dic₄₀₄	—	2	4	no	—
11	#15692255	—	2	4	no	—
12	C₂×C₄⋊Dic₁₀₁	—	2	2	no	—
13	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
14	#15692180	—	2	4	no	—
15	#15692162	—	2	2	no	—
16	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₀₂	—	2	2	no	—
19	C₂×Dic₂₀₂	—	2	2	no	—
20	C₂×Dic₂₀₂	—	2	2	no	—
21	C₂×Dic₂₀₂	—	2	2	no	—
22	C₂×Dic₂₀₂	—	2	2	no	—
23	C₂×Dic₂₀₂	—	2	2	no	—
24	C₂×Dic₂₀₂	—	2	2	no	—
25	C₂×Dic₂₀₂	—	2	1	no	—
26	C₂×Dic₂₀₂	D₄	1	4	yes	—
27	C₂×Dic₂₀₂	D₄	1	2	yes	—
28	C₂²×Dic₁₀₁	—	2	1	no	—
29	C₂²×Dic₁₀₁	—	2	2	no	—
30	C₂²×Dic₁₀₁	—	2	1	no	—
31	Dic₄₀₄	—	2	16	no	—
32	C₄⋊Dic₁₀₁	—	2	8	no	—
33	C₄⋊Dic₁₀₁	—	2	2	no	—
34	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
35	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₀₁	—	2	2	no	—
37	C₂×C₈×C₁₀₁	—	2	2	no	—
38	C₂×C₈×C₁₀₁	D₄	1	4	yes	—
39	C₄²×C₁₀₁	—	2	2	no	—
40	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	Q₁₆	1	4	yes	complemented
44	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
48	C₂×Dic₁₀₁	—	2	4	no	—
49	C₂×Dic₁₀₁	—	2	2	no	—
50	C₂×Dic₁₀₁	—	4	1	no	—
51	C₂×Dic₁₀₁	—	4	2	no	—
52	C₂×Dic₁₀₁	—	4	1	no	—
53	C₂×Dic₁₀₁	—	4	2	no	—
54	C₂×Dic₁₀₁	—	4	1	no	—
55	C₂×Dic₁₀₁	—	4	2	no	—
56	Dic₂₀₂	—	2	4	no	—
57	Dic₂₀₂	—	2	4	no	—
58	Dic₂₀₂	—	2	4	no	—
59	Dic₂₀₂	—	4	2	no	—
60	Dic₂₀₂	—	4	2	no	—
61	Dic₂₀₂	—	4	2	no	—
62	C₈×C₁₀₁	—	2	4	no	—
63	C₂²×C₁₀₁	#120	1	1	yes	characteristic
64	C₂²×C₁₀₁	#121	1	1	yes	characteristic
65	C₂²×C₁₀₁	#127	1	1	yes	characteristic
66	C₂²×C₁₀₁	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₁₀₁	#137	1	2	yes	—
68	C₄×C₁₀₁	—	2	4	no	—
69	C₄×C₁₀₁	#120	1	2	yes	—
70	C₄×C₁₀₁	#121	1	2	yes	—
71	Dic₁₀₁	—	4	2	no	—
72	Dic₁₀₁	—	4	4	no	—
73	Dic₁₀₁	—	8	1	no	—
74	C₂×C₁₀₁	#392	1	1	yes	characteristic
75	C₂×C₁₀₁	#451	1	2	yes	—
76	C₂×C₁₀₁	#462	1	2	yes	—
77	C₂×C₁₀₁	#494	1	1	yes	characteristic
78	C₂×C₁₀₁	#497	1	1	yes	characteristic
79	#2016	—	101	1	no	maximal, Sylow
80	C₁₀₁	#2016	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₁₆	—	101	2	no	—
82	C₂×C₄⋊Q₈	—	101	1	no	—
83	C₂×#103	—	101	2	no	—
84	#391	—	101	1	no	—
85	#336	D₁₀₁	1	1	yes	characteristic
86	C₂²×Q₈	—	101	2	no	—
87	C₂²×Q₈	—	101	1	no	—
88	C₂×Q₁₆	—	202	4	no	—
89	C₂×Q₁₆	—	202	4	no	—
90	C₂×Q₁₆	—	202	4	no	—
91	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
92	C₄⋊Q₈	—	202	4	no	—
93	C₂×C₄⋊C₄	—	101	1	no	—
94	C₂×C₄⋊C₄	—	202	2	no	—
95	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
96	#103	—	202	4	no	—
97	#95	—	202	2	no	—
98	C₂×Q₈	—	101	4	no	—
99	C₂×Q₈	—	101	2	no	—
100	C₂×Q₈	—	202	2	no	—
101	C₂×Q₈	—	202	2	no	—
102	C₂×Q₈	—	202	2	no	—
103	C₂×Q₈	—	202	2	no	—
104	C₂×Q₈	—	202	2	no	—
105	C₂×Q₈	—	202	2	no	—
106	C₂×Q₈	—	202	2	no	—
107	C₂×Q₈	—	202	1	no	—
108	C₂²×C₄	—	202	1	no	—
109	C₂²×C₄	—	202	1	no	—
110	C₂²×C₄	—	202	2	no	—
111	C₂²×C₄	D₄₀₄	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
113	Q₁₆	—	202	16	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	D₄₀₄	1	4	yes	—
117	C₄⋊C₄	—	202	8	no	—
118	C₄⋊C₄	—	202	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
121	Q₈	—	404	2	no	—
122	Q₈	—	404	2	no	—
123	Q₈	—	404	2	no	—
124	Q₈	—	202	4	no	—
125	Q₈	—	202	4	no	—
126	Q₈	—	202	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	404	2	no	—
130	C₂×C₄	—	404	1	no	—
131	C₂×C₄	—	404	1	no	—
132	C₂×C₄	—	404	2	no	—
133	C₂×C₄	—	404	1	no	—
134	C₂×C₄	—	404	2	no	—
135	C₂×C₄	—	202	4	no	—
136	C₂×C₄	—	202	2	no	—
137	C₂×C₄	Dic₄₀₄	1	4	yes	complemented
138	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
139	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
140	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
141	C₂×C₄	#13108075	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	#15692225	1	1	yes	characteristic
144	C₂²	#15692230	1	1	yes	characteristic
145	C₂²	#15692252	1	1	yes	characteristic
146	C₂²	C₂×Dic₄₀₄	1	2	yes	—
147	C₂²	#15692288	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	808	1	no	—
150	C₄	—	404	2	no	—
151	C₄	—	404	4	no	—
152	C₄	#15692225	1	2	yes	—
153	C₄	#15692230	1	2	yes	—
154	C₂	#20265087	1	1	yes	characteristic
155	C₂	#20265335	1	2	yes	—
156	C₂	#20265388	1	2	yes	—
157	C₂	#20265557	1	1	yes	characteristic
158	C₂	#20265571	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₀₄

C₂

yes

maximal

C₂×#15692255

C₂

yes

maximal, characteristic

C₂×#15692180

C₂

yes

maximal

#20265083

C₂

yes

maximal, characteristic

#336×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

characteristic

C₂²×Dic₂₀₂

C₂²

yes

—

C₂×Dic₄₀₄

—

C₂×Dic₄₀₄

—

C₂×Dic₄₀₄

—

#15692255

—

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

#15692180

—

#15692162

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

D₄

yes

—

C₂×Dic₂₀₂

D₄

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

Dic₄₀₄

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

Q₁₆

yes

complemented

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

C₂×Q₁₆

yes

—

C₂²×C₁₀₁

#137

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

—

C₄×C₁₀₁

#121

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#392

yes

characteristic

C₂×C₁₀₁

#451

yes

—

C₂×C₁₀₁

#462

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#2016

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

101

—

C₂×C₄⋊Q₈

—

101

—

C₂×#103

—

101

—

#391

—

101

—

#336

D₁₀₁

yes

characteristic

C₂²×Q₈

—

101

—

C₂²×Q₈

—

101

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

—

202

—

C₂²×C₈

C₂×D₁₀₁

yes

—

C₄⋊Q₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#103

—

202

—

#95

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

101

—

100

C₂×Q₈

—

202

—

101

C₂×Q₈

—

202

—

102

C₂×Q₈

—

202

—

103

C₂×Q₈

—

202

—

104

C₂×Q₈

—

202

—

105

C₂×Q₈

—

202

—

106

C₂×Q₈

—

202

—

107

C₂×Q₈

—

202

—

108

C₂²×C₄

—

202

—

109

C₂²×C₄

—

202

—

110

C₂²×C₄

—

202

—

111

C₂²×C₄

D₄₀₄

yes

—

112

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

113

Q₁₆

—

202

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

D₄₀₄

yes

—

117

C₄⋊C₄

—

202

—

118

C₄⋊C₄

—

202

—

119

C₄²

—

120

C₂³

C₂×D₄₀₄

yes

characteristic

121

Q₈

—

404

—

122

Q₈

—

404

—

123

Q₈

—

404

—

124

Q₈

—

202

—

125

Q₈

—

202

—

126

Q₈

—

202

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

404

—

130

C₂×C₄

—

404

—

131

C₂×C₄

—

404

—

132

C₂×C₄

—

404

—

133

C₂×C₄

—

404

—

134

C₂×C₄

—

404

—

135

C₂×C₄

—

202

—

136

C₂×C₄

—

202

—

137

C₂×C₄

Dic₄₀₄

yes

complemented

138

C₂×C₄

D₄×D₁₀₁

yes

characteristic

139

C₂×C₄

D₄×D₁₀₁

yes

—

140

C₂×C₄

C₂×D₄₀₄

yes

—

141

C₂×C₄

#13108075

yes

characteristic

142

C₈

—

143

C₂²

#15692225

yes

characteristic

144

C₂²

#15692230

yes

characteristic

145

C₂²

#15692252

yes

characteristic

146

C₂²

C₂×Dic₄₀₄

yes

—

147

C₂²

#15692288

yes

—

148

C₄

—

149

C₄

—

808

—

150

C₄

—

404

—

151

C₄

—

404

—

152

C₄

#15692225

yes

—

153

C₄

#15692230

yes

—

154

C₂

#20265087

yes

characteristic

155

C₂

#20265335

yes

—

156

C₂

#20265388

yes

—

157

C₂

#20265557

yes

characteristic

158

C₂

#20265571

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₀₄
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#336×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	#336	D₁₀₁
hypocenter	C₁₀₁	#2016
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₄₀₄

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#336×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
43. C₂×C₄×C₁₀₁	Q₁₆	semidirect
80. C₁₀₁	#2016	semidirect
137. C₂×C₄	Dic₄₀₄	semidirect

H ≅ G/N

type

43. C₂×C₄×C₁₀₁

Q₁₆

semidirect

80. C₁₀₁

#2016

semidirect

137. C₂×C₄

Dic₄₀₄

semidirect

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

subgroup	index
C₂²×Dic₄₀₄	2
C₂×#15692255	2
C₂×#15692180	2
#20265083	2
#336×C₁₀₁	2
#2016	101

subgroup

index

C₂²×Dic₄₀₄

#336×C₁₀₁

101

Subgroups — #27172234

Conjugacy classes of subgroups

Complements

Maximal subgroups