Group #27171114 — subgroups

Subgroups — #27171114

Order 12928.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15692214	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692184	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11717579	C₂	1	1	yes	maximal, characteristic
4	#20264931	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11717579	C₂²	1	2	yes	—
7	#15692214	—	2	4	no	—
8	C₂×C₄⋊Dic₁₀₁	C₄	1	2	yes	—
9	C₂×C₂²⋊Dic₁₀₁	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	#15692184	—	2	4	no	—
13	C₂²⋊#11717579	—	2	2	no	—
14	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
16	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
17	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
19	C₂×#11717579	—	2	2	no	—
20	C₂×#11717579	—	2	2	no	—
21	C₂×#11717579	—	2	2	no	—
22	C₄⋊Dic₁₀₁	—	2	4	no	—
23	C₂²⋊Dic₁₀₁	—	2	4	no	—
24	C₄×Dic₁₀₁	—	2	2	no	—
25	C₄×Dic₁₀₁	D₄	1	4	yes	—
26	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₁	—	2	2	no	—
28	C₂²×C₄×C₁₀₁	—	2	2	no	—
29	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
31	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
32	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₀₁	—	2	2	no	—
34	C₂×C₈×C₁₀₁	—	2	2	no	—
35	C₂×C₈×C₁₀₁	—	2	2	no	—
36	C₂³×C₁₀₁	—	2	1	no	—
37	C₂³×C₁₀₁	—	2	1	no	—
38	C₂³×C₁₀₁	—	2	2	no	—
39	C₂³×C₁₀₁	—	2	2	no	—
40	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
41	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₁	—	2	1	no	—
43	C₂×C₄×C₁₀₁	—	2	1	no	—
44	C₂×C₄×C₁₀₁	—	2	4	no	—
45	C₂×C₄×C₁₀₁	—	2	4	no	—
46	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	#41	1	2	yes	—
51	C₂×Dic₁₀₁	—	2	2	no	—
52	C₂×Dic₁₀₁	—	2	2	no	—
53	C₂×Dic₁₀₁	—	2	2	no	—
54	C₂×Dic₁₀₁	—	2	2	no	—
55	C₂×Dic₁₀₁	—	2	2	no	—
56	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
57	C₈×C₁₀₁	—	4	2	no	—
58	#11717579	—	4	2	no	—
59	C₂²×C₁₀₁	—	2	2	no	—
60	C₂²×C₁₀₁	—	2	2	no	—
61	C₂²×C₁₀₁	—	2	2	no	—
62	C₂²×C₁₀₁	—	4	2	no	—
63	C₂²×C₁₀₁	—	4	2	no	—
64	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₀₁	#117	1	1	yes	characteristic
67	C₂²×C₁₀₁	#131	1	2	yes	—
68	C₂²×C₁₀₁	#131	1	2	yes	—
69	C₄×C₁₀₁	—	2	2	no	—
70	C₄×C₁₀₁	#121	1	2	yes	—
71	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
72	Dic₁₀₁	—	2	4	no	—
73	Dic₁₀₁	—	4	2	no	—
74	C₂×C₁₀₁	—	4	2	no	—
75	C₂×C₁₀₁	#380	1	1	yes	characteristic
76	C₂×C₁₀₁	#408	1	2	yes	—
77	C₂×C₁₀₁	#424	1	1	yes	characteristic
78	C₂×C₁₀₁	#424	1	1	yes	characteristic
79	C₂×C₁₀₁	#433	1	2	yes	—
80	#1798	—	101	1	no	maximal, Sylow
81	C₁₀₁	#1798	1	1	yes	characteristic, complemented, Sylow
82	C₂×#117	—	101	1	no	—
83	C₂×C₄⋊C₈	—	101	2	no	—
84	C₂×C₂²⋊C₈	—	101	1	no	—
85	C₂×C₂²⋊C₈	D₁₀₁	1	1	yes	characteristic
86	#336	—	101	2	no	—
87	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
88	C₂²×C₈	—	101	2	no	—
89	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
90	#117	—	202	4	no	—
91	C₂×C₄⋊C₄	—	101	2	no	—
92	C₂×C₂²⋊C₄	—	101	2	no	—
93	C₂×C₄²	—	101	1	no	—
94	C₂×C₄²	—	101	1	no	—
95	C₄⋊C₈	—	202	4	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	202	2	no	—
98	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	101	2	no	—
102	C₂²×C₄	—	101	2	no	—
103	C₂²×C₄	Dic₂₀₂	1	2	yes	—
104	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	202	2	no	—
111	C₂×C₈	—	202	2	no	—
112	C₂×C₈	—	202	2	no	—
113	C₄⋊C₄	—	202	4	no	—
114	C₂²⋊C₄	—	202	4	no	—
115	C₄²	—	101	4	no	—
116	C₄²	—	202	2	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	#13108053	1	2	yes	—
122	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	101	4	no	—
128	C₂×C₄	—	202	2	no	—
129	C₂×C₄	—	202	2	no	—
130	C₂×C₄	—	202	2	no	—
131	C₂×C₄	—	202	2	no	—
132	C₂×C₄	—	202	2	no	—
133	C₂×C₄	#13108053	1	2	yes	—
134	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
137	C₂×C₄	#13108073	1	2	yes	—
138	C₈	—	4	2	no	—
139	C₈	—	404	2	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
146	C₂²	C₂×#13108053	1	1	yes	characteristic
147	C₂²	#15692211	1	1	yes	characteristic
148	C₂²	#15692261	1	2	yes	—
149	C₂²	#15692263	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	404	2	no	—
152	C₄	—	202	4	no	—
153	C₄	#15692227	1	2	yes	—
154	C₄	#15692232	1	2	yes	—
155	C₂	—	4	2	no	—
156	C₂	#20265035	1	1	yes	characteristic
157	C₂	#20265142	1	2	yes	—
158	C₂	#20265203	1	1	yes	characteristic
159	C₂	#20265205	1	1	yes	characteristic
160	C₂	#20265235	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692214

C₂

yes

maximal, characteristic

C₂×#15692184

C₂

yes

maximal, complemented

C₂×C₂²⋊#11717579

C₂

yes

maximal, characteristic

#20264931

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

#15692214

—

C₂×C₄⋊Dic₁₀₁

C₄

yes

—

C₂×C₂²⋊Dic₁₀₁

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

#15692184

—

C₂²⋊#11717579

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂²⋊C₈×C₁₀₁

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₂⁴×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#380

yes

characteristic

C₂×C₁₀₁

#408

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

#1798

—

101

maximal, Sylow

C₁₀₁

#1798

yes

characteristic, complemented, Sylow

C₂×#117

—

101

—

C₂×C₄⋊C₈

—

101

—

C₂×C₂²⋊C₈

—

101

—

C₂×C₂²⋊C₈

D₁₀₁

yes

characteristic

#336

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

C₂²⋊C₈

—

C₂²⋊C₈

—

202

—

C₂⁴

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

101

—

102

C₂²×C₄

—

101

—

103

C₂²×C₄

Dic₂₀₂

yes

—

104

C₂²×C₄

C₄×D₁₀₁

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

106

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

202

—

111

C₂×C₈

—

202

—

112

C₂×C₈

—

202

—

113

C₄⋊C₄

—

202

—

114

C₂²⋊C₄

—

202

—

115

C₄²

—

101

—

116

C₄²

—

202

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

#13108053

yes

—

122

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

101

—

128

C₂×C₄

—

202

—

129

C₂×C₄

—

202

—

130

C₂×C₄

—

202

—

131

C₂×C₄

—

202

—

132

C₂×C₄

—

202

—

133

C₂×C₄

#13108053

yes

—

134

C₂×C₄

D₄×D₁₀₁

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

136

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

137

C₂×C₄

#13108073

yes

—

138

C₈

—

139

C₈

—

404

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂²⋊C₄×D₁₀₁

yes

characteristic

146

C₂²

C₂×#13108053

yes

characteristic

147

C₂²

#15692211

yes

characteristic

148

C₂²

#15692261

yes

—

149

C₂²

#15692263

yes

—

150

C₄

—

151

C₄

—

404

—

152

C₄

—

202

—

153

C₄

#15692227

yes

—

154

C₄

#15692232

yes

—

155

C₂

—

156

C₂

#20265035

yes

characteristic

157

C₂

#20265142

yes

—

158

C₂

#20265203

yes

characteristic

159

C₂

#20265205

yes

characteristic

160

C₂

#20265235

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator, socle	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₁₀₁	C₂
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₀₁
hypocenter	C₁₀₁	#1798
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator, socle

C₂³×C₁₀₁

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₁₀₁

C₂

prime radical

C₂⁴×C₁₀₁

C₂×C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#15692184	C₂	semidirect
4. #20264931	C₂	semidirect
81. C₁₀₁	#1798	semidirect

H ≅ G/N

type

2. C₂×#15692184

C₂

semidirect

4. #20264931

C₂

semidirect

81. C₁₀₁

#1798

semidirect

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

subgroup	index
C₂×#15692214	2
C₂×#15692184	2
C₂×C₂²⋊#11717579	2
#20264931	2
C₂×C₂²⋊C₈×C₁₀₁	2
#1798	101

subgroup

index

C₂×C₂²⋊#11717579

C₂×C₂²⋊C₈×C₁₀₁

101

Subgroups — #27171114

Conjugacy classes of subgroups

Complements

Maximal subgroups