Group #27142982 — subgroups

Subgroups — #27142982

Order 12416.

298 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690733	C₂	1	2	yes	maximal
3	C₂×#15690733	C₂	1	2	yes	maximal
4	#20260490	C₂	1	1	yes	maximal, characteristic
5	#389×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	2	yes	—
8	#15690810	—	2	4	no	—
9	C₂×C₄⋊Dic₉₇	—	2	1	no	—
10	C₂×C₄⋊Dic₉₇	—	2	1	no	—
11	#15690733	—	2	4	no	—
12	#15690733	—	2	4	no	—
13	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	#95×C₉₇	—	2	1	no	—
18	C₂×Dic₁₉₄	—	2	2	no	—
19	C₂×Dic₁₉₄	—	2	2	no	—
20	C₂×Dic₁₉₄	—	2	2	no	—
21	C₂×Dic₁₉₄	—	2	2	no	—
22	C₂×Dic₁₉₄	—	2	2	no	—
23	C₂×Dic₁₉₄	D₄	1	4	yes	—
24	C₂²×Dic₉₇	—	2	2	no	—
25	C₂×#11715895	—	2	2	no	—
26	C₂×#11715895	—	2	2	no	—
27	C₂×#11715895	—	2	2	no	—
28	C₄⋊Dic₉₇	—	2	4	no	—
29	C₄⋊Dic₉₇	—	2	4	no	—
30	C₂²×C₄×C₉₇	—	2	1	no	—
31	C₂²×C₄×C₉₇	—	2	1	no	—
32	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₉₇	—	2	1	no	—
36	C₄⋊C₄×C₉₇	—	2	2	no	—
37	C₄⋊C₄×C₉₇	—	4	2	no	—
38	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
39	C₄²×C₉₇	—	2	2	no	—
40	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	—	2	1	no	—
42	C₂×C₄×C₉₇	—	2	1	no	—
43	C₂×C₄×C₉₇	—	2	1	no	—
44	C₂×C₄×C₉₇	—	2	1	no	—
45	C₂×C₄×C₉₇	—	2	2	no	—
46	C₂×C₄×C₉₇	—	4	1	no	—
47	C₂×C₄×C₉₇	—	4	1	no	—
48	C₂×C₄×C₉₇	—	4	1	no	—
49	C₂×C₄×C₉₇	—	4	1	no	—
50	C₂×C₄×C₉₇	—	4	1	no	—
51	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₉₇	#41	1	2	yes	—
57	C₂×Dic₉₇	—	2	4	no	—
58	C₂×Dic₉₇	—	4	2	no	—
59	C₂×Dic₉₇	—	4	2	no	—
60	Dic₁₉₄	—	2	4	no	—
61	Dic₁₉₄	—	2	4	no	—
62	Dic₁₉₄	—	4	2	no	—
63	Dic₁₉₄	—	4	2	no	—
64	#11715895	—	4	2	no	—
65	C₂²×C₉₇	#120	1	1	yes	characteristic
66	C₂²×C₉₇	#121	1	1	yes	characteristic
67	C₂²×C₉₇	#124	1	1	yes	characteristic
68	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₉₇	#137	1	1	yes	characteristic
71	C₂²×C₉₇	#137	1	1	yes	characteristic
72	C₄×C₉₇	—	2	2	no	—
73	C₄×C₉₇	—	2	2	no	—
74	C₄×C₉₇	—	4	2	no	—
75	C₄×C₉₇	—	8	1	no	—
76	C₄×C₉₇	#121	1	2	yes	—
77	C₄×C₉₇	#123	1	2	yes	—
78	Dic₉₇	—	4	4	no	—
79	C₂×C₉₇	#394	1	1	yes	characteristic
80	C₂×C₉₇	#448	1	1	yes	characteristic
81	C₂×C₉₇	#451	1	1	yes	characteristic
82	C₂×C₉₇	#461	1	1	yes	characteristic
83	C₂×C₉₇	#462	1	1	yes	characteristic
84	C₂×C₉₇	#487	1	1	yes	characteristic
85	C₂×C₉₇	#491	1	1	yes	characteristic
86	#2043	—	97	1	no	maximal, Sylow
87	C₉₇	#2043	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	—	97	1	no	—
89	C₂×#103	—	97	2	no	—
90	C₂×#103	—	97	2	no	—
91	#389	D₉₇	1	1	yes	characteristic
92	#336	—	97	1	no	—
93	C₂²×Q₈	—	97	2	no	—
94	C₂²×C₈	—	97	2	no	—
95	C₄⋊Q₈	—	194	4	no	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	194	1	no	—
98	C₂×C₄⋊C₄	—	194	1	no	—
99	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
102	#103	—	194	4	no	—
103	#103	—	194	4	no	—
104	#95	—	2	1	no	—
105	C₂×Q₈	—	97	4	no	—
106	C₂×Q₈	—	194	2	no	—
107	C₂×Q₈	—	194	2	no	—
108	C₂×Q₈	—	194	2	no	—
109	C₂×Q₈	—	194	2	no	—
110	C₂×Q₈	—	194	2	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	194	2	no	—
114	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
117	C₂×C₈	—	194	2	no	—
118	C₂×C₈	—	194	2	no	—
119	C₂×C₈	—	194	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	194	4	no	—
124	C₄⋊C₄	—	194	4	no	—
125	C₄⋊C₄	D₃₈₈	1	2	yes	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
128	Q₈	—	388	2	no	—
129	Q₈	—	388	2	no	—
130	Q₈	—	194	4	no	—
131	Q₈	—	194	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	388	2	no	—
143	C₂×C₄	—	388	2	no	—
144	C₂×C₄	—	194	4	no	—
145	C₂×C₄	#13095499	1	2	yes	—
146	C₂×C₄	#13095501	1	2	yes	—
147	C₂×C₄	D₄×D₉₇	1	2	yes	—
148	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
149	C₂×C₄	#13095511	1	1	yes	characteristic
150	C₂×C₄	#13095512	1	2	yes	—
151	C₈	—	388	2	no	—
152	C₂²	#15690778	1	1	yes	characteristic
153	C₂²	#15690784	1	1	yes	characteristic
154	C₂²	#15690799	1	1	yes	characteristic
155	C₂²	C₂×#13095499	1	1	yes	characteristic
156	C₂²	C₂×#13095501	1	1	yes	characteristic
157	C₂²	#15690840	1	1	yes	characteristic
158	C₂²	#15690840	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	8	1	no	—
163	C₄	—	388	4	no	—
164	C₄	#15690785	1	2	yes	—
165	C₄	#15690795	1	2	yes	—
166	C₂	#20260652	1	1	yes	characteristic
167	C₂	#20260873	1	1	yes	characteristic
168	C₂	#20260890	1	1	yes	characteristic
169	C₂	#20260940	1	1	yes	characteristic
170	C₂	#20260945	1	1	yes	characteristic
171	C₂	#20261087	1	1	yes	characteristic
172	C₂	#20261105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×#15690733

C₂

yes

maximal

C₂×#15690733

C₂

yes

maximal

#20260490

C₂

yes

maximal, characteristic

#389×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

C₂²×#11715895

C₂²

yes

—

#15690810

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

—

#15690733

—

#15690733

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

D₄

yes

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

C₂×C₄×C₉₇

yes

C₂×C₄×C₉₇

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×Q₁₆

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#121

yes

—

C₄×C₉₇

#123

yes

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#103

—

#389

D₉₇

yes

characteristic

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

100

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

101

C₂×C₄²

C₂×D₉₇

yes

characteristic

102

#103

—

194

—

103

#103

—

194

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

194

—

107

C₂×Q₈

—

194

—

108

C₂×Q₈

—

194

—

109

C₂×Q₈

—

194

—

110

C₂×Q₈

—

194

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

194

—

114

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

115

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

116

C₂²×C₄

C₂²×D₉₇

yes

characteristic

117

C₂×C₈

—

194

—

118

C₂×C₈

—

194

—

119

C₂×C₈

—

194

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

194

—

124

C₄⋊C₄

—

194

—

125

C₄⋊C₄

D₃₈₈

yes

—

126

C₄²

—

127

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

128

Q₈

—

388

—

129

Q₈

—

388

—

130

Q₈

—

194

—

131

Q₈

—

194

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

388

—

143

C₂×C₄

—

388

—

144

C₂×C₄

—

194

—

145

C₂×C₄

#13095499

yes

—

146

C₂×C₄

#13095501

yes

—

147

C₂×C₄

D₄×D₉₇

yes

—

148

C₂×C₄

C₂×D₃₈₈

yes

characteristic

149

C₂×C₄

#13095511

yes

characteristic

150

C₂×C₄

#13095512

yes

—

151

C₈

—

388

—

152

C₂²

#15690778

yes

characteristic

153

C₂²

#15690784

yes

characteristic

154

C₂²

#15690799

yes

characteristic

155

C₂²

C₂×#13095499

yes

characteristic

156

C₂²

C₂×#13095501

yes

characteristic

157

C₂²

#15690840

yes

characteristic

158

C₂²

#15690840

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

388

—

164

C₄

#15690785

yes

—

165

C₄

#15690795

yes

—

166

C₂

#20260652

yes

characteristic

167

C₂

#20260873

yes

characteristic

168

C₂

#20260890

yes

characteristic

169

C₂

#20260940

yes

characteristic

170

C₂

#20260945

yes

characteristic

171

C₂

#20261087

yes

characteristic

172

C₂

#20261105

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#389×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	#389	D₉₇
hypocenter	C₉₇	#2043
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#389×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
87. C₉₇	#2043	semidirect

H ≅ G/N

type

87. C₉₇

#2043

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×#15690733	2
#20260490	2
#389×C₉₇	2
#2043	97

subgroup

index

#389×C₉₇

Subgroups — #27142982

Conjugacy classes of subgroups

Complements

Maximal subgroups