Group #21425643 — subgroups

Subgroups — #21425643

Order 7552.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14273661	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273659	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
5	#16837308	C₂	1	1	yes	maximal, characteristic
6	#16837268	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	#14273661	—	2	2	no	—
13	#14273659	—	2	2	no	—
14	#14273652	—	2	2	no	—
15	#14273652	—	2	2	no	—
16	C₅₉⋊#95	—	2	2	no	—
17	C₅₉⋊#95	—	2	2	no	—
18	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	D₄	1	2	yes	—
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₉	—	2	1	no	—
24	C₂²×Dic₅₉	—	2	1	no	—
25	C₂²×Dic₅₉	—	2	1	no	—
26	C₂²×Dic₅₉	—	2	1	no	—
27	C₂×Dic₁₁₈	—	2	1	no	—
28	C₂×Dic₁₁₈	—	2	1	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₂×Dic₁₁₈	—	2	1	no	—
31	C₂×Dic₁₁₈	—	2	1	no	—
32	C₂×Dic₁₁₈	Q₈	1	2	yes	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	D₄	1	2	yes	—
38	C₂×C₈×C₅₉	D₄	1	2	yes	—
39	C₄²×C₅₉	—	2	1	no	—
40	C₄²×C₅₉	—	2	1	no	—
41	C₄⋊Dic₅₉	—	2	1	no	—
42	C₄⋊Dic₅₉	—	2	1	no	—
43	C₄⋊Dic₅₉	—	2	1	no	—
44	C₄⋊Dic₅₉	—	2	1	no	—
45	C₄⋊Dic₅₉	—	2	1	no	—
46	C₄⋊Dic₅₉	—	2	1	no	—
47	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
57	C₂×Dic₅₉	—	2	2	no	—
58	C₂×Dic₅₉	—	4	1	no	—
59	C₂×Dic₅₉	—	4	1	no	—
60	C₂×Dic₅₉	—	4	1	no	—
61	C₂×Dic₅₉	—	4	1	no	—
62	C₂×Dic₅₉	—	4	1	no	—
63	C₂×Dic₅₉	—	4	2	no	—
64	C₂×Dic₅₉	—	4	2	no	—
65	C₂×Dic₅₉	—	4	2	no	—
66	Dic₁₁₈	—	2	2	no	—
67	Dic₁₁₈	—	2	2	no	—
68	Dic₁₁₈	—	4	1	no	—
69	Dic₁₁₈	—	4	1	no	—
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂²×C₅₉	#123	1	1	yes	characteristic
74	C₂²×C₅₉	#127	1	1	yes	characteristic
75	C₂²×C₅₉	#135	1	2	yes	—
76	C₂²×C₅₉	#137	1	2	yes	—
77	C₄×C₅₉	—	4	2	no	—
78	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
79	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄×C₅₉	#123	1	1	yes	characteristic
81	C₄×C₅₉	#123	1	1	yes	characteristic
82	Dic₅₉	—	4	2	no	—
83	Dic₅₉	—	8	1	no	—
84	Dic₅₉	—	8	1	no	—
85	Dic₅₉	—	8	1	no	—
86	#2097	—	59	1	no	maximal, Sylow
87	C₂×C₅₉	#399	1	1	yes	characteristic
88	C₂×C₅₉	#479	1	2	yes	—
89	C₂×C₅₉	#485	1	2	yes	—
90	C₂×C₅₉	#495	1	1	yes	characteristic
91	C₂×C₅₉	#497	1	1	yes	characteristic
92	C₂×#107	—	59	1	no	—
93	C₂×#106	—	59	1	no	—
94	C₂×#103	—	59	1	no	—
95	C₂×#103	—	59	1	no	—
96	#391	—	59	1	no	—
97	#382	—	59	1	no	—
98	#336	D₅₉	1	1	yes	characteristic
99	C₅₉	#2097	1	1	yes	characteristic, complemented, Sylow
100	C₂²×Q₈	—	59	1	no	—
101	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	59	1	no	—
104	C₂×C₄⋊C₄	—	59	1	no	—
105	C₂×C₄⋊C₄	—	59	1	no	—
106	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
107	#107	—	118	2	no	—
108	#106	—	118	2	no	—
109	#103	—	118	2	no	—
110	#103	—	118	2	no	—
111	#95	—	118	2	no	—
112	#95	—	118	2	no	—
113	C₂×Q₈	—	59	2	no	—
114	C₂×Q₈	—	118	1	no	—
115	C₂×Q₈	—	118	1	no	—
116	C₂×Q₈	—	118	1	no	—
117	C₂×Q₈	—	118	1	no	—
118	C₂×Q₈	—	118	1	no	—
119	C₂²×C₄	—	118	1	no	—
120	C₂²×C₄	—	118	1	no	—
121	C₂²×C₄	—	118	1	no	—
122	C₂²×C₄	—	118	1	no	—
123	C₂²×C₄	D₂₃₆	1	2	yes	—
124	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	D₂₃₆	1	2	yes	—
130	C₂×C₈	D₂₃₆	1	2	yes	—
131	C₄⋊C₄	—	118	1	no	—
132	C₄⋊C₄	—	118	1	no	—
133	C₄⋊C₄	—	118	1	no	—
134	C₄⋊C₄	—	118	1	no	—
135	C₄⋊C₄	—	118	1	no	—
136	C₄⋊C₄	—	118	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
140	Q₈	—	118	2	no	—
141	Q₈	—	118	2	no	—
142	Q₈	—	236	1	no	—
143	Q₈	—	236	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	118	2	no	—
148	C₂×C₄	—	236	1	no	—
149	C₂×C₄	—	236	1	no	—
150	C₂×C₄	—	236	1	no	—
151	C₂×C₄	—	236	2	no	—
152	C₂×C₄	—	236	2	no	—
153	C₂×C₄	—	236	2	no	—
154	C₂×C₄	—	236	1	no	—
155	C₂×C₄	—	236	1	no	—
156	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
157	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
158	C₂×C₄	#11745687	1	1	yes	characteristic
159	C₂×C₄	#11745687	1	1	yes	characteristic
160	C₂×C₄	#11745687	1	1	yes	characteristic
161	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₂²	#14273707	1	1	yes	characteristic
165	C₂²	#14273711	1	1	yes	characteristic
166	C₂²	#14273724	1	1	yes	characteristic
167	C₂²	#14273749	1	2	yes	—
168	C₂²	#14273760	1	2	yes	—
169	C₄	—	4	2	no	—
170	C₄	—	472	1	no	—
171	C₄	—	472	1	no	—
172	C₄	—	472	1	no	—
173	C₄	—	236	2	no	—
174	C₄	#14273707	1	1	yes	characteristic
175	C₄	#14273707	1	1	yes	characteristic
176	C₄	#14273711	1	1	yes	characteristic
177	C₄	#14273711	1	1	yes	characteristic
178	C₂	#16837339	1	1	yes	characteristic
179	C₂	#16837710	1	2	yes	—
180	C₂	#16837744	1	2	yes	—
181	C₂	#16837786	1	1	yes	characteristic
182	C₂	#16837796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal, characteristic

C₂×#14273659

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

#16837308

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic

C₅₉×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

—

—

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#485

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#2097

yes

characteristic, complemented, Sylow

100

C₂²×Q₈

—

101

C₂²×C₈

C₂×D₅₉

yes

characteristic

102

C₂²×C₈

C₂×D₅₉

yes

characteristic

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂×D₅₉

yes

characteristic

107

#107

—

118

—

108

#106

—

118

—

109

#103

—

118

—

110

#103

—

118

—

111

#95

—

118

—

112

#95

—

118

—

113

C₂×Q₈

—

114

C₂×Q₈

—

118

—

115

C₂×Q₈

—

118

—

116

C₂×Q₈

—

118

—

117

C₂×Q₈

—

118

—

118

C₂×Q₈

—

118

—

119

C₂²×C₄

—

118

—

120

C₂²×C₄

—

118

—

121

C₂²×C₄

—

118

—

122

C₂²×C₄

—

118

—

123

C₂²×C₄

D₂₃₆

yes

—

124

C₂²×C₄

C₂²×D₅₉

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

D₂₃₆

yes

—

130

C₂×C₈

D₂₃₆

yes

—

131

C₄⋊C₄

—

118

—

132

C₄⋊C₄

—

118

—

133

C₄⋊C₄

—

118

—

134

C₄⋊C₄

—

118

—

135

C₄⋊C₄

—

118

—

136

C₄⋊C₄

—

118

—

137

C₄²

—

138

C₄²

—

139

C₂³

C₂×D₂₃₆

yes

characteristic

140

Q₈

—

118

—

141

Q₈

—

118

—

142

Q₈

—

236

—

143

Q₈

—

236

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

118

—

148

C₂×C₄

—

236

—

149

C₂×C₄

—

236

—

150

C₂×C₄

—

236

—

151

C₂×C₄

—

236

—

152

C₂×C₄

—

236

—

153

C₂×C₄

—

236

—

154

C₂×C₄

—

236

—

155

C₂×C₄

—

236

—

156

C₂×C₄

C₂×D₂₃₆

yes

characteristic

157

C₂×C₄

C₂×D₂₃₆

yes

characteristic

158

C₂×C₄

#11745687

yes

characteristic

159

C₂×C₄

#11745687

yes

characteristic

160

C₂×C₄

#11745687

yes

characteristic

161

C₂×C₄

Q₈×D₅₉

yes

characteristic

162

C₈

—

163

C₈

—

164

C₂²

#14273707

yes

characteristic

165

C₂²

#14273711

yes

characteristic

166

C₂²

#14273724

yes

characteristic

167

C₂²

#14273749

yes

—

168

C₂²

#14273760

yes

—

169

C₄

—

170

C₄

—

472

—

171

C₄

—

472

—

172

C₄

—

472

—

173

C₄

—

236

—

174

C₄

#14273707

yes

characteristic

175

C₄

#14273707

yes

characteristic

176

C₄

#14273711

yes

characteristic

177

C₄

#14273711

yes

characteristic

178

C₂

#16837339

yes

characteristic

179

C₂

#16837710

yes

—

180

C₂

#16837744

yes

—

181

C₂

#16837786

yes

characteristic

182

C₂

#16837796

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#336	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#336	D₅₉
hypocenter	C₅₉	#2097
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#336

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
99. C₅₉	#2097	semidirect

H ≅ G/N

type

99. C₅₉

#2097

semidirect

Subgroups — #21425643

Conjugacy classes of subgroups

Complements