Group #21423352 — subgroups

Subgroups — #21423352

Order 7552.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16837805	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837687	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837607	C₂	1	1	yes	maximal, characteristic
4	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#556	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745670	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273660	—	2	2	no	—
11	#14273658	—	2	1	no	—
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	#14273651	—	2	1	no	—
15	#14273650	—	2	1	no	—
16	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	#126×C₅₉	—	2	2	no	—
19	#124×C₅₉	—	2	1	no	—
20	C₂²⋊Q₈×C₅₉	—	2	1	no	—
21	C₄×Q₈×C₅₉	—	2	1	no	—
22	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
24	#117×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	—	2	1	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
36	C₂²⋊C₄×C₅₉	—	2	2	no	—
37	C₂²⋊C₄×C₅₉	—	2	2	no	—
38	C₂²⋊C₄×C₅₉	—	2	1	no	—
39	C₄²×C₅₉	—	2	2	no	—
40	C₄²×C₅₉	—	2	1	no	—
41	C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	#11745670	—	4	2	no	—
43	C₄⋊Dic₅₉	—	2	1	no	—
44	C₄⋊Dic₅₉	—	2	1	no	—
45	C₂×C₅₉⋊C₈	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	C₂³×C₅₉	—	2	1	no	—
49	Q₈×C₅₉	—	2	1	no	—
50	Q₈×C₅₉	—	2	1	no	—
51	Q₈×C₅₉	—	4	1	no	—
52	Q₈×C₅₉	—	4	1	no	—
53	D₄×C₅₉	—	2	1	no	—
54	D₄×C₅₉	—	2	1	no	—
55	C₂×C₄×C₅₉	—	2	1	no	—
56	C₂×C₄×C₅₉	—	2	1	no	—
57	C₂×C₄×C₅₉	—	2	1	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	4	1	no	—
60	C₂×C₄×C₅₉	—	4	1	no	—
61	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₅₉	—	4	1	no	—
65	Dic₁₁₈	—	4	1	no	—
66	Dic₁₁₈	—	4	1	no	—
67	C₅₉⋊C₈	—	2	2	no	—
68	C₅₉⋊C₈	—	4	1	no	—
69	C₅₉⋊C₈	—	4	1	no	—
70	C₂²×C₅₉	—	4	1	no	—
71	C₂²×C₅₉	—	4	1	no	—
72	C₂²×C₅₉	—	4	1	no	—
73	C₂²×C₅₉	#120	1	1	yes	characteristic
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	—	4	2	no	—
76	C₄×C₅₉	—	4	1	no	—
77	C₄×C₅₉	—	4	1	no	—
78	C₄×C₅₉	—	4	1	no	—
79	C₄×C₅₉	#135	1	1	yes	characteristic
80	C₄×C₅₉	#135	1	1	yes	characteristic
81	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₄×C₅₉	#137	1	1	yes	characteristic
83	Dic₅₉	—	8	1	no	—
84	#1650	—	59	1	no	maximal, Sylow
85	C₂×C₅₉	—	8	1	no	—
86	C₂×C₅₉	#449	1	1	yes	characteristic
87	C₂×C₅₉	#452	1	1	yes	characteristic
88	C₂×C₅₉	#456	1	1	yes	characteristic
89	#556	D₅₉	1	1	yes	characteristic
90	#500	—	59	1	no	—
91	#475	—	59	1	no	—
92	#461	—	59	1	no	—
93	#332	—	59	1	no	—
94	Q₈⋊C₈	—	59	1	no	—
95	D₄⋊C₈	—	59	1	no	—
96	C₅₉	#1650	1	1	yes	characteristic, complemented, Sylow
97	C₂×SD₁₆	—	118	1	no	—
98	C₄⋊Q₈	—	59	1	no	—
99	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
100	#126	—	2	2	no	—
101	#124	—	2	1	no	—
102	C₂²⋊Q₈	—	2	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
105	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
106	#117	—	2	1	no	—
107	#107	—	118	2	no	—
108	#106	—	118	1	no	—
109	C₄⋊C₈	—	59	1	no	—
110	C₄⋊C₈	—	59	1	no	—
111	#103	—	118	1	no	—
112	#103	—	118	1	no	—
113	C₄×C₈	—	59	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	118	1	no	—
116	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
117	C₂×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	SD₁₆	—	236	2	no	—
120	C₂×C₈	—	118	1	no	—
121	C₂×C₈	—	118	1	no	—
122	C₂×C₈	—	118	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	118	1	no	—
127	C₄⋊C₄	—	118	1	no	—
128	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
129	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
130	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	C₂²×D₅₉	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	236	1	no	—
143	Q₈	—	236	1	no	—
144	D₄	—	2	1	no	—
145	D₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	236	1	no	—
153	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
156	C₈	—	118	2	no	—
157	C₈	—	236	1	no	—
158	C₈	—	236	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	#14273696	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	472	1	no	—
169	C₄	#14273747	1	1	yes	characteristic
170	C₄	#14273747	1	1	yes	characteristic
171	C₄	#14273752	1	1	yes	characteristic
172	C₄	#14273757	1	1	yes	characteristic
173	C₂	—	8	1	no	—
174	C₂	#16837547	1	1	yes	characteristic
175	C₂	#16837564	1	1	yes	characteristic
176	C₂	#16837585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837805

C₂

yes

maximal, characteristic, complemented

#16837687

C₂

yes

maximal, characteristic, complemented

#16837607

C₂

yes

maximal, characteristic

#16837148

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

C₅₉×#556

C₂

yes

maximal, characteristic

C₂×#11745670

—

#14273727

C₂²

yes

characteristic

#14273660

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273651

—

#14273650

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#126×C₅₉

—

#124×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₄×C₅₉

C₂²

yes

characteristic

#117×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

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×D₄

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1650

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₅₉

#1650

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

100

#126

—

101

#124

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

C₄×Q₈

C₂×D₅₉

yes

characteristic

105

C₄×D₄

C₂×D₅₉

yes

characteristic

106

#117

—

107

#107

—

118

—

108

#106

—

118

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

118

—

112

#103

—

118

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

118

—

116

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

117

C₂×D₄

C₂²⋊D₅₉

yes

characteristic

118

C₂²×C₄

—

119

SD₁₆

—

236

—

120

C₂×C₈

—

118

—

121

C₂×C₈

—

118

—

122

C₂×C₈

—

118

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

118

—

127

C₄⋊C₄

—

118

—

128

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

129

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

130

C₄⋊C₄

C₂²⋊D₅₉

yes

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₅₉

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

236

—

143

Q₈

—

236

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

236

—

153

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

154

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

155

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

156

C₈

—

118

—

157

C₈

—

236

—

158

C₈

—

236

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#14273696

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

472

—

169

C₄

#14273747

yes

characteristic

170

C₄

#14273747

yes

characteristic

171

C₄

#14273752

yes

characteristic

172

C₄

#14273757

yes

characteristic

173

C₂

—

174

C₂

#16837547

yes

characteristic

175

C₂

#16837564

yes

characteristic

176

C₂

#16837585

yes

characteristic

	subgroup	G/H
center	C₂²	#14273696
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#556	C₂
socle	C₂²×C₅₉	#120
hypercenter	#556	D₅₉
hypocenter	C₅₉	#1650
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #556

subgroup

G/H

center

C₂²

#14273696

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#556

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #556

N	H ≅ G/N	type
1. #16837805	C₂	semidirect
2. #16837687	C₂	semidirect
4. #16837148	C₂	semidirect
5. Q₈⋊(C₅₉⋊C₈)	C₂	semidirect
96. C₅₉	#1650	semidirect

H ≅ G/N

type

1. #16837805

C₂

semidirect

2. #16837687

C₂

semidirect

4. #16837148

C₂

semidirect

5. Q₈⋊(C₅₉⋊C₈)

C₂

semidirect

96. C₅₉

#1650

semidirect

Subgroups — #21423352

Conjugacy classes of subgroups

Complements