Group #21424276 — subgroups

Subgroups — #21424276

Order 7552.

348 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂³×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#16837157	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	—	2	2	no	—
7	C₂²×C₅₉⋊C₈	—	2	4	no	—
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₅₉	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₅₉	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	C₂²	1	2	yes	—
17	#95×C₅₉	C₄	1	2	yes	—
18	C₂⁴×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	—	2	2	no	—
20	C₂²×C₄×C₅₉	—	2	2	no	—
21	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₉	D₄	1	2	yes	—
25	C₂²×C₄×C₅₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²⋊C₄×C₅₉	—	2	2	no	—
28	C₂²⋊C₄×C₅₉	C₈	1	4	yes	complemented
29	C₄²×C₅₉	—	2	2	no	—
30	C₄²×C₅₉	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	8	no	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂×C₅₉⋊C₈	—	2	2	no	—
34	C₂×C₅₉⋊C₈	—	2	1	no	—
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂×C₅₉⋊C₈	—	2	2	no	—
37	C₂×C₅₉⋊C₈	—	2	2	no	—
38	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
39	C₂³×C₅₉	—	2	2	no	—
40	C₂³×C₅₉	—	2	2	no	—
41	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂³×C₅₉	C₂×C₈	1	2	yes	—
43	C₂³×C₅₉	M₁₆	1	2	yes	—
44	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	4	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×C₈	1	4	yes	complemented
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#41	1	2	yes	—
60	C₅₉⋊C₈	—	2	4	no	—
61	C₅₉⋊C₈	—	4	1	no	—
62	C₅₉⋊C₈	—	4	1	no	—
63	C₂²×C₅₉	—	2	2	no	—
64	C₂²×C₅₉	—	2	2	no	—
65	C₂²×C₅₉	—	2	4	no	—
66	C₂²×C₅₉	—	2	4	no	—
67	C₂²×C₅₉	C₄⋊C₈	1	4	yes	complemented
68	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₅₉	C₄×D₄	1	2	yes	—
70	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
71	C₂²×C₅₉	C₂×M₁₆	1	1	yes	characteristic
72	C₂²×C₅₉	#131	1	2	yes	—
73	C₄×C₅₉	—	2	2	no	—
74	C₄×C₅₉	—	2	4	no	—
75	C₄×C₅₉	—	4	2	no	—
76	C₄×C₅₉	#120	1	1	yes	characteristic
77	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
78	C₄×C₅₉	#123	1	1	yes	characteristic
79	#1850	—	59	1	no	maximal, Sylow
80	C₂×C₅₉	—	2	4	no	—
81	C₂×C₅₉	#385	1	1	yes	characteristic
82	C₂×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₅₉	#424	1	1	yes	characteristic
84	C₂×C₅₉	C₈×D₄	1	2	yes	—
85	C₂×C₅₉	#435	1	2	yes	—
86	C₂³×C₈	—	59	1	no	—
87	C₂×C₄⋊C₈	—	59	2	no	—
88	C₂×C₂²⋊C₈	—	59	1	no	—
89	C₄×C₂²⋊C₄	D₅₉	1	1	yes	characteristic
90	#336	—	59	2	no	—
91	C₅₉	#1850	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
93	C₂²×C₈	—	59	2	no	—
94	C₂²×C₈	—	59	1	no	—
95	C₂²×C₈	—	59	1	no	—
96	C₂²×C₈	—	118	4	no	—
97	C₂²×C₈	—	118	2	no	—
98	C₂×C₂²⋊C₄	Dic₅₉	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	Dic₅₉	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₅₉	1	2	yes	—
101	C₄⋊C₈	—	118	4	no	—
102	C₂²⋊C₈	—	118	2	no	—
103	#95	Dic₅₉	1	2	yes	—
104	C₂⁴	C₂×Dic₅₉	1	1	yes	characteristic
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	Dic₁₁₈	1	2	yes	—
108	C₂²×C₄	D₂₃₆	1	2	yes	—
109	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
110	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
111	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
113	C₂×C₈	—	59	4	no	—
114	C₂×C₈	—	118	8	no	—
115	C₂×C₈	—	118	2	no	—
116	C₂×C₈	—	118	2	no	—
117	C₂×C₈	—	118	1	no	—
118	C₂×C₈	—	118	1	no	—
119	C₂×C₈	—	118	2	no	—
120	C₂×C₈	—	118	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	C₅₉⋊C₈	1	4	yes	complemented
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₂³	—	2	2	no	—
126	C₂³	—	2	2	no	—
127	C₂³	C₂×C₅₉⋊C₈	1	2	yes	—
128	C₂³	#11745664	1	2	yes	—
129	C₂³	C₄⋊Dic₅₉	1	2	yes	—
130	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	C₂×C₅₉⋊C₈	1	4	yes	complemented
141	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
142	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
143	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
144	C₂×C₄	D₄×D₅₉	1	2	yes	—
145	C₂×C₄	#11745687	1	2	yes	—
146	C₈	—	118	4	no	—
147	C₈	—	236	1	no	—
148	C₈	—	236	1	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	C₄⋊(C₅₉⋊C₈)	1	4	yes	complemented
154	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
155	C₂²	D₄×Dic₅₉	1	2	yes	—
156	C₂²	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
157	C₂²	C₂×#11745664	1	1	yes	characteristic
158	C₂²	#14273734	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	#14273697	1	1	yes	characteristic
163	C₄	#14273704	1	2	yes	—
164	C₄	#14273711	1	1	yes	characteristic
165	C₂	—	2	4	no	—
166	C₂	#16837277	1	1	yes	characteristic
167	C₂	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
168	C₂	#16837429	1	1	yes	characteristic
169	C₂	D₄×C₅₉⋊C₈	1	2	yes	—
170	C₂	#16837474	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837157

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₉

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₉

C₄

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

—

#95×C₅₉

C₄

yes

—

C₂⁴×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

C₈

yes

complemented

C₄²×C₅₉

—

C₄²×C₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂×C₈

yes

—

C₂³×C₅₉

M₁₆

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₈

yes

complemented

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄⋊C₈

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂²×C₅₉

C₂×M₁₆

yes

characteristic

C₂²×C₅₉

#131

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#120

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

#385

yes

characteristic

C₂×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₉

#424

yes

characteristic

C₂×C₅₉

C₈×D₄

yes

—

C₂×C₅₉

#435

yes

—

C₂³×C₈

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₄×C₂²⋊C₄

D₅₉

yes

characteristic

#336

—

C₅₉

#1850

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

118

—

C₂²×C₈

—

118

—

C₂×C₂²⋊C₄

Dic₅₉

yes

characteristic

C₂×C₂²⋊C₄

Dic₅₉

yes

characteristic

100

C₂×C₄²

C₂×D₅₉

yes

—

101

C₄⋊C₈

—

118

—

102

C₂²⋊C₈

—

118

—

103

#95

Dic₅₉

yes

—

104

C₂⁴

C₂×Dic₅₉

yes

characteristic

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

Dic₁₁₈

yes

—

108

C₂²×C₄

D₂₃₆

yes

—

109

C₂²×C₄

C₂×Dic₅₉

yes

—

110

C₂²×C₄

C₂×Dic₅₉

yes

—

111

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

112

C₂²×C₄

C₂²×D₅₉

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

118

—

115

C₂×C₈

—

118

—

116

C₂×C₈

—

118

—

117

C₂×C₈

—

118

—

118

C₂×C₈

—

118

—

119

C₂×C₈

—

118

—

120

C₂×C₈

—

118

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

C₅₉⋊C₈

yes

complemented

123

C₄²

—

124

C₄²

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂×C₅₉⋊C₈

yes

—

128

C₂³

#11745664

yes

—

129

C₂³

C₄⋊Dic₅₉

yes

—

130

C₂³

C₂²×Dic₅₉

yes

characteristic

131

C₂×C₄

—

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

C₂×C₅₉⋊C₈

yes

complemented

141

C₂×C₄

C₄⋊Dic₅₉

yes

—

142

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

143

C₂×C₄

C₂×D₂₃₆

yes

characteristic

144

C₂×C₄

D₄×D₅₉

yes

—

145

C₂×C₄

#11745687

yes

—

146

C₈

—

118

—

147

C₈

—

236

—

148

C₈

—

236

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₄⋊(C₅₉⋊C₈)

yes

complemented

154

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

155

C₂²

D₄×Dic₅₉

yes

—

156

C₂²

C₂²×C₅₉⋊C₈

yes

characteristic

157

C₂²

C₂×#11745664

yes

characteristic

158

C₂²

#14273734

yes

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#14273697

yes

characteristic

163

C₄

#14273704

yes

—

164

C₄

#14273711

yes

characteristic

165

C₂

—

166

C₂

#16837277

yes

characteristic

167

C₂

C₂×C₄⋊(C₅₉⋊C₈)

yes

characteristic

168

C₂

#16837429

yes

characteristic

169

C₂

D₄×C₅₉⋊C₈

yes

—

170

C₂

#16837474

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₈
Fitting	C₄×C₂²⋊C₄×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₄×C₂²⋊C₄	D₅₉
hypocenter	C₅₉	#1850
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₉

commutator

C₂²×C₅₉

C₂²×C₈

Fitting

C₄×C₂²⋊C₄×C₅₉

C₂

socle

C₂³×C₅₉

C₂²×C₄

hypercenter

C₄×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊(C₅₉⋊C₈)	C₂	semidirect
4. #16837157	C₂	semidirect
28. C₂²⋊C₄×C₅₉	C₈	semidirect
55. C₂×C₄×C₅₉	C₂×C₈	semidirect
67. C₂²×C₅₉	C₄⋊C₈	semidirect
91. C₅₉	#1850	semidirect
122. C₂²⋊C₄	C₅₉⋊C₈	semidirect
140. C₂×C₄	C₂×C₅₉⋊C₈	semidirect
153. C₂²	C₄⋊(C₅₉⋊C₈)	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₅₉⋊C₈)

C₂

semidirect

4. #16837157

C₂

semidirect

28. C₂²⋊C₄×C₅₉

C₈

semidirect

55. C₂×C₄×C₅₉

C₂×C₈

semidirect

67. C₂²×C₅₉

C₄⋊C₈

semidirect

91. C₅₉

#1850

semidirect

122. C₂²⋊C₄

C₅₉⋊C₈

semidirect

140. C₂×C₄

C₂×C₅₉⋊C₈

semidirect

153. C₂²

C₄⋊(C₅₉⋊C₈)

semidirect

Subgroups — #21424276

Conjugacy classes of subgroups

Complements