Group #21424301 — subgroups

Subgroups — #21424301

Order 7552.

278 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14273662	C₂	1	2	yes	maximal, complemented
2	#16837366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#16837166	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	1	yes	characteristic
7	C₂×#11745664	C₂²	1	1	yes	characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#14273662	—	2	4	no	—
10	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
11	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
12	C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
13	C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
14	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₅₉	—	2	1	no	—
16	C₂×M₁₆×C₅₉	—	2	4	no	—
17	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₅₉	—	2	1	no	—
19	C₂⁴×C₅₉	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	—	2	2	no	—
21	C₂²×C₄×C₅₉	—	2	2	no	—
22	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₅₉	—	2	4	no	—
29	M₁₆×C₅₉	—	2	2	no	—
30	M₁₆×C₅₉	D₄	1	4	yes	—
31	C₂×C₈×C₅₉	—	2	4	no	—
32	C₂×C₈×C₅₉	—	2	2	no	—
33	#11745664	—	2	2	no	—
34	#11745664	—	2	2	no	—
35	C₂×C₅₉⋊C₈	—	2	2	no	—
36	C₂×C₅₉⋊C₈	—	2	2	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
40	C₂³×C₅₉	—	2	2	no	—
41	C₂³×C₅₉	—	2	2	no	—
42	C₂³×C₅₉	—	4	1	no	—
43	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
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	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	—	2	1	no	—
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#41	1	2	yes	—
59	C₈×C₅₉	—	2	4	no	—
60	C₅₉⋊C₈	—	2	4	no	—
61	C₅₉⋊C₈	—	4	1	no	—
62	C₅₉⋊C₈	—	4	1	no	—
63	C₂²×C₅₉	—	2	1	no	—
64	C₂²×C₅₉	—	2	1	no	—
65	C₂²×C₅₉	—	2	1	no	—
66	C₂²×C₅₉	—	2	1	no	—
67	C₂²×C₅₉	—	2	1	no	—
68	C₂²×C₅₉	—	2	1	no	—
69	C₂²×C₅₉	—	2	1	no	—
70	C₂²×C₅₉	—	4	4	no	—
71	C₂²×C₅₉	#108	1	2	yes	complemented
72	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	—	2	1	no	—
76	C₄×C₅₉	C₄×D₄	1	2	yes	—
77	C₄×C₅₉	#120	1	1	yes	characteristic
78	C₄×C₅₉	#123	1	1	yes	characteristic
79	#1855	—	59	1	no	maximal, Sylow
80	C₂×C₅₉	—	2	2	no	—
81	C₂×C₅₉	—	2	2	no	—
82	C₂×C₅₉	—	4	1	no	—
83	C₂×C₅₉	#385	1	1	yes	characteristic
84	C₂×C₅₉	C₂×#108	1	1	yes	characteristic
85	C₂×C₅₉	#430	1	1	yes	characteristic
86	C₂²×M₁₆	D₅₉	1	1	yes	characteristic
87	C₂×#108	—	59	2	no	—
88	#407	—	59	1	no	—
89	C₂×C₂²⋊C₈	—	59	1	no	—
90	#341	—	59	2	no	—
91	C₅₉	#1855	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	4	no	—
95	C₂×M₁₆	—	59	1	no	—
96	C₂×M₁₆	—	59	1	no	—
97	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	59	2	no	—
100	#108	—	118	4	no	—
101	C₂²⋊C₈	—	59	2	no	—
102	C₂²⋊C₈	—	59	2	no	—
103	C₂²⋊C₈	—	118	1	no	—
104	C₂²⋊C₈	—	118	1	no	—
105	C₂⁴	Dic₁₁₈	1	1	yes	characteristic
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
110	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
114	M₁₆	—	2	4	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	—	118	2	no	—
117	M₁₆	—	118	2	no	—
118	M₁₆	C₂²⋊D₅₉	1	4	yes	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	59	4	no	—
122	C₂×C₈	—	118	2	no	—
123	C₂×C₈	—	118	2	no	—
124	C₂×C₈	—	118	1	no	—
125	C₂×C₈	—	118	1	no	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	4	1	no	—
129	C₂³	Dic₅₉⋊C₄	1	1	yes	characteristic
130	C₂³	Dic₅₉⋊C₄	1	1	yes	characteristic
131	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
140	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
142	C₂×C₄	#11745685	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
145	C₈	—	2	4	no	—
146	C₈	—	118	4	no	—
147	C₈	—	236	1	no	—
148	C₈	—	236	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	4	4	no	—
157	C₂²	#14273662	1	2	yes	complemented
158	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
159	C₂²	#14273706	1	2	yes	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
163	C₄	#14273696	1	1	yes	characteristic
164	C₄	#14273710	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	—	2	2	no	—
167	C₂	—	4	1	no	—
168	C₂	#16837275	1	1	yes	characteristic
169	C₂	C₂×#14273662	1	1	yes	characteristic
170	C₂	#16837450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273662

C₂

yes

maximal, complemented

#16837366

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal, complemented

C₂²×M₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

#14273662

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

C₄

yes

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

—

C₂⁴×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂³×C₅₉

C₂×Q₈

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#108

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂²×M₁₆

D₅₉

yes

characteristic

C₂×#108

—

#407

—

C₂×C₂²⋊C₈

—

#341

—

C₅₉

#1855

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

100

#108

—

118

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

118

—

104

C₂²⋊C₈

—

118

—

105

C₂⁴

Dic₁₁₈

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₁₁₈

yes

characteristic

109

C₂²×C₄

C₄×D₅₉

yes

characteristic

110

C₂²×C₄

C₄×D₅₉

yes

characteristic

111

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

112

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

113

C₂²×C₄

C₂²×D₅₉

yes

characteristic

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

118

—

117

M₁₆

—

118

—

118

M₁₆

C₂²⋊D₅₉

yes

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

118

—

123

C₂×C₈

—

118

—

124

C₂×C₈

—

118

—

125

C₂×C₈

—

118

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

Dic₅₉⋊C₄

yes

characteristic

130

C₂³

Dic₅₉⋊C₄

yes

characteristic

131

C₂³

C₂×Dic₁₁₈

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

140

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

141

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

142

C₂×C₄

#11745685

yes

—

143

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

144

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

145

C₈

—

146

C₈

—

118

—

147

C₈

—

236

—

148

C₈

—

236

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#14273662

yes

complemented

158

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

159

C₂²

#14273706

yes

—

160

C₄

—

161

C₄

—

162

C₄

C₄×C₂²⋊D₅₉

yes

—

163

C₄

#14273696

yes

characteristic

164

C₄

#14273710

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#16837275

yes

characteristic

169

C₂

C₂×#14273662

yes

characteristic

170

C₂

#16837450

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂²×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	C₂²×M₁₆	D₅₉
hypocenter	C₅₉	#1855
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂²×M₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

hypercenter

C₂²×M₁₆

D₅₉

hypocenter

C₅₉

#1855

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#14273662	C₂	semidirect
4. #16837166	C₂	semidirect
71. C₂²×C₅₉	#108	semidirect
91. C₅₉	#1855	semidirect
157. C₂²	#14273662	semidirect

H ≅ G/N

type

1. C₂×#14273662

C₂

semidirect

4. #16837166

C₂

semidirect

71. C₂²×C₅₉

#108

semidirect

91. C₅₉

#1855

semidirect

157. C₂²

#14273662

semidirect

Subgroups — #21424301

Conjugacy classes of subgroups

Complements