Group #21424859 — subgroups

Subgroups — #21424859

Order 7552.

256 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14273658	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16837272	C₂	1	1	yes	maximal, characteristic
4	#16837164	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	—	2	1	no	—
8	C₂×C₄⋊Dic₅₉	—	2	1	no	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	#14273658	—	2	4	no	—
11	C₄⋊(C₅₉⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	—	2	1	no	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	#95×C₅₉	—	2	1	no	—
17	C₂²×C₄×C₅₉	—	2	1	no	—
18	C₂²×C₄×C₅₉	—	2	1	no	—
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₉	—	2	2	no	—
23	C₄⋊C₄×C₅₉	—	2	1	no	—
24	C₄⋊C₄×C₅₉	—	2	1	no	—
25	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₉	Q₈	1	2	yes	—
27	C₄²×C₅₉	—	4	2	no	—
28	C₄²×C₅₉	C₂×C₄	1	2	yes	—
29	C₄²×C₅₉	Q₈	1	2	yes	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	—	2	2	no	—
32	C₄⋊Dic₅₉	—	4	2	no	—
33	C₄⋊Dic₅₉	—	4	2	no	—
34	C₂×C₅₉⋊C₈	—	2	2	no	—
35	C₂×C₅₉⋊C₈	—	2	2	no	—
36	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
37	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	1	no	—
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
49	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
50	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×Q₈	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×Dic₅₉	—	4	2	no	—
57	C₂×Dic₅₉	—	4	2	no	—
58	C₂×Dic₅₉	—	4	2	no	—
59	C₅₉⋊C₈	—	2	4	no	—
60	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₅₉	#121	1	1	yes	characteristic
62	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₅₉	#137	1	1	yes	characteristic
67	C₄×C₅₉	—	2	2	no	—
68	C₄×C₅₉	—	4	2	no	—
69	C₄×C₅₉	—	4	2	no	—
70	C₄×C₅₉	#106	1	4	yes	complemented
71	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
72	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
73	C₄×C₅₉	#125	1	1	yes	characteristic
74	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
75	Dic₅₉	—	8	2	no	—
76	#1948	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	#389	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#106	1	1	yes	characteristic
79	C₂×C₅₉	#428	1	1	yes	characteristic
80	C₂×C₅₉	#460	1	1	yes	characteristic
81	C₂×C₅₉	#461	1	1	yes	characteristic
82	C₂×C₅₉	#475	1	1	yes	characteristic
83	C₂×C₅₉	#476	1	1	yes	characteristic
84	C₂×#106	—	59	2	no	—
85	C₂×C₄⋊C₈	—	59	1	no	—
86	#384	—	59	1	no	—
87	C₄×C₄⋊C₄	D₅₉	1	1	yes	characteristic
88	#340	—	59	2	no	—
89	C₅₉	#1948	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	59	2	no	—
91	C₂×C₄⋊C₄	—	59	2	no	—
92	C₂×C₄⋊C₄	—	118	1	no	—
93	C₂×C₄⋊C₄	—	118	1	no	—
94	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
98	#106	—	118	4	no	—
99	C₄⋊C₈	—	59	4	no	—
100	#95	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	118	2	no	—
104	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
107	C₂×C₈	—	59	4	no	—
108	C₂×C₈	—	118	2	no	—
109	C₂×C₈	—	118	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	118	2	no	—
113	C₄⋊C₄	—	118	2	no	—
114	C₄⋊C₄	—	236	2	no	—
115	C₄⋊C₄	—	236	2	no	—
116	C₄⋊C₄	Dic₁₁₈	1	2	yes	—
117	C₄⋊C₄	D₂₃₆	1	2	yes	—
118	C₄²	—	4	2	no	—
119	C₄²	Dic₁₁₈	1	2	yes	—
120	C₄²	C₄×D₅₉	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	236	2	no	—
131	C₂×C₄	—	236	2	no	—
132	C₂×C₄	—	236	2	no	—
133	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
134	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
135	C₂×C₄	#11745670	1	2	yes	—
136	C₂×C₄	#11745671	1	2	yes	—
137	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
138	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
141	C₂×C₄	#11745685	1	1	yes	characteristic
142	C₂×C₄	#11745685	1	1	yes	characteristic
143	C₈	—	118	4	no	—
144	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
145	C₂²	#14273701	1	1	yes	characteristic
146	C₂²	#14273706	1	1	yes	characteristic
147	C₂²	C₂×#11745671	1	1	yes	characteristic
148	C₂²	C₂×#11745670	1	1	yes	characteristic
149	C₂²	#14273752	1	1	yes	characteristic
150	C₂²	#14273757	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	472	2	no	—
155	C₄	#14273658	1	4	yes	complemented
156	C₄	C₄×D₂₃₆	1	1	yes	characteristic
157	C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
158	C₄	#14273719	1	1	yes	characteristic
159	C₄	#14273727	1	1	yes	characteristic
160	C₂	#16837300	1	1	yes	characteristic
161	C₂	C₂×#14273658	1	1	yes	characteristic
162	C₂	#16837445	1	1	yes	characteristic
163	C₂	#16837602	1	1	yes	characteristic
164	C₂	#16837607	1	1	yes	characteristic
165	C₂	#16837687	1	1	yes	characteristic
166	C₂	#16837692	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273658

C₂

yes

maximal

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837272

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

C₄×C₄⋊C₄×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

Q₈

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂×D₄

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

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₂²×C₅₉

C₂×SD₁₆

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#106

yes

complemented

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

#125

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₄⋊C₈

—

#384

—

C₄×C₄⋊C₄

D₅₉

yes

characteristic

#340

—

C₅₉

#1948

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#106

—

118

—

C₄⋊C₈

—

100

#95

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

118

—

104

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

106

C₂²×C₄

C₂²×D₅₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

118

—

109

C₂×C₈

—

118

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

118

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

—

236

—

115

C₄⋊C₄

—

236

—

116

C₄⋊C₄

Dic₁₁₈

yes

—

117

C₄⋊C₄

D₂₃₆

yes

—

118

C₄²

—

119

C₄²

Dic₁₁₈

yes

—

120

C₄²

C₄×D₅₉

yes

—

121

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

236

—

131

C₂×C₄

—

236

—

132

C₂×C₄

—

236

—

133

C₂×C₄

Dic₅₉⋊C₄

yes

—

134

C₂×C₄

Dic₅₉⋊C₄

yes

—

135

C₂×C₄

#11745670

yes

—

136

C₂×C₄

#11745671

yes

—

137

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

138

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

139

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

140

C₂×C₄

C₂×D₂₃₆

yes

characteristic

141

C₂×C₄

#11745685

yes

characteristic

142

C₂×C₄

#11745685

yes

characteristic

143

C₈

—

118

—

144

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

145

C₂²

#14273701

yes

characteristic

146

C₂²

#14273706

yes

characteristic

147

C₂²

C₂×#11745671

yes

characteristic

148

C₂²

C₂×#11745670

yes

characteristic

149

C₂²

#14273752

yes

characteristic

150

C₂²

#14273757

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

472

—

155

C₄

#14273658

yes

complemented

156

C₄

C₄×D₂₃₆

yes

characteristic

157

C₄

C₄×Dic₁₁₈

yes

characteristic

158

C₄

#14273719

yes

characteristic

159

C₄

#14273727

yes

characteristic

160

C₂

#16837300

yes

characteristic

161

C₂

C₂×#14273658

yes

characteristic

162

C₂

#16837445

yes

characteristic

163

C₂

#16837602

yes

characteristic

164

C₂

#16837607

yes

characteristic

165

C₂

#16837687

yes

characteristic

166

C₂

#16837692

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₄×C₄⋊C₄×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₅₉
hypocenter	C₅₉	#1948
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₄×C₄⋊C₄×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₅₉

hypocenter

C₅₉

#1948

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
11. C₄⋊(C₅₉⋊C₈)	C₄	semidirect
70. C₄×C₅₉	#106	semidirect
89. C₅₉	#1948	semidirect
155. C₄	#14273658	semidirect

H ≅ G/N

type

11. C₄⋊(C₅₉⋊C₈)

C₄

semidirect

70. C₄×C₅₉

#106

semidirect

89. C₅₉

#1948

semidirect

155. C₄

#14273658

semidirect

Subgroups — #21424859

Conjugacy classes of subgroups

Complements