Group #21425559 — subgroups

Subgroups — #21425559

Order 7552.

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

normal subgroups only

This group has 204 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	#16837450	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837408	C₂	1	1	yes	maximal, characteristic
3	#16837406	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273643	C₂	1	1	yes	maximal, characteristic
5	#16837174	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	C₂×#11745664	C₂²	1	1	yes	characteristic
11	#14273686	C₂²	1	1	yes	characteristic
12	#14273662	—	2	2	no	—
13	#14273649	—	2	2	no	—
14	#14273647	—	2	2	no	—
15	#14273643	—	2	2	no	—
16	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	#117×C₅₉	C₂²	1	1	yes	characteristic
19	C₂³×D₅₉	—	2	1	no	—
20	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
21	C₂×D₂₃₆	—	2	1	no	—
22	C₂×D₂₃₆	—	2	1	no	—
23	C₂×D₂₃₆	—	2	1	no	—
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×D₂₃₆	D₄	1	2	yes	—
27	M₁₆×C₅₉	—	2	1	no	—
28	M₁₆×C₅₉	—	2	1	no	—
29	M₁₆×C₅₉	—	2	1	no	—
30	M₁₆×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	—	2	1	no	—
32	C₂×C₈×C₅₉	—	2	1	no	—
33	C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₂²⋊C₄×C₅₉	—	2	1	no	—
35	C₄²×C₅₉	—	2	1	no	—
36	C₂²⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₄×Dic₅₉	—	2	1	no	—
39	#11745664	—	2	1	no	—
40	#11745664	—	2	1	no	—
41	#11745664	—	2	1	no	—
42	#11745664	—	2	1	no	—
43	C₂×C₅₉⋊C₈	—	2	1	no	—
44	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
45	C₂³×C₅₉	#41	1	1	yes	characteristic
46	C₂²×D₅₉	—	2	2	no	—
47	C₂²×D₅₉	—	4	1	no	—
48	C₂²×D₅₉	—	4	1	no	—
49	C₂²×D₅₉	—	8	2	no	—
50	C₂×C₄×C₅₉	—	4	1	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	#41	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	—	4	1	no	—
58	D₂₃₆	—	2	2	no	—
59	D₂₃₆	—	2	2	no	—
60	D₂₃₆	—	4	1	no	—
61	D₂₃₆	—	4	1	no	—
62	C₈×C₅₉	—	4	1	no	—
63	C₈×C₅₉	—	4	1	no	—
64	C₅₉⋊C₈	—	2	2	no	—
65	C₅₉⋊C₈	—	4	1	no	—
66	C₂²×C₅₉	—	4	1	no	—
67	C₂²×C₅₉	#123	1	1	yes	characteristic
68	C₂²×C₅₉	#124	1	1	yes	characteristic
69	C₂²×C₅₉	#126	1	1	yes	characteristic
70	C₂×D₅₉	—	4	2	no	—
71	C₂×D₅₉	—	8	2	no	—
72	C₂×D₅₉	—	8	2	no	—
73	C₂×D₅₉	—	8	2	no	—
74	C₄×C₅₉	—	8	1	no	—
75	C₄×C₅₉	#121	1	1	yes	characteristic
76	C₄×C₅₉	#123	1	1	yes	characteristic
77	C₄×C₅₉	#123	1	1	yes	characteristic
78	C₄×C₅₉	#126	1	1	yes	characteristic
79	Dic₅₉	—	8	1	no	—
80	#2081	—	59	1	no	maximal, Sylow
81	C₂×C₅₉	—	2	1	no	—
82	C₂×C₅₉	—	2	1	no	—
83	C₂×C₅₉	—	2	1	no	—
84	C₂×C₅₉	#397	1	1	yes	characteristic
85	D₅₉	—	8	2	no	—
86	#430	—	59	1	no	—
87	#418	—	59	1	no	—
88	#418	—	59	1	no	—
89	C₂×#100	—	59	1	no	—
90	#344	—	59	1	no	—
91	#344	D₅₉	1	1	yes	characteristic, complemented
92	H(3,ℤ/4)	—	59	1	no	—
93	C₅₉	#2081	1	1	yes	characteristic, complemented, Sylow
94	C₂²×D₄	—	59	1	no	—
95	C₂×M₁₆	—	59	1	no	—
96	C₂×M₁₆	—	59	1	no	—
97	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
98	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
99	#117	—	59	1	no	—
100	#117	C₂×D₅₉	1	1	yes	characteristic
101	#108	—	118	2	no	—
102	#102	—	118	2	no	—
103	#102	—	118	2	no	—
104	#100	—	118	2	no	—
105	C₂⁴	—	118	1	no	—
106	C₂×D₄	—	59	2	no	—
107	C₂×D₄	—	118	1	no	—
108	C₂×D₄	—	118	1	no	—
109	C₂×D₄	—	118	1	no	—
110	C₂×D₄	—	118	1	no	—
111	C₂×D₄	—	118	1	no	—
112	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	118	1	no	—
118	M₁₆	—	118	1	no	—
119	M₁₆	—	118	1	no	—
120	M₁₆	—	118	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	59	2	no	—
124	C₂×C₈	—	118	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	118	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	118	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	118	1	no	—
131	C₂³	—	118	2	no	—
132	C₂³	—	472	2	no	—
133	C₂³	—	236	1	no	—
134	C₂³	—	236	1	no	—
135	C₂³	#11745685	1	1	yes	characteristic
136	D₄	—	118	2	no	—
137	D₄	—	118	2	no	—
138	D₄	—	236	1	no	—
139	D₄	—	236	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	236	1	no	—
142	C₂×C₄	#11745685	1	1	yes	characteristic
143	C₂×C₄	#11745685	1	1	yes	characteristic
144	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
145	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
146	C₂×C₄	#11745687	1	1	yes	characteristic
147	C₂×C₄	#11745689	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	118	2	no	—
151	C₈	—	236	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	472	2	no	—
154	C₂²	—	472	2	no	—
155	C₂²	—	472	2	no	—
156	C₂²	—	236	2	no	—
157	C₂²	#14273712	1	1	yes	characteristic
158	C₂²	#14273713	1	1	yes	characteristic
159	C₂²	#14273721	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	472	1	no	—
162	C₄	#14273700	1	1	yes	characteristic
163	C₄	#14273709	1	1	yes	characteristic
164	C₄	#14273712	1	1	yes	characteristic
165	C₄	#14273722	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	472	2	no	—
170	C₂	#16837331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837450

C₂

yes

maximal, characteristic, complemented

#16837408

C₂

yes

maximal, characteristic

#16837406

C₂

yes

maximal, characteristic

C₂×#14273643

C₂

yes

maximal, characteristic

#16837174

C₂

yes

maximal, characteristic, complemented

#16837158

C₂

yes

maximal, characteristic, complemented

C₅₉×#344

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

#14273686

C₂²

yes

—

—

—

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

C₂³×D₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

D₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

—

—

—

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2081

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#397

yes

characteristic

D₅₉

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

H(3,ℤ/4)

—

C₅₉

#2081

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

#117

—

100

#117

C₂×D₅₉

yes

characteristic

101

#108

—

118

—

102

#102

—

118

—

103

#102

—

118

—

104

#100

—

118

—

105

C₂⁴

—

118

—

106

C₂×D₄

—

107

C₂×D₄

—

118

—

108

C₂×D₄

—

118

—

109

C₂×D₄

—

118

—

110

C₂×D₄

—

118

—

111

C₂×D₄

—

118

—

112

C₂²×C₄

C₂²×D₅₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

—

118

M₁₆

—

118

—

119

M₁₆

—

118

—

120

M₁₆

—

118

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

118

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

118

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

118

—

129

C₄²

—

130

C₄²

—

118

—

131

C₂³

—

118

—

132

C₂³

—

472

—

133

C₂³

—

236

—

134

C₂³

—

236

—

135

C₂³

#11745685

yes

characteristic

136

D₄

—

118

—

137

D₄

—

118

—

138

D₄

—

236

—

139

D₄

—

236

—

140

C₂×C₄

—

141

C₂×C₄

—

236

—

142

C₂×C₄

#11745685

yes

characteristic

143

C₂×C₄

#11745685

yes

characteristic

144

C₂×C₄

D₄×D₅₉

yes

characteristic

145

C₂×C₄

D₄×D₅₉

yes

characteristic

146

C₂×C₄

#11745687

yes

characteristic

147

C₂×C₄

#11745689

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

118

—

151

C₈

—

236

—

152

C₂²

—

153

C₂²

—

472

—

154

C₂²

—

472

—

155

C₂²

—

472

—

156

C₂²

—

236

—

157

C₂²

#14273712

yes

characteristic

158

C₂²

#14273713

yes

characteristic

159

C₂²

#14273721

yes

characteristic

160

C₄

—

161

C₄

—

472

—

162

C₄

#14273700

yes

characteristic

163

C₄

#14273709

yes

characteristic

164

C₄

#14273712

yes

characteristic

165

C₄

#14273722

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

472

—

170

C₂

#16837331

yes

characteristic

	subgroup	G/H
center	C₂	#16837331
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#344	C₂
socle	C₂×C₅₉	#397
hypercenter	#344	D₅₉
hypocenter	C₅₉	#2081
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#16837331

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#344

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
1. #16837450	C₂	semidirect
5. #16837174	C₂	semidirect
6. #16837158	C₂	semidirect
7. C₅₉×#344	C₂	semidirect
91. #344	D₅₉	semidirect
93. C₅₉	#2081	semidirect

H ≅ G/N

type

1. #16837450

C₂

semidirect

5. #16837174

C₂

semidirect

6. #16837158

C₂

semidirect

7. C₅₉×#344

C₂

semidirect

91. #344

D₅₉

semidirect

93. C₅₉

#2081

semidirect

Subgroups — #21425559

Conjugacy classes of subgroups

Complements