Group #21425715 — subgroups

Subgroups — #21425715

Order 7552.

220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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×#14273735	C₂	1	1	yes	maximal, characteristic
2	M₃₂×D₅₉	C₂	1	2	yes	maximal, complemented
3	#16837184	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837183	C₂	1	2	yes	maximal, complemented
5	C₅₉×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	C₄	1	1	yes	characteristic
7	#14273735	—	2	2	no	—
8	#14273735	—	2	2	no	—
9	C₂×#11745659	C₄	1	1	yes	characteristic
10	C₂×#11745659	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₈×D₅₉	C₄	1	1	yes	characteristic
13	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
14	#14273667	C₂²	1	2	yes	—
15	#14273666	—	2	2	no	—
16	C₁₆×D₅₉	—	2	2	no	—
17	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	M₃₂×C₅₉	C₂²	1	2	yes	—
19	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₉	C₈	1	1	yes	characteristic
21	C₂×C₂²⋊D₅₉	C₈	1	1	yes	characteristic
22	#11745685	—	2	2	no	—
23	#11745685	—	2	2	no	—
24	C₂×D₂₃₆	C₈	1	1	yes	characteristic
25	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₁₁₈	C₈	1	1	yes	characteristic
28	M₁₆×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₅₉⋊C₈	—	2	1	no	—
32	C₂×C₅₉⋊C₈	—	2	1	no	—
33	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
35	#11745659	—	2	2	no	—
36	#11745659	—	2	2	no	—
37	C₈×D₅₉	—	2	1	no	—
38	C₈×D₅₉	—	2	1	no	—
39	C₈×D₅₉	D₄	1	2	yes	—
40	C₈×D₅₉	D₄	1	2	yes	—
41	C₁₆×C₅₉	—	2	2	no	—
42	C₅₉⋊C₁₆	—	2	2	no	—
43	C₂³×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₅₉	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊D₅₉	—	2	2	no	—
51	C₂²⋊D₅₉	—	2	2	no	—
52	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
53	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
54	D₂₃₆	—	2	2	no	—
55	C₄×D₅₉	—	2	1	no	—
56	C₄×D₅₉	—	2	1	no	—
57	C₄×D₅₉	C₂²⋊C₄	1	2	yes	—
58	C₄×D₅₉	M₁₆	1	2	yes	—
59	Dic₁₁₈	—	2	2	no	—
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	C₂×D₄	1	2	yes	—
62	C₅₉⋊C₈	—	2	1	no	—
63	C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₅₉	—	2	1	no	—
66	C₂²×C₅₉	—	4	1	no	—
67	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×D₅₉	—	2	1	no	—
69	C₂×D₅₉	—	4	1	no	—
70	C₂×D₅₉	—	4	1	no	—
71	C₂×D₅₉	C₂²⋊C₈	1	2	yes	—
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
75	Dic₅₉	—	2	1	no	—
76	Dic₅₉	C₂²⋊C₈	1	2	yes	—
77	#2115	—	59	1	no	maximal, Sylow
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	—	4	1	no	—
80	C₂×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	D₅₉	—	2	2	no	—
82	D₅₉	—	4	1	no	—
83	C₂×#131	—	59	1	no	—
84	C₂×M₃₂	—	59	2	no	—
85	#349	—	59	1	no	—
86	#349	—	59	2	no	—
87	#349	D₅₉	1	1	yes	characteristic, complemented
88	C₅₉	#2115	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	59	1	no	—
90	#131	—	118	2	no	—
91	#131	—	118	2	no	—
92	C₂×M₁₆	—	59	1	no	—
93	C₂×M₁₆	—	59	1	no	—
94	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
95	C₂²×C₈	—	59	1	no	—
96	C₂²×C₈	—	59	1	no	—
97	C₂²×C₈	—	59	1	no	—
98	M₃₂	—	59	2	no	—
99	M₃₂	—	118	2	no	—
100	M₃₂	C₂×D₅₉	1	2	yes	—
101	C₂×C₁₆	—	118	2	no	—
102	#41	—	118	2	no	—
103	#41	—	118	2	no	—
104	C₂×Q₈	—	59	1	no	—
105	C₂×D₄	—	59	1	no	—
106	C₂×D₄	—	59	1	no	—
107	C₂×D₄	—	59	1	no	—
108	C₂²×C₄	—	59	1	no	—
109	C₂²×C₄	—	59	1	no	—
110	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	118	2	no	—
113	M₁₆	—	118	2	no	—
114	C₂×C₈	—	59	1	no	—
115	C₂×C₈	—	59	2	no	—
116	C₂×C₈	—	59	2	no	—
117	C₂×C₈	—	59	1	no	—
118	C₂×C₈	—	118	1	no	—
119	C₂×C₈	—	118	1	no	—
120	C₂×C₈	—	118	1	no	—
121	C₂×C₈	—	118	1	no	—
122	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₁₆	—	2	2	no	—
125	C₁₆	—	118	2	no	—
126	C₂³	—	59	1	no	—
127	C₂³	—	59	1	no	—
128	C₂³	C₈×D₅₉	1	1	yes	characteristic
129	Q₈	—	118	2	no	—
130	D₄	—	118	2	no	—
131	D₄	—	118	2	no	—
132	D₄	—	118	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	59	1	no	—
136	C₂×C₄	—	59	2	no	—
137	C₂×C₄	—	59	2	no	—
138	C₂×C₄	—	59	1	no	—
139	C₂×C₄	—	118	1	no	—
140	C₂×C₄	—	118	1	no	—
141	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	59	1	no	—
145	C₈	—	59	1	no	—
146	C₈	—	118	1	no	—
147	C₈	D₄×D₅₉	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	59	2	no	—
151	C₂²	—	118	1	no	—
152	C₂²	—	236	1	no	—
153	C₂²	—	236	1	no	—
154	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	59	2	no	—
157	C₄	—	118	1	no	—
158	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
159	C₄	M₁₆×D₅₉	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	4	1	no	—
162	C₂	—	118	2	no	—
163	C₂	—	236	1	no	—
164	C₂	C₂²⋊C₈×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273735

C₂

yes

maximal, characteristic

M₃₂×D₅₉

C₂

yes

maximal, complemented

#16837184

C₂

yes

maximal, characteristic, complemented

#16837183

C₂

yes

maximal, complemented

C₅₉×#349

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₄

yes

—

—

C₄

yes

characteristic

C₂×#11745659

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₄

yes

characteristic

C₂×C₈×D₅₉

C₄

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273667

C₂²

yes

—

#14273666

—

C₁₆×D₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₉

C₈

yes

characteristic

C₂×C₂²⋊D₅₉

C₈

yes

characteristic

#11745685

—

#11745685

—

C₂×D₂₃₆

C₈

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×Dic₁₁₈

C₈

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

#11745659

—

#11745659

—

C₈×D₅₉

—

C₈×D₅₉

—

C₈×D₅₉

D₄

yes

—

C₈×D₅₉

D₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×C₈

yes

characteristic

C₂²×D₅₉

C₂×C₈

yes

characteristic

C₂²×D₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

C₂²⋊C₄

yes

—

C₄×D₅₉

M₁₆

yes

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

C₂²⋊C₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

C₂²⋊C₈

yes

—

#2115

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×C₂²⋊C₈

yes

characteristic

D₅₉

—

D₅₉

—

C₂×#131

—

C₂×M₃₂

—

#349

—

#349

—

#349

D₅₉

yes

characteristic, complemented

C₅₉

#2115

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

118

—

#131

—

118

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

118

—

100

M₃₂

C₂×D₅₉

yes

—

101

C₂×C₁₆

—

118

—

102

#41

—

118

—

103

#41

—

118

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₅₉

yes

characteristic

111

M₁₆

—

112

M₁₆

—

118

—

113

M₁₆

—

118

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

118

—

119

C₂×C₈

—

118

—

120

C₂×C₈

—

118

—

121

C₂×C₈

—

118

—

122

C₂×C₈

C₄×D₅₉

yes

characteristic

123

C₂×C₈

C₂²×D₅₉

yes

characteristic

124

C₁₆

—

125

C₁₆

—

118

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₈×D₅₉

yes

characteristic

129

Q₈

—

118

—

130

D₄

—

118

—

131

D₄

—

118

—

132

D₄

—

118

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

118

—

140

C₂×C₄

—

118

—

141

C₂×C₄

C₈×D₅₉

yes

characteristic

142

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

118

—

147

C₈

D₄×D₅₉

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

118

—

152

C₂²

—

236

—

153

C₂²

—

236

—

154

C₂²

C₂×C₈×D₅₉

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

118

—

158

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

159

C₄

M₁₆×D₅₉

yes

characteristic

160

C₂

—

161

C₂

—

162

C₂

—

118

—

163

C₂

—

236

—

164

C₂

C₂²⋊C₈×D₅₉

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₅₉
commutator	C₂²×C₅₉	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#349	C₂
socle	C₂×C₅₉	C₂×C₂²⋊C₈
hypercenter	#349	D₅₉
hypocenter	C₅₉	#2115
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

subgroup

G/H

center

C₄

C₂²⋊C₄×D₅₉

commutator

C₂²×C₅₉

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#349

C₂

socle

C₂×C₅₉

C₂×C₂²⋊C₈

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

N	H ≅ G/N	type
2. M₃₂×D₅₉	C₂	semidirect
3. #16837184	C₂	semidirect
4. #16837183	C₂	semidirect
5. C₅₉×#349	C₂	semidirect
87. #349	D₅₉	semidirect
88. C₅₉	#2115	semidirect

H ≅ G/N

type

2. M₃₂×D₅₉

C₂

semidirect

3. #16837184

C₂

semidirect

4. #16837183

C₂

semidirect

5. C₅₉×#349

C₂

semidirect

87. #349

D₅₉

semidirect

88. C₅₉

#2115

semidirect

Subgroups — #21425715

Conjugacy classes of subgroups

Complements