Group #21425504 — subgroups

Subgroups — #21425504

Order 7552.

238 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14273651	C₂	1	2	yes	maximal
2	#16837272	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	2	yes	maximal
4	#16837157	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#391	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	#14273651	—	2	4	no	—
11	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	#95×C₅₉	—	2	1	no	—
15	#95×C₅₉	—	2	1	no	—
16	C₂×Q₈×C₅₉	—	2	1	no	—
17	C₂×Q₈×C₅₉	—	2	2	no	—
18	C₂×Q₈×C₅₉	—	2	2	no	—
19	C₂×Q₈×C₅₉	D₄	1	2	yes	—
20	C₂²×C₄×C₅₉	—	2	1	no	—
21	C₂²×C₄×C₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	—	2	2	no	—
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₄²×C₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	—	4	2	no	—
31	C₄⋊Dic₅₉	—	4	2	no	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂×C₅₉⋊C₈	—	2	2	no	—
34	C₂×C₅₉⋊C₈	—	2	2	no	—
35	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₅₉	—	2	4	no	—
37	Q₈×C₅₉	—	4	2	no	—
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	1	no	—
40	C₂×C₄×C₅₉	—	2	1	no	—
41	C₂×C₄×C₅₉	—	2	1	no	—
42	C₂×C₄×C₅₉	—	2	2	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	—	4	1	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
49	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	2	yes	—
53	C₂×C₄×C₅₉	#41	1	2	yes	—
54	C₂×Dic₅₉	—	4	2	no	—
55	C₂×Dic₅₉	—	4	2	no	—
56	C₂×Dic₅₉	—	4	2	no	—
57	C₅₉⋊C₈	—	4	2	no	—
58	C₂²×C₅₉	#121	1	1	yes	characteristic
59	C₂²×C₅₉	#123	1	1	yes	characteristic
60	C₂²×C₅₉	#124	1	1	yes	characteristic
61	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₅₉	#137	1	1	yes	characteristic
65	C₄×C₅₉	—	2	2	no	—
66	C₄×C₅₉	—	2	2	no	—
67	C₄×C₅₉	—	4	2	no	—
68	C₄×C₅₉	—	8	1	no	—
69	C₄×C₅₉	#123	1	2	yes	—
70	C₄×C₅₉	#126	1	2	yes	—
71	Dic₅₉	—	8	2	no	—
72	#2072	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	#397	1	1	yes	characteristic
74	C₂×C₅₉	#460	1	1	yes	characteristic
75	C₂×C₅₉	#462	1	1	yes	characteristic
76	C₂×C₅₉	#483	1	1	yes	characteristic
77	C₂×C₅₉	#484	1	1	yes	characteristic
78	C₂×C₅₉	#487	1	1	yes	characteristic
79	C₂×C₅₉	#489	1	1	yes	characteristic
80	C₂×#103	—	59	2	no	—
81	#391	D₅₉	1	1	yes	characteristic
82	#384	—	59	1	no	—
83	#340	—	59	2	no	—
84	#336	—	59	1	no	—
85	C₅₉	#2072	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
87	C₂²×C₈	—	59	2	no	—
88	C₂×C₄⋊C₄	—	59	2	no	—
89	C₂×C₄⋊C₄	—	118	1	no	—
90	C₂×C₄⋊C₄	—	118	1	no	—
91	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
93	#103	—	118	4	no	—
94	#95	—	2	1	no	—
95	#95	—	2	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	D₂₃₆	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	118	2	no	—
103	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
106	C₂×C₈	—	118	2	no	—
107	C₂×C₈	—	118	2	no	—
108	C₂×C₈	—	118	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	118	2	no	—
111	C₄⋊C₄	—	118	2	no	—
112	C₄⋊C₄	—	236	2	no	—
113	C₄⋊C₄	—	236	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	236	2	no	—
129	C₂×C₄	—	236	2	no	—
130	C₂×C₄	—	236	2	no	—
131	C₂×C₄	#11745671	1	2	yes	—
132	C₂×C₄	#11745672	1	2	yes	—
133	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
134	C₂×C₄	#11745685	1	1	yes	characteristic
135	C₂×C₄	#11745687	1	2	yes	—
136	C₂×C₄	#11745687	1	2	yes	—
137	C₈	—	236	2	no	—
138	C₂²	#14273701	1	1	yes	characteristic
139	C₂²	#14273708	1	1	yes	characteristic
140	C₂²	#14273714	1	1	yes	characteristic
141	C₂²	C₂×#11745671	1	1	yes	characteristic
142	C₂²	C₂×#11745672	1	1	yes	characteristic
143	C₂²	#14273752	1	1	yes	characteristic
144	C₂²	#14273757	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	1	no	—
149	C₄	—	472	2	no	—
150	C₄	#14273711	1	2	yes	—
151	C₄	#14273720	1	2	yes	—
152	C₂	#16837327	1	1	yes	characteristic
153	C₂	#16837602	1	1	yes	characteristic
154	C₂	#16837612	1	1	yes	characteristic
155	C₂	#16837730	1	1	yes	characteristic
156	C₂	#16837735	1	1	yes	characteristic
157	C₂	#16837752	1	1	yes	characteristic
158	C₂	#16837761	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273651

C₂

yes

maximal

#16837272

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

#16837157

C₂

yes

maximal, characteristic

C₅₉×#391

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273651

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

#95×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

—

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

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×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

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₂²×C₅₉

C₂×Q₁₆

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#123

yes

—

C₄×C₅₉

#126

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

C₅₉

#2072

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

characteristic

#103

—

118

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₂₃₆

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

118

—

103

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

104

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

105

C₂²×C₄

C₂²×D₅₉

yes

characteristic

106

C₂×C₈

—

118

—

107

C₂×C₈

—

118

—

108

C₂×C₈

—

118

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

118

—

111

C₄⋊C₄

—

118

—

112

C₄⋊C₄

—

236

—

113

C₄⋊C₄

—

236

—

114

C₄²

—

115

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

236

—

129

C₂×C₄

—

236

—

130

C₂×C₄

—

236

—

131

C₂×C₄

#11745671

yes

—

132

C₂×C₄

#11745672

yes

—

133

C₂×C₄

C₂×D₂₃₆

yes

characteristic

134

C₂×C₄

#11745685

yes

characteristic

135

C₂×C₄

#11745687

yes

—

136

C₂×C₄

#11745687

yes

—

137

C₈

—

236

—

138

C₂²

#14273701

yes

characteristic

139

C₂²

#14273708

yes

characteristic

140

C₂²

#14273714

yes

characteristic

141

C₂²

C₂×#11745671

yes

characteristic

142

C₂²

C₂×#11745672

yes

characteristic

143

C₂²

#14273752

yes

characteristic

144

C₂²

#14273757

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

472

—

150

C₄

#14273711

yes

—

151

C₄

#14273720

yes

—

152

C₂

#16837327

yes

characteristic

153

C₂

#16837602

yes

characteristic

154

C₂

#16837612

yes

characteristic

155

C₂

#16837730

yes

characteristic

156

C₂

#16837735

yes

characteristic

157

C₂

#16837752

yes

characteristic

158

C₂

#16837761

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#391	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#391	D₅₉
hypocenter	C₅₉	#2072
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#391

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
85. C₅₉	#2072	semidirect

H ≅ G/N

type

85. C₅₉

#2072

semidirect

Subgroups — #21425504

Conjugacy classes of subgroups

Complements