Group #21425519 — subgroups

Subgroups — #21425519

Order 7552.

256 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14273661	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837262	C₂	1	2	yes	maximal
3	#16837165	C₂	1	2	yes	maximal, complemented
4	#16837165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₅₉	—	2	2	no	—
9	#14273661	—	2	2	no	—
10	C₅₉⋊#95	—	2	2	no	—
11	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₅₉	—	2	2	no	—
15	C₂⁴×C₅₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	—	2	1	no	—
17	C₂²×C₄×C₅₉	—	2	1	no	—
18	C₂²×C₄×C₅₉	—	2	2	no	—
19	C₂²×C₄×C₅₉	—	2	2	no	—
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	C₂²×Dic₅₉	—	2	2	no	—
24	C₂×C₈×C₅₉	—	2	2	no	—
25	C₂×C₈×C₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	2	no	—
27	C₂×C₈×C₅₉	D₄	1	2	yes	—
28	C₂²⋊Dic₅₉	—	4	4	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	—	2	2	no	—
32	C₄⋊Dic₅₉	—	2	2	no	—
33	C₂³×C₅₉	—	2	1	no	—
34	C₂³×C₅₉	—	2	2	no	—
35	C₂³×C₅₉	—	2	1	no	—
36	C₂³×C₅₉	—	2	2	no	—
37	C₂³×C₅₉	D₈	1	2	yes	—
38	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	2	1	no	—
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	4	1	no	—
42	C₂×C₄×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	—	4	2	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	2	yes	—
47	C₂×C₄×C₅₉	#41	1	2	yes	—
48	C₂×Dic₅₉	—	4	2	no	—
49	C₂×Dic₅₉	—	4	2	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	C₂×Dic₅₉	—	4	2	no	—
52	C₂×Dic₅₉	—	4	2	no	—
53	C₂×Dic₅₉	—	4	2	no	—
54	C₈×C₅₉	—	2	2	no	—
55	C₈×C₅₉	—	4	1	no	—
56	C₂²×C₅₉	—	2	2	no	—
57	C₂²×C₅₉	—	2	4	no	—
58	C₂²×C₅₉	—	4	1	no	—
59	C₂²×C₅₉	—	4	1	no	—
60	C₂²×C₅₉	—	4	2	no	—
61	C₂²×C₅₉	#121	1	1	yes	characteristic
62	C₂²×C₅₉	#123	1	2	yes	—
63	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₅₉	#135	1	1	yes	characteristic
65	C₂²×C₅₉	#137	1	2	yes	—
66	C₄×C₅₉	—	4	1	no	—
67	C₄×C₅₉	#123	1	1	yes	characteristic
68	C₄×C₅₉	#124	1	1	yes	characteristic
69	C₄×C₅₉	#126	1	2	yes	—
70	Dic₅₉	—	8	2	no	—
71	Dic₅₉	—	8	2	no	—
72	#2075	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	—	4	2	no	—
74	C₂×C₅₉	#397	1	1	yes	characteristic
75	C₂×C₅₉	#466	1	1	yes	characteristic
76	C₂×C₅₉	#470	1	1	yes	characteristic
77	C₂×C₅₉	#480	1	2	yes	—
78	C₂×C₅₉	#485	1	2	yes	—
79	C₂×#107	—	59	1	no	—
80	C₂×C₂²⋊C₈	D₅₉	1	1	yes	characteristic
81	#380	—	59	2	no	—
82	#340	—	59	1	no	—
83	#340	—	59	2	no	—
84	C₅₉	#2075	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	59	2	no	—
89	C₂×C₄⋊C₄	—	59	2	no	—
90	C₂×C₂²⋊C₄	—	118	2	no	—
91	#107	—	118	2	no	—
92	C₂²⋊C₈	—	2	2	no	—
93	#95	—	118	2	no	—
94	C₂⁴	D₂₃₆	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	118	2	no	—
100	C₂²×C₄	—	118	2	no	—
101	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
107	C₄⋊C₄	—	118	2	no	—
108	C₄⋊C₄	—	118	2	no	—
109	C₄⋊C₄	—	118	2	no	—
110	C₄⋊C₄	—	118	2	no	—
111	C₂²⋊C₄	—	236	4	no	—
112	C₂³	—	2	1	no	—
113	C₂³	—	2	2	no	—
114	C₂³	—	2	1	no	—
115	C₂³	—	2	2	no	—
116	C₂³	D₄₇₂	1	2	yes	—
117	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	236	2	no	—
124	C₂×C₄	—	236	2	no	—
125	C₂×C₄	—	236	2	no	—
126	C₂×C₄	—	236	2	no	—
127	C₂×C₄	—	236	2	no	—
128	C₂×C₄	—	236	2	no	—
129	C₂×C₄	#11745685	1	1	yes	characteristic
130	C₂×C₄	#11745687	1	2	yes	—
131	C₂×C₄	#11745687	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	1	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	2	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#14273701	1	1	yes	characteristic
141	C₂²	#14273711	1	2	yes	—
142	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
143	C₂²	#14273749	1	1	yes	characteristic
144	C₂²	#14273760	1	2	yes	—
145	C₄	—	4	1	no	—
146	C₄	—	472	2	no	—
147	C₄	—	472	2	no	—
148	C₄	#14273708	1	1	yes	characteristic
149	C₄	#14273714	1	1	yes	characteristic
150	C₄	#14273720	1	2	yes	—
151	C₂	—	4	2	no	—
152	C₂	#16837327	1	1	yes	characteristic
153	C₂	#16837638	1	1	yes	characteristic
154	C₂	#16837660	1	1	yes	characteristic
155	C₂	#16837716	1	2	yes	—
156	C₂	#16837744	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal, characteristic, complemented

#16837262

C₂

yes

maximal

#16837165

C₂

yes

maximal, complemented

#16837165

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₂²⋊Dic₅₉

—

#14273661

—

C₅₉⋊#95

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²⋊C₈×C₅₉

—

C₂⁴×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

D₈

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×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

#123

yes

—

C₂²×C₅₉

C₂×D₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

#2075

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#485

yes

—

C₂×#107

—

C₂×C₂²⋊C₈

D₅₉

yes

—

—

—

C₅₉

#2075

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

118

—

#107

—

118

—

C₂²⋊C₈

—

#95

—

118

—

C₂⁴

D₂₃₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

100

C₂²×C₄

—

118

—

101

C₂²×C₄

D₂₃₆

yes

characteristic

102

C₂²×C₄

C₂²×D₅₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²⋊D₅₉

yes

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

—

118

—

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

118

—

111

C₂²⋊C₄

—

236

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

D₄₇₂

yes

—

117

C₂³

C₂×D₂₃₆

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

236

—

124

C₂×C₄

—

236

—

125

C₂×C₄

—

236

—

126

C₂×C₄

—

236

—

127

C₂×C₄

—

236

—

128

C₂×C₄

—

236

—

129

C₂×C₄

#11745685

yes

characteristic

130

C₂×C₄

#11745687

yes

—

131

C₂×C₄

#11745687

yes

—

132

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#14273701

yes

characteristic

141

C₂²

#14273711

yes

—

142

C₂²

C₂×D₄₇₂

yes

characteristic

143

C₂²

#14273749

yes

characteristic

144

C₂²

#14273760

yes

—

145

C₄

—

146

C₄

—

472

—

147

C₄

—

472

—

148

C₄

#14273708

yes

characteristic

149

C₄

#14273714

yes

characteristic

150

C₄

#14273720

yes

—

151

C₂

—

152

C₂

#16837327

yes

characteristic

153

C₂

#16837638

yes

characteristic

154

C₂

#16837660

yes

characteristic

155

C₂

#16837716

yes

—

156

C₂

#16837744

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₂²⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₂²⋊C₈	D₅₉
hypocenter	C₅₉	#2075
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₂²⋊C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₂²⋊C₈

D₅₉

hypocenter

C₅₉

#2075

LCS

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#14273661	C₂	semidirect
3. #16837165	C₂	semidirect
4. #16837165	C₂	semidirect
84. C₅₉	#2075	semidirect

H ≅ G/N

type

1. C₂×#14273661

C₂

semidirect

3. #16837165

C₂

semidirect

4. #16837165

C₂

semidirect

84. C₅₉

#2075

semidirect

Subgroups — #21425519

Conjugacy classes of subgroups

Complements