Group #21425925 — subgroups

Subgroups — #21425925

Order 7552.

162 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273666	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₂²⋊D₅₉	C₂	1	1	yes	maximal, characteristic
4	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊D₅₉	C₄	1	1	yes	characteristic
10	#14273667	—	2	2	no	—
11	#14273666	—	2	2	no	—
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
15	C₂²⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
16	(C₂×D₅₉)⋊C₈	C₄	1	1	yes	characteristic
17	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₅₉	C₈	1	1	yes	characteristic
23	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	—	2	1	no	—
25	C₂×C₈×C₅₉	—	2	1	no	—
26	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₅₉	C₈	1	1	yes	characteristic
29	(C₂×D₅₉)⋊C₄	C₈	1	1	yes	characteristic
30	Dic₅₉⋊C₄	C₈	1	1	yes	characteristic
31	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₈×D₅₉	—	2	1	no	—
35	C₈×D₅₉	—	2	1	no	—
36	C₁₆×C₅₉	—	2	1	no	—
37	C₁₆×C₅₉	—	2	1	no	—
38	C₅₉⋊C₁₆	—	2	1	no	—
39	C₅₉⋊C₁₆	D₄	1	2	yes	complemented
40	C₂³×C₅₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₅₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	1	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₅₉	—	2	2	no	—
47	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
49	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
50	C₄×D₅₉	—	2	1	no	—
51	C₄×D₅₉	—	2	1	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₈×C₅₉	#41	1	1	yes	characteristic
55	C₅₉⋊C₈	—	2	1	no	—
56	C₅₉⋊C₈	—	2	1	no	—
57	C₂²×C₅₉	—	2	1	no	—
58	C₂²×C₅₉	—	2	1	no	—
59	C₂²×C₅₉	—	2	1	no	—
60	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₅₉	—	2	1	no	—
62	C₂×D₅₉	—	2	1	no	—
63	C₂×D₅₉	—	2	1	no	—
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
66	C₄×C₅₉	#131	1	1	yes	characteristic
67	Dic₅₉	—	2	1	no	—
68	Dic₅₉	—	2	1	no	—
69	Dic₅₉	M₃₂	1	2	yes	complemented
70	#2169	—	59	1	no	maximal, Sylow
71	C₂×C₅₉	—	4	1	no	—
72	C₂×C₅₉	C₈×D₄	1	1	yes	characteristic
73	C₂×C₅₉	C₂×M₃₂	1	1	yes	characteristic
74	C₂×C₅₉	#504	1	1	yes	characteristic
75	D₅₉	—	4	1	no	—
76	C₂×M₃₂	—	59	1	no	—
77	C₂×M₃₂	—	59	1	no	—
78	C₈×D₄	—	59	1	no	—
79	C₄⋊C₁₆	—	59	1	no	—
80	C₂²⋊C₁₆	—	59	1	no	—
81	C₂²⋊C₁₆	D₅₉	1	1	yes	characteristic, complemented
82	C₄×C₁₆	—	59	1	no	—
83	C₅₉	#2169	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	59	1	no	—
85	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
86	C₄×D₄	—	59	1	no	—
87	M₃₂	—	118	2	no	—
88	M₃₂	—	118	2	no	—
89	C₂×C₁₆	—	59	1	no	—
90	C₂×C₁₆	—	59	1	no	—
91	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
92	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
93	C₄⋊C₈	—	59	1	no	—
94	C₂²⋊C₈	—	59	1	no	—
95	C₂²⋊C₈	—	59	1	no	—
96	C₄×C₈	—	59	1	no	—
97	C₂×D₄	—	59	1	no	—
98	C₂²×C₄	—	59	1	no	—
99	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	59	1	no	—
103	C₂×C₈	—	59	1	no	—
104	C₂×C₈	—	118	1	no	—
105	C₂×C₈	—	118	1	no	—
106	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
108	C₄⋊C₄	—	59	1	no	—
109	C₂²⋊C₄	—	59	1	no	—
110	C₂²⋊C₄	—	59	1	no	—
111	C₄²	—	59	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	59	2	no	—
115	C₁₆	—	118	1	no	—
116	C₂³	—	59	1	no	—
117	C₂³	C₈×D₅₉	1	1	yes	characteristic
118	D₄	—	118	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	59	1	no	—
122	C₂×C₄	—	59	1	no	—
123	C₂×C₄	—	59	1	no	—
124	C₂×C₄	—	118	1	no	—
125	C₂×C₄	—	118	1	no	—
126	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	118	1	no	—
130	C₈	—	118	1	no	—
131	C₈	D₄×D₅₉	1	1	yes	characteristic
132	C₈	#11745687	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	118	1	no	—
137	C₂²	—	118	1	no	—
138	C₂²	—	118	1	no	—
139	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	59	2	no	—
142	C₄	—	118	1	no	—
143	C₄	—	118	1	no	—
144	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
145	C₄	#14273735	1	1	yes	characteristic
146	C₂	—	4	1	no	—
147	C₂	—	236	1	no	—
148	C₂	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
149	C₂	M₃₂×D₅₉	1	1	yes	characteristic
150	C₂	#16837818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273667

C₂

yes

maximal, characteristic, complemented

C₂×#14273666

C₂

yes

maximal, characteristic, complemented

C₈×C₂²⋊D₅₉

C₂

yes

maximal, characteristic

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₅₉×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₉

C₂²

yes

characteristic

C₄×C₂²⋊D₅₉

C₄

yes

characteristic

#14273667

—

#14273666

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

Dic₅₉⋊C₈

C₄

yes

characteristic

C₂²⋊(C₅₉⋊C₈)

C₄

yes

characteristic

(C₂×D₅₉)⋊C₈

C₄

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₉

C₈

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

C₈

yes

characteristic

(C₂×D₅₉)⋊C₄

C₈

yes

characteristic

Dic₅₉⋊C₄

C₈

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

D₄

yes

complemented

C₂³×C₅₉

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

—

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

M₃₂

yes

complemented

#2169

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₈×D₄

yes

characteristic

C₂×C₅₉

C₂×M₃₂

yes

characteristic

C₂×C₅₉

#504

yes

characteristic

D₅₉

—

C₂×M₃₂

—

C₂×M₃₂

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₅₉

yes

characteristic, complemented

C₄×C₁₆

—

C₅₉

#2169

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

M₃₂

—

118

—

M₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

118

—

105

C₂×C₈

—

118

—

106

C₂×C₈

C₄×D₅₉

yes

characteristic

107

C₂×C₈

C₂²×D₅₉

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

118

—

116

C₂³

—

117

C₂³

C₈×D₅₉

yes

characteristic

118

D₄

—

118

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

118

—

126

C₂×C₄

C₈×D₅₉

yes

characteristic

127

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

128

C₈

—

129

C₈

—

118

—

130

C₈

—

118

—

131

C₈

D₄×D₅₉

yes

characteristic

132

C₈

#11745687

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

118

—

137

C₂²

—

118

—

138

C₂²

—

118

—

139

C₂²

C₂×C₈×D₅₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

118

—

143

C₄

—

118

—

144

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

145

C₄

#14273735

yes

characteristic

146

C₂

—

147

C₂

—

236

—

148

C₂

(C₂²⋊D₅₉)⋊C₈

yes

characteristic

149

C₂

M₃₂×D₅₉

yes

characteristic

150

C₂

#16837818

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₈
Fitting	C₅₉×C₂²⋊C₁₆	C₂
hypercenter	C₂²⋊C₁₆	D₅₉
hypocenter	C₅₉	#2169
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₈

Fitting

C₅₉×C₂²⋊C₁₆

C₂

hypercenter

C₂²⋊C₁₆

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
1. C₂×#14273667	C₂	semidirect
2. C₂×#14273666	C₂	semidirect
4. Dic₅₉⋊C₁₆	C₂	semidirect
5. (C₂×D₅₉)⋊C₁₆	C₂	semidirect
6. C₁₆×Dic₅₉	C₂	semidirect
7. C₅₉×C₂²⋊C₁₆	C₂	semidirect
39. C₅₉⋊C₁₆	D₄	semidirect
69. Dic₅₉	M₃₂	semidirect
81. C₂²⋊C₁₆	D₅₉	semidirect
83. C₅₉	#2169	semidirect

H ≅ G/N

type

1. C₂×#14273667

C₂

semidirect

2. C₂×#14273666

C₂

semidirect

4. Dic₅₉⋊C₁₆

C₂

semidirect

5. (C₂×D₅₉)⋊C₁₆

C₂

semidirect

6. C₁₆×Dic₅₉

C₂

semidirect

7. C₅₉×C₂²⋊C₁₆

C₂

semidirect

39. C₅₉⋊C₁₆

D₄

semidirect

69. Dic₅₉

M₃₂

semidirect

81. C₂²⋊C₁₆

D₅₉

semidirect

83. C₅₉

#2169

semidirect

Subgroups — #21425925

Conjugacy classes of subgroups

Complements