Group #21422513 — subgroups

Subgroups — #21422513

Order 7552.

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

normal subgroups only

This group has 262 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	#16837455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₅₉⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₅₉×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
8	#14273638	—	2	1	no	—
9	C₄×C₅₉⋊C₈	—	2	1	no	—
10	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₂×#41×C₅₉	—	2	1	no	—
12	C₄⋊Q₈×C₅₉	C₄	1	1	yes	characteristic
13	#127×C₅₉	C₄	1	1	yes	characteristic
14	#124×C₅₉	—	2	1	no	—
15	#121×C₅₉	—	2	1	no	—
16	#121×C₅₉	C₄	1	2	yes	—
17	C₄×D₄×C₅₉	—	2	1	no	—
18	C₄×D₄×C₅₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
20	#41×C₅₉	—	4	4	no	—
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×D₄×C₅₉	—	2	2	no	—
23	C₂×D₄×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₉	—	2	2	no	—
27	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₅₉	—	2	2	no	—
33	C₂²⋊C₄×C₅₉	—	2	2	no	—
34	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₉	D₄	1	1	yes	characteristic
36	C₄²×C₅₉	D₄	1	1	yes	characteristic
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	#11745664	—	2	4	no	—
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	C₂×C₅₉⋊C₈	—	2	2	no	—
41	C₂³×C₅₉	—	2	2	no	—
42	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₅₉	—	4	2	no	—
44	D₄×C₅₉	—	2	4	no	—
45	D₄×C₅₉	—	4	2	no	—
46	D₄×C₅₉	—	4	4	no	—
47	D₄×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	4	4	no	—
53	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₅₉⋊C₈	—	2	4	no	—
60	C₅₉⋊C₈	—	4	2	no	—
61	C₂²×C₅₉	—	2	1	no	—
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	—	2	1	no	—
64	C₂²×C₅₉	—	4	2	no	—
65	C₂²×C₅₉	—	4	2	no	—
66	C₂²×C₅₉	—	4	2	no	—
67	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₅₉	—	2	1	no	—
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	—	4	2	no	—
72	C₄×C₅₉	#104	1	2	yes	—
73	C₄×C₅₉	#131	1	2	yes	—
74	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
75	#1484	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	—	4	1	no	—
77	C₂×C₅₉	—	8	2	no	—
78	C₂×C₅₉	#408	1	1	yes	characteristic
79	C₂×C₅₉	#418	1	1	yes	characteristic
80	C₂×C₅₉	C₂×#104	1	1	yes	characteristic
81	#532	D₅₉	1	1	yes	characteristic
82	#431	—	59	1	no	—
83	C₄×M₁₆	—	59	1	no	—
84	D₄⋊C₈	—	59	4	no	—
85	C₅₉	#1484	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	2	1	no	—
87	C₂×M₁₆	—	118	1	no	—
88	C₄⋊Q₈	Dic₅₉	1	1	yes	characteristic
89	#127	Dic₅₉	1	1	yes	characteristic
90	#124	—	2	1	no	—
91	#121	—	2	1	no	—
92	#121	Dic₅₉	1	2	yes	—
93	C₄×D₄	—	2	1	no	—
94	C₄×D₄	C₂×D₅₉	1	2	yes	—
95	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
96	C₄⋊C₈	—	59	2	no	—
97	C₂²⋊C₈	—	118	1	no	—
98	#97	—	118	1	no	—
99	C₄×C₈	—	59	2	no	—
100	C₄×C₈	—	118	1	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
111	M₁₆	—	118	4	no	—
112	C₂×C₈	—	118	2	no	—
113	C₂×C₈	—	118	2	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
119	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
120	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
121	C₄²	C₂²×D₅₉	1	1	yes	characteristic
122	C₂³	—	2	2	no	—
123	C₂³	C₂²⋊Dic₅₉	1	1	yes	characteristic
124	Q₈	—	4	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	4	2	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
135	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
136	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
137	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
140	C₈	—	118	4	no	—
141	C₈	—	236	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	4	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	#14273654	1	2	yes	—
154	C₄	#14273734	1	2	yes	—
155	C₄	#14273753	1	2	yes	—
156	C₂	—	4	1	no	—
157	C₂	—	8	2	no	—
158	C₂	#16837369	1	1	yes	characteristic
159	C₂	#16837407	1	1	yes	characteristic
160	C₂	C₂×#14273654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837455

C₂

yes

maximal, characteristic, complemented

C₄×#11745664

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, complemented

C₅₉×#532

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×#41×C₅₉

—

C₄⋊Q₈×C₅₉

C₄

yes

characteristic

#127×C₅₉

C₄

yes

characteristic

#124×C₅₉

—

#121×C₅₉

—

#121×C₅₉

C₄

yes

—

C₄×D₄×C₅₉

—

C₄×D₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#104

yes

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

#1484

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#431

—

C₄×M₁₆

—

D₄⋊C₈

—

C₅₉

#1484

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

Dic₅₉

yes

characteristic

#127

Dic₅₉

yes

—

—

Dic₅₉

yes

—

C₄×D₄

—

C₄×D₄

C₂×D₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

118

—

#97

—

118

—

C₄×C₈

—

100

C₄×C₈

—

118

—

101

#41

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×Dic₅₉

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

109

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

110

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

111

M₁₆

—

118

—

112

C₂×C₈

—

118

—

113

C₂×C₈

—

118

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×Dic₅₉

yes

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

C₂×Dic₅₉

yes

characteristic

119

C₄²

C₂²⋊D₅₉

yes

characteristic

120

C₄²

C₂²⋊D₅₉

yes

characteristic

121

C₄²

C₂²×D₅₉

yes

characteristic

122

C₂³

—

123

C₂³

C₂²⋊Dic₅₉

yes

characteristic

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

135

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

136

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

137

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

140

C₈

—

118

—

141

C₈

—

236

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#14273654

yes

—

154

C₄

#14273734

yes

—

155

C₄

#14273753

yes

—

156

C₂

—

157

C₂

—

158

C₂

#16837369

yes

characteristic

159

C₂

#16837407

yes

characteristic

160

C₂

C₂×#14273654

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#532	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#532	D₅₉
hypocenter	C₅₉	#1484
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#532

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16837455	C₂	semidirect
2. C₄×#11745664	C₂	semidirect
3. D₄⋊(C₅₉⋊C₈)	C₂	semidirect
85. C₅₉	#1484	semidirect

H ≅ G/N

type

1. #16837455

C₂

semidirect

2. C₄×#11745664

C₂

semidirect

3. D₄⋊(C₅₉⋊C₈)

C₂

semidirect

85. C₅₉

#1484

semidirect

Subgroups — #21422513

Conjugacy classes of subgroups

Complements