Group #21433193 — subgroups

Subgroups — #21433193

Order 7552.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#16837959	C₂	1	1	yes	maximal, characteristic
2	#16837530	C₂	1	4	yes	maximal, complemented
3	#16837529	C₂	1	2	yes	maximal, complemented
4	#16837526	C₂	1	2	yes	maximal, complemented
5	#16837460	C₂	1	4	yes	maximal
6	#16837455	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#432	C₂	1	1	yes	maximal, characteristic
8	#14273727	C₄	1	1	yes	characteristic
9	#14273727	C₄	1	1	yes	characteristic
10	#14273719	C₄	1	2	yes	—
11	#14273706	C₄	1	2	yes	—
12	#14273706	C₄	1	2	yes	—
13	C₄×Dic₁₁₈	C₂²	1	2	yes	—
14	C₄×Dic₁₁₈	C₂²	1	2	yes	—
15	#14273686	C₂²	1	2	yes	—
16	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
17	Dic₅₉⋊C₈	C₂²	1	4	yes	—
18	Dic₅₉⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
20	#14273637	C₂²	1	4	yes	—
21	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₅₉	C₂²	1	4	yes	—
24	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
26	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
27	#97×C₅₉	C₂²	1	2	yes	—
28	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
32	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₅₉	C₂³	1	4	yes	—
34	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₂²⋊Dic₅₉	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
40	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
43	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
44	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
45	C₄×Dic₅₉	C₂³	1	4	yes	—
46	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
47	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
48	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
59	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
60	Dic₁₁₈	—	2	4	no	—
61	Dic₁₁₈	—	4	2	no	—
62	C₈×C₅₉	—	2	4	no	—
63	C₅₉⋊C₈	—	2	2	no	—
64	C₅₉⋊C₈	—	2	2	no	—
65	C₂²×C₅₉	—	2	1	no	—
66	C₂²×C₅₉	—	2	1	no	—
67	C₂²×C₅₉	—	2	1	no	—
68	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	—	2	1	no	—
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	#131	1	2	yes	—
74	C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
75	Dic₅₉	—	2	4	no	—
76	Dic₅₉	—	2	4	no	—
77	#2985	—	59	1	no	maximal, Sylow
78	C₂×C₅₉	—	4	1	no	—
79	C₂×C₅₉	#519	1	1	yes	characteristic
80	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
81	C₂×C₅₉	#568	1	1	yes	characteristic
82	#533	—	59	1	no	—
83	#446	—	59	2	no	—
84	#446	—	59	4	no	—
85	C₈×Q₈	—	59	2	no	—
86	#433	—	59	4	no	—
87	#432	D₅₉	1	1	yes	characteristic
88	#431	—	59	1	no	—
89	C₅₉	#2985	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	59	1	no	—
91	C₄⋊Q₈	—	59	1	no	—
92	#125	—	59	2	no	—
93	C₂²⋊Q₈	—	59	2	no	—
94	C₂²⋊Q₈	—	59	2	no	—
95	C₄×Q₈	—	59	2	no	—
96	C₄×Q₈	—	59	2	no	—
97	#117	—	59	2	no	—
98	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
99	C₄⋊C₈	—	59	4	no	—
100	C₄⋊C₈	—	59	4	no	—
101	C₄⋊C₈	—	59	2	no	—
102	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
103	C₂²⋊C₈	—	59	2	no	—
104	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
105	#97	—	59	4	no	—
106	#97	C₂×D₅₉	1	2	yes	—
107	C₄×C₈	—	59	4	no	—
108	C₄×C₈	—	59	1	no	—
109	C₄×C₈	—	59	1	no	—
110	C₂×Q₈	—	59	2	no	—
111	C₂×Q₈	—	59	2	no	—
112	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
115	C₂×C₈	—	59	2	no	—
116	C₂×C₈	—	59	2	no	—
117	C₂×C₈	C₂²×D₅₉	1	4	yes	—
118	C₄⋊C₄	—	59	2	no	—
119	C₄⋊C₄	—	59	2	no	—
120	C₄⋊C₄	—	59	4	no	—
121	C₄⋊C₄	—	59	4	no	—
122	C₄⋊C₄	—	59	2	no	—
123	C₄⋊C₄	—	59	2	no	—
124	C₂²⋊C₄	—	59	4	no	—
125	C₄²	—	59	4	no	—
126	C₄²	C₄×D₅₉	1	1	yes	characteristic
127	C₄²	C₄×D₅₉	1	1	yes	characteristic
128	C₄²	C₂²×D₅₉	1	1	yes	characteristic
129	C₄²	C₂²×D₅₉	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
131	Q₈	—	118	4	no	—
132	Q₈	—	236	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	59	4	no	—
137	C₂×C₄	—	59	4	no	—
138	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	118	2	no	—
146	C₈	—	118	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	118	4	no	—
156	C₄	—	118	4	no	—
157	C₄	#14273735	1	2	yes	—
158	C₄	#14273774	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#16837879	1	1	yes	characteristic
161	C₂	C₂×#14273735	1	1	yes	characteristic
162	C₂	#16838168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

#16837530

C₂

yes

maximal, complemented

#16837529

C₂

yes

maximal, complemented

#16837526

C₂

yes

maximal, complemented

#16837460

C₂

yes

maximal

#16837455

C₂

yes

maximal, characteristic

C₅₉×#432

C₂

yes

maximal, characteristic

#14273727

C₄

yes

characteristic

#14273727

C₄

yes

characteristic

#14273719

C₄

yes

—

#14273706

C₄

yes

—

#14273706

C₄

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273637

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

C₂²

yes

—

#97×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

C₂×C₄

yes

—

Dic₅₉⋊C₄

C₂×C₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

C₂²×C₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

2₋¹⁺⁴

yes

—

Dic₅₉

—

Dic₅₉

—

#2985

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

C₈×Q₈

—

#433

—

#432

D₅₉

yes

characteristic

#431

—

C₅₉

#2985

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₅₉

yes

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

C₂×D₅₉

yes

—

105

#97

—

106

#97

C₂×D₅₉

yes

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

C₄×D₅₉

yes

characteristic

113

C₂²×C₄

C₄×D₅₉

yes

characteristic

114

C₂²×C₄

C₂²×D₅₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₅₉

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₅₉

yes

characteristic

127

C₄²

C₄×D₅₉

yes

characteristic

128

C₄²

C₂²×D₅₉

yes

characteristic

129

C₄²

C₂²×D₅₉

yes

characteristic

130

C₂³

C₂×C₄×D₅₉

yes

characteristic

131

Q₈

—

118

—

132

Q₈

—

236

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

139

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

141

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

142

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

143

C₂×C₄

C₂³×D₅₉

yes

characteristic

144

C₈

—

145

C₈

—

118

—

146

C₈

—

118

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₅₉

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

118

—

156

C₄

—

118

—

157

C₄

#14273735

yes

—

158

C₄

#14273774

yes

—

159

C₂

—

160

C₂

#16837879

yes

characteristic

161

C₂

C₂×#14273735

yes

characteristic

162

C₂

#16838168

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#432	C₂
hypercenter	#432	D₅₉
hypocenter	C₅₉	#2985
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₅₉×#432

C₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #16837530	C₂	semidirect
3. #16837529	C₂	semidirect
4. #16837526	C₂	semidirect
89. C₅₉	#2985	semidirect

H ≅ G/N

type

2. #16837530

C₂

semidirect

3. #16837529

C₂

semidirect

4. #16837526

C₂

semidirect

89. C₅₉

#2985

semidirect

Subgroups — #21433193

Conjugacy classes of subgroups

Complements