Group #21425496 — subgroups

Subgroups — #21425496

Order 7552.

196 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16837451	C₂	1	1	yes	maximal, characteristic
2	#16837450	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
6	C₅₉⋊#338	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#430	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	C₂×#11745664	C₂²	1	1	yes	characteristic
11	C₂×#11745664	C₂²	1	1	yes	characteristic
12	#14273663	—	2	2	no	—
13	#14273662	—	2	2	no	—
14	#14273645	—	2	2	no	—
15	#14273645	—	2	2	no	—
16	#14273645	—	2	2	no	—
17	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#108×C₅₉	—	2	2	no	—
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₉	Q₈	1	2	yes	—
23	C₂×Dic₁₁₈	—	2	1	no	—
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	1	no	—
27	C₂×Dic₁₁₈	—	2	1	no	—
28	C₂×Dic₁₁₈	—	2	1	no	—
29	M₁₆×C₅₉	—	2	1	no	—
30	M₁₆×C₅₉	—	2	1	no	—
31	M₁₆×C₅₉	—	2	1	no	—
32	M₁₆×C₅₉	—	2	1	no	—
33	M₁₆×C₅₉	D₄	1	2	yes	—
34	M₁₆×C₅₉	D₄	1	2	yes	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	#11745664	—	2	1	no	—
39	#11745664	—	2	1	no	—
40	#11745664	—	2	1	no	—
41	#11745664	—	2	1	no	—
42	#11745664	—	2	1	no	—
43	#11745664	D₄	1	2	yes	—
44	C₂×C₅₉⋊C₈	—	2	1	no	—
45	C₂×C₅₉⋊C₈	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	—	2	2	no	—
57	Dic₁₁₈	—	4	1	no	—
58	Dic₁₁₈	—	4	1	no	—
59	Dic₁₁₈	—	4	1	no	—
60	Dic₁₁₈	—	4	1	no	—
61	C₈×C₅₉	—	2	2	no	—
62	C₈×C₅₉	—	2	2	no	—
63	C₈×C₅₉	—	4	1	no	—
64	C₅₉⋊C₈	—	2	2	no	—
65	C₅₉⋊C₈	—	4	1	no	—
66	C₅₉⋊C₈	—	4	1	no	—
67	C₂²×C₅₉	—	4	1	no	—
68	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₅₉	#121	1	1	yes	characteristic
72	C₄×C₅₉	#121	1	1	yes	characteristic
73	C₄×C₅₉	#121	1	1	yes	characteristic
74	C₄×C₅₉	#126	1	1	yes	characteristic
75	Dic₅₉	—	4	2	no	—
76	#2070	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	—	2	1	no	—
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	—	2	1	no	—
80	C₂×C₅₉	#396	1	1	yes	characteristic
81	#430	—	59	1	no	—
82	#430	—	59	1	no	—
83	#430	D₅₉	1	1	yes	characteristic
84	C₂×#101	—	59	1	no	—
85	C₂×#101	—	59	1	no	—
86	C₂×#101	—	59	1	no	—
87	#338	—	59	1	no	—
88	C₅₉	#2070	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	59	1	no	—
90	C₂×M₁₆	—	59	1	no	—
91	C₂×M₁₆	—	59	1	no	—
92	C₂×M₁₆	—	59	1	no	—
93	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
94	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
95	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
96	#108	—	2	2	no	—
97	#108	—	118	2	no	—
98	#108	—	118	2	no	—
99	#101	—	118	2	no	—
100	#101	—	118	2	no	—
101	#101	—	118	2	no	—
102	C₂×Q₈	—	118	1	no	—
103	C₂×Q₈	—	118	1	no	—
104	C₂×Q₈	—	118	1	no	—
105	C₂×Q₈	—	118	1	no	—
106	C₂×Q₈	—	118	1	no	—
107	C₂×Q₈	—	118	1	no	—
108	C₂²×C₄	—	59	2	no	—
109	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	1	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	59	2	no	—
115	M₁₆	—	118	1	no	—
116	M₁₆	—	118	1	no	—
117	M₁₆	—	118	1	no	—
118	M₁₆	—	118	1	no	—
119	M₁₆	—	118	1	no	—
120	M₁₆	D₂₃₆	1	2	yes	—
121	M₁₆	C₂²⋊D₅₉	1	2	yes	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	118	1	no	—
126	C₂×C₈	—	118	1	no	—
127	C₂×C₈	—	118	1	no	—
128	C₂³	Q₈×D₅₉	1	1	yes	characteristic
129	Q₈	—	236	1	no	—
130	Q₈	—	236	1	no	—
131	Q₈	—	236	1	no	—
132	Q₈	—	236	1	no	—
133	C₂×C₄	—	118	2	no	—
134	C₂×C₄	—	118	2	no	—
135	C₂×C₄	—	118	2	no	—
136	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
137	C₂×C₄	#11745685	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	#11745687	1	1	yes	characteristic
140	C₂×C₄	#11745689	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	4	1	no	—
145	C₈	—	118	2	no	—
146	C₈	—	236	1	no	—
147	C₈	—	236	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#14273703	1	1	yes	characteristic
150	C₂²	#14273705	1	1	yes	characteristic
151	C₂²	#14273707	1	1	yes	characteristic
152	C₄	—	236	2	no	—
153	C₄	#14273699	1	1	yes	characteristic
154	C₄	#14273701	1	1	yes	characteristic
155	C₄	#14273702	1	1	yes	characteristic
156	C₄	#14273722	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#16837324	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837451

C₂

yes

maximal, characteristic

#16837450

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

C₅₉⋊#338

C₂

yes

maximal, characteristic

C₅₉×#430

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#108×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

Q₈

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

D₄

yes

—

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

—

—

—

—

—

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2070

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

—

D₅₉

yes

—

—

—

—

C₅₉

#2070

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

#108

—

#108

—

118

—

#108

—

118

—

#101

—

118

—

100

#101

—

118

—

101

#101

—

118

—

102

C₂×Q₈

—

118

—

103

C₂×Q₈

—

118

—

104

C₂×Q₈

—

118

—

105

C₂×Q₈

—

118

—

106

C₂×Q₈

—

118

—

107

C₂×Q₈

—

118

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₅₉

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

118

—

116

M₁₆

—

118

—

117

M₁₆

—

118

—

118

M₁₆

—

118

—

119

M₁₆

—

118

—

120

M₁₆

D₂₃₆

yes

—

121

M₁₆

C₂²⋊D₅₉

yes

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

118

—

126

C₂×C₈

—

118

—

127

C₂×C₈

—

118

—

128

C₂³

Q₈×D₅₉

yes

characteristic

129

Q₈

—

236

—

130

Q₈

—

236

—

131

Q₈

—

236

—

132

Q₈

—

236

—

133

C₂×C₄

—

118

—

134

C₂×C₄

—

118

—

135

C₂×C₄

—

118

—

136

C₂×C₄

C₂×D₂₃₆

yes

characteristic

137

C₂×C₄

#11745685

yes

characteristic

138

C₂×C₄

D₄×D₅₉

yes

characteristic

139

C₂×C₄

#11745687

yes

characteristic

140

C₂×C₄

#11745689

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

118

—

146

C₈

—

236

—

147

C₈

—

236

—

148

C₂²

—

149

C₂²

#14273703

yes

characteristic

150

C₂²

#14273705

yes

characteristic

151

C₂²

#14273707

yes

characteristic

152

C₄

—

236

—

153

C₄

#14273699

yes

characteristic

154

C₄

#14273701

yes

characteristic

155

C₄

#14273702

yes

characteristic

156

C₄

#14273722

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#16837324

yes

characteristic

	subgroup	G/H
center	C₂	#16837324
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#430	C₂
socle	C₂×C₅₉	#396
hypercenter	#430	D₅₉
hypocenter	C₅₉	#2070
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₂

#16837324

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#430

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
88. C₅₉	#2070	semidirect

H ≅ G/N

type

88. C₅₉

#2070

semidirect

Subgroups — #21425496

Conjugacy classes of subgroups

Complements