Group #29573689 — subgroups

Subgroups — #29573689

Order 15616.

342 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 147 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	#21445971	C₂	1	1	yes	maximal, characteristic
2	#21445565	C₂	1	1	yes	maximal, characteristic
3	#21443653	C₂	1	4	yes	maximal
4	C₆₁×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	—	2	2	no	—
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	2	yes	—
8	#16839517	C₄	1	2	yes	—
9	#16839517	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
11	C₄³×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#336	—	2	2	no	—
13	C₂²×#92418	—	2	2	no	—
14	C₂²×#92418	C₂×C₄	1	4	yes	—
15	C₄⋊#92418	—	2	4	no	—
16	C₄⋊#92418	—	4	4	no	—
17	C₄×#92418	—	2	2	no	—
18	C₄×#92418	—	2	2	no	—
19	C₂²×C₈×C₆₁	—	2	2	no	—
20	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	D₄	1	2	yes	—
23	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
24	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₆₁	—	2	4	no	—
26	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₆₁	—	4	2	no	—
32	C₂×C₈×C₆₁	—	4	2	no	—
33	C₂×C₈×C₆₁	—	4	2	no	—
34	C₄²×C₆₁	—	2	4	no	—
35	C₄²×C₆₁	—	2	2	no	—
36	C₄²×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	—	2	2	no	—
38	C₄²×C₆₁	—	2	2	no	—
39	C₄²×C₆₁	#41	1	2	yes	—
40	C₄²×C₆₁	#41	1	2	yes	—
41	C₂×#92418	—	2	4	no	—
42	C₂×#92418	—	2	4	no	—
43	C₂×#92418	—	2	4	no	—
44	C₂×#92418	—	4	2	no	—
45	C₂×#92418	—	4	4	no	—
46	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	#108	1	4	yes	—
54	C₂×C₄×C₆₁	#117	1	2	yes	—
55	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#125	1	2	yes	—
58	C₂×C₄×C₆₁	#131	1	2	yes	—
59	C₂×C₄×C₆₁	#131	1	2	yes	—
60	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄×C₆₁	#137	1	2	yes	—
62	C₈×C₆₁	—	8	2	no	—
63	#92418	—	4	4	no	—
64	#92418	—	8	2	no	—
65	#8789	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	#382	1	1	yes	characteristic
67	C₂²×C₆₁	#424	1	1	yes	characteristic
68	C₂²×C₆₁	#424	1	1	yes	characteristic
69	C₂²×C₆₁	#428	1	2	yes	—
70	C₂²×C₆₁	C₂×#108	1	2	yes	—
71	C₄×C₆₁	—	2	4	no	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	—	4	2	no	—
74	C₄×C₆₁	#433	1	4	yes	—
75	C₄×C₆₁	#489	1	2	yes	—
76	C₄×C₆₁	#490	1	1	yes	characteristic
77	C₄×C₆₁	#491	1	1	yes	characteristic
78	#1842	—	61	1	no	—
79	#1831	D₆₁	1	1	yes	characteristic
80	#1744	—	61	1	no	—
81	#1277	—	61	4	no	—
82	C₂×C₆₁	#1573	1	2	yes	—
83	C₂×C₆₁	#1574	1	2	yes	—
84	C₂×C₆₁	#1832	1	1	yes	characteristic
85	C₂×C₆₁	#1836	1	1	yes	characteristic
86	C₂×C₆₁	#1837	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	61	2	no	—
88	C₂×C₄⋊C₈	—	122	2	no	—
89	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
90	C₂×C₄×C₈	—	61	2	no	—
91	C₄³	C₂×D₆₁	1	1	yes	characteristic
92	#336	—	2	2	no	—
93	#336	—	61	2	no	—
94	#336	—	61	2	no	—
95	C₆₁	#8789	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	61	4	no	—
98	C₂²×C₈	—	122	2	no	—
99	C₂×C₄²	Dic₁₂₂	1	2	yes	—
100	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
101	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
103	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	122	4	no	—
106	C₄⋊C₈	—	244	4	no	—
107	C₄×C₈	—	122	2	no	—
108	C₄×C₈	—	122	2	no	—
109	C₂²×C₄	#11758196	1	2	yes	—
110	C₂²×C₄	#11758196	1	2	yes	—
111	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	122	4	no	—
118	C₂×C₈	—	122	4	no	—
119	C₂×C₈	—	122	4	no	—
120	C₂×C₈	—	244	2	no	—
121	C₂×C₈	—	244	4	no	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#11758224	1	2	yes	—
128	C₄²	#11758226	1	2	yes	—
129	C₂³	C₂×#11758196	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#14519712	1	4	yes	—
137	C₂×C₄	#14519735	1	2	yes	—
138	C₂×C₄	#14519764	1	1	yes	characteristic
139	C₂×C₄	#14519766	1	1	yes	characteristic
140	C₂×C₄	#14519768	1	2	yes	—
141	C₂×C₄	#14519783	1	2	yes	—
142	C₂×C₄	#14519785	1	2	yes	—
143	C₂×C₄	#14519802	1	2	yes	—
144	C₂×C₄	#14519807	1	2	yes	—
145	C₈	—	8	2	no	—
146	C₈	—	488	2	no	—
147	C₈	—	244	4	no	—
148	C₂²	#16839627	1	1	yes	characteristic
149	C₂²	#16839788	1	1	yes	characteristic
150	C₂²	#16839790	1	1	yes	characteristic
151	C₂²	#16839805	1	2	yes	—
152	C₂²	C₂×#14519712	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	4	no	—
155	C₄	—	4	2	no	—
156	C₄	#16839822	1	4	yes	—
157	C₄	#16840121	1	2	yes	—
158	C₄	#16840128	1	1	yes	characteristic
159	C₄	#16840132	1	1	yes	characteristic
160	C₂	#21444605	1	2	yes	—
161	C₂	#21444607	1	2	yes	—
162	C₂	#21445938	1	1	yes	characteristic
163	C₂	#21445952	1	1	yes	characteristic
164	C₂	#21445956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445971

C₂

yes

maximal, characteristic

#21445565

C₂

yes

maximal, characteristic

#21443653

C₂

yes

maximal

C₆₁×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

—

C₂×C₄⋊#92418

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

—

#16839517

C₄

yes

—

#16839517

C₄

yes

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₆₁×#336

—

C₂²×#92418

—

C₂²×#92418

C₂×C₄

yes

—

—

—

—

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

Q₈

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

#41

yes

—

C₄²×C₆₁

#41

yes

—

—

—

—

—

—

C₂³×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#108

yes

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#108

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

#489

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

—

D₆₁

yes

characteristic

#1744

—

#1277

—

C₂×C₆₁

#1573

yes

—

C₂×C₆₁

#1574

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

122

—

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₆₁

yes

—

—

—

C₆₁

#8789

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

122

—

C₂×C₄²

Dic₁₂₂

yes

—

100

C₂×C₄²

C₄×D₆₁

yes

characteristic

101

C₂×C₄²

C₄×D₆₁

yes

characteristic

102

C₂×C₄²

C₂²⋊D₆₁

yes

—

103

C₂×C₄²

C₂²×D₆₁

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

122

—

106

C₄⋊C₈

—

244

—

107

C₄×C₈

—

122

—

108

C₄×C₈

—

122

—

109

C₂²×C₄

#11758196

yes

—

110

C₂²×C₄

#11758196

yes

—

111

C₂²×C₄

C₂×Dic₁₂₂

yes

characteristic

112

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

113

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

122

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

244

—

121

C₂×C₈

—

244

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11758224

yes

—

128

C₄²

#11758226

yes

—

129

C₂³

C₂×#11758196

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14519712

yes

—

137

C₂×C₄

#14519735

yes

—

138

C₂×C₄

#14519764

yes

characteristic

139

C₂×C₄

#14519766

yes

characteristic

140

C₂×C₄

#14519768

yes

—

141

C₂×C₄

#14519783

yes

—

142

C₂×C₄

#14519785

yes

—

143

C₂×C₄

#14519802

yes

—

144

C₂×C₄

#14519807

yes

—

145

C₈

—

146

C₈

—

488

—

147

C₈

—

244

—

148

C₂²

#16839627

yes

characteristic

149

C₂²

#16839788

yes

characteristic

150

C₂²

#16839790

yes

characteristic

151

C₂²

#16839805

yes

—

152

C₂²

C₂×#14519712

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#16839822

yes

—

157

C₄

#16840121

yes

—

158

C₄

#16840128

yes

characteristic

159

C₄

#16840132

yes

characteristic

160

C₂

#21444605

yes

—

161

C₂

#21444607

yes

—

162

C₂

#21445938

yes

characteristic

163

C₂

#21445952

yes

characteristic

164

C₂

#21445956

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1831	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1831	D₆₁
hypocenter	C₆₁	#8789
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1831

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
95. C₆₁	#8789	semidirect

H ≅ G/N

type

95. C₆₁

#8789

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21445971	2
#21445565	2
#21443653	2
C₆₁×#1831	2
#8789	61

subgroup

index

C₆₁×#1831

Subgroups — #29573689

Conjugacy classes of subgroups

Complements

Maximal subgroups