Group #29596401 — subgroups

Subgroups — #29596401

Order 15616.

282 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 119 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×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	#21445989	C₂	1	2	yes	maximal
3	#21443913	C₂	1	1	yes	maximal, characteristic
4	#21443912	C₂	1	2	yes	maximal
5	C₆₁×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂×#14519713	C₂²	1	2	yes	—
8	C₂×#14519712	—	2	2	no	—
9	#16839785	—	2	2	no	—
10	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₆₁	C₄	1	2	yes	—
12	C₁₆×Dic₆₁	C₄	1	2	yes	—
13	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₁	C₂²	1	2	yes	—
15	C₂×#11758194	—	2	2	no	—
16	C₂²×#92418	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
18	C₂×#11758185	—	2	2	no	—
19	C₂×#11758185	C₂×C₄	1	2	yes	—
20	#14519713	—	2	2	no	—
21	#14519713	—	2	2	no	—
22	#14519712	—	4	4	no	—
23	#14519706	—	4	4	no	—
24	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
27	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	—	2	2	no	—
31	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
32	#108×C₆₁	—	2	2	no	—
33	#108×C₆₁	—	2	2	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₆₁	Q₁₆	1	2	yes	—
36	M₁₆×C₆₁	—	4	2	no	—
37	M₁₆×C₆₁	—	4	2	no	—
38	C₂×C₈×C₆₁	—	4	2	no	—
39	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	#41	1	2	yes	—
42	C₂×C₈×C₆₁	#41	1	2	yes	—
43	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₆₁	D₈	1	2	yes	—
46	#11758194	—	4	2	no	—
47	#11758194	—	4	2	no	—
48	C₂×#92418	—	4	2	no	—
49	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#92418	SD₁₆	1	4	yes	—
52	C₁₆×C₆₁	—	2	2	no	—
53	#11758185	—	2	2	no	—
54	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#124	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
59	C₂×Dic₆₁	—	2	4	no	—
60	C₂×Dic₆₁	#107	1	2	yes	—
61	C₂×Dic₆₁	#107	1	2	yes	—
62	C₈×C₆₁	—	8	2	no	—
63	C₈×C₆₁	#117	1	1	yes	characteristic
64	C₈×C₆₁	#117	1	1	yes	characteristic
65	C₈×C₆₁	#125	1	2	yes	—
66	#92418	—	8	2	no	—
67	#92418	#106	1	2	yes	—
68	#92418	#106	1	2	yes	—
69	#12891	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
72	C₂²×C₆₁	#487	1	2	yes	—
73	C₄×C₆₁	#382	1	1	yes	characteristic
74	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
75	C₄×C₆₁	#486	1	2	yes	—
76	Dic₆₁	—	2	2	no	—
77	Dic₆₁	#368	1	4	yes	—
78	C₂×C₄×C₁₆	—	61	1	no	—
79	#1848	—	61	2	no	—
80	#1381	—	61	1	no	—
81	#1381	—	61	2	no	—
82	#1381	D₆₁	1	1	yes	characteristic
83	C₂×C₆₁	—	2	2	no	—
84	C₂×C₆₁	#1834	1	1	yes	characteristic
85	C₂×C₆₁	C₂×#368	1	2	yes	—
86	C₂²×C₁₆	—	61	1	no	—
87	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
88	C₂×#108	—	61	2	no	—
89	C₂×#108	—	122	2	no	—
90	C₂×#108	C₂×D₆₁	1	2	yes	—
91	#423	—	122	2	no	—
92	C₂×C₄×C₈	—	61	1	no	—
93	C₄×C₁₆	—	61	2	no	—
94	C₄×C₁₆	—	61	2	no	—
95	C₆₁	#12891	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	122	2	no	—
98	C₂²×C₈	—	61	1	no	—
99	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
100	C₂×C₄²	—	61	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	61	2	no	—
103	C₂×C₁₆	—	122	2	no	—
104	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
105	#108	—	2	2	no	—
106	#108	—	2	2	no	—
107	#108	—	122	2	no	—
108	#108	—	122	2	no	—
109	#108	—	244	4	no	—
110	C₄⋊C₈	—	244	4	no	—
111	C₄×C₈	—	61	1	no	—
112	C₄×C₈	—	61	1	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	—	61	1	no	—
115	C₂²×C₄	—	61	2	no	—
116	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	244	2	no	—
120	M₁₆	—	244	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	61	4	no	—
125	C₂×C₈	—	244	2	no	—
126	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
127	C₂×C₈	#11758222	1	2	yes	—
128	C₂×C₈	#11758224	1	2	yes	—
129	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
130	C₄²	—	61	1	no	—
131	C₄²	—	61	1	no	—
132	C₄²	—	61	2	no	—
133	C₁₆	—	2	2	no	—
134	C₁₆	—	122	2	no	—
135	C₂³	Q₁₆×D₆₁	1	1	yes	characteristic
136	C₂×C₄	—	61	2	no	—
137	C₂×C₄	—	61	2	no	—
138	C₂×C₄	—	122	4	no	—
139	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
140	C₂×C₄	#14519763	1	2	yes	—
141	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
142	C₂×C₄	SD₁₆×D₆₁	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	61	2	no	—
145	C₈	—	61	2	no	—
146	C₈	—	488	2	no	—
147	C₈	#14519733	1	1	yes	characteristic
148	C₈	#14519734	1	1	yes	characteristic
149	C₈	#14519767	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	#107×D₆₁	1	1	yes	characteristic
152	C₂²	#16840111	1	2	yes	—
153	C₄	—	61	4	no	—
154	C₄	—	122	2	no	—
155	C₄	#16839626	1	1	yes	characteristic
156	C₄	#106×D₆₁	1	1	yes	characteristic
157	C₄	#16840107	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#21445945	1	1	yes	characteristic
160	C₂	#368×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₆₁

C₂

yes

maximal, characteristic

#21445989

C₂

yes

maximal

#21443913

C₂

yes

maximal, characteristic

#21443912

C₂

yes

maximal

C₆₁×#1381

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

C₂×#14519712

—

#16839785

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

C₁₆×Dic₆₁

C₄

yes

—

C₁₆×Dic₆₁

C₄

yes

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₂×#11758194

—

C₂²×#92418

D₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

#108×C₆₁

—

#108×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

Q₁₆

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

D₈

yes

—

—

—

—

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

SD₁₆

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#107

yes

—

C₂×Dic₆₁

#107

yes

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

#92418

—

#92418

#106

yes

—

#92418

#106

yes

—

#12891

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#107

yes

characteristic

C₂²×C₆₁

#487

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

#368

yes

—

C₂×C₄×C₁₆

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1834

yes

characteristic

C₂×C₆₁

C₂×#368

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂×#108

—

C₂×#108

—

122

—

C₂×#108

C₂×D₆₁

yes

—

#423

—

122

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₆₁

#12891

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₆₁

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

122

—

104

C₂×C₁₆

C₄×D₆₁

yes

—

105

#108

—

106

#108

—

107

#108

—

122

—

108

#108

—

122

—

109

#108

—

244

—

110

C₄⋊C₈

—

244

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₂²×C₄

—

116

C₂²×C₄

D₄×D₆₁

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

244

—

120

M₁₆

—

244

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

244

—

126

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

127

C₂×C₈

#11758222

yes

—

128

C₂×C₈

#11758224

yes

—

129

C₂×C₈

Q₈×D₆₁

yes

characteristic

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₁₆

—

134

C₁₆

—

122

—

135

C₂³

Q₁₆×D₆₁

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

122

—

139

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

140

C₂×C₄

#14519763

yes

—

141

C₂×C₄

D₈×D₆₁

yes

characteristic

142

C₂×C₄

SD₁₆×D₆₁

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

488

—

147

C₈

#14519733

yes

characteristic

148

C₈

#14519734

yes

characteristic

149

C₈

#14519767

yes

—

150

C₂²

—

151

C₂²

#107×D₆₁

yes

characteristic

152

C₂²

#16840111

yes

—

153

C₄

—

154

C₄

—

122

—

155

C₄

#16839626

yes

characteristic

156

C₄

#106×D₆₁

yes

characteristic

157

C₄

#16840107

yes

—

158

C₂

—

159

C₂

#21445945

yes

characteristic

160

C₂

#368×D₆₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1381	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×C₆₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1381	D₆₁
hypocenter	C₆₁	#12891
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1381

C₂

socle

C₂²×C₆₁

C₂×#107

prime radical

C₂³×C₆₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
95. C₆₁	#12891	semidirect

H ≅ G/N

type

95. C₆₁

#12891

semidirect

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

subgroup	index
C₂×C₁₆×Dic₆₁	2
#21445989	2
#21443913	2
#21443912	2
C₆₁×#1381	2
#12891	61

subgroup

index

C₂×C₁₆×Dic₆₁

C₆₁×#1381

Subgroups — #29596401

Conjugacy classes of subgroups

Complements

Maximal subgroups