Group #29566906 — subgroups

Subgroups — #29566906

Order 15616.

212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
2	#21444108	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	—	2	1	no	—
5	#16839815	C₄	1	1	yes	characteristic
6	C₄×#11758194	C₄	1	1	yes	characteristic
7	#16839538	—	2	1	no	—
8	C₄×#11758185	C₄	1	2	yes	—
9	#16839484	—	2	1	no	—
10	#16839483	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×C₄⋊C₁₆	C₄	1	2	yes	—
13	C₆₁×C₂²⋊C₁₆	—	2	1	no	—
14	C₆₁×#346	—	2	1	no	—
15	C₂×#11758194	—	2	1	no	—
16	#14519717	—	2	4	no	—
17	C₂×#11758185	—	2	2	no	—
18	C₄⋊#92418	—	2	1	no	—
19	C₂²⋊#92418	C₈	1	2	yes	—
20	#14519688	C₈	1	2	yes	—
21	C₄×#92418	—	2	1	no	—
22	C₄×#92418	—	2	1	no	—
23	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	—	2	2	no	—
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₄²	1	1	yes	characteristic
41	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
44	#11758194	—	4	2	no	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	—	2	2	no	—
47	C₁₆×C₆₁	—	4	2	no	—
48	#11758185	—	2	4	no	—
49	C₂³×C₆₁	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	4	1	no	—
54	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	—	2	1	no	—
62	C₈×C₆₁	#104	1	2	yes	—
63	C₈×C₆₁	#108	1	2	yes	—
64	#92418	—	4	2	no	—
65	#92418	—	4	2	no	—
66	#7164	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	2	1	no	—
68	C₂²×C₆₁	—	2	1	no	—
69	C₂²×C₆₁	—	2	1	no	—
70	C₂²×C₆₁	#336	1	1	yes	characteristic
71	C₄×C₆₁	—	2	1	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	#339	1	1	yes	characteristic
76	C₄×C₆₁	#341	1	1	yes	characteristic
77	C₄×C₆₁	#347	1	2	yes	—
78	#2163	D₆₁	1	1	yes	characteristic
79	C₄×M₃₂	—	61	1	no	—
80	#1451	—	61	1	no	—
81	C₂×C₆₁	—	4	1	no	—
82	C₂×C₆₁	#1278	1	1	yes	characteristic
83	C₂×C₆₁	#1375	1	1	yes	characteristic
84	C₂×C₆₁	#1378	1	1	yes	characteristic
85	C₂×M₃₂	—	122	1	no	—
86	#431	—	61	1	no	—
87	C₄×M₁₆	—	61	1	no	—
88	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
89	C₄⋊C₁₆	Dic₆₁	1	2	yes	—
90	C₂²⋊C₁₆	—	2	1	no	—
91	#346	—	2	1	no	—
92	#346	—	122	1	no	—
93	C₄×C₁₆	—	61	2	no	—
94	#322	—	122	1	no	—
95	#322	—	122	1	no	—
96	C₆₁	#7164	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	122	1	no	—
98	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
100	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
101	M₃₂	—	122	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	122	2	no	—
104	C₄⋊C₈	—	122	1	no	—
105	C₂²⋊C₈	—	61	2	no	—
106	#97	—	61	2	no	—
107	C₄×C₈	—	122	1	no	—
108	C₄×C₈	—	122	1	no	—
109	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
110	C₄×C₈	D₂₄₄	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
113	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
114	C₂²×C₄	#11758189	1	1	yes	characteristic
115	C₂²×C₄	#11758196	1	1	yes	characteristic
116	M₁₆	—	244	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	122	2	no	—
120	C₂×C₈	—	122	2	no	—
121	C₂×C₈	#11758196	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
123	C₂×C₈	#11758198	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
125	C₄²	C₈×D₆₁	1	1	yes	characteristic
126	C₄²	#11758189	1	1	yes	characteristic
127	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
128	C₄²	#11758198	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	C₁₆	—	122	4	no	—
131	C₂³	#14519706	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	#14519684	1	1	yes	characteristic
137	C₂×C₄	C₈×Dic₆₁	1	1	yes	characteristic
138	C₂×C₄	#14519687	1	1	yes	characteristic
139	C₂×C₄	#14519689	1	1	yes	characteristic
140	C₂×C₄	#14519689	1	1	yes	characteristic
141	C₂×C₄	#14519706	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	244	2	no	—
145	C₈	—	244	2	no	—
146	C₈	#14519704	1	2	yes	—
147	C₈	#14519712	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#16839516	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#16839521	1	1	yes	characteristic
157	C₄	#16839526	1	1	yes	characteristic
158	C₄	#16839539	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#21443655	1	1	yes	characteristic
161	C₂	#21443898	1	1	yes	characteristic
162	C₂	#21443906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519717

C₂

yes

maximal, characteristic

#21444108

C₂

yes

maximal, characteristic

C₆₁×#2163

C₂

yes

maximal, characteristic

C₂×#14519717

—

#16839815

C₄

yes

characteristic

C₄×#11758194

C₄

yes

characteristic

#16839538

—

C₄×#11758185

C₄

yes

—

#16839484

—

#16839483

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

C₄

yes

—

C₆₁×C₂²⋊C₁₆

—

C₆₁×#346

—

—

—

—

—

C₈

yes

—

#14519688

C₈

yes

—

C₄×#92418

—

C₄×#92418

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

Q₈

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

M₁₆

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

—

—

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

#97

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₈

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#104

yes

—

C₈×C₆₁

#108

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#336

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2163

D₆₁

yes

characteristic

C₄×M₃₂

—

#1451

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×M₃₂

—

122

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

Dic₆₁

yes

—

C₂²⋊C₁₆

—

#346

—

#346

—

122

—

C₄×C₁₆

—

#322

—

122

—

#322

—

122

—

C₆₁

#7164

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

Dic₁₂₂

yes

characteristic

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

100

C₂×C₄²

C₄×D₆₁

yes

characteristic

101

M₃₂

—

122

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

122

—

104

C₄⋊C₈

—

122

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

—

122

—

108

C₄×C₈

—

122

—

109

C₄×C₈

C₄×D₆₁

yes

characteristic

110

C₄×C₈

D₂₄₄

yes

characteristic

111

C₄×C₈

C₂×Dic₆₁

yes

characteristic

112

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

113

C₂²×C₄

C₈×D₆₁

yes

characteristic

114

C₂²×C₄

#11758189

yes

characteristic

115

C₂²×C₄

#11758196

yes

characteristic

116

M₁₆

—

244

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₂×C₈

#11758196

yes

characteristic

122

C₂×C₈

C₄⋊Dic₆₁

yes

characteristic

123

C₂×C₈

#11758198

yes

characteristic

124

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

125

C₄²

C₈×D₆₁

yes

characteristic

126

C₄²

#11758189

yes

characteristic

127

C₄²

C₄×Dic₆₁

yes

characteristic

128

C₄²

#11758198

yes

characteristic

129

C₁₆

—

130

C₁₆

—

122

—

131

C₂³

#14519706

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14519684

yes

characteristic

137

C₂×C₄

C₈×Dic₆₁

yes

characteristic

138

C₂×C₄

#14519687

yes

characteristic

139

C₂×C₄

#14519689

yes

characteristic

140

C₂×C₄

#14519689

yes

characteristic

141

C₂×C₄

#14519706

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

244

—

145

C₈

—

244

—

146

C₈

#14519704

yes

—

147

C₈

#14519712

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#16839516

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#16839521

yes

characteristic

157

C₄

#16839526

yes

characteristic

158

C₄

#16839539

yes

—

159

C₂

—

160

C₂

#21443655

yes

characteristic

161

C₂

#21443898

yes

characteristic

162

C₂

#21443906

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519684
commutator	C₂×C₄×C₆₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#2163	C₂
socle	C₂²×C₆₁	#336
prime radical	C₂³×C₆₁	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#2163	D₆₁
hypocenter	C₆₁	#7164
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂×C₄

#14519684

commutator

C₂×C₄×C₆₁

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₆₁

Fitting

C₆₁×#2163

C₂

socle

C₂²×C₆₁

#336

prime radical

C₂³×C₆₁

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
96. C₆₁	#7164	semidirect

H ≅ G/N

type

96. C₆₁

#7164

semidirect

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

subgroup	index
C₄×#14519717	2
#21444108	2
C₆₁×#2163	2
#7164	61

subgroup

index

C₄×#14519717

#21444108

C₆₁×#2163

#7164

Subgroups — #29566906

Conjugacy classes of subgroups

Complements

Maximal subgroups