Group #29596362 — subgroups

Subgroups — #29596362

Order 15616.

322 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 151 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×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
2	#21445505	C₂	1	1	yes	maximal, characteristic
3	#21443896	C₂	1	4	yes	maximal
4	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×#14519688	C₄	1	1	yes	characteristic
7	C₂×#14519687	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
9	#16839608	C₄	1	1	yes	characteristic
10	C₄⋊#11758185	—	2	2	no	—
11	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×#92418	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	—	2	4	no	—
18	#14519688	—	2	2	no	—
19	#14519687	—	2	4	no	—
20	C₈×Dic₆₁	—	2	2	no	—
21	C₈×Dic₆₁	—	2	2	no	—
22	#14519684	C₈	1	2	yes	—
23	#14519684	C₈	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	—	2	4	no	—
29	C₄×C₈×C₆₁	D₄	1	2	yes	—
30	C₄×C₈×C₆₁	Q₈	1	2	yes	—
31	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₆₁	#41	1	2	yes	—
40	C₂×C₈×C₆₁	#41	1	2	yes	—
41	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
42	C₄²×C₆₁	M₁₆	1	2	yes	—
43	C₄×Dic₆₁	—	2	2	no	—
44	C₄×Dic₆₁	—	2	2	no	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	—	2	2	no	—
47	C₂×#92418	M₁₆	1	4	yes	—
48	C₁₆×C₆₁	—	4	2	no	—
49	#11758185	—	4	2	no	—
50	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#117	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
59	C₂×Dic₆₁	—	2	4	no	—
60	C₂×Dic₆₁	—	2	4	no	—
61	C₂×Dic₆₁	—	2	4	no	—
62	C₈×C₆₁	—	2	2	no	—
63	C₈×C₆₁	#124	1	1	yes	characteristic
64	C₈×C₆₁	#124	1	1	yes	characteristic
65	C₈×C₆₁	#125	1	2	yes	—
66	#92418	—	2	4	no	—
67	#12880	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₆₁	#431	1	2	yes	—
70	C₂²×C₆₁	#504	1	2	yes	—
71	C₂²×C₆₁	#504	1	2	yes	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	—	2	2	no	—
74	C₄×C₆₁	#382	1	1	yes	characteristic
75	C₄×C₆₁	#423	1	1	yes	characteristic
76	C₄×C₆₁	#432	1	2	yes	—
77	Dic₆₁	—	4	4	no	—
78	C₂×C₄⋊C₁₆	—	61	1	no	—
79	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
80	#1726	—	61	1	no	—
81	#1374	—	61	4	no	—
82	C₂×C₆₁	#1831	1	1	yes	characteristic
83	C₂×C₆₁	#2151	1	1	yes	characteristic
84	C₂×C₆₁	#2151	1	1	yes	characteristic
85	C₂×C₆₁	#2164	1	4	yes	—
86	C₂²×C₁₆	—	61	2	no	—
87	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
88	C₂×#97	—	61	2	no	—
89	C₂×#97	—	61	1	no	—
90	C₂×C₄×C₈	—	61	2	no	—
91	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
92	#376	—	61	1	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	C₄⋊C₁₆	—	122	2	no	—
95	C₆₁	#12880	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	61	2	no	—
97	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
99	C₂×C₄²	—	61	2	no	—
100	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	122	4	no	—
104	C₂×C₁₆	—	122	2	no	—
105	#97	—	122	4	no	—
106	#97	—	122	2	no	—
107	C₄×C₈	—	122	2	no	—
108	C₄×C₈	—	122	2	no	—
109	C₄×C₈	Dic₁₂₂	1	2	yes	—
110	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
111	#95	—	61	2	no	—
112	#95	—	61	2	no	—
113	C₂²×C₄	—	61	4	no	—
114	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
115	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	61	4	no	—
119	C₂×C₈	—	122	2	no	—
120	C₂×C₈	—	122	2	no	—
121	C₂×C₈	#11758196	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
123	C₂×C₈	#11758224	1	2	yes	—
124	C₂×C₈	#11758226	1	2	yes	—
125	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
126	C₄²	—	122	2	no	—
127	C₄²	—	122	2	no	—
128	C₄²	#11758189	1	2	yes	—
129	C₄²	#11758196	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	244	2	no	—
132	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	122	4	no	—
136	C₂×C₄	—	122	4	no	—
137	C₂×C₄	—	122	4	no	—
138	C₂×C₄	#14519706	1	2	yes	—
139	C₂×C₄	#14519706	1	2	yes	—
140	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
141	C₂×C₄	#14519735	1	2	yes	—
142	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
143	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	122	4	no	—
146	C₈	#14519764	1	1	yes	characteristic
147	C₈	#14519766	1	1	yes	characteristic
148	C₈	#14519768	1	2	yes	—
149	C₂²	C₂×#14519706	1	1	yes	characteristic
150	C₂²	#16839814	1	2	yes	—
151	C₂²	#16840178	1	2	yes	—
152	C₂²	#16840180	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	244	4	no	—
156	C₄	#16839627	1	1	yes	characteristic
157	C₄	#16839785	1	1	yes	characteristic
158	C₄	#16839818	1	2	yes	—
159	C₂	#21445935	1	1	yes	characteristic
160	C₂	#21447577	1	1	yes	characteristic
161	C₂	#21447579	1	1	yes	characteristic
162	C₂	#21447625	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11758185

C₂

yes

maximal, characteristic

#21445505

C₂

yes

maximal, characteristic

#21443896

C₂

yes

maximal

C₂×C₆₁×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

—

C₂×#14519688

C₄

yes

characteristic

C₂×#14519687

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

—

#16839608

C₄

yes

characteristic

C₄⋊#11758185

—

C₂²×C₁₆×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₆₁

—

C₈×Dic₆₁

—

#14519684

C₈

yes

—

#14519684

C₈

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

M₁₆

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

#92418

—

#12880

—

maximal, Sylow

C₂²×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₁

#431

yes

—

C₂²×C₆₁

#504

yes

—

C₂²×C₆₁

#504

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

D₆₁

yes

characteristic

#1726

—

#1374

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₁

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

C₆₁

#12880

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₆₁

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₄×D₆₁

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

122

—

104

C₂×C₁₆

—

122

—

105

#97

—

122

—

106

#97

—

122

—

107

C₄×C₈

—

122

—

108

C₄×C₈

—

122

—

109

C₄×C₈

Dic₁₂₂

yes

—

110

C₄×C₈

C₂²⋊D₆₁

yes

—

111

#95

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₈×D₆₁

yes

characteristic

115

C₂²×C₄

C₈×D₆₁

yes

characteristic

116

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₂×C₈

#11758196

yes

—

122

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

123

C₂×C₈

#11758224

yes

—

124

C₂×C₈

#11758226

yes

—

125

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

126

C₄²

—

122

—

127

C₄²

—

122

—

128

C₄²

#11758189

yes

—

129

C₄²

#11758196

yes

—

130

C₁₆

—

131

C₁₆

—

244

—

132

C₂³

C₂×C₈×D₆₁

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

122

—

136

C₂×C₄

—

122

—

137

C₂×C₄

—

122

—

138

C₂×C₄

#14519706

yes

—

139

C₂×C₄

#14519706

yes

—

140

C₂×C₄

C₂×#11758196

yes

characteristic

141

C₂×C₄

#14519735

yes

—

142

C₂×C₄

M₁₆×D₆₁

yes

—

143

C₂×C₄

C₂×#11758189

yes

characteristic

144

C₈

—

145

C₈

—

122

—

146

C₈

#14519764

yes

characteristic

147

C₈

#14519766

yes

characteristic

148

C₈

#14519768

yes

—

149

C₂²

C₂×#14519706

yes

characteristic

150

C₂²

#16839814

yes

—

151

C₂²

#16840178

yes

—

152

C₂²

#16840180

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

244

—

156

C₄

#16839627

yes

characteristic

157

C₄

#16839785

yes

characteristic

158

C₄

#16839818

yes

—

159

C₂

#21445935

yes

characteristic

160

C₂

#21447577

yes

characteristic

161

C₂

#21447579

yes

characteristic

162

C₂

#21447625

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂²×C₈
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#12880
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₁

commutator, socle, prime radical

C₂³×C₆₁

C₂²×C₈

Fitting

C₂×C₆₁×C₄⋊C₁₆

C₂

cosocle

C₂²×C₈×C₆₁

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
95. C₆₁	#12880	semidirect

H ≅ G/N

type

95. C₆₁

#12880

semidirect

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

subgroup	index
C₂×C₄⋊#11758185	2
#21445505	2
#21443896	2
C₂×C₆₁×C₄⋊C₁₆	2
#12880	61

subgroup

index

C₂×C₆₁×C₄⋊C₁₆

Subgroups — #29596362

Conjugacy classes of subgroups

Complements

Maximal subgroups