Group #29596398 — subgroups

Subgroups — #29596398

Order 15616.

318 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16839538	C₂	1	1	yes	maximal, characteristic
2	#21445979	C₂	1	1	yes	maximal, characteristic
3	#21443910	C₂	1	4	yes	maximal
4	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×#14519711	—	2	1	no	—
7	C₂×#14519711	C₂²	1	2	yes	—
8	#16839632	—	2	1	no	—
9	#16839538	C₄	1	4	yes	complemented
10	C₂×#107×C₆₁	C₂²	1	2	yes	—
11	C₂×#106×C₆₁	—	2	1	no	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#382	—	2	1	no	—
14	C₂×C₄⋊Dic₆₁	—	2	2	no	—
15	C₂×C₄⋊Dic₆₁	—	4	2	no	—
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	—	2	2	no	—
18	C₂×#11758185	C₂×C₄	1	4	yes	—
19	#14519711	—	2	4	no	—
20	#14519711	—	4	2	no	—
21	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
25	#107×C₆₁	—	2	4	no	—
26	#106×C₆₁	—	4	2	no	—
27	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₁	Q₈	1	2	yes	—
29	#95×C₆₁	—	4	2	no	—
30	C₂²×C₄×C₆₁	—	4	2	no	—
31	C₂²×C₄×C₆₁	D₈	1	2	yes	—
32	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₆₁	—	4	2	no	—
34	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
36	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
37	C₂×C₈×C₆₁	C₂²×C₄	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	C₄⋊C₄×C₆₁	—	4	2	no	—
42	C₄⋊C₄×C₆₁	—	4	2	no	—
43	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₄²×C₆₁	Q₁₆	1	2	yes	—
45	C₄⋊Dic₆₁	—	4	2	no	—
46	C₄⋊Dic₆₁	—	4	2	no	—
47	C₄⋊Dic₆₁	—	8	4	no	—
48	#11758185	—	2	4	no	—
49	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	8	2	no	—
53	C₂×C₄×C₆₁	—	8	2	no	—
54	C₂×C₄×C₆₁	—	8	2	no	—
55	C₂×C₄×C₆₁	#107	1	4	yes	—
56	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×Dic₆₁	—	8	2	no	—
63	C₂×Dic₆₁	—	8	4	no	—
64	C₈×C₆₁	#106	1	4	yes	—
65	C₈×C₆₁	#117	1	2	yes	—
66	C₈×C₆₁	#125	1	2	yes	—
67	#12890	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
69	C₂²×C₆₁	#486	1	1	yes	characteristic
70	C₂²×C₆₁	#486	1	1	yes	characteristic
71	C₂²×C₆₁	#509	1	2	yes	—
72	C₂²×C₆₁	#510	1	2	yes	—
73	C₄×C₆₁	—	2	4	no	—
74	C₄×C₆₁	—	16	2	no	—
75	C₄×C₆₁	#382	1	1	yes	characteristic
76	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
77	C₄×C₆₁	#487	1	1	yes	characteristic
78	C₄×C₆₁	#487	1	1	yes	characteristic
79	Dic₆₁	—	16	2	no	—
80	C₂×#346	—	61	1	no	—
81	#1846	D₆₁	1	1	yes	characteristic
82	#1845	—	61	1	no	—
83	#1380	—	61	4	no	—
84	C₂×C₆₁	#1834	1	1	yes	characteristic
85	C₂×C₆₁	#2159	1	2	yes	—
86	C₂×C₆₁	#2243	1	2	yes	—
87	C₂×C₆₁	#2244	1	1	yes	characteristic
88	C₂×C₆₁	#2245	1	1	yes	characteristic
89	C₂²×C₁₆	—	61	2	no	—
90	C₂×#107	—	61	2	no	—
91	C₂×#107	—	122	1	no	—
92	C₂×#107	C₂×D₆₁	1	2	yes	—
93	C₂×#106	—	2	1	no	—
94	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
95	#384	—	122	1	no	—
96	#382	—	2	1	no	—
97	#346	—	61	4	no	—
98	C₆₁	#12890	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	122	2	no	—
103	C₂×C₄⋊C₄	—	244	2	no	—
104	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₂×C₁₆	—	61	4	no	—
106	C₂×C₁₆	—	122	2	no	—
107	C₂×C₁₆	—	122	2	no	—
108	#107	—	2	4	no	—
109	#107	—	122	4	no	—
110	#107	—	244	2	no	—
111	#106	—	4	2	no	—
112	C₄×C₈	Dic₁₂₂	1	2	yes	—
113	C₄×C₈	C₄×D₆₁	1	2	yes	—
114	#95	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	244	2	no	—
117	C₂²×C₄	#11758199	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
119	C₂×C₈	#11758196	1	2	yes	—
120	C₂×C₈	#11758200	1	2	yes	—
121	C₂×C₈	#11758201	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
124	C₂×C₈	#11758224	1	2	yes	—
125	C₂×C₈	#11758226	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	488	4	no	—
129	C₄⋊C₄	—	244	2	no	—
130	C₄⋊C₄	—	244	2	no	—
131	C₄²	#11758196	1	2	yes	—
132	C₄²	#11758202	1	2	yes	—
133	C₁₆	—	122	4	no	—
134	C₂³	C₂×#11758199	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	488	4	no	—
141	C₂×C₄	—	488	2	no	—
142	C₂×C₄	#14519710	1	4	yes	—
143	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
144	C₂×C₄	#14519764	1	1	yes	characteristic
145	C₂×C₄	#14519766	1	1	yes	characteristic
146	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
147	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
148	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
149	C₈	#14519708	1	4	yes	—
150	C₈	#14519735	1	2	yes	—
151	C₈	#14519768	1	2	yes	—
152	C₂²	C₂×#14519710	1	1	yes	characteristic
153	C₂²	#16840108	1	1	yes	characteristic
154	C₂²	#16840110	1	1	yes	characteristic
155	C₂²	#16840197	1	2	yes	—
156	C₂²	#16840202	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	976	2	no	—
160	C₄	#16839627	1	1	yes	characteristic
161	C₄	C₂×#14519708	1	1	yes	characteristic
162	C₄	#16840112	1	1	yes	characteristic
163	C₄	#16840114	1	1	yes	characteristic
164	C₂	#21445946	1	1	yes	characteristic
165	C₂	#21447608	1	2	yes	—
166	C₂	#21448055	1	2	yes	—
167	C₂	#21448062	1	1	yes	characteristic
168	C₂	#21448066	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839538

C₂

yes

maximal, characteristic

#21445979

C₂

yes

maximal, characteristic

#21443910

C₂

yes

maximal

C₆₁×#1846

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

—

C₂×#14519711

—

C₂×#14519711

C₂²

yes

—

#16839632

—

#16839538

C₄

yes

complemented

C₂×#107×C₆₁

C₂²

yes

—

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#382

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×#11758185

—

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

—

#14519711

—

#14519711

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#107×C₆₁

—

#106×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

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⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

Q₁₆

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758185

—

C₂³×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#107

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

#117

yes

—

C₈×C₆₁

#125

yes

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#510

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

C₂×#346

—

#1846

D₆₁

yes

characteristic

#1845

—

#1380

—

C₂×C₆₁

#1834

yes

characteristic

C₂×C₆₁

#2159

yes

—

C₂×C₆₁

#2243

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

—

122

—

C₂×#107

C₂×D₆₁

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#384

—

122

—

#382

—

#346

—

C₆₁

#12890

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

100

C₂²×C₈

C₂²×D₆₁

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

122

—

103

C₂×C₄⋊C₄

—

244

—

104

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

105

C₂×C₁₆

—

106

C₂×C₁₆

—

122

—

107

C₂×C₁₆

—

122

—

108

#107

—

109

#107

—

122

—

110

#107

—

244

—

111

#106

—

112

C₄×C₈

Dic₁₂₂

yes

—

113

C₄×C₈

C₄×D₆₁

yes

—

114

#95

—

115

C₂²×C₄

—

116

C₂²×C₄

—

244

—

117

C₂²×C₄

#11758199

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

119

C₂×C₈

#11758196

yes

—

120

C₂×C₈

#11758200

yes

—

121

C₂×C₈

#11758201

yes

—

122

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

123

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

124

C₂×C₈

#11758224

yes

—

125

C₂×C₈

#11758226

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

488

—

129

C₄⋊C₄

—

244

—

130

C₄⋊C₄

—

244

—

131

C₄²

#11758196

yes

—

132

C₄²

#11758202

yes

—

133

C₁₆

—

122

—

134

C₂³

C₂×#11758199

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

488

—

141

C₂×C₄

—

488

—

142

C₂×C₄

#14519710

yes

—

143

C₂×C₄

C₂×#11758196

yes

characteristic

144

C₂×C₄

#14519764

yes

characteristic

145

C₂×C₄

#14519766

yes

characteristic

146

C₂×C₄

C₂×#11758201

yes

characteristic

147

C₂×C₄

C₂×#11758200

yes

characteristic

148

C₂×C₄

C₂×#11758202

yes

characteristic

149

C₈

#14519708

yes

—

150

C₈

#14519735

yes

—

151

C₈

#14519768

yes

—

152

C₂²

C₂×#14519710

yes

characteristic

153

C₂²

#16840108

yes

characteristic

154

C₂²

#16840110

yes

characteristic

155

C₂²

#16840197

yes

—

156

C₂²

#16840202

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

976

—

160

C₄

#16839627

yes

characteristic

161

C₄

C₂×#14519708

yes

characteristic

162

C₄

#16840112

yes

characteristic

163

C₄

#16840114

yes

characteristic

164

C₂

#21445946

yes

characteristic

165

C₂

#21447608

yes

—

166

C₂

#21448055

yes

—

167

C₂

#21448062

yes

characteristic

168

C₂

#21448066

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1846	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1846	D₆₁
hypocenter	C₆₁	#12890
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11758199

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1846

C₂

socle, prime radical

C₂³×C₆₁

C₂×D₈

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. #16839538	C₄	semidirect
98. C₆₁	#12890	semidirect

H ≅ G/N

type

9. #16839538

C₄

semidirect

98. C₆₁

#12890

semidirect

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

subgroup	index
C₂×#16839538	2
#21445979	2
#21443910	2
C₆₁×#1846	2
#12890	61

subgroup

index

C₆₁×#1846

Subgroups — #29596398

Conjugacy classes of subgroups

Complements

Maximal subgroups