Group #29600159 — subgroups

Subgroups — #29600159

Order 15616.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21446567	C₂	1	1	yes	maximal, characteristic
2	#21445983	C₂	1	1	yes	maximal, characteristic
3	#21443925	C₂	1	2	yes	maximal
4	#21443918	C₂	1	2	yes	maximal
5	C₆₁×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	C₂²	1	1	yes	characteristic
7	#16839800	C₂²	1	1	yes	characteristic
8	C₂×#14519711	—	2	1	no	—
9	C₂×#14519709	C₂²	1	2	yes	—
10	#16839788	—	2	1	no	—
11	#16839628	—	2	1	no	—
12	#16839522	—	2	1	no	—
13	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#11758194	—	2	1	no	—
16	#14519736	—	2	1	no	—
17	C₂×C₄⋊Dic₆₁	—	2	2	no	—
18	#14519713	—	2	2	no	—
19	#14519711	—	4	2	no	—
20	#14519711	Q₈	1	2	yes	—
21	#14519709	—	2	1	no	—
22	#14519709	—	2	4	no	—
23	#14519706	—	4	2	no	—
24	#14519684	—	4	2	no	—
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
28	M₃₂×C₆₁	—	2	2	no	—
29	M₃₂×C₆₁	—	2	2	no	—
30	C₂×C₁₆×C₆₁	—	2	2	no	—
31	C₄×C₈×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	—	2	1	no	—
33	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₆₁	—	4	2	no	—
35	C₂×C₈×C₆₁	—	2	1	no	—
36	C₂×C₈×C₆₁	—	2	1	no	—
37	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
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	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
42	C₄²×C₆₁	—	2	1	no	—
43	C₄²×C₆₁	Q₁₆	1	2	yes	—
44	C₂²⋊Dic₆₁	—	4	1	no	—
45	C₄⋊Dic₆₁	—	2	2	no	—
46	C₄⋊Dic₆₁	—	4	2	no	—
47	C₄⋊Dic₆₁	—	4	2	no	—
48	C₄×Dic₆₁	—	4	1	no	—
49	#11758194	—	4	1	no	—
50	#11758194	—	4	1	no	—
51	C₂×#92418	—	4	1	no	—
52	C₁₆×C₆₁	—	4	2	no	—
53	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	4	2	no	—
56	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#123	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×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×Dic₆₁	—	4	2	no	—
63	C₂×Dic₆₁	—	8	4	no	—
64	C₂×Dic₆₁	—	8	2	no	—
65	C₈×C₆₁	—	2	2	no	—
66	C₈×C₆₁	#125	1	2	yes	—
67	C₈×C₆₁	#126	1	2	yes	—
68	#92418	—	8	1	no	—
69	#13542	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	#476	1	1	yes	characteristic
72	C₂²×C₆₁	#484	1	1	yes	characteristic
73	C₂²×C₆₁	#490	1	1	yes	characteristic
74	C₄×C₆₁	—	4	2	no	—
75	C₄×C₆₁	#400	1	1	yes	characteristic
76	C₄×C₆₁	#477	1	1	yes	characteristic
77	C₄×C₆₁	#481	1	1	yes	characteristic
78	C₄×C₆₁	#487	1	1	yes	characteristic
79	Dic₆₁	—	8	2	no	—
80	Dic₆₁	—	16	2	no	—
81	#1945	—	61	1	no	—
82	#1846	—	61	1	no	—
83	#1386	—	61	2	no	—
84	#1383	—	61	2	no	—
85	#1376	D₆₁	1	1	yes	characteristic
86	C₂×C₆₁	—	2	1	no	—
87	C₂×C₆₁	—	2	1	no	—
88	C₂×C₆₁	#2100	1	1	yes	characteristic
89	C₂×C₆₁	#2231	1	2	yes	—
90	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
91	C₂×#108	—	61	1	no	—
92	#427	—	61	1	no	—
93	C₂×#107	—	122	1	no	—
94	C₂×#106	—	61	2	no	—
95	#424	—	122	1	no	—
96	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
97	#382	—	122	1	no	—
98	#339	—	122	1	no	—
99	C₆₁	#13542	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	122	1	no	—
101	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
103	#117	—	122	1	no	—
104	C₂×C₄⋊C₄	—	122	2	no	—
105	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#108	—	122	2	no	—
110	#107	—	61	2	no	—
111	#107	—	244	2	no	—
112	#106	—	122	1	no	—
113	#106	—	122	4	no	—
114	C₄⋊C₈	—	244	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	244	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	244	2	no	—
119	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
120	M₁₆	—	244	1	no	—
121	M₁₆	—	244	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	244	1	no	—
125	C₂×C₈	#11758190	1	2	yes	—
126	C₂×C₈	#11758222	1	2	yes	—
127	C₂×C₈	#11758224	1	2	yes	—
128	C₂×C₈	#11758224	1	1	yes	characteristic
129	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
130	C₄⋊C₄	—	122	2	no	—
131	C₄⋊C₄	—	244	2	no	—
132	C₄⋊C₄	—	244	2	no	—
133	C₂²⋊C₄	—	244	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	244	1	no	—
136	C₄²	Dic₂₄₄	1	2	yes	—
137	C₁₆	—	4	2	no	—
138	C₂³	#14519805	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	488	4	no	—
142	C₂×C₄	—	488	2	no	—
143	C₂×C₄	—	244	2	no	—
144	C₂×C₄	#14519757	1	1	yes	characteristic
145	C₂×C₄	#14519761	1	1	yes	characteristic
146	C₂×C₄	#14519765	1	1	yes	characteristic
147	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
149	C₂×C₄	#14519805	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	488	1	no	—
152	C₈	#14519767	1	2	yes	—
153	C₈	#14519770	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#16840053	1	1	yes	characteristic
156	C₂²	#16840098	1	1	yes	characteristic
157	C₂²	#16840127	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	976	2	no	—
160	C₄	—	488	2	no	—
161	C₄	#16839702	1	1	yes	characteristic
162	C₄	#16840058	1	1	yes	characteristic
163	C₄	#16840081	1	1	yes	characteristic
164	C₄	#16840113	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#21447383	1	1	yes	characteristic
168	C₂	#21447996	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446567

C₂

yes

maximal, characteristic

#21445983

C₂

yes

maximal, characteristic

#21443925

C₂

yes

maximal

#21443918

C₂

yes

maximal

C₆₁×#1376

C₂

yes

maximal, characteristic

C₂×#14519713

C₂²

yes

characteristic

#16839800

C₂²

yes

characteristic

C₂×#14519711

—

C₂×#14519709

C₂²

yes

—

#16839788

—

#16839628

—

#16839522

—

C₂×M₃₂×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×#11758194

—

#14519736

—

C₂×C₄⋊Dic₆₁

—

#14519713

—

#14519711

—

#14519711

Q₈

yes

—

—

—

—

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

M₃₂×C₆₁

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

Q₁₆

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

#11758194

—

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

C₂³×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#125

yes

—

C₈×C₆₁

#126

yes

—

#92418

—

#13542

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2100

yes

characteristic

C₂×C₆₁

#2231

yes

—

C₂×M₃₂

C₂×D₆₁

yes

—

C₂×#108

—

#427

—

C₂×#107

—

122

—

C₂×#106

—

#424

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#382

—

122

—

#339

—

122

—

C₆₁

#13542

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

122

—

101

C₂²×C₈

D₂₄₄

yes

characteristic

102

C₂²×C₈

C₂²×D₆₁

yes

characteristic

103

#117

—

122

—

104

C₂×C₄⋊C₄

—

122

—

105

C₂×C₄²

D₂₄₄

yes

characteristic

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

#108

—

122

—

110

#107

—

111

#107

—

244

—

112

#106

—

122

—

113

#106

—

122

—

114

C₄⋊C₈

—

244

—

115

C₄×C₈

—

116

#95

—

244

—

117

C₂²×C₄

—

118

C₂²×C₄

—

244

—

119

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

120

M₁₆

—

244

—

121

M₁₆

—

244

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

244

—

125

C₂×C₈

#11758190

yes

—

126

C₂×C₈

#11758222

yes

—

127

C₂×C₈

#11758224

yes

—

128

C₂×C₈

#11758224

yes

characteristic

129

C₂×C₈

Q₈×D₆₁

yes

characteristic

130

C₄⋊C₄

—

122

—

131

C₄⋊C₄

—

244

—

132

C₄⋊C₄

—

244

—

133

C₂²⋊C₄

—

244

—

134

C₄²

—

135

C₄²

—

244

—

136

C₄²

Dic₂₄₄

yes

—

137

C₁₆

—

138

C₂³

#14519805

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

488

—

142

C₂×C₄

—

488

—

143

C₂×C₄

—

244

—

144

C₂×C₄

#14519757

yes

characteristic

145

C₂×C₄

#14519761

yes

characteristic

146

C₂×C₄

#14519765

yes

characteristic

147

C₂×C₄

C₂×#11758190

yes

characteristic

148

C₂×C₄

C₂×Dic₂₄₄

yes

characteristic

149

C₂×C₄

#14519805

yes

characteristic

150

C₈

—

151

C₈

—

488

—

152

C₈

#14519767

yes

—

153

C₈

#14519770

yes

—

154

C₂²

—

155

C₂²

#16840053

yes

characteristic

156

C₂²

#16840098

yes

characteristic

157

C₂²

#16840127

yes

characteristic

158

C₄

—

159

C₄

—

976

—

160

C₄

—

488

—

161

C₄

#16839702

yes

characteristic

162

C₄

#16840058

yes

characteristic

163

C₄

#16840081

yes

characteristic

164

C₄

#16840113

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#21447383

yes

characteristic

168

C₂

#21447996

yes

—

	subgroup	G/H
center	C₂²	#16840053
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1376	C₂
socle	C₂²×C₆₁	#476
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
hypercenter	#1376	D₆₁
hypocenter	C₆₁	#13542
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#16840053

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1376

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
99. C₆₁	#13542	semidirect

H ≅ G/N

type

99. C₆₁

#13542

semidirect

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

subgroup	index
#21446567	2
#21445983	2
#21443925	2
#21443918	2
C₆₁×#1376	2
#13542	61

subgroup

index

C₆₁×#1376

Subgroups — #29600159

Conjugacy classes of subgroups

Complements

Maximal subgroups