Group #29584797 — subgroups

Subgroups — #29584797

Order 15616.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	#21446800	C₂	1	1	yes	maximal, characteristic
2	#21444477	C₂	1	2	yes	maximal
3	#21443716	C₂	1	2	yes	maximal
4	#21443648	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1832	C₂	1	1	yes	maximal, characteristic
6	#16840319	C₂²	1	1	yes	characteristic
7	#16839815	C₂²	1	2	yes	—
8	#16839786	C₂²	1	1	yes	characteristic
9	#16839522	—	2	2	no	—
10	#16839510	—	2	2	no	—
11	#16839507	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#376	C₂²	1	1	yes	characteristic
14	C₆₁×#336	—	2	2	no	—
15	C₂×#11758194	—	2	1	no	—
16	#14519777	D₄	1	2	yes	—
17	#14519769	—	2	1	no	—
18	#14519756	—	2	2	no	—
19	C₄×Dic₁₂₂	—	2	2	no	—
20	#14519736	—	2	1	no	—
21	C₄⋊#92418	—	2	2	no	—
22	C₄⋊#92418	—	2	2	no	—
23	C₂²⋊#92418	—	2	2	no	—
24	C₄×#92418	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₂×C₄²×C₆₁	D₄	1	2	yes	—
27	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	2	no	—
29	C₄⋊C₈×C₆₁	—	2	2	no	—
30	#95×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	#41	1	2	yes	—
34	C₂×Dic₁₂₂	—	2	2	no	—
35	C₂×C₈×C₆₁	—	4	2	no	—
36	C₂×C₈×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	—	4	2	no	—
38	C₄²×C₆₁	—	2	2	no	—
39	C₄²×C₆₁	—	2	2	no	—
40	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₆₁	#41	1	1	yes	characteristic
42	C₄²×C₆₁	#41	1	2	yes	—
43	C₂²⋊Dic₆₁	—	4	1	no	—
44	C₄⋊Dic₆₁	—	2	2	no	—
45	C₄⋊Dic₆₁	—	2	2	no	—
46	#11758196	—	4	2	no	—
47	C₄×Dic₆₁	—	4	1	no	—
48	#11758194	—	4	2	no	—
49	C₂×#92418	—	4	2	no	—
50	C₂×#92418	—	4	1	no	—
51	C₂³×C₆₁	#123	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	4	2	no	—
53	C₂×C₄×C₆₁	—	4	4	no	—
54	C₂×C₄×C₆₁	—	4	2	no	—
55	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#123	1	2	yes	—
57	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#126	1	2	yes	—
59	C₂×C₄×C₆₁	#135	1	2	yes	—
60	C₂×C₄×C₆₁	#137	1	2	yes	—
61	C₂×Dic₆₁	—	4	2	no	—
62	Dic₁₂₂	—	8	2	no	—
63	C₈×C₆₁	—	8	2	no	—
64	#92418	—	8	2	no	—
65	#92418	—	8	1	no	—
66	#10630	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	2	2	no	—
68	C₂²×C₆₁	#397	1	1	yes	characteristic
69	C₂²×C₆₁	#485	1	2	yes	—
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	—	8	2	no	—
73	C₄×C₆₁	#464	1	2	yes	—
74	C₄×C₆₁	#488	1	1	yes	characteristic
75	C₄×C₆₁	#491	1	1	yes	characteristic
76	Dic₆₁	—	8	2	no	—
77	#1995	—	61	1	no	—
78	#1832	D₆₁	1	1	yes	characteristic
79	#1542	—	61	2	no	—
80	#1301	—	61	2	no	—
81	#1274	—	61	1	no	—
82	C₂×C₆₁	—	2	2	no	—
83	C₂×C₆₁	#2074	1	1	yes	characteristic
84	C₂×C₆₁	#2080	1	2	yes	—
85	#533	—	61	1	no	—
86	#431	—	61	2	no	—
87	#423	—	61	1	no	—
88	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
89	#376	C₂×D₆₁	1	1	yes	characteristic
90	#339	—	122	2	no	—
91	#336	—	2	2	no	—
92	#333	—	122	2	no	—
93	#332	—	122	2	no	—
94	C₆₁	#10630	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	122	1	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₄⋊Q₈	—	61	2	no	—
98	#125	—	122	1	no	—
99	C₂²⋊Q₈	—	122	2	no	—
100	C₄×Q₈	—	122	2	no	—
101	#117	—	122	1	no	—
102	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
103	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	122	2	no	—
107	C₄⋊C₈	—	122	2	no	—
108	C₂²⋊C₈	—	122	2	no	—
109	C₄×C₈	—	122	2	no	—
110	#95	—	2	2	no	—
111	C₂×Q₈	—	122	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	#11758222	1	2	yes	—
114	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
115	M₁₆	—	244	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	244	2	no	—
120	C₂×C₈	—	244	1	no	—
121	C₄⋊C₄	—	122	2	no	—
122	C₄⋊C₄	—	122	2	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₂²⋊C₄	—	244	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	244	1	no	—
128	C₄²	D₄×D₆₁	1	1	yes	characteristic
129	C₄²	#11758224	1	1	yes	characteristic
130	C₄²	#11758226	1	2	yes	—
131	C₂³	#14519760	1	1	yes	characteristic
132	Q₈	—	488	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	244	2	no	—
137	C₂×C₄	#14519748	1	1	yes	characteristic
138	C₂×C₄	#14519762	1	2	yes	—
139	C₂×C₄	#14519766	1	1	yes	characteristic
140	C₂×C₄	#14519772	1	2	yes	—
141	C₂×C₄	#14519797	1	2	yes	—
142	C₂×C₄	#14519807	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	488	2	no	—
145	C₈	—	488	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#16839688	1	1	yes	characteristic
148	C₂²	#16840103	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	488	2	no	—
153	C₄	#16839981	1	2	yes	—
154	C₄	#16840119	1	1	yes	characteristic
155	C₄	#16840132	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#21447238	1	1	yes	characteristic
158	C₂	#21447270	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446800

C₂

yes

maximal, characteristic

#21444477

C₂

yes

maximal

#21443716

C₂

yes

maximal

#21443648

C₂

yes

maximal, characteristic

C₆₁×#1832

C₂

yes

maximal, characteristic

#16840319

C₂²

yes

characteristic

#16839815

C₂²

yes

—

#16839786

C₂²

yes

—

—

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₆₁×#376

C₂²

yes

characteristic

C₆₁×#336

—

C₂×#11758194

—

#14519777

D₄

yes

—

#14519769

—

#14519756

—

C₄×Dic₁₂₂

—

—

—

—

—

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

C₄²×C₆₁

yes

C₄²×C₆₁

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

#11758194

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#123

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

#126

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#397

yes

characteristic

C₂²×C₆₁

#485

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#464

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#1995

—

#1832

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#2074

yes

characteristic

C₂×C₆₁

#2080

yes

—

#533

—

#431

—

#423

—

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

#376

C₂×D₆₁

yes

characteristic

#339

—

122

—

#336

—

#333

—

122

—

#332

—

122

—

C₆₁

#10630

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

122

—

C₂²⋊Q₈

—

122

—

100

C₄×Q₈

—

122

—

101

#117

—

122

—

102

C₂×C₄²

C₂²⋊D₆₁

yes

—

103

C₂×C₄²

C₂²×D₆₁

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

122

—

107

C₄⋊C₈

—

122

—

108

C₂²⋊C₈

—

122

—

109

C₄×C₈

—

122

—

110

#95

—

111

C₂×Q₈

—

122

—

112

C₂²×C₄

—

113

C₂²×C₄

#11758222

yes

—

114

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

115

M₁₆

—

244

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

244

—

120

C₂×C₈

—

244

—

121

C₄⋊C₄

—

122

—

122

C₄⋊C₄

—

122

—

123

C₄⋊C₄

—

244

—

124

C₂²⋊C₄

—

244

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

244

—

128

C₄²

D₄×D₆₁

yes

characteristic

129

C₄²

#11758224

yes

characteristic

130

C₄²

#11758226

yes

—

131

C₂³

#14519760

yes

characteristic

132

Q₈

—

488

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

244

—

137

C₂×C₄

#14519748

yes

characteristic

138

C₂×C₄

#14519762

yes

—

139

C₂×C₄

#14519766

yes

characteristic

140

C₂×C₄

#14519772

yes

—

141

C₂×C₄

#14519797

yes

—

142

C₂×C₄

#14519807

yes

—

143

C₈

—

144

C₈

—

488

—

145

C₈

—

488

—

146

C₂²

—

147

C₂²

#16839688

yes

characteristic

148

C₂²

#16840103

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

488

—

153

C₄

#16839981

yes

—

154

C₄

#16840119

yes

characteristic

155

C₄

#16840132

yes

characteristic

156

C₂

—

157

C₂

#21447238

yes

characteristic

158

C₂

#21447270

yes

—

	subgroup	G/H
center	C₂×C₄	#14519766
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1832	C₂
socle	C₂²×C₆₁	#397
prime radical	C₂³×C₆₁	#123
hypercenter	#1832	D₆₁
hypocenter	C₆₁	#10630
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1832

subgroup

G/H

center

C₂×C₄

#14519766

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1832

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1832

N	H ≅ G/N	type
94. C₆₁	#10630	semidirect

H ≅ G/N

type

94. C₆₁

#10630

semidirect

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

subgroup	index
#21446800	2
#21444477	2
#21443716	2
#21443648	2
C₆₁×#1832	2
#10630	61

subgroup

index

C₆₁×#1832

Subgroups — #29584797

Conjugacy classes of subgroups

Complements

Maximal subgroups