Group #29567250 — subgroups

Subgroups — #29567250

Order 15616.

198 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21447622	C₂	1	1	yes	maximal, characteristic
2	#21444575	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
4	#16839786	—	2	1	no	—
5	C₄⋊#11758185	—	2	1	no	—
6	C₂²⋊#11758185	C₄	1	2	yes	—
7	#16839538	—	2	1	no	—
8	#16839513	—	2	1	no	—
9	#16839513	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#335	—	2	1	no	—
13	C₆₁×#335	C₄	1	2	yes	—
14	C₂×#11758194	—	4	1	no	—
15	C₂×#11758185	—	2	2	no	—
16	C₄⋊#92418	—	2	2	no	—
17	C₄⋊#92418	—	4	1	no	—
18	C₂²×C₈×C₆₁	—	4	1	no	—
19	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	—	2	2	no	—
23	C₄⋊C₈×C₆₁	—	4	1	no	—
24	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
31	C₂×C₈×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	—	4	1	no	—
33	C₂×C₈×C₆₁	—	4	2	no	—
34	C₂×C₈×C₆₁	—	8	2	no	—
35	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₆₁	D₈	1	1	yes	characteristic
42	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
43	#11758194	—	8	2	no	—
44	C₂×#92418	—	4	2	no	—
45	#11758185	—	4	2	no	—
46	C₂³×C₆₁	#106	1	1	yes	characteristic
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	1	no	—
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
58	C₈×C₆₁	—	2	1	no	—
59	C₈×C₆₁	—	2	1	no	—
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	—	8	2	no	—
62	C₈×C₆₁	#108	1	2	yes	—
63	#92418	—	8	2	no	—
64	#7292	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	2	1	no	—
67	C₂²×C₆₁	#340	1	1	yes	characteristic
68	C₂²×C₆₁	#365	1	2	yes	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	2	1	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	#341	1	1	yes	characteristic
74	C₄×C₆₁	#344	1	1	yes	characteristic
75	C₄×C₆₁	#361	1	1	yes	characteristic
76	C₄×C₆₁	#362	1	1	yes	characteristic
77	#2163	—	61	1	no	—
78	#1566	—	61	1	no	—
79	C₂×#335	D₆₁	1	1	yes	characteristic
80	C₂×C₆₁	—	2	2	no	—
81	C₂×C₆₁	#1301	1	1	yes	characteristic
82	C₂×C₆₁	#1383	1	1	yes	characteristic
83	C₂×C₆₁	#1387	1	1	yes	characteristic
84	#423	—	122	1	no	—
85	C₂×C₄⋊C₈	—	2	1	no	—
86	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₄⋊C₁₆	—	122	1	no	—
88	C₂²⋊C₁₆	—	61	2	no	—
89	#346	—	122	1	no	—
90	#335	—	2	1	no	—
91	#335	Dic₆₁	1	2	yes	—
92	#334	—	61	2	no	—
93	#334	—	122	1	no	—
94	C₆₁	#7292	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	244	1	no	—
96	C₂²×C₈	—	4	1	no	—
97	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
98	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
100	C₂×C₁₆	—	122	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	4	1	no	—
103	C₄⋊C₈	—	122	2	no	—
104	C₄⋊C₈	—	244	1	no	—
105	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
106	C₄×C₈	D₂₄₄	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
109	C₂²×C₄	#11758196	1	1	yes	characteristic
110	C₂²×C₄	#11758200	1	1	yes	characteristic
111	C₂²×C₄	#11758201	1	1	yes	characteristic
112	M₁₆	—	488	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	—	244	2	no	—
118	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
119	C₂×C₈	#11758196	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
121	C₂×C₈	#11758198	1	1	yes	characteristic
122	C₄²	#11758198	1	1	yes	characteristic
123	C₄²	#11758199	1	1	yes	characteristic
124	C₄²	#11758202	1	1	yes	characteristic
125	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
126	C₁₆	—	244	2	no	—
127	C₂³	#14519708	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	#14519684	1	1	yes	characteristic
134	C₂×C₄	#14519697	1	1	yes	characteristic
135	C₂×C₄	#14519698	1	1	yes	characteristic
136	C₂×C₄	#14519700	1	1	yes	characteristic
137	C₂×C₄	#14519701	1	1	yes	characteristic
138	C₂×C₄	#14519710	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	488	2	no	—
144	C₈	#14519713	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16839524	1	1	yes	characteristic
148	C₂²	#16839586	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#16839527	1	1	yes	characteristic
154	C₄	#16839534	1	1	yes	characteristic
155	C₄	#16839575	1	1	yes	characteristic
156	C₄	#16839578	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	#21443716	1	1	yes	characteristic
159	C₂	#21443919	1	1	yes	characteristic
160	C₂	#21443927	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447622

C₂

yes

maximal, characteristic

#21444575

C₂

yes

maximal, characteristic

C₂×C₆₁×#335

C₂

yes

maximal, characteristic

#16839786

—

C₄⋊#11758185

—

C₂²⋊#11758185

C₄

yes

—

#16839538

—

#16839513

—

#16839513

C₄

yes

—

C₂×C₄⋊C₈×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#335

—

C₆₁×#335

C₄

yes

—

—

—

—

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

Q₈

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×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

SD₁₆

yes

characteristic

C₂²×C₄×C₆₁

SD₁₆

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

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

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄²×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

Q₁₆

yes

—

—

—

C₂³×C₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#108

yes

—

#92418

—

#7292

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₂²×C₆₁

#365

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

122

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

122

—

C₂²⋊C₁₆

—

#346

—

122

—

#335

—

#335

Dic₆₁

yes

—

#334

—

#334

—

122

—

C₆₁

#7292

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

244

—

C₂²×C₈

—

C₂²×C₈

Dic₁₂₂

yes

characteristic

C₂²×C₈

C₄×D₆₁

yes

characteristic

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

100

C₂×C₁₆

—

122

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

122

—

104

C₄⋊C₈

—

244

—

105

C₄×C₈

C₄×D₆₁

yes

characteristic

106

C₄×C₈

D₂₄₄

yes

characteristic

107

C₄×C₈

C₂×Dic₆₁

yes

characteristic

108

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

109

C₂²×C₄

#11758196

yes

characteristic

110

C₂²×C₄

#11758200

yes

characteristic

111

C₂²×C₄

#11758201

yes

characteristic

112

M₁₆

—

488

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

244

—

118

C₂×C₈

C₄×Dic₆₁

yes

characteristic

119

C₂×C₈

#11758196

yes

characteristic

120

C₂×C₈

C₄⋊Dic₆₁

yes

characteristic

121

C₂×C₈

#11758198

yes

characteristic

122

C₄²

#11758198

yes

characteristic

123

C₄²

#11758199

yes

characteristic

124

C₄²

#11758202

yes

characteristic

125

C₄²

C₂²⋊Dic₆₁

yes

characteristic

126

C₁₆

—

244

—

127

C₂³

#14519708

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#14519684

yes

characteristic

134

C₂×C₄

#14519697

yes

characteristic

135

C₂×C₄

#14519698

yes

characteristic

136

C₂×C₄

#14519700

yes

characteristic

137

C₂×C₄

#14519701

yes

characteristic

138

C₂×C₄

#14519710

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

488

—

144

C₈

#14519713

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#16839524

yes

characteristic

148

C₂²

#16839586

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#16839527

yes

characteristic

154

C₄

#16839534

yes

characteristic

155

C₄

#16839575

yes

characteristic

156

C₄

#16839578

yes

characteristic

157

C₂

—

158

C₂

#21443716

yes

characteristic

159

C₂

#21443919

yes

characteristic

160

C₂

#21443927

yes

characteristic

	subgroup	G/H
center	C₂²	#16839524
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#106
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#7292
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

subgroup

G/H

center

C₂²

#16839524

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₁

Fitting

C₂×C₆₁×#335

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄×C₈×C₆₁

C₂²

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

N	H ≅ G/N	type
94. C₆₁	#7292	semidirect

H ≅ G/N

type

94. C₆₁

#7292

semidirect

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

subgroup	index
#21447622	2
#21444575	2
C₂×C₆₁×#335	2
#7292	61

subgroup

index

#21447622

#21444575

C₂×C₆₁×#335

#7292

Subgroups — #29567250

Conjugacy classes of subgroups

Complements

Maximal subgroups