Group #29567273 — subgroups

Subgroups — #29567273

Order 15616.

212 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
2	#21444575	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519713	—	2	1	no	—
5	#16839786	—	2	1	no	—
6	C₄×#11758194	—	2	1	no	—
7	#16839527	C₄	1	2	yes	—
8	#16839513	—	2	1	no	—
9	#16839513	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₆₁×C₂²⋊C₁₆	—	2	1	no	—
13	C₆₁×#346	—	2	1	no	—
14	C₂×#11758194	—	2	2	no	—
15	C₂×#11758194	—	4	1	no	—
16	#14519713	—	2	4	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₄⋊#92418	—	4	1	no	—
19	#14519688	—	4	1	no	—
20	C₄×#92418	—	4	1	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	—	2	2	no	—
25	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	—	4	1	no	—
34	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₆₁	D₈	1	1	yes	characteristic
41	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
42	#11758194	—	2	4	no	—
43	#11758194	—	4	2	no	—
44	#11758194	—	8	2	no	—
45	C₂×#92418	—	4	2	no	—
46	C₂×#92418	—	4	2	no	—
47	C₁₆×C₆₁	—	4	2	no	—
48	C₂³×C₆₁	#103	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
59	C₈×C₆₁	—	2	1	no	—
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	—	2	1	no	—
62	C₈×C₆₁	#104	1	2	yes	—
63	#92418	—	4	4	no	—
64	#92418	—	8	2	no	—
65	#7300	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	—	2	1	no	—
67	C₂²×C₆₁	—	2	1	no	—
68	C₂²×C₆₁	—	2	1	no	—
69	C₂²×C₆₁	#340	1	1	yes	characteristic
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	2	1	no	—
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	#339	1	1	yes	characteristic
75	C₄×C₆₁	#344	1	1	yes	characteristic
76	C₄×C₆₁	#365	1	2	yes	—
77	#2163	D₆₁	1	1	yes	characteristic
78	C₄×#108	—	61	1	no	—
79	#1566	—	61	1	no	—
80	C₂×C₆₁	—	4	1	no	—
81	C₂×C₆₁	#1303	1	1	yes	characteristic
82	C₂×C₆₁	#1383	1	1	yes	characteristic
83	C₂×C₆₁	#1388	1	1	yes	characteristic
84	C₂×#108	—	122	1	no	—
85	#423	—	122	1	no	—
86	C₄×M₁₆	—	122	1	no	—
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₄⋊C₁₆	Dic₆₁	1	2	yes	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	#346	—	2	1	no	—
91	#341	—	61	2	no	—
92	#334	—	61	2	no	—
93	#334	—	122	1	no	—
94	C₆₁	#7300	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	122	2	no	—
96	C₂×M₁₆	—	244	1	no	—
97	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
98	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
99	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
100	C₂×C₁₆	—	2	2	no	—
101	#108	—	122	4	no	—
102	C₄⋊C₈	—	122	2	no	—
103	C₄⋊C₈	—	244	1	no	—
104	#97	—	244	1	no	—
105	C₄×C₈	—	244	1	no	—
106	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
107	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂²×C₄	#11758190	1	1	yes	characteristic
111	C₂²×C₄	Dic₂₄₄	1	1	yes	characteristic
112	C₂²×C₄	#11758198	1	1	yes	characteristic
113	M₁₆	—	122	4	no	—
114	M₁₆	—	488	2	no	—
115	M₁₆	—	244	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	244	2	no	—
119	C₂×C₈	—	244	2	no	—
120	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
121	C₂×C₈	#11758196	1	1	yes	characteristic
122	C₂×C₈	#11758196	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
124	C₄²	#11758190	1	1	yes	characteristic
125	C₄²	D₄₈₈	1	1	yes	characteristic
126	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
127	C₄²	#11758198	1	1	yes	characteristic
128	C₁₆	—	4	2	no	—
129	C₂³	#14519702	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	#14519684	1	1	yes	characteristic
135	C₂×C₄	#14519699	1	1	yes	characteristic
136	C₂×C₄	#14519699	1	1	yes	characteristic
137	C₂×C₄	#14519702	1	1	yes	characteristic
138	C₂×C₄	#14519709	1	1	yes	characteristic
139	C₂×C₄	#14519711	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	488	2	no	—
144	C₈	—	244	4	no	—
145	C₈	#14519704	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#16839525	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#16839521	1	1	yes	characteristic
155	C₄	#16839534	1	1	yes	characteristic
156	C₄	#16839585	1	2	yes	—
157	C₂	—	4	1	no	—
158	C₂	#21443721	1	1	yes	characteristic
159	C₂	#21443918	1	1	yes	characteristic
160	C₂	#21443931	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519713

C₂

yes

maximal, characteristic

#21444575

C₂

yes

maximal, characteristic

C₆₁×#2163

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#16839513

—

#16839513

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

C₄

yes

—

C₆₁×C₂²⋊C₁₆

—

C₆₁×#346

—

—

—

—

—

—

—

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

Q₈

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

SD₁₆

yes

characteristic

C₂²×C₄×C₆₁

Q₁₆

yes

characteristic

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

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

SD₁₆

yes

—

—

—

—

—

C₁₆×C₆₁

—

C₂³×C₆₁

#103

yes

characteristic

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

#104

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2163

D₆₁

yes

characteristic

C₄×#108

—

#1566

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

122

—

#423

—

122

—

C₄×M₁₆

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

Dic₆₁

yes

—

C₂²⋊C₁₆

—

—

—

—

—

122

—

C₆₁

#7300

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂×M₁₆

—

244

—

C₂²×C₈

C₄×D₆₁

yes

characteristic

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

C₂×C₄²

D₂₄₄

yes

characteristic

100

C₂×C₁₆

—

101

#108

—

122

—

102

C₄⋊C₈

—

122

—

103

C₄⋊C₈

—

244

—

104

#97

—

244

—

105

C₄×C₈

—

244

—

106

C₄×C₈

Dic₁₂₂

yes

characteristic

107

C₄×C₈

C₄×D₆₁

yes

characteristic

108

C₄×C₈

C₂×Dic₆₁

yes

characteristic

109

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

110

C₂²×C₄

#11758190

yes

characteristic

111

C₂²×C₄

Dic₂₄₄

yes

characteristic

112

C₂²×C₄

#11758198

yes

characteristic

113

M₁₆

—

122

—

114

M₁₆

—

488

—

115

M₁₆

—

244

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

244

—

119

C₂×C₈

—

244

—

120

C₂×C₈

C₄×Dic₆₁

yes

characteristic

121

C₂×C₈

#11758196

yes

characteristic

122

C₂×C₈

#11758196

yes

characteristic

123

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

124

C₄²

#11758190

yes

characteristic

125

C₄²

D₄₈₈

yes

characteristic

126

C₄²

C₄⋊Dic₆₁

yes

characteristic

127

C₄²

#11758198

yes

characteristic

128

C₁₆

—

129

C₂³

#14519702

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#14519684

yes

characteristic

135

C₂×C₄

#14519699

yes

characteristic

136

C₂×C₄

#14519699

yes

characteristic

137

C₂×C₄

#14519702

yes

characteristic

138

C₂×C₄

#14519709

yes

characteristic

139

C₂×C₄

#14519711

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

488

—

144

C₈

—

244

—

145

C₈

#14519704

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#16839525

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#16839521

yes

characteristic

155

C₄

#16839534

yes

characteristic

156

C₄

#16839585

yes

—

157

C₂

—

158

C₂

#21443721

yes

characteristic

159

C₂

#21443918

yes

characteristic

160

C₂

#21443931

yes

characteristic

	subgroup	G/H
center	C₂²	#16839525
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#2163	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#103
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#2163	D₆₁
hypocenter	C₆₁	#7300
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

#16839525

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₁

Fitting

C₆₁×#2163

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 ◁ #2163

N	H ≅ G/N	type
94. C₆₁	#7300	semidirect

H ≅ G/N

type

94. C₆₁

#7300

semidirect

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

subgroup	index
C₄×#14519713	2
#21444575	2
C₆₁×#2163	2
#7300	61

subgroup

index

C₄×#14519713

#21444575

C₆₁×#2163

#7300

Subgroups — #29567273

Conjugacy classes of subgroups

Complements

Maximal subgroups