Group #29596382 — subgroups

Subgroups — #29596382

Order 15616.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#21447578	C₂	1	1	yes	maximal, characteristic
2	#21445574	C₂	1	1	yes	maximal, characteristic
3	#21443909	C₂	1	2	yes	maximal
4	#21443904	C₂	1	2	yes	maximal
5	C₆₁×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	1	yes	characteristic
7	C₂²×#11758185	C₂²	1	1	yes	characteristic
8	#16839717	C₂²	1	2	yes	—
9	C₄⋊#11758185	—	2	2	no	—
10	#16839520	C₄	1	1	yes	characteristic
11	#16839519	C₄	1	1	yes	characteristic
12	#16839518	C₄	1	2	yes	—
13	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#405	C₂²	1	1	yes	characteristic
15	C₆₁×#364	—	2	2	no	—
16	C₂×#11758194	C₂×C₄	1	2	yes	—
17	#14519736	C₂×C₄	1	2	yes	—
18	#14519717	—	2	2	no	—
19	C₂×#11758185	—	2	1	no	—
20	C₂×#11758185	—	2	1	no	—
21	C₂×#11758185	—	2	2	no	—
22	#14519687	—	2	2	no	—
23	C₈×Dic₆₁	—	2	2	no	—
24	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₆₁	—	2	2	no	—
28	M₃₂×C₆₁	—	2	2	no	—
29	C₂×C₁₆×C₆₁	—	2	2	no	—
30	#97×C₆₁	Q₈	1	2	yes	—
31	C₄×C₈×C₆₁	D₄	1	2	yes	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₆₁	#41	1	2	yes	—
38	C₂×C₈×C₆₁	#41	1	2	yes	—
39	C₄⋊C₄×C₆₁	—	2	1	no	—
40	C₂²⋊C₄×C₆₁	—	2	1	no	—
41	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
42	C₂²⋊Dic₆₁	—	2	2	no	—
43	C₄⋊Dic₆₁	—	2	2	no	—
44	C₄×Dic₆₁	—	2	2	no	—
45	#11758194	—	2	2	no	—
46	#11758194	—	2	2	no	—
47	C₂×#92418	—	2	2	no	—
48	C₁₆×C₆₁	—	4	2	no	—
49	#11758185	—	4	1	no	—
50	#11758185	—	4	1	no	—
51	C₂³×C₆₁	#131	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#117	1	2	yes	—
55	C₂×C₄×C₆₁	#131	1	2	yes	—
56	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
57	C₂×Dic₆₁	—	4	2	no	—
58	C₈×C₆₁	—	2	2	no	—
59	C₈×C₆₁	#124	1	1	yes	characteristic
60	C₈×C₆₁	#124	1	1	yes	characteristic
61	C₈×C₆₁	#125	1	2	yes	—
62	#92418	—	4	2	no	—
63	#12885	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	—	4	1	no	—
65	C₂²×C₆₁	#424	1	1	yes	characteristic
66	C₂²×C₆₁	#433	1	2	yes	—
67	C₄×C₆₁	—	4	2	no	—
68	C₄×C₆₁	#382	1	1	yes	characteristic
69	C₄×C₆₁	#424	1	1	yes	characteristic
70	C₄×C₆₁	#433	1	2	yes	—
71	Dic₆₁	—	8	2	no	—
72	#2153	D₆₁	1	1	yes	characteristic
73	#2151	—	61	1	no	—
74	#1747	—	61	1	no	—
75	#1379	—	61	2	no	—
76	#1378	—	61	2	no	—
77	C₂×C₆₁	—	2	2	no	—
78	C₂×C₆₁	—	2	1	no	—
79	C₂×C₆₁	#1832	1	1	yes	characteristic
80	C₂×M₃₂	—	61	1	no	—
81	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
82	C₂²×C₁₆	—	61	1	no	—
83	#405	—	61	2	no	—
84	#405	C₂×D₆₁	1	1	yes	characteristic
85	#364	—	2	2	no	—
86	C₄⋊C₁₆	—	122	2	no	—
87	#338	—	61	1	no	—
88	H(3,ℤ/4)	—	61	1	no	—
89	H(3,ℤ/4)	—	61	2	no	—
90	C₆₁	#12885	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	61	2	no	—
92	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
94	#117	—	61	2	no	—
95	#117	C₄×D₆₁	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	122	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	122	2	no	—
101	C₂×C₁₆	—	122	1	no	—
102	C₂×C₁₆	—	122	1	no	—
103	#97	—	122	2	no	—
104	#97	Dic₁₂₂	1	2	yes	—
105	C₄×C₈	—	122	2	no	—
106	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	122	2	no	—
110	M₁₆	—	122	2	no	—
111	C₂×C₈	—	122	2	no	—
112	C₂×C₈	#11758196	1	2	yes	—
113	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
114	C₂×C₈	#11758224	1	2	yes	—
115	C₂×C₈	#11758226	1	2	yes	—
116	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	122	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	122	2	no	—
121	C₄²	—	122	2	no	—
122	C₄²	#11758196	1	2	yes	—
123	C₁₆	—	4	2	no	—
124	C₁₆	—	244	1	no	—
125	C₁₆	—	244	1	no	—
126	C₂³	#14519785	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	244	2	no	—
129	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
130	C₂×C₄	#14519735	1	2	yes	—
131	C₂×C₄	#14519783	1	2	yes	—
132	C₂×C₄	#14519785	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	244	2	no	—
135	C₈	#14519764	1	1	yes	characteristic
136	C₈	#14519766	1	1	yes	characteristic
137	C₈	#14519768	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#16839790	1	1	yes	characteristic
140	C₂²	#16839822	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	488	2	no	—
143	C₄	#16839627	1	1	yes	characteristic
144	C₄	#16839788	1	1	yes	characteristic
145	C₄	#16839822	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#21445938	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447578

C₂

yes

maximal, characteristic

#21445574

C₂

yes

maximal, characteristic

#21443909

C₂

yes

maximal

#21443904

C₂

yes

maximal

C₆₁×#2153

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

characteristic

C₂²×#11758185

C₂²

yes

characteristic

#16839717

C₂²

yes

—

C₄⋊#11758185

—

#16839520

C₄

yes

characteristic

#16839519

C₄

yes

characteristic

#16839518

C₄

yes

—

C₂×M₃₂×C₆₁

C₂²

yes

—

C₆₁×#405

C₂²

yes

characteristic

C₆₁×#364

—

C₂×#11758194

C₂×C₄

yes

—

#14519736

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

#97×C₆₁

Q₈

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

#11758194

—

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

#11758185

—

#11758185

—

C₂³×C₆₁

#131

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

#92418

—

#12885

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#424

yes

characteristic

C₂²×C₆₁

#433

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

#2153

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1832

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₁

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₆₁

yes

characteristic

#364

—

C₄⋊C₁₆

—

122

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₆₁

#12885

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₄×D₆₁

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

#117

—

#117

C₄×D₆₁

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

122

—

C₂×C₁₆

—

100

C₂×C₁₆

—

122

—

101

C₂×C₁₆

—

122

—

102

C₂×C₁₆

—

122

—

103

#97

—

122

—

104

#97

Dic₁₂₂

yes

—

105

C₄×C₈

—

122

—

106

C₄×C₈

C₂²⋊D₆₁

yes

—

107

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

108

M₁₆

—

109

M₁₆

—

122

—

110

M₁₆

—

122

—

111

C₂×C₈

—

122

—

112

C₂×C₈

#11758196

yes

—

113

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

114

C₂×C₈

#11758224

yes

—

115

C₂×C₈

#11758226

yes

—

116

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

122

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

122

—

121

C₄²

—

122

—

122

C₄²

#11758196

yes

—

123

C₁₆

—

124

C₁₆

—

244

—

125

C₁₆

—

244

—

126

C₂³

#14519785

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

244

—

129

C₂×C₄

C₂×#11758196

yes

characteristic

130

C₂×C₄

#14519735

yes

—

131

C₂×C₄

#14519783

yes

—

132

C₂×C₄

#14519785

yes

characteristic

133

C₈

—

134

C₈

—

244

—

135

C₈

#14519764

yes

characteristic

136

C₈

#14519766

yes

characteristic

137

C₈

#14519768

yes

—

138

C₂²

—

139

C₂²

#16839790

yes

characteristic

140

C₂²

#16839822

yes

—

141

C₄

—

142

C₄

—

488

—

143

C₄

#16839627

yes

characteristic

144

C₄

#16839788

yes

characteristic

145

C₄

#16839822

yes

—

146

C₂

—

147

C₂

—

148

C₂

#21445938

yes

characteristic

	subgroup	G/H
center	C₈	#14519764
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2153	C₂
socle	C₂×C₆₁	#1832
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2153	D₆₁
hypocenter	C₆₁	#12885
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#14519764

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2153

C₂

C₂×C₆₁

C₂³×C₆₁

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
90. C₆₁	#12885	semidirect

H ≅ G/N

type

90. C₆₁

#12885

semidirect

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

subgroup	index
#21447578	2
#21445574	2
#21443909	2
#21443904	2
C₆₁×#2153	2
#12885	61

subgroup

index

C₆₁×#2153

Subgroups — #29596382

Conjugacy classes of subgroups

Complements

Maximal subgroups