Group #29596434 — subgroups

Subgroups — #29596434

Order 15616.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21447585	C₂	1	1	yes	maximal, characteristic
2	#21447584	C₂	1	2	yes	maximal
3	#21447577	C₂	1	2	yes	maximal
4	#21445574	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	#16839717	C₂²	1	2	yes	—
8	#16839583	—	2	2	no	—
9	#16839582	—	2	4	no	—
10	#16839580	—	2	4	no	—
11	#16839520	C₄	1	1	yes	characteristic
12	#16839519	C₄	1	1	yes	characteristic
13	#16839518	C₄	1	2	yes	—
14	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#405	C₂²	1	1	yes	characteristic
17	C₆₁×C₄⋊C₁₆	—	2	2	no	—
18	C₂×#11758194	C₂×C₄	1	2	yes	—
19	#14519736	C₂×C₄	1	2	yes	—
20	#14519717	—	2	2	no	—
21	#14519717	—	2	2	no	—
22	C₂×#11758185	—	2	2	no	—
23	#14519687	Q₈	1	4	yes	—
24	C₈×Dic₆₁	D₄	1	4	yes	—
25	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₆₁	—	2	2	no	—
29	C₂×C₁₆×C₆₁	—	2	1	no	—
30	C₂×C₁₆×C₆₁	—	2	1	no	—
31	C₂×C₁₆×C₆₁	—	2	2	no	—
32	#97×C₆₁	Q₈	1	2	yes	—
33	C₄×C₈×C₆₁	D₄	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
41	C₄⋊C₄×C₆₁	—	2	1	no	—
42	C₂²⋊C₄×C₆₁	—	2	1	no	—
43	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂²⋊Dic₆₁	—	2	2	no	—
45	C₄⋊Dic₆₁	—	2	2	no	—
46	C₄×Dic₆₁	C₄⋊C₄	1	4	yes	—
47	#11758194	—	2	2	no	—
48	#11758194	—	2	2	no	—
49	C₂×#92418	C₄⋊C₄	1	4	yes	—
50	C₁₆×C₆₁	—	4	1	no	—
51	C₁₆×C₆₁	—	4	1	no	—
52	#11758185	—	4	2	no	—
53	C₂³×C₆₁	#131	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#131	1	2	yes	—
59	C₂×Dic₆₁	—	2	4	no	—
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	#127	1	1	yes	characteristic
62	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
64	#92418	—	2	4	no	—
65	#12902	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	—	4	1	no	—
67	C₂²×C₆₁	#424	1	2	yes	—
68	C₂²×C₆₁	#424	1	1	yes	characteristic
69	C₄×C₆₁	—	4	2	no	—
70	C₄×C₆₁	#384	1	1	yes	characteristic
71	C₄×C₆₁	#424	1	2	yes	—
72	C₄×C₆₁	#424	1	1	yes	characteristic
73	Dic₆₁	—	4	4	no	—
74	#2153	—	61	2	no	—
75	#2153	—	61	1	no	—
76	#2151	—	61	2	no	—
77	#2151	D₆₁	1	1	yes	characteristic
78	#1747	—	61	1	no	—
79	C₂×C₆₁	—	2	2	no	—
80	C₂×C₆₁	—	2	1	no	—
81	C₂×C₆₁	#1843	1	1	yes	characteristic
82	C₂×M₃₂	—	61	2	no	—
83	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
84	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
85	#405	—	61	2	no	—
86	#405	C₂×D₆₁	1	1	yes	characteristic
87	#364	—	122	4	no	—
88	#364	—	122	2	no	—
89	C₄⋊C₁₆	—	2	2	no	—
90	C₄⋊C₁₆	—	122	4	no	—
91	#338	—	61	1	no	—
92	H(3,ℤ/4)	—	61	1	no	—
93	H(3,ℤ/4)	—	61	2	no	—
94	C₆₁	#12902	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	61	2	no	—
96	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
98	#117	—	61	2	no	—
99	#117	C₄×D₆₁	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	122	2	no	—
102	M₃₂	—	122	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	122	2	no	—
107	#97	—	61	4	no	—
108	#97	Dic₁₂₂	1	2	yes	—
109	C₄×C₈	—	61	4	no	—
110	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	122	2	no	—
114	M₁₆	—	122	2	no	—
115	C₂×C₈	—	61	4	no	—
116	C₂×C₈	#11758196	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
118	C₂×C₈	D₄×D₆₁	1	2	yes	—
119	C₂×C₈	Q₈×D₆₁	1	2	yes	—
120	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	122	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	122	2	no	—
125	C₄²	—	61	4	no	—
126	C₄²	#11758196	1	2	yes	—
127	C₁₆	—	4	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	—	244	2	no	—
130	C₂³	#14519783	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	122	4	no	—
133	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
134	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
135	C₂×C₄	#14519783	1	1	yes	characteristic
136	C₂×C₄	#14519785	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	122	4	no	—
139	C₈	#14519773	1	1	yes	characteristic
140	C₈	#14519775	1	1	yes	characteristic
141	C₈	#14519776	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#16839787	1	2	yes	—
144	C₂²	#16839788	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	244	4	no	—
147	C₄	#16839631	1	1	yes	characteristic
148	C₄	#16839787	1	2	yes	—
149	C₄	#16839790	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#21445972	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447585

C₂

yes

maximal, characteristic

#21447584

C₂

yes

maximal

#21447577

C₂

yes

maximal

#21445574

C₂

yes

maximal, characteristic

C₆₁×#2151

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

—

#16839717

C₂²

yes

—

—

—

—

C₄

yes

characteristic

#16839519

C₄

yes

characteristic

#16839518

C₄

yes

—

C₂×M₃₂×C₆₁

C₂²

yes

characteristic

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×#405

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₂×#11758194

C₂×C₄

yes

—

#14519736

C₂×C₄

yes

—

—

—

—

Q₈

yes

—

C₈×Dic₆₁

D₄

yes

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×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

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

C₄⋊C₄

yes

—

#11758194

—

#11758194

—

C₂×#92418

C₄⋊C₄

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#131

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#127

yes

characteristic

C₈×C₆₁

C₄⋊Q₈

yes

characteristic

C₈×C₆₁

C₄⋊Q₈

yes

—

#92418

—

#12902

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#424

yes

—

C₂²×C₆₁

#424

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#384

yes

characteristic

C₄×C₆₁

#424

yes

—

C₄×C₆₁

#424

yes

characteristic

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

#1747

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1843

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₁

yes

characteristic

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

#405

—

#405

C₂×D₆₁

yes

characteristic

#364

—

122

—

#364

—

122

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₆₁

#12902

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₄×D₆₁

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

#117

—

#117

C₄×D₆₁

yes

characteristic

100

M₃₂

—

101

M₃₂

—

122

—

102

M₃₂

—

122

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

122

—

107

#97

—

108

#97

Dic₁₂₂

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₆₁

yes

—

111

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

112

M₁₆

—

113

M₁₆

—

122

—

114

M₁₆

—

122

—

115

C₂×C₈

—

116

C₂×C₈

#11758196

yes

—

117

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

118

C₂×C₈

D₄×D₆₁

yes

—

119

C₂×C₈

Q₈×D₆₁

yes

—

120

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

122

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

122

—

125

C₄²

—

126

C₄²

#11758196

yes

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

244

—

130

C₂³

#14519783

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

122

—

133

C₂×C₄

C₄⋊C₄×D₆₁

yes

—

134

C₂×C₄

C₂×#11758196

yes

characteristic

135

C₂×C₄

#14519783

yes

characteristic

136

C₂×C₄

#14519785

yes

—

137

C₈

—

138

C₈

—

122

—

139

C₈

#14519773

yes

characteristic

140

C₈

#14519775

yes

characteristic

141

C₈

#14519776

yes

—

142

C₂²

—

143

C₂²

#16839787

yes

—

144

C₂²

#16839788

yes

characteristic

145

C₄

—

146

C₄

—

244

—

147

C₄

#16839631

yes

characteristic

148

C₄

#16839787

yes

—

149

C₄

#16839790

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#21445972

yes

characteristic

	subgroup	G/H
center	C₈	#14519773
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2151	C₂
socle	C₂×C₆₁	#1843
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2151	D₆₁
hypocenter	C₆₁	#12902
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#14519773

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2151

C₂

C₂×C₆₁

C₂³×C₆₁

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
94. C₆₁	#12902	semidirect

H ≅ G/N

type

94. C₆₁

#12902

semidirect

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

subgroup	index
#21447585	2
#21447584	2
#21447577	2
#21445574	2
C₆₁×#2151	2
#12902	61

subgroup

index

C₆₁×#2151

Subgroups — #29596434

Conjugacy classes of subgroups

Complements

Maximal subgroups