Group #29592016 — subgroups

Subgroups — #29592016

Order 15616.

206 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21444716	C₂	1	2	yes	maximal
2	#21443893	C₂	1	1	yes	maximal, characteristic
3	#21443783	C₂	1	2	yes	maximal
4	#21443735	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#16839893	—	2	2	no	—
7	C₄×#11758185	C₂²	1	1	yes	characteristic
8	#16839515	C₂²	1	2	yes	—
9	Q₈⋊#92418	—	2	2	no	—
10	C₈×#92418	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#366	C₄	1	1	yes	characteristic
14	C₆₁×#358	C₄	1	2	yes	—
15	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#11758185	—	2	1	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₄⋊#92418	—	4	2	no	—
19	#14519688	—	4	2	no	—
20	C₄×#92418	D₄	1	1	yes	characteristic
21	C₄×#92418	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₆₁	—	2	2	no	—
24	Q₃₂×C₆₁	—	2	2	no	—
25	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
27	#107×C₆₁	C₂×C₄	1	2	yes	—
28	#103×C₆₁	—	2	2	no	—
29	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₆₁	—	2	2	no	—
31	Q₁₆×C₆₁	—	2	2	no	—
32	Q₁₆×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₄⋊C₄×C₆₁	—	4	2	no	—
37	C₄²×C₆₁	—	4	2	no	—
38	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	—	4	2	no	—
41	C₂×#92418	D₈	1	2	yes	—
42	C₁₆×C₆₁	—	2	1	no	—
43	C₁₆×C₆₁	M₁₆	1	2	yes	—
44	#11758185	—	2	2	no	—
45	Q₈×C₆₁	—	4	2	no	—
46	Q₈×C₆₁	—	4	2	no	—
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	—	4	2	no	—
49	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₆₁	#131	1	1	yes	characteristic
55	#92418	—	2	2	no	—
56	#92418	—	4	1	no	—
57	#92418	—	8	2	no	—
58	#12085	—	61	1	no	maximal, Sylow
59	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
60	C₄×C₆₁	—	2	1	no	—
61	C₄×C₆₁	—	8	2	no	—
62	C₄×C₆₁	—	8	2	no	—
63	C₄×C₆₁	#436	1	1	yes	characteristic
64	C₄×C₆₁	#443	1	1	yes	characteristic
65	C₄×C₆₁	#510	1	2	yes	—
66	C₄×Q₃₂	D₆₁	1	1	yes	characteristic
67	#1591	—	61	2	no	—
68	#1372	—	61	1	no	—
69	#1332	—	61	2	no	—
70	#1311	—	61	1	no	—
71	C₂×C₆₁	#1589	1	1	yes	characteristic
72	C₂×C₆₁	#2176	1	1	yes	characteristic
73	C₂×C₆₁	#2179	1	1	yes	characteristic
74	C₂×Q₃₂	Dic₆₁	1	1	yes	characteristic
75	#446	—	122	2	no	—
76	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
77	#366	Dic₆₁	1	1	yes	characteristic
78	#358	Dic₆₁	1	2	yes	—
79	C₄×C₁₆	—	61	1	no	—
80	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
81	#335	—	61	2	no	—
82	Q₈⋊C₈	—	122	2	no	—
83	C₈²	—	61	1	no	—
84	C₆₁	#12085	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	C₂×Dic₆₁	1	2	yes	—
86	C₄×Q₈	—	2	2	no	—
87	Q₃₂	—	2	2	no	—
88	C₂×C₁₆	—	122	1	no	—
89	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
90	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
91	#107	C₂×Dic₆₁	1	2	yes	—
92	C₄⋊C₈	—	122	2	no	—
93	C₄⋊C₈	—	244	2	no	—
94	#103	—	2	2	no	—
95	#97	—	244	2	no	—
96	C₄×C₈	—	61	1	no	—
97	C₄×C₈	—	61	1	no	—
98	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	2	2	no	—
102	C₂×C₈	—	61	2	no	—
103	C₂×C₈	—	122	1	no	—
104	C₂×C₈	—	244	2	no	—
105	C₂×C₈	#11758224	1	1	yes	characteristic
106	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	D₄×D₆₁	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	122	2	no	—
113	C₁₆	#11758194	1	2	yes	—
114	Q₈	—	4	2	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
119	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
120	C₂×C₄	#14519796	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	122	2	no	—
123	C₈	—	488	2	no	—
124	C₈	—	244	1	no	—
125	C₈	C₂×#11758194	1	1	yes	characteristic
126	C₈	#14519784	1	1	yes	characteristic
127	C₂²	D₈×Dic₆₁	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	8	2	no	—
130	C₄	—	8	2	no	—
131	C₄	#16839840	1	1	yes	characteristic
132	C₄	#16839879	1	1	yes	characteristic
133	C₄	#16840204	1	2	yes	—
134	C₂	#21444704	1	1	yes	characteristic
135	C₂	#21447690	1	1	yes	characteristic
136	C₂	#21447705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21444716

C₂

yes

maximal

#21443893

C₂

yes

maximal, characteristic

#21443783

C₂

yes

maximal

#21443735

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₆₁

C₂

yes

maximal, characteristic

#16839893

—

C₄×#11758185

C₂²

yes

characteristic

#16839515

C₂²

yes

—

Q₈⋊#92418

—

C₈×#92418

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

C₄

yes

characteristic

C₄×Q₁₆×C₆₁

C₂²

yes

—

C₆₁×#366

C₄

yes

characteristic

C₆₁×#358

C₄

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₂×Q₁₆×C₆₁

C₂×C₄

yes

—

C₄×Q₈×C₆₁

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

#107×C₆₁

C₂×C₄

yes

—

#103×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

—

D₈

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

M₁₆

yes

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

C₂×M₁₆

yes

C₈×C₆₁

yes

—

—

—

—

C₂²×C₆₁

C₄×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×Q₃₂

D₆₁

yes

—

—

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

Dic₆₁

yes

characteristic

#446

—

122

—

C₄×Q₁₆

C₂×D₆₁

yes

—

#366

Dic₆₁

yes

characteristic

#358

Dic₆₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

#335

—

Q₈⋊C₈

—

122

—

C₈²

—

C₆₁

#12085

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×Dic₆₁

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×Dic₆₁

yes

characteristic

C₂×C₁₆

C₂×Dic₆₁

yes

characteristic

#107

C₂×Dic₆₁

yes

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

#97

—

244

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

122

—

104

C₂×C₈

—

244

—

105

C₂×C₈

#11758224

yes

characteristic

106

C₂×C₈

C₂²×Dic₆₁

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

D₄×D₆₁

yes

characteristic

111

C₁₆

—

112

C₁₆

—

122

—

113

C₁₆

#11758194

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₄×Dic₆₁

yes

characteristic

119

C₂×C₄

D₈×D₆₁

yes

characteristic

120

C₂×C₄

#14519796

yes

characteristic

121

C₈

—

122

C₈

—

122

—

123

C₈

—

488

—

124

C₈

—

244

—

125

C₈

C₂×#11758194

yes

characteristic

126

C₈

#14519784

yes

characteristic

127

C₂²

D₈×Dic₆₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#16839840

yes

characteristic

132

C₄

#16839879

yes

characteristic

133

C₄

#16840204

yes

—

134

C₂

#21444704

yes

characteristic

135

C₂

#21447690

yes

characteristic

136

C₂

#21447705

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₄×Q₃₂×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	C₄×D₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₄×Q₃₂	D₆₁
hypocenter	C₆₁	#12085
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₄×Q₃₂×C₆₁

C₂

socle, prime radical

C₂²×C₆₁

C₄×D₈

cosocle

C₄×C₈×C₆₁

C₂³

hypercenter

C₄×Q₃₂

D₆₁

hypocenter

C₆₁

#12085

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

N	H ≅ G/N	type
84. C₆₁	#12085	semidirect

H ≅ G/N

type

84. C₆₁

#12085

semidirect

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

subgroup	index
#21444716	2
#21443893	2
#21443783	2
#21443735	2
C₄×Q₃₂×C₆₁	2
#12085	61

subgroup

index

C₄×Q₃₂×C₆₁

Subgroups — #29592016

Conjugacy classes of subgroups

Complements

Maximal subgroups