Group #29596404 — subgroups

Subgroups — #29596404

Order 15616.

274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×#16839537	C₂	1	1	yes	maximal, characteristic
2	#21445989	C₂	1	2	yes	maximal
3	#21443913	C₂	1	1	yes	maximal, characteristic
4	#21443912	C₂	1	2	yes	maximal
5	C₆₁×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂×#14519713	C₂²	1	2	yes	—
8	C₂×#14519712	—	2	2	no	—
9	#16839785	—	2	2	no	—
10	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	#16839537	C₄	1	4	yes	—
12	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×#108×C₆₁	C₂²	1	2	yes	—
14	C₂×#11758194	—	2	2	no	—
15	C₂²×#92418	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
17	C₂×#11758185	—	2	2	no	—
18	C₂×#11758185	C₂×C₄	1	2	yes	—
19	#14519713	—	2	2	no	—
20	#14519713	—	2	2	no	—
21	#14519712	—	4	4	no	—
22	#14519706	—	4	4	no	—
23	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
24	C₈×Dic₆₁	Q₈	1	2	yes	—
25	C₂×M₁₆×C₆₁	—	2	2	no	—
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
29	#108×C₆₁	—	2	2	no	—
30	#108×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₆₁	Q₁₆	1	2	yes	—
33	M₁₆×C₆₁	—	4	2	no	—
34	M₁₆×C₆₁	—	4	2	no	—
35	C₂×C₈×C₆₁	—	4	2	no	—
36	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₆₁	#41	1	2	yes	—
39	C₂×C₈×C₆₁	#41	1	2	yes	—
40	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
41	C₄×Dic₆₁	D₈	1	2	yes	—
42	#11758194	—	4	2	no	—
43	#11758194	—	4	2	no	—
44	C₂×#92418	—	4	2	no	—
45	C₂×#92418	C₄⋊C₄	1	2	yes	—
46	C₂×#92418	SD₁₆	1	4	yes	—
47	C₁₆×C₆₁	—	2	2	no	—
48	#11758185	—	2	2	no	—
49	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#124	1	2	yes	—
52	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
54	C₂×Dic₆₁	—	4	2	no	—
55	C₂×Dic₆₁	#107	1	4	yes	—
56	C₈×C₆₁	—	8	2	no	—
57	C₈×C₆₁	#117	1	1	yes	characteristic
58	C₈×C₆₁	#117	1	1	yes	characteristic
59	C₈×C₆₁	#125	1	2	yes	—
60	#92418	—	8	2	no	—
61	#92418	#106	1	4	yes	—
62	#12892	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	2	no	—
64	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
65	C₂²×C₆₁	#487	1	2	yes	—
66	C₄×C₆₁	#382	1	1	yes	characteristic
67	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
68	C₄×C₆₁	#486	1	2	yes	—
69	Dic₆₁	—	2	4	no	—
70	C₂×#346	—	61	1	no	—
71	#1848	—	61	2	no	—
72	#1381	—	61	1	no	—
73	#1381	—	61	2	no	—
74	#1381	D₆₁	1	1	yes	characteristic
75	C₂×C₆₁	—	2	2	no	—
76	C₂×C₆₁	#1834	1	1	yes	characteristic
77	C₂×C₆₁	#2161	1	2	yes	—
78	C₂²×C₁₆	—	61	1	no	—
79	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
80	C₂×#108	—	61	2	no	—
81	C₂×#108	—	122	2	no	—
82	C₂×#108	C₂×D₆₁	1	2	yes	—
83	#423	—	122	2	no	—
84	C₂×C₄×C₈	—	61	1	no	—
85	#346	—	61	4	no	—
86	C₆₁	#12892	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	122	2	no	—
89	C₂²×C₈	—	61	1	no	—
90	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
91	C₂×C₄²	—	61	1	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	61	2	no	—
94	C₂×C₁₆	—	122	2	no	—
95	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	122	2	no	—
99	#108	—	122	2	no	—
100	#108	—	244	4	no	—
101	C₄⋊C₈	—	244	4	no	—
102	C₄×C₈	—	61	2	no	—
103	C₄×C₈	—	61	2	no	—
104	C₂²×C₄	—	61	2	no	—
105	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	244	2	no	—
109	M₁₆	—	244	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	—	61	4	no	—
113	C₂×C₈	—	244	2	no	—
114	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
115	C₂×C₈	#11758222	1	2	yes	—
116	C₂×C₈	#11758224	1	2	yes	—
117	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
118	C₄²	—	61	2	no	—
119	C₄²	—	61	2	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	122	2	no	—
122	C₂³	Q₁₆×D₆₁	1	1	yes	characteristic
123	C₂×C₄	—	61	4	no	—
124	C₂×C₄	—	244	2	no	—
125	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
126	C₂×C₄	#14519763	1	2	yes	—
127	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
128	C₂×C₄	SD₁₆×D₆₁	1	2	yes	—
129	C₈	—	8	2	no	—
130	C₈	—	61	4	no	—
131	C₈	—	488	2	no	—
132	C₈	#14519733	1	1	yes	characteristic
133	C₈	#14519734	1	1	yes	characteristic
134	C₈	#14519767	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#107×D₆₁	1	1	yes	characteristic
137	C₂²	#16840111	1	2	yes	—
138	C₄	—	122	4	no	—
139	C₄	#16839626	1	1	yes	characteristic
140	C₄	#106×D₆₁	1	1	yes	characteristic
141	C₄	#16840107	1	2	yes	—
142	C₂	—	2	2	no	—
143	C₂	#21445945	1	1	yes	characteristic
144	C₂	#21447612	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839537

C₂

yes

maximal, characteristic

#21445989

C₂

yes

maximal

#21443913

C₂

yes

maximal, characteristic

#21443912

C₂

yes

maximal

C₆₁×#1381

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

C₂×#14519712

—

#16839785

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839537

C₄

yes

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₂×#11758194

—

C₂²×#92418

D₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

#108×C₆₁

—

#108×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

Q₁₆

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄×Dic₆₁

C₄⋊C₄

yes

—

C₄×Dic₆₁

D₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#92418

SD₁₆

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#107

yes

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

#92418

—

#92418

#106

yes

—

#12892

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#107

yes

characteristic

C₂²×C₆₁

#487

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1834

yes

characteristic

C₂×C₆₁

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂×#108

—

C₂×#108

—

122

—

C₂×#108

C₂×D₆₁

yes

—

#423

—

122

—

C₂×C₄×C₈

—

#346

—

C₆₁

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₆₁

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₄×D₆₁

yes

—

#108

—

#108

—

#108

—

122

—

#108

—

122

—

100

#108

—

244

—

101

C₄⋊C₈

—

244

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₂²×C₄

—

105

C₂²×C₄

D₄×D₆₁

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

244

—

109

M₁₆

—

244

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

244

—

114

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

115

C₂×C₈

#11758222

yes

—

116

C₂×C₈

#11758224

yes

—

117

C₂×C₈

Q₈×D₆₁

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

122

—

122

C₂³

Q₁₆×D₆₁

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

244

—

125

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

126

C₂×C₄

#14519763

yes

—

127

C₂×C₄

D₈×D₆₁

yes

characteristic

128

C₂×C₄

SD₁₆×D₆₁

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

488

—

132

C₈

#14519733

yes

characteristic

133

C₈

#14519734

yes

characteristic

134

C₈

#14519767

yes

—

135

C₂²

—

136

C₂²

#107×D₆₁

yes

characteristic

137

C₂²

#16840111

yes

—

138

C₄

—

122

—

139

C₄

#16839626

yes

characteristic

140

C₄

#106×D₆₁

yes

characteristic

141

C₄

#16840107

yes

—

142

C₂

—

143

C₂

#21445945

yes

characteristic

144

C₂

#21447612

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1381	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×C₆₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1381	D₆₁
hypocenter	C₆₁	#12892
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1381

C₂

socle

C₂²×C₆₁

C₂×#107

prime radical

C₂³×C₆₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
86. C₆₁	#12892	semidirect

H ≅ G/N

type

86. C₆₁

#12892

semidirect

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

subgroup	index
C₂×#16839537	2
#21445989	2
#21443913	2
#21443912	2
C₆₁×#1381	2
#12892	61

subgroup

index

C₆₁×#1381

Subgroups — #29596404

Conjugacy classes of subgroups

Complements

Maximal subgroups