Group #29595424 — subgroups

Subgroups — #29595424

Order 15616.

314 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16839591	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₆₁	C₂	1	2	yes	maximal
4	#21443913	C₂	1	1	yes	maximal, characteristic
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	M₁₆×Dic₆₁	—	2	2	no	—
9	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839591	C₄	1	8	yes	complemented
11	C₁₆×Dic₆₁	C₄	1	4	yes	—
12	#16839526	—	2	2	no	—
13	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₁	C₂²	1	2	yes	—
15	C₂×#11758194	—	2	2	no	—
16	C₂²×#92418	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
18	C₂×#11758185	C₂×C₄	1	2	yes	—
19	C₂×#11758185	C₂×C₄	1	4	yes	—
20	#14519713	—	2	2	no	—
21	#14519713	C₂×C₄	1	4	yes	—
22	#14519687	—	4	2	no	—
23	C₈×Dic₆₁	—	4	2	no	—
24	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
25	C₈×Dic₆₁	Q₈	1	2	yes	—
26	C₂×M₁₆×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₆₁	—	2	2	no	—
29	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
30	#108×C₆₁	—	2	2	no	—
31	#108×C₆₁	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₆₁	Q₁₆	1	2	yes	—
34	M₁₆×C₆₁	—	2	4	no	—
35	M₁₆×C₆₁	—	4	2	no	—
36	C₂×C₈×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
39	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₆₁	#41	1	2	yes	—
41	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₆₁	D₈	1	1	yes	characteristic
43	C₄×Dic₆₁	D₈	1	1	yes	characteristic
44	#11758194	—	2	4	no	—
45	#11758194	—	4	2	no	—
46	C₂×#92418	—	2	2	no	—
47	C₂×#92418	—	4	2	no	—
48	C₂×#92418	C₄⋊C₄	1	2	yes	—
49	C₁₆×C₆₁	—	2	2	no	—
50	#11758185	C₄²	1	4	yes	—
51	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#135	1	2	yes	—
56	C₂×Dic₆₁	—	2	4	no	—
57	C₂×Dic₆₁	#107	1	2	yes	—
58	C₂×Dic₆₁	#107	1	2	yes	—
59	C₈×C₆₁	—	4	4	no	—
60	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
61	C₈×C₆₁	#117	1	1	yes	characteristic
62	C₈×C₆₁	C₄×Q₈	1	2	yes	—
63	#92418	—	2	2	no	—
64	#92418	—	4	4	no	—
65	C₄×#368	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	—	2	2	no	—
67	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
68	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
69	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₆₁	#427	1	1	yes	characteristic
71	C₄×C₆₁	C₄×D₈	1	2	yes	—
72	Dic₆₁	—	2	2	no	—
73	Dic₆₁	#368	1	4	yes	complemented
74	C₂×#368	—	61	2	no	—
75	C₂×C₄×C₁₆	—	61	1	no	—
76	C₄×#108	—	61	2	no	—
77	#1381	—	61	1	no	—
78	#1381	D₆₁	1	1	yes	characteristic
79	C₂×C₆₁	—	2	2	no	—
80	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
81	C₂×C₆₁	C₂×#368	1	2	yes	—
82	C₂²×C₁₆	—	61	1	no	—
83	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
84	C₂×#108	—	61	2	no	—
85	C₂×#108	C₂×D₆₁	1	2	yes	—
86	C₄×M₁₆	—	122	2	no	—
87	C₂×C₄×C₈	—	61	1	no	—
88	#368	—	61	8	no	—
89	C₄×C₁₆	—	61	4	no	—
90	#341	—	122	2	no	—
91	C₆₁	C₄×#368	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	122	2	no	—
94	C₂²×C₈	—	61	1	no	—
95	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
96	C₂×C₄²	—	61	1	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	61	2	no	—
99	C₂×C₁₆	—	61	4	no	—
100	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
101	#108	—	2	2	no	—
102	#108	—	61	4	no	—
103	#108	—	122	2	no	—
104	#108	C₄×D₆₁	1	4	yes	—
105	#97	—	244	2	no	—
106	C₄×C₈	—	61	2	no	—
107	C₄×C₈	—	61	2	no	—
108	C₄×C₈	—	244	2	no	—
109	C₂²×C₄	—	61	2	no	—
110	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
111	M₁₆	—	2	4	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	122	4	no	—
114	M₁₆	—	244	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	—	122	2	no	—
118	C₂×C₈	—	244	2	no	—
119	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
121	C₂×C₈	#11758224	1	2	yes	—
122	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
123	C₄²	—	61	2	no	—
124	C₄²	—	61	1	no	—
125	C₄²	—	61	1	no	—
126	C₁₆	—	2	2	no	—
127	C₁₆	—	61	4	no	—
128	C₂³	Q₁₆×D₆₁	1	1	yes	characteristic
129	C₂×C₄	—	61	2	no	—
130	C₂×C₄	—	61	2	no	—
131	C₂×C₄	—	122	4	no	—
132	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄	#14519737	1	2	yes	—
134	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
135	C₂×C₄	#14519798	1	2	yes	—
136	C₈	—	4	4	no	—
137	C₈	—	122	2	no	—
138	C₈	—	244	4	no	—
139	C₈	C₄²×D₆₁	1	1	yes	characteristic
140	C₈	#14519733	1	1	yes	characteristic
141	C₈	#14519742	1	2	yes	—
142	C₂²	—	2	2	no	—
143	C₂²	#107×D₆₁	1	1	yes	characteristic
144	C₂²	#16839854	1	2	yes	—
145	C₄	—	61	4	no	—
146	C₄	—	122	2	no	—
147	C₄	#16839613	1	1	yes	characteristic
148	C₄	#16839797	1	1	yes	characteristic
149	C₄	#16839842	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#21445711	1	1	yes	characteristic
152	C₂	#368×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839591

C₂

yes

maximal

C₂×C₁₆×Dic₆₁

C₂

yes

maximal, characteristic

#108×Dic₆₁

C₂

yes

maximal

#21443913

C₂

yes

maximal, characteristic

C₆₁×#1381

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

M₁₆×Dic₆₁

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839591

C₄

yes

complemented

C₁₆×Dic₆₁

C₄

yes

—

#16839526

—

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×C₄

yes

—

C₂×#11758185

C₂×C₄

yes

—

#14519713

—

#14519713

C₂×C₄

yes

—

#14519687

—

C₈×Dic₆₁

—

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

C₂×C₄

yes

—

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

C₂²×C₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₄×Dic₆₁

C₄⋊C₄

yes

—

C₄×Dic₆₁

D₈

yes

characteristic

C₄×Dic₆₁

D₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₁₆×C₆₁

—

#11758185

C₄²

yes

—

C₂³×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#107

yes

—

C₂×Dic₆₁

#107

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₄²

yes

characteristic

C₈×C₆₁

#117

yes

characteristic

C₈×C₆₁

C₄×Q₈

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#107

yes

characteristic

C₂²×C₆₁

C₄×Q₁₆

yes

—

C₄×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₄×C₆₁

#427

yes

characteristic

C₄×C₆₁

C₄×D₈

yes

—

Dic₆₁

—

Dic₆₁

#368

yes

complemented

C₂×#368

—

C₂×C₄×C₁₆

—

C₄×#108

—

#1381

—

#1381

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

C₄×#107

yes

characteristic

C₂×C₆₁

C₂×#368

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₆₁

yes

—

C₄×M₁₆

—

122

—

C₂×C₄×C₈

—

#368

—

C₄×C₁₆

—

#341

—

122

—

C₆₁

C₄×#368

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

—

100

C₂×C₁₆

C₄×D₆₁

yes

—

101

#108

—

102

#108

—

103

#108

—

122

—

104

#108

C₄×D₆₁

yes

—

105

#97

—

244

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

244

—

109

C₂²×C₄

—

110

C₂²×C₄

D₄×D₆₁

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

122

—

114

M₁₆

—

244

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

122

—

118

C₂×C₈

—

244

—

119

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

120

C₂×C₈

C₂×C₄×D₆₁

yes

—

121

C₂×C₈

#11758224

yes

—

122

C₂×C₈

Q₈×D₆₁

yes

characteristic

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₁₆

—

127

C₁₆

—

128

C₂³

Q₁₆×D₆₁

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

122

—

132

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

133

C₂×C₄

#14519737

yes

—

134

C₂×C₄

D₈×D₆₁

yes

characteristic

135

C₂×C₄

#14519798

yes

—

136

C₈

—

137

C₈

—

122

—

138

C₈

—

244

—

139

C₈

C₄²×D₆₁

yes

characteristic

140

C₈

#14519733

yes

characteristic

141

C₈

#14519742

yes

—

142

C₂²

—

143

C₂²

#107×D₆₁

yes

characteristic

144

C₂²

#16839854

yes

—

145

C₄

—

146

C₄

—

122

—

147

C₄

#16839613

yes

characteristic

148

C₄

#16839797

yes

characteristic

149

C₄

#16839842

yes

—

150

C₂

—

151

C₂

#21445711

yes

characteristic

152

C₂

#368×D₆₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₈×C₆₁	C₂×C₄²
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	C₄×#368
LCS	G ▷ 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

C₈×C₆₁

C₂×C₄²

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 ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
10. #16839591	C₄	semidirect
73. Dic₆₁	#368	semidirect
91. C₆₁	C₄×#368	semidirect

H ≅ G/N

type

10. #16839591

C₄

semidirect

73. Dic₆₁

#368

semidirect

91. C₆₁

C₄×#368

semidirect

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

subgroup	index
C₂×#16839591	2
C₂×C₁₆×Dic₆₁	2
#108×Dic₆₁	2
#21443913	2
C₆₁×#1381	2
C₄×#368	61

subgroup

index

C₂×#16839591

C₂×C₁₆×Dic₆₁

#108×Dic₆₁

#21443913

C₆₁×#1381

C₄×#368

Subgroups — #29595424

Conjugacy classes of subgroups

Complements

Maximal subgroups