Group #29573669 — subgroups

Subgroups — #29573669

Order 15616.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21445571	C₂	1	1	yes	maximal, characteristic
2	#21444565	C₂	1	2	yes	maximal
3	#21443663	C₂	1	2	yes	maximal
4	#21443648	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1274	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519688	C₂²	1	1	yes	characteristic
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	#16839517	C₄	1	2	yes	—
10	#16839517	C₄	1	2	yes	—
11	#16839514	—	2	2	no	—
12	#16839512	—	2	2	no	—
13	C₆₁×#431	C₂²	1	2	yes	—
14	C₆₁×#376	C₂²	1	1	yes	characteristic
15	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
16	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
17	C₂²×#92418	C₂×C₄	1	2	yes	—
18	C₄⋊#92418	—	2	2	no	—
19	C₂²⋊#92418	—	2	2	no	—
20	#14519688	—	2	2	no	—
21	C₄×#92418	—	2	2	no	—
22	C₄×#92418	D₄	1	2	yes	—
23	C₄×#92418	Q₈	1	2	yes	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₆₁	—	2	2	no	—
27	C₂²⋊C₈×C₆₁	—	2	2	no	—
28	C₄×C₈×C₆₁	—	2	2	no	—
29	#95×C₆₁	—	2	2	no	—
30	C₂²×C₄×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	#41	1	2	yes	—
33	C₂×C₈×C₆₁	—	4	2	no	—
34	C₄²×C₆₁	—	2	2	no	—
35	C₄²×C₆₁	—	2	2	no	—
36	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₆₁	#41	1	2	yes	—
39	C₂×#92418	—	2	2	no	—
40	C₂×#92418	—	2	2	no	—
41	C₂×#92418	—	2	1	no	—
42	C₂×#92418	—	2	1	no	—
43	C₂×#92418	—	2	2	no	—
44	C₂×#92418	—	2	2	no	—
45	C₂×#92418	—	4	2	no	—
46	C₂×#92418	C₄⋊C₄	1	4	yes	—
47	C₂³×C₆₁	#117	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	4	2	no	—
49	C₂×C₄×C₆₁	—	4	4	no	—
50	C₂×C₄×C₆₁	—	4	2	no	—
51	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#124	1	2	yes	—
54	C₂×C₄×C₆₁	#125	1	2	yes	—
55	C₂×C₄×C₆₁	#131	1	2	yes	—
56	C₂×C₄×C₆₁	#131	1	2	yes	—
57	C₈×C₆₁	—	8	2	no	—
58	#92418	—	2	4	no	—
59	#92418	—	4	1	no	—
60	#92418	—	4	1	no	—
61	#92418	—	8	2	no	—
62	#8783	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	2	2	no	—
64	C₂²×C₆₁	#382	1	1	yes	characteristic
65	C₂²×C₆₁	#433	1	2	yes	—
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	—	8	2	no	—
69	C₄×C₆₁	#424	1	2	yes	—
70	C₄×C₆₁	#433	1	1	yes	characteristic
71	C₄×C₆₁	#433	1	1	yes	characteristic
72	#1746	—	61	1	no	—
73	#1564	—	61	2	no	—
74	#1281	—	61	2	no	—
75	#1274	—	61	1	no	—
76	#1274	D₆₁	1	1	yes	characteristic
77	C₂×C₆₁	—	2	2	no	—
78	C₂×C₆₁	#1832	1	1	yes	characteristic
79	C₂×C₆₁	#1833	1	2	yes	—
80	#431	—	61	2	no	—
81	#431	C₂×D₆₁	1	2	yes	—
82	C₂×#97	—	61	1	no	—
83	C₂×C₄×C₈	—	61	1	no	—
84	#376	C₂×D₆₁	1	1	yes	characteristic
85	#336	—	61	2	no	—
86	#336	—	61	2	no	—
87	#335	—	122	2	no	—
88	#334	—	122	2	no	—
89	C₆₁	#8783	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	61	2	no	—
91	C₂²×C₈	—	61	1	no	—
92	C₂²×C₈	—	61	1	no	—
93	C₂×C₄²	C₄×D₆₁	1	2	yes	—
94	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	122	2	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₂²⋊C₈	—	122	2	no	—
99	#97	—	122	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	61	2	no	—
102	C₄×C₈	—	61	2	no	—
103	C₄×C₈	—	122	2	no	—
104	#95	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
107	C₂²×C₄	#11758224	1	2	yes	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	61	4	no	—
110	C₂×C₈	—	122	2	no	—
111	C₂×C₈	—	122	2	no	—
112	C₂×C₈	—	122	1	no	—
113	C₂×C₈	—	122	1	no	—
114	C₂×C₈	—	122	2	no	—
115	C₂×C₈	—	122	2	no	—
116	C₂×C₈	—	244	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	#11758222	1	2	yes	—
120	C₄²	D₄×D₆₁	1	1	yes	characteristic
121	C₄²	Q₈×D₆₁	1	1	yes	characteristic
122	C₂³	#14519733	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
127	C₂×C₄	#14519734	1	1	yes	characteristic
128	C₂×C₄	#14519763	1	2	yes	—
129	C₂×C₄	#14519767	1	2	yes	—
130	C₂×C₄	#14519783	1	2	yes	—
131	C₂×C₄	#14519785	1	2	yes	—
132	C₈	—	8	2	no	—
133	C₈	—	122	4	no	—
134	C₈	—	488	2	no	—
135	C₈	—	244	1	no	—
136	C₈	—	244	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	#16839626	1	1	yes	characteristic
139	C₂²	#16839820	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	8	2	no	—
143	C₄	#16839787	1	2	yes	—
144	C₄	#16839821	1	1	yes	characteristic
145	C₄	#16839824	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#21445937	1	1	yes	characteristic
148	C₂	#21445941	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445571

C₂

yes

maximal, characteristic

#21444565

C₂

yes

maximal

#21443663

C₂

yes

maximal

#21443648

C₂

yes

maximal, characteristic

C₆₁×#1274

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×#14519688

C₂²

yes

characteristic

C₂×C₄×#92418

C₂²

yes

characteristic

#16839517

C₄

yes

—

#16839517

C₄

yes

—

#16839514

—

#16839512

—

C₆₁×#431

C₂²

yes

—

C₆₁×#376

C₂²

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#92418

Q₈

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂³×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#125

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₈×C₆₁

—

—

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#382

yes

characteristic

C₂²×C₆₁

#433

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#424

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1832

yes

characteristic

C₂×C₆₁

#1833

yes

—

#431

—

#431

C₂×D₆₁

yes

—

C₂×#97

—

C₂×C₄×C₈

—

#376

C₂×D₆₁

yes

—

—

—

122

—

#334

—

122

—

C₆₁

#8783

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₆₁

yes

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

C₂²⋊C₈

—

122

—

#97

—

122

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

122

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

107

C₂²×C₄

#11758224

yes

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

122

—

111

C₂×C₈

—

122

—

112

C₂×C₈

—

122

—

113

C₂×C₈

—

122

—

114

C₂×C₈

—

122

—

115

C₂×C₈

—

122

—

116

C₂×C₈

—

244

—

117

C₄²

—

118

C₄²

—

119

C₄²

#11758222

yes

—

120

C₄²

D₄×D₆₁

yes

characteristic

121

C₄²

Q₈×D₆₁

yes

characteristic

122

C₂³

#14519733

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

127

C₂×C₄

#14519734

yes

characteristic

128

C₂×C₄

#14519763

yes

—

129

C₂×C₄

#14519767

yes

—

130

C₂×C₄

#14519783

yes

—

131

C₂×C₄

#14519785

yes

—

132

C₈

—

133

C₈

—

122

—

134

C₈

—

488

—

135

C₈

—

244

—

136

C₈

—

244

—

137

C₂²

—

138

C₂²

#16839626

yes

characteristic

139

C₂²

#16839820

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#16839787

yes

—

144

C₄

#16839821

yes

characteristic

145

C₄

#16839824

yes

characteristic

146

C₂

—

147

C₂

#21445937

yes

characteristic

148

C₂

#21445941

yes

—

	subgroup	G/H
center	C₂×C₄	#14519734
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1274	C₂
socle	C₂²×C₆₁	#382
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1274	D₆₁
hypocenter	C₆₁	#8783
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#14519734

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1274

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
89. C₆₁	#8783	semidirect

H ≅ G/N

type

89. C₆₁

#8783

semidirect

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

subgroup	index
#21445571	2
#21444565	2
#21443663	2
#21443648	2
C₆₁×#1274	2
#8783	61

subgroup

index

C₆₁×#1274

Subgroups — #29573669

Conjugacy classes of subgroups

Complements

Maximal subgroups