Group #29594289 — subgroups

Subgroups — #29594289

Order 15616.

212 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 67 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	#21447727	C₂	1	1	yes	maximal, characteristic
2	#21445186	C₂	1	2	yes	maximal
3	#21443893	C₂	1	1	yes	maximal, characteristic
4	#21443868	C₂	1	2	yes	maximal
5	C₆₁×#2247	C₂	1	1	yes	maximal, characteristic
6	#16840044	—	2	2	no	—
7	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
8	C₄⋊#11758185	C₂²	1	1	yes	characteristic
9	#16839580	—	2	1	no	—
10	#16839537	—	2	1	no	—
11	#16839515	C₂²	1	2	yes	—
12	#16839508	—	2	2	no	—
13	C₂×Q₃₂×C₆₁	—	2	1	no	—
14	C₂×Q₃₂×C₆₁	—	2	1	no	—
15	C₆₁×#494	C₂²	1	2	yes	—
16	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
18	C₂×#11758185	—	2	1	no	—
19	#14519708	—	4	2	no	—
20	C₄⋊#92418	—	2	2	no	—
21	C₄⋊#92418	D₄	1	1	yes	characteristic
22	#14519706	—	2	1	no	—
23	#14519700	—	4	2	no	—
24	C₈×Dic₆₁	—	2	1	no	—
25	C₂×Q₁₆×C₆₁	—	2	2	no	—
26	C₂×Q₁₆×C₆₁	—	2	2	no	—
27	C₄⋊Q₈×C₆₁	—	2	2	no	—
28	Q₃₂×C₆₁	—	4	2	no	—
29	Q₃₂×C₆₁	—	4	2	no	—
30	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₆₁	—	4	2	no	—
34	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
35	Q₁₆×C₆₁	—	4	2	no	—
36	Q₁₆×C₆₁	—	4	2	no	—
37	Q₁₆×C₆₁	—	8	2	no	—
38	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₆₁	—	4	2	no	—
41	C₄⋊C₄×C₆₁	—	4	2	no	—
42	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
44	#11758196	—	2	1	no	—
45	C₄×Dic₆₁	—	2	1	no	—
46	C₂×#92418	—	2	1	no	—
47	C₂×#92418	—	4	2	no	—
48	C₁₆×C₆₁	—	2	1	no	—
49	C₁₆×C₆₁	SD₁₆	1	2	yes	—
50	#11758185	—	4	1	no	—
51	Q₈×C₆₁	—	8	2	no	—
52	Q₈×C₆₁	—	8	2	no	—
53	C₂×C₄×C₆₁	—	8	2	no	—
54	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
57	C₂×Dic₆₁	—	2	1	no	—
58	C₂×Dic₆₁	—	2	1	no	—
59	Dic₁₂₂	—	2	2	no	—
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	#92418	—	4	1	no	—
64	#92418	—	8	2	no	—
65	#12443	—	61	1	no	maximal, Sylow
66	C₂²×C₆₁	#466	1	1	yes	characteristic
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	16	2	no	—
69	C₄×C₆₁	#460	1	1	yes	characteristic
70	C₄×C₆₁	#472	1	1	yes	characteristic
71	C₄×C₆₁	#510	1	2	yes	—
72	Dic₆₁	—	4	1	no	—
73	Dic₆₁	—	4	1	no	—
74	#2247	D₆₁	1	1	yes	characteristic
75	#2183	—	61	1	no	—
76	#1670	—	61	2	no	—
77	#1372	—	61	1	no	—
78	#1362	—	61	2	no	—
79	C₂×C₆₁	#1668	1	1	yes	characteristic
80	C₂×C₆₁	#2219	1	1	yes	characteristic
81	C₂×C₆₁	#2222	1	1	yes	characteristic
82	C₂×Q₃₂	—	2	1	no	—
83	C₂×Q₃₂	—	2	1	no	—
84	#494	C₂×D₆₁	1	2	yes	—
85	#475	—	122	2	no	—
86	C₈×Q₈	—	61	1	no	—
87	C₄⋊C₁₆	—	61	1	no	—
88	C₄⋊C₁₆	—	122	1	no	—
89	#346	—	122	1	no	—
90	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
91	#335	—	61	2	no	—
92	#332	—	122	2	no	—
93	C₆₁	#12443	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	2	2	no	—
96	C₄⋊Q₈	—	2	2	no	—
97	C₄×Q₈	—	61	1	no	—
98	Q₃₂	—	4	2	no	—
99	Q₃₂	—	4	2	no	—
100	C₂×C₁₆	—	122	1	no	—
101	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
103	#106	—	244	2	no	—
104	C₄⋊C₈	—	61	1	no	—
105	C₄⋊C₈	—	122	1	no	—
106	C₄⋊C₈	—	122	2	no	—
107	#103	—	244	2	no	—
108	C₄×C₈	—	122	1	no	—
109	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
110	C₂×Q₈	—	4	2	no	—
111	C₂×Q₈	—	61	1	no	—
112	Q₁₆	—	4	2	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	8	2	no	—
115	C₂×C₈	—	122	1	no	—
116	C₂×C₈	—	244	2	no	—
117	C₂×C₈	#11758224	1	1	yes	characteristic
118	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	61	1	no	—
122	C₄⋊C₄	—	122	1	no	—
123	C₄²	—	122	1	no	—
124	C₄²	D₄×D₆₁	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	244	1	no	—
127	C₁₆	#11758200	1	2	yes	—
128	Q₈	—	8	2	no	—
129	Q₈	—	8	2	no	—
130	Q₈	—	122	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	122	1	no	—
133	C₂×C₄	—	122	1	no	—
134	C₂×C₄	#14519748	1	1	yes	characteristic
135	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
136	C₂×C₄	#14519796	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	488	2	no	—
139	C₈	—	244	1	no	—
140	C₈	C₂×#11758200	1	1	yes	characteristic
141	C₈	#14519803	1	1	yes	characteristic
142	C₂²	#16839995	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	16	2	no	—
145	C₄	—	244	1	no	—
146	C₄	—	244	1	no	—
147	C₄	#16839959	1	1	yes	characteristic
148	C₄	#16840029	1	1	yes	characteristic
149	C₄	#16840204	1	2	yes	—
150	C₂	#21445176	1	1	yes	characteristic
151	C₂	#21447924	1	1	yes	characteristic
152	C₂	#21447941	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447727

C₂

yes

maximal, characteristic

#21445186

C₂

yes

maximal

#21443893

C₂

yes

maximal, characteristic

#21443868

C₂

yes

maximal

C₆₁×#2247

C₂

yes

maximal, characteristic

#16840044

—

C₈×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊#11758185

C₂²

yes

—

—

C₂²

yes

—

#16839508

—

C₂×Q₃₂×C₆₁

—

C₂×Q₃₂×C₆₁

—

C₆₁×#494

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₄×Dic₁₂₂

D₄

yes

—

—

—

D₄

yes

characteristic

#14519706

—

#14519700

—

C₈×Dic₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

—

Q₃₂×C₆₁

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

D₈

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

D₈

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₁₆×C₆₁

—

C₁₆×C₆₁

SD₁₆

yes

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×SD₁₆

yes

C₈×C₆₁

yes

—

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

#2247

D₆₁

yes

—

—

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₆₁

yes

—

#475

—

122

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

#346

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

#335

—

#332

—

122

—

C₆₁

#12443

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

100

C₂×C₁₆

—

122

—

101

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

103

#106

—

244

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

122

—

106

C₄⋊C₈

—

122

—

107

#103

—

244

—

108

C₄×C₈

—

122

—

109

C₄×C₈

C₂²×D₆₁

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

115

C₂×C₈

—

122

—

116

C₂×C₈

—

244

—

117

C₂×C₈

#11758224

yes

characteristic

118

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

122

—

123

C₄²

—

122

—

124

C₄²

D₄×D₆₁

yes

characteristic

125

C₁₆

—

126

C₁₆

—

244

—

127

C₁₆

#11758200

yes

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

122

—

131

C₂×C₄

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

122

—

134

C₂×C₄

#14519748

yes

characteristic

135

C₂×C₄

D₈×D₆₁

yes

characteristic

136

C₂×C₄

#14519796

yes

characteristic

137

C₈

—

138

C₈

—

488

—

139

C₈

—

244

—

140

C₈

C₂×#11758200

yes

characteristic

141

C₈

#14519803

yes

characteristic

142

C₂²

#16839995

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

244

—

146

C₄

—

244

—

147

C₄

#16839959

yes

characteristic

148

C₄

#16840029

yes

characteristic

149

C₄

#16840204

yes

—

150

C₂

#21445176

yes

characteristic

151

C₂

#21447924

yes

characteristic

152

C₂

#21447941

yes

characteristic

	subgroup	G/H
center	C₂²	#16839995
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2247	C₂
socle, prime radical	C₂²×C₆₁	#466
hypercenter	#2247	D₆₁
hypocenter	C₆₁	#12443
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16839995

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2247

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
93. C₆₁	#12443	semidirect

H ≅ G/N

type

93. C₆₁

#12443

semidirect

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

subgroup	index
#21447727	2
#21445186	2
#21443893	2
#21443868	2
C₆₁×#2247	2
#12443	61

subgroup

index

C₆₁×#2247

Subgroups — #29594289

Conjugacy classes of subgroups

Complements

Maximal subgroups