Group #29591818 — subgroups

Subgroups — #29591818

Order 15616.

206 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 87 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	#21447724	C₂	1	2	yes	maximal
2	C₈×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#21443797	C₂	1	2	yes	maximal
4	C₈×#11758185	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1370	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₂₂	C₂²	1	2	yes	—
7	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic
8	#16839580	—	2	2	no	—
9	#16839537	—	2	2	no	—
10	C₄×#11758185	C₂²	1	1	yes	characteristic
11	C₄×#11758185	C₂²	1	1	yes	characteristic
12	#16839515	C₄	1	1	yes	characteristic
13	#16839494	C₄	1	2	yes	—
14	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₄₄	C₈	1	1	yes	characteristic
17	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
18	C₂×#11758185	—	2	1	no	—
19	C₂×#11758185	D₄	1	1	yes	characteristic
20	C₂×#11758185	D₄	1	1	yes	characteristic
21	#14519711	C₈	1	1	yes	characteristic
22	C₄⋊#92418	C₂×C₄	1	2	yes	—
23	#14519706	—	2	2	no	—
24	#14519702	C₈	1	2	yes	—
25	C₈×Dic₆₁	—	2	2	no	—
26	C₂×C₁₆×C₆₁	—	2	2	no	—
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₆₁	—	2	1	no	—
32	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
36	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
38	#11758196	—	2	2	no	—
39	C₄×Dic₆₁	—	2	2	no	—
40	C₂×#92418	—	2	2	no	—
41	Dic₂₄₄	—	2	2	no	—
42	C₁₆×C₆₁	—	4	2	no	—
43	#11758185	—	2	2	no	—
44	#11758185	—	2	1	no	—
45	#11758185	Q₁₆	1	2	yes	complemented
46	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
49	C₂×Dic₆₁	—	2	2	no	—
50	C₂×Dic₆₁	—	2	2	no	—
51	Dic₁₂₂	—	2	2	no	—
52	Dic₁₂₂	—	2	2	no	—
53	C₈×C₆₁	—	2	2	no	—
54	C₈×C₆₁	—	2	1	no	—
55	C₈×C₆₁	—	2	1	no	—
56	C₈×C₆₁	M₃₂	1	2	yes	—
57	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₆₁	#135	1	1	yes	characteristic
59	#92418	—	4	2	no	—
60	#12054	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
64	C₄×C₆₁	#443	1	1	yes	characteristic
65	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
66	C₄×C₆₁	#504	1	1	yes	characteristic
67	Dic₆₁	—	4	2	no	—
68	Dic₆₁	—	4	2	no	—
69	#2183	—	61	2	no	—
70	C₈×Q₁₆	—	61	1	no	—
71	#1370	D₆₁	1	1	yes	characteristic
72	Q₈⋊C₁₆	—	61	2	no	—
73	C₈×C₁₆	—	61	1	no	—
74	C₂×C₆₁	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₆₁	#2169	1	1	yes	characteristic
76	C₂×C₆₁	#2170	1	1	yes	characteristic
77	C₈×Q₈	—	61	2	no	—
78	C₄×Q₁₆	—	61	1	no	—
79	C₄⋊C₁₆	—	122	2	no	—
80	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
81	#346	—	122	2	no	—
82	C₄×C₁₆	—	61	1	no	—
83	C₄×C₁₆	—	61	1	no	—
84	#335	—	61	1	no	—
85	Q₈⋊C₈	—	61	2	no	—
86	C₈²	C₂×D₆₁	1	1	yes	characteristic
87	C₆₁	#12054	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	61	1	no	—
89	C₄×Q₈	—	61	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	61	1	no	—
92	C₂×C₁₆	—	61	1	no	—
93	C₂×C₁₆	—	122	1	no	—
94	#107	—	61	1	no	—
95	C₄⋊C₈	—	61	2	no	—
96	C₄⋊C₈	—	122	2	no	—
97	#103	—	61	2	no	—
98	C₄×C₈	—	122	2	no	—
99	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
100	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
102	C₂×Q₈	—	61	2	no	—
103	Q₁₆	—	122	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	122	2	no	—
106	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
107	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
108	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
109	C₂×C₈	#11758226	1	1	yes	characteristic
110	C₄⋊C₄	—	61	2	no	—
111	C₄⋊C₄	—	122	2	no	—
112	C₄²	—	122	2	no	—
113	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
114	C₁₆	—	4	2	no	—
115	C₁₆	—	61	2	no	—
116	C₁₆	—	122	2	no	—
117	C₁₆	—	122	1	no	—
118	Q₈	—	122	2	no	—
119	Q₈	—	122	2	no	—
120	C₂×C₄	—	122	2	no	—
121	C₂×C₄	—	122	2	no	—
122	C₂×C₄	#14519739	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
124	C₂×C₄	#14519785	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	244	2	no	—
129	C₈	#14519716	1	2	yes	—
130	C₈	Q₁₆×D₆₁	1	1	yes	characteristic
131	C₈	#14519796	1	1	yes	characteristic
132	C₂²	#16839827	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	244	2	no	—
135	C₄	—	244	2	no	—
136	C₄	#16839856	1	1	yes	characteristic
137	C₄	#16839878	1	1	yes	characteristic
138	C₄	C₂×#14519716	1	1	yes	characteristic
139	C₄	#16840180	1	1	yes	characteristic
140	C₂	#21444625	1	1	yes	characteristic
141	C₂	#21447650	1	1	yes	characteristic
142	C₂	#21447655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447724

C₂

yes

maximal

C₈×Dic₂₄₄

C₂

yes

maximal, characteristic

#21443797

C₂

yes

maximal

C₈×#11758185

C₂

yes

maximal, characteristic

C₆₁×#1370

C₂

yes

maximal, characteristic

C₈×Dic₁₂₂

C₂²

yes

—

C₄×Dic₂₄₄

C₄

yes

—

—

C₂²

yes

characteristic

C₄×#11758185

C₂²

yes

characteristic

#16839515

C₄

yes

characteristic

#16839494

C₄

yes

—

C₆₁×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

C₈

yes

characteristic

C₄×Dic₁₂₂

C₂×C₄

yes

—

C₂×#11758185

—

C₂×#11758185

D₄

yes

characteristic

C₂×#11758185

D₄

yes

characteristic

#14519711

C₈

yes

characteristic

C₄⋊#92418

C₂×C₄

yes

—

#14519706

—

#14519702

C₈

yes

—

C₈×Dic₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊Dic₆₁

C₂×C₈

yes

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

C₁₆×C₆₁

—

#11758185

—

#11758185

—

#11758185

Q₁₆

yes

complemented

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

#131

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

M₃₂

yes

—

C₈×C₆₁

C₂×Q₁₆

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₁₆

yes

characteristic

C₄×C₆₁

#443

yes

characteristic

C₄×C₆₁

C₂×M₃₂

yes

characteristic

C₄×C₆₁

#504

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2183

—

C₈×Q₁₆

—

#1370

D₆₁

yes

characteristic

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₆₁

C₈×Q₁₆

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

122

—

C₄⋊C₁₆

C₂×D₆₁

yes

—

#346

—

122

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12054

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

#103

—

C₄×C₈

—

122

—

C₄×C₈

C₄×D₆₁

yes

characteristic

100

C₄×C₈

C₄×D₆₁

yes

characteristic

101

C₄×C₈

C₂²×D₆₁

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

122

—

104

C₂×C₈

—

105

C₂×C₈

—

122

—

106

C₂×C₈

C₈×D₆₁

yes

characteristic

107

C₂×C₈

C₈×D₆₁

yes

characteristic

108

C₂×C₈

D₄×D₆₁

yes

characteristic

109

C₂×C₈

#11758226

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

122

—

112

C₄²

—

122

—

113

C₄²

C₂×C₄×D₆₁

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

122

—

117

C₁₆

—

122

—

118

Q₈

—

122

—

119

Q₈

—

122

—

120

C₂×C₄

—

122

—

121

C₂×C₄

—

122

—

122

C₂×C₄

#14519739

yes

characteristic

123

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

124

C₂×C₄

#14519785

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

244

—

129

C₈

#14519716

yes

—

130

C₈

Q₁₆×D₆₁

yes

characteristic

131

C₈

#14519796

yes

characteristic

132

C₂²

#16839827

yes

characteristic

133

C₄

—

134

C₄

—

244

—

135

C₄

—

244

—

136

C₄

#16839856

yes

characteristic

137

C₄

#16839878

yes

characteristic

138

C₄

C₂×#14519716

yes

characteristic

139

C₄

#16840180

yes

characteristic

140

C₂

#21444625

yes

characteristic

141

C₂

#21447650

yes

characteristic

142

C₂

#21447655

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1370	C₂
socle, prime radical	C₂²×C₆₁	C₈×D₄
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1370	D₆₁
hypocenter	C₆₁	#12054
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1370

C₂

socle, prime radical

C₂²×C₆₁

C₈×D₄

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
45. #11758185	Q₁₆	semidirect
87. C₆₁	#12054	semidirect

H ≅ G/N

type

45. #11758185

Q₁₆

semidirect

87. C₆₁

#12054

semidirect

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

subgroup	index
#21447724	2
C₈×Dic₂₄₄	2
#21443797	2
C₈×#11758185	2
C₆₁×#1370	2
#12054	61

subgroup

index

C₈×Dic₂₄₄

C₆₁×#1370

Subgroups — #29591818

Conjugacy classes of subgroups

Complements

Maximal subgroups