Group #29594663 — subgroups

Subgroups — #29594663

Order 15616.

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

normal subgroups only

This group has 306 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	#21448094	C₂	1	1	yes	maximal, characteristic, complemented
2	#21448042	C₂	1	2	yes	maximal, complemented
3	#21445408	C₂	1	2	yes	maximal, complemented
4	C₁₆×#92418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2246	C₂	1	1	yes	maximal, characteristic
6	#16840167	C₂²	1	2	yes	—
7	#16840134	—	2	2	no	—
8	#16840108	—	2	2	no	—
9	#16839588	—	2	2	no	—
10	#16839563	—	2	4	no	—
11	C₄×#11758185	C₂²	1	1	yes	characteristic
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₆₁	—	2	2	no	—
14	C₆₁×#493	C₂²	1	2	yes	—
15	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#11758200	—	4	4	no	—
17	#14519777	—	2	2	no	—
18	C₂×#11758185	D₄	1	2	yes	—
19	#14519711	—	2	2	no	—
20	#14519711	—	2	2	no	—
21	#14519698	—	4	4	no	—
22	C₄×#92418	D₄	1	1	yes	characteristic
23	C₄×#92418	D₄	1	1	yes	characteristic
24	C₂×D₈×C₆₁	—	2	2	no	—
25	C₂×D₈×C₆₁	—	2	2	no	—
26	#127×C₆₁	—	2	2	no	—
27	D₁₆×C₆₁	—	4	4	no	—
28	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
29	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₆₁	—	4	2	no	—
31	C₂×D₄×C₆₁	—	4	2	no	—
32	C₂×D₄×C₆₁	—	4	2	no	—
33	C₂×Dic₁₂₂	—	4	2	no	—
34	D₈×C₆₁	—	4	4	no	—
35	D₈×C₆₁	—	4	4	no	—
36	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
39	#11758200	—	8	8	no	—
40	C₄⋊Dic₆₁	—	4	2	no	—
41	C₄⋊Dic₆₁	—	4	2	no	—
42	C₂×#92418	D₈	1	2	yes	—
43	C₂×#92418	D₈	1	2	yes	—
44	C₁₆×C₆₁	SD₁₆	1	4	yes	complemented
45	#11758185	—	2	2	no	—
46	C₂³×C₆₁	—	8	2	no	—
47	D₄×C₆₁	—	8	2	no	—
48	D₄×C₆₁	—	8	2	no	—
49	D₄×C₆₁	—	8	4	no	—
50	D₄×C₆₁	—	8	4	no	—
51	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₆₁	—	8	2	no	—
55	Dic₁₂₂	—	8	4	no	—
56	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
57	C₈×C₆₁	#135	1	2	yes	—
58	#92418	—	2	2	no	—
59	#92418	D₁₆	1	4	yes	complemented
60	#12503	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	—	16	2	no	—
62	C₂²×C₆₁	—	16	2	no	—
63	C₂²×C₆₁	—	16	2	no	—
64	C₂²×C₆₁	#493	1	1	yes	characteristic
65	C₄×C₆₁	#492	1	1	yes	characteristic
66	C₄×C₆₁	#495	1	1	yes	characteristic
67	C₄×C₆₁	C₂×D₁₆	1	2	yes	—
68	C₄×C₆₁	#508	1	2	yes	—
69	Dic₆₁	—	16	2	no	—
70	#2252	—	61	1	no	—
71	#2246	D₆₁	1	1	yes	characteristic
72	#2240	—	61	2	no	—
73	#1706	—	61	2	no	—
74	C₈×C₁₆	—	61	1	no	—
75	C₂×C₆₁	—	32	2	no	—
76	C₂×C₆₁	#1706	1	1	yes	characteristic
77	C₂×C₆₁	#2246	1	1	yes	characteristic
78	C₂×C₆₁	#2249	1	1	yes	characteristic
79	C₂×D₁₆	—	2	2	no	—
80	#500	—	61	2	no	—
81	#493	C₂×D₆₁	1	2	yes	—
82	#492	—	122	2	no	—
83	#486	—	122	2	no	—
84	#366	—	122	2	no	—
85	#357	—	122	4	no	—
86	C₄×C₁₆	—	61	1	no	—
87	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
88	C₈²	—	61	1	no	—
89	C₆₁	#12503	1	1	yes	characteristic, complemented, Sylow
90	C₂×SD₁₆	—	244	4	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	2	2	no	—
93	C₄⋊Q₈	—	122	2	no	—
94	#127	—	2	2	no	—
95	D₁₆	—	4	4	no	—
96	C₂×C₁₆	—	61	2	no	—
97	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
98	#107	—	122	2	no	—
99	#107	—	122	2	no	—
100	#102	—	244	4	no	—
101	C₄×C₈	—	61	1	no	—
102	C₄×C₈	—	61	1	no	—
103	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
104	C₂×Q₈	—	244	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	—	4	2	no	—
108	SD₁₆	—	488	8	no	—
109	D₈	—	4	4	no	—
110	D₈	—	4	4	no	—
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	—	61	2	no	—
113	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
115	C₄⋊C₄	—	244	2	no	—
116	C₄⋊C₄	—	244	2	no	—
117	C₄²	D₄×D₆₁	1	1	yes	characteristic
118	C₁₆	—	122	2	no	—
119	C₁₆	#11758200	1	4	yes	complemented
120	C₂³	—	8	2	no	—
121	Q₈	—	488	4	no	—
122	D₄	—	8	2	no	—
123	D₄	—	8	2	no	—
124	D₄	—	8	4	no	—
125	D₄	—	8	4	no	—
126	C₂×C₄	—	488	2	no	—
127	C₂×C₄	#14519773	1	1	yes	characteristic
128	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
129	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
130	C₈	—	61	4	no	—
131	C₈	—	122	2	no	—
132	C₈	C₂×#11758200	1	2	yes	—
133	C₈	#14519796	1	2	yes	—
134	C₂²	—	16	2	no	—
135	C₂²	—	16	2	no	—
136	C₂²	—	16	2	no	—
137	C₂²	#16840137	1	1	yes	characteristic
138	C₄	—	976	2	no	—
139	C₄	#16840134	1	1	yes	characteristic
140	C₄	#16840145	1	1	yes	characteristic
141	C₄	D₁₆×D₆₁	1	2	yes	—
142	C₄	#16840191	1	2	yes	—
143	C₂	—	32	2	no	—
144	C₂	#21445408	1	1	yes	characteristic
145	C₂	#21448067	1	1	yes	characteristic
146	C₂	#21448077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448094

C₂

yes

maximal, characteristic, complemented

#21448042

C₂

yes

maximal, complemented

#21445408

C₂

yes

maximal, complemented

C₁₆×#92418

C₂

yes

maximal, characteristic, complemented

C₆₁×#2246

C₂

yes

maximal, characteristic

#16840167

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₂×D₁₆×C₆₁

—

C₆₁×#493

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

—

—

D₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

#127×C₆₁

—

D₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×Dic₁₂₂

—

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

#11758200

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

D₈

yes

—

C₂×#92418

D₈

yes

—

C₁₆×C₆₁

SD₁₆

yes

complemented

#11758185

—

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#135

yes

—

#92418

—

#92418

D₁₆

yes

complemented

#12503

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×D₁₆

yes

—

C₄×C₆₁

#508

yes

—

Dic₆₁

—

#2252

—

#2246

D₆₁

yes

characteristic

#2240

—

#1706

—

C₈×C₁₆

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×D₁₆

—

#500

—

#493

C₂×D₆₁

yes

—

#492

—

122

—

#486

—

122

—

#366

—

122

—

#357

—

122

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

—

C₆₁

#12503

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

244

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

122

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₁

yes

—

#107

—

122

—

#107

—

122

—

100

#102

—

244

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₆₁

yes

characteristic

104

C₂×Q₈

—

244

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

488

—

109

D₈

—

110

D₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₆₁

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

115

C₄⋊C₄

—

244

—

116

C₄⋊C₄

—

244

—

117

C₄²

D₄×D₆₁

yes

characteristic

118

C₁₆

—

122

—

119

C₁₆

#11758200

yes

complemented

120

C₂³

—

121

Q₈

—

488

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

488

—

127

C₂×C₄

#14519773

yes

characteristic

128

C₂×C₄

D₈×D₆₁

yes

characteristic

129

C₂×C₄

D₈×D₆₁

yes

characteristic

130

C₈

—

131

C₈

—

122

—

132

C₈

C₂×#11758200

yes

—

133

C₈

#14519796

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#16840137

yes

characteristic

138

C₄

—

976

—

139

C₄

#16840134

yes

characteristic

140

C₄

#16840145

yes

characteristic

141

C₄

D₁₆×D₆₁

yes

—

142

C₄

#16840191

yes

—

143

C₂

—

144

C₂

#21445408

yes

characteristic

145

C₂

#21448067

yes

characteristic

146

C₂

#21448077

yes

characteristic

	subgroup	G/H
center	C₂²	#16840137
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting, prime radical	C₆₁×#2246	C₂
socle	C₂²×C₆₁	#493
hypercenter	#2246	D₆₁
hypocenter	C₆₁	#12503
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#16840137

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting, prime radical

C₆₁×#2246

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #21448094	C₂	semidirect
2. #21448042	C₂	semidirect
3. #21445408	C₂	semidirect
4. C₁₆×#92418	C₂	semidirect
44. C₁₆×C₆₁	SD₁₆	semidirect
59. #92418	D₁₆	semidirect
89. C₆₁	#12503	semidirect
119. C₁₆	#11758200	semidirect

H ≅ G/N

type

1. #21448094

C₂

semidirect

2. #21448042

C₂

semidirect

3. #21445408

C₂

semidirect

4. C₁₆×#92418

C₂

semidirect

44. C₁₆×C₆₁

SD₁₆

semidirect

59. #92418

D₁₆

semidirect

89. C₆₁

#12503

semidirect

119. C₁₆

#11758200

semidirect

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

subgroup	index
#21448094	2
#21448042	2
#21445408	2
C₁₆×#92418	2
C₆₁×#2246	2
#12503	61

subgroup

index

#21448094

#21448042

#21445408

C₁₆×#92418

C₆₁×#2246

#12503

Subgroups — #29594663

Conjugacy classes of subgroups

Complements

Maximal subgroups