Group #29594712 — subgroups

Subgroups — #29594712

Order 15616.

268 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21448047	C₂	1	2	yes	maximal, complemented
2	#21448043	C₂	1	2	yes	maximal
3	#21445437	C₂	1	1	yes	maximal, characteristic, complemented
4	#21445425	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840167	—	2	1	no	—
7	#16840167	C₂²	1	2	yes	—
8	#16840142	—	2	1	no	—
9	#16840142	C₂²	1	1	yes	characteristic
10	#16840139	C₂²	1	1	yes	characteristic
11	#16840109	—	2	1	no	—
12	#16840109	—	2	1	no	—
13	#16839567	—	2	4	no	—
14	#16839564	—	2	4	no	—
15	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
16	C₈²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₄₄	—	2	2	no	—
18	C₂×Dic₂₄₄	—	4	2	no	—
19	C₂×D₄₈₈	—	2	2	no	—
20	#14519777	—	2	1	no	—
21	#14519777	—	2	2	no	—
22	#14519774	—	2	1	no	—
23	#14519711	—	2	2	no	—
24	#14519711	—	2	2	no	—
25	#14519711	—	4	2	no	—
26	#14519699	—	4	2	no	—
27	#14519699	—	4	2	no	—
28	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
29	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×D₂₄₄	—	4	1	no	—
33	C₂×D₂₄₄	—	4	1	no	—
34	C₂×D₂₄₄	—	4	1	no	—
35	C₂×Dic₁₂₂	—	4	1	no	—
36	C₂×Dic₁₂₂	—	4	2	no	—
37	C₂×C₈×C₆₁	D₈	1	2	yes	—
38	C₂×C₈×C₆₁	D₈	1	2	yes	—
39	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
40	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₆₁	—	4	2	no	—
42	C₄⋊Dic₆₁	—	4	1	no	—
43	C₄⋊Dic₆₁	—	4	1	no	—
44	C₄⋊Dic₆₁	—	4	2	no	—
45	Dic₂₄₄	—	4	4	no	—
46	Dic₂₄₄	—	8	4	no	—
47	D₄₈₈	—	4	4	no	—
48	C₁₆×C₆₁	—	2	4	no	—
49	C₂²×D₆₁	—	8	1	no	—
50	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₆₁	—	8	1	no	—
54	C₂×Dic₆₁	—	8	2	no	—
55	D₂₄₄	—	8	2	no	—
56	D₂₄₄	—	8	2	no	—
57	D₂₄₄	—	8	2	no	—
58	Dic₁₂₂	—	8	2	no	—
59	Dic₁₂₂	—	8	4	no	—
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	—	2	2	no	—
62	C₈×C₆₁	#135	1	4	yes	—
63	#12512	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	#493	1	1	yes	characteristic
65	C₂×D₆₁	—	16	1	no	—
66	C₂×D₆₁	—	16	1	no	—
67	C₂×D₆₁	—	16	1	no	—
68	C₄×C₆₁	#495	1	2	yes	—
69	C₄×C₆₁	#509	1	2	yes	—
70	C₄×C₆₁	#510	1	2	yes	—
71	Dic₆₁	—	16	1	no	—
72	Dic₆₁	—	16	2	no	—
73	#2241	—	61	2	no	—
74	#2240	—	61	2	no	—
75	#1713	—	61	1	no	—
76	#1709	—	61	1	no	—
77	#1311	D₆₁	1	1	yes	characteristic, complemented
78	C₂×C₆₁	#1709	1	1	yes	characteristic
79	C₂×C₆₁	#2250	1	1	yes	characteristic
80	C₂×C₆₁	#2251	1	1	yes	characteristic
81	D₆₁	—	32	1	no	—
82	#500	—	61	2	no	—
83	#500	—	122	1	no	—
84	#494	—	61	1	no	—
85	#494	—	122	1	no	—
86	#493	—	61	1	no	—
87	#486	—	122	1	no	—
88	#486	—	122	1	no	—
89	#358	—	122	4	no	—
90	#357	—	122	4	no	—
91	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
92	C₈²	C₂×D₆₁	1	1	yes	characteristic
93	C₆₁	#12512	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	122	2	no	—
95	C₂×Q₁₆	—	244	2	no	—
96	C₂×D₈	—	122	2	no	—
97	C₄⋊Q₈	—	122	1	no	—
98	C₄⋊Q₈	—	122	2	no	—
99	#127	—	122	1	no	—
100	C₂×C₁₆	D₂₄₄	1	4	yes	—
101	#107	—	122	2	no	—
102	#107	—	122	2	no	—
103	#107	—	244	2	no	—
104	#102	—	244	2	no	—
105	#102	—	244	2	no	—
106	C₄×C₈	D₂₄₄	1	1	yes	characteristic
107	C₄×C₈	D₂₄₄	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	C₂×Q₈	—	244	1	no	—
110	C₂×Q₈	—	244	2	no	—
111	C₂×D₄	—	244	1	no	—
112	C₂×D₄	—	244	1	no	—
113	C₂×D₄	—	244	1	no	—
114	Q₁₆	—	488	4	no	—
115	Q₁₆	—	244	4	no	—
116	D₈	—	244	4	no	—
117	C₂×C₈	D₄₈₈	1	2	yes	—
118	C₂×C₈	D₄₈₈	1	2	yes	—
119	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
120	C₄⋊C₄	—	244	2	no	—
121	C₄⋊C₄	—	244	1	no	—
122	C₄⋊C₄	—	244	1	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	488	1	no	—
127	Q₈	—	488	4	no	—
128	Q₈	—	488	2	no	—
129	D₄	—	488	2	no	—
130	D₄	—	488	2	no	—
131	D₄	—	488	2	no	—
132	C₂×C₄	—	488	1	no	—
133	C₂×C₄	—	488	2	no	—
134	C₂×C₄	#14519774	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
136	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#14519799	1	4	yes	—
140	C₂²	—	976	1	no	—
141	C₂²	—	976	1	no	—
142	C₂²	—	976	1	no	—
143	C₂²	#16840139	1	1	yes	characteristic
144	C₄	—	976	1	no	—
145	C₄	—	976	2	no	—
146	C₄	#16840146	1	2	yes	—
147	C₄	#16840200	1	2	yes	—
148	C₄	#16840205	1	2	yes	—
149	C₂	—	1952	1	no	—
150	C₂	#21445425	1	1	yes	characteristic
151	C₂	#21448085	1	1	yes	characteristic
152	C₂	#21448090	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448047

C₂

yes

maximal, complemented

#21448043

C₂

yes

maximal

#21445437

C₂

yes

maximal, characteristic, complemented

#21445425

C₂

yes

maximal, characteristic

C₆₁×#1311

C₂

yes

maximal, characteristic, complemented

#16840167

—

#16840167

C₂²

yes

—

#16840142

—

#16840142

C₂²

yes

characteristic

#16840139

C₂²

yes

—

—

—

—

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×D₄₈₈

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

D₄₈₈

—

C₁₆×C₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#509

yes

—

C₄×C₆₁

#510

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#500

—

#500

—

122

—

#494

—

#494

—

122

—

#493

—

#486

—

122

—

#486

—

122

—

#358

—

122

—

#357

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12512

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

244

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

#127

—

122

—

100

C₂×C₁₆

D₂₄₄

yes

—

101

#107

—

122

—

102

#107

—

122

—

103

#107

—

244

—

104

#102

—

244

—

105

#102

—

244

—

106

C₄×C₈

D₂₄₄

yes

characteristic

107

C₄×C₈

D₂₄₄

yes

characteristic

108

C₄×C₈

C₂²×D₆₁

yes

characteristic

109

C₂×Q₈

—

244

—

110

C₂×Q₈

—

244

—

111

C₂×D₄

—

244

—

112

C₂×D₄

—

244

—

113

C₂×D₄

—

244

—

114

Q₁₆

—

488

—

115

Q₁₆

—

244

—

116

D₈

—

244

—

117

C₂×C₈

D₄₈₈

yes

—

118

C₂×C₈

D₄₈₈

yes

—

119

C₂×C₈

C₂×D₂₄₄

yes

—

120

C₄⋊C₄

—

244

—

121

C₄⋊C₄

—

244

—

122

C₄⋊C₄

—

244

—

123

C₄⋊C₄

—

244

—

124

C₄²

C₂×D₂₄₄

yes

characteristic

125

C₁₆

—

126

C₂³

—

488

—

127

Q₈

—

488

—

128

Q₈

—

488

—

129

D₄

—

488

—

130

D₄

—

488

—

131

D₄

—

488

—

132

C₂×C₄

—

488

—

133

C₂×C₄

—

488

—

134

C₂×C₄

#14519774

yes

characteristic

135

C₂×C₄

C₂×D₄₈₈

yes

characteristic

136

C₂×C₄

C₂×D₄₈₈

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#14519799

yes

—

140

C₂²

—

976

—

141

C₂²

—

976

—

142

C₂²

—

976

—

143

C₂²

#16840139

yes

characteristic

144

C₄

—

976

—

145

C₄

—

976

—

146

C₄

#16840146

yes

—

147

C₄

#16840200

yes

—

148

C₄

#16840205

yes

—

149

C₂

—

1952

—

150

C₂

#21445425

yes

characteristic

151

C₂

#21448085

yes

characteristic

152

C₂

#21448090

yes

characteristic

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1311	C₂
socle	C₂²×C₆₁	#493
prime radical	#16840139	C₂²
hypercenter	#1311	D₆₁
hypocenter	C₆₁	#12512
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16840139

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1311

C₂

C₂²×C₆₁

C₂²

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #21448047	C₂	semidirect
3. #21445437	C₂	semidirect
5. C₆₁×#1311	C₂	semidirect
77. #1311	D₆₁	semidirect
93. C₆₁	#12512	semidirect

H ≅ G/N

type

1. #21448047

C₂

semidirect

3. #21445437

C₂

semidirect

5. C₆₁×#1311

C₂

semidirect

77. #1311

D₆₁

semidirect

93. C₆₁

#12512

semidirect

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

subgroup	index
#21448047	2
#21448043	2
#21445437	2
#21445425	2
C₆₁×#1311	2
#12512	61

subgroup

index

C₆₁×#1311

Subgroups — #29594712

Conjugacy classes of subgroups

Complements

Maximal subgroups