Group #29579405 — subgroups

Subgroups — #29579405

Order 15616.

370 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 115 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	#21456333	C₂	1	1	yes	maximal, characteristic
2	#21445950	C₂	1	1	yes	maximal, characteristic
3	#21444384	C₂	1	4	yes	maximal
4	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic
5	#16840319	C₂²	1	2	yes	—
6	#16840167	C₄	1	2	yes	complemented
7	#16840164	—	2	2	no	—
8	#16840160	C₄	1	2	yes	complemented
9	#16839800	—	2	2	no	—
10	#16839628	C₂²	1	2	yes	—
11	#16839525	—	2	2	no	—
12	#16839500	—	2	8	no	—
13	C₆₁×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#322	—	2	2	no	—
16	#14519777	C₂×C₄	1	4	yes	—
17	#14519769	—	2	2	no	—
18	#14519756	—	2	4	no	—
19	C₄×Dic₁₂₂	—	2	4	no	—
20	#14519736	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	#14519711	—	2	4	no	—
23	#14519709	—	2	4	no	—
24	#14519684	—	2	4	no	—
25	C₂²×C₈×C₆₁	D₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₁	—	2	2	no	—
28	C₂²⋊C₈×C₆₁	D₄	1	4	yes	—
29	C₄×C₈×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	D₄	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₆₁	—	4	2	no	—
36	C₂×Dic₁₂₂	—	2	4	no	—
37	C₂×C₈×C₆₁	—	2	4	no	—
38	C₂×C₈×C₆₁	—	2	4	no	—
39	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
41	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₆₁	#41	1	2	yes	—
44	C₂²⋊Dic₆₁	—	4	2	no	—
45	C₄⋊Dic₆₁	—	2	4	no	—
46	C₄⋊Dic₆₁	—	2	4	no	—
47	C₄⋊Dic₆₁	—	4	4	no	—
48	#11758196	—	4	4	no	—
49	C₄×Dic₆₁	—	4	2	no	—
50	C₂³×C₆₁	#127	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	4	2	no	—
52	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₆₁	#121	1	2	yes	—
55	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#137	1	4	yes	—
57	C₂×Dic₆₁	—	4	4	no	—
58	C₂×Dic₆₁	—	8	4	no	—
59	C₂×Dic₆₁	—	8	2	no	—
60	Dic₁₂₂	—	8	4	no	—
61	C₈×C₆₁	—	2	2	no	—
62	C₈×C₆₁	—	2	2	no	—
63	C₈×C₆₁	—	4	4	no	—
64	#9759	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	—	2	2	no	—
66	C₂²×C₆₁	#390	1	1	yes	characteristic
67	C₂²×C₆₁	#497	1	2	yes	—
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	#419	1	2	yes	—
71	C₄×C₆₁	#491	1	2	yes	—
72	Dic₆₁	—	8	4	no	—
73	Dic₆₁	—	16	2	no	—
74	#3160	—	61	1	no	—
75	#1835	—	61	1	no	—
76	#1518	—	61	4	no	—
77	#1452	D₆₁	1	1	yes	characteristic
78	C₂×C₆₁	—	2	2	no	—
79	C₂×C₆₁	#1960	1	1	yes	characteristic
80	C₂×C₆₁	#1976	1	2	yes	—
81	#533	—	61	2	no	—
82	#500	—	61	2	no	—
83	#499	—	122	2	no	—
84	#498	—	61	2	no	—
85	#431	C₂×D₆₁	1	2	yes	—
86	#427	—	122	2	no	—
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	#382	—	61	2	no	—
89	#340	—	122	2	no	—
90	#328	—	122	8	no	—
91	#322	—	2	2	no	—
92	C₆₁	#9759	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	D₂₄₄	1	2	yes	—
94	C₄⋊Q₈	—	61	4	no	—
95	#125	—	122	2	no	—
96	C₂²⋊Q₈	—	122	4	no	—
97	C₄×Q₈	—	122	4	no	—
98	#117	—	122	2	no	—
99	C₂×C₄⋊C₄	—	122	2	no	—
100	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
101	#107	—	122	4	no	—
102	#106	—	122	4	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	D₂₄₄	1	4	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
107	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
109	#95	—	122	4	no	—
110	C₂×Q₈	—	122	4	no	—
111	C₂²×C₄	—	244	2	no	—
112	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
113	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	#11758198	1	2	yes	—
117	C₂×C₈	#11758198	1	2	yes	—
118	C₄⋊C₄	—	122	4	no	—
119	C₄⋊C₄	—	122	4	no	—
120	C₄⋊C₄	—	244	4	no	—
121	C₄⋊C₄	—	244	4	no	—
122	C₂²⋊C₄	—	244	2	no	—
123	C₄²	—	244	2	no	—
124	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
125	C₄²	#11758222	1	2	yes	—
126	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
127	C₂³	#14519774	1	1	yes	characteristic
128	Q₈	—	488	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	488	4	no	—
131	C₂×C₄	—	488	2	no	—
132	C₂×C₄	—	244	4	no	—
133	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
134	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
135	C₂×C₄	#14519751	1	2	yes	—
136	C₂×C₄	#14519765	1	1	yes	characteristic
137	C₂×C₄	#14519810	1	4	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	4	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#16839664	1	1	yes	characteristic
143	C₂²	#16840156	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	976	2	no	—
147	C₄	—	488	4	no	—
148	C₄	#16839772	1	2	yes	—
149	C₄	#16840131	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#21446639	1	1	yes	characteristic
152	C₂	#21446717	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21456333

C₂

yes

maximal, characteristic

#21445950

C₂

yes

maximal, characteristic

#21444384

C₂

yes

maximal

C₆₁×#1452

C₂

yes

maximal, characteristic

#16840319

C₂²

yes

—

#16840167

C₄

yes

complemented

#16840164

—

#16840160

C₄

yes

complemented

#16839800

—

#16839628

C₂²

yes

—

#16839525

—

#16839500

—

C₆₁×#431

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

—

#14519777

C₂×C₄

yes

—

#14519769

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

—

C₂×C₄⋊Dic₆₁

—

#14519711

—

#14519709

—

#14519684

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#9759

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

#497

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#419

yes

—

C₄×C₆₁

#491

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1960

yes

characteristic

C₂×C₆₁

#1976

yes

—

#533

—

#500

—

#499

—

122

—

#498

—

#431

C₂×D₆₁

yes

—

#427

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#382

—

#340

—

122

—

#328

—

122

—

#322

—

C₆₁

#9759

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₄₄

yes

—

C₄⋊Q₈

—

#125

—

122

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

122

—

C₂×C₄⋊C₄

—

122

—

100

C₂×C₄²

C₂²×D₆₁

yes

characteristic

101

#107

—

122

—

102

#106

—

122

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

D₂₄₄

yes

—

105

C₄×C₈

—

106

C₄×C₈

C₄×D₆₁

yes

characteristic

107

C₄×C₈

C₄×D₆₁

yes

characteristic

108

C₄×C₈

C₂²⋊D₆₁

yes

—

109

#95

—

122

—

110

C₂×Q₈

—

122

—

111

C₂²×C₄

—

244

—

112

C₂²×C₄

C₂×D₂₄₄

yes

—

113

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

#11758198

yes

—

117

C₂×C₈

#11758198

yes

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

—

122

—

120

C₄⋊C₄

—

244

—

121

C₄⋊C₄

—

244

—

122

C₂²⋊C₄

—

244

—

123

C₄²

—

244

—

124

C₄²

C₂×C₄×D₆₁

yes

characteristic

125

C₄²

#11758222

yes

—

126

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

127

C₂³

#14519774

yes

characteristic

128

Q₈

—

488

—

129

C₂×C₄

—

130

C₂×C₄

—

488

—

131

C₂×C₄

—

488

—

132

C₂×C₄

—

244

—

133

C₂×C₄

C₂×#11758198

yes

characteristic

134

C₂×C₄

C₄×D₂₄₄

yes

—

135

C₂×C₄

#14519751

yes

—

136

C₂×C₄

#14519765

yes

characteristic

137

C₂×C₄

#14519810

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

#16839664

yes

characteristic

143

C₂²

#16840156

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

976

—

147

C₄

—

488

—

148

C₄

#16839772

yes

—

149

C₄

#16840131

yes

—

150

C₂

—

151

C₂

#21446639

yes

characteristic

152

C₂

#21446717

yes

—

	subgroup	G/H
center	C₂²	#16839664
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1452	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1452	D₆₁
hypocenter	C₆₁	#9759
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16839664

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1452

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
6. #16840167	C₄	semidirect
8. #16840160	C₄	semidirect
92. C₆₁	#9759	semidirect

H ≅ G/N

type

6. #16840167

C₄

semidirect

8. #16840160

C₄

semidirect

92. C₆₁

#9759

semidirect

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

subgroup	index
#21456333	2
#21445950	2
#21444384	2
C₆₁×#1452	2
#9759	61

subgroup

index

C₆₁×#1452

Subgroups — #29579405

Conjugacy classes of subgroups

Complements

Maximal subgroups