Group #29594672 — subgroups

Subgroups — #29594672

Order 15616.

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

normal subgroups only

This group has 262 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	#21448072	C₂	1	1	yes	maximal, characteristic
2	#21448048	C₂	1	2	yes	maximal
3	#21445417	C₂	1	2	yes	maximal
4	C₁₆×#92418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2252	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₈₈	—	2	2	no	—
7	#16840158	—	2	2	no	—
8	#16840142	C₂²	1	2	yes	—
9	#16840112	—	2	2	no	—
10	#16839565	—	2	4	no	—
11	C₄×#11758185	C₂²	1	1	yes	characteristic
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₆₁×#500	C₂²	1	2	yes	—
14	C₆₁×#366	—	2	2	no	—
15	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₄₄	—	2	2	no	—
17	C₂×Dic₂₄₄	—	2	2	no	—
18	#14519777	—	2	2	no	—
19	Dic₄₈₈	—	4	4	no	—
20	C₂×#11758185	D₄	1	2	yes	—
21	#14519708	—	4	4	no	—
22	#14519701	—	4	4	no	—
23	C₄×#92418	D₄	1	1	yes	characteristic
24	C₄×#92418	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₆₁	—	2	2	no	—
26	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
27	#107×C₆₁	—	2	2	no	—
28	#107×C₆₁	—	2	2	no	—
29	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₆₁	—	4	2	no	—
31	C₂×Dic₁₂₂	—	4	2	no	—
32	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₆₁	—	4	2	no	—
35	C₄⋊C₄×C₆₁	—	4	2	no	—
36	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₄⋊Dic₆₁	—	4	2	no	—
39	C₂×#92418	D₈	1	2	yes	—
40	C₂×#92418	D₈	1	2	yes	—
41	Dic₂₄₄	—	4	4	no	—
42	Dic₂₄₄	—	4	4	no	—
43	C₁₆×C₆₁	SD₁₆	1	4	yes	—
44	#11758185	—	2	2	no	—
45	Q₈×C₆₁	—	8	4	no	—
46	C₂×C₄×C₆₁	—	8	2	no	—
47	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₆₁	—	8	2	no	—
51	Dic₁₂₂	—	8	2	no	—
52	Dic₁₂₂	—	8	2	no	—
53	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
54	C₈×C₆₁	#135	1	2	yes	—
55	#92418	—	2	2	no	—
56	#92418	Q₃₂	1	4	yes	complemented
57	#12504	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	#493	1	1	yes	characteristic
59	C₄×C₆₁	—	16	2	no	—
60	C₄×C₆₁	#492	1	1	yes	characteristic
61	C₄×C₆₁	#495	1	1	yes	characteristic
62	C₄×C₆₁	C₂×Q₃₂	1	2	yes	—
63	C₄×C₆₁	#508	1	2	yes	—
64	Dic₆₁	—	16	2	no	—
65	#2252	D₆₁	1	1	yes	characteristic
66	#2247	—	61	1	no	—
67	#2241	—	61	2	no	—
68	#1707	—	61	2	no	—
69	C₈×C₁₆	—	61	1	no	—
70	C₂×C₆₁	#1706	1	1	yes	characteristic
71	C₂×C₆₁	#2247	1	1	yes	characteristic
72	C₂×C₆₁	#2249	1	1	yes	characteristic
73	C₂×Q₃₂	—	122	2	no	—
74	#500	C₂×D₆₁	1	2	yes	—
75	#498	—	122	2	no	—
76	#494	—	61	2	no	—
77	#487	—	122	2	no	—
78	#366	—	2	2	no	—
79	#358	—	122	4	no	—
80	C₄×C₁₆	—	61	1	no	—
81	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
82	C₈²	—	61	1	no	—
83	C₆₁	#12504	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	122	2	no	—
85	C₂×Q₁₆	—	122	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	122	2	no	—
88	Q₃₂	—	244	4	no	—
89	C₂×C₁₆	—	61	2	no	—
90	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
91	#107	—	2	2	no	—
92	#107	—	2	2	no	—
93	#106	—	244	4	no	—
94	#103	—	244	4	no	—
95	C₄×C₈	—	61	1	no	—
96	C₄×C₈	—	61	1	no	—
97	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	244	2	no	—
100	Q₁₆	—	244	4	no	—
101	Q₁₆	—	244	4	no	—
102	C₂×C₈	—	61	2	no	—
103	C₂×C₈	—	61	2	no	—
104	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	244	2	no	—
109	C₄⋊C₄	—	244	2	no	—
110	C₄²	D₄×D₆₁	1	1	yes	characteristic
111	C₁₆	—	122	2	no	—
112	C₁₆	#11758201	1	4	yes	—
113	Q₈	—	8	4	no	—
114	Q₈	—	488	2	no	—
115	Q₈	—	488	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	488	2	no	—
118	C₂×C₄	#14519773	1	1	yes	characteristic
119	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
120	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
121	C₈	—	61	4	no	—
122	C₈	—	122	2	no	—
123	C₈	C₂×#11758201	1	2	yes	—
124	C₈	#14519800	1	2	yes	—
125	C₂²	#16840137	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	976	2	no	—
128	C₄	#16840136	1	1	yes	characteristic
129	C₄	#16840147	1	1	yes	characteristic
130	C₄	Q₃₂×D₆₁	1	2	yes	—
131	C₄	#16840191	1	2	yes	—
132	C₂	#21445412	1	1	yes	characteristic
133	C₂	#21448070	1	1	yes	characteristic
134	C₂	#21448077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448072

C₂

yes

maximal, characteristic

#21448048

C₂

yes

maximal

#21445417

C₂

yes

maximal

C₁₆×#92418

C₂

yes

maximal, characteristic

C₆₁×#2252

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

—

#16840158

—

#16840142

C₂²

yes

—

#16840112

—

#16839565

—

C₄×#11758185

C₂²

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₆₁×#500

C₂²

yes

—

C₆₁×#366

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

Dic₄₈₈

—

C₂×#11758185

D₄

yes

—

#14519708

—

#14519701

—

C₄×#92418

D₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₄⋊Q₈×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#107×C₆₁

—

#107×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

D₈

yes

—

C₂×#92418

D₈

yes

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

SD₁₆

yes

—

#11758185

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#135

yes

—

#92418

—

#92418

Q₃₂

yes

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×Q₃₂

yes

—

C₄×C₆₁

#508

yes

—

Dic₆₁

—

#2252

D₆₁

yes

—

—

—

C₈×C₁₆

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

—

122

—

#500

C₂×D₆₁

yes

—

#498

—

122

—

#494

—

#487

—

122

—

#366

—

#358

—

122

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

—

C₆₁

#12504

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

Q₃₂

—

244

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₁

yes

—

#107

—

#107

—

#106

—

244

—

#103

—

244

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

244

—

100

Q₁₆

—

244

—

101

Q₁₆

—

244

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₆₁

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

244

—

109

C₄⋊C₄

—

244

—

110

C₄²

D₄×D₆₁

yes

characteristic

111

C₁₆

—

122

—

112

C₁₆

#11758201

yes

—

113

Q₈

—

114

Q₈

—

488

—

115

Q₈

—

488

—

116

C₂×C₄

—

117

C₂×C₄

—

488

—

118

C₂×C₄

#14519773

yes

characteristic

119

C₂×C₄

D₈×D₆₁

yes

characteristic

120

C₂×C₄

D₈×D₆₁

yes

characteristic

121

C₈

—

122

C₈

—

122

—

123

C₈

C₂×#11758201

yes

—

124

C₈

#14519800

yes

—

125

C₂²

#16840137

yes

characteristic

126

C₄

—

127

C₄

—

976

—

128

C₄

#16840136

yes

characteristic

129

C₄

#16840147

yes

characteristic

130

C₄

Q₃₂×D₆₁

yes

—

131

C₄

#16840191

yes

—

132

C₂

#21445412

yes

characteristic

133

C₂

#21448070

yes

characteristic

134

C₂

#21448077

yes

characteristic

	subgroup	G/H
center	C₂²	#16840137
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2252	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	#2252	D₆₁
hypocenter	C₆₁	#12504
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16840137

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2252

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
56. #92418	Q₃₂	semidirect
83. C₆₁	#12504	semidirect

H ≅ G/N

type

56. #92418

Q₃₂

semidirect

83. C₆₁

#12504

semidirect

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

subgroup	index
#21448072	2
#21448048	2
#21445417	2
C₁₆×#92418	2
C₆₁×#2252	2
#12504	61

subgroup

index

#21448072

#21448048

#21445417

C₁₆×#92418

C₆₁×#2252

#12504

Subgroups — #29594672

Conjugacy classes of subgroups

Complements

Maximal subgroups