Group #29594689 — subgroups

Subgroups — #29594689

Order 15616.

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

normal subgroups only

This group has 258 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	#21448072	C₂	1	1	yes	maximal, characteristic
2	#21448048	C₂	1	2	yes	maximal
3	#21445417	C₂	1	2	yes	maximal
4	#21443735	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2254	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×#367	—	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	2	yes	—
24	C₄⋊Q₈×C₆₁	—	2	2	no	—
25	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
26	#107×C₆₁	—	2	2	no	—
27	#107×C₆₁	—	2	2	no	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₆₁	—	4	2	no	—
30	C₂×Dic₁₂₂	—	4	2	no	—
31	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	—	4	2	no	—
34	C₄⋊C₄×C₆₁	—	4	2	no	—
35	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₆₁	—	4	2	no	—
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₂×#92418	D₈	1	4	yes	—
39	Dic₂₄₄	—	4	4	no	—
40	Dic₂₄₄	—	4	4	no	—
41	C₁₆×C₆₁	SD₁₆	1	4	yes	—
42	#11758185	—	2	2	no	—
43	Q₈×C₆₁	—	8	4	no	—
44	C₂×C₄×C₆₁	—	8	2	no	—
45	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
47	C₂×Dic₆₁	—	8	2	no	—
48	Dic₁₂₂	—	8	2	no	—
49	Dic₁₂₂	—	8	2	no	—
50	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
51	C₈×C₆₁	#135	1	2	yes	—
52	#92418	—	2	4	no	—
53	#12507	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	#493	1	1	yes	characteristic
55	C₄×C₆₁	—	16	2	no	—
56	C₄×C₆₁	#492	1	1	yes	characteristic
57	C₄×C₆₁	#495	1	1	yes	characteristic
58	C₄×C₆₁	#510	1	4	yes	—
59	Dic₆₁	—	16	2	no	—
60	#2254	D₆₁	1	1	yes	characteristic
61	#2247	—	61	1	no	—
62	#2241	—	61	2	no	—
63	#1707	—	61	2	no	—
64	#1311	—	61	1	no	—
65	C₂×C₆₁	#1706	1	1	yes	characteristic
66	C₂×C₆₁	#2251	1	2	yes	—
67	C₂×Q₃₂	—	122	2	no	—
68	#500	C₂×D₆₁	1	2	yes	—
69	#498	—	122	2	no	—
70	#494	—	61	2	no	—
71	#487	—	122	2	no	—
72	#367	—	2	2	no	—
73	#358	—	122	4	no	—
74	C₄×C₁₆	—	61	1	no	—
75	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
76	C₈²	—	61	1	no	—
77	C₆₁	#12507	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	122	2	no	—
79	C₂×Q₁₆	—	122	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	122	2	no	—
82	Q₃₂	—	244	4	no	—
83	C₂×C₁₆	—	61	2	no	—
84	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
85	#107	—	2	2	no	—
86	#107	—	2	2	no	—
87	#106	—	244	4	no	—
88	#103	—	244	4	no	—
89	C₄×C₈	—	61	2	no	—
90	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	244	2	no	—
93	Q₁₆	—	244	4	no	—
94	Q₁₆	—	244	4	no	—
95	C₂×C₈	—	61	4	no	—
96	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
97	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	244	2	no	—
101	C₄⋊C₄	—	244	2	no	—
102	C₄²	D₄×D₆₁	1	1	yes	characteristic
103	C₁₆	—	122	2	no	—
104	C₁₆	#11758201	1	4	yes	—
105	Q₈	—	8	4	no	—
106	Q₈	—	488	2	no	—
107	Q₈	—	488	2	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	488	2	no	—
110	C₂×C₄	#14519773	1	1	yes	characteristic
111	C₂×C₄	D₈×D₆₁	1	2	yes	—
112	C₈	—	122	4	no	—
113	C₈	C₂×#11758201	1	2	yes	—
114	C₈	#14519800	1	2	yes	—
115	C₂²	#16840137	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	976	2	no	—
118	C₄	#16840136	1	1	yes	characteristic
119	C₄	#16840147	1	1	yes	characteristic
120	C₄	#16840201	1	4	yes	—
121	C₂	#21445412	1	1	yes	characteristic
122	C₂	#21448089	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448072

C₂

yes

maximal, characteristic

#21448048

C₂

yes

maximal

#21445417

C₂

yes

maximal

#21443735

C₂

yes

maximal, characteristic

C₆₁×#2254

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×#367

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

Dic₄₈₈

—

C₂×#11758185

D₄

yes

—

#14519708

—

#14519701

—

C₄×#92418

D₄

yes

—

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

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

SD₁₆

yes

—

#11758185

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#135

yes

—

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

#2254

D₆₁

yes

—

—

—

—

C₂×C₆₁

#1706

yes

characteristic

C₂×C₆₁

#2251

yes

—

C₂×Q₃₂

—

122

—

#500

C₂×D₆₁

yes

—

#498

—

122

—

#494

—

#487

—

122

—

#367

—

#358

—

122

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₈²

—

C₆₁

#12507

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

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

244

—

Q₁₆

—

244

—

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

D₄×D₆₁

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

244

—

101

C₄⋊C₄

—

244

—

102

C₄²

D₄×D₆₁

yes

characteristic

103

C₁₆

—

122

—

104

C₁₆

#11758201

yes

—

105

Q₈

—

106

Q₈

—

488

—

107

Q₈

—

488

—

108

C₂×C₄

—

109

C₂×C₄

—

488

—

110

C₂×C₄

#14519773

yes

characteristic

111

C₂×C₄

D₈×D₆₁

yes

—

112

C₈

—

122

—

113

C₈

C₂×#11758201

yes

—

114

C₈

#14519800

yes

—

115

C₂²

#16840137

yes

characteristic

116

C₄

—

117

C₄

—

976

—

118

C₄

#16840136

yes

characteristic

119

C₄

#16840147

yes

characteristic

120

C₄

#16840201

yes

—

121

C₂

#21445412

yes

characteristic

122

C₂

#21448089

yes

—

	subgroup	G/H
center	C₂²	#16840137
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2254	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	#2254	D₆₁
hypocenter	C₆₁	#12507
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#16840137

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2254

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
77. C₆₁	#12507	semidirect

H ≅ G/N

type

77. C₆₁

#12507

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
#21443735	2
C₆₁×#2254	2
#12507	61

subgroup

index

C₆₁×#2254

Subgroups — #29594689

Conjugacy classes of subgroups

Complements

Maximal subgroups