Group #29591894 — subgroups

Subgroups — #29591894

Order 15616.

214 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21447691	C₂	1	1	yes	maximal, characteristic
2	#21444639	C₂	1	2	yes	maximal
3	#21443885	C₂	1	1	yes	maximal, characteristic
4	#21443790	C₂	1	2	yes	maximal
5	C₆₁×#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic
7	C₈×Dic₁₂₂	—	2	2	no	—
8	C₄×Dic₂₄₄	C₂²	1	2	yes	—
9	#16839590	C₄	1	1	yes	characteristic
10	#16839567	C₄	1	2	yes	—
11	#16839513	C₂²	1	2	yes	—
12	#16839494	—	2	2	no	—
13	C₆₁×#346	C₂²	1	1	yes	characteristic
14	C₆₁×#346	C₂²	1	1	yes	characteristic
15	C₆₁×#322	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₂₂	—	2	2	no	—
18	Dic₄₈₈	C₈	1	4	yes	complemented
19	#14519711	C₂×C₄	1	2	yes	—
20	C₄⋊#92418	—	2	2	no	—
21	#14519706	—	4	2	no	—
22	#14519702	—	2	2	no	—
23	C₈×Dic₆₁	—	4	2	no	—
24	C₂×C₁₆×C₆₁	—	2	1	no	—
25	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₂₂	—	2	2	no	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
33	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₆₁	—	2	2	no	—
35	#11758196	—	4	2	no	—
36	C₄×Dic₆₁	—	4	2	no	—
37	C₂×#92418	—	4	2	no	—
38	Dic₂₄₄	—	2	2	no	—
39	Dic₂₄₄	C₂×C₈	1	4	yes	—
40	C₁₆×C₆₁	—	2	1	no	—
41	C₁₆×C₆₁	—	4	1	no	—
42	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×Dic₆₁	—	4	2	no	—
46	C₂×Dic₆₁	—	4	2	no	—
47	Dic₁₂₂	—	2	4	no	—
48	Dic₁₂₂	—	4	2	no	—
49	C₈×C₆₁	—	2	1	no	—
50	C₈×C₆₁	—	4	2	no	—
51	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₆₁	#131	1	1	yes	characteristic
53	#92418	—	8	2	no	—
54	#12066	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	#442	1	1	yes	characteristic
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
58	C₄×C₆₁	#444	1	1	yes	characteristic
59	Dic₆₁	—	4	4	no	—
60	Dic₆₁	—	8	2	no	—
61	#2176	—	61	1	no	—
62	#1579	—	61	2	no	—
63	#1368	—	61	1	no	—
64	#1334	—	61	2	no	—
65	#1313	D₆₁	1	1	yes	characteristic
66	C₂×C₆₁	#1580	1	1	yes	characteristic
67	C₂×C₆₁	#2180	1	2	yes	—
68	C₂×Q₃₂	—	61	1	no	—
69	C₈×Q₈	—	122	2	no	—
70	C₄×Q₁₆	—	61	2	no	—
71	#367	—	61	1	no	—
72	#358	—	61	2	no	—
73	#346	C₂×D₆₁	1	1	yes	characteristic
74	#346	C₂×D₆₁	1	1	yes	characteristic
75	#334	—	61	2	no	—
76	Q₈⋊C₈	—	122	2	no	—
77	#322	C₂×D₆₁	1	1	yes	characteristic
78	C₆₁	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	61	2	no	—
80	C₄×Q₈	—	122	2	no	—
81	Q₃₂	—	61	4	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
85	#107	—	61	2	no	—
86	C₄⋊C₈	—	122	2	no	—
87	C₄⋊C₈	—	244	2	no	—
88	#103	—	122	2	no	—
89	C₄×C₈	—	244	2	no	—
90	C₄×C₈	D₂₄₄	1	2	yes	—
91	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
92	C₂×Q₈	—	122	2	no	—
93	Q₁₆	—	61	4	no	—
94	Q₁₆	—	122	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	244	2	no	—
97	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
98	C₂×C₈	#11758222	1	1	yes	characteristic
99	C₄⋊C₄	—	122	2	no	—
100	C₄⋊C₄	—	244	2	no	—
101	C₄²	—	244	2	no	—
102	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	C₈×D₆₁	1	2	yes	—
106	Q₈	—	122	4	no	—
107	Q₈	—	244	2	no	—
108	C₂×C₄	—	244	2	no	—
109	C₂×C₄	—	244	2	no	—
110	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
111	C₂×C₄	#14519805	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	4	2	no	—
114	C₈	—	488	2	no	—
115	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
116	C₈	#14519783	1	1	yes	characteristic
117	C₂²	#16839875	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	488	2	no	—
120	C₄	—	244	4	no	—
121	C₄	C₈×D₂₄₄	1	1	yes	characteristic
122	C₄	#16839885	1	1	yes	characteristic
123	C₂	#21444644	1	1	yes	characteristic
124	C₂	#21447712	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447691

C₂

yes

maximal, characteristic

#21444639

C₂

yes

maximal

#21443885

C₂

yes

maximal, characteristic

#21443790

C₂

yes

maximal

C₆₁×#1313

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

C₄

yes

characteristic

C₈×Dic₁₂₂

—

C₄×Dic₂₄₄

C₂²

yes

—

#16839590

C₄

yes

characteristic

#16839567

C₄

yes

—

#16839513

C₂²

yes

—

#16839494

—

C₆₁×#346

C₂²

yes

characteristic

C₆₁×#346

C₂²

yes

characteristic

C₆₁×#322

C₂²

yes

characteristic

C₂×Dic₂₄₄

C₂×C₄

yes

—

C₄×Dic₁₂₂

—

Dic₄₈₈

C₈

yes

complemented

#14519711

C₂×C₄

yes

—

C₄⋊#92418

—

#14519706

—

#14519702

—

C₈×Dic₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

C₂×C₈

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

#444

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×Q₃₂

—

C₈×Q₈

—

122

—

C₄×Q₁₆

—

#367

—

#358

—

#346

C₂×D₆₁

yes

characteristic

#346

C₂×D₆₁

yes

characteristic

#334

—

Q₈⋊C₈

—

122

—

#322

C₂×D₆₁

yes

characteristic

C₆₁

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

122

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₁

yes

characteristic

C₂×C₁₆

C₄×D₆₁

yes

characteristic

#107

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

122

—

C₄×C₈

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

Q₁₆

—

Q₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

C₂×C₈

#11758222

yes

characteristic

C₄⋊C₄

—

122

—

100

C₄⋊C₄

—

244

—

101

C₄²

—

244

—

102

C₄²

C₂×D₂₄₄

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

C₈×D₆₁

yes

—

106

Q₈

—

122

—

107

Q₈

—

244

—

108

C₂×C₄

—

244

—

109

C₂×C₄

—

244

—

110

C₂×C₄

C₄×D₂₄₄

yes

characteristic

111

C₂×C₄

#14519805

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

488

—

115

C₈

C₂×C₈×D₆₁

yes

characteristic

116

C₈

#14519783

yes

characteristic

117

C₂²

#16839875

yes

characteristic

118

C₄

—

119

C₄

—

488

—

120

C₄

—

244

—

121

C₄

C₈×D₂₄₄

yes

characteristic

122

C₄

#16839885

yes

characteristic

123

C₂

#21444644

yes

characteristic

124

C₂

#21447712

yes

—

	subgroup	G/H
center	C₂²	#16839875
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1313	C₂
socle, prime radical	C₂²×C₆₁	#442
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1313	D₆₁
hypocenter	C₆₁	#12066
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#16839875

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1313

C₂

socle, prime radical

C₂²×C₆₁

#442

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
18. Dic₄₈₈	C₈	semidirect
78. C₆₁	#12066	semidirect

H ≅ G/N

type

18. Dic₄₈₈

C₈

semidirect

78. C₆₁

#12066

semidirect

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

subgroup	index
#21447691	2
#21444639	2
#21443885	2
#21443790	2
C₆₁×#1313	2
#12066	61

subgroup

index

C₆₁×#1313

Subgroups — #29591894

Conjugacy classes of subgroups

Complements

Maximal subgroups