Group #29573710 — subgroups

Subgroups — #29573710

Order 15616.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#21445991	C₂	1	1	yes	maximal, characteristic
2	#21444111	C₂	1	1	yes	maximal, characteristic
3	#21443677	C₂	1	4	yes	maximal
4	C₆₁×#1835	C₂	1	1	yes	maximal, characteristic
5	#16839815	C₂²	1	2	yes	—
6	C₂×#14519713	—	2	2	no	—
7	#16839786	C₂²	1	2	yes	—
8	#16839484	C₄	1	4	yes	complemented
9	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#382	C₂²	1	2	yes	—
11	C₆₁×#340	—	2	2	no	—
12	C₂×#11758194	—	2	2	no	—
13	#14519713	—	4	4	no	—
14	C₄⋊#92418	—	2	2	no	—
15	C₄⋊#92418	—	2	4	no	—
16	C₂²⋊#92418	—	2	2	no	—
17	C₄×#92418	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₆₁	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
20	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₁	Q₈	1	2	yes	—
23	#95×C₆₁	—	2	4	no	—
24	C₂²×C₄×C₆₁	—	4	2	no	—
25	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	#41	1	2	yes	—
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
30	C₄⋊C₄×C₆₁	—	4	2	no	—
31	C₄⋊C₄×C₆₁	—	4	2	no	—
32	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₆₁	#41	1	2	yes	—
35	#11758194	—	4	4	no	—
36	C₂×#92418	—	2	4	no	—
37	C₂×#92418	—	4	2	no	—
38	C₂³×C₆₁	#124	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	4	2	no	—
40	C₂×C₄×C₆₁	—	8	4	no	—
41	C₂×C₄×C₆₁	—	8	2	no	—
42	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	#117	1	2	yes	—
44	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#125	1	2	yes	—
46	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×C₄×C₆₁	#137	1	2	yes	—
48	C₈×C₆₁	—	2	4	no	—
49	#92418	—	4	4	no	—
50	#92418	—	8	2	no	—
51	#8794	—	61	1	no	maximal, Sylow
52	C₂²×C₆₁	—	2	2	no	—
53	C₂²×C₆₁	#382	1	1	yes	characteristic
54	C₂²×C₆₁	#489	1	2	yes	—
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	—	16	2	no	—
58	C₄×C₆₁	#428	1	2	yes	—
59	C₄×C₆₁	#490	1	1	yes	characteristic
60	C₄×C₆₁	#491	1	1	yes	characteristic
61	#1848	—	61	1	no	—
62	#1835	D₆₁	1	1	yes	characteristic
63	#1452	—	61	1	no	—
64	#1286	—	61	4	no	—
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	#1833	1	2	yes	—
67	C₂×C₆₁	#1836	1	1	yes	characteristic
68	#431	—	61	2	no	—
69	C₂×#108	—	122	2	no	—
70	#423	—	61	2	no	—
71	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
72	#382	C₂×D₆₁	1	2	yes	—
73	#340	—	2	2	no	—
74	#322	—	61	4	no	—
75	C₆₁	#8794	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	122	2	no	—
77	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
80	#108	—	244	4	no	—
81	C₄⋊C₈	—	122	2	no	—
82	C₄⋊C₈	—	122	4	no	—
83	C₂²⋊C₈	—	122	2	no	—
84	C₄×C₈	—	61	4	no	—
85	C₄×C₈	Dic₁₂₂	1	2	yes	—
86	C₄×C₈	C₄×D₆₁	1	2	yes	—
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	#11758224	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
91	M₁₆	—	244	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	122	4	no	—
95	C₂×C₈	—	244	2	no	—
96	C₂×C₈	#11758196	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
100	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
101	C₄²	#11758226	1	2	yes	—
102	C₂³	#14519764	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
107	C₂×C₄	#14519735	1	2	yes	—
108	C₂×C₄	#14519766	1	1	yes	characteristic
109	C₂×C₄	#14519768	1	2	yes	—
110	C₂×C₄	#14519802	1	2	yes	—
111	C₂×C₄	#14519807	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	488	2	no	—
114	C₈	—	244	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#16839627	1	1	yes	characteristic
117	C₂²	#16840121	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#16839805	1	2	yes	—
122	C₄	#16840128	1	1	yes	characteristic
123	C₄	#16840132	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#21445942	1	2	yes	—
126	C₂	#21445952	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445991

C₂

yes

maximal, characteristic

#21444111

C₂

yes

maximal, characteristic

#21443677

C₂

yes

maximal

C₆₁×#1835

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×#14519713

—

#16839786

C₂²

yes

—

#16839484

C₄

yes

complemented

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#382

C₂²

yes

—

C₆₁×#340

—

—

—

—

—

—

C₂×C₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

#11758194

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

#125

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#382

yes

characteristic

C₂²×C₆₁

#489

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#428

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

—

D₆₁

yes

characteristic

#1452

—

#1286

—

C₂×C₆₁

—

C₂×C₆₁

#1833

yes

—

C₂×C₆₁

yes

—

—

122

—

#423

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#382

C₂×D₆₁

yes

—

#340

—

#322

—

C₆₁

#8794

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

C₂²⋊D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#108

—

244

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

C₄×D₆₁

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11758224

yes

—

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

#11758196

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₁₂₂

yes

characteristic

100

C₄²

C₂×C₄×D₆₁

yes

characteristic

101

C₄²

#11758226

yes

—

102

C₂³

#14519764

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×#11758196

yes

characteristic

107

C₂×C₄

#14519735

yes

—

108

C₂×C₄

#14519766

yes

characteristic

109

C₂×C₄

#14519768

yes

—

110

C₂×C₄

#14519802

yes

—

111

C₂×C₄

#14519807

yes

—

112

C₈

—

113

C₈

—

488

—

114

C₈

—

244

—

115

C₂²

—

116

C₂²

#16839627

yes

characteristic

117

C₂²

#16840121

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#16839805

yes

—

122

C₄

#16840128

yes

characteristic

123

C₄

#16840132

yes

characteristic

124

C₂

—

125

C₂

#21445942

yes

—

126

C₂

#21445952

yes

characteristic

	subgroup	G/H
center	C₂²	#16839627
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1835	C₂
socle	C₂²×C₆₁	#382
prime radical	C₂³×C₆₁	#124
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1835	D₆₁
hypocenter	C₆₁	#8794
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#16839627

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1835

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #16839484	C₄	semidirect
75. C₆₁	#8794	semidirect

H ≅ G/N

type

8. #16839484

C₄

semidirect

75. C₆₁

#8794

semidirect

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

subgroup	index
#21445991	2
#21444111	2
#21443677	2
C₆₁×#1835	2
#8794	61

subgroup

index

C₆₁×#1835

Subgroups — #29573710

Conjugacy classes of subgroups

Complements

Maximal subgroups