Group #29573711 — subgroups

Subgroups — #29573711

Order 15616.

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

normal subgroups only

This group has 270 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	#21445991	C₂	1	1	yes	maximal, characteristic
2	#21445950	C₂	1	1	yes	maximal, characteristic
3	#21443678	C₂	1	4	yes	maximal
4	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519713	—	2	2	no	—
6	#16839786	C₂²	1	2	yes	—
7	#16839628	C₂²	1	2	yes	—
8	#16839525	—	2	2	no	—
9	C₆₁×#431	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#322	C₄	1	4	yes	complemented
12	C₂×#11758194	—	2	2	no	—
13	C₂×C₄⋊Dic₆₁	—	2	2	no	—
14	#14519713	—	4	4	no	—
15	C₄⋊#92418	—	2	4	no	—
16	#14519684	—	2	4	no	—
17	C₂²×C₈×C₆₁	D₄	1	2	yes	—
18	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₆₁	—	2	2	no	—
20	C₂²⋊C₈×C₆₁	—	2	2	no	—
21	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₆₁	Q₈	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	#41	1	2	yes	—
26	C₂²×Dic₆₁	—	4	2	no	—
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	—	2	4	no	—
30	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
31	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₆₁	#41	1	2	yes	—
34	C₄⋊Dic₆₁	—	4	2	no	—
35	C₄⋊Dic₆₁	—	4	2	no	—
36	#11758194	—	4	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	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#117	1	2	yes	—
42	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
43	C₂×C₄×C₆₁	#125	1	2	yes	—
44	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×C₄×C₆₁	#137	1	2	yes	—
46	C₂×Dic₆₁	—	8	4	no	—
47	C₂×Dic₆₁	—	8	2	no	—
48	C₈×C₆₁	—	2	4	no	—
49	C₈×C₆₁	—	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	#428	1	2	yes	—
58	C₄×C₆₁	#490	1	1	yes	characteristic
59	C₄×C₆₁	#491	1	1	yes	characteristic
60	Dic₆₁	—	16	2	no	—
61	#1848	—	61	1	no	—
62	#1835	—	61	1	no	—
63	#1452	D₆₁	1	1	yes	characteristic
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	C₂×D₆₁	1	2	yes	—
69	C₂×#108	—	122	2	no	—
70	#423	—	61	2	no	—
71	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
72	#382	—	61	2	no	—
73	#340	—	122	2	no	—
74	#322	Dic₆₁	1	4	yes	complemented
75	C₆₁	#8794	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	122	2	no	—
77	C₂²×C₈	D₂₄₄	1	2	yes	—
78	C₂×C₄⋊C₄	—	122	2	no	—
79	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
80	#108	—	244	4	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	122	4	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₄×C₈	Dic₁₂₂	1	2	yes	—
85	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
86	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
87	#95	—	122	4	no	—
88	C₂²×C₄	—	244	2	no	—
89	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
90	C₂²×C₄	#11758222	1	2	yes	—
91	M₁₆	—	244	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	244	2	no	—
96	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
97	C₄⋊C₄	—	244	2	no	—
98	C₄⋊C₄	—	244	2	no	—
99	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
100	C₄²	#11758222	1	2	yes	—
101	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
102	C₂³	#14519765	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	488	4	no	—
105	C₂×C₄	—	488	2	no	—
106	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
107	C₂×C₄	#14519736	1	2	yes	—
108	C₂×C₄	#14519765	1	1	yes	characteristic
109	C₂×C₄	#14519769	1	2	yes	—
110	C₂×C₄	#14519805	1	2	yes	—
111	C₂×C₄	#14519810	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	4	4	no	—
114	C₈	—	488	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#16839628	1	1	yes	characteristic
117	C₂²	#16840122	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	976	2	no	—
121	C₄	#16839804	1	2	yes	—
122	C₄	#16840127	1	1	yes	characteristic
123	C₄	#16840131	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#21445943	1	2	yes	—
126	C₂	#21445953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445991

C₂

yes

maximal, characteristic

#21445950

C₂

yes

maximal, characteristic

#21443678

C₂

yes

maximal

C₆₁×#1452

C₂

yes

maximal, characteristic

C₂×#14519713

—

#16839786

C₂²

yes

—

#16839628

C₂²

yes

—

#16839525

—

C₆₁×#431

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

C₄

yes

complemented

C₂×#11758194

—

C₂×C₄⋊Dic₆₁

—

#14519713

—

C₄⋊#92418

—

#14519684

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

#124

yes

characteristic

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

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#8794

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#382

yes

characteristic

C₂²×C₆₁

#489

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#428

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#1848

—

#1835

—

#1452

D₆₁

yes

characteristic

#1286

—

C₂×C₆₁

—

C₂×C₆₁

#1833

yes

—

C₂×C₆₁

#1836

yes

characteristic

#431

C₂×D₆₁

yes

—

C₂×#108

—

122

—

#423

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#382

—

#340

—

122

—

#322

Dic₆₁

yes

complemented

C₆₁

#8794

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

D₂₄₄

yes

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#108

—

244

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

#95

—

122

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

C₂²×C₄

#11758222

yes

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

C₂×Dic₁₂₂

yes

characteristic

100

C₄²

#11758222

yes

—

101

C₄²

C₂²×Dic₆₁

yes

characteristic

102

C₂³

#14519765

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

488

—

105

C₂×C₄

—

488

—

106

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

107

C₂×C₄

#14519736

yes

—

108

C₂×C₄

#14519765

yes

characteristic

109

C₂×C₄

#14519769

yes

—

110

C₂×C₄

#14519805

yes

—

111

C₂×C₄

#14519810

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

488

—

115

C₂²

—

116

C₂²

#16839628

yes

characteristic

117

C₂²

#16840122

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

976

—

121

C₄

#16839804

yes

—

122

C₄

#16840127

yes

characteristic

123

C₄

#16840131

yes

characteristic

124

C₂

—

125

C₂

#21445943

yes

—

126

C₂

#21445953

yes

characteristic

	subgroup	G/H
center	C₂²	#16839628
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1452	C₂
socle	C₂²×C₆₁	#382
prime radical	C₂³×C₆₁	#124
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1452	D₆₁
hypocenter	C₆₁	#8794
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16839628

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
11. C₆₁×#322	C₄	semidirect
74. #322	Dic₆₁	semidirect
75. C₆₁	#8794	semidirect

H ≅ G/N

type

11. C₆₁×#322

C₄

semidirect

74. #322

Dic₆₁

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
#21445950	2
#21443678	2
C₆₁×#1452	2
#8794	61

subgroup

index

C₆₁×#1452

Subgroups — #29573711

Conjugacy classes of subgroups

Complements

Maximal subgroups