Group #29569327 — subgroups

Subgroups — #29569327

Order 15616.

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

normal subgroups only

This group has 322 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	#21447382	C₂	1	3	yes	maximal
2	#21443642	C₂	1	3	yes	maximal
3	C₆₁×#2810	C₂	1	1	yes	maximal, characteristic
4	#16839632	C₂²	1	1	yes	characteristic
5	#16839524	—	2	3	no	—
6	#16839524	—	2	3	no	—
7	#16839517	C₂²	1	3	yes	—
8	C₆₁×#400	—	2	3	no	—
9	C₆₁×#389	C₂²	1	3	yes	—
10	C₆₁×#387	—	2	3	no	—
11	C₂²×#92418	—	2	3	no	—
12	C₂×C₄⋊Dic₆₁	—	2	3	no	—
13	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
14	C₂×C₄⋊C₄×C₆₁	D₄	1	3	yes	—
15	C₂×C₄⋊C₄×C₆₁	D₄	1	3	yes	—
16	C₂×C₄²×C₆₁	—	2	1	no	—
17	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
18	#95×C₆₁	—	2	3	no	—
19	#95×C₆₁	—	2	3	no	—
20	#95×C₆₁	—	2	1	no	—
21	#95×C₆₁	—	2	3	no	—
22	C₂²×C₄×C₆₁	—	2	3	no	—
23	C₂²×C₄×C₆₁	—	2	3	no	—
24	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
25	C₂²×Dic₆₁	—	4	1	no	—
26	C₄⋊C₄×C₆₁	—	2	3	no	—
27	C₄⋊C₄×C₆₁	—	2	3	no	—
28	C₄⋊C₄×C₆₁	—	4	3	no	—
29	C₄⋊C₄×C₆₁	—	4	3	no	—
30	C₄⋊C₄×C₆₁	—	4	3	no	—
31	C₄²×C₆₁	—	4	1	no	—
32	C₄²×C₆₁	—	8	1	no	—
33	C₄⋊Dic₆₁	—	4	6	no	—
34	C₂×#92418	—	4	3	no	—
35	C₂×#92418	—	4	3	no	—
36	C₂×#92418	—	4	3	no	—
37	C₂³×C₆₁	#120	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	3	no	—
39	C₂×C₄×C₆₁	—	2	3	no	—
40	C₂×C₄×C₆₁	—	4	3	no	—
41	C₂×C₄×C₆₁	—	4	3	no	—
42	C₂×C₄×C₆₁	—	4	3	no	—
43	C₂×C₄×C₆₁	—	4	3	no	—
44	C₂×C₄×C₆₁	—	4	3	no	—
45	C₂×C₄×C₆₁	—	4	3	no	—
46	C₂×C₄×C₆₁	#135	1	3	yes	—
47	C₂×C₄×C₆₁	#137	1	3	yes	—
48	C₂×Dic₆₁	—	8	3	no	—
49	#92418	—	8	3	no	—
50	#7832	—	61	1	no	maximal, Sylow
51	C₂²×C₆₁	#452	1	3	yes	—
52	C₂²×C₆₁	#454	1	3	yes	—
53	C₂²×C₆₁	#458	1	1	yes	characteristic
54	C₄×C₆₁	—	2	3	no	—
55	C₄×C₆₁	—	2	3	no	—
56	C₄×C₆₁	—	8	3	no	—
57	C₄×C₆₁	—	8	3	no	—
58	Dic₆₁	—	16	1	no	—
59	#2810	D₆₁	1	1	yes	characteristic
60	#2100	—	61	3	no	—
61	#1271	—	61	3	no	—
62	C₂×C₆₁	#1615	1	3	yes	—
63	C₂×C₆₁	#1638	1	3	yes	—
64	C₂×C₆₁	#1658	1	1	yes	characteristic
65	#400	—	2	3	no	—
66	#389	C₂×D₆₁	1	3	yes	—
67	#387	—	2	3	no	—
68	#384	—	61	1	no	—
69	#340	—	122	3	no	—
70	#340	—	122	3	no	—
71	#336	—	61	3	no	—
72	C₆₁	#7832	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	122	3	no	—
74	C₂×C₄⋊C₄	—	2	3	no	—
75	C₂×C₄⋊C₄	—	122	3	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	3	yes	—
77	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	3	yes	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
80	#95	—	2	3	no	—
81	#95	—	2	3	no	—
82	#95	—	2	1	no	—
83	#95	—	2	3	no	—
84	C₂²×C₄	—	2	3	no	—
85	C₂²×C₄	—	2	3	no	—
86	C₂²×C₄	—	244	1	no	—
87	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
88	C₂×C₈	—	244	3	no	—
89	C₂×C₈	—	244	3	no	—
90	C₂×C₈	—	244	3	no	—
91	C₄⋊C₄	—	2	3	no	—
92	C₄⋊C₄	—	2	3	no	—
93	C₄⋊C₄	—	4	3	no	—
94	C₄⋊C₄	—	4	3	no	—
95	C₄⋊C₄	—	4	3	no	—
96	C₄⋊C₄	—	244	6	no	—
97	C₄²	—	4	1	no	—
98	C₄²	—	8	1	no	—
99	C₂³	#14519746	1	1	yes	characteristic
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	4	3	no	—
103	C₂×C₄	—	4	3	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	—	4	3	no	—
106	C₂×C₄	—	4	3	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	—	488	3	no	—
109	C₂×C₄	#14519797	1	3	yes	—
110	C₂×C₄	#14519807	1	3	yes	—
111	C₈	—	488	3	no	—
112	C₂²	#16839924	1	3	yes	—
113	C₂²	#16839935	1	3	yes	—
114	C₂²	#16839952	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	2	3	no	—
117	C₄	—	8	3	no	—
118	C₄	—	8	3	no	—
119	C₄	—	976	1	no	—
120	C₂	#21444850	1	3	yes	—
121	C₂	#21444991	1	3	yes	—
122	C₂	#21445121	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447382

C₂

yes

maximal

#21443642

C₂

yes

maximal

C₆₁×#2810

C₂

yes

maximal, characteristic

#16839632

C₂²

yes

—

—

C₂²

yes

—

C₆₁×#400

—

C₆₁×#389

C₂²

yes

—

C₆₁×#387

—

C₂²×#92418

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

#95×C₆₁

—

#95×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

—

—

C₂²×C₆₁

yes

—

C₂²×C₆₁

#454

yes

—

C₂²×C₆₁

#458

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

#2810

D₆₁

yes

characteristic

#2100

—

#1271

—

C₂×C₆₁

#1615

yes

—

C₂×C₆₁

#1638

yes

—

C₂×C₆₁

yes

—

C₂×D₆₁

yes

—

#387

—

#384

—

#340

—

122

—

#340

—

122

—

#336

—

C₆₁

#7832

yes

characteristic, complemented, Sylow

C₂²×C₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₂×C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₁

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

244

—

C₄²

—

C₄²

—

C₂³

#14519746

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

488

—

109

C₂×C₄

#14519797

yes

—

110

C₂×C₄

#14519807

yes

—

111

C₈

—

488

—

112

C₂²

#16839924

yes

—

113

C₂²

#16839935

yes

—

114

C₂²

#16839952

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

976

—

120

C₂

#21444850

yes

—

121

C₂

#21444991

yes

—

122

C₂

#21445121

yes

characteristic

	subgroup	G/H
center	C₂³	#14519746
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2810	C₂
socle, prime radical	C₂³×C₆₁	#120
hypercenter	#2810	D₆₁
hypocenter	C₆₁	#7832
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2810

subgroup

G/H

center

C₂³

#14519746

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#2810

C₂

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂³×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2810

N	H ≅ G/N	type
72. C₆₁	#7832	semidirect

H ≅ G/N

type

72. C₆₁

#7832

semidirect

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

subgroup	index
#21447382	2
#21443642	2
C₆₁×#2810	2
#7832	61

subgroup

index

#21447382

#21443642

C₆₁×#2810

#7832

Subgroups — #29569327

Conjugacy classes of subgroups

Complements

Maximal subgroups