Group #29585119 — subgroups

Subgroups — #29585119

Order 15616.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#21446625	C₂	1	3	yes	maximal
2	#21445976	C₂	1	3	yes	maximal, complemented
3	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14519774	C₂²	1	1	yes	characteristic
5	C₂×#14519709	—	2	6	no	—
6	C₂×#14519699	—	2	6	no	—
7	#16839632	C₂²	1	3	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
9	C₂²×D₂₄₄	—	2	3	no	—
10	#14519774	—	2	3	no	—
11	#14519774	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	—	2	3	no	—
13	C₂×C₄⋊Dic₆₁	—	2	6	no	—
14	#14519709	—	4	12	no	—
15	#14519699	—	4	12	no	—
16	C₂²×C₈×C₆₁	D₄	1	6	yes	—
17	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₆₁	—	2	6	no	—
19	C₂³×D₆₁	—	4	1	no	—
20	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
21	C₂²×Dic₆₁	—	4	3	no	—
22	C₂×D₂₄₄	—	2	6	no	—
23	C₂×D₂₄₄	—	4	6	no	—
24	C₂×D₂₄₄	—	4	6	no	—
25	C₂×D₂₄₄	—	4	3	no	—
26	C₂×C₈×C₆₁	—	2	6	no	—
27	C₂×C₈×C₆₁	—	2	6	no	—
28	C₂×C₈×C₆₁	SD₁₆	1	12	yes	—
29	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₆₁	#41	1	3	yes	—
31	C₄⋊Dic₆₁	—	4	6	no	—
32	C₄⋊Dic₆₁	—	4	12	no	—
33	C₂³×C₆₁	#127	1	1	yes	characteristic
34	C₂²×D₆₁	—	4	2	no	—
35	C₂²×D₆₁	—	8	3	no	—
36	C₂²×D₆₁	—	8	3	no	—
37	C₂×C₄×C₆₁	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₆₁	#123	1	3	yes	—
39	C₂×C₄×C₆₁	C₂×SD₁₆	1	6	yes	—
40	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	6	yes	—
41	C₂×Dic₆₁	—	8	3	no	—
42	C₂×Dic₆₁	—	8	6	no	—
43	D₂₄₄	—	4	12	no	—
44	D₂₄₄	—	8	6	no	—
45	C₈×C₆₁	—	2	12	no	—
46	#10683	—	61	1	no	maximal, Sylow
47	C₂²×C₆₁	#399	1	1	yes	characteristic
48	C₂²×C₆₁	#492	1	3	yes	—
49	C₂²×C₆₁	#496	1	3	yes	—
50	C₂×D₆₁	—	8	6	no	—
51	C₂×D₆₁	—	16	1	no	—
52	C₂×D₆₁	—	16	3	no	—
53	C₄×C₆₁	#476	1	6	yes	—
54	C₄×C₆₁	#481	1	6	yes	—
55	Dic₆₁	—	16	3	no	—
56	#1957	—	61	3	no	—
57	#1844	—	61	3	no	—
58	#1273	D₆₁	1	1	yes	characteristic, complemented
59	C₂×C₆₁	#1705	1	1	yes	characteristic
60	C₂×C₆₁	#1715	1	3	yes	—
61	C₂×C₆₁	#2094	1	3	yes	—
62	D₆₁	—	16	2	no	—
63	C₂×#127	—	61	1	no	—
64	C₂×#106	—	122	6	no	—
65	C₂×#102	—	122	6	no	—
66	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
67	#384	—	61	3	no	—
68	C₆₁	#10683	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	122	3	no	—
70	C₂²×C₈	D₂₄₄	1	6	yes	—
71	#127	—	61	2	no	—
72	#127	—	122	3	no	—
73	C₂×C₄⋊C₄	—	122	3	no	—
74	C₂×C₄⋊C₄	—	122	6	no	—
75	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
76	#106	—	244	12	no	—
77	#102	—	244	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	244	1	no	—
80	C₂×D₄	—	122	6	no	—
81	C₂×D₄	—	244	3	no	—
82	C₂×D₄	—	244	6	no	—
83	C₂×D₄	—	244	6	no	—
84	C₂²×C₄	—	244	3	no	—
85	C₂²×C₄	C₂×D₂₄₄	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	#11758190	1	12	yes	—
89	C₄⋊C₄	—	244	6	no	—
90	C₄⋊C₄	—	244	12	no	—
91	C₄²	#11758224	1	3	yes	—
92	C₄²	Q₈×D₆₁	1	1	yes	characteristic
93	C₂³	—	488	3	no	—
94	C₂³	—	488	3	no	—
95	C₂³	—	244	2	no	—
96	C₂³	#14519774	1	1	yes	characteristic
97	D₄	—	488	6	no	—
98	D₄	—	244	12	no	—
99	C₂×C₄	—	488	3	no	—
100	C₂×C₄	—	488	6	no	—
101	C₂×C₄	#14519757	1	3	yes	—
102	C₂×C₄	#14519761	1	3	yes	—
103	C₂×C₄	C₂×#11758190	1	6	yes	—
104	C₂×C₄	#14519805	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	976	1	no	—
107	C₂²	—	976	3	no	—
108	C₂²	—	488	6	no	—
109	C₂²	#16839699	1	1	yes	characteristic
110	C₂²	#16840135	1	3	yes	—
111	C₂²	#16840151	1	3	yes	—
112	C₄	—	976	3	no	—
113	C₄	#16840053	1	6	yes	—
114	C₄	#16840081	1	6	yes	—
115	C₂	—	976	2	no	—
116	C₂	#21445406	1	1	yes	characteristic
117	C₂	#21445447	1	3	yes	—
118	C₂	#21447351	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446625

C₂

yes

maximal

#21445976

C₂

yes

maximal, complemented

C₆₁×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14519774

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂²×D₂₄₄

—

#14519774

—

#14519774

Q₈

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519709

—

#14519699

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₂³×D₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

#127

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#496

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#476

yes

—

C₄×C₆₁

#481

yes

—

Dic₆₁

—

#1957

—

#1844

—

#1273

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1705

yes

characteristic

C₂×C₆₁

#1715

yes

—

C₂×C₆₁

#2094

yes

—

D₆₁

—

C₂×#127

—

C₂×#106

—

122

—

C₂×#102

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

#384

—

C₆₁

#10683

yes

characteristic, complemented, Sylow

C₂²×D₄

—

122

—

C₂²×C₈

D₂₄₄

yes

—

#127

—

#127

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#106

—

244

—

#102

—

244

—

C₄×C₈

—

C₂⁴

—

244

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂×D₂₄₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#11758190

yes

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

#11758224

yes

—

C₄²

Q₈×D₆₁

yes

characteristic

C₂³

—

488

—

C₂³

—

488

—

C₂³

—

244

—

C₂³

#14519774

yes

characteristic

D₄

—

488

—

D₄

—

244

—

C₂×C₄

—

488

—

100

C₂×C₄

—

488

—

101

C₂×C₄

#14519757

yes

—

102

C₂×C₄

#14519761

yes

—

103

C₂×C₄

C₂×#11758190

yes

—

104

C₂×C₄

#14519805

yes

—

105

C₈

—

106

C₂²

—

976

—

107

C₂²

—

976

—

108

C₂²

—

488

—

109

C₂²

#16839699

yes

characteristic

110

C₂²

#16840135

yes

—

111

C₂²

#16840151

yes

—

112

C₄

—

976

—

113

C₄

#16840053

yes

—

114

C₄

#16840081

yes

—

115

C₂

—

976

—

116

C₂

#21445406

yes

characteristic

117

C₂

#21445447

yes

—

118

C₂

#21447351

yes

—

	subgroup	G/H
center	C₂³	#14519774
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1273	C₂
socle	C₂³×C₆₁	#127
prime radical	C₂×#14519774	C₂²
hypercenter	#1273	D₆₁
hypocenter	C₆₁	#10683
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14519774

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1273

C₂

C₂³×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #21445976	C₂	semidirect
3. C₆₁×#1273	C₂	semidirect
58. #1273	D₆₁	semidirect
68. C₆₁	#10683	semidirect

H ≅ G/N

type

2. #21445976

C₂

semidirect

3. C₆₁×#1273

C₂

semidirect

58. #1273

D₆₁

semidirect

68. C₆₁

#10683

semidirect

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

subgroup	index
#21446625	2
#21445976	2
C₆₁×#1273	2
#10683	61

subgroup

index

#21446625

#21445976

C₆₁×#1273

#10683

Subgroups — #29585119

Conjugacy classes of subgroups

Complements

Maximal subgroups