Group #29567124 — subgroups

Subgroups — #29567124

Order 15616.

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

normal subgroups only

This group has 184 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	#21444582	C₂	1	2	yes	maximal
2	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic
3	#16839819	—	2	2	no	—
4	#16839515	—	2	2	no	—
5	#16839513	C₄	1	4	yes	—
6	C₂×M₃₂×C₆₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×C₄⋊C₁₆	—	2	2	no	—
9	C₄⋊#92418	—	2	4	no	—
10	C₂²⋊#92418	—	4	2	no	—
11	#14519688	—	4	2	no	—
12	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
15	M₃₂×C₆₁	—	4	2	no	—
16	C₂×C₁₆×C₆₁	—	2	2	no	—
17	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₁	D₄	1	2	yes	—
19	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
21	C₂×C₈×C₆₁	—	2	1	no	—
22	C₂×C₈×C₆₁	—	2	1	no	—
23	C₂×C₈×C₆₁	—	4	1	no	—
24	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
27	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
28	C₄²×C₆₁	D₈	1	1	yes	characteristic
29	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
30	C₂×#92418	—	4	4	no	—
31	C₁₆×C₆₁	—	4	2	no	—
32	C₂³×C₆₁	#106	1	1	yes	characteristic
33	C₂×C₄×C₆₁	—	2	1	no	—
34	C₂×C₄×C₆₁	—	2	1	no	—
35	C₂×C₄×C₆₁	—	4	2	no	—
36	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
37	C₂×C₄×C₆₁	#102	1	2	yes	—
38	C₂×C₄×C₆₁	#103	1	2	yes	—
39	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
40	C₈×C₆₁	—	2	2	no	—
41	C₈×C₆₁	—	2	1	no	—
42	C₈×C₆₁	#100	1	1	yes	characteristic
43	C₈×C₆₁	#101	1	1	yes	characteristic
44	#92418	—	8	4	no	—
45	#7246	—	61	1	no	maximal, Sylow
46	C₂²×C₆₁	—	2	1	no	—
47	C₂²×C₆₁	—	2	2	no	—
48	C₂²×C₆₁	#340	1	1	yes	characteristic
49	C₄×C₆₁	—	2	1	no	—
50	C₄×C₆₁	—	2	2	no	—
51	C₄×C₆₁	—	2	2	no	—
52	C₄×C₆₁	#338	1	1	yes	characteristic
53	C₄×C₆₁	#343	1	1	yes	characteristic
54	#2150	D₆₁	1	1	yes	characteristic
55	#1568	—	61	2	no	—
56	C₂×C₆₁	—	4	1	no	—
57	C₂×C₆₁	#1292	1	1	yes	characteristic
58	C₂×C₆₁	#1389	1	2	yes	—
59	C₂×M₃₂	Dic₆₁	1	2	yes	—
60	#432	—	122	2	no	—
61	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
62	C₄⋊C₁₆	—	2	2	no	—
63	#335	—	122	2	no	—
64	#334	—	61	4	no	—
65	C₆₁	#7246	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
67	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
68	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
69	M₃₂	—	4	2	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₄⋊C₈	—	122	4	no	—
72	C₂²⋊C₈	—	244	2	no	—
73	#97	—	244	2	no	—
74	C₄×C₈	C₄×D₆₁	1	2	yes	—
75	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
76	C₂²×C₄	#11758190	1	2	yes	—
77	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	244	4	no	—
82	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
83	C₂×C₈	#11758196	1	2	yes	—
84	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
85	C₄²	D₄₈₈	1	1	yes	characteristic
86	C₄²	Dic₂₄₄	1	1	yes	characteristic
87	C₄²	#11758198	1	2	yes	—
88	C₁₆	—	4	2	no	—
89	C₂³	#14519709	1	1	yes	characteristic
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	#14519684	1	1	yes	characteristic
94	C₂×C₄	#14519699	1	2	yes	—
95	C₂×C₄	#14519702	1	2	yes	—
96	C₂×C₄	#14519711	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	488	4	no	—
100	C₈	#14519694	1	1	yes	characteristic
101	C₈	#14519696	1	1	yes	characteristic
102	C₂²	—	2	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#16839525	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	2	1	no	—
108	C₄	#16839520	1	1	yes	characteristic
109	C₄	#16839531	1	1	yes	characteristic
110	C₂	—	4	1	no	—
111	C₂	#21443694	1	1	yes	characteristic
112	C₂	#21443934	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21444582

C₂

yes

maximal

C₆₁×#2150

C₂

yes

maximal, characteristic

#16839819

—

#16839515

—

#16839513

C₄

yes

—

C₂×M₃₂×C₆₁

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₄⋊#92418

—

C₂²⋊#92418

—

#14519688

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

Q₈

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

Q₁₆

yes

characteristic

C₂×#92418

—

C₁₆×C₆₁

—

C₂³×C₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

#102

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

#107

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

D₆₁

yes

characteristic

#1568

—

C₂×C₆₁

—

C₂×C₆₁

#1292

yes

characteristic

C₂×C₆₁

#1389

yes

—

C₂×M₃₂

Dic₆₁

yes

—

#432

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

#335

—

122

—

#334

—

C₆₁

#7246

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₂₂

yes

characteristic

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

C₂×C₄²

D₂₄₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

244

—

#97

—

244

—

C₄×C₈

C₄×D₆₁

yes

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₂²×C₄

#11758190

yes

—

C₂²×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₄×Dic₆₁

yes

characteristic

C₂×C₈

#11758196

yes

—

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

C₄²

D₄₈₈

yes

characteristic

C₄²

Dic₂₄₄

yes

characteristic

C₄²

#11758198

yes

—

C₁₆

—

C₂³

#14519709

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14519684

yes

characteristic

C₂×C₄

#14519699

yes

—

C₂×C₄

#14519702

yes

—

C₂×C₄

#14519711

yes

characteristic

C₈

—

C₈

—

C₈

—

488

—

100

C₈

#14519694

yes

characteristic

101

C₈

#14519696

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

#16839525

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#16839520

yes

characteristic

109

C₄

#16839531

yes

characteristic

110

C₂

—

111

C₂

#21443694

yes

characteristic

112

C₂

#21443934

yes

—

	subgroup	G/H
center	C₂²	#16839525
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#2150	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#106
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#2150	D₆₁
hypocenter	C₆₁	#7246
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

#16839525

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₁

Fitting

C₆₁×#2150

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄×C₈×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
65. C₆₁	#7246	semidirect

H ≅ G/N

type

65. C₆₁

#7246

semidirect

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

subgroup	index
#21444582	2
C₆₁×#2150	2
#7246	61

subgroup

index

#21444582

C₆₁×#2150

#7246

Subgroups — #29567124

Conjugacy classes of subgroups

Complements

Maximal subgroups