Group #41013545 — subgroups

Subgroups — #41013545

Order 27008.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#27212916	C₂	1	2	yes	maximal
3	#27212905	C₂	1	2	yes	maximal
4	C₂₁₁⋊#338	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂×#16809946	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	#20269887	C₄	1	2	yes	complemented
10	C₈×Dic₂₁₁	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
12	#117×C₂₁₁	C₂²	1	2	yes	—
13	C₂²×Dic₂₁₁	—	2	1	no	—
14	#16809946	—	2	2	no	—
15	#16809946	—	2	2	no	—
16	#16809946	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₄×Dic₂₁₁	—	2	1	no	—
20	C₄×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₁₁	—	2	2	no	—
24	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
25	C₄⋊C₄×C₂₁₁	—	2	2	no	—
26	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
27	C₄²×C₂₁₁	—	2	2	no	—
28	C₂×Dic₂₁₁	—	2	1	no	—
29	C₂×Dic₂₁₁	—	2	1	no	—
30	C₂×Dic₂₁₁	—	4	2	no	—
31	C₂₁₁⋊C₈	—	2	2	no	—
32	C₂₁₁⋊C₈	—	4	2	no	—
33	C₂³×C₂₁₁	#41	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	—	4	2	no	—
35	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	#41	1	2	yes	—
37	C₂×C₄×C₂₁₁	#41	1	2	yes	—
38	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
39	C₈×C₂₁₁	—	2	2	no	—
40	C₂²×C₂₁₁	—	4	1	no	—
41	C₂²×C₂₁₁	#117	1	1	yes	characteristic
42	C₂²×C₂₁₁	#126	1	2	yes	—
43	C₄×C₂₁₁	—	8	2	no	—
44	C₄×C₂₁₁	#117	1	1	yes	characteristic
45	C₄×C₂₁₁	#117	1	1	yes	characteristic
46	C₄×C₂₁₁	#126	1	2	yes	—
47	Dic₂₁₁	—	2	2	no	—
48	Dic₂₁₁	—	4	1	no	—
49	C₂×C₂₁₁	—	2	1	no	—
50	C₂×C₂₁₁	—	2	2	no	—
51	C₂×C₂₁₁	#383	1	1	yes	characteristic
52	C₂₁₁	#1841	1	1	yes	characteristic, complemented, Sylow
53	#1841	—	211	1	no	maximal, Sylow
54	C₄×M₁₆	—	211	1	no	—
55	#344	—	211	2	no	—
56	#339	—	211	2	no	—
57	#338	—	211	1	no	—
58	H(3,ℤ/4)	D₂₁₁	1	1	yes	characteristic
59	C₂×M₁₆	—	211	2	no	—
60	C₂×M₁₆	—	211	1	no	—
61	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
62	#117	C₂×D₂₁₁	1	2	yes	—
63	C₂×C₄²	—	211	1	no	—
64	#97	—	211	2	no	—
65	C₄×C₈	—	211	2	no	—
66	C₂²×C₄	—	422	1	no	—
67	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	422	2	no	—
70	M₁₆	—	422	2	no	—
71	M₁₆	—	422	2	no	—
72	C₂×C₈	—	422	2	no	—
73	C₂×C₈	—	211	2	no	—
74	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	422	1	no	—
79	C₄²	—	211	1	no	—
80	C₄²	—	211	1	no	—
81	C₂³	#16809961	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	844	2	no	—
84	C₂×C₄	—	422	1	no	—
85	C₂×C₄	—	422	1	no	—
86	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
87	C₂×C₄	#16809961	1	2	yes	—
88	C₂×C₄	#16809962	1	2	yes	—
89	C₂×C₄	#16809963	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	844	2	no	—
92	C₈	—	422	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#20269935	1	1	yes	characteristic
95	C₂²	#20269970	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	844	1	no	—
98	C₄	—	422	2	no	—
99	C₄	#20269933	1	1	yes	characteristic
100	C₄	#20269934	1	1	yes	characteristic
101	C₄	#20269972	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#27213013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₂₁₁

C₂

yes

maximal, characteristic

#27212916

C₂

yes

maximal

#27212905

C₂

yes

maximal

C₂₁₁⋊#338

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂×#16809946

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

#20269887

C₄

yes

complemented

C₈×Dic₂₁₁

C₄

yes

complemented

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

—

C₂²×Dic₂₁₁

—

#16809946

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

C₂×C₄

yes

characteristic

C₄×Dic₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#117

yes

characteristic

C₂²×C₂₁₁

#126

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#383

yes

characteristic

C₂₁₁

#1841

yes

characteristic, complemented, Sylow

#1841

—

211

maximal, Sylow

C₄×M₁₆

—

211

—

#344

—

211

—

#339

—

211

—

#338

—

211

—

H(3,ℤ/4)

D₂₁₁

yes

characteristic

C₂×M₁₆

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

#117

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

#97

—

211

—

C₄×C₈

—

211

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

422

—

C₄²

—

211

—

C₄²

—

211

—

C₂³

#16809961

yes

characteristic

C₂×C₄

—

C₂×C₄

—

844

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

#16809961

yes

—

C₂×C₄

#16809962

yes

—

C₂×C₄

#16809963

yes

characteristic

C₈

—

C₈

—

844

—

C₈

—

422

—

C₂²

—

C₂²

#20269935

yes

characteristic

C₂²

#20269970

yes

—

C₄

—

C₄

—

844

—

C₄

—

422

—

C₄

#20269933

yes

characteristic

100

C₄

#20269934

yes

characteristic

101

C₄

#20269972

yes

—

102

C₂

—

103

C₂

—

104

C₂

#27213013

yes

characteristic

	subgroup	G/H
center	C₄	#20269934
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	H(3,ℤ/4)×C₂₁₁	C₂
socle	C₂×C₂₁₁	#383
prime radical	C₂³×C₂₁₁	#41
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	H(3,ℤ/4)	D₂₁₁
hypocenter	C₂₁₁	#1841
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#20269934

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

H(3,ℤ/4)×C₂₁₁

C₂

C₂×C₂₁₁

C₂³×C₂₁₁

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #20269887	C₄	semidirect
10. C₈×Dic₂₁₁	C₄	semidirect
52. C₂₁₁	#1841	semidirect

H ≅ G/N

type

9. #20269887

C₄

semidirect

10. C₈×Dic₂₁₁

C₄

semidirect

52. C₂₁₁

#1841

semidirect

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

subgroup	index
M₁₆×Dic₂₁₁	2
#27212916	2
#27212905	2
C₂₁₁⋊#338	2
H(3,ℤ/4)×C₂₁₁	2
#1841	211

subgroup

index

M₁₆×Dic₂₁₁

#27212916

#27212905

C₂₁₁⋊#338

H(3,ℤ/4)×C₂₁₁

#1841

211

Subgroups — #41013545

Conjugacy classes of subgroups

Complements

Maximal subgroups