Group #41013152 — subgroups

Subgroups — #41013152

Order 27008.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#27213104	C₂	1	1	yes	maximal, characteristic
2	#27213102	C₂	1	2	yes	maximal
3	C₂₁₁⋊#338	C₂	1	1	yes	maximal, characteristic
4	#27212902	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	#20269936	C₂²	1	1	yes	characteristic
9	#20269888	C₄	1	4	yes	complemented
10	#20269887	C₄	1	2	yes	complemented
11	C₄×C₂₁₁⋊C₈	C₄	1	4	yes	complemented
12	C₈×Dic₂₁₁	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
14	#117×C₂₁₁	C₂²	1	2	yes	—
15	#16809946	—	2	2	no	—
16	#16809946	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₂₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₂₁₁	—	2	1	no	—
23	C₂²⋊Dic₂₁₁	—	2	1	no	—
24	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₁₁	—	2	2	no	—
27	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₂₁₁	—	2	2	no	—
29	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
30	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
31	C₂×Dic₂₁₁	—	2	2	no	—
32	C₂₁₁⋊C₈	—	2	4	no	—
33	C₂₁₁⋊C₈	C₄²	1	1	yes	characteristic
34	C₂₁₁⋊C₈	C₄²	1	1	yes	characteristic
35	C₂₁₁⋊C₈	C₄²	1	2	yes	—
36	C₂³×C₂₁₁	#41	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	—	2	4	no	—
38	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₁₁	#41	1	2	yes	—
41	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
42	C₈×C₂₁₁	—	2	2	no	—
43	C₂²×C₂₁₁	—	4	1	no	—
44	C₂²×C₂₁₁	#117	1	2	yes	—
45	C₂²×C₂₁₁	#117	1	1	yes	characteristic
46	C₄×C₂₁₁	—	4	4	no	—
47	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₂₁₁	#117	1	1	yes	characteristic
49	C₄×C₂₁₁	#117	1	2	yes	—
50	Dic₂₁₁	—	4	2	no	—
51	C₂×C₂₁₁	—	2	1	no	—
52	C₂×C₂₁₁	—	2	2	no	—
53	C₂×C₂₁₁	#376	1	1	yes	characteristic
54	C₂₁₁	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	211	1	no	maximal, Sylow
56	#405	—	211	1	no	—
57	#405	—	211	2	no	—
58	#338	—	211	1	no	—
59	H(3,ℤ/4)	—	211	2	no	—
60	H(3,ℤ/4)	D₂₁₁	1	1	yes	characteristic
61	C₂×M₁₆	—	211	2	no	—
62	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
63	C₂²×C₈	—	211	1	no	—
64	#117	—	211	1	no	—
65	#117	C₂×D₂₁₁	1	2	yes	—
66	#97	—	211	2	no	—
67	#97	—	211	4	no	—
68	C₄×C₈	—	211	2	no	—
69	C₄×C₈	—	211	4	no	—
70	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	422	2	no	—
73	M₁₆	—	422	2	no	—
74	C₂×C₈	—	211	4	no	—
75	C₂×C₈	—	211	1	no	—
76	C₂×C₈	—	211	2	no	—
77	C₂×C₈	—	211	1	no	—
78	C₂×C₈	—	211	2	no	—
79	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	422	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	422	1	no	—
84	C₄²	—	211	2	no	—
85	C₄²	C₄×D₂₁₁	1	4	yes	—
86	C₂³	#16809961	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	422	2	no	—
89	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
91	C₂×C₄	#16809961	1	2	yes	—
92	C₂×C₄	#16809963	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	422	4	no	—
95	C₈	—	211	1	no	—
96	C₈	—	211	1	no	—
97	C₈	—	211	2	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#20269933	1	2	yes	—
100	C₂²	#20269935	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	844	2	no	—
103	C₄	C₄²×D₂₁₁	1	1	yes	characteristic
104	C₄	#20269933	1	1	yes	characteristic
105	C₄	#20269935	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#27212991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213104

C₂

yes

maximal, characteristic

#27213102

C₂

yes

maximal

C₂₁₁⋊#338

C₂

yes

maximal, characteristic

#27212902

C₂

yes

maximal

H(3,ℤ/4)×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

#20269936

C₂²

yes

characteristic

#20269888

C₄

yes

complemented

#20269887

C₄

yes

complemented

C₄×C₂₁₁⋊C₈

C₄

yes

complemented

C₈×Dic₂₁₁

C₄

yes

complemented

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₂₁₁

—

C₂²⋊Dic₂₁₁

—

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

C₄²

yes

characteristic

C₂₁₁⋊C₈

C₄²

yes

characteristic

C₂₁₁⋊C₈

C₄²

yes

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#117

yes

—

C₂²×C₂₁₁

#117

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₄²

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#376

yes

characteristic

C₂₁₁

#1747

yes

characteristic, complemented, Sylow

#1747

—

211

maximal, Sylow

#405

—

211

—

#405

—

211

—

#338

—

211

—

H(3,ℤ/4)

—

211

—

H(3,ℤ/4)

D₂₁₁

yes

characteristic

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

#117

—

211

—

#117

C₂×D₂₁₁

yes

—

#97

—

211

—

#97

—

211

—

C₄×C₈

—

211

—

C₄×C₈

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₂²⋊C₄

—

C₂²⋊C₄

—

422

—

C₄²

—

211

—

C₄²

C₄×D₂₁₁

yes

—

C₂³

#16809961

yes

characteristic

C₂×C₄

—

C₂×C₄

—

422

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂×C₄×D₂₁₁

yes

—

C₂×C₄

#16809961

yes

—

C₂×C₄

#16809963

yes

characteristic

C₈

—

C₈

—

422

—

C₈

—

211

—

C₈

—

211

—

C₈

—

211

—

C₂²

—

C₂²

#20269933

yes

—

100

C₂²

#20269935

yes

characteristic

101

C₄

—

102

C₄

—

844

—

103

C₄

C₄²×D₂₁₁

yes

characteristic

104

C₄

#20269933

yes

characteristic

105

C₄

#20269935

yes

—

106

C₂

—

107

C₂

—

108

C₂

#27212991

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₂₁₁
commutator	C₄×C₂₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	H(3,ℤ/4)×C₂₁₁	C₂
socle	C₂×C₂₁₁	#376
prime radical	C₂³×C₂₁₁	#41
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	H(3,ℤ/4)	D₂₁₁
hypocenter	C₂₁₁	#1747
LCS	G ▷ C₄×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₂₁₁

commutator

C₄×C₂₁₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

H(3,ℤ/4)×C₂₁₁

C₂

C₂×C₂₁₁

C₂³×C₂₁₁

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₄×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #20269888	C₄	semidirect
10. #20269887	C₄	semidirect
11. C₄×C₂₁₁⋊C₈	C₄	semidirect
12. C₈×Dic₂₁₁	C₄	semidirect
54. C₂₁₁	#1747	semidirect

H ≅ G/N

type

9. #20269888

C₄

semidirect

10. #20269887

C₄

semidirect

11. C₄×C₂₁₁⋊C₈

C₄

semidirect

12. C₈×Dic₂₁₁

C₄

semidirect

54. C₂₁₁

#1747

semidirect

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

subgroup	index
#27213104	2
#27213102	2
C₂₁₁⋊#338	2
#27212902	2
H(3,ℤ/4)×C₂₁₁	2
#1747	211

subgroup

index

C₂₁₁⋊#338

H(3,ℤ/4)×C₂₁₁

211

Subgroups — #41013152

Conjugacy classes of subgroups

Complements

Maximal subgroups