Group #41011306 — subgroups

Subgroups — #41011306

Order 27008.

192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27213110	C₂	1	2	yes	maximal
2	#381×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16809946	—	2	2	no	—
4	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₂₁₁⋊C₈)	C₄	1	4	yes	complemented
6	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₂₁₁	C₄	1	2	yes	—
8	#95×C₂₁₁	—	2	2	no	—
9	#16809946	—	4	4	no	—
10	C₂×C₂₁₁⋊C₈	—	2	4	no	—
11	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₁₁	—	2	2	no	—
13	C₂²×C₄×C₂₁₁	—	4	2	no	—
14	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
16	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₁₁	Q₈	1	1	yes	characteristic
18	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
19	C₂₁₁⋊C₈	—	4	4	no	—
20	C₂³×C₂₁₁	—	2	2	no	—
21	C₂³×C₂₁₁	—	4	2	no	—
22	C₂³×C₂₁₁	C₄²	1	1	yes	characteristic
23	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂³×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	—	2	4	no	—
26	C₂×C₄×C₂₁₁	—	4	2	no	—
27	C₂×C₄×C₂₁₁	—	4	1	no	—
28	C₂×C₄×C₂₁₁	—	4	1	no	—
29	C₂×C₄×C₂₁₁	—	4	4	no	—
30	C₂×C₄×C₂₁₁	—	4	2	no	—
31	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
33	C₂²×C₂₁₁	—	2	2	no	—
34	C₂²×C₂₁₁	—	2	1	no	—
35	C₂²×C₂₁₁	—	2	1	no	—
36	C₂²×C₂₁₁	—	2	2	no	—
37	C₂²×C₂₁₁	—	2	2	no	—
38	C₂²×C₂₁₁	—	4	4	no	—
39	C₂²×C₂₁₁	#95	1	1	yes	characteristic
40	C₂²×C₂₁₁	#99	1	2	yes	—
41	C₄×C₂₁₁	—	2	2	no	—
42	C₄×C₂₁₁	—	2	2	no	—
43	C₄×C₂₁₁	—	8	2	no	—
44	C₂×C₂₁₁	—	2	2	no	—
45	C₂×C₂₁₁	—	4	1	no	—
46	C₂×C₂₁₁	—	4	1	no	—
47	C₂×C₂₁₁	#342	1	1	yes	characteristic
48	C₂×C₂₁₁	#344	1	2	yes	—
49	C₂₁₁	#1306	1	1	yes	characteristic, complemented, Sylow
50	#1306	—	211	1	no	maximal, Sylow
51	#407	—	211	2	no	—
52	#381	D₂₁₁	1	1	yes	characteristic
53	C₂³×C₄	C₂×D₂₁₁	1	1	yes	characteristic
54	C₂×M₁₆	—	422	2	no	—
55	C₂×C₂²⋊C₄	Dic₂₁₁	1	2	yes	—
56	C₂²⋊C₈	—	422	2	no	—
57	C₂²⋊C₈	—	211	4	no	—
58	#95	—	2	2	no	—
59	C₂⁴	C₂×Dic₂₁₁	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
63	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
64	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
65	C₂²×C₄	Dic₄₂₂	1	1	yes	characteristic
66	M₁₆	—	844	4	no	—
67	C₂×C₈	—	422	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₄×Dic₂₁₁	1	1	yes	characteristic
72	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
73	C₂³	C₄⋊Dic₂₁₁	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
81	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
82	C₈	—	844	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	C₂₁₁⋊#95	1	1	yes	characteristic
90	C₂²	#20269892	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#27212913	1	1	yes	characteristic
98	C₂	#27212916	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213110

C₂

yes

maximal

#381×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

—

C₂²⋊(C₂₁₁⋊C₈)

—

C₂²⋊(C₂₁₁⋊C₈)

C₄

yes

complemented

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₁₁

C₄

yes

—

#95×C₂₁₁

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂⁴×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

Q₈

yes

characteristic

C₂²⋊C₄×C₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₄²

yes

characteristic

C₂³×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂³×C₂₁₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#95

yes

characteristic

C₂²×C₂₁₁

#99

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#342

yes

characteristic

C₂×C₂₁₁

#344

yes

—

C₂₁₁

#1306

yes

characteristic, complemented, Sylow

#1306

—

211

maximal, Sylow

#407

—

211

—

#381

D₂₁₁

yes

characteristic

C₂³×C₄

C₂×D₂₁₁

yes

characteristic

C₂×M₁₆

—

422

—

C₂×C₂²⋊C₄

Dic₂₁₁

yes

—

C₂²⋊C₈

—

422

—

C₂²⋊C₈

—

211

—

#95

—

C₂⁴

C₂×Dic₂₁₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₁₁

yes

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂²×C₄

D₈₄₄

yes

characteristic

C₂²×C₄

Dic₄₂₂

yes

characteristic

M₁₆

—

844

—

C₂×C₈

—

422

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₄×Dic₂₁₁

yes

characteristic

C₂³

C₂²⋊Dic₂₁₁

yes

characteristic

C₂³

C₄⋊Dic₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

C₈

—

844

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂₁₁⋊#95

yes

characteristic

C₂²

#20269892

yes

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

C₂

#27212913

yes

characteristic

C₂

#27212916

yes

—

	subgroup	G/H
center	C₂²	C₂₁₁⋊#95
commutator	C₂³×C₂₁₁	C₄²
Frattini	C₂³×C₄	C₂×D₂₁₁
Fitting	#381×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#95
prime radical	C₂⁴×C₂₁₁	C₂×C₄
cosocle	C₂³×C₄×C₂₁₁	C₂²
hypercenter	#381	D₂₁₁
hypocenter	C₂₁₁	#1306
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

subgroup

G/H

center

C₂²

C₂₁₁⋊#95

commutator

C₂³×C₂₁₁

C₄²

Frattini

C₂³×C₄

C₂×D₂₁₁

Fitting

#381×C₂₁₁

C₂

socle

C₂²×C₂₁₁

#95

prime radical

C₂⁴×C₂₁₁

C₂×C₄

cosocle

C₂³×C₄×C₂₁₁

C₂²

D₂₁₁

C₂₁₁

G ▷ C₂³×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

N	H ≅ G/N	type
5. C₂²⋊(C₂₁₁⋊C₈)	C₄	semidirect
49. C₂₁₁	#1306	semidirect

H ≅ G/N

type

5. C₂²⋊(C₂₁₁⋊C₈)

C₄

semidirect

49. C₂₁₁

#1306

semidirect

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

subgroup	index
#27213110	2
#381×C₂₁₁	2
#1306	211

subgroup

index

#27213110

#381×C₂₁₁

#1306

211

Subgroups — #41011306

Conjugacy classes of subgroups

Complements

Maximal subgroups