Group #41013534 — subgroups

Subgroups — #41013534

Order 27008.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#27213169	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#27212911	C₂	1	1	yes	maximal, characteristic
4	#27212910	C₂	1	2	yes	maximal
5	#341×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	Dic₂₁₁⋊C₈	—	2	4	no	—
10	C₈×Dic₂₁₁	C₄	1	2	yes	—
11	C₈×Dic₂₁₁	C₄	1	2	yes	—
12	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
13	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₁₁	Q₈	1	2	yes	—
15	#16809946	—	2	2	no	—
16	#16809946	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₂₁₁	D₄	1	2	yes	—
23	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	M₁₆×C₂₁₁	—	2	2	no	—
25	M₁₆×C₂₁₁	—	2	2	no	—
26	C₂×C₈×C₂₁₁	—	2	2	no	—
27	C₂×C₈×C₂₁₁	—	2	2	no	—
28	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₁₁	—	2	4	no	—
30	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
32	C₂₁₁⋊C₈	—	2	2	no	—
33	C₂₁₁⋊C₈	—	4	2	no	—
34	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	#41	1	2	yes	—
38	C₂×C₄×C₂₁₁	#41	1	2	yes	—
39	C₈×C₂₁₁	—	2	2	no	—
40	C₈×C₂₁₁	—	4	2	no	—
41	C₂²×C₂₁₁	—	2	2	no	—
42	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₁₁	#125	1	2	yes	—
44	C₄×C₂₁₁	#117	1	1	yes	characteristic
45	C₄×C₂₁₁	#117	1	1	yes	characteristic
46	C₄×C₂₁₁	#124	1	2	yes	—
47	Dic₂₁₁	—	2	2	no	—
48	Dic₂₁₁	#108	1	4	yes	—
49	C₂×C₂₁₁	—	2	2	no	—
50	C₂×C₂₁₁	#382	1	1	yes	characteristic
51	C₂×C₂₁₁	C₂×#108	1	2	yes	—
52	C₂₁₁	#1837	1	1	yes	characteristic, complemented, Sylow
53	#1837	—	211	1	no	maximal, Sylow
54	#423	—	211	2	no	—
55	C₂×C₄×C₈	—	211	1	no	—
56	#341	—	211	2	no	—
57	#341	—	211	1	no	—
58	#341	D₂₁₁	1	1	yes	characteristic
59	C₂×M₁₆	—	211	2	no	—
60	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
61	C₂²×C₈	—	211	1	no	—
62	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
63	C₂×C₄²	—	211	1	no	—
64	C₄⋊C₈	—	422	4	no	—
65	C₄×C₈	—	211	2	no	—
66	C₄×C₈	—	211	2	no	—
67	C₂²×C₄	—	211	2	no	—
68	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	422	2	no	—
72	M₁₆	—	422	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	422	2	no	—
76	C₂×C₈	—	422	2	no	—
77	C₂×C₈	—	211	2	no	—
78	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
79	C₄²	—	211	1	no	—
80	C₄²	—	211	1	no	—
81	C₄²	—	211	2	no	—
82	C₂³	Q₈×D₂₁₁	1	1	yes	characteristic
83	C₂×C₄	—	422	4	no	—
84	C₂×C₄	—	211	2	no	—
85	C₂×C₄	—	211	2	no	—
86	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
87	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
88	C₂×C₄	#16809961	1	2	yes	—
89	C₂×C₄	#16809962	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	844	2	no	—
93	C₈	—	422	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
96	C₂²	#20269967	1	2	yes	—
97	C₄	—	422	2	no	—
98	C₄	—	211	4	no	—
99	C₄	#20269933	1	1	yes	characteristic
100	C₄	#20269934	1	1	yes	characteristic
101	C₄	#20269963	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#27213010	1	1	yes	characteristic
104	C₂	#108×D₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213169

C₂

yes

maximal

C₂×C₈×Dic₂₁₁

C₂

yes

maximal, characteristic

#27212911

C₂

yes

maximal, characteristic

#27212910

C₂

yes

maximal

#341×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

Dic₂₁₁⋊C₈

—

C₈×Dic₂₁₁

C₄

yes

—

C₈×Dic₂₁₁

C₄

yes

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

Q₈

yes

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₁₁

C₂×C₄

yes

characteristic

C₄×Dic₂₁₁

C₂×C₄

yes

characteristic

C₄×Dic₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

#125

yes

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

Dic₂₁₁

—

Dic₂₁₁

#108

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#382

yes

characteristic

C₂×C₂₁₁

C₂×#108

yes

—

C₂₁₁

#1837

yes

characteristic, complemented, Sylow

#1837

—

211

maximal, Sylow

#423

—

211

—

C₂×C₄×C₈

—

211

—

#341

—

211

—

#341

—

211

—

#341

D₂₁₁

yes

characteristic

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

—

211

—

C₄⋊C₈

—

422

—

C₄×C₈

—

211

—

C₄×C₈

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₄²

—

211

—

C₄²

—

211

—

C₄²

—

211

—

C₂³

Q₈×D₂₁₁

yes

characteristic

C₂×C₄

—

422

—

C₂×C₄

—

211

—

C₂×C₄

—

211

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

D₄×D₂₁₁

yes

characteristic

C₂×C₄

#16809961

yes

—

C₂×C₄

#16809962

yes

—

C₈

—

C₈

—

C₈

—

844

—

C₈

—

422

—

C₂²

—

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

C₂²

#20269967

yes

—

C₄

—

422

—

C₄

—

211

—

C₄

#20269933

yes

characteristic

100

C₄

#20269934

yes

characteristic

101

C₄

#20269963

yes

—

102

C₂

—

103

C₂

#27213010

yes

characteristic

104

C₂

#108×D₂₁₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#341×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#341	D₂₁₁
hypocenter	C₂₁₁	#1837
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#341×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₄⋊C₄

prime radical

C₂³×C₂₁₁

C₂×Q₈

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
52. C₂₁₁	#1837	semidirect

H ≅ G/N

type

52. C₂₁₁

#1837

semidirect

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

subgroup	index
#27213169	2
C₂×C₈×Dic₂₁₁	2
#27212911	2
#27212910	2
#341×C₂₁₁	2
#1837	211

subgroup

index

C₂×C₈×Dic₂₁₁

#341×C₂₁₁

211

Subgroups — #41013534

Conjugacy classes of subgroups

Complements

Maximal subgroups