Group #41013303 — subgroups

Subgroups — #41013303

Order 27008.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#20269912	C₂	1	2	yes	maximal
2	M₁₆×Dic₂₁₁	C₂	1	2	yes	maximal
3	C₂×#20269887	C₂	1	1	yes	maximal, characteristic
4	#27212911	C₂	1	1	yes	maximal, characteristic
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	#20269912	C₄	1	8	yes	complemented
10	#20269887	—	2	2	no	—
11	#20269887	C₄	1	4	yes	—
12	C₈×Dic₂₁₁	—	2	2	no	—
13	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₁₁	Q₈	1	2	yes	—
16	#16809946	—	2	2	no	—
17	#16809946	C₂×C₄	1	4	yes	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	4	yes	—
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	—	4	2	no	—
30	C₂×Dic₂₁₁	C₄⋊C₄	1	4	yes	—
31	C₂₁₁⋊C₈	—	2	4	no	—
32	C₂₁₁⋊C₈	C₄²	1	4	yes	—
33	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	#41	1	2	yes	—
38	C₈×C₂₁₁	—	2	2	no	—
39	C₈×C₂₁₁	—	2	4	no	—
40	C₂²×C₂₁₁	—	2	2	no	—
41	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₂₁₁	C₄×Q₈	1	2	yes	—
43	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₂₁₁	#117	1	1	yes	characteristic
45	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
46	Dic₂₁₁	—	2	4	no	—
47	C₂×C₂₁₁	—	2	2	no	—
48	C₂×C₂₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₂₁₁	#430	1	2	yes	—
50	C₂₁₁	#1778	1	1	yes	characteristic, complemented, Sylow
51	#1778	—	211	1	no	maximal, Sylow
52	C₂×#108	—	211	2	no	—
53	C₄×M₁₆	—	211	2	no	—
54	C₂×#97	—	211	1	no	—
55	#341	—	211	1	no	—
56	#341	D₂₁₁	1	1	yes	characteristic
57	C₂×M₁₆	—	211	2	no	—
58	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
59	C₂²×C₈	—	211	1	no	—
60	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
61	C₂×C₄²	—	211	1	no	—
62	#108	—	211	8	no	—
63	#97	—	422	2	no	—
64	#97	—	211	4	no	—
65	C₄×C₈	—	422	2	no	—
66	C₂²×C₄	—	211	2	no	—
67	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	422	2	no	—
70	M₁₆	—	211	4	no	—
71	M₁₆	C₄×D₂₁₁	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	422	2	no	—
75	C₂×C₈	—	211	2	no	—
76	C₂×C₈	—	211	4	no	—
77	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
78	C₄²	—	211	2	no	—
79	C₄²	—	211	2	no	—
80	C₂³	Q₈×D₂₁₁	1	1	yes	characteristic
81	C₂×C₄	—	844	2	no	—
82	C₂×C₄	—	211	4	no	—
83	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
85	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
86	C₂×C₄	#16809961	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	422	4	no	—
90	C₈	—	211	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
93	C₂²	#20269942	1	2	yes	—
94	C₄	—	422	4	no	—
95	C₄	C₄²×D₂₁₁	1	1	yes	characteristic
96	C₄	#20269933	1	1	yes	characteristic
97	C₄	(C₂²⋊D₂₁₁)⋊C₄	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#27212997	1	1	yes	characteristic
100	C₂	#27213193	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269912

C₂

yes

maximal

M₁₆×Dic₂₁₁

C₂

yes

maximal

C₂×#20269887

C₂

yes

maximal, characteristic

#27212911

C₂

yes

maximal, characteristic

#341×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

#20269912

C₄

yes

complemented

#20269887

—

#20269887

C₄

yes

—

C₈×Dic₂₁₁

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

Q₈

yes

—

#16809946

—

#16809946

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₄×Dic₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

C₄²

yes

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

C₂×C₄²

yes

characteristic

C₄×C₂₁₁

#117

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₁₁

#430

yes

—

C₂₁₁

#1778

yes

characteristic, complemented, Sylow

#1778

—

211

maximal, Sylow

C₂×#108

—

211

—

C₄×M₁₆

—

211

—

C₂×#97

—

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

—

#108

—

211

—

#97

—

422

—

#97

—

211

—

C₄×C₈

—

422

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

211

—

M₁₆

C₄×D₂₁₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₄²

—

211

—

C₄²

—

211

—

C₂³

Q₈×D₂₁₁

yes

characteristic

C₂×C₄

—

844

—

C₂×C₄

—

211

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂×C₄×D₂₁₁

yes

—

C₂×C₄

D₄×D₂₁₁

yes

characteristic

C₂×C₄

#16809961

yes

—

C₈

—

C₈

—

C₈

—

422

—

C₈

—

211

—

C₂²

—

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

C₂²

#20269942

yes

—

C₄

—

422

—

C₄

C₄²×D₂₁₁

yes

characteristic

C₄

#20269933

yes

characteristic

C₄

(C₂²⋊D₂₁₁)⋊C₄

yes

—

C₂

—

C₂

#27212997

yes

characteristic

100

C₂

#27213193

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₁₁
commutator	C₄×C₂₁₁	C₂×C₄²
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	#1778
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₁₁

commutator

C₄×C₂₁₁

C₂×C₄²

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 ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #20269912	C₄	semidirect
50. C₂₁₁	#1778	semidirect

H ≅ G/N

type

9. #20269912

C₄

semidirect

50. C₂₁₁

#1778

semidirect

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

subgroup	index
C₂×#20269912	2
M₁₆×Dic₂₁₁	2
C₂×#20269887	2
#27212911	2
#341×C₂₁₁	2
#1778	211

subgroup

index

M₁₆×Dic₂₁₁

#341×C₂₁₁

211

Subgroups — #41013303

Conjugacy classes of subgroups

Complements

Maximal subgroups