Group #41013299 — subgroups

Subgroups — #41013299

Order 27008.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₈×Dic₂₁₁	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	C₈×Dic₂₁₁	—	2	2	no	—
12	C₈×Dic₂₁₁	C₄	1	4	yes	—
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	1	yes	characteristic
23	C₄×Dic₂₁₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₁₁	—	2	2	no	—
26	M₁₆×C₂₁₁	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₂₁₁	—	2	2	no	—
28	C₂×C₈×C₂₁₁	—	2	2	no	—
29	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₁₁	—	2	4	no	—
31	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
32	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
33	C₂₁₁⋊C₈	—	2	4	no	—
34	C₂₁₁⋊C₈	C₄²	1	4	yes	—
35	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	#41	1	2	yes	—
40	C₈×C₂₁₁	—	2	2	no	—
41	C₈×C₂₁₁	—	2	4	no	—
42	C₂²×C₂₁₁	—	2	2	no	—
43	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₂₁₁	C₄×Q₈	1	2	yes	—
45	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₂₁₁	#117	1	1	yes	characteristic
47	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
48	Dic₂₁₁	—	2	2	no	—
49	Dic₂₁₁	#108	1	4	yes	complemented
50	C₂×C₂₁₁	—	2	2	no	—
51	C₂×C₂₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₂₁₁	C₂×#108	1	2	yes	—
53	C₂₁₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
54	C₄×#108	—	211	1	no	maximal, Sylow
55	C₂×#108	—	211	2	no	—
56	C₄×M₁₆	—	211	2	no	—
57	C₂×C₄×C₈	—	211	1	no	—
58	#341	—	211	1	no	—
59	#341	D₂₁₁	1	1	yes	characteristic
60	C₂×M₁₆	—	211	2	no	—
61	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
62	C₂²×C₈	—	211	1	no	—
63	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
64	C₂×C₄²	—	211	1	no	—
65	#108	—	211	8	no	—
66	#97	—	422	2	no	—
67	C₄×C₈	—	422	2	no	—
68	C₄×C₈	—	211	4	no	—
69	C₂²×C₄	—	211	2	no	—
70	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	422	2	no	—
73	M₁₆	—	211	4	no	—
74	M₁₆	C₄×D₂₁₁	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	422	2	no	—
78	C₂×C₈	—	211	2	no	—
79	C₂×C₈	—	211	4	no	—
80	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
81	C₄²	—	211	2	no	—
82	C₄²	—	211	1	no	—
83	C₄²	—	211	1	no	—
84	C₂³	Q₈×D₂₁₁	1	1	yes	characteristic
85	C₂×C₄	—	422	4	no	—
86	C₂×C₄	—	211	2	no	—
87	C₂×C₄	—	211	2	no	—
88	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
90	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
91	C₂×C₄	#16809961	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	422	4	no	—
95	C₈	—	211	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
98	C₂²	#20269942	1	2	yes	—
99	C₄	—	422	2	no	—
100	C₄	—	211	4	no	—
101	C₄	C₄²×D₂₁₁	1	1	yes	characteristic
102	C₄	#20269933	1	1	yes	characteristic
103	C₄	(C₂²⋊D₂₁₁)⋊C₄	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#27212997	1	1	yes	characteristic
106	C₂	#108×D₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269912

C₂

yes

maximal

M₁₆×Dic₂₁₁

C₂

yes

maximal

C₂×C₈×Dic₂₁₁

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

—

C₈×Dic₂₁₁

—

C₈×Dic₂₁₁

C₄

yes

—

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

characteristic

C₄×Dic₂₁₁

D₄

yes

characteristic

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

—

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

—

Dic₂₁₁

#108

yes

complemented

C₂×C₂₁₁

—

C₂×C₂₁₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₁₁

C₂×#108

yes

—

C₂₁₁

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

211

maximal, Sylow

C₂×#108

—

211

—

C₄×M₁₆

—

211

—

C₂×C₄×C₈

—

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

—

C₄×C₈

—

422

—

C₄×C₈

—

211

—

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

—

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

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

—

100

C₄

—

211

—

101

C₄

C₄²×D₂₁₁

yes

characteristic

102

C₄

#20269933

yes

characteristic

103

C₄

(C₂²⋊D₂₁₁)⋊C₄

yes

—

104

C₂

—

105

C₂

#27212997

yes

characteristic

106

C₂

#108×D₂₁₁

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	C₄×#108
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
49. Dic₂₁₁	#108	semidirect
53. C₂₁₁	C₄×#108	semidirect

H ≅ G/N

type

9. #20269912

C₄

semidirect

49. Dic₂₁₁

#108

semidirect

53. C₂₁₁

C₄×#108

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×C₈×Dic₂₁₁	2
#27212911	2
#341×C₂₁₁	2
C₄×#108	211

subgroup

index

C₂×#20269912

M₁₆×Dic₂₁₁

C₂×C₈×Dic₂₁₁

#27212911

#341×C₂₁₁

C₄×#108

211

Subgroups — #41013299

Conjugacy classes of subgroups

Complements

Maximal subgroups