Group #41009841 — subgroups

Subgroups — #2221×C₂₁₁ #41009841

Order 27008.

148 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 60 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×D₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#504×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#472×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#368×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#361×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	AGL(1,ℤ/8)×C₂₁₁	—	2	2	no	—
7	C₂×D₈×C₂₁₁	C₂²	1	2	yes	—
8	#131×C₂₁₁	C₂²	1	1	yes	characteristic
9	D₁₆×C₂₁₁	—	2	1	no	—
10	D₁₆×C₂₁₁	—	2	1	no	—
11	M₃₂×C₂₁₁	—	2	1	no	—
12	M₃₂×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₂₁₁	—	2	1	no	—
14	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
15	#108×C₂₁₁	C₂²	1	2	yes	—
16	#100×C₂₁₁	—	2	2	no	—
17	#41×C₂₁₁	D₄	1	1	yes	characteristic
18	C₂×D₄×C₂₁₁	—	2	2	no	—
19	SD₁₆×C₂₁₁	—	4	2	no	—
20	D₈×C₂₁₁	—	2	2	no	—
21	D₈×C₂₁₁	—	2	2	no	—
22	D₈×C₂₁₁	—	4	2	no	—
23	M₁₆×C₂₁₁	—	2	1	no	—
24	M₁₆×C₂₁₁	—	2	2	no	—
25	M₁₆×C₂₁₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₂₁₁	—	2	1	no	—
27	C₂×C₈×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₁₆×C₂₁₁	—	2	1	no	—
29	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
30	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
31	C₂³×C₂₁₁	—	4	2	no	—
32	Q₈×C₂₁₁	D₈	1	1	yes	characteristic
33	D₄×C₂₁₁	—	2	1	no	—
34	D₄×C₂₁₁	—	4	2	no	—
35	D₄×C₂₁₁	—	4	2	no	—
36	D₄×C₂₁₁	D₈	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	—	2	1	no	—
38	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₈×C₂₁₁	—	2	1	no	—
40	C₈×C₂₁₁	—	4	2	no	—
41	C₈×C₂₁₁	C₂×D₄	1	1	yes	characteristic
42	C₈×C₂₁₁	#41	1	1	yes	characteristic
43	C₂²×C₂₁₁	—	2	1	no	—
44	C₂²×C₂₁₁	—	8	2	no	—
45	C₂²×C₂₁₁	—	16	2	no	—
46	C₂²×C₂₁₁	#135	1	1	yes	characteristic
47	C₄×C₂₁₁	—	2	1	no	—
48	C₄×C₂₁₁	#121	1	1	yes	characteristic
49	C₄×C₂₁₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₂₁₁	—	2	1	no	—
51	C₂×C₂₁₁	—	4	1	no	—
52	C₂×C₂₁₁	—	16	2	no	—
53	C₂×C₂₁₁	#466	1	1	yes	characteristic
54	C₂₁₁	#2221	1	1	yes	characteristic, direct product factor, Sylow
55	#2221	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×D₁₆	C₂×C₂₁₁	1	1	yes	characteristic, complemented
57	#504	C₂×C₂₁₁	1	1	yes	characteristic, complemented
58	#472	C₂×C₂₁₁	1	2	yes	complemented
59	#368	C₂×C₂₁₁	1	1	yes	characteristic, complemented
60	#361	C₂×C₂₁₁	1	2	yes	complemented
61	AGL(1,ℤ/8)	—	2	2	no	—
62	C₂×D₈	C₂²×C₂₁₁	1	2	yes	—
63	#131	C₂²×C₂₁₁	1	1	yes	characteristic
64	D₁₆	—	2	1	no	—
65	D₁₆	—	2	1	no	—
66	M₃₂	—	2	1	no	—
67	M₃₂	C₂²×C₂₁₁	1	1	yes	characteristic
68	C₂×C₁₆	—	2	1	no	—
69	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
70	#108	C₂²×C₂₁₁	1	2	yes	—
71	#100	—	2	2	no	—
72	#41	D₄×C₂₁₁	1	1	yes	characteristic
73	C₂×D₄	—	2	2	no	—
74	SD₁₆	—	4	2	no	—
75	D₈	—	2	2	no	—
76	D₈	—	2	2	no	—
77	D₈	—	4	2	no	—
78	M₁₆	—	2	1	no	—
79	M₁₆	—	2	2	no	—
80	M₁₆	D₄×C₂₁₁	1	1	yes	characteristic
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	C₂³×C₂₁₁	1	1	yes	characteristic
83	C₁₆	—	2	1	no	—
84	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
85	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
86	C₂³	—	4	2	no	—
87	Q₈	D₈×C₂₁₁	1	1	yes	characteristic
88	D₄	—	2	1	no	—
89	D₄	—	4	2	no	—
90	D₄	—	4	2	no	—
91	D₄	D₈×C₂₁₁	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	C₂×D₄×C₂₁₁	1	1	yes	characteristic
97	C₈	#41×C₂₁₁	1	1	yes	characteristic
98	C₂²	—	2	1	no	—
99	C₂²	—	8	2	no	—
100	C₂²	—	16	2	no	—
101	C₂²	#135×C₂₁₁	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	#121×C₂₁₁	1	1	yes	characteristic
104	C₄	C₂×D₈×C₂₁₁	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	4	1	no	—
107	C₂	—	16	2	no	—
108	C₂	#466×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₆×C₂₁₁

C₂

yes

maximal, characteristic, complemented

#504×C₂₁₁

C₂

yes

maximal, characteristic, complemented

#472×C₂₁₁

C₂

yes

maximal, complemented

#368×C₂₁₁

C₂

yes

maximal, characteristic, complemented

#361×C₂₁₁

C₂

yes

maximal, complemented

AGL(1,ℤ/8)×C₂₁₁

—

C₂×D₈×C₂₁₁

C₂²

yes

—

#131×C₂₁₁

C₂²

yes

characteristic

D₁₆×C₂₁₁

—

D₁₆×C₂₁₁

—

M₃₂×C₂₁₁

—

M₃₂×C₂₁₁

C₂²

yes

characteristic

C₂×C₁₆×C₂₁₁

—

C₂×C₁₆×C₂₁₁

C₂²

yes

characteristic

#108×C₂₁₁

C₂²

yes

—

#100×C₂₁₁

—

#41×C₂₁₁

D₄

yes

characteristic

C₂×D₄×C₂₁₁

—

SD₁₆×C₂₁₁

—

D₈×C₂₁₁

—

D₈×C₂₁₁

—

D₈×C₂₁₁

—

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂³

yes

characteristic

C₁₆×C₂₁₁

—

C₁₆×C₂₁₁

D₄

yes

characteristic

C₁₆×C₂₁₁

D₄

yes

characteristic

C₂³×C₂₁₁

—

Q₈×C₂₁₁

D₈

yes

characteristic

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

D₈

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

C₂×D₄

yes

characteristic

C₈×C₂₁₁

#41

yes

characteristic

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#135

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

#121

yes

characteristic

C₄×C₂₁₁

C₂×D₈

yes

characteristic

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#466

yes

characteristic

C₂₁₁

#2221

yes

characteristic, direct product factor, Sylow

#2221

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×D₁₆

C₂×C₂₁₁

yes

characteristic, complemented

#504

C₂×C₂₁₁

yes

characteristic, complemented

#472

C₂×C₂₁₁

yes

complemented

#368

C₂×C₂₁₁

yes

characteristic, complemented

#361

C₂×C₂₁₁

yes

complemented

AGL(1,ℤ/8)

—

C₂×D₈

C₂²×C₂₁₁

yes

—

#131

C₂²×C₂₁₁

yes

characteristic

D₁₆

—

D₁₆

—

M₃₂

—

M₃₂

C₂²×C₂₁₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₂₁₁

yes

characteristic

#108

C₂²×C₂₁₁

yes

—

#100

—

#41

D₄×C₂₁₁

yes

characteristic

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

D₄×C₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂³×C₂₁₁

yes

characteristic

C₁₆

—

C₁₆

D₄×C₂₁₁

yes

characteristic

C₁₆

D₄×C₂₁₁

yes

characteristic

C₂³

—

Q₈

D₈×C₂₁₁

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

D₈×C₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

C₈

—

C₈

—

C₈

C₂×D₄×C₂₁₁

yes

characteristic

C₈

#41×C₂₁₁

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#135×C₂₁₁

yes

characteristic

102

C₄

—

103

C₄

#121×C₂₁₁

yes

characteristic

104

C₄

C₂×D₈×C₂₁₁

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

#466×C₂₁₁

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₂₁₁	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₁₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₁₁ ◁ C₂×C₄×C₂₁₁ ◁ #131×C₂₁₁ ◁ G

subgroup

G/H

center, socle

C₂×C₂₁₁

#466

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₂₁₁

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₂₁₁ ◁ C₂×C₄×C₂₁₁ ◁ #131×C₂₁₁ ◁ G

N	H ≅ G/N	type
1. C₂×D₁₆×C₂₁₁	C₂	semidirect
2. #504×C₂₁₁	C₂	semidirect
3. #472×C₂₁₁	C₂	semidirect
4. #368×C₂₁₁	C₂	semidirect
5. #361×C₂₁₁	C₂	semidirect
54. C₂₁₁	#2221	direct (N × H)
55. #2221	C₂₁₁	direct (N × H)
56. C₂×D₁₆	C₂×C₂₁₁	semidirect
57. #504	C₂×C₂₁₁	semidirect
58. #472	C₂×C₂₁₁	semidirect
59. #368	C₂×C₂₁₁	semidirect
60. #361	C₂×C₂₁₁	semidirect

H ≅ G/N

type

1. C₂×D₁₆×C₂₁₁

C₂

semidirect

2. #504×C₂₁₁

C₂

semidirect

3. #472×C₂₁₁

C₂

semidirect

4. #368×C₂₁₁

C₂

semidirect

5. #361×C₂₁₁

C₂

semidirect

54. C₂₁₁

#2221

direct (N × H)

55. #2221

C₂₁₁

direct (N × H)

56. C₂×D₁₆

C₂×C₂₁₁

semidirect

57. #504

C₂×C₂₁₁

semidirect

58. #472

C₂×C₂₁₁

semidirect

59. #368

C₂×C₂₁₁

semidirect

60. #361

C₂×C₂₁₁

semidirect

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

subgroup	index
C₂×D₁₆×C₂₁₁	2
#504×C₂₁₁	2
#472×C₂₁₁	2
#368×C₂₁₁	2
#361×C₂₁₁	2
#2221	211

subgroup

index

C₂×D₁₆×C₂₁₁

#504×C₂₁₁

#472×C₂₁₁

#368×C₂₁₁

#361×C₂₁₁

#2221

211

Subgroups — #2221×C₂₁₁ #41009841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2221×C211 #41009841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2221×C₂₁₁ #41009841