Group #41009211 — subgroups

Subgroups — #1591×C₂₁₁ #41009211

Order 27008.

152 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 88 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	#446×C₂₁₁	C₂	1	2	yes	maximal
2	C₄×Q₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#335×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₈×C₂₁₁	C₂	1	2	yes	maximal
5	C₈²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₂₁₁	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₂₁₁	C₂²	1	2	yes	—
8	#107×C₂₁₁	C₄	1	1	yes	characteristic
9	C₄⋊C₈×C₂₁₁	—	2	2	no	—
10	C₄⋊C₈×C₂₁₁	C₂²	1	2	yes	—
11	#103×C₂₁₁	C₄	1	2	yes	—
12	#97×C₂₁₁	—	2	2	no	—
13	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₁₁	C₂×C₄	1	2	yes	—
17	Q₁₆×C₂₁₁	—	2	2	no	—
18	C₂×C₈×C₂₁₁	—	2	1	no	—
19	C₂×C₈×C₂₁₁	—	2	2	no	—
20	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₂₁₁	—	2	2	no	—
25	C₄⋊C₄×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₄²×C₂₁₁	—	2	2	no	—
27	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
28	Q₈×C₂₁₁	—	2	2	no	—
29	Q₈×C₂₁₁	—	2	2	no	—
30	C₂×C₄×C₂₁₁	—	2	2	no	—
31	C₂×C₄×C₂₁₁	—	2	2	no	—
32	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
35	C₈×C₂₁₁	—	2	2	no	—
36	C₈×C₂₁₁	—	2	1	no	—
37	C₈×C₂₁₁	—	2	1	no	—
38	C₈×C₂₁₁	—	4	2	no	—
39	C₈×C₂₁₁	M₁₆	1	2	yes	—
40	C₈×C₂₁₁	Q₁₆	1	2	yes	complemented
41	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
42	C₄×C₂₁₁	—	2	1	no	—
43	C₄×C₂₁₁	—	4	2	no	—
44	C₄×C₂₁₁	—	4	2	no	—
45	C₄×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
46	C₄×C₂₁₁	#131	1	1	yes	characteristic
47	C₄×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
48	C₄×C₂₁₁	#135	1	1	yes	characteristic
49	C₂×C₂₁₁	#436	1	1	yes	characteristic
50	C₂×C₂₁₁	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₂₁₁	#443	1	1	yes	characteristic
52	C₂₁₁	#1591	1	1	yes	characteristic, direct product factor, Sylow
53	#1591	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	#446	C₂×C₂₁₁	1	2	yes	—
55	C₄×Q₁₆	C₂×C₂₁₁	1	1	yes	characteristic
56	#335	C₂×C₂₁₁	1	1	yes	characteristic
57	Q₈⋊C₈	C₂×C₂₁₁	1	2	yes	—
58	C₈²	C₂×C₂₁₁	1	1	yes	characteristic
59	C₂×Q₁₆	C₄×C₂₁₁	1	1	yes	characteristic
60	C₄×Q₈	C₂²×C₂₁₁	1	2	yes	—
61	#107	C₄×C₂₁₁	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	C₂²×C₂₁₁	1	2	yes	—
64	#103	C₄×C₂₁₁	1	2	yes	—
65	#97	—	2	2	no	—
66	C₄×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
67	C₄×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
68	C₄×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
69	C₂×Q₈	C₂×C₄×C₂₁₁	1	2	yes	—
70	Q₁₆	—	2	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
74	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
75	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
76	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	C₂×C₄×C₂₁₁	1	2	yes	—
79	C₄²	—	2	2	no	—
80	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
81	Q₈	—	2	2	no	—
82	Q₈	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
87	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	4	2	no	—
92	C₈	M₁₆×C₂₁₁	1	2	yes	—
93	C₈	Q₁₆×C₂₁₁	1	2	yes	complemented
94	C₂²	C₄×D₄×C₂₁₁	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	C₂×M₁₆×C₂₁₁	1	1	yes	characteristic
99	C₄	#131×C₂₁₁	1	1	yes	characteristic
100	C₄	C₂×Q₁₆×C₂₁₁	1	1	yes	characteristic
101	C₄	#135×C₂₁₁	1	1	yes	characteristic
102	C₂	#436×C₂₁₁	1	1	yes	characteristic
103	C₂	C₄×Q₁₆×C₂₁₁	1	1	yes	characteristic
104	C₂	#443×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#446×C₂₁₁

C₂

yes

maximal

C₄×Q₁₆×C₂₁₁

C₂

yes

maximal, characteristic

#335×C₂₁₁

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₂₁₁

C₂

yes

maximal

C₈²×C₂₁₁

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₂₁₁

C₄

yes

characteristic

C₄×Q₈×C₂₁₁

C₂²

yes

—

#107×C₂₁₁

C₄

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₄⋊C₈×C₂₁₁

C₂²

yes

—

#103×C₂₁₁

C₄

yes

—

#97×C₂₁₁

—

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

Q₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

M₁₆

yes

—

C₈×C₂₁₁

Q₁₆

yes

complemented

C₂²×C₂₁₁

C₄×D₄

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×M₁₆

yes

characteristic

C₄×C₂₁₁

#131

yes

characteristic

C₄×C₂₁₁

C₂×Q₁₆

yes

C₄×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

C₄×Q₁₆

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, direct product factor, Sylow

#1591

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

#446

C₂×C₂₁₁

yes

—

C₄×Q₁₆

C₂×C₂₁₁

yes

characteristic

#335

C₂×C₂₁₁

yes

characteristic

Q₈⋊C₈

C₂×C₂₁₁

yes

—

C₈²

C₂×C₂₁₁

yes

characteristic

C₂×Q₁₆

C₄×C₂₁₁

yes

characteristic

C₄×Q₈

C₂²×C₂₁₁

yes

—

#107

C₄×C₂₁₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₁₁

yes

—

#103

C₄×C₂₁₁

yes

—

#97

—

C₄×C₈

C₂²×C₂₁₁

yes

characteristic

C₄×C₈

C₂²×C₂₁₁

yes

characteristic

C₄×C₈

C₂²×C₂₁₁

yes

characteristic

C₂×Q₈

C₂×C₄×C₂₁₁

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₁₁

yes

characteristic

C₂×C₈

C₂×C₄×C₂₁₁

yes

characteristic

C₂×C₈

D₄×C₂₁₁

yes

characteristic

C₂×C₈

D₄×C₂₁₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₂₁₁

yes

—

C₄²

—

C₄²

C₂³×C₂₁₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

C₂×C₄

#41×C₂₁₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

M₁₆×C₂₁₁

yes

—

C₈

Q₁₆×C₂₁₁

yes

complemented

C₂²

C₄×D₄×C₂₁₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×M₁₆×C₂₁₁

yes

characteristic

C₄

#131×C₂₁₁

yes

characteristic

100

C₄

C₂×Q₁₆×C₂₁₁

yes

characteristic

101

C₄

#135×C₂₁₁

yes

characteristic

102

C₂

#436×C₂₁₁

yes

characteristic

103

C₂

C₄×Q₁₆×C₂₁₁

yes

characteristic

104

C₂

#443×C₂₁₁

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₂₁₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₄²	C₂³×C₂₁₁
socle, prime radical	C₂²×C₂₁₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₁₁ ◁ C₄×C₈×C₂₁₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₁₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₄²

C₂³×C₂₁₁

socle, prime radical

C₂²×C₂₁₁

C₄×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₁₁ ◁ C₄×C₈×C₂₁₁ ◁ G

N	H ≅ G/N	type
40. C₈×C₂₁₁	Q₁₆	semidirect
52. C₂₁₁	#1591	direct (N × H)
53. #1591	C₂₁₁	direct (N × H)
93. C₈	Q₁₆×C₂₁₁	semidirect

H ≅ G/N

type

40. C₈×C₂₁₁

Q₁₆

semidirect

52. C₂₁₁

#1591

direct (N × H)

53. #1591

C₂₁₁

direct (N × H)

93. C₈

Q₁₆×C₂₁₁

semidirect

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

subgroup	index
#446×C₂₁₁	2
C₄×Q₁₆×C₂₁₁	2
#335×C₂₁₁	2
Q₈⋊C₈×C₂₁₁	2
C₈²×C₂₁₁	2
#1591	211

subgroup

index

#446×C₂₁₁

C₄×Q₁₆×C₂₁₁

#335×C₂₁₁

Q₈⋊C₈×C₂₁₁

C₈²×C₂₁₁

#1591

211

Subgroups — #1591×C₂₁₁ #41009211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1591×C211 #41009211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1591×C₂₁₁ #41009211