Group #41008913 — subgroups

Subgroups — #1293×C₂₁₁ #41008913

Order 27008.

184 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 68 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	#407×C₂₁₁	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
4	C₂×M₁₆×C₂₁₁	—	2	2	no	—
5	C₂²×C₈×C₂₁₁	C₄	1	2	yes	—
6	C₂²⋊C₈×C₂₁₁	—	2	2	no	—
7	C₂²⋊C₈×C₂₁₁	—	2	2	no	—
8	C₂²⋊C₈×C₂₁₁	C₄	1	4	yes	—
9	C₂⁴×C₂₁₁	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₂₁₁	—	4	2	no	—
11	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
14	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
15	M₁₆×C₂₁₁	—	4	4	no	—
16	C₂×C₈×C₂₁₁	—	2	2	no	—
17	C₂×C₈×C₂₁₁	—	2	4	no	—
18	C₂×C₈×C₂₁₁	—	4	2	no	—
19	C₂³×C₂₁₁	—	2	2	no	—
20	C₂³×C₂₁₁	—	4	1	no	—
21	C₂³×C₂₁₁	—	4	1	no	—
22	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
23	C₂³×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₁₁	—	2	2	no	—
25	C₂×C₄×C₂₁₁	—	2	4	no	—
26	C₂×C₄×C₂₁₁	—	4	2	no	—
27	C₂×C₄×C₂₁₁	—	4	1	no	—
28	C₂×C₄×C₂₁₁	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₂₁₁	—	4	4	no	—
32	C₈×C₂₁₁	—	4	2	no	—
33	C₂²×C₂₁₁	—	2	1	no	—
34	C₂²×C₂₁₁	—	2	1	no	—
35	C₂²×C₂₁₁	—	2	2	no	—
36	C₂²×C₂₁₁	—	2	2	no	—
37	C₂²×C₂₁₁	—	2	2	no	—
38	C₂²×C₂₁₁	—	4	4	no	—
39	C₂²×C₂₁₁	#95	1	1	yes	characteristic
40	C₂²×C₂₁₁	#108	1	2	yes	—
41	C₄×C₂₁₁	—	2	2	no	—
42	C₄×C₂₁₁	—	2	1	no	—
43	C₄×C₂₁₁	#99	1	2	yes	—
44	C₂×C₂₁₁	—	2	2	no	—
45	C₂×C₂₁₁	—	4	2	no	—
46	C₂×C₂₁₁	#338	1	1	yes	characteristic
47	C₂×C₂₁₁	#341	1	1	yes	characteristic
48	C₂×C₂₁₁	#342	1	1	yes	characteristic
49	C₂₁₁	#1293	1	1	yes	characteristic, direct product factor, Sylow
50	#1293	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	#407	C₂×C₂₁₁	1	2	yes	—
52	C₂×C₂²⋊C₈	C₂×C₂₁₁	1	1	yes	characteristic
53	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
54	C₂×M₁₆	—	2	2	no	—
55	C₂²×C₈	C₄×C₂₁₁	1	2	yes	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	C₂²⋊C₈	C₄×C₂₁₁	1	4	yes	—
59	C₂⁴	Q₈×C₂₁₁	1	1	yes	characteristic
60	C₂²×C₄	—	4	2	no	—
61	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
62	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
63	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
64	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
65	M₁₆	—	4	4	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	1	no	—
71	C₂³	—	4	1	no	—
72	C₂³	C₄⋊C₄×C₂₁₁	1	2	yes	—
73	C₂³	C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₄²×C₂₁₁	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
80	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
81	C₈	—	4	4	no	—
82	C₈	—	4	2	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#95×C₂₁₁	1	1	yes	characteristic
90	C₂²	#108×C₂₁₁	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	1	no	—
93	C₄	#99×C₂₁₁	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	4	2	no	—
96	C₂	#338×C₂₁₁	1	1	yes	characteristic
97	C₂	#341×C₂₁₁	1	1	yes	characteristic
98	C₂	#342×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#407×C₂₁₁

C₂

yes

maximal

C₂×C₂²⋊C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₁₁

—

C₂²×C₈×C₂₁₁

C₄

yes

—

C₂²⋊C₈×C₂₁₁

—

C₂²⋊C₈×C₂₁₁

—

C₂²⋊C₈×C₂₁₁

C₄

yes

—

C₂⁴×C₂₁₁

Q₈

yes

characteristic

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄²

yes

characteristic

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#95

yes

characteristic

C₂²×C₂₁₁

#108

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#99

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, direct product factor, Sylow

#1293

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

#407

C₂×C₂₁₁

yes

—

C₂×C₂²⋊C₈

C₂×C₂₁₁

yes

characteristic

C₂³×C₄

C₂²×C₂₁₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₄×C₂₁₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₄×C₂₁₁

yes

—

C₂⁴

Q₈×C₂₁₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

D₄×C₂₁₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄⋊C₄×C₂₁₁

yes

—

C₂³

C₄⋊C₄×C₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄²×C₂₁₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#95×C₂₁₁

yes

characteristic

C₂²

#108×C₂₁₁

yes

—

C₄

—

C₄

—

C₄

#99×C₂₁₁

yes

—

C₂

—

C₂

—

C₂

#338×C₂₁₁

yes

characteristic

C₂

#341×C₂₁₁

yes

characteristic

C₂

#342×C₂₁₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₁₁	#95
commutator	C₂×C₄	C₄²×C₂₁₁
Frattini, cosocle	C₂³×C₄	C₂²×C₂₁₁
prime radical	C₂⁴×C₂₁₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ C₂³×C₄×C₂₁₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₁₁

#95

commutator

C₂×C₄

C₄²×C₂₁₁

Frattini, cosocle

C₂³×C₄

C₂²×C₂₁₁

prime radical

C₂⁴×C₂₁₁

Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₁₁ ◁ C₂³×C₄×C₂₁₁ ◁ G

N	H ≅ G/N	type
49. C₂₁₁	#1293	direct (N × H)
50. #1293	C₂₁₁	direct (N × H)

H ≅ G/N

type

49. C₂₁₁

#1293

direct (N × H)

50. #1293

C₂₁₁

direct (N × H)

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

subgroup	index
#407×C₂₁₁	2
C₂×C₂²⋊C₈×C₂₁₁	2
#1293	211

subgroup

index

#407×C₂₁₁

C₂×C₂²⋊C₈×C₂₁₁

#1293

211

Subgroups — #1293×C₂₁₁ #41008913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1293×C211 #41008913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1293×C₂₁₁ #41008913