Group #41010463 — subgroups

Subgroups — #2843×C₂₁₁ #41010463

Order 27008.

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

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 184 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	#400×C₂₁₁	C₂	1	3	yes	maximal
2	#398×C₂₁₁	C₂	1	6	yes	maximal
3	#395×C₂₁₁	C₂	1	3	yes	maximal
4	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#383×C₂₁₁	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	3	yes	—
8	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	3	yes	—
9	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	6	yes	—
10	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	3	yes	—
11	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
12	#95×C₂₁₁	C₂²	1	6	yes	—
13	#95×C₂₁₁	C₂²	1	6	yes	—
14	#95×C₂₁₁	C₂²	1	3	yes	—
15	#95×C₂₁₁	C₂²	1	3	yes	—
16	C₂×Q₈×C₂₁₁	—	4	3	no	—
17	C₂²×C₄×C₂₁₁	C₂³	1	6	yes	—
18	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
19	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
21	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
22	C₄⋊C₄×C₂₁₁	—	4	3	no	—
23	C₄⋊C₄×C₂₁₁	—	4	3	no	—
24	C₄⋊C₄×C₂₁₁	—	4	3	no	—
25	C₄⋊C₄×C₂₁₁	—	4	6	no	—
26	C₄²×C₂₁₁	Q₈	1	4	yes	complemented
27	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
28	Q₈×C₂₁₁	—	4	4	no	—
29	C₂×C₄×C₂₁₁	—	2	3	no	—
30	C₂×C₄×C₂₁₁	—	2	3	no	—
31	C₂×C₄×C₂₁₁	—	2	6	no	—
32	C₂×C₄×C₂₁₁	—	2	3	no	—
33	C₂×C₄×C₂₁₁	—	2	6	no	—
34	C₂×C₄×C₂₁₁	—	2	3	no	—
35	C₂×C₄×C₂₁₁	—	2	6	no	—
36	C₂×C₄×C₂₁₁	—	2	12	no	—
37	C₂×C₄×C₂₁₁	C₂×Q₈	1	6	yes	—
38	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₂₁₁	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₂₁₁	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₂₁₁	—	2	6	no	—
42	C₄×C₂₁₁	—	4	3	no	—
43	C₄×C₂₁₁	—	4	1	no	—
44	C₄×C₂₁₁	—	4	2	no	—
45	C₄×C₂₁₁	—	4	6	no	—
46	C₂×C₂₁₁	#544	1	3	yes	—
47	C₂×C₂₁₁	#560	1	1	yes	characteristic
48	C₂×C₂₁₁	#563	1	3	yes	—
49	C₂₁₁	#2843	1	1	yes	characteristic, direct product factor, Sylow
50	#2843	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	#400	C₂×C₂₁₁	1	3	yes	—
52	#398	C₂×C₂₁₁	1	6	yes	—
53	#395	C₂×C₂₁₁	1	3	yes	—
54	#384	C₂×C₂₁₁	1	1	yes	characteristic
55	#383	C₂×C₂₁₁	1	2	yes	—
56	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
57	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	3	yes	—
58	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	3	yes	—
59	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	6	yes	—
60	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	3	yes	—
61	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
62	#95	C₂²×C₂₁₁	1	6	yes	—
63	#95	C₂²×C₂₁₁	1	6	yes	—
64	#95	C₂²×C₂₁₁	1	3	yes	—
65	#95	C₂²×C₂₁₁	1	3	yes	—
66	C₂×Q₈	—	4	3	no	—
67	C₂²×C₄	C₂³×C₂₁₁	1	6	yes	—
68	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
69	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
70	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
71	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	4	3	no	—
75	C₄⋊C₄	—	4	6	no	—
76	C₄²	Q₈×C₂₁₁	1	4	yes	complemented
77	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
78	Q₈	—	4	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	6	no	—
84	C₂×C₄	—	2	3	no	—
85	C₂×C₄	—	2	6	no	—
86	C₂×C₄	—	2	12	no	—
87	C₂×C₄	C₂×Q₈×C₂₁₁	1	6	yes	—
88	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
89	C₂²	H(5,𝔽₂)×C₂₁₁	1	3	yes	—
90	C₂²	2₋¹⁺⁴×C₂₁₁	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	3	no	—
95	C₄	—	4	6	no	—
96	C₂	#544×C₂₁₁	1	3	yes	—
97	C₂	#560×C₂₁₁	1	1	yes	characteristic
98	C₂	#563×C₂₁₁	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#400×C₂₁₁

C₂

yes

maximal

#398×C₂₁₁

C₂

yes

maximal

#395×C₂₁₁

C₂

yes

maximal

#384×C₂₁₁

C₂

yes

maximal, characteristic

#383×C₂₁₁

C₂

yes

maximal

C₂²×Q₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#95×C₂₁₁

C₂²

yes

—

#95×C₂₁₁

C₂²

yes

—

#95×C₂₁₁

C₂²

yes

—

#95×C₂₁₁

C₂²

yes

—

C₂×Q₈×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

Q₈

yes

complemented

C₂³×C₂₁₁

C₂⁴

yes

characteristic

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂²×C₂₁₁

C₂²×Q₈

yes

characteristic

C₂²×C₂₁₁

H(5,𝔽₂)

yes

—

C₂²×C₂₁₁

2₋¹⁺⁴

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₂×C₂₁₁

#544

yes

—

C₂×C₂₁₁

#560

yes

characteristic

C₂×C₂₁₁

#563

yes

—

C₂₁₁

#2843

yes

characteristic, direct product factor, Sylow

#2843

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

#400

C₂×C₂₁₁

yes

—

#398

C₂×C₂₁₁

yes

—

#395

C₂×C₂₁₁

yes

—

#384

C₂×C₂₁₁

yes

characteristic

#383

C₂×C₂₁₁

yes

—

C₂²×Q₈

C₂²×C₂₁₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

#95

C₂²×C₂₁₁

yes

—

#95

C₂²×C₂₁₁

yes

—

#95

C₂²×C₂₁₁

yes

—

#95

C₂²×C₂₁₁

yes

—

C₂×Q₈

—

C₂²×C₄

C₂³×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

C₂²×C₄

C₂³×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×C₂₁₁

yes

complemented

C₂³

C₂⁴×C₂₁₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Q₈×C₂₁₁

yes

—

C₂²

C₂²×Q₈×C₂₁₁

yes

characteristic

C₂²

H(5,𝔽₂)×C₂₁₁

yes

—

C₂²

2₋¹⁺⁴×C₂₁₁

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

#544×C₂₁₁

yes

—

C₂

#560×C₂₁₁

yes

characteristic

C₂

#563×C₂₁₁

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₁₁

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₂₁₁

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ G

N	H ≅ G/N	type
26. C₄²×C₂₁₁	Q₈	semidirect
49. C₂₁₁	#2843	direct (N × H)
50. #2843	C₂₁₁	direct (N × H)
76. C₄²	Q₈×C₂₁₁	semidirect

H ≅ G/N

type

26. C₄²×C₂₁₁

Q₈

semidirect

49. C₂₁₁

#2843

direct (N × H)

50. #2843

C₂₁₁

direct (N × H)

76. C₄²

Q₈×C₂₁₁

semidirect

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

subgroup	index
#400×C₂₁₁	2
#398×C₂₁₁	2
#395×C₂₁₁	2
#384×C₂₁₁	2
#383×C₂₁₁	2
#2843	211

subgroup

index

#400×C₂₁₁

#398×C₂₁₁

#395×C₂₁₁

#384×C₂₁₁

#383×C₂₁₁

#2843

211

Subgroups — #2843×C₂₁₁ #41010463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2843×C211 #41010463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2843×C₂₁₁ #41010463