Group #27213957 — subgroups

Subgroups — #27213957

Order 13504.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20270024	C₂	1	1	yes	maximal, characteristic, complemented
2	#20270007	C₂	1	3	yes	maximal, complemented
3	#20270003	C₂	1	3	yes	maximal, complemented
4	#20269997	C₂	1	3	yes	maximal, complemented
5	C₂×#16809950	C₂	1	3	yes	maximal, complemented
6	#20269984	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#16809962	C₂²	1	3	yes	complemented
9	#16809961	—	2	3	no	—
10	C₂×Dic₄₂₂	C₂²	1	3	yes	complemented
11	Q₈×D₂₁₁	—	2	1	no	—
12	#16809952	C₂²	1	3	yes	complemented
13	#16809950	C₂²	1	3	yes	complemented
14	#16809950	C₂²	1	6	yes	complemented
15	#16809949	C₂²	1	1	yes	characteristic, complemented
16	#16809947	C₂²	1	3	yes	complemented
17	#16809946	C₂²	1	3	yes	complemented
18	C₂×C₂₁₁⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₂₁₁	—	2	1	no	—
20	#41×C₂₁₁	C₂²	1	3	yes	—
21	#41×C₂₁₁	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₂₁₁	—	2	3	no	—
23	C₂×D₄×C₂₁₁	C₂²	1	3	yes	—
24	Dic₄₂₂	—	2	3	no	—
25	Dic₄₂₂	C₂³	1	3	yes	—
26	D₈₄₄	C₂³	1	1	yes	characteristic
27	C₂²⋊D₂₁₁	—	2	3	no	—
28	C₂×Dic₂₁₁	—	2	3	no	—
29	C₄×D₂₁₁	—	2	3	no	—
30	C₂₁₁⋊C₈	C₂³	1	1	yes	characteristic
31	C₂₁₁⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₂₁₁	—	2	3	no	—
33	Q₈×C₂₁₁	D₄	1	1	yes	characteristic
34	Q₈×C₂₁₁	C₂³	1	1	yes	characteristic
35	D₄×C₂₁₁	—	2	3	no	—
36	D₄×C₂₁₁	—	2	3	no	—
37	D₄×C₂₁₁	D₄	1	3	yes	—
38	D₄×C₂₁₁	C₂³	1	3	yes	—
39	C₂×C₄×C₂₁₁	—	2	3	no	—
40	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
41	C₂²×C₂₁₁	—	2	3	no	—
42	C₂²×C₂₁₁	—	8	3	no	—
43	C₂²×C₂₁₁	C₂×D₄	1	3	yes	—
44	C₂×D₂₁₁	—	2	1	no	—
45	C₄×C₂₁₁	—	2	1	no	—
46	C₄×C₂₁₁	C₂×D₄	1	3	yes	—
47	C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
48	Dic₂₁₁	—	2	3	no	—
49	C₂×C₂₁₁	—	2	3	no	—
50	C₂×C₂₁₁	—	4	3	no	—
51	C₂×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
52	D₂₁₁	—	4	1	no	—
53	C₂₁₁	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	211	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	211	1	no	—
56	H(5,𝔽₂)	D₂₁₁	1	1	yes	characteristic, complemented
57	#137	—	211	3	no	—
58	AGL(1,ℤ/8)	—	211	3	no	—
59	#135	—	211	3	no	—
60	C₂×SD₁₆	—	211	3	no	—
61	#131	—	211	1	no	—
62	#41	—	2	1	no	—
63	#41	—	422	3	no	—
64	#41	—	211	3	no	—
65	#41	C₂×D₂₁₁	1	3	yes	—
66	#41	C₂×D₂₁₁	1	1	yes	characteristic
67	C₂×Q₈	—	422	1	no	—
68	C₂×Q₈	—	211	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₂₁₁	1	3	yes	—
71	Q₁₆	—	211	3	no	—
72	SD₁₆	—	211	1	no	—
73	SD₁₆	—	211	3	no	—
74	SD₁₆	—	211	6	no	—
75	D₈	—	211	3	no	—
76	M₁₆	—	211	3	no	—
77	C₂×C₈	—	211	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	422	3	no	—
80	Q₈	—	211	3	no	—
81	Q₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
82	Q₈	C₂²×D₂₁₁	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	422	3	no	—
86	D₄	—	211	1	no	—
87	D₄	C₂²⋊D₂₁₁	1	3	yes	—
88	D₄	C₂²×D₂₁₁	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	422	3	no	—
91	C₂×C₄	—	422	3	no	—
92	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
93	C₈	—	211	1	no	—
94	C₈	—	211	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	422	1	no	—
98	C₂²	C₂×C₂²⋊D₂₁₁	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	422	3	no	—
101	C₄	C₂×C₂²⋊D₂₁₁	1	3	yes	—
102	C₄	C₂³×D₂₁₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	844	1	no	—
106	C₂	C₂²×C₂²⋊D₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20270024

C₂

yes

maximal, characteristic, complemented

#20270007

C₂

yes

maximal, complemented

#20270003

C₂

yes

maximal, complemented

#20269997

C₂

yes

maximal, complemented

C₂×#16809950

C₂

yes

maximal, complemented

#20269984

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₂₁₁

C₂

yes

maximal, characteristic, complemented

#16809962

C₂²

yes

complemented

#16809961

—

C₂×Dic₄₂₂

C₂²

yes

complemented

Q₈×D₂₁₁

—

#16809952

C₂²

yes

complemented

#16809950

C₂²

yes

complemented

#16809950

C₂²

yes

complemented

#16809949

C₂²

yes

characteristic, complemented

#16809947

C₂²

yes

complemented

#16809946

C₂²

yes

complemented

C₂×C₂₁₁⋊C₈

C₂²

yes

complemented

#41×C₂₁₁

—

#41×C₂₁₁

C₂²

yes

—

#41×C₂₁₁

C₂²

yes

characteristic

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

C₂²

yes

—

Dic₄₂₂

—

Dic₄₂₂

C₂³

yes

—

D₈₄₄

C₂³

yes

characteristic

C₂²⋊D₂₁₁

—

C₂×Dic₂₁₁

—

C₄×D₂₁₁

—

C₂₁₁⋊C₈

C₂³

yes

characteristic

C₂₁₁⋊C₈

C₂³

yes

—

C₂³×C₂₁₁

—

Q₈×C₂₁₁

D₄

yes

characteristic

Q₈×C₂₁₁

C₂³

yes

characteristic

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

D₄

yes

—

D₄×C₂₁₁

C₂³

yes

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂³

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×D₄

yes

—

C₂×D₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×D₄

yes

—

C₄×C₂₁₁

C₂⁴

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂²×D₄

yes

characteristic

D₂₁₁

—

C₂₁₁

#577

yes

characteristic, complemented, Sylow

#577

—

211

maximal, Sylow

2₋¹⁺⁴

—

211

—

H(5,𝔽₂)

D₂₁₁

yes

characteristic, complemented

#137

—

211

—

AGL(1,ℤ/8)

—

211

—

#135

—

211

—

C₂×SD₁₆

—

211

—

#131

—

211

—

#41

—

#41

—

422

—

#41

—

211

—

#41

C₂×D₂₁₁

yes

—

#41

C₂×D₂₁₁

yes

characteristic

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂×D₄

—

C₂×D₄

C₂×D₂₁₁

yes

—

Q₁₆

—

211

—

SD₁₆

—

211

—

SD₁₆

—

211

—

SD₁₆

—

211

—

D₈

—

211

—

M₁₆

—

211

—

C₂×C₈

—

211

—

C₂³

—

Q₈

—

422

—

Q₈

—

211

—

Q₈

C₂²⋊D₂₁₁

yes

characteristic

Q₈

C₂²×D₂₁₁

yes

characteristic

D₄

—

D₄

—

D₄

—

422

—

D₄

—

211

—

D₄

C₂²⋊D₂₁₁

yes

—

D₄

C₂²×D₂₁₁

yes

—

C₂×C₄

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

C₂²×D₂₁₁

yes

—

C₈

—

211

—

C₈

—

211

—

C₂²

—

C₂²

—

C₂²

—

422

—

C₂²

C₂×C₂²⋊D₂₁₁

yes

—

C₄

—

100

C₄

—

422

—

101

C₄

C₂×C₂²⋊D₂₁₁

yes

—

102

C₄

C₂³×D₂₁₁

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

844

—

106

C₂

C₂²×C₂²⋊D₂₁₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₂₁₁
commutator, cosocle	C₄×C₂₁₁	C₂⁴
Frattini	C₄	C₂³×D₂₁₁
Fitting	H(5,𝔽₂)×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₂₁₁
hypocenter	C₂₁₁	#577
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₂₁₁

commutator, cosocle

C₄×C₂₁₁

C₂⁴

Frattini

C₄

C₂³×D₂₁₁

Fitting

H(5,𝔽₂)×C₂₁₁

C₂

socle

C₂×C₂₁₁

C₂²×D₄

D₂₁₁

C₂₁₁

G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #20270024	C₂	semidirect
2. #20270007	C₂	semidirect
3. #20270003	C₂	semidirect
4. #20269997	C₂	semidirect
5. C₂×#16809950	C₂	semidirect
6. #20269984	C₂	semidirect
7. H(5,𝔽₂)×C₂₁₁	C₂	semidirect
8. #16809962	C₂²	semidirect
10. C₂×Dic₄₂₂	C₂²	semidirect
12. #16809952	C₂²	semidirect
13. #16809950	C₂²	semidirect
14. #16809950	C₂²	semidirect
15. #16809949	C₂²	semidirect
16. #16809947	C₂²	semidirect
17. #16809946	C₂²	semidirect
18. C₂×C₂₁₁⋊C₈	C₂²	semidirect
53. C₂₁₁	#577	semidirect
56. H(5,𝔽₂)	D₂₁₁	semidirect

H ≅ G/N

type

1. #20270024

C₂

semidirect

2. #20270007

C₂

semidirect

3. #20270003

C₂

semidirect

4. #20269997

C₂

semidirect

5. C₂×#16809950

C₂

semidirect

6. #20269984

C₂

semidirect

7. H(5,𝔽₂)×C₂₁₁

C₂

semidirect

8. #16809962

C₂²

semidirect

10. C₂×Dic₄₂₂

C₂²

semidirect

12. #16809952

C₂²

semidirect

13. #16809950

C₂²

semidirect

14. #16809950

C₂²

semidirect

15. #16809949

C₂²

semidirect

16. #16809947

C₂²

semidirect

17. #16809946

C₂²

semidirect

18. C₂×C₂₁₁⋊C₈

C₂²

semidirect

53. C₂₁₁

#577

semidirect

56. H(5,𝔽₂)

D₂₁₁

semidirect

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

subgroup	index
#20270024	2
#20270007	2
#20270003	2
#20269997	2
C₂×#16809950	2
#20269984	2
H(5,𝔽₂)×C₂₁₁	2
#577	211

subgroup

index

H(5,𝔽₂)×C₂₁₁

211

Subgroups — #27213957

Conjugacy classes of subgroups

Complements

Maximal subgroups