Group #20299409 — subgroups

Subgroups — #20299409

Order 7232.

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	#16813669	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813652	C₂	1	3	yes	maximal, complemented
3	#16813648	C₂	1	3	yes	maximal, complemented
4	#16813642	C₂	1	3	yes	maximal, complemented
5	C₂×#14268635	C₂	1	3	yes	maximal, complemented
6	#16813629	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14268647	C₂²	1	3	yes	complemented
9	#14268646	—	2	3	no	—
10	C₂×Dic₂₂₆	C₂²	1	3	yes	complemented
11	Q₈×D₁₁₃	—	2	1	no	—
12	#14268637	C₂²	1	3	yes	complemented
13	#14268635	C₂²	1	3	yes	complemented
14	#14268635	C₂²	1	6	yes	complemented
15	#14268634	C₂²	1	1	yes	characteristic, complemented
16	#14268632	C₂²	1	3	yes	complemented
17	#14268631	C₂²	1	3	yes	complemented
18	C₂×#11744751	C₂²	1	3	yes	complemented
19	#41×C₁₁₃	—	2	1	no	—
20	#41×C₁₁₃	C₂²	1	3	yes	—
21	#41×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₁₃	—	2	3	no	—
23	C₂×D₄×C₁₁₃	C₂²	1	3	yes	—
24	C₂³×C₁₁₃	—	2	3	no	—
25	Q₈×C₁₁₃	D₄	1	1	yes	characteristic
26	Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
27	D₄×C₁₁₃	—	2	3	no	—
28	D₄×C₁₁₃	—	2	3	no	—
29	D₄×C₁₁₃	D₄	1	3	yes	—
30	D₄×C₁₁₃	C₂³	1	3	yes	—
31	C₂×C₄×C₁₁₃	—	2	3	no	—
32	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
33	C₂²⋊D₁₁₃	—	2	3	no	—
34	C₂×Dic₁₁₃	—	2	3	no	—
35	D₄₅₂	C₂³	1	1	yes	characteristic
36	C₄×D₁₁₃	—	2	3	no	—
37	Dic₂₂₆	—	2	3	no	—
38	Dic₂₂₆	C₂³	1	3	yes	—
39	#11744751	C₂³	1	1	yes	characteristic
40	#11744751	C₂³	1	3	yes	—
41	C₂²×C₁₁₃	—	2	3	no	—
42	C₂²×C₁₁₃	—	8	3	no	—
43	C₂²×C₁₁₃	C₂×D₄	1	3	yes	—
44	C₂×D₁₁₃	—	2	1	no	—
45	C₄×C₁₁₃	—	2	1	no	—
46	C₄×C₁₁₃	C₂×D₄	1	3	yes	—
47	C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
48	Dic₁₁₃	—	2	3	no	—
49	C₂×C₁₁₃	—	2	3	no	—
50	C₂×C₁₁₃	—	4	3	no	—
51	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
52	D₁₁₃	—	4	1	no	—
53	C₁₁₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	113	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	113	1	no	—
56	H(5,𝔽₂)	D₁₁₃	1	1	yes	characteristic, complemented
57	#137	—	113	3	no	—
58	AGL(1,ℤ/8)	—	113	3	no	—
59	#135	—	113	3	no	—
60	C₂×SD₁₆	—	113	3	no	—
61	#131	—	113	1	no	—
62	#41	—	2	1	no	—
63	#41	—	113	3	no	—
64	#41	—	226	3	no	—
65	#41	C₂×D₁₁₃	1	3	yes	—
66	#41	C₂×D₁₁₃	1	1	yes	characteristic
67	C₂×Q₈	—	113	3	no	—
68	C₂×Q₈	—	226	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₁₃	1	3	yes	—
71	Q₁₆	—	113	3	no	—
72	SD₁₆	—	113	1	no	—
73	SD₁₆	—	113	3	no	—
74	SD₁₆	—	113	6	no	—
75	D₈	—	113	3	no	—
76	M₁₆	—	113	3	no	—
77	C₂×C₈	—	113	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	113	3	no	—
80	Q₈	—	226	3	no	—
81	Q₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
82	Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	113	1	no	—
86	D₄	—	226	3	no	—
87	D₄	C₂²⋊D₁₁₃	1	3	yes	—
88	D₄	C₂²×D₁₁₃	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	226	3	no	—
91	C₂×C₄	—	226	3	no	—
92	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
93	C₈	—	113	1	no	—
94	C₈	—	113	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	226	1	no	—
98	C₂²	C₂×C₂²⋊D₁₁₃	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	226	3	no	—
101	C₄	C₂×C₂²⋊D₁₁₃	1	3	yes	—
102	C₄	C₂³×D₁₁₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	452	1	no	—
106	C₂	C₂²×C₂²⋊D₁₁₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813669

C₂

yes

maximal, characteristic, complemented

#16813652

C₂

yes

maximal, complemented

#16813648

C₂

yes

maximal, complemented

#16813642

C₂

yes

maximal, complemented

C₂×#14268635

C₂

yes

maximal, complemented

#16813629

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₁₃

C₂

yes

maximal, characteristic, complemented

#14268647

C₂²

yes

complemented

#14268646

—

C₂×Dic₂₂₆

C₂²

yes

complemented

Q₈×D₁₁₃

—

#14268637

C₂²

yes

complemented

#14268635

C₂²

yes

complemented

#14268635

C₂²

yes

complemented

#14268634

C₂²

yes

characteristic, complemented

#14268632

C₂²

yes

complemented

#14268631

C₂²

yes

complemented

C₂×#11744751

C₂²

yes

complemented

#41×C₁₁₃

—

#41×C₁₁₃

C₂²

yes

—

#41×C₁₁₃

C₂²

yes

characteristic

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

C₂²

yes

—

C₂³×C₁₁₃

—

Q₈×C₁₁₃

D₄

yes

characteristic

Q₈×C₁₁₃

C₂³

yes

characteristic

D₄×C₁₁₃

—

D₄×C₁₁₃

—

D₄×C₁₁₃

D₄

yes

—

D₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

—

D₄₅₂

C₂³

yes

characteristic

C₄×D₁₁₃

—

Dic₂₂₆

—

Dic₂₂₆

C₂³

yes

—

#11744751

C₂³

yes

characteristic

#11744751

C₂³

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

—

C₂×D₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×D₄

yes

—

C₄×C₁₁₃

C₂⁴

yes

characteristic

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×D₄

yes

characteristic

D₁₁₃

—

C₁₁₃

#577

yes

characteristic, complemented, Sylow

#577

—

113

maximal, Sylow

2₋¹⁺⁴

—

113

—

H(5,𝔽₂)

D₁₁₃

yes

characteristic, complemented

#137

—

113

—

AGL(1,ℤ/8)

—

113

—

#135

—

113

—

C₂×SD₁₆

—

113

—

#131

—

113

—

#41

—

#41

—

113

—

#41

—

226

—

#41

C₂×D₁₁₃

yes

—

#41

C₂×D₁₁₃

yes

characteristic

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

C₂×D₄

—

C₂×D₄

C₂×D₁₁₃

yes

—

Q₁₆

—

113

—

SD₁₆

—

113

—

SD₁₆

—

113

—

SD₁₆

—

113

—

D₈

—

113

—

M₁₆

—

113

—

C₂×C₈

—

113

—

C₂³

—

Q₈

—

113

—

Q₈

—

226

—

Q₈

C₂²⋊D₁₁₃

yes

characteristic

Q₈

C₂²×D₁₁₃

yes

characteristic

D₄

—

D₄

—

D₄

—

113

—

D₄

—

226

—

D₄

C₂²⋊D₁₁₃

yes

—

D₄

C₂²×D₁₁₃

yes

—

C₂×C₄

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

C₂²×D₁₁₃

yes

—

C₈

—

113

—

C₈

—

113

—

C₂²

—

C₂²

—

C₂²

—

226

—

C₂²

C₂×C₂²⋊D₁₁₃

yes

—

C₄

—

100

C₄

—

226

—

101

C₄

C₂×C₂²⋊D₁₁₃

yes

—

102

C₄

C₂³×D₁₁₃

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

452

—

106

C₂

C₂²×C₂²⋊D₁₁₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₁₃
commutator	C₄×C₁₁₃	C₂⁴
Frattini	C₄	C₂³×D₁₁₃
Fitting	H(5,𝔽₂)×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₁₃
hypocenter	C₁₁₃	#577
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₁₃

commutator

C₄×C₁₁₃

C₂⁴

Frattini

C₄

C₂³×D₁₁₃

Fitting

H(5,𝔽₂)×C₁₁₃

C₂

socle

C₂×C₁₁₃

C₂²×D₄

D₁₁₃

C₁₁₃

G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #16813669	C₂	semidirect
2. #16813652	C₂	semidirect
3. #16813648	C₂	semidirect
4. #16813642	C₂	semidirect
5. C₂×#14268635	C₂	semidirect
6. #16813629	C₂	semidirect
7. H(5,𝔽₂)×C₁₁₃	C₂	semidirect
8. #14268647	C₂²	semidirect
10. C₂×Dic₂₂₆	C₂²	semidirect
12. #14268637	C₂²	semidirect
13. #14268635	C₂²	semidirect
14. #14268635	C₂²	semidirect
15. #14268634	C₂²	semidirect
16. #14268632	C₂²	semidirect
17. #14268631	C₂²	semidirect
18. C₂×#11744751	C₂²	semidirect
53. C₁₁₃	#577	semidirect
56. H(5,𝔽₂)	D₁₁₃	semidirect

H ≅ G/N

type

1. #16813669

C₂

semidirect

2. #16813652

C₂

semidirect

3. #16813648

C₂

semidirect

4. #16813642

C₂

semidirect

5. C₂×#14268635

C₂

semidirect

6. #16813629

C₂

semidirect

7. H(5,𝔽₂)×C₁₁₃

C₂

semidirect

8. #14268647

C₂²

semidirect

10. C₂×Dic₂₂₆

C₂²

semidirect

12. #14268637

C₂²

semidirect

13. #14268635

C₂²

semidirect

14. #14268635

C₂²

semidirect

15. #14268634

C₂²

semidirect

16. #14268632

C₂²

semidirect

17. #14268631

C₂²

semidirect

18. C₂×#11744751

C₂²

semidirect

53. C₁₁₃

#577

semidirect

56. H(5,𝔽₂)

D₁₁₃

semidirect

Subgroups — #20299409

Conjugacy classes of subgroups

Complements