Group #20299410 — subgroups

Subgroups — #20299410

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	#16813668	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813655	C₂	1	3	yes	maximal, complemented
4	#16813650	C₂	1	3	yes	maximal, complemented
5	#16813644	C₂	1	3	yes	maximal, complemented
6	C₂×#14268636	C₂	1	3	yes	maximal, complemented
7	#131×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14268648	—	2	1	no	—
9	#14268647	C₂²	1	3	yes	—
10	#14268647	C₂²	1	3	yes	complemented
11	#14268646	—	2	3	no	—
12	C₂×Dic₂₂₆	C₂²	1	3	yes	complemented
13	Q₈×D₁₁₃	—	2	1	no	—
14	C₂×D₄₅₂	C₂²	1	3	yes	—
15	D₄×D₁₁₃	—	2	3	no	—
16	Dic₄₅₂	C₂²	1	3	yes	complemented
17	#14268636	C₂²	1	3	yes	complemented
18	#14268636	C₂²	1	6	yes	complemented
19	#14268636	C₂²	1	1	yes	characteristic, complemented
20	D₉₀₄	C₂²	1	3	yes	complemented
21	#41×C₁₁₃	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₁₃	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₁₃	C₂²	1	3	yes	complemented
24	C₂²×D₁₁₃	—	2	3	no	—
25	Q₈×C₁₁₃	D₄	1	1	yes	characteristic
26	D₄×C₁₁₃	D₄	1	3	yes	—
27	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
28	C₂²⋊D₁₁₃	—	2	3	no	—
29	C₂²⋊D₁₁₃	—	2	3	no	—
30	C₂×Dic₁₁₃	—	2	3	no	—
31	D₄₅₂	—	2	3	no	—
32	D₄₅₂	C₂³	1	1	yes	characteristic
33	D₄₅₂	C₂³	1	3	yes	—
34	C₄×D₁₁₃	—	2	3	no	—
35	C₄×D₁₁₃	—	2	3	no	—
36	Dic₂₂₆	—	2	3	no	—
37	Dic₂₂₆	C₂³	1	3	yes	—
38	Dic₂₂₆	C₂³	1	1	yes	characteristic
39	C₈×C₁₁₃	C₂³	1	1	yes	characteristic
40	C₈×C₁₁₃	C₂³	1	3	yes	—
41	C₂²×C₁₁₃	C₂×D₄	1	3	yes	—
42	C₂×D₁₁₃	—	2	1	no	—
43	C₂×D₁₁₃	—	2	3	no	—
44	C₂×D₁₁₃	—	8	3	no	—
45	C₄×C₁₁₃	C₂×D₄	1	3	yes	—
46	C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
47	Dic₁₁₃	—	2	3	no	—
48	Dic₁₁₃	—	2	1	no	—
49	C₂×C₁₁₃	—	2	3	no	—
50	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
51	D₁₁₃	—	4	1	no	—
52	D₁₁₃	—	4	3	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)	—	113	1	no	—
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	D₁₁₃	1	1	yes	characteristic, complemented
62	#41	—	113	3	no	—
63	#41	—	113	3	no	—
64	#41	—	226	1	no	—
65	#41	—	226	3	no	—
66	#41	C₂×D₁₁₃	1	1	yes	characteristic
67	C₂×Q₈	—	113	3	no	—
68	C₂×Q₈	—	226	1	no	—
69	C₂×D₄	—	113	3	no	—
70	C₂×D₄	—	226	3	no	—
71	Q₁₆	—	113	3	no	—
72	SD₁₆	—	113	3	no	—
73	SD₁₆	—	113	6	no	—
74	SD₁₆	—	113	1	no	—
75	D₈	—	113	3	no	—
76	M₁₆	C₂×D₁₁₃	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₁₃	1	3	yes	complemented
78	C₂³	—	226	3	no	—
79	Q₈	—	113	3	no	—
80	Q₈	—	113	1	no	—
81	Q₈	—	226	3	no	—
82	Q₈	D₄₅₂	1	1	yes	characteristic
83	D₄	—	113	1	no	—
84	D₄	—	113	3	no	—
85	D₄	—	226	3	no	—
86	D₄	—	226	3	no	—
87	D₄	—	226	3	no	—
88	D₄	D₄₅₂	1	3	yes	—
89	C₂×C₄	—	226	3	no	—
90	C₂×C₄	—	226	3	no	—
91	C₂×C₄	—	226	3	no	—
92	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
93	C₈	C₂²×D₁₁₃	1	1	yes	characteristic
94	C₈	C₂²×D₁₁₃	1	3	yes	—
95	C₂²	—	904	3	no	—
96	C₂²	—	226	1	no	—
97	C₂²	—	226	3	no	—
98	C₂²	C₂×D₄₅₂	1	3	yes	—
99	C₄	—	226	1	no	—
100	C₄	—	226	3	no	—
101	C₄	C₂×D₄₅₂	1	3	yes	—
102	C₄	C₂³×D₁₁₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	452	3	no	—
105	C₂	—	452	1	no	—
106	C₂	C₂²×D₄₅₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813669

C₂

yes

maximal, characteristic, complemented

#16813668

C₂

yes

maximal, characteristic, complemented

#16813655

C₂

yes

maximal, complemented

#16813650

C₂

yes

maximal, complemented

#16813644

C₂

yes

maximal, complemented

C₂×#14268636

C₂

yes

maximal, complemented

#131×C₁₁₃

C₂

yes

maximal, characteristic, complemented

#14268648

—

#14268647

C₂²

yes

—

#14268647

C₂²

yes

complemented

#14268646

—

C₂×Dic₂₂₆

C₂²

yes

complemented

Q₈×D₁₁₃

—

C₂×D₄₅₂

C₂²

yes

—

D₄×D₁₁₃

—

Dic₄₅₂

C₂²

yes

complemented

#14268636

C₂²

yes

complemented

#14268636

C₂²

yes

complemented

#14268636

C₂²

yes

characteristic, complemented

D₉₀₄

C₂²

yes

complemented

#41×C₁₁₃

C₂²

yes

characteristic

M₁₆×C₁₁₃

C₂²

yes

complemented

C₂×C₈×C₁₁₃

C₂²

yes

complemented

C₂²×D₁₁₃

—

Q₈×C₁₁₃

D₄

yes

characteristic

D₄×C₁₁₃

D₄

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂²⋊D₁₁₃

—

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

—

D₄₅₂

—

D₄₅₂

C₂³

yes

characteristic

D₄₅₂

C₂³

yes

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

Dic₂₂₆

—

Dic₂₂₆

C₂³

yes

—

Dic₂₂₆

C₂³

yes

characteristic

C₈×C₁₁₃

C₂³

yes

characteristic

C₈×C₁₁₃

C₂³

yes

—

C₂²×C₁₁₃

C₂×D₄

yes

—

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

C₂×D₄

yes

—

C₄×C₁₁₃

C₂⁴

yes

characteristic

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×D₄

yes

characteristic

D₁₁₃

—

D₁₁₃

—

C₁₁₃

#577

yes

characteristic, complemented, Sylow

#577

—

113

maximal, Sylow

2₋¹⁺⁴

—

113

—

H(5,𝔽₂)

—

113

—

#137

—

113

—

AGL(1,ℤ/8)

—

113

—

#135

—

113

—

C₂×SD₁₆

—

113

—

#131

D₁₁₃

yes

characteristic, complemented

#41

—

113

—

#41

—

113

—

#41

—

226

—

#41

—

226

—

#41

C₂×D₁₁₃

yes

characteristic

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

C₂×D₄

—

113

—

C₂×D₄

—

226

—

Q₁₆

—

113

—

SD₁₆

—

113

—

SD₁₆

—

113

—

SD₁₆

—

113

—

D₈

—

113

—

M₁₆

C₂×D₁₁₃

yes

complemented

C₂×C₈

C₂×D₁₁₃

yes

complemented

C₂³

—

226

—

Q₈

—

113

—

Q₈

—

113

—

Q₈

—

226

—

Q₈

D₄₅₂

yes

characteristic

D₄

—

113

—

D₄

—

113

—

D₄

—

226

—

D₄

—

226

—

D₄

—

226

—

D₄

D₄₅₂

yes

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

C₂²×D₁₁₃

yes

—

C₈

C₂²×D₁₁₃

yes

characteristic

C₈

C₂²×D₁₁₃

yes

—

C₂²

—

904

—

C₂²

—

226

—

C₂²

—

226

—

C₂²

C₂×D₄₅₂

yes

—

C₄

—

226

—

100

C₄

—

226

—

101

C₄

C₂×D₄₅₂

yes

—

102

C₄

C₂³×D₁₁₃

yes

characteristic

103

C₂

—

104

C₂

—

452

—

105

C₂

—

452

—

106

C₂

C₂²×D₄₅₂

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₅₂
commutator	C₄×C₁₁₃	C₂⁴
Frattini	C₄	C₂³×D₁₁₃
Fitting	#131×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₂²×D₄
hypercenter	#131	D₁₁₃
hypocenter	C₁₁₃	#577
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₄₅₂

commutator

C₄×C₁₁₃

C₂⁴

Frattini

C₄

C₂³×D₁₁₃

Fitting

#131×C₁₁₃

C₂

socle

C₂×C₁₁₃

C₂²×D₄

D₁₁₃

C₁₁₃

G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #16813669	C₂	semidirect
2. #16813668	C₂	semidirect
3. #16813655	C₂	semidirect
4. #16813650	C₂	semidirect
5. #16813644	C₂	semidirect
6. C₂×#14268636	C₂	semidirect
7. #131×C₁₁₃	C₂	semidirect
10. #14268647	C₂²	semidirect
12. C₂×Dic₂₂₆	C₂²	semidirect
16. Dic₄₅₂	C₂²	semidirect
17. #14268636	C₂²	semidirect
18. #14268636	C₂²	semidirect
19. #14268636	C₂²	semidirect
20. D₉₀₄	C₂²	semidirect
22. M₁₆×C₁₁₃	C₂²	semidirect
23. C₂×C₈×C₁₁₃	C₂²	semidirect
53. C₁₁₃	#577	semidirect
61. #131	D₁₁₃	semidirect
76. M₁₆	C₂×D₁₁₃	semidirect
77. C₂×C₈	C₂×D₁₁₃	semidirect

H ≅ G/N

type

1. #16813669

C₂

semidirect

2. #16813668

C₂

semidirect

3. #16813655

C₂

semidirect

4. #16813650

C₂

semidirect

5. #16813644

C₂

semidirect

6. C₂×#14268636

C₂

semidirect

7. #131×C₁₁₃

C₂

semidirect

10. #14268647

C₂²

semidirect

12. C₂×Dic₂₂₆

C₂²

semidirect

16. Dic₄₅₂

C₂²

semidirect

17. #14268636

C₂²

semidirect

18. #14268636

C₂²

semidirect

19. #14268636

C₂²

semidirect

20. D₉₀₄

C₂²

semidirect

22. M₁₆×C₁₁₃

C₂²

semidirect

23. C₂×C₈×C₁₁₃

C₂²

semidirect

53. C₁₁₃

#577

semidirect

61. #131

D₁₁₃

semidirect

76. M₁₆

C₂×D₁₁₃

semidirect

77. C₂×C₈

C₂×D₁₁₃

semidirect

Subgroups — #20299410

Conjugacy classes of subgroups

Complements