Group #38786926 — subgroups

Subgroups — #38786926

Order 25664.

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	#27165784	C₂	1	1	yes	maximal, characteristic, complemented
2	#27165783	C₂	1	1	yes	maximal, characteristic, complemented
3	#27165770	C₂	1	3	yes	maximal, complemented
4	#27165765	C₂	1	3	yes	maximal, complemented
5	#27165759	C₂	1	3	yes	maximal, complemented
6	C₂×#20264355	C₂	1	3	yes	maximal, complemented
7	#131×C₄₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20264367	—	2	1	no	—
9	#20264366	C₂²	1	3	yes	—
10	#20264366	C₂²	1	3	yes	complemented
11	#20264365	—	2	3	no	—
12	C₂×Dic₈₀₂	C₂²	1	3	yes	complemented
13	Q₈×D₄₀₁	—	2	1	no	—
14	C₂×D₁₆₀₄	C₂²	1	3	yes	—
15	D₄×D₄₀₁	—	2	3	no	—
16	Dic₁₆₀₄	C₂²	1	3	yes	complemented
17	#20264355	C₂²	1	3	yes	complemented
18	#20264355	C₂²	1	6	yes	complemented
19	#20264355	C₂²	1	1	yes	characteristic, complemented
20	D₃₂₀₈	C₂²	1	3	yes	complemented
21	#41×C₄₀₁	C₂²	1	1	yes	characteristic
22	M₁₆×C₄₀₁	C₂²	1	3	yes	complemented
23	C₂×C₈×C₄₀₁	C₂²	1	3	yes	complemented
24	C₂²×D₄₀₁	—	2	3	no	—
25	Dic₈₀₂	—	2	3	no	—
26	Dic₈₀₂	C₂³	1	3	yes	—
27	Dic₈₀₂	C₂³	1	1	yes	characteristic
28	D₁₆₀₄	—	2	3	no	—
29	D₁₆₀₄	C₂³	1	1	yes	characteristic
30	D₁₆₀₄	C₂³	1	3	yes	—
31	C₂²⋊D₄₀₁	—	2	3	no	—
32	C₂²⋊D₄₀₁	—	2	3	no	—
33	C₂×Dic₄₀₁	—	2	3	no	—
34	C₄×D₄₀₁	—	2	3	no	—
35	C₄×D₄₀₁	—	2	3	no	—
36	Q₈×C₄₀₁	D₄	1	1	yes	characteristic
37	D₄×C₄₀₁	D₄	1	3	yes	—
38	C₂×C₄×C₄₀₁	C₂³	1	3	yes	—
39	C₈×C₄₀₁	C₂³	1	1	yes	characteristic
40	C₈×C₄₀₁	C₂³	1	3	yes	—
41	C₂²×C₄₀₁	C₂×D₄	1	3	yes	—
42	C₂×D₄₀₁	—	2	1	no	—
43	C₂×D₄₀₁	—	2	3	no	—
44	C₂×D₄₀₁	—	8	3	no	—
45	C₄×C₄₀₁	C₂×D₄	1	3	yes	—
46	C₄×C₄₀₁	C₂⁴	1	1	yes	characteristic
47	Dic₄₀₁	—	2	1	no	—
48	Dic₄₀₁	—	2	3	no	—
49	C₂×C₄₀₁	—	2	3	no	—
50	C₂×C₄₀₁	C₂²×D₄	1	1	yes	characteristic
51	D₄₀₁	—	4	1	no	—
52	D₄₀₁	—	4	3	no	—
53	C₄₀₁	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	401	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	401	1	no	—
56	H(5,𝔽₂)	—	401	1	no	—
57	#137	—	401	3	no	—
58	AGL(1,ℤ/8)	—	401	3	no	—
59	#135	—	401	3	no	—
60	C₂×SD₁₆	—	401	3	no	—
61	#131	D₄₀₁	1	1	yes	characteristic, complemented
62	#41	—	802	1	no	—
63	#41	—	802	3	no	—
64	#41	—	401	3	no	—
65	#41	—	401	3	no	—
66	#41	C₂×D₄₀₁	1	1	yes	characteristic
67	C₂×Q₈	—	802	1	no	—
68	C₂×Q₈	—	401	3	no	—
69	C₂×D₄	—	802	3	no	—
70	C₂×D₄	—	401	3	no	—
71	Q₁₆	—	401	3	no	—
72	SD₁₆	—	401	3	no	—
73	SD₁₆	—	401	6	no	—
74	SD₁₆	—	401	1	no	—
75	D₈	—	401	3	no	—
76	M₁₆	C₂×D₄₀₁	1	3	yes	complemented
77	C₂×C₈	C₂×D₄₀₁	1	3	yes	complemented
78	C₂³	—	802	3	no	—
79	Q₈	—	802	3	no	—
80	Q₈	—	401	3	no	—
81	Q₈	—	401	1	no	—
82	Q₈	D₁₆₀₄	1	1	yes	characteristic
83	D₄	—	802	3	no	—
84	D₄	—	802	3	no	—
85	D₄	—	802	3	no	—
86	D₄	—	401	1	no	—
87	D₄	—	401	3	no	—
88	D₄	D₁₆₀₄	1	3	yes	—
89	C₂×C₄	—	802	3	no	—
90	C₂×C₄	—	802	3	no	—
91	C₂×C₄	—	802	3	no	—
92	C₂×C₄	C₂²×D₄₀₁	1	3	yes	—
93	C₈	C₂²×D₄₀₁	1	1	yes	characteristic
94	C₈	C₂²×D₄₀₁	1	3	yes	—
95	C₂²	—	802	1	no	—
96	C₂²	—	802	3	no	—
97	C₂²	—	3208	3	no	—
98	C₂²	C₂×D₁₆₀₄	1	3	yes	—
99	C₄	—	802	1	no	—
100	C₄	—	802	3	no	—
101	C₄	C₂×D₁₆₀₄	1	3	yes	—
102	C₄	C₂³×D₄₀₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	1604	3	no	—
105	C₂	—	1604	1	no	—
106	C₂	C₂²×D₁₆₀₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27165784

C₂

yes

maximal, characteristic, complemented

#27165783

C₂

yes

maximal, characteristic, complemented

#27165770

C₂

yes

maximal, complemented

#27165765

C₂

yes

maximal, complemented

#27165759

C₂

yes

maximal, complemented

C₂×#20264355

C₂

yes

maximal, complemented

#131×C₄₀₁

C₂

yes

maximal, characteristic, complemented

#20264367

—

#20264366

C₂²

yes

—

#20264366

C₂²

yes

complemented

#20264365

—

C₂×Dic₈₀₂

C₂²

yes

complemented

Q₈×D₄₀₁

—

C₂×D₁₆₀₄

C₂²

yes

—

D₄×D₄₀₁

—

Dic₁₆₀₄

C₂²

yes

complemented

#20264355

C₂²

yes

complemented

#20264355

C₂²

yes

complemented

#20264355

C₂²

yes

characteristic, complemented

D₃₂₀₈

C₂²

yes

complemented

#41×C₄₀₁

C₂²

yes

characteristic

M₁₆×C₄₀₁

C₂²

yes

complemented

C₂×C₈×C₄₀₁

C₂²

yes

complemented

C₂²×D₄₀₁

—

Dic₈₀₂

—

Dic₈₀₂

C₂³

yes

—

Dic₈₀₂

C₂³

yes

characteristic

D₁₆₀₄

—

D₁₆₀₄

C₂³

yes

characteristic

D₁₆₀₄

C₂³

yes

—

C₂²⋊D₄₀₁

—

C₂²⋊D₄₀₁

—

C₂×Dic₄₀₁

—

C₄×D₄₀₁

—

C₄×D₄₀₁

—

Q₈×C₄₀₁

D₄

yes

characteristic

D₄×C₄₀₁

D₄

yes

—

C₂×C₄×C₄₀₁

C₂³

yes

—

C₈×C₄₀₁

C₂³

yes

characteristic

C₈×C₄₀₁

C₂³

yes

—

C₂²×C₄₀₁

C₂×D₄

yes

—

C₂×D₄₀₁

—

C₂×D₄₀₁

—

C₂×D₄₀₁

—

C₄×C₄₀₁

C₂×D₄

yes

—

C₄×C₄₀₁

C₂⁴

yes

characteristic

Dic₄₀₁

—

Dic₄₀₁

—

C₂×C₄₀₁

—

C₂×C₄₀₁

C₂²×D₄

yes

characteristic

D₄₀₁

—

D₄₀₁

—

C₄₀₁

#577

yes

characteristic, complemented, Sylow

#577

—

401

maximal, Sylow

2₋¹⁺⁴

—

401

—

H(5,𝔽₂)

—

401

—

#137

—

401

—

AGL(1,ℤ/8)

—

401

—

#135

—

401

—

C₂×SD₁₆

—

401

—

#131

D₄₀₁

yes

characteristic, complemented

#41

—

802

—

#41

—

802

—

#41

—

401

—

#41

—

401

—

#41

C₂×D₄₀₁

yes

characteristic

C₂×Q₈

—

802

—

C₂×Q₈

—

401

—

C₂×D₄

—

802

—

C₂×D₄

—

401

—

Q₁₆

—

401

—

SD₁₆

—

401

—

SD₁₆

—

401

—

SD₁₆

—

401

—

D₈

—

401

—

M₁₆

C₂×D₄₀₁

yes

complemented

C₂×C₈

C₂×D₄₀₁

yes

complemented

C₂³

—

802

—

Q₈

—

802

—

Q₈

—

401

—

Q₈

—

401

—

Q₈

D₁₆₀₄

yes

characteristic

D₄

—

802

—

D₄

—

802

—

D₄

—

802

—

D₄

—

401

—

D₄

—

401

—

D₄

D₁₆₀₄

yes

—

C₂×C₄

—

802

—

C₂×C₄

—

802

—

C₂×C₄

—

802

—

C₂×C₄

C₂²×D₄₀₁

yes

—

C₈

C₂²×D₄₀₁

yes

characteristic

C₈

C₂²×D₄₀₁

yes

—

C₂²

—

802

—

C₂²

—

802

—

C₂²

—

3208

—

C₂²

C₂×D₁₆₀₄

yes

—

C₄

—

802

—

100

C₄

—

802

—

101

C₄

C₂×D₁₆₀₄

yes

—

102

C₄

C₂³×D₄₀₁

yes

characteristic

103

C₂

—

104

C₂

—

1604

—

105

C₂

—

1604

—

106

C₂

C₂²×D₁₆₀₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₆₀₄
commutator, cosocle	C₄×C₄₀₁	C₂⁴
Frattini	C₄	C₂³×D₄₀₁
Fitting	#131×C₄₀₁	C₂
socle	C₂×C₄₀₁	C₂²×D₄
hypercenter	#131	D₄₀₁
hypocenter	C₄₀₁	#577
LCS	G ▷ C₄×C₄₀₁ ▷ C₂×C₄₀₁ ▷ C₄₀₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₁₆₀₄

commutator, cosocle

C₄×C₄₀₁

C₂⁴

Frattini

C₄

C₂³×D₄₀₁

Fitting

#131×C₄₀₁

C₂

socle

C₂×C₄₀₁

C₂²×D₄

D₄₀₁

C₄₀₁

G ▷ C₄×C₄₀₁ ▷ C₂×C₄₀₁ ▷ C₄₀₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #27165784	C₂	semidirect
2. #27165783	C₂	semidirect
3. #27165770	C₂	semidirect
4. #27165765	C₂	semidirect
5. #27165759	C₂	semidirect
6. C₂×#20264355	C₂	semidirect
7. #131×C₄₀₁	C₂	semidirect
10. #20264366	C₂²	semidirect
12. C₂×Dic₈₀₂	C₂²	semidirect
16. Dic₁₆₀₄	C₂²	semidirect
17. #20264355	C₂²	semidirect
18. #20264355	C₂²	semidirect
19. #20264355	C₂²	semidirect
20. D₃₂₀₈	C₂²	semidirect
22. M₁₆×C₄₀₁	C₂²	semidirect
23. C₂×C₈×C₄₀₁	C₂²	semidirect
53. C₄₀₁	#577	semidirect
61. #131	D₄₀₁	semidirect
76. M₁₆	C₂×D₄₀₁	semidirect
77. C₂×C₈	C₂×D₄₀₁	semidirect

H ≅ G/N

type

1. #27165784

C₂

semidirect

2. #27165783

C₂

semidirect

3. #27165770

C₂

semidirect

4. #27165765

C₂

semidirect

5. #27165759

C₂

semidirect

6. C₂×#20264355

C₂

semidirect

7. #131×C₄₀₁

C₂

semidirect

10. #20264366

C₂²

semidirect

12. C₂×Dic₈₀₂

C₂²

semidirect

16. Dic₁₆₀₄

C₂²

semidirect

17. #20264355

C₂²

semidirect

18. #20264355

C₂²

semidirect

19. #20264355

C₂²

semidirect

20. D₃₂₀₈

C₂²

semidirect

22. M₁₆×C₄₀₁

C₂²

semidirect

23. C₂×C₈×C₄₀₁

C₂²

semidirect

53. C₄₀₁

#577

semidirect

61. #131

D₄₀₁

semidirect

76. M₁₆

C₂×D₄₀₁

semidirect

77. C₂×C₈

C₂×D₄₀₁

semidirect

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

subgroup	index
#27165784	2
#27165783	2
#27165770	2
#27165765	2
#27165759	2
C₂×#20264355	2
#131×C₄₀₁	2
#577	401

subgroup

index

#131×C₄₀₁

401

Subgroups — #38786926

Conjugacy classes of subgroups

Complements

Maximal subgroups