Group #38786925 — subgroups

Subgroups — #38786925

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	#27165767	C₂	1	3	yes	maximal, complemented
3	#27165763	C₂	1	3	yes	maximal, complemented
4	#27165757	C₂	1	3	yes	maximal, complemented
5	C₂×#20264354	C₂	1	3	yes	maximal, complemented
6	#27165744	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₄₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20264366	C₂²	1	3	yes	complemented
9	#20264365	—	2	3	no	—
10	C₂×Dic₈₀₂	C₂²	1	3	yes	complemented
11	Q₈×D₄₀₁	—	2	1	no	—
12	#20264356	C₂²	1	3	yes	complemented
13	#20264354	C₂²	1	3	yes	complemented
14	#20264354	C₂²	1	6	yes	complemented
15	#20264353	C₂²	1	1	yes	characteristic, complemented
16	#20264351	C₂²	1	3	yes	complemented
17	#20264350	C₂²	1	3	yes	complemented
18	C₂×#15691999	C₂²	1	3	yes	complemented
19	#41×C₄₀₁	—	2	1	no	—
20	#41×C₄₀₁	C₂²	1	3	yes	—
21	#41×C₄₀₁	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₄₀₁	—	2	3	no	—
23	C₂×D₄×C₄₀₁	C₂²	1	3	yes	—
24	Dic₈₀₂	—	2	3	no	—
25	Dic₈₀₂	C₂³	1	3	yes	—
26	D₁₆₀₄	C₂³	1	1	yes	characteristic
27	C₂²⋊D₄₀₁	—	2	3	no	—
28	C₂×Dic₄₀₁	—	2	3	no	—
29	C₄×D₄₀₁	—	2	3	no	—
30	#15691999	C₂³	1	1	yes	characteristic
31	#15691999	C₂³	1	3	yes	—
32	C₂³×C₄₀₁	—	2	3	no	—
33	Q₈×C₄₀₁	D₄	1	1	yes	characteristic
34	Q₈×C₄₀₁	C₂³	1	1	yes	characteristic
35	D₄×C₄₀₁	—	2	3	no	—
36	D₄×C₄₀₁	—	2	3	no	—
37	D₄×C₄₀₁	D₄	1	3	yes	—
38	D₄×C₄₀₁	C₂³	1	3	yes	—
39	C₂×C₄×C₄₀₁	—	2	3	no	—
40	C₂×C₄×C₄₀₁	C₂³	1	3	yes	—
41	C₂²×C₄₀₁	—	2	3	no	—
42	C₂²×C₄₀₁	—	8	3	no	—
43	C₂²×C₄₀₁	C₂×D₄	1	3	yes	—
44	C₂×D₄₀₁	—	2	1	no	—
45	C₄×C₄₀₁	—	2	1	no	—
46	C₄×C₄₀₁	C₂×D₄	1	3	yes	—
47	C₄×C₄₀₁	C₂⁴	1	1	yes	characteristic
48	Dic₄₀₁	—	2	3	no	—
49	C₂×C₄₀₁	—	2	3	no	—
50	C₂×C₄₀₁	—	4	3	no	—
51	C₂×C₄₀₁	C₂²×D₄	1	1	yes	characteristic
52	D₄₀₁	—	4	1	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)	D₄₀₁	1	1	yes	characteristic, complemented
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	—	401	1	no	—
62	#41	—	2	1	no	—
63	#41	—	802	3	no	—
64	#41	—	401	3	no	—
65	#41	C₂×D₄₀₁	1	3	yes	—
66	#41	C₂×D₄₀₁	1	1	yes	characteristic
67	C₂×Q₈	—	802	1	no	—
68	C₂×Q₈	—	401	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₄₀₁	1	3	yes	—
71	Q₁₆	—	401	3	no	—
72	SD₁₆	—	401	1	no	—
73	SD₁₆	—	401	3	no	—
74	SD₁₆	—	401	6	no	—
75	D₈	—	401	3	no	—
76	M₁₆	—	401	3	no	—
77	C₂×C₈	—	401	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	802	3	no	—
80	Q₈	—	401	3	no	—
81	Q₈	C₂²⋊D₄₀₁	1	1	yes	characteristic
82	Q₈	C₂²×D₄₀₁	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	802	3	no	—
86	D₄	—	401	1	no	—
87	D₄	C₂²⋊D₄₀₁	1	3	yes	—
88	D₄	C₂²×D₄₀₁	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	802	3	no	—
91	C₂×C₄	—	802	3	no	—
92	C₂×C₄	C₂²×D₄₀₁	1	3	yes	—
93	C₈	—	401	1	no	—
94	C₈	—	401	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	802	1	no	—
98	C₂²	C₂×C₂²⋊D₄₀₁	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	802	3	no	—
101	C₄	C₂×C₂²⋊D₄₀₁	1	3	yes	—
102	C₄	C₂³×D₄₀₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	1604	1	no	—
106	C₂	C₂²×C₂²⋊D₄₀₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27165784

C₂

yes

maximal, characteristic, complemented

#27165767

C₂

yes

maximal, complemented

#27165763

C₂

yes

maximal, complemented

#27165757

C₂

yes

maximal, complemented

C₂×#20264354

C₂

yes

maximal, complemented

#27165744

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₄₀₁

C₂

yes

maximal, characteristic, complemented

#20264366

C₂²

yes

complemented

#20264365

—

C₂×Dic₈₀₂

C₂²

yes

complemented

Q₈×D₄₀₁

—

#20264356

C₂²

yes

complemented

#20264354

C₂²

yes

complemented

#20264354

C₂²

yes

complemented

#20264353

C₂²

yes

characteristic, complemented

#20264351

C₂²

yes

complemented

#20264350

C₂²

yes

complemented

C₂×#15691999

C₂²

yes

complemented

#41×C₄₀₁

—

#41×C₄₀₁

C₂²

yes

—

#41×C₄₀₁

C₂²

yes

characteristic

C₂×D₄×C₄₀₁

—

C₂×D₄×C₄₀₁

C₂²

yes

—

Dic₈₀₂

—

Dic₈₀₂

C₂³

yes

—

D₁₆₀₄

C₂³

yes

characteristic

C₂²⋊D₄₀₁

—

C₂×Dic₄₀₁

—

C₄×D₄₀₁

—

#15691999

C₂³

yes

characteristic

#15691999

C₂³

yes

—

C₂³×C₄₀₁

—

Q₈×C₄₀₁

D₄

yes

characteristic

Q₈×C₄₀₁

C₂³

yes

characteristic

D₄×C₄₀₁

—

D₄×C₄₀₁

—

D₄×C₄₀₁

D₄

yes

—

D₄×C₄₀₁

C₂³

yes

—

C₂×C₄×C₄₀₁

—

C₂×C₄×C₄₀₁

C₂³

yes

—

C₂²×C₄₀₁

—

C₂²×C₄₀₁

—

C₂²×C₄₀₁

C₂×D₄

yes

—

C₂×D₄₀₁

—

C₄×C₄₀₁

—

C₄×C₄₀₁

C₂×D₄

yes

—

C₄×C₄₀₁

C₂⁴

yes

characteristic

Dic₄₀₁

—

C₂×C₄₀₁

—

C₂×C₄₀₁

—

C₂×C₄₀₁

C₂²×D₄

yes

characteristic

D₄₀₁

—

C₄₀₁

#577

yes

characteristic, complemented, Sylow

#577

—

401

maximal, Sylow

2₋¹⁺⁴

—

401

—

H(5,𝔽₂)

D₄₀₁

yes

characteristic, complemented

#137

—

401

—

AGL(1,ℤ/8)

—

401

—

#135

—

401

—

C₂×SD₁₆

—

401

—

#131

—

401

—

#41

—

#41

—

802

—

#41

—

401

—

#41

C₂×D₄₀₁

yes

—

#41

C₂×D₄₀₁

yes

characteristic

C₂×Q₈

—

802

—

C₂×Q₈

—

401

—

C₂×D₄

—

C₂×D₄

C₂×D₄₀₁

yes

—

Q₁₆

—

401

—

SD₁₆

—

401

—

SD₁₆

—

401

—

SD₁₆

—

401

—

D₈

—

401

—

M₁₆

—

401

—

C₂×C₈

—

401

—

C₂³

—

Q₈

—

802

—

Q₈

—

401

—

Q₈

C₂²⋊D₄₀₁

yes

characteristic

Q₈

C₂²×D₄₀₁

yes

characteristic

D₄

—

D₄

—

D₄

—

802

—

D₄

—

401

—

D₄

C₂²⋊D₄₀₁

yes

—

D₄

C₂²×D₄₀₁

yes

—

C₂×C₄

—

C₂×C₄

—

802

—

C₂×C₄

—

802

—

C₂×C₄

C₂²×D₄₀₁

yes

—

C₈

—

401

—

C₈

—

401

—

C₂²

—

C₂²

—

C₂²

—

802

—

C₂²

C₂×C₂²⋊D₄₀₁

yes

—

C₄

—

100

C₄

—

802

—

101

C₄

C₂×C₂²⋊D₄₀₁

yes

—

102

C₄

C₂³×D₄₀₁

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

1604

—

106

C₂

C₂²×C₂²⋊D₄₀₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₄₀₁
commutator, cosocle	C₄×C₄₀₁	C₂⁴
Frattini	C₄	C₂³×D₄₀₁
Fitting	H(5,𝔽₂)×C₄₀₁	C₂
socle	C₂×C₄₀₁	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₄₀₁
hypocenter	C₄₀₁	#577
LCS	G ▷ C₄×C₄₀₁ ▷ C₂×C₄₀₁ ▷ C₄₀₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₄₀₁

commutator, cosocle

C₄×C₄₀₁

C₂⁴

Frattini

C₄

C₂³×D₄₀₁

Fitting

H(5,𝔽₂)×C₄₀₁

C₂

socle

C₂×C₄₀₁

C₂²×D₄

D₄₀₁

C₄₀₁

G ▷ C₄×C₄₀₁ ▷ C₂×C₄₀₁ ▷ C₄₀₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #27165784	C₂	semidirect
2. #27165767	C₂	semidirect
3. #27165763	C₂	semidirect
4. #27165757	C₂	semidirect
5. C₂×#20264354	C₂	semidirect
6. #27165744	C₂	semidirect
7. H(5,𝔽₂)×C₄₀₁	C₂	semidirect
8. #20264366	C₂²	semidirect
10. C₂×Dic₈₀₂	C₂²	semidirect
12. #20264356	C₂²	semidirect
13. #20264354	C₂²	semidirect
14. #20264354	C₂²	semidirect
15. #20264353	C₂²	semidirect
16. #20264351	C₂²	semidirect
17. #20264350	C₂²	semidirect
18. C₂×#15691999	C₂²	semidirect
53. C₄₀₁	#577	semidirect
56. H(5,𝔽₂)	D₄₀₁	semidirect

H ≅ G/N

type

1. #27165784

C₂

semidirect

2. #27165767

C₂

semidirect

3. #27165763

C₂

semidirect

4. #27165757

C₂

semidirect

5. C₂×#20264354

C₂

semidirect

6. #27165744

C₂

semidirect

7. H(5,𝔽₂)×C₄₀₁

C₂

semidirect

8. #20264366

C₂²

semidirect

10. C₂×Dic₈₀₂

C₂²

semidirect

12. #20264356

C₂²

semidirect

13. #20264354

C₂²

semidirect

14. #20264354

C₂²

semidirect

15. #20264353

C₂²

semidirect

16. #20264351

C₂²

semidirect

17. #20264350

C₂²

semidirect

18. C₂×#15691999

C₂²

semidirect

53. C₄₀₁

#577

semidirect

56. H(5,𝔽₂)

D₄₀₁

semidirect

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

subgroup	index
#27165784	2
#27165767	2
#27165763	2
#27165757	2
C₂×#20264354	2
#27165744	2
H(5,𝔽₂)×C₄₀₁	2
#577	401

subgroup

index

H(5,𝔽₂)×C₄₀₁

401

Subgroups — #38786925

Conjugacy classes of subgroups

Complements

Maximal subgroups