Group #37560225 — subgroups

Subgroups — #37560225

Order 23872.

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

normal subgroups only

This group has 268 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	#26045838	C₂	1	1	yes	maximal, characteristic, complemented
2	#26045818	C₂	1	3	yes	maximal, complemented
3	#26045817	C₂	1	3	yes	maximal, complemented
4	#26045812	C₂	1	3	yes	maximal, complemented
5	C₂×#20238595	C₂	1	3	yes	maximal, complemented
6	#26045799	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20238611	—	2	1	no	—
9	#20238610	C₂²	1	3	yes	complemented
10	C₂×D₁₄₉₂	C₂²	1	3	yes	complemented
11	D₄×D₃₇₃	—	2	3	no	—
12	#20238600	C₂²	1	1	yes	characteristic, complemented
13	#20238598	C₂²	1	3	yes	complemented
14	#20238597	C₂²	1	3	yes	complemented
15	#20238595	C₂²	1	3	yes	complemented
16	#20238595	C₂²	1	6	yes	complemented
17	#20238594	C₂²	1	3	yes	complemented
18	C₂×#15688412	C₂²	1	3	yes	complemented
19	#41×C₃₇₃	—	2	1	no	—
20	#41×C₃₇₃	C₂²	1	1	yes	characteristic
21	#41×C₃₇₃	C₂²	1	3	yes	complemented
22	C₂×D₄×C₃₇₃	—	2	3	no	—
23	C₂×D₄×C₃₇₃	C₂²	1	3	yes	complemented
24	C₂²×D₃₇₃	—	2	3	no	—
25	Dic₇₄₆	C₂³	1	1	yes	characteristic
26	D₁₄₉₂	—	2	3	no	—
27	D₁₄₉₂	C₂³	1	3	yes	—
28	C₂²⋊D₃₇₃	—	2	3	no	—
29	C₄×D₃₇₃	—	2	3	no	—
30	#15688412	C₂³	1	1	yes	characteristic
31	#15688412	C₂³	1	3	yes	—
32	C₂³×C₃₇₃	—	2	3	no	—
33	Q₈×C₃₇₃	D₄	1	1	yes	characteristic
34	Q₈×C₃₇₃	C₂³	1	1	yes	characteristic
35	D₄×C₃₇₃	—	2	3	no	—
36	D₄×C₃₇₃	—	2	3	no	—
37	D₄×C₃₇₃	D₄	1	3	yes	—
38	D₄×C₃₇₃	C₂³	1	3	yes	—
39	C₂×C₄×C₃₇₃	—	2	3	no	—
40	C₂×C₄×C₃₇₃	C₂³	1	3	yes	—
41	C₂²×C₃₇₃	—	2	3	no	—
42	C₂²×C₃₇₃	—	8	3	no	—
43	C₂²×C₃₇₃	C₂×D₄	1	3	yes	—
44	C₂×D₃₇₃	—	2	3	no	—
45	C₂×D₃₇₃	—	8	3	no	—
46	C₄×C₃₇₃	—	2	1	no	—
47	C₄×C₃₇₃	C₂×D₄	1	3	yes	—
48	C₄×C₃₇₃	C₂⁴	1	1	yes	characteristic
49	Dic₃₇₃	—	2	1	no	—
50	C₂×C₃₇₃	—	2	3	no	—
51	C₂×C₃₇₃	—	4	3	no	—
52	C₂×C₃₇₃	C₂²×D₄	1	1	yes	characteristic
53	D₃₇₃	—	4	3	no	—
54	C₃₇₃	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	373	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	373	1	no	—
57	H(5,𝔽₂)	D₃₇₃	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	373	3	no	—
59	AGL(1,ℤ/8)	—	373	3	no	—
60	#135	—	373	3	no	—
61	C₂×D₈	—	373	3	no	—
62	#131	—	373	1	no	—
63	#41	—	2	1	no	—
64	#41	—	373	3	no	—
65	#41	—	746	1	no	—
66	#41	C₂×D₃₇₃	1	1	yes	characteristic
67	#41	C₂×D₃₇₃	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	373	3	no	—
70	C₂×D₄	—	746	3	no	—
71	C₂×D₄	C₂×D₃₇₃	1	3	yes	complemented
72	Q₁₆	—	373	1	no	—
73	SD₁₆	—	373	3	no	—
74	SD₁₆	—	373	3	no	—
75	D₈	—	373	3	no	—
76	D₈	—	373	6	no	—
77	M₁₆	—	373	3	no	—
78	C₂×C₈	—	373	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	746	3	no	—
81	Q₈	—	373	1	no	—
82	Q₈	C₂²⋊D₃₇₃	1	1	yes	characteristic
83	Q₈	C₂²×D₃₇₃	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	373	3	no	—
87	D₄	—	746	3	no	—
88	D₄	—	746	3	no	—
89	D₄	C₂²⋊D₃₇₃	1	3	yes	—
90	D₄	C₂²×D₃₇₃	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	746	3	no	—
93	C₂×C₄	C₂²×D₃₇₃	1	3	yes	—
94	C₈	—	373	1	no	—
95	C₈	—	373	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	2984	3	no	—
99	C₂²	—	746	3	no	—
100	C₂²	C₂×C₂²⋊D₃₇₃	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	746	1	no	—
103	C₄	C₂×C₂²⋊D₃₇₃	1	3	yes	—
104	C₄	C₂³×D₃₇₃	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	1492	3	no	—
108	C₂	C₂²×C₂²⋊D₃₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26045838

C₂

yes

maximal, characteristic, complemented

#26045818

C₂

yes

maximal, complemented

#26045817

C₂

yes

maximal, complemented

#26045812

C₂

yes

maximal, complemented

C₂×#20238595

C₂

yes

maximal, complemented

#26045799

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₃₇₃

C₂

yes

maximal, characteristic, complemented

#20238611

—

#20238610

C₂²

yes

complemented

C₂×D₁₄₉₂

C₂²

yes

complemented

D₄×D₃₇₃

—

#20238600

C₂²

yes

characteristic, complemented

#20238598

C₂²

yes

complemented

#20238597

C₂²

yes

complemented

#20238595

C₂²

yes

complemented

#20238595

C₂²

yes

complemented

#20238594

C₂²

yes

complemented

C₂×#15688412

C₂²

yes

complemented

#41×C₃₇₃

—

#41×C₃₇₃

C₂²

yes

characteristic

#41×C₃₇₃

C₂²

yes

complemented

C₂×D₄×C₃₇₃

—

C₂×D₄×C₃₇₃

C₂²

yes

complemented

C₂²×D₃₇₃

—

Dic₇₄₆

C₂³

yes

characteristic

D₁₄₉₂

—

D₁₄₉₂

C₂³

yes

—

C₂²⋊D₃₇₃

—

C₄×D₃₇₃

—

#15688412

C₂³

yes

characteristic

#15688412

C₂³

yes

—

C₂³×C₃₇₃

—

Q₈×C₃₇₃

D₄

yes

characteristic

Q₈×C₃₇₃

C₂³

yes

characteristic

D₄×C₃₇₃

—

D₄×C₃₇₃

—

D₄×C₃₇₃

D₄

yes

—

D₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

C₂×D₄

yes

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

C₂×D₄

yes

—

C₄×C₃₇₃

C₂⁴

yes

characteristic

Dic₃₇₃

—

C₂×C₃₇₃

—

C₂×C₃₇₃

—

C₂×C₃₇₃

C₂²×D₄

yes

characteristic

D₃₇₃

—

C₃₇₃

#576

yes

characteristic, complemented, Sylow

#576

—

373

maximal, Sylow

H(5,𝔽₂)

—

373

—

H(5,𝔽₂)

D₃₇₃

yes

characteristic, complemented

AGL(1,ℤ/8)

—

373

—

AGL(1,ℤ/8)

—

373

—

#135

—

373

—

C₂×D₈

—

373

—

#131

—

373

—

#41

—

#41

—

373

—

#41

—

746

—

#41

C₂×D₃₇₃

yes

characteristic

#41

C₂×D₃₇₃

yes

complemented

C₂×D₄

—

C₂×D₄

—

373

—

C₂×D₄

—

746

—

C₂×D₄

C₂×D₃₇₃

yes

complemented

Q₁₆

—

373

—

SD₁₆

—

373

—

SD₁₆

—

373

—

D₈

—

373

—

D₈

—

373

—

M₁₆

—

373

—

C₂×C₈

—

373

—

C₂³

—

C₂³

—

746

—

Q₈

—

373

—

Q₈

C₂²⋊D₃₇₃

yes

characteristic

Q₈

C₂²×D₃₇₃

yes

characteristic

D₄

—

D₄

—

D₄

—

373

—

D₄

—

746

—

D₄

—

746

—

D₄

C₂²⋊D₃₇₃

yes

—

D₄

C₂²×D₃₇₃

yes

—

C₂×C₄

—

C₂×C₄

—

746

—

C₂×C₄

C₂²×D₃₇₃

yes

—

C₈

—

373

—

C₈

—

373

—

C₂²

—

C₂²

—

C₂²

—

2984

—

C₂²

—

746

—

100

C₂²

C₂×C₂²⋊D₃₇₃

yes

—

101

C₄

—

102

C₄

—

746

—

103

C₄

C₂×C₂²⋊D₃₇₃

yes

—

104

C₄

C₂³×D₃₇₃

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

1492

—

108

C₂

C₂²×C₂²⋊D₃₇₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₃₇₃
commutator, cosocle	C₄×C₃₇₃	C₂⁴
Frattini	C₄	C₂³×D₃₇₃
Fitting	H(5,𝔽₂)×C₃₇₃	C₂
socle	C₂×C₃₇₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₃₇₃
hypocenter	C₃₇₃	#576
LCS	G ▷ C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₃₇₃

commutator, cosocle

C₄×C₃₇₃

C₂⁴

Frattini

C₄

C₂³×D₃₇₃

Fitting

H(5,𝔽₂)×C₃₇₃

C₂

socle

C₂×C₃₇₃

C₂²×D₄

D₃₇₃

C₃₇₃

G ▷ C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #26045838	C₂	semidirect
2. #26045818	C₂	semidirect
3. #26045817	C₂	semidirect
4. #26045812	C₂	semidirect
5. C₂×#20238595	C₂	semidirect
6. #26045799	C₂	semidirect
7. H(5,𝔽₂)×C₃₇₃	C₂	semidirect
9. #20238610	C₂²	semidirect
10. C₂×D₁₄₉₂	C₂²	semidirect
12. #20238600	C₂²	semidirect
13. #20238598	C₂²	semidirect
14. #20238597	C₂²	semidirect
15. #20238595	C₂²	semidirect
16. #20238595	C₂²	semidirect
17. #20238594	C₂²	semidirect
18. C₂×#15688412	C₂²	semidirect
21. #41×C₃₇₃	C₂²	semidirect
23. C₂×D₄×C₃₇₃	C₂²	semidirect
54. C₃₇₃	#576	semidirect
57. H(5,𝔽₂)	D₃₇₃	semidirect
67. #41	C₂×D₃₇₃	semidirect
71. C₂×D₄	C₂×D₃₇₃	semidirect

H ≅ G/N

type

1. #26045838

C₂

semidirect

2. #26045818

C₂

semidirect

3. #26045817

C₂

semidirect

4. #26045812

C₂

semidirect

5. C₂×#20238595

C₂

semidirect

6. #26045799

C₂

semidirect

7. H(5,𝔽₂)×C₃₇₃

C₂

semidirect

9. #20238610

C₂²

semidirect

10. C₂×D₁₄₉₂

C₂²

semidirect

12. #20238600

C₂²

semidirect

13. #20238598

C₂²

semidirect

14. #20238597

C₂²

semidirect

15. #20238595

C₂²

semidirect

16. #20238595

C₂²

semidirect

17. #20238594

C₂²

semidirect

18. C₂×#15688412

C₂²

semidirect

21. #41×C₃₇₃

C₂²

semidirect

23. C₂×D₄×C₃₇₃

C₂²

semidirect

54. C₃₇₃

#576

semidirect

57. H(5,𝔽₂)

D₃₇₃

semidirect

67. #41

C₂×D₃₇₃

semidirect

71. C₂×D₄

C₂×D₃₇₃

semidirect

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

subgroup	index
#26045838	2
#26045818	2
#26045817	2
#26045812	2
C₂×#20238595	2
#26045799	2
H(5,𝔽₂)×C₃₇₃	2
#576	373

subgroup

index

H(5,𝔽₂)×C₃₇₃

373

Subgroups — #37560225

Conjugacy classes of subgroups

Complements

Maximal subgroups