Group #34092497 — subgroups

Subgroups — #34092497

Order 19904.

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	#23755754	C₂	1	1	yes	maximal, characteristic, complemented
2	#23755734	C₂	1	3	yes	maximal, complemented
3	#23755733	C₂	1	3	yes	maximal, complemented
4	#23755728	C₂	1	3	yes	maximal, complemented
5	C₂×#19097300	C₂	1	3	yes	maximal, complemented
6	#23755715	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#19097316	—	2	1	no	—
9	#19097315	C₂²	1	3	yes	complemented
10	C₂×D₁₂₄₄	C₂²	1	3	yes	complemented
11	D₄×D₃₁₁	—	2	3	no	—
12	#19097305	C₂²	1	1	yes	characteristic, complemented
13	#19097303	C₂²	1	3	yes	complemented
14	#19097302	C₂²	1	3	yes	complemented
15	#19097300	C₂²	1	3	yes	complemented
16	#19097300	C₂²	1	6	yes	complemented
17	#19097299	C₂²	1	3	yes	complemented
18	C₂×C₃₁₁⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₃₁₁	—	2	1	no	—
20	#41×C₃₁₁	C₂²	1	1	yes	characteristic
21	#41×C₃₁₁	C₂²	1	3	yes	complemented
22	C₂×D₄×C₃₁₁	—	2	3	no	—
23	C₂×D₄×C₃₁₁	C₂²	1	3	yes	complemented
24	C₂²×D₃₁₁	—	2	3	no	—
25	Dic₆₂₂	C₂³	1	1	yes	characteristic
26	D₁₂₄₄	—	2	3	no	—
27	D₁₂₄₄	C₂³	1	3	yes	—
28	C₂²⋊D₃₁₁	—	2	3	no	—
29	C₄×D₃₁₁	—	2	3	no	—
30	C₃₁₁⋊C₈	C₂³	1	1	yes	characteristic
31	C₃₁₁⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₃₁₁	—	2	3	no	—
33	Q₈×C₃₁₁	D₄	1	1	yes	characteristic
34	Q₈×C₃₁₁	C₂³	1	1	yes	characteristic
35	D₄×C₃₁₁	—	2	3	no	—
36	D₄×C₃₁₁	—	2	3	no	—
37	D₄×C₃₁₁	D₄	1	3	yes	—
38	D₄×C₃₁₁	C₂³	1	3	yes	—
39	C₂×C₄×C₃₁₁	—	2	3	no	—
40	C₂×C₄×C₃₁₁	C₂³	1	3	yes	—
41	C₂²×C₃₁₁	—	2	3	no	—
42	C₂²×C₃₁₁	—	8	3	no	—
43	C₂²×C₃₁₁	C₂×D₄	1	3	yes	—
44	C₂×D₃₁₁	—	2	3	no	—
45	C₂×D₃₁₁	—	8	3	no	—
46	C₄×C₃₁₁	—	2	1	no	—
47	C₄×C₃₁₁	C₂×D₄	1	3	yes	—
48	C₄×C₃₁₁	C₂⁴	1	1	yes	characteristic
49	Dic₃₁₁	—	2	1	no	—
50	C₂×C₃₁₁	—	2	3	no	—
51	C₂×C₃₁₁	—	4	3	no	—
52	C₂×C₃₁₁	C₂²×D₄	1	1	yes	characteristic
53	D₃₁₁	—	4	3	no	—
54	C₃₁₁	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	311	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	311	1	no	—
57	H(5,𝔽₂)	D₃₁₁	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	311	3	no	—
59	AGL(1,ℤ/8)	—	311	3	no	—
60	#135	—	311	3	no	—
61	C₂×D₈	—	311	3	no	—
62	#131	—	311	1	no	—
63	#41	—	2	1	no	—
64	#41	—	311	3	no	—
65	#41	—	622	1	no	—
66	#41	C₂×D₃₁₁	1	1	yes	characteristic
67	#41	C₂×D₃₁₁	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	311	3	no	—
70	C₂×D₄	—	622	3	no	—
71	C₂×D₄	C₂×D₃₁₁	1	3	yes	complemented
72	Q₁₆	—	311	1	no	—
73	SD₁₆	—	311	3	no	—
74	SD₁₆	—	311	3	no	—
75	D₈	—	311	3	no	—
76	D₈	—	311	6	no	—
77	M₁₆	—	311	3	no	—
78	C₂×C₈	—	311	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	622	3	no	—
81	Q₈	—	311	1	no	—
82	Q₈	C₂²⋊D₃₁₁	1	1	yes	characteristic
83	Q₈	C₂²×D₃₁₁	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	311	3	no	—
87	D₄	—	622	3	no	—
88	D₄	—	622	3	no	—
89	D₄	C₂²⋊D₃₁₁	1	3	yes	—
90	D₄	C₂²×D₃₁₁	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	622	3	no	—
93	C₂×C₄	C₂²×D₃₁₁	1	3	yes	—
94	C₈	—	311	1	no	—
95	C₈	—	311	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	622	3	no	—
99	C₂²	—	2488	3	no	—
100	C₂²	C₂×C₂²⋊D₃₁₁	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	622	1	no	—
103	C₄	C₂×C₂²⋊D₃₁₁	1	3	yes	—
104	C₄	C₂³×D₃₁₁	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	1244	3	no	—
108	C₂	C₂²×C₂²⋊D₃₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23755754

C₂

yes

maximal, characteristic, complemented

#23755734

C₂

yes

maximal, complemented

#23755733

C₂

yes

maximal, complemented

#23755728

C₂

yes

maximal, complemented

C₂×#19097300

C₂

yes

maximal, complemented

#23755715

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₃₁₁

C₂

yes

maximal, characteristic, complemented

#19097316

—

#19097315

C₂²

yes

complemented

C₂×D₁₂₄₄

C₂²

yes

complemented

D₄×D₃₁₁

—

#19097305

C₂²

yes

characteristic, complemented

#19097303

C₂²

yes

complemented

#19097302

C₂²

yes

complemented

#19097300

C₂²

yes

complemented

#19097300

C₂²

yes

complemented

#19097299

C₂²

yes

complemented

C₂×C₃₁₁⋊C₈

C₂²

yes

complemented

#41×C₃₁₁

—

#41×C₃₁₁

C₂²

yes

characteristic

#41×C₃₁₁

C₂²

yes

complemented

C₂×D₄×C₃₁₁

—

C₂×D₄×C₃₁₁

C₂²

yes

complemented

C₂²×D₃₁₁

—

Dic₆₂₂

C₂³

yes

characteristic

D₁₂₄₄

—

D₁₂₄₄

C₂³

yes

—

C₂²⋊D₃₁₁

—

C₄×D₃₁₁

—

C₃₁₁⋊C₈

C₂³

yes

characteristic

C₃₁₁⋊C₈

C₂³

yes

—

C₂³×C₃₁₁

—

Q₈×C₃₁₁

D₄

yes

characteristic

Q₈×C₃₁₁

C₂³

yes

characteristic

D₄×C₃₁₁

—

D₄×C₃₁₁

—

D₄×C₃₁₁

D₄

yes

—

D₄×C₃₁₁

C₂³

yes

—

C₂×C₄×C₃₁₁

—

C₂×C₄×C₃₁₁

C₂³

yes

—

C₂²×C₃₁₁

—

C₂²×C₃₁₁

—

C₂²×C₃₁₁

C₂×D₄

yes

—

C₂×D₃₁₁

—

C₂×D₃₁₁

—

C₄×C₃₁₁

—

C₄×C₃₁₁

C₂×D₄

yes

—

C₄×C₃₁₁

C₂⁴

yes

characteristic

Dic₃₁₁

—

C₂×C₃₁₁

—

C₂×C₃₁₁

—

C₂×C₃₁₁

C₂²×D₄

yes

characteristic

D₃₁₁

—

C₃₁₁

#576

yes

characteristic, complemented, Sylow

#576

—

311

maximal, Sylow

H(5,𝔽₂)

—

311

—

H(5,𝔽₂)

D₃₁₁

yes

characteristic, complemented

AGL(1,ℤ/8)

—

311

—

AGL(1,ℤ/8)

—

311

—

#135

—

311

—

C₂×D₈

—

311

—

#131

—

311

—

#41

—

#41

—

311

—

#41

—

622

—

#41

C₂×D₃₁₁

yes

characteristic

#41

C₂×D₃₁₁

yes

complemented

C₂×D₄

—

C₂×D₄

—

311

—

C₂×D₄

—

622

—

C₂×D₄

C₂×D₃₁₁

yes

complemented

Q₁₆

—

311

—

SD₁₆

—

311

—

SD₁₆

—

311

—

D₈

—

311

—

D₈

—

311

—

M₁₆

—

311

—

C₂×C₈

—

311

—

C₂³

—

C₂³

—

622

—

Q₈

—

311

—

Q₈

C₂²⋊D₃₁₁

yes

characteristic

Q₈

C₂²×D₃₁₁

yes

characteristic

D₄

—

D₄

—

D₄

—

311

—

D₄

—

622

—

D₄

—

622

—

D₄

C₂²⋊D₃₁₁

yes

—

D₄

C₂²×D₃₁₁

yes

—

C₂×C₄

—

C₂×C₄

—

622

—

C₂×C₄

C₂²×D₃₁₁

yes

—

C₈

—

311

—

C₈

—

311

—

C₂²

—

C₂²

—

C₂²

—

622

—

C₂²

—

2488

—

100

C₂²

C₂×C₂²⋊D₃₁₁

yes

—

101

C₄

—

102

C₄

—

622

—

103

C₄

C₂×C₂²⋊D₃₁₁

yes

—

104

C₄

C₂³×D₃₁₁

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

1244

—

108

C₂

C₂²×C₂²⋊D₃₁₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₃₁₁
commutator, cosocle	C₄×C₃₁₁	C₂⁴
Frattini	C₄	C₂³×D₃₁₁
Fitting	H(5,𝔽₂)×C₃₁₁	C₂
socle	C₂×C₃₁₁	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₃₁₁
hypocenter	C₃₁₁	#576
LCS	G ▷ C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₃₁₁

commutator, cosocle

C₄×C₃₁₁

C₂⁴

Frattini

C₄

C₂³×D₃₁₁

Fitting

H(5,𝔽₂)×C₃₁₁

C₂

socle

C₂×C₃₁₁

C₂²×D₄

D₃₁₁

C₃₁₁

G ▷ C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #23755754	C₂	semidirect
2. #23755734	C₂	semidirect
3. #23755733	C₂	semidirect
4. #23755728	C₂	semidirect
5. C₂×#19097300	C₂	semidirect
6. #23755715	C₂	semidirect
7. H(5,𝔽₂)×C₃₁₁	C₂	semidirect
9. #19097315	C₂²	semidirect
10. C₂×D₁₂₄₄	C₂²	semidirect
12. #19097305	C₂²	semidirect
13. #19097303	C₂²	semidirect
14. #19097302	C₂²	semidirect
15. #19097300	C₂²	semidirect
16. #19097300	C₂²	semidirect
17. #19097299	C₂²	semidirect
18. C₂×C₃₁₁⋊C₈	C₂²	semidirect
21. #41×C₃₁₁	C₂²	semidirect
23. C₂×D₄×C₃₁₁	C₂²	semidirect
54. C₃₁₁	#576	semidirect
57. H(5,𝔽₂)	D₃₁₁	semidirect
67. #41	C₂×D₃₁₁	semidirect
71. C₂×D₄	C₂×D₃₁₁	semidirect

H ≅ G/N

type

1. #23755754

C₂

semidirect

2. #23755734

C₂

semidirect

3. #23755733

C₂

semidirect

4. #23755728

C₂

semidirect

5. C₂×#19097300

C₂

semidirect

6. #23755715

C₂

semidirect

7. H(5,𝔽₂)×C₃₁₁

C₂

semidirect

9. #19097315

C₂²

semidirect

10. C₂×D₁₂₄₄

C₂²

semidirect

12. #19097305

C₂²

semidirect

13. #19097303

C₂²

semidirect

14. #19097302

C₂²

semidirect

15. #19097300

C₂²

semidirect

16. #19097300

C₂²

semidirect

17. #19097299

C₂²

semidirect

18. C₂×C₃₁₁⋊C₈

C₂²

semidirect

21. #41×C₃₁₁

C₂²

semidirect

23. C₂×D₄×C₃₁₁

C₂²

semidirect

54. C₃₁₁

#576

semidirect

57. H(5,𝔽₂)

D₃₁₁

semidirect

67. #41

C₂×D₃₁₁

semidirect

71. C₂×D₄

C₂×D₃₁₁

semidirect

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

subgroup	index
#23755754	2
#23755734	2
#23755733	2
#23755728	2
C₂×#19097300	2
#23755715	2
H(5,𝔽₂)×C₃₁₁	2
#576	311

subgroup

index

H(5,𝔽₂)×C₃₁₁

311

Subgroups — #34092497

Conjugacy classes of subgroups

Complements

Maximal subgroups