Group #26012302 — subgroups

Subgroups — #26012302

Order 11072.

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	#19149726	C₂	1	1	yes	maximal, characteristic, complemented
2	#19149709	C₂	1	3	yes	maximal, complemented
3	#19149705	C₂	1	3	yes	maximal, complemented
4	#19149699	C₂	1	3	yes	maximal, complemented
5	C₂×#14583814	C₂	1	3	yes	maximal, complemented
6	#19149686	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583826	C₂²	1	3	yes	complemented
9	#14583825	—	2	3	no	—
10	C₂×Dic₃₄₆	C₂²	1	3	yes	complemented
11	Q₈×D₁₇₃	—	2	1	no	—
12	#14583816	C₂²	1	3	yes	complemented
13	#14583814	C₂²	1	3	yes	complemented
14	#14583814	C₂²	1	6	yes	complemented
15	#14583813	C₂²	1	1	yes	characteristic, complemented
16	#14583811	C₂²	1	3	yes	complemented
17	#14583810	C₂²	1	3	yes	complemented
18	C₂×#13063436	C₂²	1	3	yes	complemented
19	#41×C₁₇₃	—	2	1	no	—
20	#41×C₁₇₃	C₂²	1	3	yes	—
21	#41×C₁₇₃	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₇₃	—	2	3	no	—
23	C₂×D₄×C₁₇₃	C₂²	1	3	yes	—
24	Dic₃₄₆	—	2	3	no	—
25	Dic₃₄₆	C₂³	1	3	yes	—
26	D₆₉₂	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₇₃	—	2	3	no	—
28	C₂×Dic₁₇₃	—	2	3	no	—
29	C₄×D₁₇₃	—	2	3	no	—
30	#13063436	C₂³	1	1	yes	characteristic
31	#13063436	C₂³	1	3	yes	—
32	C₂³×C₁₇₃	—	2	3	no	—
33	Q₈×C₁₇₃	D₄	1	1	yes	characteristic
34	Q₈×C₁₇₃	C₂³	1	1	yes	characteristic
35	D₄×C₁₇₃	—	2	3	no	—
36	D₄×C₁₇₃	—	2	3	no	—
37	D₄×C₁₇₃	D₄	1	3	yes	—
38	D₄×C₁₇₃	C₂³	1	3	yes	—
39	C₂×C₄×C₁₇₃	—	2	3	no	—
40	C₂×C₄×C₁₇₃	C₂³	1	3	yes	—
41	C₂²×C₁₇₃	—	2	3	no	—
42	C₂²×C₁₇₃	—	8	3	no	—
43	C₂²×C₁₇₃	C₂×D₄	1	3	yes	—
44	C₂×D₁₇₃	—	2	1	no	—
45	C₄×C₁₇₃	—	2	1	no	—
46	C₄×C₁₇₃	C₂×D₄	1	3	yes	—
47	C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
48	Dic₁₇₃	—	2	3	no	—
49	C₂×C₁₇₃	—	2	3	no	—
50	C₂×C₁₇₃	—	4	3	no	—
51	C₂×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
52	D₁₇₃	—	4	1	no	—
53	C₁₇₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	173	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	173	1	no	—
56	H(5,𝔽₂)	D₁₇₃	1	1	yes	characteristic, complemented
57	#137	—	173	3	no	—
58	AGL(1,ℤ/8)	—	173	3	no	—
59	#135	—	173	3	no	—
60	C₂×SD₁₆	—	173	3	no	—
61	#131	—	173	1	no	—
62	#41	—	2	1	no	—
63	#41	—	346	3	no	—
64	#41	—	173	3	no	—
65	#41	C₂×D₁₇₃	1	3	yes	—
66	#41	C₂×D₁₇₃	1	1	yes	characteristic
67	C₂×Q₈	—	346	1	no	—
68	C₂×Q₈	—	173	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₇₃	1	3	yes	—
71	Q₁₆	—	173	3	no	—
72	SD₁₆	—	173	1	no	—
73	SD₁₆	—	173	3	no	—
74	SD₁₆	—	173	6	no	—
75	D₈	—	173	3	no	—
76	M₁₆	—	173	3	no	—
77	C₂×C₈	—	173	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	346	3	no	—
80	Q₈	—	173	3	no	—
81	Q₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
82	Q₈	C₂²×D₁₇₃	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	346	3	no	—
86	D₄	—	173	1	no	—
87	D₄	C₂²⋊D₁₇₃	1	3	yes	—
88	D₄	C₂²×D₁₇₃	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	346	3	no	—
91	C₂×C₄	—	346	3	no	—
92	C₂×C₄	C₂²×D₁₇₃	1	3	yes	—
93	C₈	—	173	1	no	—
94	C₈	—	173	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	346	1	no	—
98	C₂²	C₂×C₂²⋊D₁₇₃	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	346	3	no	—
101	C₄	C₂×C₂²⋊D₁₇₃	1	3	yes	—
102	C₄	C₂³×D₁₇₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	692	1	no	—
106	C₂	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19149726

C₂

yes

maximal, characteristic, complemented

#19149709

C₂

yes

maximal, complemented

#19149705

C₂

yes

maximal, complemented

#19149699

C₂

yes

maximal, complemented

C₂×#14583814

C₂

yes

maximal, complemented

#19149686

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₇₃

C₂

yes

maximal, characteristic, complemented

#14583826

C₂²

yes

complemented

#14583825

—

C₂×Dic₃₄₆

C₂²

yes

complemented

Q₈×D₁₇₃

—

#14583816

C₂²

yes

complemented

#14583814

C₂²

yes

complemented

#14583814

C₂²

yes

complemented

#14583813

C₂²

yes

characteristic, complemented

#14583811

C₂²

yes

complemented

#14583810

C₂²

yes

complemented

C₂×#13063436

C₂²

yes

complemented

#41×C₁₇₃

—

#41×C₁₇₃

C₂²

yes

—

#41×C₁₇₃

C₂²

yes

characteristic

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

C₂²

yes

—

Dic₃₄₆

—

Dic₃₄₆

C₂³

yes

—

D₆₉₂

C₂³

yes

characteristic

C₂²⋊D₁₇₃

—

C₂×Dic₁₇₃

—

C₄×D₁₇₃

—

#13063436

C₂³

yes

characteristic

#13063436

C₂³

yes

—

C₂³×C₁₇₃

—

Q₈×C₁₇₃

D₄

yes

characteristic

Q₈×C₁₇₃

C₂³

yes

characteristic

D₄×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

D₄

yes

—

D₄×C₁₇₃

C₂³

yes

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂×D₄

yes

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×D₄

yes

—

C₄×C₁₇₃

C₂⁴

yes

characteristic

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

C₂²×D₄

yes

characteristic

D₁₇₃

—

C₁₇₃

#577

yes

characteristic, complemented, Sylow

#577

—

173

maximal, Sylow

2₋¹⁺⁴

—

173

—

H(5,𝔽₂)

D₁₇₃

yes

characteristic, complemented

#137

—

173

—

AGL(1,ℤ/8)

—

173

—

#135

—

173

—

C₂×SD₁₆

—

173

—

#131

—

173

—

#41

—

#41

—

346

—

#41

—

173

—

#41

C₂×D₁₇₃

yes

—

#41

C₂×D₁₇₃

yes

characteristic

C₂×Q₈

—

346

—

C₂×Q₈

—

173

—

C₂×D₄

—

C₂×D₄

C₂×D₁₇₃

yes

—

Q₁₆

—

173

—

SD₁₆

—

173

—

SD₁₆

—

173

—

SD₁₆

—

173

—

D₈

—

173

—

M₁₆

—

173

—

C₂×C₈

—

173

—

C₂³

—

Q₈

—

346

—

Q₈

—

173

—

Q₈

C₂²⋊D₁₇₃

yes

characteristic

Q₈

C₂²×D₁₇₃

yes

characteristic

D₄

—

D₄

—

D₄

—

346

—

D₄

—

173

—

D₄

C₂²⋊D₁₇₃

yes

—

D₄

C₂²×D₁₇₃

yes

—

C₂×C₄

—

C₂×C₄

—

346

—

C₂×C₄

—

346

—

C₂×C₄

C₂²×D₁₇₃

yes

—

C₈

—

173

—

C₈

—

173

—

C₂²

—

C₂²

—

C₂²

—

346

—

C₂²

C₂×C₂²⋊D₁₇₃

yes

—

C₄

—

100

C₄

—

346

—

101

C₄

C₂×C₂²⋊D₁₇₃

yes

—

102

C₄

C₂³×D₁₇₃

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

692

—

106

C₂

C₂²×C₂²⋊D₁₇₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₇₃
commutator, cosocle	C₄×C₁₇₃	C₂⁴
Frattini	C₄	C₂³×D₁₇₃
Fitting	H(5,𝔽₂)×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₇₃
hypocenter	C₁₇₃	#577
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₇₃

commutator, cosocle

C₄×C₁₇₃

C₂⁴

Frattini

C₄

C₂³×D₁₇₃

Fitting

H(5,𝔽₂)×C₁₇₃

C₂

socle

C₂×C₁₇₃

C₂²×D₄

D₁₇₃

C₁₇₃

G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #19149726	C₂	semidirect
2. #19149709	C₂	semidirect
3. #19149705	C₂	semidirect
4. #19149699	C₂	semidirect
5. C₂×#14583814	C₂	semidirect
6. #19149686	C₂	semidirect
7. H(5,𝔽₂)×C₁₇₃	C₂	semidirect
8. #14583826	C₂²	semidirect
10. C₂×Dic₃₄₆	C₂²	semidirect
12. #14583816	C₂²	semidirect
13. #14583814	C₂²	semidirect
14. #14583814	C₂²	semidirect
15. #14583813	C₂²	semidirect
16. #14583811	C₂²	semidirect
17. #14583810	C₂²	semidirect
18. C₂×#13063436	C₂²	semidirect
53. C₁₇₃	#577	semidirect
56. H(5,𝔽₂)	D₁₇₃	semidirect

H ≅ G/N

type

1. #19149726

C₂

semidirect

2. #19149709

C₂

semidirect

3. #19149705

C₂

semidirect

4. #19149699

C₂

semidirect

5. C₂×#14583814

C₂

semidirect

6. #19149686

C₂

semidirect

7. H(5,𝔽₂)×C₁₇₃

C₂

semidirect

8. #14583826

C₂²

semidirect

10. C₂×Dic₃₄₆

C₂²

semidirect

12. #14583816

C₂²

semidirect

13. #14583814

C₂²

semidirect

14. #14583814

C₂²

semidirect

15. #14583813

C₂²

semidirect

16. #14583811

C₂²

semidirect

17. #14583810

C₂²

semidirect

18. C₂×#13063436

C₂²

semidirect

53. C₁₇₃

#577

semidirect

56. H(5,𝔽₂)

D₁₇₃

semidirect

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

subgroup	index
#19149726	2
#19149709	2
#19149705	2
#19149699	2
C₂×#14583814	2
#19149686	2
H(5,𝔽₂)×C₁₇₃	2
#577	173

subgroup

index

H(5,𝔽₂)×C₁₇₃

173

Subgroups — #26012302

Conjugacy classes of subgroups

Complements

Maximal subgroups