Group #26035064 — subgroups

Subgroups — #26035064

Order 11392.

5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 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	#19152435	C₂	1	16	yes	maximal, complemented
2	#19152434	C₂	1	20	yes	maximal, complemented
3	C₂×#15668060	C₂	1	10	yes	maximal, complemented
4	2₋¹⁺⁴×D₈₉	C₂	1	16	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668060	C₂²	1	80	yes	complemented
7	#15668060	C₂²	1	40	yes	complemented
8	#15668059	C₂²	1	80	yes	complemented
9	#15668058	C₂²	1	80	yes	complemented
10	#15668057	C₂²	1	40	yes	complemented
11	C₂×#13083575	C₂²	1	40	yes	complemented
12	C₂×#13083574	C₂²	1	60	yes	complemented
13	#41×D₈₉	C₂²	1	160	yes	complemented
14	C₂×Q₈×D₈₉	C₂²	1	40	yes	complemented
15	2₋¹⁺⁴×C₈₉	C₂²	1	16	yes	complemented
16	C₂×#41×C₈₉	C₂²	1	10	yes	complemented
17	C₂²×Q₈×C₈₉	C₂²	1	5	yes	complemented
18	#13083575	C₂³	1	160	yes	—
19	#13083574	C₂³	1	240	yes	—
20	#13083574	C₂³	1	80	yes	—
21	#13083573	C₂³	1	160	yes	—
22	C₂×Dic₁₇₈	C₂³	1	60	yes	—
23	Q₈×D₈₉	C₂³	1	160	yes	—
24	C₂×D₃₅₆	C₂³	1	60	yes	—
25	C₂×C₂²⋊D₈₉	C₂³	1	40	yes	—
26	D₄×D₈₉	C₂³	1	160	yes	—
27	C₂×C₄×D₈₉	C₂³	1	120	yes	—
28	#41×C₈₉	C₂³	1	80	yes	—
29	C₂×Q₈×C₈₉	C₂³	1	40	yes	—
30	C₂×Q₈×C₈₉	C₂³	1	10	yes	—
31	C₂×D₄×C₈₉	C₂³	1	10	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	15	yes	—
33	C₂³×C₈₉	C₂⁴	1	5	yes	—
34	C₂²×D₈₉	C₂⁴	1	40	yes	—
35	Q₈×C₈₉	C₂⁴	1	40	yes	—
36	D₄×C₈₉	C₂⁴	1	40	yes	—
37	C₂×C₄×C₈₉	C₂⁴	1	10	yes	—
38	C₂×C₄×C₈₉	C₂⁴	1	60	yes	—
39	C₂²⋊D₈₉	C₂⁴	1	80	yes	—
40	C₂²⋊D₈₉	C₂⁴	1	16	yes	—
41	C₂×Dic₈₉	C₂⁴	1	40	yes	—
42	D₃₅₆	C₂⁴	1	80	yes	—
43	C₄×D₈₉	C₂⁴	1	160	yes	—
44	Dic₁₇₈	C₂⁴	1	80	yes	—
45	C₂²×C₈₉	—	4	5	no	—
46	C₂²×C₈₉	C₂⁵	1	10	yes	—
47	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
48	C₂×D₈₉	—	4	40	no	—
49	C₂×D₈₉	C₂⁵	1	16	yes	—
50	C₄×C₈₉	C₂⁵	1	20	yes	—
51	Dic₈₉	C₂⁵	1	16	yes	—
52	C₂×C₈₉	—	2	10	no	—
53	C₂×C₈₉	—	2	1	no	—
54	C₂×C₈₉	C₂⁶	1	1	yes	characteristic
55	D₈₉	—	2	16	no	—
56	2₋¹⁺⁶	—	89	1	no	maximal, Sylow
57	C₈₉	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
58	#585	—	89	16	no	—
59	#585	—	89	20	no	—
60	C₂×2₋¹⁺⁴	—	89	16	no	—
61	C₂×2₋¹⁺⁴	—	89	10	no	—
62	C₂×2₋¹⁺⁴	D₈₉	1	1	yes	characteristic, complemented
63	2₋¹⁺⁴	—	89	80	no	—
64	2₋¹⁺⁴	—	89	80	no	—
65	2₋¹⁺⁴	—	89	40	no	—
66	2₋¹⁺⁴	C₂×D₈₉	1	16	yes	complemented
67	H(5,𝔽₂)	—	89	80	no	—
68	H(5,𝔽₂)	—	89	40	no	—
69	C₂×#41	—	89	160	no	—
70	C₂×#41	—	89	40	no	—
71	C₂×#41	—	89	60	no	—
72	C₂×#41	C₂×D₈₉	1	10	yes	complemented
73	C₂²×Q₈	—	89	40	no	—
74	C₂²×Q₈	C₂×D₈₉	1	5	yes	complemented
75	#41	—	89	160	no	—
76	#41	—	89	160	no	—
77	#41	—	89	240	no	—
78	#41	—	89	80	no	—
79	#41	C₂²×D₈₉	1	80	yes	—
80	C₂×Q₈	—	89	160	no	—
81	C₂×Q₈	—	89	60	no	—
82	C₂×Q₈	C₂²×D₈₉	1	40	yes	—
83	C₂×Q₈	C₂²×D₈₉	1	10	yes	—
84	C₂×D₄	—	89	40	no	—
85	C₂×D₄	—	89	160	no	—
86	C₂×D₄	—	89	60	no	—
87	C₂×D₄	C₂²×D₈₉	1	10	yes	—
88	C₂²×C₄	—	89	120	no	—
89	C₂²×C₄	C₂²×D₈₉	1	15	yes	—
90	C₂³	—	89	40	no	—
91	C₂³	C₂³×D₈₉	1	5	yes	—
92	Q₈	—	89	80	no	—
93	Q₈	C₂³×D₈₉	1	40	yes	—
94	D₄	—	89	80	no	—
95	D₄	—	89	16	no	—
96	D₄	—	89	80	no	—
97	D₄	C₂³×D₈₉	1	40	yes	—
98	C₂×C₄	—	89	40	no	—
99	C₂×C₄	—	89	160	no	—
100	C₂×C₄	C₂³×D₈₉	1	60	yes	—
101	C₂×C₄	C₂³×D₈₉	1	10	yes	—
102	C₂²	—	4	5	no	—
103	C₂²	—	89	16	no	—
104	C₂²	—	356	40	no	—
105	C₂²	C₂⁴×D₈₉	1	10	yes	—
106	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
107	C₄	—	89	16	no	—
108	C₄	C₂⁴×D₈₉	1	20	yes	—
109	C₂	—	2	1	no	—
110	C₂	—	2	10	no	—
111	C₂	—	178	16	no	—
112	C₂	C₂⁵×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152435

C₂

yes

maximal, complemented

#19152434

C₂

yes

maximal, complemented

C₂×#15668060

C₂

yes

maximal, complemented

2₋¹⁺⁴×D₈₉

C₂

yes

maximal, complemented

C₂×2₋¹⁺⁴×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668060

C₂²

yes

complemented

#15668060

C₂²

yes

complemented

#15668059

C₂²

yes

complemented

#15668058

C₂²

yes

complemented

#15668057

C₂²

yes

complemented

C₂×#13083575

C₂²

yes

complemented

C₂×#13083574

C₂²

yes

complemented

#41×D₈₉

C₂²

160

yes

complemented

C₂×Q₈×D₈₉

C₂²

yes

complemented

2₋¹⁺⁴×C₈₉

C₂²

yes

complemented

C₂×#41×C₈₉

C₂²

yes

complemented

C₂²×Q₈×C₈₉

C₂²

yes

complemented

#13083575

C₂³

160

yes

—

#13083574

C₂³

240

yes

—

#13083574

C₂³

yes

—

#13083573

C₂³

160

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

Q₈×D₈₉

C₂³

160

yes

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

C₂³

yes

—

D₄×D₈₉

C₂³

160

yes

—

C₂×C₄×D₈₉

C₂³

120

yes

—

#41×C₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂⁴

yes

—

C₂²×D₈₉

C₂⁴

yes

—

Q₈×C₈₉

C₂⁴

yes

—

D₄×C₈₉

C₂⁴

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

—

C₂²⋊D₈₉

C₂⁴

yes

—

C₂²⋊D₈₉

C₂⁴

yes

—

C₂×Dic₈₉

C₂⁴

yes

—

D₃₅₆

C₂⁴

yes

—

C₄×D₈₉

C₂⁴

160

yes

—

Dic₁₇₈

C₂⁴

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂⁵

yes

—

C₂²×C₈₉

C₂⁵

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

C₂⁵

yes

—

C₄×C₈₉

C₂⁵

yes

—

Dic₈₉

C₂⁵

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂⁶

yes

characteristic

D₈₉

—

2₋¹⁺⁶

—

maximal, Sylow

C₈₉

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

#585

—

#585

—

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

D₈₉

yes

characteristic, complemented

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₈₉

yes

complemented

H(5,𝔽₂)

—

H(5,𝔽₂)

—

C₂×#41

—

160

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₈₉

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

complemented

#41

—

160

—

#41

—

160

—

#41

—

240

—

#41

—

#41

C₂²×D₈₉

yes

—

C₂×Q₈

—

160

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₈₉

yes

—

C₂×Q₈

C₂²×D₈₉

yes

—

C₂×D₄

—

C₂×D₄

—

160

—

C₂×D₄

—

C₂×D₄

C₂²×D₈₉

yes

—

C₂²×C₄

—

120

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂³

—

C₂³

C₂³×D₈₉

yes

—

Q₈

—

Q₈

C₂³×D₈₉

yes

—

D₄

—

D₄

—

D₄

—

D₄

C₂³×D₈₉

yes

—

C₂×C₄

—

C₂×C₄

—

160

—

100

C₂×C₄

C₂³×D₈₉

yes

—

101

C₂×C₄

C₂³×D₈₉

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

356

—

105

C₂²

C₂⁴×D₈₉

yes

—

106

C₂²

C₂⁴×D₈₉

yes

characteristic

107

C₄

—

108

C₄

C₂⁴×D₈₉

yes

—

109

C₂

—

110

C₂

—

111

C₂

—

178

—

112

C₂

C₂⁵×D₈₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₈₉
commutator, socle, cosocle	C₂×C₈₉	C₂⁶
Fitting	C₂×2₋¹⁺⁴×C₈₉	C₂
hypercenter	C₂×2₋¹⁺⁴	D₈₉
hypocenter	C₈₉	2₋¹⁺⁶
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×2₋¹⁺⁴

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₈₉

commutator, socle, cosocle

C₂×C₈₉

C₂⁶

Fitting

C₂×2₋¹⁺⁴×C₈₉

C₂

hypercenter

C₂×2₋¹⁺⁴

D₈₉

hypocenter

C₈₉

2₋¹⁺⁶

LCS

G ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×2₋¹⁺⁴

N	H ≅ G/N	type
1. #19152435	C₂	semidirect
2. #19152434	C₂	semidirect
3. C₂×#15668060	C₂	semidirect
4. 2₋¹⁺⁴×D₈₉	C₂	semidirect
5. C₂×2₋¹⁺⁴×C₈₉	C₂	semidirect
6. #15668060	C₂²	semidirect
7. #15668060	C₂²	semidirect
8. #15668059	C₂²	semidirect
9. #15668058	C₂²	semidirect
10. #15668057	C₂²	semidirect
11. C₂×#13083575	C₂²	semidirect
12. C₂×#13083574	C₂²	semidirect
13. #41×D₈₉	C₂²	semidirect
14. C₂×Q₈×D₈₉	C₂²	semidirect
15. 2₋¹⁺⁴×C₈₉	C₂²	semidirect
16. C₂×#41×C₈₉	C₂²	semidirect
17. C₂²×Q₈×C₈₉	C₂²	semidirect
57. C₈₉	2₋¹⁺⁶	semidirect
62. C₂×2₋¹⁺⁴	D₈₉	semidirect
66. 2₋¹⁺⁴	C₂×D₈₉	semidirect
72. C₂×#41	C₂×D₈₉	semidirect
74. C₂²×Q₈	C₂×D₈₉	semidirect

H ≅ G/N

type

1. #19152435

C₂

semidirect

2. #19152434

C₂

semidirect

3. C₂×#15668060

C₂

semidirect

4. 2₋¹⁺⁴×D₈₉

C₂

semidirect

5. C₂×2₋¹⁺⁴×C₈₉

C₂

semidirect

6. #15668060

C₂²

semidirect

7. #15668060

C₂²

semidirect

8. #15668059

C₂²

semidirect

9. #15668058

C₂²

semidirect

10. #15668057

C₂²

semidirect

11. C₂×#13083575

C₂²

semidirect

12. C₂×#13083574

C₂²

semidirect

13. #41×D₈₉

C₂²

semidirect

14. C₂×Q₈×D₈₉

C₂²

semidirect

15. 2₋¹⁺⁴×C₈₉

C₂²

semidirect

16. C₂×#41×C₈₉

C₂²

semidirect

17. C₂²×Q₈×C₈₉

C₂²

semidirect

57. C₈₉

2₋¹⁺⁶

semidirect

62. C₂×2₋¹⁺⁴

D₈₉

semidirect

66. 2₋¹⁺⁴

C₂×D₈₉

semidirect

72. C₂×#41

C₂×D₈₉

semidirect

74. C₂²×Q₈

C₂×D₈₉

semidirect

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

subgroup	index
#19152435	2
#19152434	2
C₂×#15668060	2
2₋¹⁺⁴×D₈₉	2
C₂×2₋¹⁺⁴×C₈₉	2
2₋¹⁺⁶	89

subgroup

index

#19152435

#19152434

C₂×#15668060

2₋¹⁺⁴×D₈₉

C₂×2₋¹⁺⁴×C₈₉

2₋¹⁺⁶

Subgroups — #26035064

Conjugacy classes of subgroups

Complements

Maximal subgroups