Group #26035065 — subgroups

Subgroups — #26035065

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	#19152434	C₂	1	15	yes	maximal, complemented
2	#19152433	C₂	1	20	yes	maximal, complemented
3	C₂×#15668059	C₂	1	15	yes	maximal, complemented
4	2₋¹⁺⁴×D₈₉	C₂	1	12	yes	maximal, complemented
5	#585×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668060	C₂²	1	30	yes	complemented
7	#15668059	C₂²	1	120	yes	complemented
8	#15668059	C₂²	1	60	yes	complemented
9	#15668058	C₂²	1	20	yes	complemented
10	#15668057	C₂²	1	90	yes	complemented
11	C₂×#13083574	C₂²	1	45	yes	complemented
12	C₂×#13083573	C₂²	1	90	yes	complemented
13	#41×D₈₉	C₂²	1	120	yes	complemented
14	C₂²×Dic₁₇₈	C₂²	1	15	yes	complemented
15	C₂×Q₈×D₈₉	C₂²	1	30	yes	complemented
16	2₋¹⁺⁴×C₈₉	C₂²	1	6	yes	complemented
17	H(5,𝔽₂)×C₈₉	C₂²	1	10	yes	complemented
18	C₂×#41×C₈₉	C₂²	1	15	yes	complemented
19	#13083575	C₂³	1	40	yes	—
20	#13083574	C₂³	1	180	yes	—
21	#13083574	C₂³	1	60	yes	—
22	#13083573	C₂³	1	360	yes	—
23	C₂×Dic₁₇₈	C₂³	1	90	yes	—
24	C₂×Dic₁₇₈	C₂³	1	45	yes	—
25	Q₈×D₈₉	C₂³	1	120	yes	—
26	C₂×D₃₅₆	C₂³	1	15	yes	—
27	C₂×C₂²⋊D₈₉	C₂³	1	90	yes	—
28	D₄×D₈₉	C₂³	1	120	yes	—
29	C₂²×Dic₈₉	C₂³	1	30	yes	—
30	C₂×C₄×D₈₉	C₂³	1	90	yes	—
31	#41×C₈₉	C₂³	1	60	yes	—
32	#41×C₈₉	C₂³	1	20	yes	—
33	C₂×Q₈×C₈₉	C₂³	1	15	yes	—
34	C₂×D₄×C₈₉	C₂³	1	45	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	15	yes	—
36	C₂³×C₈₉	C₂⁴	1	15	yes	—
37	C₂²×D₈₉	C₂⁴	1	30	yes	—
38	Q₈×C₈₉	C₂⁴	1	20	yes	—
39	D₄×C₈₉	C₂⁴	1	60	yes	—
40	C₂×C₄×C₈₉	C₂⁴	1	15	yes	—
41	C₂×C₄×C₈₉	C₂⁴	1	45	yes	—
42	C₂²⋊D₈₉	C₂⁴	1	120	yes	—
43	C₂×Dic₈₉	C₂⁴	1	90	yes	—
44	D₃₅₆	C₂⁴	1	30	yes	—
45	D₃₅₆	C₂⁴	1	6	yes	—
46	C₄×D₈₉	C₂⁴	1	120	yes	—
47	Dic₁₇₈	C₂⁴	1	90	yes	—
48	Dic₁₇₈	C₂⁴	1	10	yes	—
49	C₂²×C₈₉	—	4	15	no	—
50	C₂²×C₈₉	C₂⁵	1	15	yes	—
51	C₂×D₈₉	—	4	30	no	—
52	C₂×D₈₉	C₂⁵	1	12	yes	—
53	C₄×C₈₉	C₂⁵	1	1	yes	characteristic
54	C₄×C₈₉	C₂⁵	1	15	yes	—
55	Dic₈₉	C₂⁵	1	20	yes	—
56	C₂×C₈₉	—	2	15	no	—
57	C₂×C₈₉	C₂⁶	1	1	yes	characteristic
58	D₈₉	—	2	12	no	—
59	2₋¹⁺⁶	—	89	1	no	maximal, Sylow
60	C₈₉	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
61	#585	—	89	20	no	—
62	#585	—	89	15	no	—
63	#585	D₈₉	1	1	yes	characteristic, complemented
64	C₂×2₋¹⁺⁴	—	89	12	no	—
65	C₂×2₋¹⁺⁴	—	89	15	no	—
66	2₋¹⁺⁴	—	89	120	no	—
67	2₋¹⁺⁴	—	89	60	no	—
68	2₋¹⁺⁴	—	89	30	no	—
69	2₋¹⁺⁴	C₂×D₈₉	1	6	yes	complemented
70	H(5,𝔽₂)	—	89	20	no	—
71	H(5,𝔽₂)	—	89	90	no	—
72	H(5,𝔽₂)	C₂×D₈₉	1	10	yes	complemented
73	C₂×#41	—	89	120	no	—
74	C₂×#41	—	89	90	no	—
75	C₂×#41	—	89	45	no	—
76	C₂×#41	C₂×D₈₉	1	15	yes	complemented
77	C₂²×Q₈	—	89	30	no	—
78	C₂²×Q₈	—	89	15	no	—
79	#41	—	89	40	no	—
80	#41	—	89	360	no	—
81	#41	—	89	180	no	—
82	#41	—	89	60	no	—
83	#41	C₂²×D₈₉	1	60	yes	—
84	#41	C₂²×D₈₉	1	20	yes	—
85	C₂×Q₈	—	89	120	no	—
86	C₂×Q₈	—	89	90	no	—
87	C₂×Q₈	—	89	45	no	—
88	C₂×Q₈	C₂²×D₈₉	1	15	yes	—
89	C₂×D₄	—	89	90	no	—
90	C₂×D₄	—	89	120	no	—
91	C₂×D₄	—	89	15	no	—
92	C₂×D₄	C₂²×D₈₉	1	45	yes	—
93	C₂²×C₄	—	89	30	no	—
94	C₂²×C₄	—	89	90	no	—
95	C₂²×C₄	C₂²×D₈₉	1	15	yes	—
96	C₂³	—	89	30	no	—
97	C₂³	C₂³×D₈₉	1	15	yes	—
98	Q₈	—	89	90	no	—
99	Q₈	—	89	10	no	—
100	Q₈	C₂³×D₈₉	1	20	yes	—
101	D₄	—	89	120	no	—
102	D₄	—	89	30	no	—
103	D₄	—	89	6	no	—
104	D₄	C₂³×D₈₉	1	60	yes	—
105	C₂×C₄	—	89	90	no	—
106	C₂×C₄	—	89	120	no	—
107	C₂×C₄	C₂³×D₈₉	1	45	yes	—
108	C₂×C₄	C₂³×D₈₉	1	15	yes	—
109	C₂²	—	4	15	no	—
110	C₂²	—	89	12	no	—
111	C₂²	—	356	30	no	—
112	C₂²	C₂⁴×D₈₉	1	15	yes	—
113	C₄	—	89	20	no	—
114	C₄	C₂⁴×D₈₉	1	15	yes	—
115	C₄	C₂⁴×D₈₉	1	1	yes	characteristic
116	C₂	—	2	15	no	—
117	C₂	—	178	12	no	—
118	C₂	C₂⁵×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152434

C₂

yes

maximal, complemented

#19152433

C₂

yes

maximal, complemented

C₂×#15668059

C₂

yes

maximal, complemented

2₋¹⁺⁴×D₈₉

C₂

yes

maximal, complemented

#585×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668060

C₂²

yes

complemented

#15668059

C₂²

120

yes

complemented

#15668059

C₂²

yes

complemented

#15668058

C₂²

yes

complemented

#15668057

C₂²

yes

complemented

C₂×#13083574

C₂²

yes

complemented

C₂×#13083573

C₂²

yes

complemented

#41×D₈₉

C₂²

120

yes

complemented

C₂²×Dic₁₇₈

C₂²

yes

complemented

C₂×Q₈×D₈₉

C₂²

yes

complemented

2₋¹⁺⁴×C₈₉

C₂²

yes

complemented

H(5,𝔽₂)×C₈₉

C₂²

yes

complemented

C₂×#41×C₈₉

C₂²

yes

complemented

#13083575

C₂³

yes

—

#13083574

C₂³

180

yes

—

#13083574

C₂³

yes

—

#13083573

C₂³

360

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

Q₈×D₈₉

C₂³

120

yes

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

C₂³

yes

—

D₄×D₈₉

C₂³

120

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

#41×C₈₉

C₂³

yes

—

#41×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

120

yes

—

C₂×Dic₈₉

C₂⁴

yes

—

D₃₅₆

C₂⁴

yes

—

D₃₅₆

C₂⁴

yes

—

C₄×D₈₉

C₂⁴

120

yes

—

Dic₁₇₈

C₂⁴

yes

—

Dic₁₇₈

C₂⁴

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂⁵

yes

—

C₂×D₈₉

—

C₂×D₈₉

C₂⁵

yes

—

C₄×C₈₉

C₂⁵

yes

characteristic

C₄×C₈₉

C₂⁵

yes

—

Dic₈₉

C₂⁵

yes

—

C₂×C₈₉

—

C₂×C₈₉

C₂⁶

yes

characteristic

D₈₉

—

2₋¹⁺⁶

—

maximal, Sylow

C₈₉

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

#585

—

#585

—

#585

D₈₉

yes

characteristic, complemented

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

2₋¹⁺⁴

—

120

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₈₉

yes

—

—

C₂×D₈₉

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₈₉

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

#41

—

#41

—

360

—

#41

—

180

—

#41

—

#41

C₂²×D₈₉

yes

—

#41

C₂²×D₈₉

yes

—

C₂×Q₈

—

120

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₈₉

yes

—

C₂×D₄

—

C₂×D₄

—

120

—

C₂×D₄

—

C₂×D₄

C₂²×D₈₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂³

—

C₂³

C₂³×D₈₉

yes

—

Q₈

—

Q₈

—

100

Q₈

C₂³×D₈₉

yes

—

101

D₄

—

120

—

102

D₄

—

103

D₄

—

104

D₄

C₂³×D₈₉

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

120

—

107

C₂×C₄

C₂³×D₈₉

yes

—

108

C₂×C₄

C₂³×D₈₉

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

356

—

112

C₂²

C₂⁴×D₈₉

yes

—

113

C₄

—

114

C₄

C₂⁴×D₈₉

yes

—

115

C₄

C₂⁴×D₈₉

yes

characteristic

116

C₂

—

117

C₂

—

178

—

118

C₂

C₂⁵×D₈₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₈₉
commutator, socle, cosocle	C₂×C₈₉	C₂⁶
Fitting	#585×C₈₉	C₂
hypercenter	#585	D₈₉
hypocenter	C₈₉	2₋¹⁺⁶
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₈₉

commutator, socle, cosocle

C₂×C₈₉

C₂⁶

Fitting

#585×C₈₉

C₂

hypercenter

#585

D₈₉

hypocenter

C₈₉

2₋¹⁺⁶

LCS

G ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. #19152434	C₂	semidirect
2. #19152433	C₂	semidirect
3. C₂×#15668059	C₂	semidirect
4. 2₋¹⁺⁴×D₈₉	C₂	semidirect
5. #585×C₈₉	C₂	semidirect
6. #15668060	C₂²	semidirect
7. #15668059	C₂²	semidirect
8. #15668059	C₂²	semidirect
9. #15668058	C₂²	semidirect
10. #15668057	C₂²	semidirect
11. C₂×#13083574	C₂²	semidirect
12. C₂×#13083573	C₂²	semidirect
13. #41×D₈₉	C₂²	semidirect
14. C₂²×Dic₁₇₈	C₂²	semidirect
15. C₂×Q₈×D₈₉	C₂²	semidirect
16. 2₋¹⁺⁴×C₈₉	C₂²	semidirect
17. H(5,𝔽₂)×C₈₉	C₂²	semidirect
18. C₂×#41×C₈₉	C₂²	semidirect
60. C₈₉	2₋¹⁺⁶	semidirect
63. #585	D₈₉	semidirect
69. 2₋¹⁺⁴	C₂×D₈₉	semidirect
72. H(5,𝔽₂)	C₂×D₈₉	semidirect
76. C₂×#41	C₂×D₈₉	semidirect

H ≅ G/N

type

1. #19152434

C₂

semidirect

2. #19152433

C₂

semidirect

3. C₂×#15668059

C₂

semidirect

4. 2₋¹⁺⁴×D₈₉

C₂

semidirect

5. #585×C₈₉

C₂

semidirect

6. #15668060

C₂²

semidirect

7. #15668059

C₂²

semidirect

8. #15668059

C₂²

semidirect

9. #15668058

C₂²

semidirect

10. #15668057

C₂²

semidirect

11. C₂×#13083574

C₂²

semidirect

12. C₂×#13083573

C₂²

semidirect

13. #41×D₈₉

C₂²

semidirect

14. C₂²×Dic₁₇₈

C₂²

semidirect

15. C₂×Q₈×D₈₉

C₂²

semidirect

16. 2₋¹⁺⁴×C₈₉

C₂²

semidirect

17. H(5,𝔽₂)×C₈₉

C₂²

semidirect

18. C₂×#41×C₈₉

C₂²

semidirect

60. C₈₉

2₋¹⁺⁶

semidirect

63. #585

D₈₉

semidirect

69. 2₋¹⁺⁴

C₂×D₈₉

semidirect

72. H(5,𝔽₂)

C₂×D₈₉

semidirect

76. C₂×#41

C₂×D₈₉

semidirect

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

subgroup	index
#19152434	2
#19152433	2
C₂×#15668059	2
2₋¹⁺⁴×D₈₉	2
#585×C₈₉	2
2₋¹⁺⁶	89

subgroup

index

#19152434

#19152433

C₂×#15668059

2₋¹⁺⁴×D₈₉

#585×C₈₉

2₋¹⁺⁶

Subgroups — #26035065

Conjugacy classes of subgroups

Complements

Maximal subgroups