Group #21438907 — subgroups

Subgroups — #21438907

Order 7552.

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 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	#16838239	C₂	1	12	yes	maximal, complemented
2	#16838238	C₂	1	15	yes	maximal, complemented
3	C₂×#14273773	C₂	1	15	yes	maximal, complemented
4	H(5,𝔽₂)×D₅₉	C₂	1	20	yes	maximal, complemented
5	C₅₉×#585	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273775	C₂²	1	30	yes	complemented
7	#14273774	C₂²	1	20	yes	complemented
8	#14273773	C₂²	1	120	yes	complemented
9	#14273773	C₂²	1	60	yes	complemented
10	#14273772	C₂²	1	90	yes	complemented
11	C₂×#11745689	C₂²	1	30	yes	complemented
12	C₂×#11745685	C₂²	1	45	yes	complemented
13	#41×D₅₉	C₂²	1	120	yes	complemented
14	C₂²×D₂₃₆	C₂²	1	15	yes	complemented
15	C₂×D₄×D₅₉	C₂²	1	90	yes	complemented
16	2₋¹⁺⁴×C₅₉	C₂²	1	6	yes	complemented
17	H(5,𝔽₂)×C₅₉	C₂²	1	10	yes	complemented
18	C₂×#41×C₅₉	C₂²	1	15	yes	complemented
19	C₂³×D₅₉	C₂³	1	30	yes	complemented
20	#41×C₅₉	C₂³	1	20	yes	complemented
21	#41×C₅₉	C₂³	1	60	yes	complemented
22	C₂×Q₈×C₅₉	C₂³	1	15	yes	complemented
23	C₂×D₄×C₅₉	C₂³	1	45	yes	complemented
24	C₂²×C₄×C₅₉	C₂³	1	15	yes	complemented
25	C₂×C₂²⋊D₅₉	C₂³	1	90	yes	complemented
26	#11745689	C₂³	1	120	yes	complemented
27	Q₈×D₅₉	C₂³	1	40	yes	complemented
28	#11745687	C₂³	1	120	yes	complemented
29	D₄×D₅₉	C₂³	1	360	yes	complemented
30	#11745685	C₂³	1	180	yes	complemented
31	#11745685	C₂³	1	60	yes	complemented
32	C₂×D₂₃₆	C₂³	1	90	yes	complemented
33	C₂×D₂₃₆	C₂³	1	45	yes	complemented
34	C₂×C₄×D₅₉	C₂³	1	90	yes	complemented
35	C₂×Dic₁₁₈	C₂³	1	15	yes	complemented
36	C₂³×C₅₉	C₂⁴	1	15	yes	—
37	C₂²×D₅₉	—	8	30	no	—
38	C₂²×D₅₉	C₂⁴	1	90	yes	—
39	Q₈×C₅₉	C₂⁴	1	20	yes	—
40	D₄×C₅₉	C₂⁴	1	60	yes	—
41	C₂×C₄×C₅₉	C₂⁴	1	15	yes	—
42	C₂×C₄×C₅₉	C₂⁴	1	45	yes	—
43	C₂²⋊D₅₉	C₂⁴	1	120	yes	—
44	C₂×Dic₅₉	C₂⁴	1	30	yes	—
45	D₂₃₆	C₂⁴	1	10	yes	—
46	D₂₃₆	C₂⁴	1	90	yes	—
47	C₄×D₅₉	C₂⁴	1	120	yes	—
48	Dic₁₁₈	C₂⁴	1	6	yes	—
49	Dic₁₁₈	C₂⁴	1	30	yes	—
50	C₂²×C₅₉	—	4	15	no	—
51	C₂²×C₅₉	C₂⁵	1	15	yes	—
52	C₂×D₅₉	—	4	90	no	—
53	C₂×D₅₉	C₂⁵	1	20	yes	—
54	C₄×C₅₉	C₂⁵	1	1	yes	characteristic
55	C₄×C₅₉	C₂⁵	1	15	yes	—
56	Dic₅₉	C₂⁵	1	12	yes	—
57	H(7,𝔽₂)	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	2	15	no	—
59	C₂×C₅₉	C₂⁶	1	1	yes	characteristic
60	D₅₉	—	2	20	no	—
61	#585	—	59	12	no	—
62	#585	—	59	15	no	—
63	#585	D₅₉	1	1	yes	characteristic, complemented
64	C₂×H(5,𝔽₂)	—	59	20	no	—
65	C₂×H(5,𝔽₂)	—	59	15	no	—
66	C₅₉	H(7,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
67	2₋¹⁺⁴	—	59	20	no	—
68	2₋¹⁺⁴	—	59	30	no	—
69	2₋¹⁺⁴	C₂×D₅₉	1	6	yes	complemented
70	H(5,𝔽₂)	—	59	120	no	—
71	H(5,𝔽₂)	—	59	60	no	—
72	H(5,𝔽₂)	—	59	90	no	—
73	H(5,𝔽₂)	C₂×D₅₉	1	10	yes	complemented
74	C₂×#41	—	59	120	no	—
75	C₂×#41	—	59	30	no	—
76	C₂×#41	—	59	45	no	—
77	C₂×#41	C₂×D₅₉	1	15	yes	complemented
78	C₂²×D₄	—	59	90	no	—
79	C₂²×D₄	—	59	15	no	—
80	C₂⁴	—	59	30	no	—
81	#41	—	59	120	no	—
82	#41	—	59	120	no	—
83	#41	—	59	180	no	—
84	#41	—	59	60	no	—
85	#41	C₂²×D₅₉	1	60	yes	complemented
86	#41	C₂²×D₅₉	1	20	yes	complemented
87	C₂×Q₈	—	59	40	no	—
88	C₂×Q₈	—	59	15	no	—
89	C₂×Q₈	C₂²×D₅₉	1	15	yes	complemented
90	C₂×D₄	—	59	90	no	—
91	C₂×D₄	—	59	360	no	—
92	C₂×D₄	—	59	90	no	—
93	C₂×D₄	—	59	45	no	—
94	C₂×D₄	C₂²×D₅₉	1	45	yes	complemented
95	C₂²×C₄	—	59	90	no	—
96	C₂²×C₄	C₂²×D₅₉	1	15	yes	complemented
97	C₂³	—	59	90	no	—
98	C₂³	—	472	30	no	—
99	C₂³	C₂³×D₅₉	1	15	yes	—
100	Q₈	—	59	30	no	—
101	Q₈	—	59	6	no	—
102	Q₈	C₂³×D₅₉	1	20	yes	—
103	D₄	—	59	120	no	—
104	D₄	—	59	90	no	—
105	D₄	—	59	10	no	—
106	D₄	C₂³×D₅₉	1	60	yes	—
107	C₂×C₄	—	59	30	no	—
108	C₂×C₄	—	59	120	no	—
109	C₂×C₄	C₂³×D₅₉	1	45	yes	—
110	C₂×C₄	C₂³×D₅₉	1	15	yes	—
111	C₂²	—	4	15	no	—
112	C₂²	—	59	20	no	—
113	C₂²	—	236	90	no	—
114	C₂²	C₂⁴×D₅₉	1	15	yes	—
115	C₄	—	59	12	no	—
116	C₄	C₂⁴×D₅₉	1	15	yes	—
117	C₄	C₂⁴×D₅₉	1	1	yes	characteristic
118	C₂	—	2	15	no	—
119	C₂	—	118	20	no	—
120	C₂	C₂⁵×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838239

C₂

yes

maximal, complemented

#16838238

C₂

yes

maximal, complemented

C₂×#14273773

C₂

yes

maximal, complemented

H(5,𝔽₂)×D₅₉

C₂

yes

maximal, complemented

C₅₉×#585

C₂

yes

maximal, characteristic, complemented

#14273775

C₂²

yes

complemented

#14273774

C₂²

yes

complemented

#14273773

C₂²

120

yes

complemented

#14273773

C₂²

yes

complemented

#14273772

C₂²

yes

complemented

C₂×#11745689

C₂²

yes

complemented

C₂×#11745685

C₂²

yes

complemented

#41×D₅₉

C₂²

120

yes

complemented

C₂²×D₂₃₆

C₂²

yes

complemented

C₂×D₄×D₅₉

C₂²

yes

complemented

2₋¹⁺⁴×C₅₉

C₂²

yes

complemented

H(5,𝔽₂)×C₅₉

C₂²

yes

complemented

C₂×#41×C₅₉

C₂²

yes

complemented

C₂³×D₅₉

C₂³

yes

complemented

#41×C₅₉

C₂³

yes

complemented

#41×C₅₉

C₂³

yes

complemented

C₂×Q₈×C₅₉

C₂³

yes

complemented

C₂×D₄×C₅₉

C₂³

yes

complemented

C₂²×C₄×C₅₉

C₂³

yes

complemented

C₂×C₂²⋊D₅₉

C₂³

yes

complemented

#11745689

C₂³

120

yes

complemented

Q₈×D₅₉

C₂³

yes

complemented

#11745687

C₂³

120

yes

complemented

D₄×D₅₉

C₂³

360

yes

complemented

#11745685

C₂³

180

yes

complemented

#11745685

C₂³

yes

complemented

C₂×D₂₃₆

C₂³

yes

complemented

C₂×D₂₃₆

C₂³

yes

complemented

C₂×C₄×D₅₉

C₂³

yes

complemented

C₂×Dic₁₁₈

C₂³

yes

complemented

C₂³×C₅₉

C₂⁴

yes

—

C₂²×D₅₉

—

C₂²×D₅₉

C₂⁴

yes

—

Q₈×C₅₉

C₂⁴

yes

—

D₄×C₅₉

C₂⁴

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

—

C₂²⋊D₅₉

C₂⁴

120

yes

—

C₂×Dic₅₉

C₂⁴

yes

—

D₂₃₆

C₂⁴

yes

—

D₂₃₆

C₂⁴

yes

—

C₄×D₅₉

C₂⁴

120

yes

—

Dic₁₁₈

C₂⁴

yes

—

Dic₁₁₈

C₂⁴

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂⁵

yes

—

C₂×D₅₉

—

C₂×D₅₉

C₂⁵

yes

—

C₄×C₅₉

C₂⁵

yes

characteristic

C₄×C₅₉

C₂⁵

yes

—

Dic₅₉

C₂⁵

yes

—

H(7,𝔽₂)

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂⁶

yes

characteristic

D₅₉

—

#585

—

#585

—

#585

D₅₉

yes

characteristic, complemented

C₂×H(5,𝔽₂)

—

C₂×H(5,𝔽₂)

—

C₅₉

H(7,𝔽₂)

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₅₉

yes

complemented

H(5,𝔽₂)

—

120

—

H(5,𝔽₂)

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₅₉

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₅₉

yes

complemented

C₂²×D₄

—

C₂²×D₄

—

C₂⁴

—

#41

—

120

—

#41

—

120

—

#41

—

180

—

#41

—

#41

C₂²×D₅₉

yes

complemented

#41

C₂²×D₅₉

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₅₉

yes

complemented

C₂×D₄

—

C₂×D₄

—

360

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₅₉

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

complemented

C₂³

—

C₂³

—

472

—

C₂³

C₂³×D₅₉

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

C₂³×D₅₉

yes

—

103

D₄

—

120

—

104

D₄

—

105

D₄

—

106

D₄

C₂³×D₅₉

yes

—

107

C₂×C₄

—

108

C₂×C₄

—

120

—

109

C₂×C₄

C₂³×D₅₉

yes

—

110

C₂×C₄

C₂³×D₅₉

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

236

—

114

C₂²

C₂⁴×D₅₉

yes

—

115

C₄

—

116

C₄

C₂⁴×D₅₉

yes

—

117

C₄

C₂⁴×D₅₉

yes

characteristic

118

C₂

—

119

C₂

—

118

—

120

C₂

C₂⁵×D₅₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₅₉
commutator, socle	C₂×C₅₉	C₂⁶
Fitting	C₅₉×#585	C₂
hypercenter	#585	D₅₉
hypocenter	C₅₉	H(7,𝔽₂)
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₅₉

commutator, socle

C₂×C₅₉

C₂⁶

Fitting

C₅₉×#585

C₂

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. #16838239	C₂	semidirect
2. #16838238	C₂	semidirect
3. C₂×#14273773	C₂	semidirect
4. H(5,𝔽₂)×D₅₉	C₂	semidirect
5. C₅₉×#585	C₂	semidirect
6. #14273775	C₂²	semidirect
7. #14273774	C₂²	semidirect
8. #14273773	C₂²	semidirect
9. #14273773	C₂²	semidirect
10. #14273772	C₂²	semidirect
11. C₂×#11745689	C₂²	semidirect
12. C₂×#11745685	C₂²	semidirect
13. #41×D₅₉	C₂²	semidirect
14. C₂²×D₂₃₆	C₂²	semidirect
15. C₂×D₄×D₅₉	C₂²	semidirect
16. 2₋¹⁺⁴×C₅₉	C₂²	semidirect
17. H(5,𝔽₂)×C₅₉	C₂²	semidirect
18. C₂×#41×C₅₉	C₂²	semidirect
19. C₂³×D₅₉	C₂³	semidirect
20. #41×C₅₉	C₂³	semidirect
21. #41×C₅₉	C₂³	semidirect
22. C₂×Q₈×C₅₉	C₂³	semidirect
23. C₂×D₄×C₅₉	C₂³	semidirect
24. C₂²×C₄×C₅₉	C₂³	semidirect
25. C₂×C₂²⋊D₅₉	C₂³	semidirect
26. #11745689	C₂³	semidirect
27. Q₈×D₅₉	C₂³	semidirect
28. #11745687	C₂³	semidirect
29. D₄×D₅₉	C₂³	semidirect
30. #11745685	C₂³	semidirect
31. #11745685	C₂³	semidirect
32. C₂×D₂₃₆	C₂³	semidirect
33. C₂×D₂₃₆	C₂³	semidirect
34. C₂×C₄×D₅₉	C₂³	semidirect
35. C₂×Dic₁₁₈	C₂³	semidirect
63. #585	D₅₉	semidirect
66. C₅₉	H(7,𝔽₂)	semidirect
69. 2₋¹⁺⁴	C₂×D₅₉	semidirect
73. H(5,𝔽₂)	C₂×D₅₉	semidirect
77. C₂×#41	C₂×D₅₉	semidirect
85. #41	C₂²×D₅₉	semidirect
86. #41	C₂²×D₅₉	semidirect
89. C₂×Q₈	C₂²×D₅₉	semidirect
94. C₂×D₄	C₂²×D₅₉	semidirect
96. C₂²×C₄	C₂²×D₅₉	semidirect

H ≅ G/N

type

1. #16838239

C₂

semidirect

2. #16838238

C₂

semidirect

3. C₂×#14273773

C₂

semidirect

4. H(5,𝔽₂)×D₅₉

C₂

semidirect

5. C₅₉×#585

C₂

semidirect

6. #14273775

C₂²

semidirect

7. #14273774

C₂²

semidirect

8. #14273773

C₂²

semidirect

9. #14273773

C₂²

semidirect

10. #14273772

C₂²

semidirect

11. C₂×#11745689

C₂²

semidirect

12. C₂×#11745685

C₂²

semidirect

13. #41×D₅₉

C₂²

semidirect

14. C₂²×D₂₃₆

C₂²

semidirect

15. C₂×D₄×D₅₉

C₂²

semidirect

16. 2₋¹⁺⁴×C₅₉

C₂²

semidirect

17. H(5,𝔽₂)×C₅₉

C₂²

semidirect

18. C₂×#41×C₅₉

C₂²

semidirect

19. C₂³×D₅₉

C₂³

semidirect

20. #41×C₅₉

C₂³

semidirect

21. #41×C₅₉

C₂³

semidirect

22. C₂×Q₈×C₅₉

C₂³

semidirect

23. C₂×D₄×C₅₉

C₂³

semidirect

24. C₂²×C₄×C₅₉

C₂³

semidirect

25. C₂×C₂²⋊D₅₉

C₂³

semidirect

26. #11745689

C₂³

semidirect

27. Q₈×D₅₉

C₂³

semidirect

28. #11745687

C₂³

semidirect

29. D₄×D₅₉

C₂³

semidirect

30. #11745685

C₂³

semidirect

31. #11745685

C₂³

semidirect

32. C₂×D₂₃₆

C₂³

semidirect

33. C₂×D₂₃₆

C₂³

semidirect

34. C₂×C₄×D₅₉

C₂³

semidirect

35. C₂×Dic₁₁₈

C₂³

semidirect

63. #585

D₅₉

semidirect

66. C₅₉

H(7,𝔽₂)

semidirect

69. 2₋¹⁺⁴

C₂×D₅₉

semidirect

73. H(5,𝔽₂)

C₂×D₅₉

semidirect

77. C₂×#41

C₂×D₅₉

semidirect

85. #41

C₂²×D₅₉

semidirect

86. #41

C₂²×D₅₉

semidirect

89. C₂×Q₈

C₂²×D₅₉

semidirect

94. C₂×D₄

C₂²×D₅₉

semidirect

96. C₂²×C₄

C₂²×D₅₉

semidirect

Subgroups — #21438907

Conjugacy classes of subgroups

Complements