Group #21438906 — subgroups

Subgroups — #21438906

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	#16838238	C₂	1	12	yes	maximal, complemented
2	#16838237	C₂	1	16	yes	maximal, complemented
3	C₂×#14273772	C₂	1	18	yes	maximal, complemented
4	H(5,𝔽₂)×D₅₉	C₂	1	16	yes	maximal, complemented
5	C₂×H(5,𝔽₂)×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273775	C₂²	1	8	yes	complemented
7	#14273774	C₂²	1	48	yes	complemented
8	#14273773	C₂²	1	48	yes	complemented
9	#14273772	C₂²	1	144	yes	complemented
10	#14273772	C₂²	1	72	yes	complemented
11	C₂×#11745685	C₂²	1	36	yes	complemented
12	C₂×#11745687	C₂²	1	72	yes	complemented
13	#41×D₅₉	C₂²	1	96	yes	complemented
14	C₂²×C₂²⋊D₅₉	C₂²	1	24	yes	complemented
15	C₂×D₄×D₅₉	C₂²	1	72	yes	complemented
16	H(5,𝔽₂)×C₅₉	C₂²	1	16	yes	complemented
17	C₂×#41×C₅₉	C₂²	1	6	yes	complemented
18	C₂²×D₄×C₅₉	C₂²	1	9	yes	complemented
19	C₂⁴×C₅₉	C₂³	1	6	yes	complemented
20	C₂³×D₅₉	C₂³	1	24	yes	complemented
21	#41×C₅₉	C₂³	1	48	yes	complemented
22	C₂×Q₈×C₅₉	C₂³	1	2	yes	complemented
23	C₂×D₄×C₅₉	C₂³	1	18	yes	complemented
24	C₂×D₄×C₅₉	C₂³	1	72	yes	complemented
25	C₂²×C₄×C₅₉	C₂³	1	9	yes	complemented
26	C₂×C₂²⋊D₅₉	C₂³	1	144	yes	complemented
27	C₂×C₂²⋊D₅₉	C₂³	1	72	yes	complemented
28	C₂²×Dic₅₉	C₂³	1	24	yes	complemented
29	#11745689	C₂³	1	32	yes	complemented
30	Q₈×D₅₉	C₂³	1	32	yes	complemented
31	#11745687	C₂³	1	288	yes	complemented
32	D₄×D₅₉	C₂³	1	288	yes	complemented
33	#11745685	C₂³	1	144	yes	complemented
34	#11745685	C₂³	1	48	yes	complemented
35	C₂×D₂₃₆	C₂³	1	36	yes	complemented
36	C₂×C₄×D₅₉	C₂³	1	72	yes	complemented
37	C₂×Dic₁₁₈	C₂³	1	36	yes	complemented
38	C₂³×C₅₉	—	8	6	no	—
39	C₂³×C₅₉	C₂⁴	1	24	yes	—
40	C₂³×C₅₉	C₂⁴	1	9	yes	—
41	C₂²×D₅₉	—	8	24	no	—
42	C₂²×D₅₉	C₂⁴	1	72	yes	—
43	Q₈×C₅₉	C₂⁴	1	8	yes	—
44	D₄×C₅₉	C₂⁴	1	72	yes	—
45	C₂×C₄×C₅₉	C₂⁴	1	36	yes	—
46	C₂×C₄×C₅₉	C₂⁴	1	6	yes	—
47	C₂²⋊D₅₉	C₂⁴	1	16	yes	—
48	C₂²⋊D₅₉	C₂⁴	1	144	yes	—
49	C₂×Dic₅₉	C₂⁴	1	72	yes	—
50	D₂₃₆	C₂⁴	1	48	yes	—
51	C₄×D₅₉	C₂⁴	1	96	yes	—
52	Dic₁₁₈	C₂⁴	1	48	yes	—
53	C₂²×C₅₉	—	4	24	no	—
54	C₂²×C₅₉	—	4	9	no	—
55	C₂²×C₅₉	C₂⁵	1	18	yes	—
56	C₂²×C₅₉	C₂⁵	1	1	yes	characteristic
57	C₂×D₅₉	—	4	72	no	—
58	C₂×D₅₉	C₂⁵	1	16	yes	—
59	C₄×C₅₉	C₂⁵	1	12	yes	—
60	Dic₅₉	C₂⁵	1	16	yes	—
61	H(7,𝔽₂)	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	18	no	—
63	C₂×C₅₉	—	2	1	no	—
64	C₂×C₅₉	C₂⁶	1	1	yes	characteristic
65	D₅₉	—	2	16	no	—
66	#585	—	59	16	no	—
67	#585	—	59	12	no	—
68	C₂×H(5,𝔽₂)	—	59	16	no	—
69	C₂×H(5,𝔽₂)	—	59	18	no	—
70	C₂×H(5,𝔽₂)	D₅₉	1	1	yes	characteristic, complemented
71	C₅₉	H(7,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
72	2₋¹⁺⁴	—	59	48	no	—
73	2₋¹⁺⁴	—	59	8	no	—
74	H(5,𝔽₂)	—	59	48	no	—
75	H(5,𝔽₂)	—	59	144	no	—
76	H(5,𝔽₂)	—	59	72	no	—
77	H(5,𝔽₂)	C₂×D₅₉	1	16	yes	complemented
78	C₂×#41	—	59	96	no	—
79	C₂×#41	—	59	72	no	—
80	C₂×#41	—	59	36	no	—
81	C₂×#41	C₂×D₅₉	1	6	yes	complemented
82	C₂²×D₄	—	59	24	no	—
83	C₂²×D₄	—	59	72	no	—
84	C₂²×D₄	C₂×D₅₉	1	9	yes	complemented
85	C₂⁴	—	59	24	no	—
86	C₂⁴	C₂²×D₅₉	1	6	yes	complemented
87	#41	—	59	32	no	—
88	#41	—	59	288	no	—
89	#41	—	59	144	no	—
90	#41	—	59	48	no	—
91	#41	C₂²×D₅₉	1	48	yes	complemented
92	C₂×Q₈	—	59	32	no	—
93	C₂×Q₈	—	59	36	no	—
94	C₂×Q₈	C₂²×D₅₉	1	2	yes	complemented
95	C₂×D₄	—	59	144	no	—
96	C₂×D₄	—	59	72	no	—
97	C₂×D₄	—	59	288	no	—
98	C₂×D₄	—	59	36	no	—
99	C₂×D₄	C₂²×D₅₉	1	72	yes	complemented
100	C₂×D₄	C₂²×D₅₉	1	18	yes	complemented
101	C₂²×C₄	—	59	24	no	—
102	C₂²×C₄	—	59	72	no	—
103	C₂²×C₄	C₂²×D₅₉	1	9	yes	complemented
104	C₂³	—	8	6	no	—
105	C₂³	—	59	72	no	—
106	C₂³	—	472	24	no	—
107	C₂³	C₂³×D₅₉	1	24	yes	—
108	C₂³	C₂³×D₅₉	1	9	yes	—
109	Q₈	—	59	48	no	—
110	Q₈	C₂³×D₅₉	1	8	yes	—
111	D₄	—	59	144	no	—
112	D₄	—	59	16	no	—
113	D₄	—	59	48	no	—
114	D₄	C₂³×D₅₉	1	72	yes	—
115	C₂×C₄	—	59	72	no	—
116	C₂×C₄	—	59	96	no	—
117	C₂×C₄	C₂³×D₅₉	1	36	yes	—
118	C₂×C₄	C₂³×D₅₉	1	6	yes	—
119	C₂²	—	4	9	no	—
120	C₂²	—	4	24	no	—
121	C₂²	—	59	16	no	—
122	C₂²	—	236	72	no	—
123	C₂²	C₂⁴×D₅₉	1	18	yes	—
124	C₂²	C₂⁴×D₅₉	1	1	yes	characteristic
125	C₄	—	59	16	no	—
126	C₄	C₂⁴×D₅₉	1	12	yes	—
127	C₂	—	2	1	no	—
128	C₂	—	2	18	no	—
129	C₂	—	118	16	no	—
130	C₂	C₂⁵×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838238

C₂

yes

maximal, complemented

#16838237

C₂

yes

maximal, complemented

C₂×#14273772

C₂

yes

maximal, complemented

H(5,𝔽₂)×D₅₉

C₂

yes

maximal, complemented

C₂×H(5,𝔽₂)×C₅₉

C₂

yes

maximal, characteristic, complemented

#14273775

C₂²

yes

complemented

#14273774

C₂²

yes

complemented

#14273773

C₂²

yes

complemented

#14273772

C₂²

144

yes

complemented

#14273772

C₂²

yes

complemented

C₂×#11745685

C₂²

yes

complemented

C₂×#11745687

C₂²

yes

complemented

#41×D₅₉

C₂²

yes

complemented

C₂²×C₂²⋊D₅₉

C₂²

yes

complemented

C₂×D₄×D₅₉

C₂²

yes

complemented

H(5,𝔽₂)×C₅₉

C₂²

yes

complemented

C₂×#41×C₅₉

C₂²

yes

complemented

C₂²×D₄×C₅₉

C₂²

yes

complemented

C₂⁴×C₅₉

C₂³

yes

complemented

C₂³×D₅₉

C₂³

yes

complemented

#41×C₅₉

C₂³

yes

complemented

C₂×Q₈×C₅₉

C₂³

yes

complemented

C₂×D₄×C₅₉

C₂³

yes

complemented

C₂×D₄×C₅₉

C₂³

yes

complemented

C₂²×C₄×C₅₉

C₂³

yes

complemented

C₂×C₂²⋊D₅₉

C₂³

144

yes

complemented

C₂×C₂²⋊D₅₉

C₂³

yes

complemented

C₂²×Dic₅₉

C₂³

yes

complemented

#11745689

C₂³

yes

complemented

Q₈×D₅₉

C₂³

yes

complemented

#11745687

C₂³

288

yes

complemented

D₄×D₅₉

C₂³

288

yes

complemented

#11745685

C₂³

144

yes

complemented

#11745685

C₂³

yes

complemented

C₂×D₂₃₆

C₂³

yes

complemented

C₂×C₄×D₅₉

C₂³

yes

complemented

C₂×Dic₁₁₈

C₂³

yes

complemented

C₂³×C₅₉

—

C₂³×C₅₉

C₂⁴

yes

—

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

yes

—

C₂²⋊D₅₉

C₂⁴

144

yes

—

C₂×Dic₅₉

C₂⁴

yes

—

D₂₃₆

C₂⁴

yes

—

C₄×D₅₉

C₂⁴

yes

—

Dic₁₁₈

C₂⁴

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂⁵

yes

—

C₂²×C₅₉

C₂⁵

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

C₂⁵

yes

—

C₄×C₅₉

C₂⁵

yes

—

Dic₅₉

C₂⁵

yes

—

H(7,𝔽₂)

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂⁶

yes

characteristic

D₅₉

—

#585

—

#585

—

C₂×H(5,𝔽₂)

—

C₂×H(5,𝔽₂)

—

C₂×H(5,𝔽₂)

D₅₉

yes

characteristic, complemented

C₅₉

H(7,𝔽₂)

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

—

144

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₅₉

yes

complemented

C₂×#41

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₅₉

yes

complemented

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₅₉

yes

complemented

C₂⁴

—

C₂⁴

C₂²×D₅₉

yes

complemented

#41

—

#41

—

288

—

#41

—

144

—

#41

—

#41

C₂²×D₅₉

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₅₉

yes

complemented

C₂×D₄

—

144

—

C₂×D₄

—

C₂×D₄

—

288

—

C₂×D₄

—

C₂×D₄

C₂²×D₅₉

yes

complemented

100

C₂×D₄

C₂²×D₅₉

yes

complemented

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₅₉

yes

complemented

104

C₂³

—

105

C₂³

—

106

C₂³

—

472

—

107

C₂³

C₂³×D₅₉

yes

—

108

C₂³

C₂³×D₅₉

yes

—

109

Q₈

—

110

Q₈

C₂³×D₅₉

yes

—

111

D₄

—

144

—

112

D₄

—

113

D₄

—

114

D₄

C₂³×D₅₉

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂³×D₅₉

yes

—

118

C₂×C₄

C₂³×D₅₉

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

236

—

123

C₂²

C₂⁴×D₅₉

yes

—

124

C₂²

C₂⁴×D₅₉

yes

characteristic

125

C₄

—

126

C₄

C₂⁴×D₅₉

yes

—

127

C₂

—

128

C₂

—

129

C₂

—

118

—

130

C₂

C₂⁵×D₅₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₅₉
commutator, socle	C₂×C₅₉	C₂⁶
Fitting	C₂×H(5,𝔽₂)×C₅₉	C₂
hypercenter	C₂×H(5,𝔽₂)	D₅₉
hypocenter	C₅₉	H(7,𝔽₂)
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₅₉

commutator, socle

C₂×C₅₉

C₂⁶

Fitting

C₂×H(5,𝔽₂)×C₅₉

C₂

hypercenter

C₂×H(5,𝔽₂)

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂ ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. #16838238	C₂	semidirect
2. #16838237	C₂	semidirect
3. C₂×#14273772	C₂	semidirect
4. H(5,𝔽₂)×D₅₉	C₂	semidirect
5. C₂×H(5,𝔽₂)×C₅₉	C₂	semidirect
6. #14273775	C₂²	semidirect
7. #14273774	C₂²	semidirect
8. #14273773	C₂²	semidirect
9. #14273772	C₂²	semidirect
10. #14273772	C₂²	semidirect
11. C₂×#11745685	C₂²	semidirect
12. C₂×#11745687	C₂²	semidirect
13. #41×D₅₉	C₂²	semidirect
14. C₂²×C₂²⋊D₅₉	C₂²	semidirect
15. C₂×D₄×D₅₉	C₂²	semidirect
16. H(5,𝔽₂)×C₅₉	C₂²	semidirect
17. C₂×#41×C₅₉	C₂²	semidirect
18. C₂²×D₄×C₅₉	C₂²	semidirect
19. C₂⁴×C₅₉	C₂³	semidirect
20. C₂³×D₅₉	C₂³	semidirect
21. #41×C₅₉	C₂³	semidirect
22. C₂×Q₈×C₅₉	C₂³	semidirect
23. C₂×D₄×C₅₉	C₂³	semidirect
24. C₂×D₄×C₅₉	C₂³	semidirect
25. C₂²×C₄×C₅₉	C₂³	semidirect
26. C₂×C₂²⋊D₅₉	C₂³	semidirect
27. C₂×C₂²⋊D₅₉	C₂³	semidirect
28. C₂²×Dic₅₉	C₂³	semidirect
29. #11745689	C₂³	semidirect
30. Q₈×D₅₉	C₂³	semidirect
31. #11745687	C₂³	semidirect
32. D₄×D₅₉	C₂³	semidirect
33. #11745685	C₂³	semidirect
34. #11745685	C₂³	semidirect
35. C₂×D₂₃₆	C₂³	semidirect
36. C₂×C₄×D₅₉	C₂³	semidirect
37. C₂×Dic₁₁₈	C₂³	semidirect
70. C₂×H(5,𝔽₂)	D₅₉	semidirect
71. C₅₉	H(7,𝔽₂)	semidirect
77. H(5,𝔽₂)	C₂×D₅₉	semidirect
81. C₂×#41	C₂×D₅₉	semidirect
84. C₂²×D₄	C₂×D₅₉	semidirect
86. C₂⁴	C₂²×D₅₉	semidirect
91. #41	C₂²×D₅₉	semidirect
94. C₂×Q₈	C₂²×D₅₉	semidirect
99. C₂×D₄	C₂²×D₅₉	semidirect
100. C₂×D₄	C₂²×D₅₉	semidirect
103. C₂²×C₄	C₂²×D₅₉	semidirect

H ≅ G/N

type

1. #16838238

C₂

semidirect

2. #16838237

C₂

semidirect

3. C₂×#14273772

C₂

semidirect

4. H(5,𝔽₂)×D₅₉

C₂

semidirect

5. C₂×H(5,𝔽₂)×C₅₉

C₂

semidirect

6. #14273775

C₂²

semidirect

7. #14273774

C₂²

semidirect

8. #14273773

C₂²

semidirect

9. #14273772

C₂²

semidirect

10. #14273772

C₂²

semidirect

11. C₂×#11745685

C₂²

semidirect

12. C₂×#11745687

C₂²

semidirect

13. #41×D₅₉

C₂²

semidirect

14. C₂²×C₂²⋊D₅₉

C₂²

semidirect

15. C₂×D₄×D₅₉

C₂²

semidirect

16. H(5,𝔽₂)×C₅₉

C₂²

semidirect

17. C₂×#41×C₅₉

C₂²

semidirect

18. C₂²×D₄×C₅₉

C₂²

semidirect

19. C₂⁴×C₅₉

C₂³

semidirect

20. C₂³×D₅₉

C₂³

semidirect

21. #41×C₅₉

C₂³

semidirect

22. C₂×Q₈×C₅₉

C₂³

semidirect

23. C₂×D₄×C₅₉

C₂³

semidirect

24. C₂×D₄×C₅₉

C₂³

semidirect

25. C₂²×C₄×C₅₉

C₂³

semidirect

26. C₂×C₂²⋊D₅₉

C₂³

semidirect

27. C₂×C₂²⋊D₅₉

C₂³

semidirect

28. C₂²×Dic₅₉

C₂³

semidirect

29. #11745689

C₂³

semidirect

30. Q₈×D₅₉

C₂³

semidirect

31. #11745687

C₂³

semidirect

32. D₄×D₅₉

C₂³

semidirect

33. #11745685

C₂³

semidirect

34. #11745685

C₂³

semidirect

35. C₂×D₂₃₆

C₂³

semidirect

36. C₂×C₄×D₅₉

C₂³

semidirect

37. C₂×Dic₁₁₈

C₂³

semidirect

70. C₂×H(5,𝔽₂)

D₅₉

semidirect

71. C₅₉

H(7,𝔽₂)

semidirect

77. H(5,𝔽₂)

C₂×D₅₉

semidirect

81. C₂×#41

C₂×D₅₉

semidirect

84. C₂²×D₄

C₂×D₅₉

semidirect

86. C₂⁴

C₂²×D₅₉

semidirect

91. #41

C₂²×D₅₉

semidirect

94. C₂×Q₈

C₂²×D₅₉

semidirect

99. C₂×D₄

C₂²×D₅₉

semidirect

100. C₂×D₄

C₂²×D₅₉

semidirect

103. C₂²×C₄

C₂²×D₅₉

semidirect

Subgroups — #21438906

Conjugacy classes of subgroups

Complements