Group #22603151 — subgroups

Subgroups — #22603151

Order 9088.

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	#17986836	C₂	1	12	yes	maximal, complemented
2	#17986835	C₂	1	15	yes	maximal, complemented
3	C₂×#14556174	C₂	1	15	yes	maximal, complemented
4	H(5,𝔽₂)×D₇₁	C₂	1	20	yes	maximal, complemented
5	#585×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556176	C₂²	1	30	yes	complemented
7	#14556175	C₂²	1	20	yes	complemented
8	#14556174	C₂²	1	120	yes	complemented
9	#14556174	C₂²	1	60	yes	complemented
10	#14556173	C₂²	1	90	yes	complemented
11	C₂×#11765866	C₂²	1	30	yes	complemented
12	C₂×#11765865	C₂²	1	45	yes	complemented
13	#41×D₇₁	C₂²	1	120	yes	complemented
14	C₂²×D₂₈₄	C₂²	1	15	yes	complemented
15	C₂×D₄×D₇₁	C₂²	1	90	yes	complemented
16	2₋¹⁺⁴×C₇₁	C₂²	1	6	yes	complemented
17	H(5,𝔽₂)×C₇₁	C₂²	1	10	yes	complemented
18	C₂×#41×C₇₁	C₂²	1	15	yes	complemented
19	C₂³×D₇₁	C₂³	1	30	yes	complemented
20	#11765866	C₂³	1	120	yes	complemented
21	#11765865	C₂³	1	180	yes	complemented
22	#11765865	C₂³	1	60	yes	complemented
23	#11765864	C₂³	1	120	yes	complemented
24	C₂×Dic₁₄₂	C₂³	1	15	yes	complemented
25	Q₈×D₇₁	C₂³	1	40	yes	complemented
26	C₂×D₂₈₄	C₂³	1	90	yes	complemented
27	C₂×D₂₈₄	C₂³	1	45	yes	complemented
28	C₂×C₂²⋊D₇₁	C₂³	1	90	yes	complemented
29	D₄×D₇₁	C₂³	1	360	yes	complemented
30	C₂×C₄×D₇₁	C₂³	1	90	yes	complemented
31	#41×C₇₁	C₂³	1	60	yes	complemented
32	#41×C₇₁	C₂³	1	20	yes	complemented
33	C₂×Q₈×C₇₁	C₂³	1	15	yes	complemented
34	C₂×D₄×C₇₁	C₂³	1	45	yes	complemented
35	C₂²×C₄×C₇₁	C₂³	1	15	yes	complemented
36	C₂³×C₇₁	C₂⁴	1	15	yes	—
37	C₂²×D₇₁	—	8	30	no	—
38	C₂²×D₇₁	C₂⁴	1	90	yes	—
39	Q₈×C₇₁	C₂⁴	1	20	yes	—
40	D₄×C₇₁	C₂⁴	1	60	yes	—
41	C₂×C₄×C₇₁	C₂⁴	1	15	yes	—
42	C₂×C₄×C₇₁	C₂⁴	1	45	yes	—
43	C₂²⋊D₇₁	C₂⁴	1	120	yes	—
44	C₂×Dic₇₁	C₂⁴	1	30	yes	—
45	D₂₈₄	C₂⁴	1	90	yes	—
46	D₂₈₄	C₂⁴	1	10	yes	—
47	C₄×D₇₁	C₂⁴	1	120	yes	—
48	Dic₁₄₂	C₂⁴	1	30	yes	—
49	Dic₁₄₂	C₂⁴	1	6	yes	—
50	C₂²×C₇₁	—	4	15	no	—
51	C₂²×C₇₁	C₂⁵	1	15	yes	—
52	C₂×D₇₁	—	4	90	no	—
53	C₂×D₇₁	C₂⁵	1	20	yes	—
54	C₄×C₇₁	C₂⁵	1	1	yes	characteristic
55	C₄×C₇₁	C₂⁵	1	15	yes	—
56	Dic₇₁	C₂⁵	1	12	yes	—
57	C₂×C₇₁	—	2	15	no	—
58	C₂×C₇₁	C₂⁶	1	1	yes	characteristic
59	D₇₁	—	2	20	no	—
60	H(7,𝔽₂)	—	71	1	no	maximal, Sylow
61	C₇₁	H(7,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
62	#585	—	71	12	no	—
63	#585	—	71	15	no	—
64	#585	D₇₁	1	1	yes	characteristic, complemented
65	C₂×H(5,𝔽₂)	—	71	20	no	—
66	C₂×H(5,𝔽₂)	—	71	15	no	—
67	2₋¹⁺⁴	—	71	20	no	—
68	2₋¹⁺⁴	—	71	30	no	—
69	2₋¹⁺⁴	C₂×D₇₁	1	6	yes	complemented
70	H(5,𝔽₂)	—	71	120	no	—
71	H(5,𝔽₂)	—	71	60	no	—
72	H(5,𝔽₂)	—	71	90	no	—
73	H(5,𝔽₂)	C₂×D₇₁	1	10	yes	complemented
74	C₂×#41	—	71	120	no	—
75	C₂×#41	—	71	30	no	—
76	C₂×#41	—	71	45	no	—
77	C₂×#41	C₂×D₇₁	1	15	yes	complemented
78	C₂²×D₄	—	71	90	no	—
79	C₂²×D₄	—	71	15	no	—
80	C₂⁴	—	71	30	no	—
81	#41	—	71	120	no	—
82	#41	—	71	120	no	—
83	#41	—	71	180	no	—
84	#41	—	71	60	no	—
85	#41	C₂²×D₇₁	1	60	yes	complemented
86	#41	C₂²×D₇₁	1	20	yes	complemented
87	C₂×Q₈	—	71	40	no	—
88	C₂×Q₈	—	71	15	no	—
89	C₂×Q₈	C₂²×D₇₁	1	15	yes	complemented
90	C₂×D₄	—	71	90	no	—
91	C₂×D₄	—	71	360	no	—
92	C₂×D₄	—	71	90	no	—
93	C₂×D₄	—	71	45	no	—
94	C₂×D₄	C₂²×D₇₁	1	45	yes	complemented
95	C₂²×C₄	—	71	90	no	—
96	C₂²×C₄	C₂²×D₇₁	1	15	yes	complemented
97	C₂³	—	568	30	no	—
98	C₂³	—	71	90	no	—
99	C₂³	C₂³×D₇₁	1	15	yes	—
100	Q₈	—	71	30	no	—
101	Q₈	—	71	6	no	—
102	Q₈	C₂³×D₇₁	1	20	yes	—
103	D₄	—	71	120	no	—
104	D₄	—	71	90	no	—
105	D₄	—	71	10	no	—
106	D₄	C₂³×D₇₁	1	60	yes	—
107	C₂×C₄	—	71	30	no	—
108	C₂×C₄	—	71	120	no	—
109	C₂×C₄	C₂³×D₇₁	1	45	yes	—
110	C₂×C₄	C₂³×D₇₁	1	15	yes	—
111	C₂²	—	4	15	no	—
112	C₂²	—	284	90	no	—
113	C₂²	—	71	20	no	—
114	C₂²	C₂⁴×D₇₁	1	15	yes	—
115	C₄	—	71	12	no	—
116	C₄	C₂⁴×D₇₁	1	15	yes	—
117	C₄	C₂⁴×D₇₁	1	1	yes	characteristic
118	C₂	—	2	15	no	—
119	C₂	—	142	20	no	—
120	C₂	C₂⁵×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986836

C₂

yes

maximal, complemented

#17986835

C₂

yes

maximal, complemented

C₂×#14556174

C₂

yes

maximal, complemented

H(5,𝔽₂)×D₇₁

C₂

yes

maximal, complemented

#585×C₇₁

C₂

yes

maximal, characteristic, complemented

#14556176

C₂²

yes

complemented

#14556175

C₂²

yes

complemented

#14556174

C₂²

120

yes

complemented

#14556174

C₂²

yes

complemented

#14556173

C₂²

yes

complemented

C₂×#11765866

C₂²

yes

complemented

C₂×#11765865

C₂²

yes

complemented

#41×D₇₁

C₂²

120

yes

complemented

C₂²×D₂₈₄

C₂²

yes

complemented

C₂×D₄×D₇₁

C₂²

yes

complemented

2₋¹⁺⁴×C₇₁

C₂²

yes

complemented

H(5,𝔽₂)×C₇₁

C₂²

yes

complemented

C₂×#41×C₇₁

C₂²

yes

complemented

C₂³×D₇₁

C₂³

yes

complemented

#11765866

C₂³

120

yes

complemented

#11765865

C₂³

180

yes

complemented

#11765865

C₂³

yes

complemented

#11765864

C₂³

120

yes

complemented

C₂×Dic₁₄₂

C₂³

yes

complemented

Q₈×D₇₁

C₂³

yes

complemented

C₂×D₂₈₄

C₂³

yes

complemented

C₂×D₂₈₄

C₂³

yes

complemented

C₂×C₂²⋊D₇₁

C₂³

yes

complemented

D₄×D₇₁

C₂³

360

yes

complemented

C₂×C₄×D₇₁

C₂³

yes

complemented

#41×C₇₁

C₂³

yes

complemented

#41×C₇₁

C₂³

yes

complemented

C₂×Q₈×C₇₁

C₂³

yes

complemented

C₂×D₄×C₇₁

C₂³

yes

complemented

C₂²×C₄×C₇₁

C₂³

yes

complemented

C₂³×C₇₁

C₂⁴

yes

—

C₂²×D₇₁

—

C₂²×D₇₁

C₂⁴

yes

—

Q₈×C₇₁

C₂⁴

yes

—

D₄×C₇₁

C₂⁴

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂²⋊D₇₁

C₂⁴

120

yes

—

C₂×Dic₇₁

C₂⁴

yes

—

D₂₈₄

C₂⁴

yes

—

D₂₈₄

C₂⁴

yes

—

C₄×D₇₁

C₂⁴

120

yes

—

Dic₁₄₂

C₂⁴

yes

—

Dic₁₄₂

C₂⁴

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂⁵

yes

—

C₂×D₇₁

—

C₂×D₇₁

C₂⁵

yes

—

C₄×C₇₁

C₂⁵

yes

characteristic

C₄×C₇₁

C₂⁵

yes

—

Dic₇₁

C₂⁵

yes

—

C₂×C₇₁

—

C₂×C₇₁

C₂⁶

yes

characteristic

D₇₁

—

H(7,𝔽₂)

—

maximal, Sylow

C₇₁

H(7,𝔽₂)

yes

characteristic, complemented, Sylow

#585

—

#585

—

#585

D₇₁

yes

characteristic, complemented

C₂×H(5,𝔽₂)

—

C₂×H(5,𝔽₂)

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₇₁

yes

complemented

H(5,𝔽₂)

—

120

—

H(5,𝔽₂)

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₇₁

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₇₁

yes

complemented

C₂²×D₄

—

C₂²×D₄

—

C₂⁴

—

#41

—

120

—

#41

—

120

—

#41

—

180

—

#41

—

#41

C₂²×D₇₁

yes

complemented

#41

C₂²×D₇₁

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₇₁

yes

complemented

C₂×D₄

—

C₂×D₄

—

360

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₇₁

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₁

yes

complemented

C₂³

—

568

—

C₂³

—

C₂³

C₂³×D₇₁

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

C₂³×D₇₁

yes

—

103

D₄

—

120

—

104

D₄

—

105

D₄

—

106

D₄

C₂³×D₇₁

yes

—

107

C₂×C₄

—

108

C₂×C₄

—

120

—

109

C₂×C₄

C₂³×D₇₁

yes

—

110

C₂×C₄

C₂³×D₇₁

yes

—

111

C₂²

—

112

C₂²

—

284

—

113

C₂²

—

114

C₂²

C₂⁴×D₇₁

yes

—

115

C₄

—

116

C₄

C₂⁴×D₇₁

yes

—

117

C₄

C₂⁴×D₇₁

yes

characteristic

118

C₂

—

119

C₂

—

142

—

120

C₂

C₂⁵×D₇₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₇₁
commutator, socle	C₂×C₇₁	C₂⁶
Fitting	#585×C₇₁	C₂
hypercenter	#585	D₇₁
hypocenter	C₇₁	H(7,𝔽₂)
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₇₁

commutator, socle

C₂×C₇₁

C₂⁶

Fitting

#585×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₂×C₇₁ ▷ C₇₁

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. #17986836	C₂	semidirect
2. #17986835	C₂	semidirect
3. C₂×#14556174	C₂	semidirect
4. H(5,𝔽₂)×D₇₁	C₂	semidirect
5. #585×C₇₁	C₂	semidirect
6. #14556176	C₂²	semidirect
7. #14556175	C₂²	semidirect
8. #14556174	C₂²	semidirect
9. #14556174	C₂²	semidirect
10. #14556173	C₂²	semidirect
11. C₂×#11765866	C₂²	semidirect
12. C₂×#11765865	C₂²	semidirect
13. #41×D₇₁	C₂²	semidirect
14. C₂²×D₂₈₄	C₂²	semidirect
15. C₂×D₄×D₇₁	C₂²	semidirect
16. 2₋¹⁺⁴×C₇₁	C₂²	semidirect
17. H(5,𝔽₂)×C₇₁	C₂²	semidirect
18. C₂×#41×C₇₁	C₂²	semidirect
19. C₂³×D₇₁	C₂³	semidirect
20. #11765866	C₂³	semidirect
21. #11765865	C₂³	semidirect
22. #11765865	C₂³	semidirect
23. #11765864	C₂³	semidirect
24. C₂×Dic₁₄₂	C₂³	semidirect
25. Q₈×D₇₁	C₂³	semidirect
26. C₂×D₂₈₄	C₂³	semidirect
27. C₂×D₂₈₄	C₂³	semidirect
28. C₂×C₂²⋊D₇₁	C₂³	semidirect
29. D₄×D₇₁	C₂³	semidirect
30. C₂×C₄×D₇₁	C₂³	semidirect
31. #41×C₇₁	C₂³	semidirect
32. #41×C₇₁	C₂³	semidirect
33. C₂×Q₈×C₇₁	C₂³	semidirect
34. C₂×D₄×C₇₁	C₂³	semidirect
35. C₂²×C₄×C₇₁	C₂³	semidirect
61. C₇₁	H(7,𝔽₂)	semidirect
64. #585	D₇₁	semidirect
69. 2₋¹⁺⁴	C₂×D₇₁	semidirect
73. H(5,𝔽₂)	C₂×D₇₁	semidirect
77. C₂×#41	C₂×D₇₁	semidirect
85. #41	C₂²×D₇₁	semidirect
86. #41	C₂²×D₇₁	semidirect
89. C₂×Q₈	C₂²×D₇₁	semidirect
94. C₂×D₄	C₂²×D₇₁	semidirect
96. C₂²×C₄	C₂²×D₇₁	semidirect

H ≅ G/N

type

1. #17986836

C₂

semidirect

2. #17986835

C₂

semidirect

3. C₂×#14556174

C₂

semidirect

4. H(5,𝔽₂)×D₇₁

C₂

semidirect

5. #585×C₇₁

C₂

semidirect

6. #14556176

C₂²

semidirect

7. #14556175

C₂²

semidirect

8. #14556174

C₂²

semidirect

9. #14556174

C₂²

semidirect

10. #14556173

C₂²

semidirect

11. C₂×#11765866

C₂²

semidirect

12. C₂×#11765865

C₂²

semidirect

13. #41×D₇₁

C₂²

semidirect

14. C₂²×D₂₈₄

C₂²

semidirect

15. C₂×D₄×D₇₁

C₂²

semidirect

16. 2₋¹⁺⁴×C₇₁

C₂²

semidirect

17. H(5,𝔽₂)×C₇₁

C₂²

semidirect

18. C₂×#41×C₇₁

C₂²

semidirect

19. C₂³×D₇₁

C₂³

semidirect

20. #11765866

C₂³

semidirect

21. #11765865

C₂³

semidirect

22. #11765865

C₂³

semidirect

23. #11765864

C₂³

semidirect

24. C₂×Dic₁₄₂

C₂³

semidirect

25. Q₈×D₇₁

C₂³

semidirect

26. C₂×D₂₈₄

C₂³

semidirect

27. C₂×D₂₈₄

C₂³

semidirect

28. C₂×C₂²⋊D₇₁

C₂³

semidirect

29. D₄×D₇₁

C₂³

semidirect

30. C₂×C₄×D₇₁

C₂³

semidirect

31. #41×C₇₁

C₂³

semidirect

32. #41×C₇₁

C₂³

semidirect

33. C₂×Q₈×C₇₁

C₂³

semidirect

34. C₂×D₄×C₇₁

C₂³

semidirect

35. C₂²×C₄×C₇₁

C₂³

semidirect

61. C₇₁

H(7,𝔽₂)

semidirect

64. #585

D₇₁

semidirect

69. 2₋¹⁺⁴

C₂×D₇₁

semidirect

73. H(5,𝔽₂)

C₂×D₇₁

semidirect

77. C₂×#41

C₂×D₇₁

semidirect

85. #41

C₂²×D₇₁

semidirect

86. #41

C₂²×D₇₁

semidirect

89. C₂×Q₈

C₂²×D₇₁

semidirect

94. C₂×D₄

C₂²×D₇₁

semidirect

96. C₂²×C₄

C₂²×D₇₁

semidirect

Subgroups — #22603151

Conjugacy classes of subgroups

Complements