Group #22575020 — subgroups

Subgroups — #22575020

Order 8576.

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	#16864233	C₂	1	12	yes	maximal, complemented
2	#16864232	C₂	1	15	yes	maximal, complemented
3	C₂×#14524089	C₂	1	15	yes	maximal, complemented
4	H(5,𝔽₂)×D₆₇	C₂	1	20	yes	maximal, complemented
5	#585×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14524091	C₂²	1	30	yes	complemented
7	#14524090	C₂²	1	20	yes	complemented
8	#14524089	C₂²	1	120	yes	complemented
9	#14524089	C₂²	1	60	yes	complemented
10	#14524088	C₂²	1	90	yes	complemented
11	C₂×#11761629	C₂²	1	30	yes	complemented
12	C₂×#11761628	C₂²	1	45	yes	complemented
13	#41×D₆₇	C₂²	1	120	yes	complemented
14	C₂²×D₂₆₈	C₂²	1	15	yes	complemented
15	C₂×D₄×D₆₇	C₂²	1	90	yes	complemented
16	2₋¹⁺⁴×C₆₇	C₂²	1	6	yes	complemented
17	H(5,𝔽₂)×C₆₇	C₂²	1	10	yes	complemented
18	C₂×#41×C₆₇	C₂²	1	15	yes	complemented
19	C₂³×D₆₇	C₂³	1	30	yes	complemented
20	#11761629	C₂³	1	120	yes	complemented
21	#11761628	C₂³	1	180	yes	complemented
22	#11761628	C₂³	1	60	yes	complemented
23	#11761627	C₂³	1	120	yes	complemented
24	C₂×Dic₁₃₄	C₂³	1	15	yes	complemented
25	Q₈×D₆₇	C₂³	1	40	yes	complemented
26	C₂×D₂₆₈	C₂³	1	90	yes	complemented
27	C₂×D₂₆₈	C₂³	1	45	yes	complemented
28	C₂×C₂²⋊D₆₇	C₂³	1	90	yes	complemented
29	D₄×D₆₇	C₂³	1	360	yes	complemented
30	C₂×C₄×D₆₇	C₂³	1	90	yes	complemented
31	#41×C₆₇	C₂³	1	20	yes	complemented
32	#41×C₆₇	C₂³	1	60	yes	complemented
33	C₂×Q₈×C₆₇	C₂³	1	15	yes	complemented
34	C₂×D₄×C₆₇	C₂³	1	45	yes	complemented
35	C₂²×C₄×C₆₇	C₂³	1	15	yes	complemented
36	C₂³×C₆₇	C₂⁴	1	15	yes	—
37	C₂²×D₆₇	—	8	30	no	—
38	C₂²×D₆₇	C₂⁴	1	90	yes	—
39	Q₈×C₆₇	C₂⁴	1	20	yes	—
40	D₄×C₆₇	C₂⁴	1	60	yes	—
41	C₂×C₄×C₆₇	C₂⁴	1	15	yes	—
42	C₂×C₄×C₆₇	C₂⁴	1	45	yes	—
43	C₂²⋊D₆₇	C₂⁴	1	120	yes	—
44	C₂×Dic₆₇	C₂⁴	1	30	yes	—
45	D₂₆₈	C₂⁴	1	90	yes	—
46	D₂₆₈	C₂⁴	1	10	yes	—
47	C₄×D₆₇	C₂⁴	1	120	yes	—
48	Dic₁₃₄	C₂⁴	1	30	yes	—
49	Dic₁₃₄	C₂⁴	1	6	yes	—
50	C₂²×C₆₇	—	4	15	no	—
51	C₂²×C₆₇	C₂⁵	1	15	yes	—
52	C₂×D₆₇	—	4	90	no	—
53	C₂×D₆₇	C₂⁵	1	20	yes	—
54	C₄×C₆₇	C₂⁵	1	1	yes	characteristic
55	C₄×C₆₇	C₂⁵	1	15	yes	—
56	Dic₆₇	C₂⁵	1	12	yes	—
57	C₂×C₆₇	—	2	15	no	—
58	C₂×C₆₇	C₂⁶	1	1	yes	characteristic
59	D₆₇	—	2	20	no	—
60	H(7,𝔽₂)	—	67	1	no	maximal, Sylow
61	C₆₇	H(7,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
62	#585	—	67	12	no	—
63	#585	—	67	15	no	—
64	#585	D₆₇	1	1	yes	characteristic, complemented
65	C₂×H(5,𝔽₂)	—	67	20	no	—
66	C₂×H(5,𝔽₂)	—	67	15	no	—
67	2₋¹⁺⁴	—	67	20	no	—
68	2₋¹⁺⁴	—	67	30	no	—
69	2₋¹⁺⁴	C₂×D₆₇	1	6	yes	complemented
70	H(5,𝔽₂)	—	67	120	no	—
71	H(5,𝔽₂)	—	67	60	no	—
72	H(5,𝔽₂)	—	67	90	no	—
73	H(5,𝔽₂)	C₂×D₆₇	1	10	yes	complemented
74	C₂×#41	—	67	120	no	—
75	C₂×#41	—	67	30	no	—
76	C₂×#41	—	67	45	no	—
77	C₂×#41	C₂×D₆₇	1	15	yes	complemented
78	C₂²×D₄	—	67	90	no	—
79	C₂²×D₄	—	67	15	no	—
80	C₂⁴	—	67	30	no	—
81	#41	—	67	120	no	—
82	#41	—	67	120	no	—
83	#41	—	67	180	no	—
84	#41	—	67	60	no	—
85	#41	C₂²×D₆₇	1	60	yes	complemented
86	#41	C₂²×D₆₇	1	20	yes	complemented
87	C₂×Q₈	—	67	40	no	—
88	C₂×Q₈	—	67	15	no	—
89	C₂×Q₈	C₂²×D₆₇	1	15	yes	complemented
90	C₂×D₄	—	67	90	no	—
91	C₂×D₄	—	67	360	no	—
92	C₂×D₄	—	67	90	no	—
93	C₂×D₄	—	67	45	no	—
94	C₂×D₄	C₂²×D₆₇	1	45	yes	complemented
95	C₂²×C₄	—	67	90	no	—
96	C₂²×C₄	C₂²×D₆₇	1	15	yes	complemented
97	C₂³	—	536	30	no	—
98	C₂³	—	67	90	no	—
99	C₂³	C₂³×D₆₇	1	15	yes	—
100	Q₈	—	67	30	no	—
101	Q₈	—	67	6	no	—
102	Q₈	C₂³×D₆₇	1	20	yes	—
103	D₄	—	67	120	no	—
104	D₄	—	67	90	no	—
105	D₄	—	67	10	no	—
106	D₄	C₂³×D₆₇	1	60	yes	—
107	C₂×C₄	—	67	30	no	—
108	C₂×C₄	—	67	120	no	—
109	C₂×C₄	C₂³×D₆₇	1	45	yes	—
110	C₂×C₄	C₂³×D₆₇	1	15	yes	—
111	C₂²	—	4	15	no	—
112	C₂²	—	268	90	no	—
113	C₂²	—	67	20	no	—
114	C₂²	C₂⁴×D₆₇	1	15	yes	—
115	C₄	—	67	12	no	—
116	C₄	C₂⁴×D₆₇	1	15	yes	—
117	C₄	C₂⁴×D₆₇	1	1	yes	characteristic
118	C₂	—	2	15	no	—
119	C₂	—	134	20	no	—
120	C₂	C₂⁵×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864233

C₂

yes

maximal, complemented

#16864232

C₂

yes

maximal, complemented

C₂×#14524089

C₂

yes

maximal, complemented

H(5,𝔽₂)×D₆₇

C₂

yes

maximal, complemented

#585×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524091

C₂²

yes

complemented

#14524090

C₂²

yes

complemented

#14524089

C₂²

120

yes

complemented

#14524089

C₂²

yes

complemented

#14524088

C₂²

yes

complemented

C₂×#11761629

C₂²

yes

complemented

C₂×#11761628

C₂²

yes

complemented

#41×D₆₇

C₂²

120

yes

complemented

C₂²×D₂₆₈

C₂²

yes

complemented

C₂×D₄×D₆₇

C₂²

yes

complemented

2₋¹⁺⁴×C₆₇

C₂²

yes

complemented

H(5,𝔽₂)×C₆₇

C₂²

yes

complemented

C₂×#41×C₆₇

C₂²

yes

complemented

C₂³×D₆₇

C₂³

yes

complemented

#11761629

C₂³

120

yes

complemented

#11761628

C₂³

180

yes

complemented

#11761628

C₂³

yes

complemented

#11761627

C₂³

120

yes

complemented

C₂×Dic₁₃₄

C₂³

yes

complemented

Q₈×D₆₇

C₂³

yes

complemented

C₂×D₂₆₈

C₂³

yes

complemented

C₂×D₂₆₈

C₂³

yes

complemented

C₂×C₂²⋊D₆₇

C₂³

yes

complemented

D₄×D₆₇

C₂³

360

yes

complemented

C₂×C₄×D₆₇

C₂³

yes

complemented

#41×C₆₇

C₂³

yes

complemented

#41×C₆₇

C₂³

yes

complemented

C₂×Q₈×C₆₇

C₂³

yes

complemented

C₂×D₄×C₆₇

C₂³

yes

complemented

C₂²×C₄×C₆₇

C₂³

yes

complemented

C₂³×C₆₇

C₂⁴

yes

—

C₂²×D₆₇

—

C₂²×D₆₇

C₂⁴

yes

—

Q₈×C₆₇

C₂⁴

yes

—

D₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂²⋊D₆₇

C₂⁴

120

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

D₂₆₈

C₂⁴

yes

—

D₂₆₈

C₂⁴

yes

—

C₄×D₆₇

C₂⁴

120

yes

—

Dic₁₃₄

C₂⁴

yes

—

Dic₁₃₄

C₂⁴

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂⁵

yes

—

C₂×D₆₇

—

C₂×D₆₇

C₂⁵

yes

—

C₄×C₆₇

C₂⁵

yes

characteristic

C₄×C₆₇

C₂⁵

yes

—

Dic₆₇

C₂⁵

yes

—

C₂×C₆₇

—

C₂×C₆₇

C₂⁶

yes

characteristic

D₆₇

—

H(7,𝔽₂)

—

maximal, Sylow

C₆₇

H(7,𝔽₂)

yes

characteristic, complemented, Sylow

#585

—

#585

—

#585

D₆₇

yes

characteristic, complemented

C₂×H(5,𝔽₂)

—

C₂×H(5,𝔽₂)

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₆₇

yes

complemented

H(5,𝔽₂)

—

120

—

H(5,𝔽₂)

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₆₇

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₆₇

yes

complemented

C₂²×D₄

—

C₂²×D₄

—

C₂⁴

—

#41

—

120

—

#41

—

120

—

#41

—

180

—

#41

—

#41

C₂²×D₆₇

yes

complemented

#41

C₂²×D₆₇

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₆₇

yes

complemented

C₂×D₄

—

C₂×D₄

—

360

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₆₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

complemented

C₂³

—

536

—

C₂³

—

C₂³

C₂³×D₆₇

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

C₂³×D₆₇

yes

—

103

D₄

—

120

—

104

D₄

—

105

D₄

—

106

D₄

C₂³×D₆₇

yes

—

107

C₂×C₄

—

108

C₂×C₄

—

120

—

109

C₂×C₄

C₂³×D₆₇

yes

—

110

C₂×C₄

C₂³×D₆₇

yes

—

111

C₂²

—

112

C₂²

—

268

—

113

C₂²

—

114

C₂²

C₂⁴×D₆₇

yes

—

115

C₄

—

116

C₄

C₂⁴×D₆₇

yes

—

117

C₄

C₂⁴×D₆₇

yes

characteristic

118

C₂

—

119

C₂

—

134

—

120

C₂

C₂⁵×D₆₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₆₇
commutator, socle	C₂×C₆₇	C₂⁶
Fitting	#585×C₆₇	C₂
hypercenter	#585	D₆₇
hypocenter	C₆₇	H(7,𝔽₂)
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₆₇

commutator, socle

C₂×C₆₇

C₂⁶

Fitting

#585×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂×C₆₇ ▷ C₆₇

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. #16864233	C₂	semidirect
2. #16864232	C₂	semidirect
3. C₂×#14524089	C₂	semidirect
4. H(5,𝔽₂)×D₆₇	C₂	semidirect
5. #585×C₆₇	C₂	semidirect
6. #14524091	C₂²	semidirect
7. #14524090	C₂²	semidirect
8. #14524089	C₂²	semidirect
9. #14524089	C₂²	semidirect
10. #14524088	C₂²	semidirect
11. C₂×#11761629	C₂²	semidirect
12. C₂×#11761628	C₂²	semidirect
13. #41×D₆₇	C₂²	semidirect
14. C₂²×D₂₆₈	C₂²	semidirect
15. C₂×D₄×D₆₇	C₂²	semidirect
16. 2₋¹⁺⁴×C₆₇	C₂²	semidirect
17. H(5,𝔽₂)×C₆₇	C₂²	semidirect
18. C₂×#41×C₆₇	C₂²	semidirect
19. C₂³×D₆₇	C₂³	semidirect
20. #11761629	C₂³	semidirect
21. #11761628	C₂³	semidirect
22. #11761628	C₂³	semidirect
23. #11761627	C₂³	semidirect
24. C₂×Dic₁₃₄	C₂³	semidirect
25. Q₈×D₆₇	C₂³	semidirect
26. C₂×D₂₆₈	C₂³	semidirect
27. C₂×D₂₆₈	C₂³	semidirect
28. C₂×C₂²⋊D₆₇	C₂³	semidirect
29. D₄×D₆₇	C₂³	semidirect
30. C₂×C₄×D₆₇	C₂³	semidirect
31. #41×C₆₇	C₂³	semidirect
32. #41×C₆₇	C₂³	semidirect
33. C₂×Q₈×C₆₇	C₂³	semidirect
34. C₂×D₄×C₆₇	C₂³	semidirect
35. C₂²×C₄×C₆₇	C₂³	semidirect
61. C₆₇	H(7,𝔽₂)	semidirect
64. #585	D₆₇	semidirect
69. 2₋¹⁺⁴	C₂×D₆₇	semidirect
73. H(5,𝔽₂)	C₂×D₆₇	semidirect
77. C₂×#41	C₂×D₆₇	semidirect
85. #41	C₂²×D₆₇	semidirect
86. #41	C₂²×D₆₇	semidirect
89. C₂×Q₈	C₂²×D₆₇	semidirect
94. C₂×D₄	C₂²×D₆₇	semidirect
96. C₂²×C₄	C₂²×D₆₇	semidirect

H ≅ G/N

type

1. #16864233

C₂

semidirect

2. #16864232

C₂

semidirect

3. C₂×#14524089

C₂

semidirect

4. H(5,𝔽₂)×D₆₇

C₂

semidirect

5. #585×C₆₇

C₂

semidirect

6. #14524091

C₂²

semidirect

7. #14524090

C₂²

semidirect

8. #14524089

C₂²

semidirect

9. #14524089

C₂²

semidirect

10. #14524088

C₂²

semidirect

11. C₂×#11761629

C₂²

semidirect

12. C₂×#11761628

C₂²

semidirect

13. #41×D₆₇

C₂²

semidirect

14. C₂²×D₂₆₈

C₂²

semidirect

15. C₂×D₄×D₆₇

C₂²

semidirect

16. 2₋¹⁺⁴×C₆₇

C₂²

semidirect

17. H(5,𝔽₂)×C₆₇

C₂²

semidirect

18. C₂×#41×C₆₇

C₂²

semidirect

19. C₂³×D₆₇

C₂³

semidirect

20. #11761629

C₂³

semidirect

21. #11761628

C₂³

semidirect

22. #11761628

C₂³

semidirect

23. #11761627

C₂³

semidirect

24. C₂×Dic₁₃₄

C₂³

semidirect

25. Q₈×D₆₇

C₂³

semidirect

26. C₂×D₂₆₈

C₂³

semidirect

27. C₂×D₂₆₈

C₂³

semidirect

28. C₂×C₂²⋊D₆₇

C₂³

semidirect

29. D₄×D₆₇

C₂³

semidirect

30. C₂×C₄×D₆₇

C₂³

semidirect

31. #41×C₆₇

C₂³

semidirect

32. #41×C₆₇

C₂³

semidirect

33. C₂×Q₈×C₆₇

C₂³

semidirect

34. C₂×D₄×C₆₇

C₂³

semidirect

35. C₂²×C₄×C₆₇

C₂³

semidirect

61. C₆₇

H(7,𝔽₂)

semidirect

64. #585

D₆₇

semidirect

69. 2₋¹⁺⁴

C₂×D₆₇

semidirect

73. H(5,𝔽₂)

C₂×D₆₇

semidirect

77. C₂×#41

C₂×D₆₇

semidirect

85. #41

C₂²×D₆₇

semidirect

86. #41

C₂²×D₆₇

semidirect

89. C₂×Q₈

C₂²×D₆₇

semidirect

94. C₂×D₄

C₂²×D₆₇

semidirect

96. C₂²×C₄

C₂²×D₆₇

semidirect

Subgroups — #22575020

Conjugacy classes of subgroups

Complements