Group #33999737 — subgroups

Subgroups — C₂⁴×C₈×D₇₃ #33999737

Order 18688.

15454 conjugacy classes of subgroups; only the 8715 normal subgroups are shown.

normal subgroups only

This group has 15454 conjugacy classes of subgroups — too many to draw. Only its 8715 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	C₂⁴×C₄×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₈×D₇₃	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₇₃	C₄	1	1	yes	characteristic
6	C₂⁴×Dic₇₃	C₄	1	1	yes	characteristic
7	C₂³×C₄×D₇₃	C₄	1	30	yes	—
8	C₂³×C₄×D₇₃	C₂²	1	30	yes	—
9	C₂³×#10596831	C₂²	1	30	yes	complemented
10	C₂²×C₈×D₇₃	C₂²	1	560	yes	direct product factor
11	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₇₃	C₂²	1	30	yes	complemented
13	C₂⁴×D₇₃	C₈	1	32	yes	direct product factor
14	C₂⁴×D₇₃	C₂×C₄	1	30	yes	—
15	C₂³×Dic₇₃	C₂×C₄	1	30	yes	—
16	C₂²×C₄×D₇₃	C₂×C₄	1	420	yes	—
17	C₂²×C₄×D₇₃	C₂³	1	140	yes	—
18	C₂²×#10596831	C₂³	1	140	yes	complemented
19	C₂×C₈×D₇₃	C₂³	1	960	yes	direct product factor
20	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇₃	C₂×C₄	1	15	yes	—
22	C₂³×C₄×C₇₃	C₂³	1	15	yes	—
23	C₂²×C₈×C₇₃	C₂³	1	140	yes	complemented
24	C₂³×D₇₃	C₂×C₈	1	480	yes	direct product factor
25	C₂³×D₇₃	C₂²×C₄	1	140	yes	—
26	C₂²×Dic₇₃	C₂²×C₄	1	140	yes	—
27	C₂×C₄×D₇₃	C₂²×C₄	1	840	yes	—
28	C₂×C₄×D₇₃	C₂⁴	1	120	yes	—
29	C₂×#10596831	C₂⁴	1	120	yes	complemented
30	C₈×D₇₃	C₂⁴	1	256	yes	direct product factor
31	C₂⁴×C₇₃	C₂×C₈	1	16	yes	complemented
32	C₂⁴×C₇₃	C₂²×C₄	1	15	yes	—
33	C₂²×C₄×C₇₃	C₂²×C₄	1	105	yes	—
34	C₂²×C₄×C₇₃	C₂⁴	1	35	yes	—
35	C₂×C₈×C₇₃	C₂⁴	1	120	yes	complemented
36	C₂³×C₇₃	C₂²×C₈	1	120	yes	complemented
37	C₂³×C₇₃	C₂³×C₄	1	35	yes	—
38	C₂²×D₇₃	C₂²×C₈	1	1120	yes	direct product factor
39	C₂²×D₇₃	C₂³×C₄	1	120	yes	—
40	C₂×C₄×C₇₃	C₂³×C₄	1	105	yes	—
41	C₂×C₄×C₇₃	C₂⁵	1	15	yes	—
42	C₂×Dic₇₃	C₂³×C₄	1	120	yes	—
43	C₄×D₇₃	C₂³×C₄	1	240	yes	—
44	C₄×D₇₃	C₂⁵	1	16	yes	—
45	C₈×C₇₃	C₂⁵	1	16	yes	complemented
46	#10596831	C₂⁵	1	16	yes	complemented
47	C₂²×C₇₃	C₂³×C₈	1	140	yes	complemented
48	C₂²×C₇₃	C₂⁴×C₄	1	15	yes	—
49	C₂×D₇₃	C₂³×C₈	1	480	yes	direct product factor
50	C₂×D₇₃	C₂⁴×C₄	1	16	yes	—
51	C₄×C₇₃	C₂⁴×C₄	1	15	yes	—
52	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
53	Dic₇₃	C₂⁴×C₄	1	16	yes	—
54	C₂⁵×C₈	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	C₂⁴×C₈	1	30	yes	complemented
56	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
57	D₇₃	C₂⁴×C₈	1	32	yes	direct product factor
58	C₂⁵×C₄	—	73	1	no	—
59	C₂⁴×C₈	—	73	1	no	—
60	C₂⁴×C₈	—	73	60	no	—
61	C₂⁴×C₈	D₇₃	1	1	yes	characteristic, direct product factor
62	C₇₃	C₂⁵×C₈	1	1	yes	characteristic, complemented, Sylow
63	C₂⁶	—	73	1	no	—
64	C₂⁴×C₄	—	73	1	no	—
65	C₂⁴×C₄	—	73	30	no	—
66	C₂⁴×C₄	—	73	30	no	—
67	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
68	C₂³×C₈	—	73	30	no	—
69	C₂³×C₈	—	73	560	no	—
70	C₂³×C₈	C₂×D₇₃	1	30	yes	direct product factor
71	C₂⁵	—	73	32	no	—
72	C₂⁵	—	73	30	no	—
73	C₂⁵	C₄×D₇₃	1	1	yes	characteristic
74	C₂³×C₄	—	73	30	no	—
75	C₂³×C₄	—	73	420	no	—
76	C₂³×C₄	—	73	140	no	—
77	C₂³×C₄	C₄×D₇₃	1	15	yes	—
78	C₂³×C₄	C₂²×D₇₃	1	15	yes	—
79	C₂²×C₈	—	73	140	no	—
80	C₂²×C₈	—	73	960	no	—
81	C₂²×C₈	C₂²×D₇₃	1	140	yes	direct product factor
82	C₂⁴	—	73	480	no	—
83	C₂⁴	—	73	140	no	—
84	C₂⁴	C₈×D₇₃	1	16	yes	direct product factor
85	C₂⁴	C₂×C₄×D₇₃	1	15	yes	—
86	C₂²×C₄	—	73	140	no	—
87	C₂²×C₄	—	73	840	no	—
88	C₂²×C₄	—	73	120	no	—
89	C₂²×C₄	C₂×C₄×D₇₃	1	105	yes	—
90	C₂²×C₄	C₂³×D₇₃	1	35	yes	—
91	C₂×C₈	—	73	120	no	—
92	C₂×C₈	—	73	256	no	—
93	C₂×C₈	C₂³×D₇₃	1	120	yes	direct product factor
94	C₂³	—	73	120	no	—
95	C₂³	—	73	1120	no	—
96	C₂³	C₂×C₈×D₇₃	1	120	yes	direct product factor
97	C₂³	C₂²×C₄×D₇₃	1	35	yes	—
98	C₂×C₄	—	73	120	no	—
99	C₂×C₄	—	73	240	no	—
100	C₂×C₄	—	73	16	no	—
101	C₂×C₄	C₂²×C₄×D₇₃	1	105	yes	—
102	C₂×C₄	C₂⁴×D₇₃	1	15	yes	—
103	C₈	—	73	16	no	—
104	C₈	C₂⁴×D₇₃	1	16	yes	direct product factor
105	C₂²	—	73	480	no	—
106	C₂²	—	73	16	no	—
107	C₂²	C₂²×C₈×D₇₃	1	140	yes	direct product factor
108	C₂²	C₂³×C₄×D₇₃	1	15	yes	—
109	C₄	—	73	16	no	—
110	C₄	C₂³×C₄×D₇₃	1	15	yes	—
111	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
112	C₂	—	73	32	no	—
113	C₂	C₂³×C₈×D₇₃	1	30	yes	direct product factor
114	C₂	C₂⁴×C₄×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₄×D₇₃

C₂

yes

maximal, characteristic

C₂⁴×#10596831

C₂

yes

maximal, characteristic, complemented

C₂³×C₈×D₇₃

C₂

yes

maximal, direct product factor

C₂⁴×C₈×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂⁵×D₇₃

C₄

yes

characteristic

C₂⁴×Dic₇₃

C₄

yes

characteristic

C₂³×C₄×D₇₃

C₄

yes

—

C₂³×C₄×D₇₃

C₂²

yes

—

C₂³×#10596831

C₂²

yes

complemented

C₂²×C₈×D₇₃

C₂²

560

yes

direct product factor

C₂⁴×C₄×C₇₃

C₂²

yes

characteristic

C₂³×C₈×C₇₃

C₂²

yes

complemented

C₂⁴×D₇₃

C₈

yes

direct product factor

C₂⁴×D₇₃

C₂×C₄

yes

—

C₂³×Dic₇₃

C₂×C₄

yes

—

C₂²×C₄×D₇₃

C₂×C₄

420

yes

—

C₂²×C₄×D₇₃

C₂³

140

yes

—

C₂²×#10596831

C₂³

140

yes

complemented

C₂×C₈×D₇₃

C₂³

960

yes

direct product factor

C₂⁵×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₄×C₇₃

C₂³

yes

—

C₂²×C₈×C₇₃

C₂³

140

yes

complemented

C₂³×D₇₃

C₂×C₈

480

yes

direct product factor

C₂³×D₇₃

C₂²×C₄

140

yes

—

C₂²×Dic₇₃

C₂²×C₄

140

yes

—

C₂×C₄×D₇₃

C₂²×C₄

840

yes

—

C₂×C₄×D₇₃

C₂⁴

120

yes

—

C₂×#10596831

C₂⁴

120

yes

complemented

C₈×D₇₃

C₂⁴

256

yes

direct product factor

C₂⁴×C₇₃

C₂×C₈

yes

complemented

C₂⁴×C₇₃

C₂²×C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

105

yes

—

C₂²×C₄×C₇₃

C₂⁴

yes

—

C₂×C₈×C₇₃

C₂⁴

120

yes

complemented

C₂³×C₇₃

C₂²×C₈

120

yes

complemented

C₂³×C₇₃

C₂³×C₄

yes

—

C₂²×D₇₃

C₂²×C₈

1120

yes

direct product factor

C₂²×D₇₃

C₂³×C₄

120

yes

—

C₂×C₄×C₇₃

C₂³×C₄

105

yes

—

C₂×C₄×C₇₃

C₂⁵

yes

—

C₂×Dic₇₃

C₂³×C₄

120

yes

—

C₄×D₇₃

C₂³×C₄

240

yes

—

C₄×D₇₃

C₂⁵

yes

—

C₈×C₇₃

C₂⁵

yes

complemented

#10596831

C₂⁵

yes

complemented

C₂²×C₇₃

C₂³×C₈

140

yes

complemented

C₂²×C₇₃

C₂⁴×C₄

yes

—

C₂×D₇₃

C₂³×C₈

480

yes

direct product factor

C₂×D₇₃

C₂⁴×C₄

yes

—

C₄×C₇₃

C₂⁴×C₄

yes

—

C₄×C₇₃

C₂⁶

yes

characteristic

Dic₇₃

C₂⁴×C₄

yes

—

C₂⁵×C₈

—

maximal, Sylow

C₂×C₇₃

C₂⁴×C₈

yes

complemented

C₂×C₇₃

C₂⁵×C₄

yes

characteristic

D₇₃

C₂⁴×C₈

yes

direct product factor

C₂⁵×C₄

—

C₂⁴×C₈

—

C₂⁴×C₈

—

C₂⁴×C₈

D₇₃

yes

characteristic, direct product factor

C₇₃

C₂⁵×C₈

yes

characteristic, complemented, Sylow

C₂⁶

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

C₂×D₇₃

yes

characteristic

C₂³×C₈

—

C₂³×C₈

—

560

—

C₂³×C₈

C₂×D₇₃

yes

direct product factor

C₂⁵

—

C₂⁵

—

C₂⁵

C₄×D₇₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

420

—

C₂³×C₄

—

140

—

C₂³×C₄

C₄×D₇₃

yes

—

C₂³×C₄

C₂²×D₇₃

yes

—

C₂²×C₈

—

140

—

C₂²×C₈

—

960

—

C₂²×C₈

C₂²×D₇₃

140

yes

direct product factor

C₂⁴

—

480

—

C₂⁴

—

140

—

C₂⁴

C₈×D₇₃

yes

direct product factor

C₂⁴

C₂×C₄×D₇₃

yes

—

C₂²×C₄

—

140

—

C₂²×C₄

—

840

—

C₂²×C₄

—

120

—

C₂²×C₄

C₂×C₄×D₇₃

105

yes

—

C₂²×C₄

C₂³×D₇₃

yes

—

C₂×C₈

—

120

—

C₂×C₈

—

256

—

C₂×C₈

C₂³×D₇₃

120

yes

direct product factor

C₂³

—

120

—

C₂³

—

1120

—

C₂³

C₂×C₈×D₇₃

120

yes

direct product factor

C₂³

C₂²×C₄×D₇₃

yes

—

C₂×C₄

—

120

—

C₂×C₄

—

240

—

100

C₂×C₄

—

101

C₂×C₄

C₂²×C₄×D₇₃

105

yes

—

102

C₂×C₄

C₂⁴×D₇₃

yes

—

103

C₈

—

104

C₈

C₂⁴×D₇₃

yes

direct product factor

105

C₂²

—

480

—

106

C₂²

—

107

C₂²

C₂²×C₈×D₇₃

140

yes

direct product factor

108

C₂²

C₂³×C₄×D₇₃

yes

—

109

C₄

—

110

C₄

C₂³×C₄×D₇₃

yes

—

111

C₄

C₂⁵×D₇₃

yes

characteristic

112

C₂

—

113

C₂

C₂³×C₈×D₇₃

yes

direct product factor

114

C₂

C₂⁴×C₄×D₇₃

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₈	D₇₃
commutator, hypocenter	C₇₃	C₂⁵×C₈
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂⁴×C₈×C₇₃	C₂
socle	C₂⁵×C₇₃	C₂×C₄
prime radical	C₂⁵×D₇₃	C₄
cosocle	C₄×C₇₃	C₂⁶
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂⁴×C₈

subgroup

G/H

center, hypercenter

C₂⁴×C₈

D₇₃

commutator, hypocenter

C₇₃

C₂⁵×C₈

Frattini

C₄

C₂⁵×D₇₃

Fitting

C₂⁴×C₈×C₇₃

C₂

socle

C₂⁵×C₇₃

C₂×C₄

prime radical

C₂⁵×D₇₃

C₄

cosocle

C₄×C₇₃

C₂⁶

LCS

G ▷ C₇₃

UCS

C₁ ◁ C₂⁴×C₈

N	H ≅ G/N	type
2. C₂⁴×#10596831	C₂	semidirect
3. C₂³×C₈×D₇₃	C₂	direct (N × H)
4. C₂⁴×C₈×C₇₃	C₂	semidirect
9. C₂³×#10596831	C₂²	semidirect
10. C₂²×C₈×D₇₃	C₂²	direct (N × H)
12. C₂³×C₈×C₇₃	C₂²	semidirect
13. C₂⁴×D₇₃	C₈	direct (N × H)
18. C₂²×#10596831	C₂³	semidirect
19. C₂×C₈×D₇₃	C₂³	direct (N × H)
23. C₂²×C₈×C₇₃	C₂³	semidirect
24. C₂³×D₇₃	C₂×C₈	direct (N × H)
29. C₂×#10596831	C₂⁴	semidirect
30. C₈×D₇₃	C₂⁴	direct (N × H)
31. C₂⁴×C₇₃	C₂×C₈	semidirect
35. C₂×C₈×C₇₃	C₂⁴	semidirect
36. C₂³×C₇₃	C₂²×C₈	semidirect
38. C₂²×D₇₃	C₂²×C₈	direct (N × H)
45. C₈×C₇₃	C₂⁵	semidirect
46. #10596831	C₂⁵	semidirect
47. C₂²×C₇₃	C₂³×C₈	semidirect
49. C₂×D₇₃	C₂³×C₈	direct (N × H)
55. C₂×C₇₃	C₂⁴×C₈	semidirect
57. D₇₃	C₂⁴×C₈	direct (N × H)
61. C₂⁴×C₈	D₇₃	direct (N × H)
62. C₇₃	C₂⁵×C₈	semidirect
70. C₂³×C₈	C₂×D₇₃	direct (N × H)
81. C₂²×C₈	C₂²×D₇₃	direct (N × H)
84. C₂⁴	C₈×D₇₃	direct (N × H)
93. C₂×C₈	C₂³×D₇₃	direct (N × H)
96. C₂³	C₂×C₈×D₇₃	direct (N × H)
104. C₈	C₂⁴×D₇₃	direct (N × H)
107. C₂²	C₂²×C₈×D₇₃	direct (N × H)
113. C₂	C₂³×C₈×D₇₃	direct (N × H)

H ≅ G/N

type

2. C₂⁴×#10596831

C₂

semidirect

3. C₂³×C₈×D₇₃

C₂

direct (N × H)

4. C₂⁴×C₈×C₇₃

C₂

semidirect

9. C₂³×#10596831

C₂²

semidirect

10. C₂²×C₈×D₇₃

C₂²

direct (N × H)

12. C₂³×C₈×C₇₃

C₂²

semidirect

13. C₂⁴×D₇₃

C₈

direct (N × H)

18. C₂²×#10596831

C₂³

semidirect

19. C₂×C₈×D₇₃

C₂³

direct (N × H)

23. C₂²×C₈×C₇₃

C₂³

semidirect

24. C₂³×D₇₃

C₂×C₈

direct (N × H)

29. C₂×#10596831

C₂⁴

semidirect

30. C₈×D₇₃

C₂⁴

direct (N × H)

31. C₂⁴×C₇₃

C₂×C₈

semidirect

35. C₂×C₈×C₇₃

C₂⁴

semidirect

36. C₂³×C₇₃

C₂²×C₈

semidirect

38. C₂²×D₇₃

C₂²×C₈

direct (N × H)

45. C₈×C₇₃

C₂⁵

semidirect

46. #10596831

C₂⁵

semidirect

47. C₂²×C₇₃

C₂³×C₈

semidirect

49. C₂×D₇₃

C₂³×C₈

direct (N × H)

55. C₂×C₇₃

C₂⁴×C₈

semidirect

57. D₇₃

C₂⁴×C₈

direct (N × H)

61. C₂⁴×C₈

D₇₃

direct (N × H)

62. C₇₃

C₂⁵×C₈

semidirect

70. C₂³×C₈

C₂×D₇₃

direct (N × H)

81. C₂²×C₈

C₂²×D₇₃

direct (N × H)

84. C₂⁴

C₈×D₇₃

direct (N × H)

93. C₂×C₈

C₂³×D₇₃

direct (N × H)

96. C₂³

C₂×C₈×D₇₃

direct (N × H)

104. C₈

C₂⁴×D₇₃

direct (N × H)

107. C₂²

C₂²×C₈×D₇₃

direct (N × H)

113. C₂

C₂³×C₈×D₇₃

direct (N × H)

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

subgroup	index
C₂⁴×C₄×D₇₃	2
C₂⁴×#10596831	2
C₂³×C₈×D₇₃	2
C₂⁴×C₈×C₇₃	2
C₂⁵×C₈	73

subgroup

index

C₂⁴×C₄×D₇₃

C₂⁴×#10596831

C₂³×C₈×D₇₃

C₂⁴×C₈×C₇₃

C₂⁵×C₈

Subgroups — C₂⁴×C₈×D₇₃ #33999737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C24×C8×D73 #33999737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₈×D₇₃ #33999737