Group #26102337 — subgroups

Subgroups — C₂⁴×C₂²⋊C₄×C₄₇ #26102337

Order 12032.

43678 conjugacy classes of subgroups; only the 15454 normal subgroups are shown.

normal subgroups only

This group has 43678 conjugacy classes of subgroups — too many to draw. Only its 15454 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×C₄×C₄₇	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊C₄×C₄₇	C₂	1	60	yes	maximal, direct product factor
4	C₂⁶×C₄₇	—	2	32	no	—
5	C₂⁶×C₄₇	C₄	1	32	yes	complemented
6	C₂⁶×C₄₇	C₂²	1	30	yes	—
7	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₄₇	—	2	32	no	—
9	C₂⁴×C₄×C₄₇	C₂²	1	60	yes	complemented
10	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	560	yes	direct product factor
11	C₂⁵×C₄₇	—	2	512	no	—
12	C₂⁵×C₄₇	—	2	480	no	—
13	C₂⁵×C₄₇	C₂×C₄	1	16	yes	—
14	C₂⁵×C₄₇	C₂×C₄	1	480	yes	complemented
15	C₂⁵×C₄₇	D₄	1	32	yes	—
16	C₂⁵×C₄₇	C₂³	1	140	yes	—
17	C₂⁵×C₄₇	C₂³	1	15	yes	—
18	C₂³×C₄×C₄₇	—	2	480	no	—
19	C₂³×C₄×C₄₇	C₂³	1	280	yes	complemented
20	C₂×C₂²⋊C₄×C₄₇	C₂³	1	960	yes	direct product factor
21	C₂⁴×C₄₇	—	2	3840	no	—
22	C₂⁴×C₄₇	—	2	1120	no	—
23	C₂⁴×C₄₇	C₂²⋊C₄	1	256	yes	direct product factor
24	C₂⁴×C₄₇	C₂²×C₄	1	120	yes	—
25	C₂⁴×C₄₇	C₂²×C₄	1	1120	yes	complemented
26	C₂⁴×C₄₇	C₂×D₄	1	240	yes	—
27	C₂⁴×C₄₇	C₂⁴	1	120	yes	—
28	C₂⁴×C₄₇	C₂⁴	1	35	yes	—
29	C₂²×C₄×C₄₇	—	2	1120	no	—
30	C₂²×C₄×C₄₇	C₂⁴	1	240	yes	complemented
31	C₂²⋊C₄×C₄₇	C₂⁴	1	256	yes	direct product factor
32	C₂³×C₄₇	—	2	4480	no	—
33	C₂³×C₄₇	—	2	480	no	—
34	C₂³×C₄₇	C₂×C₂²⋊C₄	1	960	yes	direct product factor
35	C₂³×C₄₇	C₂³×C₄	1	140	yes	—
36	C₂³×C₄₇	C₂³×C₄	1	480	yes	complemented
37	C₂³×C₄₇	C₂²×D₄	1	280	yes	—
38	C₂³×C₄₇	C₂⁵	1	16	yes	—
39	C₂³×C₄₇	C₂⁵	1	15	yes	—
40	C₂×C₄×C₄₇	—	2	480	no	—
41	C₂×C₄×C₄₇	C₂⁵	1	32	yes	complemented
42	C₂⁴×C₂²⋊C₄	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₄₇	—	2	32	no	—
44	C₂²×C₄₇	—	2	960	no	—
45	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	560	yes	direct product factor
46	C₂²×C₄₇	C₂⁴×C₄	1	30	yes	—
47	C₂²×C₄₇	C₂⁴×C₄	1	32	yes	complemented
48	C₂²×C₄₇	C₂³×D₄	1	60	yes	—
49	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
50	C₄×C₄₇	—	2	32	no	—
51	C₂⁷	C₂×C₄₇	1	1	yes	characteristic
52	C₂⁵×C₄	C₂×C₄₇	1	2	yes	complemented
53	C₂³×C₂²⋊C₄	C₂×C₄₇	1	60	yes	direct product factor
54	C₂×C₄₇	—	2	32	no	—
55	C₂×C₄₇	C₂³×C₂²⋊C₄	1	60	yes	direct product factor
56	C₂×C₄₇	C₂⁵×C₄	1	1	yes	characteristic
57	C₂×C₄₇	C₂⁴×D₄	1	2	yes	—
58	C₂⁶	—	2	32	no	—
59	C₂⁶	C₄×C₄₇	1	32	yes	complemented
60	C₂⁶	C₂²×C₄₇	1	30	yes	—
61	C₂⁶	C₂²×C₄₇	1	1	yes	characteristic
62	C₂⁴×C₄	—	2	32	no	—
63	C₂⁴×C₄	C₂²×C₄₇	1	60	yes	complemented
64	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	560	yes	direct product factor
65	C₄₇	C₂⁴×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
66	C₂⁵	—	2	512	no	—
67	C₂⁵	—	2	480	no	—
68	C₂⁵	C₂×C₄×C₄₇	1	16	yes	—
69	C₂⁵	C₂×C₄×C₄₇	1	480	yes	complemented
70	C₂⁵	D₄×C₄₇	1	32	yes	—
71	C₂⁵	C₂³×C₄₇	1	140	yes	—
72	C₂⁵	C₂³×C₄₇	1	15	yes	—
73	C₂³×C₄	—	2	480	no	—
74	C₂³×C₄	C₂³×C₄₇	1	280	yes	complemented
75	C₂×C₂²⋊C₄	C₂³×C₄₇	1	960	yes	direct product factor
76	C₂⁴	—	2	3840	no	—
77	C₂⁴	—	2	1120	no	—
78	C₂⁴	C₂²⋊C₄×C₄₇	1	256	yes	direct product factor
79	C₂⁴	C₂²×C₄×C₄₇	1	120	yes	—
80	C₂⁴	C₂²×C₄×C₄₇	1	1120	yes	complemented
81	C₂⁴	C₂×D₄×C₄₇	1	240	yes	—
82	C₂⁴	C₂⁴×C₄₇	1	120	yes	—
83	C₂⁴	C₂⁴×C₄₇	1	35	yes	—
84	C₂²×C₄	—	2	1120	no	—
85	C₂²×C₄	C₂⁴×C₄₇	1	240	yes	complemented
86	C₂²⋊C₄	C₂⁴×C₄₇	1	256	yes	direct product factor
87	C₂³	—	2	4480	no	—
88	C₂³	—	2	480	no	—
89	C₂³	C₂×C₂²⋊C₄×C₄₇	1	960	yes	direct product factor
90	C₂³	C₂³×C₄×C₄₇	1	140	yes	—
91	C₂³	C₂³×C₄×C₄₇	1	480	yes	complemented
92	C₂³	C₂²×D₄×C₄₇	1	280	yes	—
93	C₂³	C₂⁵×C₄₇	1	16	yes	—
94	C₂³	C₂⁵×C₄₇	1	15	yes	—
95	C₂×C₄	—	2	480	no	—
96	C₂×C₄	C₂⁵×C₄₇	1	32	yes	complemented
97	C₂²	—	2	32	no	—
98	C₂²	—	2	960	no	—
99	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	560	yes	direct product factor
100	C₂²	C₂⁴×C₄×C₄₇	1	30	yes	—
101	C₂²	C₂⁴×C₄×C₄₇	1	32	yes	complemented
102	C₂²	C₂³×D₄×C₄₇	1	60	yes	—
103	C₂²	C₂⁶×C₄₇	1	1	yes	characteristic
104	C₄	—	2	32	no	—
105	C₂	—	2	32	no	—
106	C₂	C₂³×C₂²⋊C₄×C₄₇	1	60	yes	direct product factor
107	C₂	C₂⁵×C₄×C₄₇	1	1	yes	characteristic
108	C₂	C₂⁴×D₄×C₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁷×C₄₇

C₂

yes

maximal, characteristic

C₂⁵×C₄×C₄₇

C₂

yes

maximal, complemented

C₂³×C₂²⋊C₄×C₄₇

C₂

yes

maximal, direct product factor

C₂⁶×C₄₇

—

C₂⁶×C₄₇

C₄

yes

complemented

C₂⁶×C₄₇

C₂²

yes

—

C₂⁶×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄×C₄₇

—

C₂⁴×C₄×C₄₇

C₂²

yes

complemented

C₂²×C₂²⋊C₄×C₄₇

C₂²

560

yes

direct product factor

C₂⁵×C₄₇

—

512

—

C₂⁵×C₄₇

—

480

—

C₂⁵×C₄₇

C₂×C₄

yes

—

C₂⁵×C₄₇

C₂×C₄

480

yes

complemented

C₂⁵×C₄₇

D₄

yes

—

C₂⁵×C₄₇

C₂³

140

yes

—

C₂⁵×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

—

480

—

C₂³×C₄×C₄₇

C₂³

280

yes

complemented

C₂×C₂²⋊C₄×C₄₇

C₂³

960

yes

direct product factor

C₂⁴×C₄₇

—

3840

—

C₂⁴×C₄₇

—

1120

—

C₂⁴×C₄₇

C₂²⋊C₄

256

yes

direct product factor

C₂⁴×C₄₇

C₂²×C₄

120

yes

—

C₂⁴×C₄₇

C₂²×C₄

1120

yes

complemented

C₂⁴×C₄₇

C₂×D₄

240

yes

—

C₂⁴×C₄₇

C₂⁴

120

yes

—

C₂⁴×C₄₇

C₂⁴

yes

—

C₂²×C₄×C₄₇

—

1120

—

C₂²×C₄×C₄₇

C₂⁴

240

yes

complemented

C₂²⋊C₄×C₄₇

C₂⁴

256

yes

direct product factor

C₂³×C₄₇

—

4480

—

C₂³×C₄₇

—

480

—

C₂³×C₄₇

C₂×C₂²⋊C₄

960

yes

direct product factor

C₂³×C₄₇

C₂³×C₄

140

yes

—

C₂³×C₄₇

C₂³×C₄

480

yes

complemented

C₂³×C₄₇

C₂²×D₄

280

yes

—

C₂³×C₄₇

C₂⁵

yes

—

C₂³×C₄₇

C₂⁵

yes

—

C₂×C₄×C₄₇

—

480

—

C₂×C₄×C₄₇

C₂⁵

yes

complemented

C₂⁴×C₂²⋊C₄

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

960

—

C₂²×C₄₇

C₂²×C₂²⋊C₄

560

yes

direct product factor

C₂²×C₄₇

C₂⁴×C₄

yes

—

C₂²×C₄₇

C₂⁴×C₄

yes

complemented

C₂²×C₄₇

C₂³×D₄

yes

—

C₂²×C₄₇

C₂⁶

yes

characteristic

C₄×C₄₇

—

C₂⁷

C₂×C₄₇

yes

characteristic

C₂⁵×C₄

C₂×C₄₇

yes

complemented

C₂³×C₂²⋊C₄

C₂×C₄₇

yes

direct product factor

C₂×C₄₇

—

C₂×C₄₇

C₂³×C₂²⋊C₄

yes

direct product factor

C₂×C₄₇

C₂⁵×C₄

yes

characteristic

C₂×C₄₇

C₂⁴×D₄

yes

—

C₂⁶

—

C₂⁶

C₄×C₄₇

yes

complemented

C₂⁶

C₂²×C₄₇

yes

—

C₂⁶

C₂²×C₄₇

yes

characteristic

C₂⁴×C₄

—

C₂⁴×C₄

C₂²×C₄₇

yes

complemented

C₂²×C₂²⋊C₄

C₂²×C₄₇

560

yes

direct product factor

C₄₇

C₂⁴×C₂²⋊C₄

yes

characteristic, direct product factor, Sylow

C₂⁵

—

512

—

C₂⁵

—

480

—

C₂⁵

C₂×C₄×C₄₇

yes

—

C₂⁵

C₂×C₄×C₄₇

480

yes

complemented

C₂⁵

D₄×C₄₇

yes

—

C₂⁵

C₂³×C₄₇

140

yes

—

C₂⁵

C₂³×C₄₇

yes

—

C₂³×C₄

—

480

—

C₂³×C₄

C₂³×C₄₇

280

yes

complemented

C₂×C₂²⋊C₄

C₂³×C₄₇

960

yes

direct product factor

C₂⁴

—

3840

—

C₂⁴

—

1120

—

C₂⁴

C₂²⋊C₄×C₄₇

256

yes

direct product factor

C₂⁴

C₂²×C₄×C₄₇

120

yes

—

C₂⁴

C₂²×C₄×C₄₇

1120

yes

complemented

C₂⁴

C₂×D₄×C₄₇

240

yes

—

C₂⁴

C₂⁴×C₄₇

120

yes

—

C₂⁴

C₂⁴×C₄₇

yes

—

C₂²×C₄

—

1120

—

C₂²×C₄

C₂⁴×C₄₇

240

yes

complemented

C₂²⋊C₄

C₂⁴×C₄₇

256

yes

direct product factor

C₂³

—

4480

—

C₂³

—

480

—

C₂³

C₂×C₂²⋊C₄×C₄₇

960

yes

direct product factor

C₂³

C₂³×C₄×C₄₇

140

yes

—

C₂³

C₂³×C₄×C₄₇

480

yes

complemented

C₂³

C₂²×D₄×C₄₇

280

yes

—

C₂³

C₂⁵×C₄₇

yes

—

C₂³

C₂⁵×C₄₇

yes

—

C₂×C₄

—

480

—

C₂×C₄

C₂⁵×C₄₇

yes

complemented

C₂²

—

C₂²

—

960

—

C₂²

C₂²×C₂²⋊C₄×C₄₇

560

yes

direct product factor

100

C₂²

C₂⁴×C₄×C₄₇

yes

—

101

C₂²

C₂⁴×C₄×C₄₇

yes

complemented

102

C₂²

C₂³×D₄×C₄₇

yes

—

103

C₂²

C₂⁶×C₄₇

yes

characteristic

104

C₄

—

105

C₂

—

106

C₂

C₂³×C₂²⋊C₄×C₄₇

yes

direct product factor

107

C₂

C₂⁵×C₄×C₄₇

yes

characteristic

108

C₂

C₂⁴×D₄×C₄₇

yes

—

	subgroup	G/H
center, socle	C₂⁶×C₄₇	C₂²
commutator	C₂	C₂⁵×C₄×C₄₇
Frattini, cosocle	C₂²	C₂⁶×C₄₇
prime radical	C₂⁷×C₄₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂⁶×C₄₇

C₂²

commutator

C₂

C₂⁵×C₄×C₄₇

Frattini, cosocle

C₂²

C₂⁶×C₄₇

prime radical

C₂⁷×C₄₇

C₂

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂⁶×C₄₇ ◁ G

Complements

N	H ≅ G/N	type
2. C₂⁵×C₄×C₄₇	C₂	semidirect
3. C₂³×C₂²⋊C₄×C₄₇	C₂	direct (N × H)
5. C₂⁶×C₄₇	C₄	semidirect
9. C₂⁴×C₄×C₄₇	C₂²	semidirect
10. C₂²×C₂²⋊C₄×C₄₇	C₂²	direct (N × H)
14. C₂⁵×C₄₇	C₂×C₄	semidirect
19. C₂³×C₄×C₄₇	C₂³	semidirect
20. C₂×C₂²⋊C₄×C₄₇	C₂³	direct (N × H)
23. C₂⁴×C₄₇	C₂²⋊C₄	direct (N × H)
25. C₂⁴×C₄₇	C₂²×C₄	semidirect
30. C₂²×C₄×C₄₇	C₂⁴	semidirect
31. C₂²⋊C₄×C₄₇	C₂⁴	direct (N × H)
34. C₂³×C₄₇	C₂×C₂²⋊C₄	direct (N × H)
36. C₂³×C₄₇	C₂³×C₄	semidirect
41. C₂×C₄×C₄₇	C₂⁵	semidirect
42. C₂⁴×C₂²⋊C₄	C₄₇	direct (N × H)
45. C₂²×C₄₇	C₂²×C₂²⋊C₄	direct (N × H)
47. C₂²×C₄₇	C₂⁴×C₄	semidirect
52. C₂⁵×C₄	C₂×C₄₇	semidirect
53. C₂³×C₂²⋊C₄	C₂×C₄₇	direct (N × H)
55. C₂×C₄₇	C₂³×C₂²⋊C₄	direct (N × H)
59. C₂⁶	C₄×C₄₇	semidirect
63. C₂⁴×C₄	C₂²×C₄₇	semidirect
64. C₂²×C₂²⋊C₄	C₂²×C₄₇	direct (N × H)
65. C₄₇	C₂⁴×C₂²⋊C₄	direct (N × H)
69. C₂⁵	C₂×C₄×C₄₇	semidirect
74. C₂³×C₄	C₂³×C₄₇	semidirect
75. C₂×C₂²⋊C₄	C₂³×C₄₇	direct (N × H)
78. C₂⁴	C₂²⋊C₄×C₄₇	direct (N × H)
80. C₂⁴	C₂²×C₄×C₄₇	semidirect
85. C₂²×C₄	C₂⁴×C₄₇	semidirect
86. C₂²⋊C₄	C₂⁴×C₄₇	direct (N × H)
89. C₂³	C₂×C₂²⋊C₄×C₄₇	direct (N × H)
91. C₂³	C₂³×C₄×C₄₇	semidirect
96. C₂×C₄	C₂⁵×C₄₇	semidirect
99. C₂²	C₂²×C₂²⋊C₄×C₄₇	direct (N × H)
101. C₂²	C₂⁴×C₄×C₄₇	semidirect
106. C₂	C₂³×C₂²⋊C₄×C₄₇	direct (N × H)

H ≅ G/N

type

2. C₂⁵×C₄×C₄₇

C₂

semidirect

3. C₂³×C₂²⋊C₄×C₄₇

C₂

direct (N × H)

5. C₂⁶×C₄₇

C₄

semidirect

9. C₂⁴×C₄×C₄₇

C₂²

semidirect

10. C₂²×C₂²⋊C₄×C₄₇

C₂²

direct (N × H)

14. C₂⁵×C₄₇

C₂×C₄

semidirect

19. C₂³×C₄×C₄₇

C₂³

semidirect

20. C₂×C₂²⋊C₄×C₄₇

C₂³

direct (N × H)

23. C₂⁴×C₄₇

C₂²⋊C₄

direct (N × H)

25. C₂⁴×C₄₇

C₂²×C₄

semidirect

30. C₂²×C₄×C₄₇

C₂⁴

semidirect

31. C₂²⋊C₄×C₄₇

C₂⁴

direct (N × H)

34. C₂³×C₄₇

C₂×C₂²⋊C₄

direct (N × H)

36. C₂³×C₄₇

C₂³×C₄

semidirect

41. C₂×C₄×C₄₇

C₂⁵

semidirect

42. C₂⁴×C₂²⋊C₄

C₄₇

direct (N × H)

45. C₂²×C₄₇

C₂²×C₂²⋊C₄

direct (N × H)

47. C₂²×C₄₇

C₂⁴×C₄

semidirect

52. C₂⁵×C₄

C₂×C₄₇

semidirect

53. C₂³×C₂²⋊C₄

C₂×C₄₇

direct (N × H)

55. C₂×C₄₇

C₂³×C₂²⋊C₄

direct (N × H)

59. C₂⁶

C₄×C₄₇

semidirect

63. C₂⁴×C₄

C₂²×C₄₇

semidirect

64. C₂²×C₂²⋊C₄

C₂²×C₄₇

direct (N × H)

65. C₄₇

C₂⁴×C₂²⋊C₄

direct (N × H)

69. C₂⁵

C₂×C₄×C₄₇

semidirect

74. C₂³×C₄

C₂³×C₄₇

semidirect

75. C₂×C₂²⋊C₄

C₂³×C₄₇

direct (N × H)

78. C₂⁴

C₂²⋊C₄×C₄₇

direct (N × H)

80. C₂⁴

C₂²×C₄×C₄₇

semidirect

85. C₂²×C₄

C₂⁴×C₄₇

semidirect

86. C₂²⋊C₄

C₂⁴×C₄₇

direct (N × H)

89. C₂³

C₂×C₂²⋊C₄×C₄₇

direct (N × H)

91. C₂³

C₂³×C₄×C₄₇

semidirect

96. C₂×C₄

C₂⁵×C₄₇

semidirect

99. C₂²

C₂²×C₂²⋊C₄×C₄₇

direct (N × H)

101. C₂²

C₂⁴×C₄×C₄₇

semidirect

106. C₂

C₂³×C₂²⋊C₄×C₄₇

direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁷×C₄₇	2
C₂⁵×C₄×C₄₇	2
C₂³×C₂²⋊C₄×C₄₇	2
C₂⁴×C₂²⋊C₄	47

subgroup

index

C₂⁷×C₄₇

C₂⁵×C₄×C₄₇

C₂³×C₂²⋊C₄×C₄₇

C₂⁴×C₂²⋊C₄

Subgroups — C24×C22⋊C4×C47 #26102337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂²⋊C₄×C₄₇ #26102337