Group #14267675 — subgroups

Subgroups — C₂⁴×Q₈×D₇ #14267675

Order 1792.

63326 conjugacy classes of subgroups; only the 34795 normal subgroups are shown.

normal subgroups only

This group has 63326 conjugacy classes of subgroups — too many to draw. Only its 34795 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	C₂⁴×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₈×D₇	C₂	1	120	yes	maximal, direct product factor
3	C₂⁴×C₄×D₇	C₂	1	3	yes	maximal
4	C₂⁴×Dic₁₄	C₂	1	3	yes	maximal, complemented
5	C₂⁵×D₇	C₂²	1	1	yes	characteristic
6	C₂³×C₇×Q₈	C₂²	1	60	yes	complemented
7	C₂⁴×C₄×C₇	C₂²	1	3	yes	—
8	C₂⁴×Dic₇	C₂²	1	3	yes	—
9	C₂²×Q₈×D₇	C₂²	1	2240	yes	direct product factor
10	C₂³×C₄×D₇	C₂²	1	180	yes	—
11	C₂³×Dic₁₄	C₂²	1	180	yes	complemented
12	C₂⁵×Q₈	—	7	1	no	maximal, Sylow
13	C₂⁵×C₇	C₂³	1	1	yes	characteristic
14	C₂⁴×D₇	Q₈	1	32	yes	direct product factor
15	C₂⁴×D₇	C₂³	1	30	yes	—
16	C₂²×C₇×Q₈	C₂³	1	560	yes	complemented
17	C₂³×C₄×C₇	C₂³	1	90	yes	—
18	C₂³×Dic₇	C₂³	1	90	yes	—
19	C₂×Q₈×D₇	C₂³	1	7680	yes	direct product factor
20	C₂²×C₄×D₇	C₂³	1	1680	yes	—
21	C₂²×Dic₁₄	C₂³	1	1680	yes	complemented
22	C₂⁴×Q₈	—	7	120	no	—
23	C₂⁴×Q₈	—	7	3	no	—
24	C₂⁴×Q₈	D₇	1	1	yes	characteristic, direct product factor
25	C₂⁵×C₄	—	7	3	no	—
26	C₂⁴×C₇	C₂×Q₈	1	16	yes	complemented
27	C₂⁴×C₇	C₂⁴	1	15	yes	—
28	C₂³×D₇	C₂×Q₈	1	480	yes	direct product factor
29	C₂³×D₇	C₂⁴	1	140	yes	—
30	C₂×C₇×Q₈	C₂⁴	1	960	yes	complemented
31	C₂²×C₄×C₇	C₂⁴	1	420	yes	—
32	C₂²×Dic₇	C₂⁴	1	420	yes	—
33	Q₈×D₇	C₂⁴	1	4096	yes	direct product factor
34	C₂×C₄×D₇	C₂⁴	1	2880	yes	—
35	C₂×Dic₁₄	C₂⁴	1	2880	yes	complemented
36	C₂⁶	—	7	1	no	—
37	C₂³×Q₈	—	7	2240	no	—
38	C₂³×Q₈	—	7	180	no	—
39	C₂³×Q₈	C₂×D₇	1	60	yes	direct product factor
40	C₂⁴×C₄	—	7	3	no	—
41	C₂⁴×C₄	—	7	180	no	—
42	C₂⁴×C₄	C₂×D₇	1	3	yes	—
43	C₂³×C₇	C₂²×Q₈	1	120	yes	complemented
44	C₂³×C₇	C₂⁵	1	35	yes	—
45	C₂²×D₇	C₂²×Q₈	1	1120	yes	direct product factor
46	C₂²×D₇	C₂⁵	1	120	yes	—
47	C₇×Q₈	C₂⁵	1	256	yes	complemented
48	C₂×C₄×C₇	C₂⁵	1	360	yes	—
49	C₂×Dic₇	C₂⁵	1	360	yes	—
50	C₄×D₇	C₂⁵	1	768	yes	—
51	Dic₁₄	C₂⁵	1	768	yes	complemented
52	C₂⁵	—	7	32	no	—
53	C₂⁵	—	7	30	no	—
54	C₂⁵	C₂²×D₇	1	1	yes	characteristic
55	C₂²×Q₈	—	7	7680	no	—
56	C₂²×Q₈	—	7	1680	no	—
57	C₂²×Q₈	C₂²×D₇	1	560	yes	direct product factor
58	C₂³×C₄	—	7	90	no	—
59	C₂³×C₄	—	7	1680	no	—
60	C₂³×C₄	C₂²×D₇	1	90	yes	—
61	C₂²×C₇	C₂³×Q₈	1	140	yes	complemented
62	C₂²×C₇	C₂⁶	1	15	yes	—
63	C₂×D₇	C₂³×Q₈	1	480	yes	direct product factor
64	C₂×D₇	C₂⁶	1	16	yes	—
65	C₄×C₇	C₂⁶	1	48	yes	—
66	Dic₇	C₂⁶	1	48	yes	—
67	C₂⁴	—	7	480	no	—
68	C₂⁴	—	7	140	no	—
69	C₂⁴	Q₈×D₇	1	16	yes	direct product factor
70	C₂⁴	C₂³×D₇	1	15	yes	—
71	C₂×Q₈	—	7	4096	no	—
72	C₂×Q₈	—	7	2880	no	—
73	C₂×Q₈	C₂³×D₇	1	960	yes	direct product factor
74	C₂²×C₄	—	7	420	no	—
75	C₂²×C₄	—	7	2880	no	—
76	C₂²×C₄	C₂³×D₇	1	420	yes	—
77	C₂×C₇	C₂⁴×Q₈	1	30	yes	complemented
78	C₂×C₇	C₂⁷	1	1	yes	characteristic
79	D₇	C₂⁴×Q₈	1	32	yes	direct product factor
80	C₂³	—	7	120	no	—
81	C₂³	—	7	1120	no	—
82	C₂³	C₂×Q₈×D₇	1	120	yes	direct product factor
83	C₂³	C₂⁴×D₇	1	35	yes	—
84	Q₈	—	7	768	no	—
85	Q₈	C₂⁴×D₇	1	256	yes	direct product factor
86	C₂×C₄	—	7	360	no	—
87	C₂×C₄	—	7	768	no	—
88	C₂×C₄	C₂⁴×D₇	1	360	yes	—
89	C₇	C₂⁵×Q₈	1	1	yes	characteristic, complemented, Sylow
90	C₂²	—	7	480	no	—
91	C₂²	—	7	16	no	—
92	C₂²	C₂²×Q₈×D₇	1	140	yes	direct product factor
93	C₂²	C₂⁵×D₇	1	15	yes	—
94	C₄	—	7	48	no	—
95	C₄	C₂⁵×D₇	1	48	yes	—
96	C₂	—	7	32	no	—
97	C₂	C₂³×Q₈×D₇	1	30	yes	direct product factor
98	C₂	C₂⁶×D₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₇×Q₈

C₂

yes

maximal, characteristic, complemented

C₂³×Q₈×D₇

C₂

120

yes

maximal, direct product factor

C₂⁴×C₄×D₇

C₂

yes

maximal

C₂⁴×Dic₁₄

C₂

yes

maximal, complemented

C₂⁵×D₇

C₂²

yes

characteristic

C₂³×C₇×Q₈

C₂²

yes

complemented

C₂⁴×C₄×C₇

C₂²

yes

—

C₂⁴×Dic₇

C₂²

yes

—

C₂²×Q₈×D₇

C₂²

2240

yes

direct product factor

C₂³×C₄×D₇

C₂²

180

yes

—

C₂³×Dic₁₄

C₂²

180

yes

complemented

C₂⁵×Q₈

—

maximal, Sylow

C₂⁵×C₇

C₂³

yes

characteristic

C₂⁴×D₇

Q₈

yes

direct product factor

C₂⁴×D₇

C₂³

yes

—

C₂²×C₇×Q₈

C₂³

560

yes

complemented

C₂³×C₄×C₇

C₂³

yes

—

C₂³×Dic₇

C₂³

yes

—

C₂×Q₈×D₇

C₂³

7680

yes

direct product factor

C₂²×C₄×D₇

C₂³

1680

yes

—

C₂²×Dic₁₄

C₂³

1680

yes

complemented

C₂⁴×Q₈

—

120

—

C₂⁴×Q₈

—

C₂⁴×Q₈

D₇

yes

characteristic, direct product factor

C₂⁵×C₄

—

C₂⁴×C₇

C₂×Q₈

yes

complemented

C₂⁴×C₇

C₂⁴

yes

—

C₂³×D₇

C₂×Q₈

480

yes

direct product factor

C₂³×D₇

C₂⁴

140

yes

—

C₂×C₇×Q₈

C₂⁴

960

yes

complemented

C₂²×C₄×C₇

C₂⁴

420

yes

—

C₂²×Dic₇

C₂⁴

420

yes

—

Q₈×D₇

C₂⁴

4096

yes

direct product factor

C₂×C₄×D₇

C₂⁴

2880

yes

—

C₂×Dic₁₄

C₂⁴

2880

yes

complemented

C₂⁶

—

C₂³×Q₈

—

2240

—

C₂³×Q₈

—

180

—

C₂³×Q₈

C₂×D₇

yes

direct product factor

C₂⁴×C₄

—

C₂⁴×C₄

—

180

—

C₂⁴×C₄

C₂×D₇

yes

—

C₂³×C₇

C₂²×Q₈

120

yes

complemented

C₂³×C₇

C₂⁵

yes

—

C₂²×D₇

C₂²×Q₈

1120

yes

direct product factor

C₂²×D₇

C₂⁵

120

yes

—

C₇×Q₈

C₂⁵

256

yes

complemented

C₂×C₄×C₇

C₂⁵

360

yes

—

C₂×Dic₇

C₂⁵

360

yes

—

C₄×D₇

C₂⁵

768

yes

—

Dic₁₄

C₂⁵

768

yes

complemented

C₂⁵

—

C₂⁵

—

C₂⁵

C₂²×D₇

yes

characteristic

C₂²×Q₈

—

7680

—

C₂²×Q₈

—

1680

—

C₂²×Q₈

C₂²×D₇

560

yes

direct product factor

C₂³×C₄

—

C₂³×C₄

—

1680

—

C₂³×C₄

C₂²×D₇

yes

—

C₂²×C₇

C₂³×Q₈

140

yes

complemented

C₂²×C₇

C₂⁶

yes

—

C₂×D₇

C₂³×Q₈

480

yes

direct product factor

C₂×D₇

C₂⁶

yes

—

C₄×C₇

C₂⁶

yes

—

Dic₇

C₂⁶

yes

—

C₂⁴

—

480

—

C₂⁴

—

140

—

C₂⁴

Q₈×D₇

yes

direct product factor

C₂⁴

C₂³×D₇

yes

—

C₂×Q₈

—

4096

—

C₂×Q₈

—

2880

—

C₂×Q₈

C₂³×D₇

960

yes

direct product factor

C₂²×C₄

—

420

—

C₂²×C₄

—

2880

—

C₂²×C₄

C₂³×D₇

420

yes

—

C₂×C₇

C₂⁴×Q₈

yes

complemented

C₂×C₇

C₂⁷

yes

characteristic

D₇

C₂⁴×Q₈

yes

direct product factor

C₂³

—

120

—

C₂³

—

1120

—

C₂³

C₂×Q₈×D₇

120

yes

direct product factor

C₂³

C₂⁴×D₇

yes

—

Q₈

—

768

—

Q₈

C₂⁴×D₇

256

yes

direct product factor

C₂×C₄

—

360

—

C₂×C₄

—

768

—

C₂×C₄

C₂⁴×D₇

360

yes

—

C₇

C₂⁵×Q₈

yes

characteristic, complemented, Sylow

C₂²

—

480

—

C₂²

—

C₂²

C₂²×Q₈×D₇

140

yes

direct product factor

C₂²

C₂⁵×D₇

yes

—

C₄

—

C₄

C₂⁵×D₇

yes

—

C₂

—

C₂

C₂³×Q₈×D₇

yes

direct product factor

C₂

C₂⁶×D₇

yes

characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂⁷
Frattini	C₂	C₂⁶×D₇
Fitting	C₂⁴×C₇×Q₈	C₂
socle	C₂⁵×C₇	C₂³
prime radical	C₂⁵×D₇	C₂²
hypercenter	C₂⁴×Q₈	D₇
hypocenter	C₇	C₂⁵×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×Q₈

subgroup

G/H

center

C₂⁵

C₂²×D₇

commutator, cosocle

C₂×C₇

C₂⁷

Frattini

C₂

C₂⁶×D₇

Fitting

C₂⁴×C₇×Q₈

C₂

socle

C₂⁵×C₇

C₂³

prime radical

C₂⁵×D₇

C₂²

hypercenter

C₂⁴×Q₈

D₇

hypocenter

C₇

C₂⁵×Q₈

LCS

G ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂⁵ ◁ C₂⁴×Q₈

Complements

N	H ≅ G/N	type
1. C₂⁴×C₇×Q₈	C₂	semidirect
2. C₂³×Q₈×D₇	C₂	direct (N × H)
4. C₂⁴×Dic₁₄	C₂	semidirect
6. C₂³×C₇×Q₈	C₂²	semidirect
9. C₂²×Q₈×D₇	C₂²	direct (N × H)
11. C₂³×Dic₁₄	C₂²	semidirect
14. C₂⁴×D₇	Q₈	direct (N × H)
16. C₂²×C₇×Q₈	C₂³	semidirect
19. C₂×Q₈×D₇	C₂³	direct (N × H)
21. C₂²×Dic₁₄	C₂³	semidirect
24. C₂⁴×Q₈	D₇	direct (N × H)
26. C₂⁴×C₇	C₂×Q₈	semidirect
28. C₂³×D₇	C₂×Q₈	direct (N × H)
30. C₂×C₇×Q₈	C₂⁴	semidirect
33. Q₈×D₇	C₂⁴	direct (N × H)
35. C₂×Dic₁₄	C₂⁴	semidirect
39. C₂³×Q₈	C₂×D₇	direct (N × H)
43. C₂³×C₇	C₂²×Q₈	semidirect
45. C₂²×D₇	C₂²×Q₈	direct (N × H)
47. C₇×Q₈	C₂⁵	semidirect
51. Dic₁₄	C₂⁵	semidirect
57. C₂²×Q₈	C₂²×D₇	direct (N × H)
61. C₂²×C₇	C₂³×Q₈	semidirect
63. C₂×D₇	C₂³×Q₈	direct (N × H)
69. C₂⁴	Q₈×D₇	direct (N × H)
73. C₂×Q₈	C₂³×D₇	direct (N × H)
77. C₂×C₇	C₂⁴×Q₈	semidirect
79. D₇	C₂⁴×Q₈	direct (N × H)
82. C₂³	C₂×Q₈×D₇	direct (N × H)
85. Q₈	C₂⁴×D₇	direct (N × H)
89. C₇	C₂⁵×Q₈	semidirect
92. C₂²	C₂²×Q₈×D₇	direct (N × H)
97. C₂	C₂³×Q₈×D₇	direct (N × H)

H ≅ G/N

type

1. C₂⁴×C₇×Q₈

C₂

semidirect

2. C₂³×Q₈×D₇

C₂

direct (N × H)

4. C₂⁴×Dic₁₄

C₂

semidirect

6. C₂³×C₇×Q₈

C₂²

semidirect

9. C₂²×Q₈×D₇

C₂²

direct (N × H)

11. C₂³×Dic₁₄

C₂²

semidirect

14. C₂⁴×D₇

Q₈

direct (N × H)

16. C₂²×C₇×Q₈

C₂³

semidirect

19. C₂×Q₈×D₇

C₂³

direct (N × H)

21. C₂²×Dic₁₄

C₂³

semidirect

24. C₂⁴×Q₈

D₇

direct (N × H)

26. C₂⁴×C₇

C₂×Q₈

semidirect

28. C₂³×D₇

C₂×Q₈

direct (N × H)

30. C₂×C₇×Q₈

C₂⁴

semidirect

33. Q₈×D₇

C₂⁴

direct (N × H)

35. C₂×Dic₁₄

C₂⁴

semidirect

39. C₂³×Q₈

C₂×D₇

direct (N × H)

43. C₂³×C₇

C₂²×Q₈

semidirect

45. C₂²×D₇

C₂²×Q₈

direct (N × H)

47. C₇×Q₈

C₂⁵

semidirect

51. Dic₁₄

C₂⁵

semidirect

57. C₂²×Q₈

C₂²×D₇

direct (N × H)

61. C₂²×C₇

C₂³×Q₈

semidirect

63. C₂×D₇

C₂³×Q₈

direct (N × H)

69. C₂⁴

Q₈×D₇

direct (N × H)

73. C₂×Q₈

C₂³×D₇

direct (N × H)

77. C₂×C₇

C₂⁴×Q₈

semidirect

79. D₇

C₂⁴×Q₈

direct (N × H)

82. C₂³

C₂×Q₈×D₇

direct (N × H)

85. Q₈

C₂⁴×D₇

direct (N × H)

89. C₇

C₂⁵×Q₈

semidirect

92. C₂²

C₂²×Q₈×D₇

direct (N × H)

97. C₂

C₂³×Q₈×D₇

direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×Q₈	2
C₂³×Q₈×D₇	2
C₂⁴×C₄×D₇	2
C₂⁴×Dic₁₄	2
C₂⁵×Q₈	7

subgroup

index

C₂⁴×C₇×Q₈

C₂³×Q₈×D₇

C₂⁴×C₄×D₇

C₂⁴×Dic₁₄

C₂⁵×Q₈

Subgroups — C24×Q8×D7 #14267675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Q₈×D₇ #14267675