Group #41003608 — subgroups

Subgroups — C₂×D₈×D₄₂₁ #41003608

Order 26944.

338 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×D₄×D₄₂₁	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20269756	C₂	1	2	yes	maximal, complemented
4	D₈×D₄₂₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄₂₁	—	2	2	no	—
8	C₂×D₁₆₈₄	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₄₂₁	—	2	2	no	—
10	D₄×D₄₂₁	—	2	4	no	—
11	D₄×D₄₂₁	C₂²	1	8	yes	complemented
12	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
13	D₃₃₆₈	C₂²	1	4	yes	complemented
14	#20269756	C₂²	1	8	yes	complemented
15	C₂×#16809890	C₂²	1	1	yes	characteristic, complemented
16	C₈×D₄₂₁	C₂²	1	4	yes	complemented
17	C₂×D₄×C₄₂₁	C₂²	1	2	yes	complemented
18	D₈×C₄₂₁	C₂²	1	4	yes	complemented
19	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄₂₁	—	2	2	no	—
21	C₂²×D₄₂₁	—	2	8	no	—
22	C₂²×D₄₂₁	—	4	8	no	—
23	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
24	D₁₆₈₄	—	2	2	no	—
25	D₁₆₈₄	C₂³	1	4	yes	complemented
26	C₂²⋊D₄₂₁	—	2	8	no	—
27	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
28	C₄×D₄₂₁	D₄	1	2	yes	—
29	C₄×D₄₂₁	C₂³	1	2	yes	—
30	#16809890	C₂³	1	2	yes	complemented
31	C₂³×C₄₂₁	—	2	2	no	—
32	D₄×C₄₂₁	—	2	2	no	—
33	D₄×C₄₂₁	C₂³	1	4	yes	complemented
34	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
35	C₈×C₄₂₁	C₂³	1	2	yes	complemented
36	C₂²×C₄₂₁	—	2	4	no	—
37	C₂²×C₄₂₁	—	4	4	no	—
38	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
39	C₂×D₄₂₁	—	2	4	no	—
40	C₂×D₄₂₁	—	4	4	no	—
41	C₂×D₄₂₁	—	4	16	no	—
42	C₂×D₄₂₁	D₈	1	4	yes	direct product factor
43	C₂×D₄₂₁	C₂×D₄	1	2	yes	—
44	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
45	C₄×C₄₂₁	C₂⁴	1	1	yes	characteristic
46	Dic₄₂₁	C₂×D₄	1	2	yes	—
47	C₂×C₄₂₁	—	4	4	no	—
48	C₂×C₄₂₁	C₂×D₈	1	2	yes	complemented
49	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
50	D₄₂₁	—	4	4	no	—
51	D₄₂₁	C₂×D₈	1	4	yes	direct product factor
52	C₄₂₁	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
53	C₂²×D₈	—	421	1	no	maximal, Sylow
54	C₂²×D₄	—	421	2	no	—
55	C₂×D₈	—	421	2	no	—
56	C₂×D₈	—	421	8	no	—
57	C₂×D₈	—	421	1	no	—
58	C₂×D₈	D₄₂₁	1	1	yes	characteristic, direct product factor
59	C₂²×C₈	—	421	1	no	—
60	C₂⁴	—	842	2	no	—
61	C₂×D₄	—	842	4	no	—
62	C₂×D₄	—	842	2	no	—
63	C₂×D₄	—	421	8	no	—
64	C₂×D₄	—	421	2	no	—
65	C₂×D₄	C₂×D₄₂₁	1	2	yes	complemented
66	C₂²×C₄	—	421	1	no	—
67	D₈	—	421	8	no	—
68	D₈	—	421	4	no	—
69	D₈	C₂×D₄₂₁	1	4	yes	direct product factor
70	C₂×C₈	—	421	1	no	—
71	C₂×C₈	—	421	4	no	—
72	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic, complemented
73	C₂³	—	2	2	no	—
74	C₂³	—	842	8	no	—
75	C₂³	—	842	2	no	—
76	C₂³	—	1684	8	no	—
77	C₂³	—	421	1	no	—
78	D₄	—	2	2	no	—
79	D₄	—	842	8	no	—
80	D₄	—	842	2	no	—
81	D₄	—	421	4	no	—
82	D₄	C₂²×D₄₂₁	1	4	yes	complemented
83	C₂×C₄	—	421	1	no	—
84	C₂×C₄	—	421	2	no	—
85	C₂×C₄	—	421	2	no	—
86	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
87	C₈	—	421	2	no	—
88	C₈	C₂²×D₄₂₁	1	2	yes	complemented
89	C₂²	—	2	4	no	—
90	C₂²	—	4	4	no	—
91	C₂²	—	842	4	no	—
92	C₂²	—	1684	16	no	—
93	C₂²	—	1684	4	no	—
94	C₂²	—	421	4	no	—
95	C₂²	—	421	2	no	—
96	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
97	C₄	—	421	2	no	—
98	C₄	D₄×D₄₂₁	1	1	yes	characteristic
99	C₄	C₂³×D₄₂₁	1	1	yes	characteristic
100	C₂	—	4	4	no	—
101	C₂	—	1684	4	no	—
102	C₂	—	421	4	no	—
103	C₂	D₈×D₄₂₁	1	2	yes	direct product factor
104	C₂	C₂×D₄×D₄₂₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₄₂₁

C₂

yes

maximal, complemented

C₂×D₃₃₆₈

C₂

yes

maximal, characteristic, complemented

C₂×#20269756

C₂

yes

maximal, complemented

D₈×D₄₂₁

C₂

yes

maximal, direct product factor

C₂×C₈×D₄₂₁

C₂

yes

maximal, characteristic, complemented

C₂×D₈×C₄₂₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₄₂₁

—

C₂×D₁₆₈₄

C₂²

yes

complemented

C₂×C₂²⋊D₄₂₁

—

D₄×D₄₂₁

—

D₄×D₄₂₁

C₂²

yes

complemented

C₂×C₄×D₄₂₁

C₂²

yes

characteristic

D₃₃₆₈

C₂²

yes

complemented

#20269756

C₂²

yes

complemented

C₂×#16809890

C₂²

yes

characteristic, complemented

C₈×D₄₂₁

C₂²

yes

complemented

C₂×D₄×C₄₂₁

C₂²

yes

complemented

D₈×C₄₂₁

C₂²

yes

complemented

C₂×C₈×C₄₂₁

C₂²

yes

characteristic, complemented

C₂²×D₄₂₁

—

C₂²×D₄₂₁

—

C₂²×D₄₂₁

—

C₂²×D₄₂₁

D₄

yes

characteristic

D₁₆₈₄

—

D₁₆₈₄

C₂³

yes

complemented

C₂²⋊D₄₂₁

—

C₂×Dic₄₂₁

D₄

yes

characteristic

C₄×D₄₂₁

D₄

yes

—

C₄×D₄₂₁

C₂³

yes

—

#16809890

C₂³

yes

complemented

C₂³×C₄₂₁

—

D₄×C₄₂₁

—

D₄×C₄₂₁

C₂³

yes

complemented

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₈×C₄₂₁

C₂³

yes

complemented

C₂²×C₄₂₁

—

C₂²×C₄₂₁

—

C₂²×C₄₂₁

C₂×D₄

yes

characteristic

C₂×D₄₂₁

—

C₂×D₄₂₁

—

C₂×D₄₂₁

—

C₂×D₄₂₁

D₈

yes

direct product factor

C₂×D₄₂₁

C₂×D₄

yes

—

C₄×C₄₂₁

C₂×D₄

yes

characteristic

C₄×C₄₂₁

C₂⁴

yes

characteristic

Dic₄₂₁

C₂×D₄

yes

—

C₂×C₄₂₁

—

C₂×C₄₂₁

C₂×D₈

yes

complemented

C₂×C₄₂₁

C₂²×D₄

yes

characteristic

D₄₂₁

—

D₄₂₁

C₂×D₈

yes

direct product factor

C₄₂₁

C₂²×D₈

yes

characteristic, complemented, Sylow

C₂²×D₈

—

421

maximal, Sylow

C₂²×D₄

—

421

—

C₂×D₈

—

421

—

C₂×D₈

—

421

—

C₂×D₈

—

421

—

C₂×D₈

D₄₂₁

yes

characteristic, direct product factor

C₂²×C₈

—

421

—

C₂⁴

—

842

—

C₂×D₄

—

842

—

C₂×D₄

—

842

—

C₂×D₄

—

421

—

C₂×D₄

—

421

—

C₂×D₄

C₂×D₄₂₁

yes

complemented

C₂²×C₄

—

421

—

D₈

—

421

—

D₈

—

421

—

D₈

C₂×D₄₂₁

yes

direct product factor

C₂×C₈

—

421

—

C₂×C₈

—

421

—

C₂×C₈

C₂×D₄₂₁

yes

characteristic, complemented

C₂³

—

C₂³

—

842

—

C₂³

—

842

—

C₂³

—

1684

—

C₂³

—

421

—

D₄

—

D₄

—

842

—

D₄

—

842

—

D₄

—

421

—

D₄

C₂²×D₄₂₁

yes

complemented

C₂×C₄

—

421

—

C₂×C₄

—

421

—

C₂×C₄

—

421

—

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

C₈

—

421

—

C₈

C₂²×D₄₂₁

yes

complemented

C₂²

—

C₂²

—

C₂²

—

842

—

C₂²

—

1684

—

C₂²

—

1684

—

C₂²

—

421

—

C₂²

—

421

—

C₂²

D₄×D₄₂₁

yes

characteristic

C₄

—

421

—

C₄

D₄×D₄₂₁

yes

characteristic

C₄

C₂³×D₄₂₁

yes

characteristic

100

C₂

—

101

C₂

—

1684

—

102

C₂

—

421

—

103

C₂

D₈×D₄₂₁

yes

direct product factor

104

C₂

C₂×D₄×D₄₂₁

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₄×C₄₂₁	C₂⁴
Frattini	C₄	C₂³×D₄₂₁
Fitting	C₂×D₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₂×D₈	D₄₂₁
hypocenter	C₄₂₁	C₂²×D₈
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×D₄₂₁

commutator, cosocle

C₄×C₄₂₁

C₂⁴

Frattini

C₄

C₂³×D₄₂₁

Fitting

C₂×D₈×C₄₂₁

C₂

socle

C₂²×C₄₂₁

C₂×D₄

hypercenter

C₂×D₈

D₄₂₁

hypocenter

C₄₂₁

C₂²×D₈

LCS

G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. C₂×D₄×D₄₂₁	C₂	semidirect
2. C₂×D₃₃₆₈	C₂	semidirect
3. C₂×#20269756	C₂	semidirect
4. D₈×D₄₂₁	C₂	direct (N × H)
5. C₂×C₈×D₄₂₁	C₂	semidirect
6. C₂×D₈×C₄₂₁	C₂	semidirect
8. C₂×D₁₆₈₄	C₂²	semidirect
11. D₄×D₄₂₁	C₂²	semidirect
13. D₃₃₆₈	C₂²	semidirect
14. #20269756	C₂²	semidirect
15. C₂×#16809890	C₂²	semidirect
16. C₈×D₄₂₁	C₂²	semidirect
17. C₂×D₄×C₄₂₁	C₂²	semidirect
18. D₈×C₄₂₁	C₂²	semidirect
19. C₂×C₈×C₄₂₁	C₂²	semidirect
25. D₁₆₈₄	C₂³	semidirect
30. #16809890	C₂³	semidirect
33. D₄×C₄₂₁	C₂³	semidirect
35. C₈×C₄₂₁	C₂³	semidirect
42. C₂×D₄₂₁	D₈	direct (N × H)
48. C₂×C₄₂₁	C₂×D₈	semidirect
51. D₄₂₁	C₂×D₈	direct (N × H)
52. C₄₂₁	C₂²×D₈	semidirect
58. C₂×D₈	D₄₂₁	direct (N × H)
65. C₂×D₄	C₂×D₄₂₁	semidirect
69. D₈	C₂×D₄₂₁	direct (N × H)
72. C₂×C₈	C₂×D₄₂₁	semidirect
82. D₄	C₂²×D₄₂₁	semidirect
88. C₈	C₂²×D₄₂₁	semidirect
103. C₂	D₈×D₄₂₁	direct (N × H)

H ≅ G/N

type

1. C₂×D₄×D₄₂₁

C₂

semidirect

2. C₂×D₃₃₆₈

C₂

semidirect

3. C₂×#20269756

C₂

semidirect

4. D₈×D₄₂₁

C₂

direct (N × H)

5. C₂×C₈×D₄₂₁

C₂

semidirect

6. C₂×D₈×C₄₂₁

C₂

semidirect

8. C₂×D₁₆₈₄

C₂²

semidirect

11. D₄×D₄₂₁

C₂²

semidirect

13. D₃₃₆₈

C₂²

semidirect

14. #20269756

C₂²

semidirect

15. C₂×#16809890

C₂²

semidirect

16. C₈×D₄₂₁

C₂²

semidirect

17. C₂×D₄×C₄₂₁

C₂²

semidirect

18. D₈×C₄₂₁

C₂²

semidirect

19. C₂×C₈×C₄₂₁

C₂²

semidirect

25. D₁₆₈₄

C₂³

semidirect

30. #16809890

C₂³

semidirect

33. D₄×C₄₂₁

C₂³

semidirect

35. C₈×C₄₂₁

C₂³

semidirect

42. C₂×D₄₂₁

D₈

direct (N × H)

48. C₂×C₄₂₁

C₂×D₈

semidirect

51. D₄₂₁

C₂×D₈

direct (N × H)

52. C₄₂₁

C₂²×D₈

semidirect

58. C₂×D₈

D₄₂₁

direct (N × H)

65. C₂×D₄

C₂×D₄₂₁

semidirect

69. D₈

C₂×D₄₂₁

direct (N × H)

72. C₂×C₈

C₂×D₄₂₁

semidirect

82. D₄

C₂²×D₄₂₁

semidirect

88. C₈

C₂²×D₄₂₁

semidirect

103. C₂

D₈×D₄₂₁

direct (N × H)

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

subgroup	index
C₂×D₄×D₄₂₁	2
C₂×D₃₃₆₈	2
C₂×#20269756	2
D₈×D₄₂₁	2
C₂×C₈×D₄₂₁	2
C₂×D₈×C₄₂₁	2
C₂²×D₈	421

subgroup

index

C₂×D₄×D₄₂₁

C₂×D₃₃₆₈

C₂×#20269756

D₈×D₄₂₁

C₂×C₈×D₄₂₁

C₂×D₈×C₄₂₁

C₂²×D₈

421

Subgroups — C₂×D₈×D₄₂₁ #41003608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D8×D421 #41003608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₈×D₄₂₁ #41003608