Group #63708 — subgroups

Subgroups — C₂×D₄×D₉ #63708

Order 288.

354 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 116 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	C₂³×D₉	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#611	C₂	1	2	yes	maximal, complemented
4	D₄×D₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×D₃₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₃×D₄	—	3	1	no	maximal
8	C₂³×C₉	C₂²	1	2	yes	complemented
9	C₂²×D₉	—	2	8	no	—
10	C₂²×D₉	C₂²	1	1	yes	characteristic
11	C₂²×D₉	C₂²	1	8	yes	complemented
12	C₂²×D₉	C₂²	1	2	yes	complemented
13	D₄×C₉	C₂²	1	4	yes	complemented
14	C₂×C₄×C₉	C₂²	1	1	yes	characteristic, complemented
15	#611	C₂²	1	8	yes	complemented
16	C₂×Dic₉	C₂²	1	1	yes	characteristic, complemented
17	D₃₆	C₂²	1	4	yes	complemented
18	C₄×D₉	C₂²	1	4	yes	complemented
19	C₂³×D₃	—	3	2	no	—
20	C₂×C₃×D₄	D₃	1	1	yes	characteristic
21	C₂×#67	—	3	2	no	—
22	D₃×D₄	—	3	8	no	—
23	C₂×D₁₂	—	3	1	no	—
24	C₂×C₄×D₃	—	3	1	no	—
25	C₂²×C₉	—	2	4	no	—
26	C₂²×C₉	C₂³	1	1	yes	characteristic
27	C₂²×C₉	C₂³	1	4	yes	complemented
28	C₂×D₉	—	2	4	no	—
29	C₂×D₉	—	2	16	no	—
30	C₂×D₉	D₄	1	4	yes	direct product factor
31	C₂×D₉	C₂³	1	2	yes	—
32	C₂×D₉	C₂³	1	4	yes	complemented
33	C₄×C₉	C₂³	1	2	yes	complemented
34	Dic₉	C₂³	1	2	yes	complemented
35	C₂²×D₄	—	9	1	no	Sylow
36	C₂³×C₃	C₂×D₃	1	2	yes	—
37	C₂²×D₃	—	3	1	no	—
38	C₂²×D₃	—	3	2	no	—
39	C₂²×D₃	—	3	8	no	—
40	C₂²×D₃	—	6	8	no	—
41	C₃×D₄	C₂×D₃	1	4	yes	—
42	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
43	#67	—	3	8	no	—
44	C₂×Dic₃	—	3	1	no	—
45	D₁₂	—	3	4	no	—
46	C₄×D₃	—	3	4	no	—
47	C₂×C₉	—	2	4	no	—
48	C₂×C₉	C₂×D₄	1	2	yes	complemented
49	C₂×C₉	C₂⁴	1	1	yes	characteristic
50	D₉	—	2	4	no	—
51	D₉	C₂×D₄	1	4	yes	direct product factor
52	C₂⁴	—	9	2	no	—
53	C₂×D₄	—	9	2	no	—
54	C₂×D₄	—	9	8	no	—
55	C₂×D₄	—	9	1	no	—
56	C₂×D₄	D₉	1	1	yes	characteristic, direct product factor
57	C₂²×C₄	—	9	1	no	—
58	C₂²×C₃	—	2	4	no	—
59	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
60	C₂²×C₃	C₂²×D₃	1	4	yes	—
61	C₂×D₃	—	3	4	no	—
62	C₂×D₃	—	3	2	no	—
63	C₂×D₃	—	3	4	no	—
64	C₂×D₃	—	6	16	no	—
65	C₂×D₃	—	6	4	no	—
66	C₃×C₄	C₂²×D₃	1	2	yes	—
67	Dic₃	—	3	2	no	—
68	C₉	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
69	C₂³	—	9	1	no	—
70	C₂³	—	9	2	no	—
71	C₂³	—	9	8	no	—
72	C₂³	—	18	8	no	—
73	C₂³	C₂×D₉	1	2	yes	complemented
74	D₄	—	9	8	no	—
75	D₄	—	9	4	no	—
76	D₄	C₂×D₉	1	4	yes	direct product factor
77	C₂×C₄	—	9	1	no	—
78	C₂×C₄	—	9	4	no	—
79	C₂×C₄	C₂×D₉	1	1	yes	characteristic, complemented
80	C₂×C₃	—	2	4	no	—
81	C₂×C₃	D₃×D₄	1	2	yes	—
82	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
83	D₃	—	3	4	no	—
84	D₃	—	6	4	no	—
85	C₂²	—	2	4	no	—
86	C₂²	—	9	4	no	—
87	C₂²	—	9	2	no	—
88	C₂²	—	9	4	no	—
89	C₂²	—	18	16	no	—
90	C₂²	—	18	4	no	—
91	C₂²	C₂²×D₉	1	1	yes	characteristic
92	C₂²	C₂²×D₉	1	4	yes	complemented
93	C₄	—	9	2	no	—
94	C₄	C₂²×D₉	1	2	yes	complemented
95	C₃	C₂×D₃×D₄	1	1	yes	characteristic
96	C₂	—	2	4	no	—
97	C₂	—	9	4	no	—
98	C₂	—	18	4	no	—
99	C₂	D₄×D₉	1	2	yes	direct product factor
100	C₂	C₂³×D₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×D₉

C₂

yes

maximal, complemented

C₂×D₄×C₉

C₂

yes

maximal, characteristic, complemented

C₂×#611

C₂

yes

maximal, complemented

D₄×D₉

C₂

yes

maximal, direct product factor

C₂×D₃₆

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₉

C₂

yes

maximal, characteristic, complemented

C₂×D₃×D₄

—

maximal

C₂³×C₉

C₂²

yes

complemented

C₂²×D₉

—

C₂²×D₉

C₂²

yes

characteristic

C₂²×D₉

C₂²

yes

complemented

C₂²×D₉

C₂²

yes

complemented

D₄×C₉

C₂²

yes

complemented

C₂×C₄×C₉

C₂²

yes

characteristic, complemented

#611

C₂²

yes

complemented

C₂×Dic₉

C₂²

yes

characteristic, complemented

D₃₆

C₂²

yes

complemented

C₄×D₉

C₂²

yes

complemented

C₂³×D₃

—

C₂×C₃×D₄

D₃

yes

characteristic

C₂×#67

—

D₃×D₄

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂²×C₉

—

C₂²×C₉

C₂³

yes

characteristic

C₂²×C₉

C₂³

yes

complemented

C₂×D₉

—

C₂×D₉

—

C₂×D₉

D₄

yes

direct product factor

C₂×D₉

C₂³

yes

—

C₂×D₉

C₂³

yes

complemented

C₄×C₉

C₂³

yes

complemented

Dic₉

C₂³

yes

complemented

C₂²×D₄

—

Sylow

C₂³×C₃

C₂×D₃

yes

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₃×D₄

C₂×D₃

yes

—

C₂×C₃×C₄

C₂×D₃

yes

characteristic

#67

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

C₂×C₉

—

C₂×C₉

C₂×D₄

yes

complemented

C₂×C₉

C₂⁴

yes

characteristic

D₉

—

D₉

C₂×D₄

yes

direct product factor

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₉

yes

characteristic, direct product factor

C₂²×C₄

—

C₂²×C₃

—

C₂²×C₃

C₂²×D₃

yes

characteristic

C₂²×C₃

C₂²×D₃

yes

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

C₂²×D₃

yes

—

Dic₃

—

C₉

C₂²×D₄

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₉

yes

complemented

D₄

—

D₄

—

D₄

C₂×D₉

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₉

yes

characteristic, complemented

C₂×C₃

—

C₂×C₃

D₃×D₄

yes

—

C₂×C₃

C₂³×D₃

yes

characteristic

D₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₉

yes

characteristic

C₂²

C₂²×D₉

yes

complemented

C₄

—

C₄

C₂²×D₉

yes

complemented

C₃

C₂×D₃×D₄

yes

characteristic

C₂

—

C₂

—

C₂

—

C₂

D₄×D₉

yes

direct product factor

100

C₂

C₂³×D₉

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₉
commutator, cosocle	C₂×C₉	C₂⁴
Frattini	C₂×C₃	C₂³×D₃
Fitting	C₂×D₄×C₉	C₂
socle	C₂²×C₃	C₂²×D₃
hypercenter	C₂×D₄	D₉
hypocenter	C₉	C₂²×D₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center

C₂²

C₂²×D₉

commutator, cosocle

C₂×C₉

C₂⁴

Frattini

C₂×C₃

C₂³×D₃

Fitting

C₂×D₄×C₉

C₂

socle

C₂²×C₃

C₂²×D₃

hypercenter

C₂×D₄

D₉

hypocenter

C₉

C₂²×D₄

LCS

G ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. C₂³×D₉	C₂	semidirect
2. C₂×D₄×C₉	C₂	semidirect
3. C₂×#611	C₂	semidirect
4. D₄×D₉	C₂	direct (N × H)
5. C₂×D₃₆	C₂	semidirect
6. C₂×C₄×D₉	C₂	semidirect
8. C₂³×C₉	C₂²	semidirect
11. C₂²×D₉	C₂²	semidirect
12. C₂²×D₉	C₂²	semidirect
13. D₄×C₉	C₂²	semidirect
14. C₂×C₄×C₉	C₂²	semidirect
15. #611	C₂²	semidirect
16. C₂×Dic₉	C₂²	semidirect
17. D₃₆	C₂²	semidirect
18. C₄×D₉	C₂²	semidirect
27. C₂²×C₉	C₂³	semidirect
30. C₂×D₉	D₄	direct (N × H)
32. C₂×D₉	C₂³	semidirect
33. C₄×C₉	C₂³	semidirect
34. Dic₉	C₂³	semidirect
48. C₂×C₉	C₂×D₄	semidirect
51. D₉	C₂×D₄	direct (N × H)
56. C₂×D₄	D₉	direct (N × H)
68. C₉	C₂²×D₄	semidirect
73. C₂³	C₂×D₉	semidirect
76. D₄	C₂×D₉	direct (N × H)
79. C₂×C₄	C₂×D₉	semidirect
92. C₂²	C₂²×D₉	semidirect
94. C₄	C₂²×D₉	semidirect
99. C₂	D₄×D₉	direct (N × H)

H ≅ G/N

type

1. C₂³×D₉

C₂

semidirect

2. C₂×D₄×C₉

C₂

semidirect

3. C₂×#611

C₂

semidirect

4. D₄×D₉

C₂

direct (N × H)

5. C₂×D₃₆

C₂

semidirect

6. C₂×C₄×D₉

C₂

semidirect

8. C₂³×C₉

C₂²

semidirect

11. C₂²×D₉

C₂²

semidirect

12. C₂²×D₉

C₂²

semidirect

13. D₄×C₉

C₂²

semidirect

14. C₂×C₄×C₉

C₂²

semidirect

15. #611

C₂²

semidirect

16. C₂×Dic₉

C₂²

semidirect

17. D₃₆

C₂²

semidirect

18. C₄×D₉

C₂²

semidirect

27. C₂²×C₉

C₂³

semidirect

30. C₂×D₉

D₄

direct (N × H)

32. C₂×D₉

C₂³

semidirect

33. C₄×C₉

C₂³

semidirect

34. Dic₉

C₂³

semidirect

48. C₂×C₉

C₂×D₄

semidirect

51. D₉

C₂×D₄

direct (N × H)

56. C₂×D₄

D₉

direct (N × H)

68. C₉

C₂²×D₄

semidirect

73. C₂³

C₂×D₉

semidirect

76. D₄

C₂×D₉

direct (N × H)

79. C₂×C₄

C₂×D₉

semidirect

92. C₂²

C₂²×D₉

semidirect

94. C₄

C₂²×D₉

semidirect

99. C₂

D₄×D₉

direct (N × H)

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

subgroup	index
C₂³×D₉	2
C₂×D₄×C₉	2
C₂×#611	2
D₄×D₉	2
C₂×D₃₆	2
C₂×C₄×D₉	2
C₂×D₃×D₄	3

subgroup

index

C₂³×D₉

C₂×D₄×C₉

C₂×#611

D₄×D₉

C₂×D₃₆

C₂×C₄×D₉

C₂×D₃×D₄

Subgroups — C₂×D₄×D₉ #63708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D4×D9 #63708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₉ #63708