Subgroups — C₂×C₉×C₂²⋊C₄ #63517

Order 288.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₉	C₂	1	2	yes	maximal, complemented
3	C₉×C₂²⋊C₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₉	—	2	4	no	—
6	C₂³×C₉	C₄	1	4	yes	complemented
7	C₂³×C₉	C₂²	1	2	yes	—
8	C₂³×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₉	—	2	4	no	—
10	C₂×C₄×C₉	C₂²	1	4	yes	complemented
11	C₂⁴×C₃	C₂×C₃	1	1	yes	characteristic
12	C₂²×C₃×C₄	C₂×C₃	1	2	yes	—
13	C₃×C₂²⋊C₄	C₂×C₃	1	4	yes	—
14	C₂²×C₉	—	2	8	no	—
15	C₂²×C₉	—	2	4	no	—
16	C₂²×C₉	C₂×C₄	1	2	yes	—
17	C₂²×C₉	C₂×C₄	1	4	yes	complemented
18	C₂²×C₉	D₄	1	4	yes	—
19	C₂²×C₉	C₂³	1	1	yes	characteristic
20	C₄×C₉	—	2	4	no	—
21	C₂×C₂²⋊C₄	C₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₂³×C₃	—	2	4	no	—
23	C₂³×C₃	C₃×C₄	1	4	yes	—
24	C₂³×C₃	C₂²×C₃	1	2	yes	—
25	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₃×C₄	—	2	4	no	—
27	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
28	C₂×C₉	—	2	4	no	—
29	C₂×C₉	C₂²⋊C₄	1	4	yes	direct product factor
30	C₂×C₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₉	C₂×D₄	1	2	yes	—
32	C₂⁴	C₂×C₉	1	1	yes	characteristic
33	C₂²×C₄	C₂×C₉	1	2	yes	complemented
34	C₂²⋊C₄	C₂×C₉	1	4	yes	direct product factor
35	C₂²×C₃	—	2	8	no	—
36	C₂²×C₃	—	2	4	no	—
37	C₂²×C₃	C₂×C₃×C₄	1	2	yes	—
38	C₂²×C₃	C₂×C₃×C₄	1	4	yes	—
39	C₂²×C₃	C₃×D₄	1	4	yes	—
40	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
41	C₃×C₄	—	2	4	no	—
42	C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³	—	2	4	no	—
44	C₂³	C₄×C₉	1	4	yes	complemented
45	C₂³	C₂²×C₉	1	2	yes	—
46	C₂³	C₂²×C₉	1	1	yes	characteristic
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	C₂²×C₉	1	4	yes	complemented
49	C₂×C₃	—	2	4	no	—
50	C₂×C₃	C₃×C₂²⋊C₄	1	4	yes	—
51	C₂×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
52	C₂×C₃	C₂×C₃×D₄	1	2	yes	—
53	C₂²	—	2	8	no	—
54	C₂²	—	2	4	no	—
55	C₂²	C₂×C₄×C₉	1	2	yes	—
56	C₂²	C₂×C₄×C₉	1	4	yes	complemented
57	C₂²	D₄×C₉	1	4	yes	—
58	C₂²	C₂³×C₉	1	1	yes	characteristic
59	C₄	—	2	4	no	—
60	C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
61	C₂	—	2	4	no	—
62	C₂	C₉×C₂²⋊C₄	1	4	yes	direct product factor
63	C₂	C₂²×C₄×C₉	1	1	yes	characteristic
64	C₂	C₂×D₄×C₉	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂²
commutator	C₂	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃
prime radical	C₂⁴×C₃	C₂×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₄×C₉	C₂	semidirect
3. C₉×C₂²⋊C₄	C₂	direct (N × H)
6. C₂³×C₉	C₄	semidirect
10. C₂×C₄×C₉	C₂²	semidirect
17. C₂²×C₉	C₂×C₄	semidirect
21. C₂×C₂²⋊C₄	C₉	direct (N × H)
29. C₂×C₉	C₂²⋊C₄	direct (N × H)
33. C₂²×C₄	C₂×C₉	semidirect
34. C₂²⋊C₄	C₂×C₉	direct (N × H)
42. C₉	C₂×C₂²⋊C₄	direct (N × H)
44. C₂³	C₄×C₉	semidirect
48. C₂×C₄	C₂²×C₉	semidirect
56. C₂²	C₂×C₄×C₉	semidirect
62. C₂	C₉×C₂²⋊C₄	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×C₉	2
C₂²×C₄×C₉	2
C₉×C₂²⋊C₄	2
C₂×C₃×C₂²⋊C₄	3

Subgroups — C2×C9×C22⋊C4 #63517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×C₂²⋊C₄ #63517