Subgroups — C₉×#102 #63404

Order 288.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#102	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₉	—	2	1	no	—
6	D₄×C₉	—	2	1	no	—
7	D₄×C₉	C₄	1	2	yes	complemented
8	C₂×C₄×C₉	—	2	1	no	—
9	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₈×C₉	—	2	1	no	—
11	C₂×C₃×D₄	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×C₈	C₂×C₃	1	1	yes	characteristic
13	C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₉	—	2	2	no	—
15	C₂²×C₉	—	4	1	no	—
16	C₂²×C₉	—	4	1	no	—
17	C₂²×C₉	D₄	1	1	yes	characteristic
18	C₄×C₉	—	4	1	no	—
19	C₄×C₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₉	D₄	1	1	yes	characteristic
21	#102	C₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₂³×C₃	—	2	1	no	—
23	C₃×D₄	—	2	1	no	—
24	C₃×D₄	C₃×C₄	1	2	yes	—
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
27	C₃×C₈	—	2	1	no	—
28	C₂×C₉	—	4	2	no	—
29	C₂×C₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₉	D₈	1	1	yes	characteristic
31	C₂×C₉	SD₁₆	1	1	yes	characteristic
32	C₂×D₄	C₂×C₉	1	1	yes	characteristic
33	C₂×C₈	C₂×C₉	1	1	yes	characteristic, complemented
34	C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
35	C₂²×C₃	—	2	2	no	—
36	C₂²×C₃	—	4	1	no	—
37	C₂²×C₃	—	4	1	no	—
38	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
39	C₃×C₄	—	4	1	no	—
40	C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
41	C₃×C₄	C₃×D₄	1	1	yes	characteristic
42	C₉	#102	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³	—	2	1	no	—
44	D₄	—	2	1	no	—
45	D₄	C₄×C₉	1	2	yes	complemented
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
48	C₈	—	2	1	no	—
49	C₂×C₃	—	4	2	no	—
50	C₂×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₃	C₃×D₈	1	1	yes	characteristic
52	C₂×C₃	C₃×SD₁₆	1	1	yes	characteristic
53	C₂²	—	2	2	no	—
54	C₂²	—	4	1	no	—
55	C₂²	—	4	1	no	—
56	C₂²	D₄×C₉	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	C₂×C₄×C₉	1	1	yes	characteristic
59	C₄	D₄×C₉	1	1	yes	characteristic
60	C₃	C₃×#102	1	1	yes	characteristic
61	C₂	—	4	2	no	—
62	C₂	C₉×C₂²⋊C₄	1	1	yes	characteristic
63	C₂	C₉×D₈	1	1	yes	characteristic
64	C₂	C₉×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₄
commutator	C₄	C₂×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂²×C₃
socle	C₂²×C₃	C₃×D₄
prime radical	C₂×C₃×D₄	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×C₉	C₂	semidirect
3. C₉×C₄⋊C₄	C₂	semidirect
7. D₄×C₉	C₄	semidirect
21. #102	C₉	direct (N × H)
33. C₂×C₈	C₂×C₉	semidirect
34. C₄⋊C₄	C₂×C₉	semidirect
42. C₉	#102	direct (N × H)
45. D₄	C₄×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₉	2
C₂×C₈×C₉	2
C₉×C₄⋊C₄	2
C₃×#102	3

Subgroups — C9×#102 #63404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#102 #63404