Subgroups — #63434

Order 288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₉	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₉)⋊C₄	C₂	1	2	yes	maximal
4	Dic₉⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#872	—	3	1	no	maximal
7	C₂²×D₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₉	C₂²	1	2	yes	—
9	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₉	C₂²	1	2	yes	—
11	C₂×Dic₉	C₂²	1	1	yes	characteristic
12	C₄×D₉	C₄	1	4	yes	complemented
13	C₂×C₄×D₃	—	3	1	no	—
14	C₃×C₄²	D₃	1	1	yes	characteristic
15	(C₂×D₃)⋊C₄	—	3	2	no	—
16	Dic₃⋊C₄	—	3	2	no	—
17	C₄×Dic₃	—	3	1	no	—
18	C₂²×C₉	C₂³	1	1	yes	characteristic
19	C₂×D₉	—	2	2	no	—
20	C₂×D₉	C₂×C₄	1	2	yes	—
21	C₄×C₉	—	2	2	no	—
22	C₄×C₉	C₂×C₄	1	2	yes	—
23	Dic₉	—	2	2	no	—
24	Dic₉	C₂×C₄	1	2	yes	—
25	#117	—	9	1	no	Sylow
26	C₂²×D₃	—	3	1	no	—
27	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
28	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂×Dic₃	—	3	1	no	—
30	C₂×Dic₃	—	3	2	no	—
31	C₄×D₃	—	3	4	no	—
32	C₂×C₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₉	#41	1	2	yes	—
34	D₉	—	2	2	no	—
35	C₂²×C₄	—	9	1	no	—
36	C₄⋊C₄	—	9	2	no	—
37	C₂²⋊C₄	—	9	2	no	—
38	C₄²	—	9	1	no	—
39	C₄²	D₉	1	1	yes	characteristic, complemented
40	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
41	C₂×D₃	—	3	2	no	—
42	C₂×D₃	—	6	2	no	—
43	C₃×C₄	—	2	2	no	—
44	C₃×C₄	C₄×D₃	1	2	yes	—
45	Dic₃	—	3	2	no	—
46	Dic₃	—	6	2	no	—
47	C₉	#117	1	1	yes	characteristic, complemented, Sylow
48	C₂³	—	9	1	no	—
49	C₂×C₄	—	9	1	no	—
50	C₂×C₄	—	9	4	no	—
51	C₂×C₄	—	9	2	no	—
52	C₂×C₄	C₂×D₉	1	2	yes	—
53	C₂×C₄	C₂×D₉	1	1	yes	characteristic
54	C₂×C₃	C₂×C₄×D₃	1	1	yes	characteristic
55	C₂×C₃	#235	1	2	yes	—
56	D₃	—	6	2	no	—
57	C₂²	—	9	2	no	—
58	C₂²	—	18	2	no	—
59	C₂²	C₂²×D₉	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	9	2	no	—
62	C₄	—	18	2	no	—
63	C₄	C₄×D₉	1	2	yes	—
64	C₃	#872	1	1	yes	characteristic
65	C₂	—	18	2	no	—
66	C₂	C₂×C₄×D₉	1	1	yes	characteristic
67	C₂	#3697	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₉
commutator	C₂×C₉	C₂²×C₄
Frattini, socle	C₂²×C₃	C₂²×D₃
Fitting	C₄²×C₉	C₂
prime radical	C₂²×D₉	C₂²
cosocle	C₂²×C₉	C₂³
hypercenter	C₄²	D₉
hypocenter	C₉	#117
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×C₉	C₂	semidirect
4. Dic₉⋊C₄	C₂	semidirect
5. C₄×Dic₉	C₂	semidirect
12. C₄×D₉	C₄	semidirect
39. C₄²	D₉	semidirect
47. C₉	#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₉	2
C₄²×C₉	2
(C₂×D₉)⋊C₄	2
Dic₉⋊C₄	2
C₄×Dic₉	2
#872	3