Subgroups — #63737

Order 288.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#157	C₂	1	1	yes	maximal, characteristic
2	C₂×#624	C₂	1	2	yes	maximal
3	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
4	C₂×#157	—	2	2	no	—
5	C₂×#157	C₄	1	2	yes	complemented
6	C₂×C₃×Dic₃	—	2	2	no	—
7	#624	—	2	2	no	—
8	#624	C₄	1	4	yes	complemented
9	C₂×C₄×D₃	—	6	2	no	—
10	C₂²×C₃²	D₄	1	1	yes	characteristic
11	C₂×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	—
13	C₂×C₃⋊D₃	D₄	1	2	yes	—
14	C₂×C₃⋊D₃	Q₈	1	1	yes	characteristic
15	#157	—	2	2	no	—
16	C₃×Dic₃	—	2	4	no	—
17	#95	—	9	1	no	maximal, Sylow
18	C₂²×D₃	—	6	2	no	—
19	C₂×C₃×C₄	—	6	2	no	—
20	C₂×Dic₃	—	2	2	no	—
21	C₄×D₃	—	6	4	no	—
22	C₄×D₃	—	6	4	no	—
23	C₂×C₃²	C₂²⋊C₄	1	2	yes	—
24	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
25	C₃⋊D₃	C₄²	1	1	yes	characteristic
26	C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₃⋊D₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄	—	9	2	no	—
29	C₂²×C₄	—	9	1	no	—
30	C₂²×C₃	—	2	2	no	—
31	C₂×D₃	—	6	2	no	—
32	C₂×D₃	—	6	2	no	—
33	C₂×D₃	—	6	2	no	—
34	C₂×D₃	—	6	2	no	—
35	C₂×D₃	—	6	2	no	—
36	C₂×D₃	—	6	2	no	—
37	C₃×C₄	—	6	4	no	—
38	Dic₃	—	2	4	no	—
39	C₃²	#95	1	1	yes	characteristic, complemented, Sylow
40	C₂³	—	9	1	no	—
41	C₂×C₄	—	6	2	no	—
42	C₂×C₄	—	9	4	no	—
43	C₂×C₄	—	9	2	no	—
44	C₂×C₄	—	18	2	no	—
45	C₂×C₄	—	18	2	no	—
46	C₂×C₃	—	2	2	no	—
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	—	2	2	no	—
49	D₃	—	6	2	no	—
50	D₃	—	6	2	no	—
51	D₃	—	6	2	no	—
52	D₃	—	6	2	no	—
53	C₂²	—	9	2	no	—
54	C₂²	—	9	2	no	—
55	C₂²	—	9	1	no	—
56	C₂²	—	9	1	no	—
57	C₂²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
58	C₄	—	6	4	no	—
59	C₄	—	18	2	no	—
60	C₃	—	2	2	no	—
61	C₂	—	9	1	no	—
62	C₂	—	9	1	no	—
63	C₂	—	9	2	no	—
64	C₂	D₃²⋊C₄	1	2	yes	—
65	C₂	#3773	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₄²
Fitting, socle	C₂²×C₃²	D₄
prime radical, cosocle	C₂²×C₃⋊D₃	C₂²
hypocenter	C₃²	#95
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
5. C₂×#157	C₄	semidirect
8. #624	C₄	semidirect
39. C₃²	#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#157	2
C₂×#624	2
#95	9