Subgroups — C₂×#3773 #64234

Order 288.

146 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	#3773	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂×#157	C₄	1	4	yes	complemented
6	C₂×#157	C₂²	1	2	yes	—
7	C₂×C₃×Dic₃	—	2	2	no	—
8	#624	—	2	2	no	—
9	#624	C₂²	1	4	yes	—
10	C₂×C₄×D₃	—	6	2	no	—
11	C₂²×C₃²	D₄	1	1	yes	characteristic
12	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	—
13	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
14	C₂×C₃⋊D₃	Q₈	1	2	yes	—
15	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
16	#157	C₂×C₄	1	4	yes	complemented
17	C₃×Dic₃	—	2	4	no	—
18	C₂×C₄⋊C₄	—	9	1	no	maximal, Sylow
19	C₂²×D₃	—	6	2	no	—
20	C₂×C₃×C₄	—	6	2	no	—
21	C₂×Dic₃	—	2	2	no	—
22	C₄×D₃	—	6	4	no	—
23	C₄×D₃	—	6	4	no	—
24	C₂×C₃²	C₄⋊C₄	1	2	yes	complemented
25	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
26	C₃⋊D₃	C₄⋊C₄	1	2	yes	complemented
27	C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
28	C₃⋊D₃	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₄	—	9	2	no	—
30	C₂²×C₄	—	9	1	no	—
31	C₄⋊C₄	—	9	4	no	—
32	C₂²×C₃	—	2	2	no	—
33	C₂×D₃	—	6	2	no	—
34	C₂×D₃	—	6	2	no	—
35	C₂×D₃	—	6	4	no	—
36	C₂×D₃	—	6	4	no	—
37	C₃×C₄	—	6	4	no	—
38	Dic₃	—	2	4	no	—
39	C₃²	C₂×C₄⋊C₄	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	—	9	4	no	—
45	C₂×C₄	—	18	2	no	—
46	C₂×C₃	—	2	2	no	—
47	C₂×C₃	—	2	4	no	—
48	D₃	—	6	4	no	—
49	D₃	—	6	2	no	—
50	D₃	—	6	2	no	—
51	C₂²	—	9	1	no	—
52	C₂²	—	9	1	no	—
53	C₂²	—	9	2	no	—
54	C₂²	—	9	2	no	—
55	C₂²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
56	C₄	—	6	4	no	—
57	C₄	—	9	4	no	—
58	C₃	—	2	2	no	—
59	C₂	—	9	1	no	—
60	C₂	—	9	1	no	—
61	C₂	—	9	2	no	—
62	C₂	#3773	1	2	yes	direct product factor
63	C₂	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₂²×C₄
Frattini	C₂	C₂×GO⁺(4,𝔽₂)
Fitting, socle	C₂²×C₃²	D₄
prime radical	C₂²×C₃⋊D₃	C₂²
cosocle	C₂×C₃⋊D₃	C₂³
hypocenter	C₃²	C₂×C₄⋊C₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
3. #3773	C₂	direct (N × H)
5. C₂×#157	C₄	semidirect
16. #157	C₂×C₄	semidirect
24. C₂×C₃²	C₄⋊C₄	semidirect
26. C₃⋊D₃	C₄⋊C₄	semidirect
39. C₃²	C₂×C₄⋊C₄	semidirect
62. C₂	#3773	direct (N × H)

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
#3773	2
C₂×C₄⋊C₄	9

Subgroups — C2×#3773 #64234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3773 #64234