Subgroups — #64283

Order 288.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	#3792	C₂	1	2	yes	maximal, complemented
3	#3788	C₂	1	2	yes	maximal, complemented
4	#3787	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃⋊D₃	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₄	1	1	yes	characteristic
7	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₄×C₃⋊D₃	C₄	1	1	yes	characteristic
9	C₄×C₃⋊D₃	C₄	1	1	yes	characteristic
10	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	#622	C₂²	1	4	yes	complemented
13	C₂×C₄×D₃	—	6	2	no	—
14	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃⋊D₃	—	2	2	no	—
16	C₂×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
18	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
21	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆	—	9	1	no	maximal, Sylow
23	C₂²×D₃	—	6	2	no	—
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×Dic₃	—	6	2	no	—
26	C₄×D₃	—	6	2	no	—
27	C₄×D₃	—	6	2	no	—
28	C₄×D₃	—	6	2	no	—
29	C₄×D₃	—	6	2	no	—
30	C₂×C₃²	—	2	1	no	—
31	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
32	C₃⋊D₃	—	2	1	no	—
33	C₃⋊D₃	M₁₆	1	2	yes	complemented
34	C₂²×C₄	—	9	1	no	—
35	M₁₆	—	9	2	no	—
36	M₁₆	—	9	2	no	—
37	C₂×C₈	—	9	2	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂×D₃	—	6	2	no	—
40	C₂×D₃	—	6	2	no	—
41	C₂×D₃	—	6	8	no	—
42	C₃×C₄	—	2	2	no	—
43	C₃×C₄	—	2	2	no	—
44	Dic₃	—	6	2	no	—
45	Dic₃	—	6	2	no	—
46	C₃²	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₂³	—	9	1	no	—
48	C₂×C₄	—	9	1	no	—
49	C₂×C₄	—	9	1	no	—
50	C₂×C₄	—	9	1	no	—
51	C₂×C₄	—	9	1	no	—
52	C₂×C₄	—	9	1	no	—
53	C₂×C₄	#157	1	1	yes	characteristic
54	C₈	—	9	4	no	—
55	C₂×C₃	—	2	4	no	—
56	C₂×C₃	—	2	2	no	—
57	D₃	—	6	4	no	—
58	D₃	—	6	4	no	—
59	C₂²	—	9	1	no	—
60	C₂²	—	9	1	no	—
61	C₂²	—	18	2	no	—
62	C₂²	C₂×#157	1	1	yes	characteristic
63	C₄	—	9	1	no	—
64	C₄	—	9	1	no	—
65	C₄	C₂×#157	1	1	yes	characteristic
66	C₄	C₂×#157	1	1	yes	characteristic
67	C₃	—	2	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	9	2	no	—
70	C₂	—	18	1	no	—
71	C₂	C₂²×#157	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#157
commutator, socle	C₂×C₃²	C₂²×C₄
Frattini	C₂	C₂²×#157
Fitting	C₂×C₃²×C₄	C₄
prime radical	C₂²×C₃⋊D₃	C₄
cosocle	C₃⋊Dic₃	C₂³
hypercenter	C₂×C₄	#157
hypocenter	C₃²	C₂×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. #3792	C₂	semidirect
3. #3788	C₂	semidirect
4. #3787	C₂	semidirect
12. #622	C₂²	semidirect
33. C₃⋊D₃	M₁₆	semidirect
46. C₃²	C₂×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊D₃	2
#3792	2
#3788	2
#3787	2
C₂×M₁₆	9