Subgroups — #64286

Order 288.

130 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	C₂²⋊#157	C₂	1	2	yes	maximal
3	C₄⋊#157	C₂	1	2	yes	maximal, complemented
4	C₄×#157	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₂×#157	C₂²	1	4	yes	—
7	C₂×C₃²×C₄	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic, complemented
9	C₄×C₃⋊D₃	C₄	1	1	yes	characteristic, complemented
10	C₄×C₃⋊D₃	C₄	1	1	yes	characteristic, complemented
11	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
18	#157	—	2	4	no	—
19	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
22	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
23	#117	—	9	1	no	maximal, Sylow
24	C₂²×D₃	—	6	2	no	—
25	C₂×C₃×C₄	—	2	2	no	—
26	C₂×Dic₃	—	6	2	no	—
27	C₄×D₃	—	6	2	no	—
28	C₄×D₃	—	6	2	no	—
29	C₄×D₃	—	6	2	no	—
30	C₄×D₃	—	6	2	no	—
31	C₂×C₃²	—	2	1	no	—
32	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
33	C₃⋊D₃	—	2	1	no	—
34	C₃⋊D₃	#41	1	2	yes	—
35	C₂²×C₄	—	9	1	no	—
36	C₄⋊C₄	—	9	2	no	—
37	C₂²⋊C₄	—	9	2	no	—
38	C₄²	—	9	2	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂×D₃	—	6	2	no	—
41	C₂×D₃	—	6	2	no	—
42	C₂×D₃	—	6	8	no	—
43	C₃×C₄	—	2	2	no	—
44	C₃×C₄	—	2	2	no	—
45	Dic₃	—	6	2	no	—
46	Dic₃	—	6	2	no	—
47	C₃²	#117	1	1	yes	characteristic, complemented, Sylow
48	C₂³	—	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	—	9	1	no	—
54	C₂×C₄	—	9	4	no	—
55	C₂×C₄	#157	1	1	yes	characteristic, complemented
56	C₂×C₃	—	2	4	no	—
57	C₂×C₃	—	2	2	no	—
58	D₃	—	6	4	no	—
59	D₃	—	6	4	no	—
60	C₂²	—	9	1	no	—
61	C₂²	—	9	1	no	—
62	C₂²	—	18	2	no	—
63	C₂²	C₂×#157	1	1	yes	characteristic
64	C₄	—	9	1	no	—
65	C₄	—	9	1	no	—
66	C₄	—	18	4	no	—
67	C₄	C₂×#157	1	1	yes	characteristic
68	C₄	C₂×#157	1	1	yes	characteristic
69	C₃	—	2	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	9	2	no	—
72	C₂	—	18	1	no	—
73	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	C₂²
cosocle	C₂×C₃⋊D₃	C₂³
hypercenter	C₂×C₄	#157
hypocenter	C₃²	#117
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
3. C₄⋊#157	C₂	semidirect
4. C₄×#157	C₂	semidirect
7. C₂×C₃²×C₄	C₄	semidirect
8. C₂×C₃⋊Dic₃	C₄	semidirect
9. C₄×C₃⋊D₃	C₄	semidirect
10. C₄×C₃⋊D₃	C₄	semidirect
47. C₃²	#117	semidirect
55. C₂×C₄	#157	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
C₂²⋊#157	2
C₄⋊#157	2
C₄×#157	2
#117	9