Subgroups — #30599521

Order 15616.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461740	C₂	1	6	yes	maximal
2	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#16840708	C₂²	1	4	yes	—
4	#16840636	—	2	12	no	—
5	C₄×C₈×D₆₁	C₂²	1	3	yes	—
6	C₈×#92418	C₄	1	3	yes	complemented
7	C₈²×C₆₁	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄⋊#6923	—	2	12	no	—
9	C₂×C₈×D₆₁	D₄	1	6	yes	—
10	C₄²×D₆₁	C₂³	1	1	yes	characteristic
11	C₄×#92418	C₂×C₄	1	3	yes	—
12	C₈×Dic₆₁	Q₈	1	6	yes	—
13	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
14	C₂²×#6923	—	4	4	no	—
15	C₂×C₄×D₆₁	C₂×D₄	1	3	yes	—
16	C₄⋊#6923	—	4	24	no	—
17	C₂×C₈×C₆₁	C₄⋊C₄	1	6	yes	—
18	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
19	C₄×Dic₆₁	C₂×Q₈	1	3	yes	—
20	C₂×#92418	C₄⋊C₄	1	6	yes	—
21	C₈×D₆₁	—	2	12	no	—
22	C₂²×D₆₁	#127	1	1	yes	characteristic
23	C₂×#6923	—	8	12	no	—
24	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
25	C₂×Dic₆₁	C₄⋊Q₈	1	3	yes	—
26	C₄×D₆₁	#135	1	12	yes	—
27	C₈×C₆₁	—	2	6	no	—
28	#92418	—	2	6	no	—
29	#8817	—	61	1	no	maximal, Sylow
30	C₂²×C₆₁	#384	1	1	yes	characteristic
31	C₂×D₆₁	#495	1	6	yes	—
32	#6923	—	16	4	no	—
33	C₄×C₆₁	#427	1	6	yes	—
34	Dic₆₁	#499	1	6	yes	—
35	#1834	—	61	6	no	—
36	C₂×C₈²	—	61	1	no	—
37	C₂×C₆₁	#1846	1	3	yes	—
38	D₆₁	#1711	1	4	yes	—
39	C₂×C₄×C₈	—	61	3	no	—
40	#384	—	61	4	no	—
41	#340	—	122	12	no	—
42	C₈²	—	61	3	no	—
43	C₈²	#6923	1	1	yes	characteristic, complemented
44	C₆₁	#8817	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	—	61	6	no	—
46	C₂×C₄⋊C₄	—	122	12	no	—
47	C₂×C₄²	—	61	1	no	—
48	C₄×C₈	—	61	6	no	—
49	C₄×C₈	—	61	3	no	—
50	C₄×C₈	C₂×#6923	1	3	yes	—
51	C₂²×C₄	—	61	3	no	—
52	C₂²×C₄	—	244	4	no	—
53	C₂×C₈	—	61	6	no	—
54	C₂×C₈	—	122	12	no	—
55	C₂×C₈	C₄⋊#6923	1	6	yes	—
56	C₄⋊C₄	—	244	24	no	—
57	C₄²	—	61	3	no	—
58	C₄²	C₂²×#6923	1	1	yes	characteristic
59	C₂³	—	61	1	no	—
60	C₂×C₄	—	61	3	no	—
61	C₂×C₄	—	61	12	no	—
62	C₂×C₄	—	488	12	no	—
63	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
64	C₈	—	2	6	no	—
65	C₈	—	122	6	no	—
66	C₂²	—	61	6	no	—
67	C₂²	#16840708	1	1	yes	characteristic
68	C₄	—	61	6	no	—
69	C₄	—	976	4	no	—
70	C₄	#16840772	1	6	yes	—
71	C₂	—	61	4	no	—
72	C₂	#21461756	1	3	yes	—

	subgroup	G/H
center	C₂²	#16840708
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	#384
prime radical	C₂²×D₆₁	#127
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8817
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
6. C₈×#92418	C₄	semidirect
7. C₈²×C₆₁	C₄	semidirect
43. C₈²	#6923	semidirect
44. C₆₁	#8817	semidirect

Maximal subgroups

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

subgroup	index
#21461740	2
C₈²×D₆₁	2
#8817	61