Subgroups — #564×Dic₆₁ #29842735

Order 15616.

1800 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21449005	C₂	1	15	yes	maximal
2	#125×Dic₆₁	C₂	1	15	yes	maximal
3	C₂×C₆₁×#564	C₂	1	1	yes	maximal, characteristic
4	#16839661	C₂²	1	30	yes	—
5	#16839648	C₂²	1	60	yes	—
6	#16839628	C₂²	1	15	yes	—
7	C₄⋊C₄×Dic₆₁	C₂²	1	30	yes	—
8	C₄²×Dic₆₁	C₂²	1	5	yes	—
9	C₆₁×#564	C₄	1	16	yes	complemented
10	C₂×#125×C₆₁	C₂²	1	15	yes	—
11	C₂×C₄⋊Dic₆₁	C₂³	1	30	yes	—
12	C₂×C₄×Dic₆₁	C₂³	1	15	yes	—
13	C₂×C₄×Dic₆₁	C₂³	1	15	yes	—
14	#14519684	C₂³	1	60	yes	—
15	#125×C₆₁	C₂×C₄	1	120	yes	—
16	C₂×C₄⋊C₄×C₆₁	C₂³	1	30	yes	—
17	C₂×C₄²×C₆₁	C₂³	1	5	yes	—
18	C₂²×C₄×C₆₁	C₂⁴	1	15	yes	—
19	C₂²×Dic₆₁	C₂⁴	1	15	yes	—
20	C₂²×Dic₆₁	C₂⁴	1	1	yes	characteristic
21	C₄⋊C₄×C₆₁	C₂²×C₄	1	120	yes	—
22	C₄²×C₆₁	C₂²×C₄	1	20	yes	—
23	C₄⋊Dic₆₁	—	4	30	no	—
24	C₄×Dic₆₁	—	2	30	no	—
25	C₄×Dic₆₁	—	4	15	no	—
26	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
27	C₂×C₄×C₆₁	—	2	15	no	—
28	C₂×C₄×C₆₁	—	2	15	no	—
29	C₂×C₄×C₆₁	C₂³×C₄	1	30	yes	—
30	C₂×Dic₆₁	—	2	15	no	—
31	C₂×Dic₆₁	—	2	30	no	—
32	C₂×Dic₆₁	2₋¹⁺⁴	1	6	yes	—
33	C₄×#564	—	61	1	no	maximal, Sylow
34	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
35	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	3	yes	—
36	C₂²×C₆₁	#585	1	3	yes	—
37	C₄×C₆₁	—	2	30	no	—
38	Dic₆₁	—	4	15	no	—
39	Dic₆₁	#564	1	4	yes	direct product factor
40	C₂×#564	D₆₁	1	1	yes	characteristic
41	#2382	—	61	15	no	—
42	C₄×#125	—	61	15	no	—
43	C₂×C₆₁	C₄×2₋¹⁺⁴	1	3	yes	—
44	C₂×C₆₁	C₂×#564	1	1	yes	characteristic
45	C₂×C₆₁	#3531	1	3	yes	—
46	#564	Dic₆₁	1	16	yes	direct product factor
47	C₂×#125	C₂×D₆₁	1	15	yes	—
48	#389	—	61	30	no	—
49	#387	—	61	60	no	—
50	#382	—	61	15	no	—
51	C₄×C₄⋊C₄	—	61	30	no	—
52	C₄³	—	61	5	no	—
53	C₆₁	C₄×#564	1	1	yes	characteristic, complemented, Sylow
54	#125	C₂×Dic₆₁	1	120	yes	—
55	C₂×C₄⋊C₄	—	61	30	no	—
56	C₂×C₄⋊C₄	C₂²×D₆₁	1	30	yes	—
57	C₂×C₄²	—	61	15	no	—
58	C₂×C₄²	—	61	15	no	—
59	C₂×C₄²	C₂²×D₆₁	1	5	yes	—
60	#95	—	61	60	no	—
61	C₂²×C₄	—	61	1	no	—
62	C₂²×C₄	—	61	15	no	—
63	C₂²×C₄	C₂³×D₆₁	1	15	yes	—
64	C₄⋊C₄	—	244	30	no	—
65	C₄⋊C₄	C₂²×Dic₆₁	1	120	yes	—
66	C₄²	—	122	30	no	—
67	C₄²	—	244	15	no	—
68	C₄²	C₂²×Dic₆₁	1	20	yes	—
69	C₂³	C₂⁴×D₆₁	1	1	yes	characteristic
70	C₂×C₄	—	2	15	no	—
71	C₂×C₄	—	2	15	no	—
72	C₂×C₄	—	61	6	no	—
73	C₂×C₄	—	122	15	no	—
74	C₂×C₄	—	122	30	no	—
75	C₂×C₄	C₂³×Dic₆₁	1	30	yes	—
76	C₂²	C₂⁴×Dic₆₁	1	1	yes	characteristic
77	C₂²	2₋¹⁺⁴×D₆₁	1	3	yes	—
78	C₂²	#16840599	1	3	yes	—
79	C₄	—	2	30	no	—
80	C₄	—	61	4	no	—
81	C₄	—	244	15	no	—
82	C₂	2₋¹⁺⁴×Dic₆₁	1	3	yes	—
83	C₂	#564×D₆₁	1	1	yes	characteristic
84	C₂	#21459967	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₆₁
commutator	C₂²×C₆₁	C₂⁴×C₄
Fitting	C₂×C₆₁×#564	C₂
socle, prime radical, cosocle	C₂³×C₆₁	C₂⁵
hypercenter	C₂×#564	D₆₁
hypocenter	C₆₁	C₄×#564
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#564

Complements

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

N	H ≅ G/N	type
9. C₆₁×#564	C₄	semidirect
39. Dic₆₁	#564	direct (N × H)
46. #564	Dic₆₁	direct (N × H)
53. C₆₁	C₄×#564	semidirect

Maximal subgroups

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

subgroup	index
#21449005	2
#125×Dic₆₁	2
C₂×C₆₁×#564	2
C₄×#564	61

Subgroups — #564×Dic61 #29842735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #564×Dic₆₁ #29842735