Subgroups — #29573744

Order 15616.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21443691	C₂	1	3	yes	maximal
2	#21443684	C₂	1	3	yes	maximal
3	C₆₁×#1455	C₂	1	1	yes	maximal, characteristic
4	#16839819	C₂²	1	3	yes	—
5	#16839630	C₂²	1	1	yes	characteristic
6	C₄×M₁₆×C₆₁	C₂²	1	3	yes	—
7	C₆₁×#322	C₄	1	3	yes	complemented
8	C₈²×C₆₁	C₄	1	1	yes	characteristic, complemented
9	C₄⋊#92418	—	2	3	no	—
10	C₂²⋊#92418	—	2	3	no	—
11	#14519688	—	2	3	no	—
12	#14519684	—	2	3	no	—
13	C₂×M₁₆×C₆₁	—	2	3	no	—
14	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
15	#97×C₆₁	—	2	3	no	—
16	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
17	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₆₁	#41	1	3	yes	—
19	C₂²×Dic₆₁	—	4	1	no	—
20	M₁₆×C₆₁	—	4	6	no	—
21	C₂×C₈×C₆₁	—	2	3	no	—
22	C₂×C₈×C₆₁	—	2	3	no	—
23	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
24	C₄²×C₆₁	#41	1	3	yes	—
25	C₂×#92418	—	4	3	no	—
26	C₂³×C₆₁	#126	1	1	yes	characteristic
27	C₂×C₄×C₆₁	—	4	3	no	—
28	C₂×C₄×C₆₁	#117	1	3	yes	—
29	C₂×C₄×C₆₁	#126	1	3	yes	—
30	C₂×Dic₆₁	—	8	3	no	—
31	C₈×C₆₁	—	2	6	no	—
32	C₈×C₆₁	—	4	3	no	—
33	#92418	—	8	3	no	—
34	#8803	—	61	1	no	maximal, Sylow
35	C₂²×C₆₁	—	2	3	no	—
36	C₂²×C₆₁	#383	1	1	yes	characteristic
37	C₄×C₆₁	—	2	3	no	—
38	C₄×C₆₁	—	2	3	no	—
39	Dic₆₁	—	16	1	no	—
40	#1455	D₆₁	1	1	yes	characteristic
41	#1291	—	61	3	no	—
42	#1288	—	61	3	no	—
43	C₂×C₆₁	—	4	1	no	—
44	C₂×C₆₁	#1841	1	3	yes	—
45	#432	—	61	3	no	—
46	C₄×M₁₆	C₂×D₆₁	1	3	yes	—
47	#383	—	61	1	no	—
48	#322	Dic₆₁	1	3	yes	complemented
49	C₈²	Dic₆₁	1	1	yes	characteristic, complemented
50	C₆₁	#8803	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₁₆	—	2	3	no	—
52	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
53	C₄⋊C₈	—	122	3	no	—
54	C₂²⋊C₈	—	122	3	no	—
55	#97	—	2	3	no	—
56	#97	—	122	3	no	—
57	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
58	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
59	#95	—	122	3	no	—
60	C₂²×C₄	—	244	1	no	—
61	C₂²×C₄	#11758222	1	3	yes	—
62	M₁₆	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	—	244	3	no	—
66	C₄²	#11758222	1	3	yes	—
67	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
68	C₂³	#14519771	1	1	yes	characteristic
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	488	3	no	—
71	C₂×C₄	#14519736	1	3	yes	—
72	C₂×C₄	#14519771	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₈	—	4	3	no	—
75	C₈	—	488	3	no	—
76	C₂²	—	2	3	no	—
77	C₂²	#16839630	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	976	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#21445968	1	3	yes	—

	subgroup	G/H
center	C₂²	#16839630
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1455	C₂
socle	C₂²×C₆₁	#383
prime radical	C₂³×C₆₁	#126
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1455	D₆₁
hypocenter	C₆₁	#8803
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
7. C₆₁×#322	C₄	semidirect
8. C₈²×C₆₁	C₄	semidirect
48. #322	Dic₆₁	semidirect
49. C₈²	Dic₆₁	semidirect
50. C₆₁	#8803	semidirect

Maximal subgroups

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

subgroup	index
#21443691	2
#21443684	2
C₆₁×#1455	2
#8803	61