Subgroups — #29573780

Order 15616.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445976	C₂	1	6	yes	maximal
2	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519709	—	2	12	no	—
4	#16839632	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
6	C₈²×C₆₁	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₆₁	—	2	12	no	—
8	#14519709	—	4	24	no	—
9	C₂²×C₈×C₆₁	D₄	1	6	yes	—
10	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
11	C₄×C₈×C₆₁	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₆₁	Q₈	1	6	yes	—
13	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
14	C₂²×Dic₆₁	—	4	4	no	—
15	C₂×C₈×C₆₁	C₄⋊C₄	1	12	yes	—
16	C₂×C₈×C₆₁	SD₁₆	1	24	yes	—
17	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₆₁	C₂×Q₈	1	3	yes	—
19	C₄⋊Dic₆₁	—	4	24	no	—
20	C₂³×C₆₁	#127	1	1	yes	characteristic
21	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
22	C₂×C₄×C₆₁	C₄⋊Q₈	1	3	yes	—
23	C₂×C₄×C₆₁	C₂×SD₁₆	1	12	yes	—
24	C₂×Dic₆₁	—	8	12	no	—
25	C₈×C₆₁	#106	1	24	yes	complemented
26	#8813	—	61	1	no	maximal, Sylow
27	C₂²×C₆₁	#384	1	1	yes	characteristic
28	C₂²×C₆₁	#492	1	6	yes	—
29	C₄×C₆₁	C₂×#106	1	6	yes	—
30	C₄×C₆₁	#498	1	6	yes	—
31	Dic₆₁	—	16	4	no	—
32	#1844	—	61	6	no	—
33	C₂×C₈²	D₆₁	1	1	yes	characteristic
34	C₂×C₆₁	#1705	1	4	yes	—
35	C₂×C₆₁	#1844	1	3	yes	—
36	C₂×#106	—	122	12	no	—
37	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
38	#384	—	61	4	no	—
39	C₈²	Dic₆₁	1	4	yes	complemented
40	C₆₁	#8813	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	D₂₄₄	1	6	yes	—
42	C₂×C₄⋊C₄	—	122	12	no	—
43	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
44	#106	—	244	24	no	—
45	C₄×C₈	Dic₁₂₂	1	6	yes	—
46	C₄×C₈	C₂×Dic₆₁	1	6	yes	—
47	C₂²×C₄	—	244	4	no	—
48	C₂²×C₄	C₂×D₂₄₄	1	3	yes	—
49	C₂×C₈	#11758190	1	24	yes	—
50	C₂×C₈	C₄⋊Dic₆₁	1	12	yes	—
51	C₄⋊C₄	—	244	24	no	—
52	C₄²	C₂×Dic₁₂₂	1	3	yes	—
53	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
54	C₂³	#14519774	1	1	yes	characteristic
55	C₂×C₄	—	488	12	no	—
56	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
57	C₂×C₄	#14519777	1	3	yes	—
58	C₂×C₄	C₂×#11758190	1	12	yes	—
59	C₈	#14519709	1	24	yes	complemented
60	C₂²	#16839632	1	1	yes	characteristic
61	C₂²	#16840135	1	6	yes	—
62	C₄	—	976	4	no	—
63	C₄	C₂×#14519709	1	6	yes	—
64	C₄	#16840160	1	6	yes	—
65	C₂	#21445406	1	4	yes	—
66	C₂	#21445976	1	3	yes	—

	subgroup	G/H
center	C₂³	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#8813
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₆₁	C₄	semidirect
25. C₈×C₆₁	#106	semidirect
39. C₈²	Dic₆₁	semidirect
40. C₆₁	#8813	semidirect
59. C₈	#14519709	semidirect

Maximal subgroups

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

subgroup	index
#21445976	2
C₂×C₈²×C₆₁	2
#8813	61