Subgroups — #29567697

Order 15616.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21443953	C₂	1	1	yes	maximal, characteristic
2	#21443895	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1373	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519683	C₄	1	1	yes	characteristic
5	C₂×#14519683	C₄	1	1	yes	characteristic
6	#16839583	—	2	1	no	—
7	C₆₁×#364	—	2	1	no	—
8	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
9	#14519717	—	4	1	no	—
10	#14519683	—	2	1	no	—
11	#14519683	C₈	1	2	yes	—
12	M₃₂×C₆₁	—	4	1	no	—
13	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
16	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
18	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
19	C₁₆×C₆₁	—	2	1	no	—
20	C₁₆×C₆₁	—	8	1	no	—
21	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₆₁	M₁₆	1	1	yes	characteristic
23	#11758185	—	8	1	no	—
24	C₂×C₄×C₆₁	—	2	1	no	—
25	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
26	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
27	C₈×C₆₁	#108	1	1	yes	characteristic
28	C₈×C₆₁	SD₃₂	1	1	yes	characteristic
29	C₈×C₆₁	SD₃₂	1	1	yes	characteristic
30	#7464	—	61	1	no	maximal, Sylow
31	C₂²×C₆₁	#368	1	1	yes	characteristic
32	C₄×C₆₁	—	4	1	no	—
33	C₄×C₆₁	#335	1	1	yes	characteristic
34	C₄×C₆₁	#367	1	1	yes	characteristic
35	#1397	—	61	1	no	—
36	#1373	—	61	1	no	—
37	#1373	D₆₁	1	1	yes	characteristic
38	C₂×C₆₁	—	2	1	no	—
39	C₂×C₆₁	#1372	1	1	yes	characteristic
40	C₂×C₃₂	—	61	1	no	—
41	C₂×C₃₂	—	61	1	no	—
42	#364	—	2	1	no	—
43	#364	—	122	1	no	—
44	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
45	C₆₁	#7464	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	4	1	no	—
47	M₃₂	—	244	1	no	—
48	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
49	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
50	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
51	C₃₂	—	61	2	no	—
52	C₃₂	—	122	1	no	—
53	C₂×C₈	#11758196	1	1	yes	characteristic
54	C₂×C₈	#11758199	1	1	yes	characteristic
55	C₄²	#11758202	1	1	yes	characteristic
56	C₁₆	—	2	1	no	—
57	C₁₆	—	8	1	no	—
58	C₁₆	—	488	1	no	—
59	C₁₆	C₈×D₆₁	1	1	yes	characteristic
60	C₁₆	#11758189	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#14519710	1	1	yes	characteristic
63	C₈	#14519706	1	1	yes	characteristic
64	C₈	#14519712	1	1	yes	characteristic
65	C₈	#14519720	1	1	yes	characteristic
66	C₈	#14519721	1	1	yes	characteristic
67	C₂²	#16839591	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#16839514	1	1	yes	characteristic
70	C₄	#16839589	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#21443892	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16839589
commutator	C₁₆×C₆₁	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₆₁
Fitting	C₆₁×#1373	C₂
socle	C₂×C₆₁	#1372
prime radical	C₂²×C₆₁	#368
cosocle	C₄×C₁₆×C₆₁	C₂²
hypercenter	#1373	D₆₁
hypocenter	C₆₁	#7464
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
45. C₆₁	#7464	semidirect

Maximal subgroups

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

subgroup	index
#21443953	2
#21443895	2
C₆₁×#1373	2
#7464	61