Subgroups — #30595304

Order 15616.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460989	C₂	1	1	yes	maximal, characteristic
2	#21460983	C₂	1	1	yes	maximal, characteristic
3	Q₆₄×D₆₁	C₂	1	1	yes	maximal, characteristic
4	#16840690	—	2	1	no	—
5	#16840601	—	2	1	no	—
6	Q₃₂×D₆₁	—	2	1	no	—
7	Dic₉₇₆	C₄	1	1	yes	characteristic
8	#16839602	—	2	1	no	—
9	C₃₂×D₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×Q₆₄	C₄	1	1	yes	characteristic
11	#14519848	—	4	1	no	—
12	Q₁₆×D₆₁	—	4	1	no	—
13	Dic₄₈₈	—	2	1	no	—
14	#14519723	—	4	1	no	—
15	C₁₆×D₆₁	D₄	1	1	yes	characteristic
16	#14519683	D₄	1	1	yes	characteristic
17	Q₃₂×C₆₁	—	2	1	no	—
18	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
19	Q₈×D₆₁	—	8	1	no	—
20	#11758213	—	8	1	no	—
21	Q₁₆×C₆₁	—	4	1	no	—
22	#11758202	—	8	1	no	—
23	Dic₂₄₄	—	4	1	no	—
24	C₈×D₆₁	D₈	1	1	yes	characteristic
25	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
26	#11758185	SD₁₆	1	1	yes	characteristic
27	Q₈×C₆₁	—	8	1	no	—
28	C₄×D₆₁	—	16	1	no	—
29	C₄×D₆₁	D₁₆	1	1	yes	characteristic
30	Dic₁₂₂	—	8	1	no	—
31	Dic₁₂₂	—	16	1	no	—
32	#92420	—	16	1	no	—
33	C₈×C₆₁	#102	1	1	yes	characteristic
34	#92418	SD₃₂	1	1	yes	characteristic
35	#7542	—	61	1	no	maximal, Sylow
36	C₂×D₆₁	SD₆₄	1	1	yes	characteristic
37	C₄×C₆₁	—	16	1	no	—
38	C₄×C₆₁	#357	1	1	yes	characteristic
39	Dic₆₁	—	16	1	no	—
40	Dic₆₁	D₃₂	1	1	yes	characteristic
41	C₂×Q₆₄	—	61	1	no	—
42	M₁₂₈	—	61	1	no	—
43	#1425	—	61	1	no	—
44	C₂×C₆₁	#1415	1	1	yes	characteristic
45	D₆₁	—	2	1	no	—
46	C₂×Q₃₂	—	122	1	no	—
47	Q₆₄	—	61	1	no	—
48	Q₆₄	—	122	1	no	—
49	Q₆₄	#6923	1	1	yes	characteristic
50	C₂×C₃₂	—	61	1	no	—
51	#368	—	122	1	no	—
52	C₆₄	—	122	1	no	—
53	C₆₁	#7542	1	1	yes	characteristic, complemented, Sylow
54	C₂×Q₁₆	—	244	1	no	—
55	Q₃₂	—	2	1	no	—
56	Q₃₂	—	122	1	no	—
57	Q₃₂	—	244	1	no	—
58	C₂×C₁₆	—	61	1	no	—
59	#108	—	244	1	no	—
60	C₃₂	—	61	1	no	—
61	C₃₂	C₂×#6923	1	1	yes	characteristic
62	C₂×Q₈	—	488	1	no	—
63	Q₁₆	—	4	1	no	—
64	Q₁₆	—	488	1	no	—
65	Q₁₆	—	244	1	no	—
66	M₁₆	—	488	1	no	—
67	C₂×C₈	—	61	1	no	—
68	C₁₆	—	61	1	no	—
69	C₁₆	C₂²⋊#6923	1	1	yes	characteristic
70	Q₈	—	8	1	no	—
71	Q₈	—	976	1	no	—
72	Q₈	—	488	1	no	—
73	C₂×C₄	—	61	1	no	—
74	C₂×C₄	—	976	1	no	—
75	C₈	—	61	1	no	—
76	C₈	—	976	1	no	—
77	C₈	#14519839	1	1	yes	characteristic
78	C₂²	—	61	1	no	—
79	C₄	—	16	1	no	—
80	C₄	—	61	1	no	—
81	C₄	—	976	1	no	—
82	C₄	#16840674	1	1	yes	characteristic
83	C₂	—	122	1	no	—
84	C₂	#21460968	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21460968
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₃₂	C₂×#6923
Fitting	C₆₁×Q₆₄	C₄
socle	C₂×C₆₁	#1415
prime radical	C₂×D₆₁	SD₆₄
cosocle	C₃₂×D₆₁	C₂²
hypercenter	Q₆₄	#6923
hypocenter	C₆₁	#7542
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
53. C₆₁	#7542	semidirect

Maximal subgroups

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

subgroup	index
#21460989	2
#21460983	2
Q₆₄×D₆₁	2
#7542	61