Subgroups — #29567674

Order 15616.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448101	C₂	1	1	yes	maximal, characteristic
2	#21444017	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1373	C₂	1	1	yes	maximal, characteristic
4	#16840167	—	2	1	no	—
5	#16839596	—	2	1	no	—
6	#16839590	—	2	1	no	—
7	#16839590	C₄	1	2	yes	—
8	C₆₁×#364	—	2	1	no	—
9	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	#14519777	—	4	1	no	—
11	#14519711	—	2	2	no	—
12	#14519711	—	4	1	no	—
13	#14519683	—	4	1	no	—
14	M₃₂×C₆₁	—	4	1	no	—
15	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
17	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
18	C₂×Dic₁₂₂	—	8	1	no	—
19	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
21	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
22	C₄⋊Dic₆₁	—	4	2	no	—
23	C₄⋊Dic₆₁	—	8	1	no	—
24	C₁₆×C₆₁	—	2	1	no	—
25	C₁₆×C₆₁	—	8	1	no	—
26	C₁₆×C₆₁	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₆₁	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₆₁	—	2	1	no	—
29	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
30	C₂×Dic₆₁	—	8	2	no	—
31	Dic₁₂₂	—	16	2	no	—
32	C₈×C₆₁	#101	1	1	yes	characteristic
33	C₈×C₆₁	#103	1	1	yes	characteristic
34	C₈×C₆₁	SD₃₂	1	1	yes	characteristic
35	C₈×C₆₁	Q₃₂	1	1	yes	characteristic
36	#7453	—	61	1	no	maximal, Sylow
37	C₂²×C₆₁	#361	1	1	yes	characteristic
38	C₄×C₆₁	—	2	2	no	—
39	C₄×C₆₁	#332	1	1	yes	characteristic
40	C₄×C₆₁	#358	1	1	yes	characteristic
41	Dic₆₁	—	16	2	no	—
42	#2254	—	61	1	no	—
43	#1422	—	61	1	no	—
44	#1373	D₆₁	1	1	yes	characteristic
45	C₂×C₆₁	—	2	1	no	—
46	C₂×C₆₁	#1365	1	1	yes	characteristic
47	#500	—	122	1	no	—
48	M₆₄	—	122	1	no	—
49	#367	—	61	2	no	—
50	#367	—	122	1	no	—
51	#364	—	2	1	no	—
52	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
53	C₆₁	#7453	1	1	yes	characteristic, complemented, Sylow
54	C₄⋊Q₈	—	244	1	no	—
55	M₃₂	—	4	1	no	—
56	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
57	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
58	#107	—	122	2	no	—
59	#107	—	244	1	no	—
60	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
61	C₃₂	—	244	1	no	—
62	C₂×Q₈	—	488	1	no	—
63	C₂×C₈	#11758198	1	1	yes	characteristic
64	C₂×C₈	#11758199	1	1	yes	characteristic
65	C₄⋊C₄	—	488	1	no	—
66	C₄⋊C₄	—	244	2	no	—
67	C₄²	#11758201	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	8	1	no	—
70	C₁₆	#11758190	1	1	yes	characteristic
71	C₁₆	Dic₂₄₄	1	1	yes	characteristic
72	Q₈	—	976	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	488	2	no	—
75	C₂×C₄	#14519697	1	1	yes	characteristic
76	C₈	#14519695	1	1	yes	characteristic
77	C₈	#14519702	1	1	yes	characteristic
78	C₈	#14519721	1	1	yes	characteristic
79	C₈	#14519723	1	1	yes	characteristic
80	C₂²	#16839574	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	976	2	no	—
83	C₄	#16839509	1	1	yes	characteristic
84	C₄	#16839565	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	#21443878	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21443878
commutator	C₂×C₁₆×C₆₁	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₆₁
Fitting	C₆₁×#1373	C₂
socle	C₂×C₆₁	#1365
prime radical	C₂²×C₆₁	#361
cosocle	C₄×C₁₆×C₆₁	C₂²
hypercenter	#1373	D₆₁
hypocenter	C₆₁	#7453
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
53. C₆₁	#7453	semidirect

Maximal subgroups

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

subgroup	index
#21448101	2
#21444017	2
C₆₁×#1373	2
#7453	61