Subgroups — #30595058

Order 15616.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460978	C₂	1	2	yes	maximal
2	#21443735	C₂	1	1	yes	maximal, characteristic
3	#16840694	—	2	2	no	—
4	C₄×#11758185	C₂²	1	1	yes	characteristic
5	C₈×#92418	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
7	#14519827	—	4	2	no	—
8	C₂×#11758185	D₄	1	1	yes	characteristic
9	C₂×#11758185	Q₈	1	1	yes	characteristic
10	C₄×#92418	—	2	1	no	—
11	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
13	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
16	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
17	C₂×#92418	—	2	2	no	—
18	C₁₆×C₆₁	—	2	1	no	—
19	C₁₆×C₆₁	C₁₆	1	2	yes	—
20	#11758185	—	2	1	no	—
21	#11758185	SD₁₆	1	2	yes	—
22	C₂×C₄×C₆₁	—	2	1	no	—
23	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₆₁	#106	1	1	yes	characteristic
25	C₈×C₆₁	#108	1	1	yes	characteristic
26	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
27	C₈×C₆₁	M₃₂	1	1	yes	characteristic
28	#92418	—	4	2	no	—
29	#7456	—	61	1	no	maximal, Sylow
30	C₂²×C₆₁	#364	1	1	yes	characteristic
31	C₄×C₆₁	—	2	2	no	—
32	C₄×C₆₁	#334	1	1	yes	characteristic
33	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
34	#1422	—	61	2	no	—
35	#1311	—	61	1	no	—
36	C₂×C₆₁	—	2	1	no	—
37	C₂×C₆₁	#1367	1	1	yes	characteristic
38	M₆₄	—	122	2	no	—
39	C₄×C₁₆	—	61	1	no	—
40	C₄×C₁₆	#6923	1	1	yes	characteristic
41	C₈²	—	61	1	no	—
42	C₆₁	#7456	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	—	61	1	no	—
44	C₂×C₁₆	—	61	1	no	—
45	C₂×C₁₆	#92420	1	1	yes	characteristic
46	C₂×C₁₆	#92420	1	1	yes	characteristic
47	C₄×C₈	—	122	1	no	—
48	C₄×C₈	C₂×#6923	1	1	yes	characteristic
49	C₃₂	—	244	2	no	—
50	C₂×C₈	—	122	2	no	—
51	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
52	C₂×C₈	C₂×#92420	1	1	yes	characteristic
53	C₄²	#11758213	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	61	2	no	—
56	C₁₆	—	122	1	no	—
57	C₁₆	#11758187	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊#92420	1	1	yes	characteristic
60	C₈	—	244	2	no	—
61	C₈	#14519845	1	1	yes	characteristic
62	C₈	#14519847	1	1	yes	characteristic
63	C₈	C₂×#11758187	1	1	yes	characteristic
64	C₈	#14519851	1	1	yes	characteristic
65	C₂²	#16840684	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#16840623	1	1	yes	characteristic
68	C₄	C₄⋊#11758187	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#21460841	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519845
commutator	C₈×C₆₁	C₂×C₁₆
Frattini	C₄×C₈	C₂×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂×C₆₁	#1367
prime radical	C₂²×C₆₁	#364
cosocle	C₄×#11758185	C₂²
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#7456
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
42. C₆₁	#7456	semidirect

Maximal subgroups

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

subgroup	index
#21460978	2
#21443735	2
#7456	61