Subgroups — #30595307

Order 15616.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460988	C₂	1	2	yes	maximal
2	C₄×#14519683	C₂	1	1	yes	maximal, characteristic
3	#16840601	—	2	2	no	—
4	C₂×#14519683	C₄	1	1	yes	characteristic
5	C₂×#14519683	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
7	#14519683	—	2	1	no	—
8	#14519683	D₄	1	1	yes	characteristic
9	#14519683	Q₈	1	1	yes	characteristic
10	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₆₁	C₈	1	1	yes	characteristic
13	C₂×C₈×C₆₁	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₆₁	C₁₆	1	1	yes	characteristic
16	C₁₆×C₆₁	—	2	1	no	—
17	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₆₁	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₆₁	—	2	1	no	—
20	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₆₁	C₃₂	1	2	yes	—
22	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₆₁	M₃₂	1	1	yes	characteristic
24	#7544	—	61	1	no	maximal, Sylow
25	C₂²×C₆₁	M₆₄	1	1	yes	characteristic
26	C₄×C₆₁	—	2	2	no	—
27	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₆₁	C₂×C₃₂	1	1	yes	characteristic
29	M₁₂₈	—	61	2	no	—
30	C₄×C₃₂	—	61	1	no	—
31	C₂×C₆₁	—	2	1	no	—
32	C₂×C₆₁	C₄⋊C₃₂	1	1	yes	characteristic
33	C₂×C₃₂	—	61	1	no	—
34	C₂×C₃₂	—	61	1	no	—
35	C₄×C₁₆	#6923	1	1	yes	characteristic
36	C₆₄	—	122	2	no	—
37	C₆₁	#7544	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	#92420	1	1	yes	characteristic
39	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
40	C₄×C₈	#92420	1	1	yes	characteristic
41	C₃₂	—	61	1	no	—
42	C₃₂	—	61	1	no	—
43	C₃₂	—	122	1	no	—
44	C₂×C₈	#11758187	1	1	yes	characteristic
45	C₂×C₈	C₂×#92420	1	1	yes	characteristic
46	C₄²	#11758187	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	#11758213	1	1	yes	characteristic
49	C₁₆	C₄⋊#6923	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×#11758187	1	1	yes	characteristic
52	C₈	#14519827	1	2	yes	—
53	C₈	C₄⋊#92420	1	1	yes	characteristic
54	C₈	#14519851	1	1	yes	characteristic
55	C₂²	#16840694	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₄⋊#11758187	1	1	yes	characteristic
58	C₄	C₂×#14519827	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₄⋊#14519827	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄⋊#6923
commutator	C₄×C₆₁	C₂×C₃₂
Frattini	C₂×C₁₆	C₂×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂×C₆₁	C₄⋊C₃₂
prime radical	C₂²×C₆₁	M₆₄
cosocle	C₂×#14519683	C₂²
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#7544
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
37. C₆₁	#7544	semidirect

Maximal subgroups

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

subgroup	index
#21460988	2
C₄×#14519683	2
#7544	61