Subgroups — #29567688

Order 15616.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21444017	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839596	—	2	2	no	—
4	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
6	C₈²×C₆₁	C₄	1	1	yes	characteristic
7	#14519683	—	4	2	no	—
8	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
12	C₄×C₈×C₆₁	—	2	1	no	—
13	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₆₁	—	2	2	no	—
15	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
18	C₁₆×C₆₁	—	2	1	no	—
19	C₁₆×C₆₁	—	2	1	no	—
20	C₁₆×C₆₁	C₁₆	1	2	yes	—
21	C₁₆×C₆₁	D₈	1	1	yes	characteristic
22	C₁₆×C₆₁	Q₁₆	1	1	yes	characteristic
23	C₂×C₄×C₆₁	—	2	1	no	—
24	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₆₁	—	2	4	no	—
26	C₈×C₆₁	#107	1	1	yes	characteristic
27	C₈×C₆₁	#108	1	1	yes	characteristic
28	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
29	C₈×C₆₁	M₃₂	1	1	yes	characteristic
30	#7459	—	61	1	no	maximal, Sylow
31	C₂²×C₆₁	#364	1	1	yes	characteristic
32	C₄×C₆₁	—	2	2	no	—
33	C₄×C₆₁	#335	1	1	yes	characteristic
34	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
35	#1422	—	61	2	no	—
36	C₈×C₁₆	D₆₁	1	1	yes	characteristic
37	C₂×C₆₁	—	2	1	no	—
38	C₂×C₆₁	#1370	1	1	yes	characteristic
39	M₆₄	—	122	2	no	—
40	C₄×C₁₆	Dic₆₁	1	1	yes	characteristic
41	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
42	C₈²	Dic₆₁	1	1	yes	characteristic
43	C₆₁	#7459	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₁₆	#92418	1	1	yes	characteristic
45	C₂×C₁₆	#92418	1	1	yes	characteristic
46	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
47	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
50	C₃₂	—	244	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	C₂×#92418	1	1	yes	characteristic
53	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
54	C₄²	#11758194	1	1	yes	characteristic
55	C₁₆	—	2	1	no	—
56	C₁₆	—	2	1	no	—
57	C₁₆	#11758185	1	2	yes	—
58	C₁₆	D₄₈₈	1	1	yes	characteristic
59	C₁₆	Dic₂₄₄	1	1	yes	characteristic
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	C₄⋊#92418	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₈	#14519711	1	1	yes	characteristic
64	C₈	#14519713	1	1	yes	characteristic
65	C₈	C₂×#11758185	1	1	yes	characteristic
66	C₈	#14519717	1	1	yes	characteristic
67	C₂²	#16839583	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	#16839515	1	1	yes	characteristic
70	C₄	C₄⋊#11758185	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#21443889	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₄₈₈
commutator	C₈×C₆₁	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₆₁
Fitting	C₈×C₁₆×C₆₁	C₂
socle	C₂×C₆₁	#1370
prime radical	C₂²×C₆₁	#364
cosocle	C₄×C₁₆×C₆₁	C₂²
hypercenter	C₈×C₁₆	D₆₁
hypocenter	C₆₁	#7459
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
43. C₆₁	#7459	semidirect

Maximal subgroups

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

subgroup	index
#21444017	2
C₈×C₁₆×C₆₁	2
#7459	61