Subgroups — #29567694

Order 15616.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21443895	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839583	—	2	2	no	—
4	C₂×C₃₂×C₆₁	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₆₁	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
7	#14519717	—	4	2	no	—
8	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
11	C₃₂×C₆₁	—	2	1	no	—
12	C₃₂×C₆₁	C₈	1	2	yes	—
13	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
15	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
16	C₁₆×C₆₁	—	2	1	no	—
17	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
18	C₁₆×C₆₁	M₁₆	1	1	yes	characteristic
19	#11758185	—	8	2	no	—
20	C₂×C₄×C₆₁	—	2	1	no	—
21	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
22	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₆₁	#108	1	1	yes	characteristic
24	C₈×C₆₁	D₁₆	1	1	yes	characteristic
25	C₈×C₆₁	Q₃₂	1	1	yes	characteristic
26	#7462	—	61	1	no	maximal, Sylow
27	C₂²×C₆₁	#368	1	1	yes	characteristic
28	C₄×C₆₁	—	2	2	no	—
29	C₄×C₆₁	#335	1	1	yes	characteristic
30	C₄×C₆₁	#366	1	1	yes	characteristic
31	C₄×C₃₂	D₆₁	1	1	yes	characteristic
32	#1373	—	61	2	no	—
33	C₂×C₆₁	—	2	1	no	—
34	C₂×C₆₁	#1371	1	1	yes	characteristic
35	C₂×C₃₂	Dic₆₁	1	1	yes	characteristic
36	C₂×C₃₂	Dic₆₁	1	1	yes	characteristic
37	#364	—	122	2	no	—
38	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
39	C₆₁	#7462	1	1	yes	characteristic, complemented, Sylow
40	M₃₂	—	244	2	no	—
41	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
42	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
43	C₄×C₈	D₂₄₄	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	#92418	1	2	yes	—
46	C₂×C₈	D₄₈₈	1	1	yes	characteristic
47	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
48	C₄²	Dic₂₄₄	1	1	yes	characteristic
49	C₁₆	—	2	1	no	—
50	C₁₆	—	488	2	no	—
51	C₁₆	C₂×#92418	1	1	yes	characteristic
52	C₁₆	#11758194	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	#14519711	1	1	yes	characteristic
55	C₈	C₄⋊#92418	1	1	yes	characteristic
56	C₈	#14519713	1	1	yes	characteristic
57	C₈	D₉₇₆	1	1	yes	characteristic
58	C₈	Dic₄₈₈	1	1	yes	characteristic
59	C₂²	#16839592	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#16839515	1	1	yes	characteristic
62	C₄	#16839588	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#21443891	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₉₇₆
commutator	C₁₆×C₆₁	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₆₁
Fitting	C₄×C₃₂×C₆₁	C₂
socle	C₂×C₆₁	#1371
prime radical	C₂²×C₆₁	#368
cosocle	C₄×C₁₆×C₆₁	C₂²
hypercenter	C₄×C₃₂	D₆₁
hypocenter	C₆₁	#7462
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
39. C₆₁	#7462	semidirect

Maximal subgroups

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

subgroup	index
#21443895	2
C₄×C₃₂×C₆₁	2
#7462	61