Subgroups — #29594728

Order 15616.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448072	C₂	1	4	yes	maximal
2	#21445437	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₈₈	—	2	8	no	—
5	#16840167	—	2	4	no	—
6	#16840142	C₂²	1	4	yes	—
7	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
8	C₈²×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₄₄	—	2	8	no	—
10	#14519777	—	2	4	no	—
11	Dic₄₈₈	—	4	16	no	—
12	#14519711	—	4	8	no	—
13	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
14	C₄×C₈×C₆₁	D₄	1	2	yes	—
15	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
16	C₂×Dic₁₂₂	—	4	4	no	—
17	C₂×C₈×C₆₁	D₈	1	4	yes	—
18	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
19	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
20	C₄⋊Dic₆₁	—	4	8	no	—
21	Dic₂₄₄	—	4	16	no	—
22	C₁₆×C₆₁	D₈	1	8	yes	—
23	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
24	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
25	C₂×Dic₆₁	—	8	4	no	—
26	Dic₁₂₂	—	8	8	no	—
27	C₈×C₆₁	Q₃₂	1	8	yes	complemented
28	C₈×C₆₁	C₂×D₈	1	4	yes	—
29	#12516	—	61	1	no	maximal, Sylow
30	C₂²×C₆₁	#493	1	1	yes	characteristic
31	C₄×C₆₁	#493	1	2	yes	—
32	C₄×C₆₁	C₂×Q₃₂	1	4	yes	—
33	Dic₆₁	—	16	4	no	—
34	#2247	—	61	4	no	—
35	#1713	—	61	2	no	—
36	C₈×C₁₆	D₆₁	1	1	yes	characteristic
37	C₂×C₆₁	#1712	1	1	yes	characteristic
38	C₂×C₆₁	#2247	1	2	yes	—
39	C₂×Q₃₂	—	122	8	no	—
40	#500	—	122	4	no	—
41	#494	—	61	4	no	—
42	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
43	C₈²	C₂×D₆₁	1	1	yes	characteristic
44	C₆₁	#12516	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₁₆	—	122	8	no	—
46	C₄⋊Q₈	—	122	4	no	—
47	Q₃₂	—	244	16	no	—
48	C₂×C₁₆	D₂₄₄	1	4	yes	—
49	#107	—	244	8	no	—
50	C₄×C₈	D₂₄₄	1	2	yes	—
51	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
52	C₂×Q₈	—	244	4	no	—
53	Q₁₆	—	244	16	no	—
54	C₂×C₈	D₄₈₈	1	4	yes	—
55	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
56	C₄⋊C₄	—	244	8	no	—
57	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
58	C₁₆	D₄₈₈	1	8	yes	—
59	Q₈	—	488	8	no	—
60	C₂×C₄	—	488	4	no	—
61	C₂×C₄	#14519774	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
63	C₈	Dic₄₈₈	1	8	yes	complemented
64	C₈	C₂×D₄₈₈	1	4	yes	—
65	C₂²	#16840139	1	1	yes	characteristic
66	C₄	—	976	4	no	—
67	C₄	#16840139	1	2	yes	—
68	C₄	C₂×Dic₄₈₈	1	4	yes	—
69	C₂	#21445434	1	1	yes	characteristic
70	C₂	#21448072	1	2	yes	—

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₈×C₁₆×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	C₈×C₁₆	D₆₁
hypocenter	C₆₁	#12516
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
27. C₈×C₆₁	Q₃₂	semidirect
44. C₆₁	#12516	semidirect
63. C₈	Dic₄₈₈	semidirect

Maximal subgroups

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

subgroup	index
#21448072	2
#21445437	2
C₈×C₁₆×C₆₁	2
#12516	61