Subgroups — #29594720

Order 15616.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448046	C₂	1	4	yes	maximal
2	#21445429	C₂	1	1	yes	maximal, characteristic
3	#21443736	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1713	C₂	1	1	yes	maximal, characteristic
5	#16840167	C₂²	1	2	yes	—
6	#16840113	—	2	2	no	—
7	#16839566	—	2	8	no	—
8	C₄×#11758185	C₂²	1	2	yes	—
9	C₆₁×#500	—	2	2	no	—
10	C₆₁×#494	C₂²	1	2	yes	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	#14519777	—	2	2	no	—
13	C₂×#11758185	D₄	1	4	yes	—
14	#14519711	—	2	4	no	—
15	#14519702	—	4	4	no	—
16	C₂×Q₁₆×C₆₁	—	2	4	no	—
17	C₄⋊Q₈×C₆₁	—	2	2	no	—
18	#107×C₆₁	—	4	4	no	—
19	C₄×C₈×C₆₁	D₄	1	2	yes	—
20	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	4	2	no	—
22	C₂×Dic₁₂₂	—	4	2	no	—
23	Q₁₆×C₆₁	—	4	8	no	—
24	C₂×C₈×C₆₁	D₈	1	4	yes	—
25	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₆₁	—	4	4	no	—
27	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
28	C₄⋊Dic₆₁	—	4	2	no	—
29	C₄⋊Dic₆₁	—	4	2	no	—
30	#11758185	—	2	4	no	—
31	Q₈×C₆₁	—	8	4	no	—
32	C₂×C₄×C₆₁	—	8	2	no	—
33	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
35	C₂×Dic₆₁	—	8	2	no	—
36	Dic₁₂₂	—	8	4	no	—
37	C₈×C₆₁	—	2	4	no	—
38	C₈×C₆₁	#135	1	4	yes	—
39	#12514	—	61	1	no	maximal, Sylow
40	C₂²×C₆₁	#493	1	1	yes	characteristic
41	C₄×C₆₁	—	16	2	no	—
42	C₄×C₆₁	#495	1	2	yes	—
43	C₄×C₆₁	#510	1	4	yes	—
44	Dic₆₁	—	16	2	no	—
45	#2241	—	61	4	no	—
46	#1713	D₆₁	1	1	yes	characteristic
47	#1710	—	61	1	no	—
48	#1311	—	61	1	no	—
49	C₂×C₆₁	#1709	1	1	yes	characteristic
50	C₂×C₆₁	#2251	1	2	yes	—
51	#500	—	2	2	no	—
52	#500	—	61	2	no	—
53	#494	C₂×D₆₁	1	2	yes	—
54	#487	—	122	2	no	—
55	#358	—	122	8	no	—
56	C₄×C₁₆	—	61	2	no	—
57	C₈²	C₂×D₆₁	1	1	yes	characteristic
58	C₆₁	#12514	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	—	2	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	122	2	no	—
62	C₂×C₁₆	—	61	4	no	—
63	#107	—	4	4	no	—
64	#107	—	122	4	no	—
65	#103	—	244	4	no	—
66	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
67	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	244	2	no	—
70	Q₁₆	—	4	8	no	—
71	C₂×C₈	#11758199	1	4	yes	—
72	C₂×C₈	D₄×D₆₁	1	2	yes	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	244	2	no	—
75	C₄⋊C₄	—	244	2	no	—
76	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
77	C₁₆	—	122	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	488	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	488	2	no	—
82	C₂×C₄	#14519773	1	1	yes	characteristic
83	C₂×C₄	C₂×#11758199	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#14519796	1	4	yes	—
86	C₂²	#16840138	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	976	2	no	—
89	C₄	#16840145	1	2	yes	—
90	C₄	#16840203	1	4	yes	—
91	C₂	#21445424	1	1	yes	characteristic
92	C₂	#21448088	1	2	yes	—

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1713	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	#1713	D₆₁
hypocenter	C₆₁	#12514
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
58. C₆₁	#12514	semidirect

Maximal subgroups

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

subgroup	index
#21448046	2
#21445429	2
#21443736	2
C₆₁×#1713	2
#12514	61