Subgroups — #29567898

Order 15616.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448128	C₂	1	1	yes	maximal, characteristic
2	#21444023	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840194	—	2	1	no	—
5	Dic₉₇₆	C₄	1	1	yes	characteristic
6	#16839601	—	2	1	no	—
7	D₁₉₅₂	C₄	1	1	yes	characteristic
8	#16839592	—	2	1	no	—
9	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×C₆₄	—	2	1	no	—
11	#14519799	—	4	1	no	—
12	Dic₄₈₈	—	2	1	no	—
13	#14519722	—	4	1	no	—
14	D₉₇₆	—	2	1	no	—
15	#14519713	—	4	1	no	—
16	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
17	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₆₁	D₄	1	1	yes	characteristic
19	#11758222	—	8	1	no	—
20	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
21	#11758194	—	8	1	no	—
22	Dic₂₄₄	—	4	1	no	—
23	D₄₈₈	—	4	1	no	—
24	#11758190	—	8	1	no	—
25	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₆₁	SD₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₆₁	—	16	1	no	—
29	D₂₄₄	—	8	1	no	—
30	C₄×D₆₁	—	16	1	no	—
31	Dic₁₂₂	—	8	1	no	—
32	C₈×C₆₁	#102	1	1	yes	characteristic
33	C₈×C₆₁	SD₃₂	1	1	yes	characteristic
34	#92418	—	16	1	no	—
35	#7539	—	61	1	no	maximal, Sylow
36	C₂²×C₆₁	SD₆₄	1	1	yes	characteristic
37	C₂×D₆₁	—	16	1	no	—
38	C₄×C₆₁	#357	1	1	yes	characteristic
39	C₄×C₆₁	D₃₂	1	1	yes	characteristic
40	Dic₆₁	—	16	1	no	—
41	#2262	—	61	1	no	—
42	C₂×C₆₄	D₆₁	1	1	yes	characteristic, complemented
43	#1425	—	61	1	no	—
44	C₂×C₆₁	—	2	1	no	—
45	C₂×C₆₁	#1415	1	1	yes	characteristic
46	D₆₁	—	32	1	no	—
47	#508	—	122	1	no	—
48	Q₆₄	—	61	1	no	—
49	SD₆₄	—	122	1	no	—
50	D₃₂	—	61	1	no	—
51	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
52	#368	—	122	1	no	—
53	C₆₄	—	2	1	no	—
54	C₆₁	#7539	1	1	yes	characteristic, complemented, Sylow
55	#135	—	244	1	no	—
56	Q₃₂	—	122	1	no	—
57	SD₃₂	—	244	1	no	—
58	D₁₆	—	122	1	no	—
59	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
60	#108	—	244	1	no	—
61	C₃₂	C₄×D₆₁	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₆₁	1	1	yes	characteristic
63	#41	—	488	1	no	—
64	Q₁₆	—	244	1	no	—
65	SD₁₆	—	488	1	no	—
66	D₈	—	244	1	no	—
67	M₁₆	—	488	1	no	—
68	C₂×C₈	D₄₈₈	1	1	yes	characteristic
69	C₁₆	#11758190	1	1	yes	characteristic
70	C₁₆	#11758198	1	1	yes	characteristic
71	Q₈	—	488	1	no	—
72	D₄	—	976	1	no	—
73	D₄	—	488	1	no	—
74	C₂×C₄	—	976	1	no	—
75	C₂×C₄	D₉₇₆	1	1	yes	characteristic
76	C₈	—	976	1	no	—
77	C₈	#14519699	1	1	yes	characteristic
78	C₈	#14519722	1	1	yes	characteristic
79	C₂²	—	976	1	no	—
80	C₂²	#16839601	1	1	yes	characteristic
81	C₄	—	976	1	no	—
82	C₄	#16839564	1	1	yes	characteristic
83	C₄	D₁₉₅₂	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	1952	1	no	—
86	C₂	#21443998	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₉₅₂
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₆₁
Fitting	C₂×C₆₁×C₆₄	C₂
socle	C₂×C₆₁	#1415
prime radical	#21448128	C₂
cosocle	C₂×C₃₂×C₆₁	C₂²
hypercenter	C₂×C₆₄	D₆₁
hypocenter	C₆₁	#7539
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #21444023	C₂	semidirect
3. C₂×C₆₁×C₆₄	C₂	semidirect
42. C₂×C₆₄	D₆₁	semidirect
54. C₆₁	#7539	semidirect

Maximal subgroups

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

subgroup	index
#21448128	2
#21444023	2
C₂×C₆₁×C₆₄	2
#7539	61