Subgroups — #29567901

Order 15616.

90 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	#21444021	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840194	—	2	1	no	—
5	Dic₉₇₆	C₄	1	1	yes	characteristic, complemented
6	#16839601	—	2	1	no	—
7	D₁₉₅₂	C₄	1	1	yes	characteristic, complemented
8	#16839588	—	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	#14519711	—	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	C₄⋊Dic₆₁	—	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	C₂×Dic₆₁	—	16	1	no	—
30	D₂₄₄	—	8	1	no	—
31	C₄×D₆₁	—	16	1	no	—
32	Dic₁₂₂	—	8	1	no	—
33	C₈×C₆₁	#102	1	1	yes	characteristic
34	C₈×C₆₁	SD₃₂	1	1	yes	characteristic
35	#7540	—	61	1	no	maximal, Sylow
36	C₂²×C₆₁	D₃₂	1	1	yes	characteristic
37	C₂×D₆₁	—	16	1	no	—
38	C₄×C₆₁	#357	1	1	yes	characteristic
39	C₄×C₆₁	SD₆₄	1	1	yes	characteristic
40	Dic₆₁	—	16	1	no	—
41	Dic₆₁	—	32	1	no	—
42	#2262	—	61	1	no	—
43	M₁₂₈	D₆₁	1	1	yes	characteristic, complemented
44	#1424	—	61	1	no	—
45	C₂×C₆₁	—	2	1	no	—
46	C₂×C₆₁	#1415	1	1	yes	characteristic
47	D₆₁	—	32	1	no	—
48	#508	—	122	1	no	—
49	Q₆₄	—	61	1	no	—
50	SD₆₄	—	122	1	no	—
51	D₃₂	—	61	1	no	—
52	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
53	#366	—	122	1	no	—
54	C₆₄	—	2	1	no	—
55	C₆₁	#7540	1	1	yes	characteristic, complemented, Sylow
56	#135	—	244	1	no	—
57	Q₃₂	—	122	1	no	—
58	SD₃₂	—	244	1	no	—
59	D₁₆	—	122	1	no	—
60	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
61	#107	—	244	1	no	—
62	C₃₂	C₄×D₆₁	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₆₁	1	1	yes	characteristic
64	#41	—	488	1	no	—
65	Q₁₆	—	244	1	no	—
66	SD₁₆	—	488	1	no	—
67	D₈	—	244	1	no	—
68	C₂×C₈	D₄₈₈	1	1	yes	characteristic
69	C₄⋊C₄	—	488	1	no	—
70	C₁₆	#11758190	1	1	yes	characteristic
71	C₁₆	#11758198	1	1	yes	characteristic
72	Q₈	—	488	1	no	—
73	D₄	—	976	1	no	—
74	D₄	—	488	1	no	—
75	C₂×C₄	—	976	1	no	—
76	C₂×C₄	—	976	1	no	—
77	C₂×C₄	D₉₇₆	1	1	yes	characteristic
78	C₈	#14519699	1	1	yes	characteristic
79	C₈	#14519722	1	1	yes	characteristic
80	C₂²	—	976	1	no	—
81	C₂²	D₁₉₅₂	1	1	yes	characteristic
82	C₄	—	1952	1	no	—
83	C₄	—	976	1	no	—
84	C₄	#16839564	1	1	yes	characteristic
85	C₄	#16839601	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1952	1	no	—
88	C₂	#21443998	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21443998
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₆₁
Fitting	C₆₁×M₁₂₈	C₂
socle	C₂×C₆₁	#1415
prime radical	#21448128	C₂
cosocle	C₂×C₃₂×C₆₁	C₂²
hypercenter	M₁₂₈	D₆₁
hypocenter	C₆₁	#7540
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #21444021	C₂	semidirect
3. C₆₁×M₁₂₈	C₂	semidirect
5. Dic₉₇₆	C₄	semidirect
7. D₁₉₅₂	C₄	semidirect
43. M₁₂₈	D₆₁	semidirect
55. C₆₁	#7540	semidirect

Maximal subgroups

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

subgroup	index
#21448128	2
#21444021	2
C₆₁×M₁₂₈	2
#7540	61