Subgroups — #28299322

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289884	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811462	C₂	1	1	yes	maximal, characteristic, complemented
3	#20277009	C₂	1	1	yes	maximal, characteristic
4	#16811462	—	2	1	no	—
5	#16811462	—	2	1	no	—
6	#16810457	—	2	1	no	—
7	#16810457	C₄	1	1	yes	characteristic
8	C₂×#13133123	—	2	1	no	—
9	C₂×#13133123	C₂²	1	1	yes	characteristic
10	C₅₃×#504	C₄	1	1	yes	characteristic
11	#13133123	—	2	1	no	—
12	#13133123	D₄	1	1	yes	characteristic
13	#13133123	D₄	1	1	yes	characteristic
14	#131×C₅₃	C₈	1	1	yes	characteristic
15	M₃₂×C₅₃	—	2	1	no	—
16	M₃₂×C₅₃	C₈	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	—	2	1	no	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#41×C₅₃	C₁₆	1	1	yes	characteristic
20	M₁₆×C₅₃	—	2	1	no	—
21	M₁₆×C₅₃	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₅₃	—	2	1	no	—
25	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
27	Q₈×C₅₃	C₃₂	1	1	yes	characteristic
28	D₄×C₅₃	—	2	1	no	—
29	D₄×C₅₃	C₃₂	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₅₃	—	2	1	no	—
33	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₅₃	M₃₂	1	1	yes	characteristic
35	#7514	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
40	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
41	#2258	—	53	1	no	—
42	M₁₂₈	—	53	1	no	—
43	C₂×C₆₄	—	53	1	no	—
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	—	4	1	no	—
46	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
47	#504	#6157	1	1	yes	characteristic
48	M₆₄	—	53	1	no	—
49	M₆₄	—	106	1	no	—
50	C₂×C₃₂	—	53	1	no	—
51	C₂×C₃₂	—	106	1	no	—
52	C₆₄	—	106	1	no	—
53	C₆₄	—	106	1	no	—
54	C₅₃	#7514	1	1	yes	characteristic, complemented, Sylow
55	#131	#88324	1	1	yes	characteristic
56	M₃₂	—	2	1	no	—
57	M₃₂	#88324	1	1	yes	characteristic
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
60	C₃₂	—	53	1	no	—
61	C₃₂	—	53	1	no	—
62	C₃₂	—	106	1	no	—
63	#41	#11719910	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	#11719910	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×#88324	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	#11719940	1	1	yes	characteristic
70	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
71	Q₈	#13133267	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	#13133267	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×#11719910	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂²⋊#88324	1	1	yes	characteristic
78	C₈	#13133291	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	#16811555	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊#11719910	1	1	yes	characteristic
83	C₄	C₂×#13133267	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂²⋊#13133267	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₅₃×#504	C₄
socle	C₂×C₅₃	C₂²⋊C₃₂
prime radical	#41×C₅₃	C₁₆
cosocle	C₂×#13133123	C₂²
hypercenter	#504	#6157
hypocenter	C₅₃	#7514
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #20289884	C₂	semidirect
2. C₂×#16811462	C₂	semidirect
54. C₅₃	#7514	semidirect

Maximal subgroups

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

subgroup	index
#20289884	2
C₂×#16811462	2
#20277009	2
#7514	53