Subgroups — #28299144

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13133267	C₂	1	2	yes	maximal
2	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133267	—	2	2	no	—
4	C₄×#11719908	C₂²	1	1	yes	characteristic
5	C₈×#88322	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	#13133267	—	4	2	no	—
8	C₂×#11719908	D₄	1	1	yes	characteristic
9	C₂×#11719908	Q₈	1	1	yes	characteristic
10	C₄×#88322	C₈	1	1	yes	characteristic
11	C₄×#88322	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	—	2	1	no	—
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
16	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₂×#88322	—	2	1	no	—
18	C₂×#88322	C₁₆	1	1	yes	characteristic
19	C₂×#88322	C₁₆	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	2	no	—
21	#11719908	—	2	1	no	—
22	#11719908	D₈	1	1	yes	characteristic
23	#11719908	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	#107	1	1	yes	characteristic
29	C₈×C₅₃	#108	1	1	yes	characteristic
30	#88322	—	2	2	no	—
31	#88322	—	2	1	no	—
32	#88322	C₃₂	1	2	yes	complemented
33	#7458	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	#335	1	1	yes	characteristic
37	C₄×C₅₃	#364	1	1	yes	characteristic
38	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
39	C₄×C₅₃	M₆₄	1	1	yes	characteristic
40	C₄⋊C₃₂	—	53	2	no	—
41	C₈×C₁₆	—	53	1	no	—
42	C₂×C₅₃	#1370	1	1	yes	characteristic
43	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
44	C₂×C₅₃	#1422	1	1	yes	characteristic
45	C₂×C₃₂	—	106	2	no	—
46	C₄×C₁₆	—	53	1	no	—
47	C₄×C₁₆	#6157	1	1	yes	characteristic
48	C₈²	—	53	1	no	—
49	C₅₃	#7458	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	53	1	no	—
52	C₂×C₁₆	—	53	1	no	—
53	C₄×C₈	—	53	1	no	—
54	C₄×C₈	—	53	1	no	—
55	C₄×C₈	C₂×#6157	1	1	yes	characteristic
56	C₃₂	—	212	2	no	—
57	C₂×C₈	—	53	1	no	—
58	C₂×C₈	—	53	1	no	—
59	C₂×C₈	—	106	1	no	—
60	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
61	C₂×C₈	#11719940	1	1	yes	characteristic
62	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	53	1	no	—
65	C₁₆	—	53	1	no	—
66	C₁₆	—	106	1	no	—
67	C₂×C₄	#13133284	1	1	yes	characteristic
68	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
69	C₂×C₄	#13133292	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	53	2	no	—
72	C₈	—	106	2	no	—
73	C₈	—	106	1	no	—
74	C₈	#13133286	1	1	yes	characteristic
75	C₈	#13133287	1	1	yes	characteristic
76	C₂²	#16811544	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#16811487	1	1	yes	characteristic
79	C₄	#16811546	1	1	yes	characteristic
80	C₄	D₅₃⋊C₃₂	1	1	yes	characteristic
81	C₄	#16811556	1	1	yes	characteristic
82	C₂	#20289745	1	1	yes	characteristic
83	C₂	#20289873	1	1	yes	characteristic
84	C₂	#20289875	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₄×C₈	C₂×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₄⋊C₁₆
cosocle	C₄×#11719908	C₂²
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#7458
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
32. #88322	C₃₂	semidirect
49. C₅₃	#7458	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13133267	2
C₁₆×#88322	2
#7458	53