Subgroups — #27305118

Order 13568.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276989	C₂	1	2	yes	maximal
2	#20272917	C₂	1	1	yes	maximal, characteristic
3	#20272916	C₂	1	2	yes	maximal
4	C₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1424	C₂	1	1	yes	maximal, characteristic
6	#16811027	—	2	2	no	—
7	C₂×#13133123	C₂²	1	1	yes	characteristic
8	#16810449	C₂²	1	2	yes	—
9	#16810448	—	2	2	no	—
10	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#366	C₂²	1	2	yes	—
13	#13133216	—	4	2	no	—
14	C₂×#11719908	D₄	1	1	yes	characteristic
15	#13133151	—	2	2	no	—
16	#13133150	—	4	2	no	—
17	C₈×Dic₅₃	D₄	1	1	yes	characteristic
18	#13133123	Q₈	1	2	yes	complemented
19	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
20	#107×C₅₃	—	2	2	no	—
21	C₃₂×C₅₃	Q₈	1	2	yes	complemented
22	C₂×Dic₁₀₆	—	8	2	no	—
23	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	4	2	no	—
25	C₄⋊Dic₅₃	—	4	2	no	—
26	#11719919	—	8	2	no	—
27	C₄×Dic₅₃	D₈	1	1	yes	characteristic
28	C₂×#88322	D₈	1	1	yes	characteristic
29	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	#11719908	Q₁₆	1	2	yes	—
32	C₂×C₄×C₅₃	—	8	2	no	—
33	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
34	C₂×Dic₅₃	—	8	2	no	—
35	C₂×Dic₅₃	D₁₆	1	2	yes	—
36	Dic₁₀₆	—	16	4	no	—
37	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
38	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
39	#88322	Q₃₂	1	2	yes	—
40	#13733	—	53	1	no	maximal, Sylow
41	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
42	C₄×C₅₃	—	16	2	no	—
43	C₄×C₅₃	#500	1	1	yes	characteristic
44	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
45	Dic₅₃	—	16	2	no	—
46	Dic₅₃	SD₆₄	1	4	yes	—
47	#2252	—	53	2	no	—
48	#1424	—	53	1	no	—
49	#1424	—	53	2	no	—
50	#1424	D₅₃	1	1	yes	characteristic
51	C₄×C₃₂	—	53	1	no	—
52	C₂×C₅₃	#2252	1	1	yes	characteristic
53	C₂×C₅₃	C₂×SD₆₄	1	2	yes	—
54	#500	—	106	2	no	—
55	C₂×C₃₂	—	53	1	no	—
56	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
57	#366	—	53	2	no	—
58	#366	—	106	2	no	—
59	#366	C₂×D₅₃	1	2	yes	—
60	C₄×C₁₆	—	53	1	no	—
61	C₅₃	#13733	1	1	yes	characteristic, complemented, Sylow
62	C₄⋊Q₈	—	212	2	no	—
63	C₂×C₁₆	—	53	1	no	—
64	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
65	#107	—	2	2	no	—
66	#107	—	106	2	no	—
67	#107	—	212	2	no	—
68	C₄×C₈	—	53	1	no	—
69	C₃₂	—	53	2	no	—
70	C₃₂	Dic₁₀₆	1	2	yes	complemented
71	C₂×Q₈	—	424	2	no	—
72	C₂×C₈	—	53	1	no	—
73	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	424	2	no	—
76	C₄⋊C₄	—	212	2	no	—
77	C₄²	—	53	1	no	—
78	C₁₆	—	53	2	no	—
79	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
80	C₁₆	Q₈×D₅₃	1	1	yes	characteristic
81	Q₈	—	848	4	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	53	2	no	—
84	C₂×C₄	—	424	2	no	—
85	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
86	C₈	—	53	2	no	—
87	C₈	#13133216	1	1	yes	characteristic
88	C₈	Q₁₆×D₅₃	1	1	yes	characteristic
89	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
90	C₄	—	16	2	no	—
91	C₄	—	53	4	no	—
92	C₄	—	848	2	no	—
93	C₄	#16811027	1	1	yes	characteristic
94	C₄	Q₃₂×D₅₃	1	1	yes	characteristic
95	C₂	#20276989	1	1	yes	characteristic
96	C₂	SD₆₄×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1424	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	#1424	D₅₃
hypocenter	C₅₃	#13733
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
18. #13133123	Q₈	semidirect
21. C₃₂×C₅₃	Q₈	semidirect
61. C₅₃	#13733	semidirect
70. C₃₂	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20276989	2
#20272917	2
#20272916	2
C₃₂×Dic₅₃	2
C₅₃×#1424	2
#13733	53