Subgroups — #27271805

Order 13568.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810453	C₂	1	2	yes	maximal
2	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133153	—	2	2	no	—
4	#16810453	—	2	2	no	—
5	#16810453	C₄	1	4	yes	—
6	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×M₆₄	—	2	2	no	—
8	C₂×C₃₂×C₅₃	C₄	1	2	yes	—
9	C₂×#11719917	—	4	2	no	—
10	#13133153	—	2	4	no	—
11	#13133153	—	4	2	no	—
12	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
16	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
17	C₃₂×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
22	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
23	#11719917	—	4	4	no	—
24	#11719917	—	8	2	no	—
25	C₂×#88322	—	8	2	no	—
26	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
27	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂³×C₅₃	Q₃₂	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#102	1	2	yes	—
31	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
32	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
34	C₈×C₅₃	#95	1	1	yes	characteristic
35	C₈×C₅₃	#103	1	2	yes	—
36	C₈×C₅₃	#106	1	1	yes	characteristic
37	#88322	—	8	4	no	—
38	#7476	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	—	4	1	no	—
40	C₂²×C₅₃	#358	1	2	yes	—
41	C₂²×C₅₃	#366	1	1	yes	characteristic
42	C₄×C₅₃	#340	1	1	yes	characteristic
43	C₄×C₅₃	#357	1	2	yes	—
44	C₄×C₅₃	#367	1	1	yes	characteristic
45	C₂×M₆₄	D₅₃	1	1	yes	characteristic
46	C₂×#368	—	53	2	no	—
47	C₂×C₅₃	—	2	2	no	—
48	C₂×C₅₃	—	2	1	no	—
49	C₂×C₅₃	#1380	1	1	yes	characteristic
50	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
51	C₂×#108	—	106	2	no	—
52	M₆₄	—	2	2	no	—
53	C₂×C₃₂	Dic₅₃	1	2	yes	—
54	#368	—	53	4	no	—
55	#368	—	106	2	no	—
56	C₅₃	#7476	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	212	2	no	—
58	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
59	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
60	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
61	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
62	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
63	#108	—	106	4	no	—
64	#108	—	212	2	no	—
65	C₃₂	—	2	2	no	—
66	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
67	M₁₆	—	424	2	no	—
68	M₁₆	—	212	4	no	—
69	C₂×C₈	—	424	2	no	—
70	C₂×C₈	#11719913	1	2	yes	—
71	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
72	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
73	C₂×C₈	#11719921	1	2	yes	—
74	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
75	C₁₆	#11719919	1	2	yes	—
76	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
77	C₂³	Dic₄₂₄	1	1	yes	characteristic
78	C₂×C₄	#13133139	1	2	yes	—
79	C₂×C₄	#13133151	1	1	yes	characteristic
80	C₂×C₄	D₈₄₈	1	1	yes	characteristic
81	C₂×C₄	#13133162	1	2	yes	—
82	C₈	—	424	4	no	—
83	C₈	#13133124	1	1	yes	characteristic
84	C₈	#13133142	1	2	yes	—
85	C₈	#13133149	1	1	yes	characteristic
86	C₂²	—	4	1	no	—
87	C₂²	#16810428	1	2	yes	—
88	C₂²	#16810449	1	1	yes	characteristic
89	C₄	#16810386	1	1	yes	characteristic
90	C₄	#16810425	1	2	yes	—
91	C₄	#16810451	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	—	2	1	no	—
94	C₂	#20272807	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810451
commutator	C₁₆×C₅₃	C₄²
Frattini	C₂²×C₁₆	C₂×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂×C₅₃	#1380
prime radical	C₂³×C₅₃	Q₃₂
cosocle	C₂²×C₁₆×C₅₃	C₂²
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	#7476
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
56. C₅₃	#7476	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
C₂×C₅₃×M₆₄	2
#7476	53