Subgroups — #27300494

Order 13568.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272793	C₂	1	6	yes	maximal
2	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719908	C₂²	1	4	yes	—
4	C₂×#97×C₅₃	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
6	C₅₃×#376	C₄	1	1	yes	characteristic
7	C₂×#11719908	—	2	12	no	—
8	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
11	#97×C₅₃	—	2	6	no	—
12	C₄×C₈×C₅₃	—	2	6	no	—
13	#95×C₅₃	C₈	1	4	yes	—
14	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
15	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	—	2	3	no	—
17	C₂×C₈×C₅₃	—	2	3	no	—
18	C₂×C₈×C₅₃	M₁₆	1	6	yes	—
19	C₂×C₈×C₅₃	#41	1	6	yes	—
20	C₄²×C₅₃	—	2	3	no	—
21	C₄²×C₅₃	—	2	3	no	—
22	#11719908	—	4	4	no	—
23	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	6	no	—
25	C₂×C₄×C₅₃	—	2	12	no	—
26	C₂×C₄×C₅₃	#117	1	3	yes	—
27	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
28	C₈×C₅₃	—	2	6	no	—
29	C₈×C₅₃	#126	1	4	yes	—
30	#12895	—	53	1	no	maximal, Sylow
31	C₂²×C₅₃	#431	1	3	yes	—
32	C₂²×C₅₃	#504	1	4	yes	—
33	C₄×C₅₃	—	4	6	no	—
34	C₄×C₅₃	#383	1	1	yes	characteristic
35	C₄×C₅₃	#432	1	3	yes	—
36	#1726	D₅₃	1	1	yes	characteristic
37	#1374	—	53	6	no	—
38	C₂×C₅₃	#1839	1	1	yes	characteristic
39	C₂×C₅₃	#2164	1	6	yes	—
40	C₂²×C₁₆	—	53	4	no	—
41	C₂×#97	Dic₅₃	1	3	yes	—
42	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
43	#376	Dic₅₃	1	1	yes	characteristic
44	C₅₃	#12895	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
46	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
47	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
48	C₂×C₁₆	—	106	12	no	—
49	#97	—	2	6	no	—
50	C₄×C₈	—	2	6	no	—
51	#95	#88322	1	4	yes	—
52	C₂²×C₄	C₂×#88322	1	6	yes	—
53	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	#11719917	1	6	yes	—
57	C₂×C₈	#11719945	1	6	yes	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	212	4	no	—
61	C₂³	C₂²×#88322	1	1	yes	characteristic
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	#13133176	1	3	yes	—
65	C₂×C₄	C₂×#11719917	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	#13133211	1	4	yes	—
68	C₂²	#16810676	1	3	yes	—
69	C₂²	#16811040	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#16810491	1	1	yes	characteristic
72	C₄	#16810680	1	3	yes	—
73	C₂	#20274858	1	1	yes	characteristic
74	C₂	#20276522	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₅₃×#1726	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12895
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
44. C₅₃	#12895	semidirect

Maximal subgroups

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

subgroup	index
#20272793	2
C₅₃×#1726	2
#12895	53