Subgroups — #27298916

Order 13568.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810399	C₂	1	3	yes	maximal
2	C₂×C₄×#11719908	C₂	1	3	yes	maximal
3	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	C₂²	1	1	yes	characteristic
5	C₂²×#11719908	C₂²	1	3	yes	—
6	#16810399	C₄	1	12	yes	complemented
7	C₄×#11719908	C₄	1	12	yes	complemented
8	C₂×#97×C₅₃	C₄	1	3	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
10	C₅₃×#376	C₄	1	1	yes	characteristic
11	C₂×#11719908	C₂×C₄	1	6	yes	—
12	C₂×#11719908	C₂×C₄	1	6	yes	—
13	C₂×#11719908	C₂×C₄	1	12	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
17	#97×C₅₃	C₈	1	12	yes	—
18	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
20	#95×C₅₃	C₈	1	4	yes	—
21	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
22	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₄²	1	6	yes	—
24	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	—
25	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
26	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
27	C₄²×C₅₃	C₄²	1	6	yes	—
28	C₄²×C₅₃	C₂×C₈	1	6	yes	—
29	#11719908	C₄²	1	4	yes	—
30	#11719908	C₄²	1	12	yes	—
31	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	12	no	—
33	C₂×C₄×C₅₃	C₄×C₈	1	12	yes	—
34	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
36	C₈×C₅₃	C₄×C₈	1	12	yes	—
37	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
38	C₈×C₅₃	C₂×C₄²	1	3	yes	—
39	#12539	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	C₂×C₄×C₈	1	3	yes	—
41	C₂²×C₅₃	#504	1	4	yes	—
42	C₄×C₅₃	—	2	12	no	—
43	C₄×C₅₃	C₄³	1	1	yes	characteristic
44	C₄×C₅₃	C₂×C₄×C₈	1	3	yes	—
45	C₂×#346	—	53	3	no	—
46	C₂×C₄×C₁₆	—	53	3	no	—
47	#1726	D₅₃	1	1	yes	characteristic
48	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
49	C₂×C₅₃	#2108	1	6	yes	—
50	C₂²×C₁₆	—	53	1	no	—
51	C₂²×C₁₆	—	53	3	no	—
52	C₂×#97	Dic₅₃	1	3	yes	—
53	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
54	#376	Dic₅₃	1	1	yes	characteristic
55	#346	—	53	12	no	—
56	C₄×C₁₆	—	53	12	no	—
57	C₅₃	#12539	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
59	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
61	C₂×C₁₆	—	53	6	no	—
62	C₂×C₁₆	—	53	6	no	—
63	C₂×C₁₆	—	53	12	no	—
64	#97	#88322	1	12	yes	—
65	C₄×C₈	C₄×D₅₃	1	6	yes	—
66	C₄×C₈	C₄×D₅₃	1	6	yes	—
67	#95	#88322	1	4	yes	—
68	C₂²×C₄	C₂×#88322	1	6	yes	—
69	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
70	C₂×C₈	C₂×#88322	1	12	yes	—
71	C₂×C₈	C₄×Dic₅₃	1	6	yes	—
72	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
73	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
74	C₄²	C₂×#88322	1	6	yes	—
75	C₄²	C₄×Dic₅₃	1	6	yes	—
76	C₁₆	—	53	4	no	—
77	C₁₆	—	53	12	no	—
78	C₂³	C₂²×#88322	1	1	yes	characteristic
79	C₂×C₄	—	2	12	no	—
80	C₂×C₄	C₄×#88322	1	12	yes	—
81	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
82	C₂×C₄	C₂²×#88322	1	3	yes	—
83	C₈	C₄×#88322	1	12	yes	—
84	C₈	C₄²×D₅₃	1	1	yes	characteristic
85	C₈	C₄²×D₅₃	1	3	yes	—
86	C₂²	C₂×C₄×#88322	1	3	yes	—
87	C₂²	#16811040	1	4	yes	—
88	C₄	—	2	12	no	—
89	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
90	C₄	C₂×C₄×#88322	1	3	yes	—
91	C₂	C₄²×#88322	1	1	yes	characteristic
92	C₂	#20276312	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₅₃×#1726	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12539
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. #16810399	C₄	semidirect
7. C₄×#11719908	C₄	semidirect
57. C₅₃	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810399	2
C₂×C₄×#11719908	2
C₅₃×#1726	2
#12539	53