Subgroups — #27300492

Order 13568.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272793	C₂	1	6	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719908	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₅₃	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
6	C₄³×C₅₃	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	C₈	1	3	yes	—
11	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₅₃	—	2	6	no	—
14	C₄×C₈×C₅₃	—	2	6	no	—
15	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	—	2	3	no	—
19	C₂×C₈×C₅₃	—	2	3	no	—
20	C₂×C₈×C₅₃	M₁₆	1	6	yes	—
21	C₂×C₈×C₅₃	#41	1	6	yes	—
22	C₄²×C₅₃	—	2	6	no	—
23	C₄²×C₅₃	—	2	3	no	—
24	C₄²×C₅₃	—	2	3	no	—
25	C₄²×C₅₃	C₁₆	1	4	yes	complemented
26	#11719908	—	4	4	no	—
27	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	3	no	—
29	C₂×C₄×C₅₃	—	2	3	no	—
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	—	2	3	no	—
32	C₂×C₄×C₅₃	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₅₃	M₃₂	1	6	yes	—
34	C₂×C₄×C₅₃	#117	1	3	yes	—
35	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
36	C₈×C₅₃	—	2	6	no	—
37	C₈×C₅₃	#126	1	4	yes	—
38	#12894	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	#431	1	3	yes	—
40	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
41	C₂²×C₅₃	C₂×M₃₂	1	3	yes	—
42	C₄×C₅₃	—	2	6	no	—
43	C₄×C₅₃	—	2	6	no	—
44	C₄×C₅₃	#383	1	1	yes	characteristic
45	C₄×C₅₃	#432	1	3	yes	—
46	C₄²×C₈	D₅₃	1	1	yes	characteristic
47	#1374	—	53	6	no	—
48	C₂×C₅₃	#1839	1	1	yes	characteristic
49	C₂×C₅₃	#2162	1	3	yes	—
50	C₂×C₅₃	#2163	1	3	yes	—
51	C₂²×C₁₆	—	53	4	no	—
52	C₂×C₄×C₈	Dic₅₃	1	3	yes	—
53	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
54	C₄³	Dic₅₃	1	1	yes	characteristic
55	C₅₃	#12894	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
57	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
58	C₂×C₄²	#88322	1	3	yes	—
59	C₂×C₄²	#88322	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
61	C₂×C₁₆	—	106	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×#88322	1	3	yes	—
65	C₂²×C₄	C₂×#88322	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#11719917	1	6	yes	—
70	C₂×C₈	#11719945	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	#11719908	1	4	yes	complemented
75	C₁₆	—	212	4	no	—
76	C₂³	C₂²×#88322	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×#11719908	1	6	yes	—
82	C₂×C₄	#13133157	1	6	yes	—
83	C₂×C₄	#13133176	1	3	yes	—
84	C₂×C₄	C₂×#11719917	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#13133211	1	4	yes	—
87	C₂²	#16810676	1	3	yes	—
88	C₂²	C₂²×#11719908	1	1	yes	characteristic
89	C₂²	C₂×#13133157	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#16810491	1	1	yes	characteristic
93	C₄	#16810680	1	3	yes	—
94	C₂	#20274858	1	1	yes	characteristic
95	C₂	#20276514	1	3	yes	—
96	C₂	#20276518	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#12894
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
25. C₄²×C₅₃	C₁₆	semidirect
55. C₅₃	#12894	semidirect
74. C₄²	#11719908	semidirect

Maximal subgroups

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

subgroup	index
#20272793	2
C₄²×C₈×C₅₃	2
#12894	53