Subgroups — #27273457

Order 13568.

526 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274461	C₂	1	2	yes	maximal
2	C₂×#16810345	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂²	1	4	yes	—
5	#16810378	C₄	1	8	yes	—
6	#16810345	C₄	1	16	yes	complemented
7	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂×C₄	1	8	yes	—
11	C₄×#88322	C₂×C₄	1	16	yes	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	—	2	8	no	—
17	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	—	2	8	no	—
22	C₂×C₈×C₅₃	C₄²	1	8	yes	—
23	C₄²×C₅₃	—	2	8	no	—
24	C₄²×C₅₃	M₁₆	1	8	yes	—
25	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
26	C₂×#88322	—	2	16	no	—
27	C₂×#88322	C₄²	1	16	yes	—
28	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	8	no	—
30	C₂×C₄×C₅₃	#97	1	8	yes	—
31	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
32	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
34	C₂×C₄×C₅₃	#131	1	8	yes	—
35	C₈×C₅₃	—	2	8	no	—
36	#88322	—	2	16	no	—
37	#7843	—	53	1	no	maximal, Sylow
38	C₂²×C₅₃	C₄³	1	1	yes	characteristic
39	C₂²×C₅₃	C₂×#97	1	2	yes	—
40	C₂²×C₅₃	#405	1	4	yes	—
41	C₄×C₅₃	—	2	8	no	—
42	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
43	C₄×C₅₃	#405	1	8	yes	—
44	#1744	—	53	2	no	—
45	C₄²×C₈	D₅₃	1	1	yes	characteristic
46	C₂×#322	—	53	4	no	—
47	C₂×C₅₃	#1452	1	4	yes	—
48	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
49	C₂×C₅₃	#1726	1	2	yes	—
50	C₂×C₄×C₈	—	53	4	no	—
51	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
52	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
53	C₄³	C₂×D₅₃	1	1	yes	characteristic
54	#336	—	53	8	no	—
55	#322	—	53	16	no	—
56	C₅₃	#7843	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	53	8	no	—
58	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
59	C₂×C₄²	C₄×D₅₃	1	4	yes	—
60	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
61	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
62	C₄×C₈	—	2	8	no	—
63	C₄×C₈	—	53	16	no	—
64	C₄×C₈	C₄×D₅₃	1	8	yes	—
65	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
66	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
67	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
68	C₂×C₈	—	2	8	no	—
69	C₂×C₈	—	53	16	no	—
70	C₂×C₈	—	106	16	no	—
71	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
72	C₄²	—	2	8	no	—
73	C₄²	#11719917	1	8	yes	—
74	C₄²	C₂×C₄×D₅₃	1	4	yes	—
75	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	#13133128	1	8	yes	—
78	C₂×C₄	C₄²×D₅₃	1	4	yes	—
79	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
80	C₂×C₄	C₂×#11719917	1	4	yes	—
81	C₂×C₄	#13133223	1	8	yes	—
82	C₈	—	2	8	no	—
83	C₈	—	106	16	no	—
84	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
85	C₂²	C₂×#13133128	1	2	yes	—
86	C₂²	#16810578	1	4	yes	—
87	C₄	—	2	8	no	—
88	C₄	C₄×#11719917	1	4	yes	—
89	C₄	#16810577	1	8	yes	—
90	C₂	#20273007	1	4	yes	—
91	C₂	C₄×#13133128	1	1	yes	characteristic
92	C₂	#20274401	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₄³
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#7843
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #16810345	C₄	semidirect
56. C₅₃	#7843	semidirect

Maximal subgroups

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

subgroup	index
#20274461	2
C₂×#16810345	2
C₄²×C₈×C₅₃	2
#7843	53