Subgroups — #27298806

Order 13568.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276967	C₂	1	4	yes	maximal
2	#20274333	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719908	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133163	—	2	8	no	—
6	#16811028	—	2	2	no	—
7	#16811003	C₂²	1	2	yes	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	C₅₃×#500	—	2	2	no	—
10	C₅₃×#494	C₂²	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₁₂	—	2	4	no	—
13	#13133217	—	2	2	no	—
14	#13133163	—	4	16	no	—
15	C₂×#11719908	D₄	1	4	yes	—
16	#13133151	—	4	4	no	—
17	C₂×Q₁₆×C₅₃	—	2	4	no	—
18	C₄⋊Q₈×C₅₃	—	2	2	no	—
19	#107×C₅₃	—	4	4	no	—
20	C₄×C₈×C₅₃	D₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×Q₈×C₅₃	—	4	2	no	—
23	C₂×Dic₁₀₆	—	4	2	no	—
24	Q₁₆×C₅₃	—	4	8	no	—
25	C₂×C₈×C₅₃	D₈	1	4	yes	—
26	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
27	C₄⋊C₄×C₅₃	—	4	4	no	—
28	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₅₃	—	4	4	no	—
30	Dic₂₁₂	—	4	8	no	—
31	#11719908	D₈	1	8	yes	—
32	Q₈×C₅₃	—	8	4	no	—
33	C₂×C₄×C₅₃	—	8	2	no	—
34	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
36	C₂×Dic₅₃	—	8	2	no	—
37	Dic₁₀₆	—	8	4	no	—
38	C₈×C₅₃	Q₃₂	1	8	yes	complemented
39	C₈×C₅₃	C₂×D₈	1	4	yes	—
40	#12516	—	53	1	no	maximal, Sylow
41	C₂²×C₅₃	#493	1	1	yes	characteristic
42	C₄×C₅₃	—	16	2	no	—
43	C₄×C₅₃	#493	1	2	yes	—
44	C₄×C₅₃	C₂×Q₃₂	1	4	yes	—
45	Dic₅₃	—	16	2	no	—
46	#2247	—	53	4	no	—
47	#1713	—	53	1	no	—
48	#1713	D₅₃	1	1	yes	characteristic
49	C₈×C₁₆	—	53	1	no	—
50	C₂×C₅₃	#1712	1	1	yes	characteristic
51	C₂×C₅₃	#2247	1	2	yes	—
52	C₂×Q₃₂	—	106	8	no	—
53	#500	—	2	2	no	—
54	#500	—	106	2	no	—
55	#494	—	53	2	no	—
56	#494	C₂×D₅₃	1	2	yes	—
57	C₄×C₁₆	—	53	2	no	—
58	C₈²	C₂×D₅₃	1	1	yes	characteristic
59	C₅₃	#12516	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₁₆	—	2	4	no	—
61	C₂×Q₁₆	—	106	4	no	—
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	106	2	no	—
64	Q₃₂	—	212	16	no	—
65	C₂×C₁₆	—	53	4	no	—
66	#107	—	4	4	no	—
67	#107	—	212	4	no	—
68	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
69	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×Q₈	—	4	2	no	—
71	C₂×Q₈	—	212	2	no	—
72	Q₁₆	—	4	8	no	—
73	Q₁₆	—	212	8	no	—
74	C₂×C₈	#11719922	1	4	yes	—
75	C₂×C₈	D₄×D₅₃	1	2	yes	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	212	4	no	—
78	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
79	C₁₆	—	53	8	no	—
80	Q₈	—	8	4	no	—
81	Q₈	—	424	4	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	424	2	no	—
84	C₂×C₄	#13133213	1	1	yes	characteristic
85	C₂×C₄	C₂×#11719922	1	2	yes	—
86	C₈	#13133163	1	8	yes	complemented
87	C₈	D₈×D₅₃	1	4	yes	—
88	C₂²	#16810999	1	1	yes	characteristic
89	C₄	—	16	2	no	—
90	C₄	—	848	2	no	—
91	C₄	#16810998	1	2	yes	—
92	C₄	C₂×#13133163	1	4	yes	—
93	C₂	#20274329	1	1	yes	characteristic
94	C₂	#20276967	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1713	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#1713	D₅₃
hypocenter	C₅₃	#12516
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
38. C₈×C₅₃	Q₃₂	semidirect
59. C₅₃	#12516	semidirect
86. C₈	#13133163	semidirect

Maximal subgroups

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

subgroup	index
#20276967	2
#20274333	2
C₈×#11719908	2
C₅₃×#1713	2
#12516	53