Subgroups — #27300506

Order 13568.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	6	yes	maximal
2	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719908	C₂²	1	4	yes	—
4	C₄⋊#11719908	—	2	12	no	—
5	C₂×#97×C₅₃	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
7	C₅₃×#376	C₄	1	1	yes	characteristic
8	C₂×#11719908	—	2	12	no	—
9	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
12	#97×C₅₃	—	2	6	no	—
13	C₄×C₈×C₅₃	D₄	1	6	yes	—
14	C₄×C₈×C₅₃	Q₈	1	6	yes	—
15	#95×C₅₃	C₈	1	4	yes	—
16	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	—	2	6	no	—
19	C₂×C₈×C₅₃	C₄⋊C₄	1	6	yes	—
20	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
22	C₄²×C₅₃	C₄⋊C₄	1	6	yes	—
23	C₄²×C₅₃	M₁₆	1	6	yes	—
24	#11719908	—	4	4	no	—
25	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	12	no	—
27	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	—
28	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
30	C₈×C₅₃	—	2	6	no	—
31	C₈×C₅₃	#127	1	1	yes	characteristic
32	C₈×C₅₃	C₄⋊Q₈	1	3	yes	—
33	#12899	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
35	C₂²×C₅₃	#504	1	4	yes	—
36	C₄×C₅₃	—	2	12	no	—
37	C₄×C₅₃	#384	1	1	yes	characteristic
38	C₄×C₅₃	#423	1	3	yes	—
39	C₂×C₄⋊C₁₆	—	53	6	no	—
40	#1726	D₅₃	1	1	yes	characteristic
41	C₂×C₅₃	#1842	1	1	yes	characteristic
42	C₂×C₅₃	#2151	1	6	yes	—
43	C₂²×C₁₆	—	53	4	no	—
44	C₂×#97	Dic₅₃	1	3	yes	—
45	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
46	#376	Dic₅₃	1	1	yes	characteristic
47	C₄⋊C₁₆	—	106	12	no	—
48	C₅₃	#12899	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
50	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
51	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
52	C₂×C₁₆	—	106	12	no	—
53	#97	—	2	6	no	—
54	C₄×C₈	Dic₁₀₆	1	6	yes	—
55	C₄×C₈	D₂₁₂	1	6	yes	—
56	#95	#88322	1	4	yes	—
57	C₂²×C₄	C₂×#88322	1	6	yes	—
58	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
59	C₂×C₈	—	2	6	no	—
60	C₂×C₈	C₄⋊Dic₅₃	1	6	yes	—
61	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
62	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
63	C₄²	#11719917	1	6	yes	—
64	C₄²	C₄⋊Dic₅₃	1	6	yes	—
65	C₁₆	—	212	4	no	—
66	C₂³	C₂²×#88322	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#88322	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
70	C₂×C₄	C₂×#11719917	1	3	yes	—
71	C₈	—	2	6	no	—
72	C₈	#13133214	1	1	yes	characteristic
73	C₈	#13133217	1	3	yes	—
74	C₂²	C₂×C₄⋊#88322	1	3	yes	—
75	C₂²	#16811040	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#16810493	1	1	yes	characteristic
78	C₄	#16810647	1	3	yes	—
79	C₂	#20274867	1	1	yes	characteristic
80	C₂	#20276474	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	#12899
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
48. C₅₃	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₅₃×#1726	2
#12899	53