Subgroups — #27271071

Order 13568.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	2	yes	maximal
2	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
4	C₂²⋊#11719908	C₄	1	4	yes	—
5	#16810399	—	2	2	no	—
6	C₅₃×#431	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#322	—	2	2	no	—
9	C₂×#11719908	—	2	4	no	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	—	2	2	no	—
13	C₂²⋊C₈×C₅₃	C₈	1	4	yes	—
14	C₄×C₈×C₅₃	—	2	2	no	—
15	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₄×C₈×C₅₃	D₄	1	2	yes	—
17	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
20	C₂×C₈×C₅₃	—	2	4	no	—
21	C₂×C₈×C₅₃	—	4	2	no	—
22	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₄²×C₅₃	M₁₆	1	2	yes	—
27	#11719908	—	4	4	no	—
28	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
32	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
34	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
35	C₈×C₅₃	—	2	2	no	—
36	C₈×C₅₃	—	2	2	no	—
37	C₈×C₅₃	—	4	4	no	—
38	#7197	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	#336	1	1	yes	characteristic
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	#344	1	2	yes	—
46	#2163	—	53	2	no	—
47	#1452	D₅₃	1	1	yes	characteristic
48	C₂×C₅₃	—	4	1	no	—
49	C₂×C₅₃	#1281	1	1	yes	characteristic
50	C₂×C₅₃	#1378	1	2	yes	—
51	#431	Dic₅₃	1	2	yes	—
52	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
53	C₄⋊C₁₆	—	106	2	no	—
54	C₂²⋊C₁₆	—	53	4	no	—
55	#346	—	106	2	no	—
56	#322	—	2	2	no	—
57	C₅₃	#7197	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	C₄×D₅₃	1	2	yes	—
59	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
60	C₂×C₁₆	—	106	4	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₂²⋊C₈	#88322	1	4	yes	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
65	C₄×C₈	D₂₁₂	1	1	yes	characteristic
66	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
67	C₂²×C₄	C₂×#88322	1	2	yes	—
68	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	#11719919	1	2	yes	—
72	C₂×C₈	#11719921	1	2	yes	—
73	C₄²	#11719917	1	2	yes	—
74	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
75	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
76	C₁₆	—	212	4	no	—
77	C₂³	C₄×#88322	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	#13133124	1	1	yes	characteristic
81	C₂×C₄	#13133128	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊#88322	1	2	yes	—
83	C₂×C₄	C₄⋊#88322	1	2	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#16810378	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	2	no	—
92	C₄	—	2	1	no	—
93	C₄	#16810394	1	2	yes	—
94	C₂	—	4	1	no	—
95	C₂	#20272560	1	1	yes	characteristic
96	C₂	#20272801	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133124
commutator, prime radical	C₂³×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	#336
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#7197
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
57. C₅₃	#7197	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
C₅₃×#1452	2
#7197	53