Subgroups — C₅₃×#13636 #27221148

Order 13568.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#368	C₂	1	2	yes	maximal
3	C₅₃×#1425	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×#108×C₅₃	—	2	2	no	—
6	C₅₃×M₆₄	C₄	1	4	yes	—
7	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
8	C₅₃×#368	—	2	2	no	—
9	C₅₃×#368	C₂²	1	4	yes	—
10	C₂×M₁₆×C₅₃	—	4	2	no	—
11	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
15	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
16	#108×C₅₃	—	2	4	no	—
17	#108×C₅₃	—	4	2	no	—
18	C₃₂×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
20	M₁₆×C₅₃	—	4	4	no	—
21	M₁₆×C₅₃	—	8	2	no	—
22	C₂×C₈×C₅₃	—	8	2	no	—
23	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
25	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₂³×C₅₃	Q₃₂	1	1	yes	characteristic
31	C₂×C₄×C₅₃	#107	1	2	yes	—
32	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
33	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
35	C₈×C₅₃	—	8	4	no	—
36	C₈×C₅₃	#107	1	2	yes	—
37	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
39	#13636	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	#366	1	2	yes	—
42	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
43	C₄×C₅₃	#366	1	2	yes	—
44	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
45	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
46	C₂×M₆₄	C₂×C₅₃	1	1	yes	characteristic
47	C₂×#368	C₂×C₅₃	1	2	yes	—
48	#1425	C₂×C₅₃	1	4	yes	complemented
49	C₂×C₅₃	—	2	2	no	—
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
52	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
53	C₂×#108	—	2	2	no	—
54	M₆₄	C₄×C₅₃	1	4	yes	—
55	C₂×C₃₂	C₂²×C₅₃	1	2	yes	—
56	#368	—	2	2	no	—
57	#368	C₂²×C₅₃	1	4	yes	—
58	C₅₃	#13636	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×M₁₆	—	4	2	no	—
60	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
61	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
62	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
63	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
64	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
65	#108	—	2	4	no	—
66	#108	—	4	2	no	—
67	C₃₂	C₂×C₄×C₅₃	1	4	yes	—
68	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
69	M₁₆	—	4	4	no	—
70	M₁₆	—	8	2	no	—
71	C₂×C₈	—	8	2	no	—
72	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
73	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
74	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
75	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
76	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
77	C₁₆	C₂²×C₄×C₅₃	1	1	yes	characteristic
78	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
79	C₂³	Q₃₂×C₅₃	1	1	yes	characteristic
80	C₂×C₄	#107×C₅₃	1	2	yes	—
81	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
82	C₂×C₄	Q₃₂×C₅₃	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
84	C₈	—	8	4	no	—
85	C₈	#107×C₅₃	1	2	yes	—
86	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
87	C₈	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
88	C₂²	—	4	1	no	—
89	C₂²	C₅₃×#366	1	2	yes	—
90	C₂²	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
91	C₄	C₅₃×#366	1	2	yes	—
92	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
93	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2	2	no	—
96	C₂	C₂×C₅₃×#366	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂×C₅₃	C₂×#366
prime radical	C₂³×C₅₃	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1425	C₂	semidirect
39. #13636	C₅₃	direct (N × H)
48. #1425	C₂×C₅₃	semidirect
58. C₅₃	#13636	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×M₆₄	2
C₂×C₅₃×#368	2
C₅₃×#1425	2
#13636	53

Subgroups — C53×#13636 #27221148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13636 #27221148