Subgroups — C₅₃×#33414 #27240926

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2916	C₂	1	2	yes	maximal
3	C₅₃×#2161	C₂	1	6	yes	maximal, complemented
4	C₂×C₅₃×#368	C₂	1	6	yes	maximal, complemented
5	C₅₃×#568	—	2	2	no	—
6	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#504	C₄	1	8	yes	—
8	C₂×M₃₂×C₅₃	C₂²	1	3	yes	—
9	C₂²×C₁₆×C₅₃	C₂²	1	3	yes	—
10	C₅₃×#430	—	2	6	no	—
11	C₂×#108×C₅₃	C₂²	1	12	yes	—
12	C₅₃×#368	C₂²	1	4	yes	—
13	C₅₃×#368	C₂²	1	12	yes	—
14	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
15	#131×C₅₃	—	4	8	no	—
16	#131×C₅₃	C₂×C₄	1	4	yes	—
17	#131×C₅₃	Q₈	1	4	yes	—
18	C₂×M₁₆×C₅₃	—	2	12	no	—
19	C₂×M₁₆×C₅₃	D₄	1	3	yes	—
20	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
21	M₃₂×C₅₃	C₂×C₄	1	12	yes	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	—
23	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	C₂³	1	3	yes	—
25	#108×C₅₃	—	2	12	no	—
26	#108×C₅₃	C₂³	1	8	yes	—
27	#41×C₅₃	C₄⋊C₄	1	4	yes	—
28	#41×C₅₃	D₈	1	4	yes	—
29	C₂×Q₈×C₅₃	Q₁₆	1	1	yes	characteristic
30	C₂×D₄×C₅₃	Q₁₆	1	3	yes	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
32	M₁₆×C₅₃	—	2	8	no	—
33	M₁₆×C₅₃	—	4	12	no	—
34	M₁₆×C₅₃	C₄⋊C₄	1	12	yes	—
35	C₂×C₈×C₅₃	—	4	12	no	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
38	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
39	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
41	C₁₆×C₅₃	C₂²×C₄	1	6	yes	—
42	C₂³×C₅₃	C₂×Q₁₆	1	3	yes	—
43	Q₈×C₅₃	#107	1	4	yes	—
44	D₄×C₅₃	#107	1	12	yes	—
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
46	C₂×C₄×C₅₃	C₂×D₈	1	6	yes	—
47	C₂×C₄×C₅₃	C₂×Q₁₆	1	3	yes	—
48	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
49	C₈×C₅₃	—	4	8	no	—
50	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
51	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
52	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
53	#33414	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂²×C₅₃	—	4	3	no	—
55	C₂²×C₅₃	C₂×#107	1	6	yes	—
56	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
57	C₄×C₅₃	C₂×#107	1	6	yes	—
58	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
60	C₂×#504	C₂×C₅₃	1	1	yes	characteristic
61	#2916	C₂×C₅₃	1	2	yes	—
62	#2161	C₂×C₅₃	1	6	yes	complemented
63	C₂×#368	C₂×C₅₃	1	6	yes	complemented
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	6	no	—
66	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
67	#568	—	2	2	no	—
68	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
69	#504	C₄×C₅₃	1	8	yes	—
70	C₂×M₃₂	C₂²×C₅₃	1	3	yes	—
71	C₂²×C₁₆	C₂²×C₅₃	1	3	yes	—
72	#430	—	2	6	no	—
73	C₂×#108	C₂²×C₅₃	1	12	yes	—
74	#368	C₂²×C₅₃	1	4	yes	—
75	#368	C₂²×C₅₃	1	12	yes	—
76	C₅₃	#33414	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
78	#131	—	4	8	no	—
79	#131	C₂×C₄×C₅₃	1	4	yes	—
80	#131	Q₈×C₅₃	1	4	yes	—
81	C₂×M₁₆	—	2	12	no	—
82	C₂×M₁₆	D₄×C₅₃	1	3	yes	—
83	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
84	M₃₂	C₂×C₄×C₅₃	1	12	yes	—
85	C₂×C₁₆	C₂×C₄×C₅₃	1	12	yes	—
86	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂³×C₅₃	1	3	yes	—
88	#108	—	2	12	no	—
89	#108	C₂³×C₅₃	1	8	yes	—
90	#41	C₄⋊C₄×C₅₃	1	4	yes	—
91	#41	D₈×C₅₃	1	4	yes	—
92	C₂×Q₈	Q₁₆×C₅₃	1	1	yes	characteristic
93	C₂×D₄	Q₁₆×C₅₃	1	3	yes	—
94	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
95	M₁₆	—	2	8	no	—
96	M₁₆	—	4	12	no	—
97	M₁₆	C₄⋊C₄×C₅₃	1	12	yes	—
98	C₂×C₈	—	4	12	no	—
99	C₂×C₈	C₂²×C₄×C₅₃	1	6	yes	—
100	C₂×C₈	C₂×D₄×C₅₃	1	3	yes	—
101	C₂×C₈	C₂×Q₈×C₅₃	1	6	yes	—
102	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
103	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
104	C₁₆	C₂²×C₄×C₅₃	1	6	yes	—
105	C₂³	C₂×Q₁₆×C₅₃	1	3	yes	—
106	Q₈	#107×C₅₃	1	4	yes	—
107	D₄	#107×C₅₃	1	12	yes	—
108	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
109	C₂×C₄	C₂×D₈×C₅₃	1	6	yes	—
110	C₂×C₄	C₂×Q₁₆×C₅₃	1	3	yes	—
111	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
112	C₈	—	4	8	no	—
113	C₈	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
114	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
115	C₈	C₂²×Q₈×C₅₃	1	1	yes	characteristic
116	C₂²	—	4	3	no	—
117	C₂²	C₂×#107×C₅₃	1	6	yes	—
118	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
119	C₄	C₂×#107×C₅₃	1	6	yes	—
120	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
121	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	6	no	—
124	C₂	C₂²×#107×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂×C₅₃	C₂²×#107
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2161	C₂	semidirect
4. C₂×C₅₃×#368	C₂	semidirect
53. #33414	C₅₃	direct (N × H)
62. #2161	C₂×C₅₃	semidirect
63. C₂×#368	C₂×C₅₃	semidirect
76. C₅₃	#33414	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×#504	2
C₅₃×#2916	2
C₅₃×#2161	2
C₂×C₅₃×#368	2
#33414	53

Subgroups — C53×#33414 #27240926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33414 #27240926