Subgroups — C₅₃×#11999 #27219511

Order 13568.

234 conjugacy classes of subgroups.

The subgroup lattice has 134 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	2	yes	maximal
3	C₅₃×#1368	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₅₃×#431	—	2	2	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#346	C₄	1	2	yes	—
9	C₅₃×#346	C₂²	1	2	yes	—
10	C₅₃×#335	—	2	2	no	—
11	C₅₃×#334	C₂²	1	4	yes	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	—	2	2	no	—
16	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
17	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₄⋊C₈×C₅₃	—	2	4	no	—
20	C₂²⋊C₈×C₅₃	—	4	2	no	—
21	C₄×C₈×C₅₃	—	4	2	no	—
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	4	4	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₃	SD₁₆	1	2	yes	—
38	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₅₃	—	2	2	no	—
40	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
41	C₂³×C₅₃	#106	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₅₃	#106	1	2	yes	—
45	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#108	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	—	2	1	no	—
51	C₈×C₅₃	—	8	4	no	—
52	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
53	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	#11999	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	#334	1	2	yes	—
58	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	#334	1	2	yes	—
62	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
64	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
65	#1564	C₂×C₅₃	1	2	yes	—
66	#1368	C₂×C₅₃	1	4	yes	complemented
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
69	C₂×C₅₃	#2155	1	2	yes	—
70	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
71	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
72	#431	—	2	2	no	—
73	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
74	#346	C₄×C₅₃	1	2	yes	—
75	#346	C₂²×C₅₃	1	2	yes	—
76	#335	—	2	2	no	—
77	#334	C₂²×C₅₃	1	4	yes	—
78	C₅₃	#11999	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
80	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
81	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₈×C₅₃	1	4	yes	complemented
84	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
85	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
86	C₄⋊C₈	—	2	4	no	—
87	C₂²⋊C₈	—	4	2	no	—
88	C₄×C₈	—	4	2	no	—
89	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
90	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
91	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
92	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
93	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
94	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
98	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
99	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
100	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
101	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
102	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
103	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
104	C₄²	SD₁₆×C₅₃	1	2	yes	—
105	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂×C₈×C₅₃	1	4	yes	—
108	C₂³	#106×C₅₃	1	1	yes	characteristic
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
111	C₂×C₄	#106×C₅₃	1	2	yes	—
112	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
113	C₂×C₄	#108×C₅₃	1	2	yes	—
114	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	8	4	no	—
119	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
120	C₈	C₂²×C₈×C₅₃	1	1	yes	characteristic
121	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	C₅₃×#334	1	2	yes	—
124	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	C₅₃×#334	1	2	yes	—
128	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
129	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×C₅₃×#334	1	1	yes	characteristic
132	C₂	C₅₃×#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#106
commutator	C₈	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1368	C₂	semidirect
16. C₂×C₁₆×C₅₃	C₈	semidirect
55. #11999	C₅₃	direct (N × H)
66. #1368	C₂×C₅₃	semidirect
78. C₅₃	#11999	direct (N × H)
83. C₂×C₁₆	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#346	2
C₅₃×#1564	2
C₅₃×#1368	2
#11999	53

Subgroups — C53×#11999 #27219511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11999 #27219511