Subgroups — C₅₃×#8295 #27215807

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1451	C₂	1	2	yes	maximal
4	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1278	C₂	1	2	yes	maximal
6	C₅₃×#431	C₂²	1	2	yes	—
7	C₂×#108×C₅₃	C₄	1	2	yes	—
8	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#341	C₄	1	2	yes	—
13	C₅₃×#335	C₄	1	2	yes	—
14	C₅₃×#334	C₄	1	2	yes	—
15	C₅₃×#322	—	2	4	no	—
16	C₅₃×#322	C₄	1	2	yes	—
17	C₅₃×#322	C₄	1	2	yes	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#108×C₅₃	—	2	4	no	—
25	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²⋊C₈×C₅₃	—	2	2	no	—
27	#97×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	1	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	M₁₆×C₅₃	—	2	4	no	—
42	M₁₆×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
54	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	1	yes	characteristic
59	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	#131	1	2	yes	—
70	C₈×C₅₃	—	2	1	no	—
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	4	no	—
73	C₈×C₅₃	—	4	2	no	—
74	C₈×C₅₃	—	4	4	no	—
75	C₈×C₅₃	—	4	4	no	—
76	C₈×C₅₃	#97	1	2	yes	—
77	C₈×C₅₃	#97	1	2	yes	—
78	#8295	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	#446	1	2	yes	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
86	C₄×C₅₃	#405	1	1	yes	characteristic
87	C₄×C₅₃	#430	1	2	yes	—
88	C₄×C₅₃	#436	1	2	yes	—
89	C₄×C₅₃	#444	1	2	yes	—
90	C₄×#108	C₂×C₅₃	1	1	yes	characteristic
91	#1564	C₂×C₅₃	1	1	yes	characteristic
92	#1451	C₂×C₅₃	1	2	yes	—
93	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
94	#1278	C₂×C₅₃	1	2	yes	—
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	#1770	1	1	yes	characteristic
97	C₂×C₅₃	#1773	1	1	yes	characteristic
98	C₂×C₅₃	#1778	1	1	yes	characteristic
99	#431	C₂²×C₅₃	1	2	yes	—
100	C₂×#108	C₄×C₅₃	1	2	yes	—
101	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
105	#341	C₄×C₅₃	1	2	yes	—
106	#335	C₄×C₅₃	1	2	yes	—
107	#334	C₄×C₅₃	1	2	yes	—
108	#322	—	2	4	no	—
109	#322	C₄×C₅₃	1	2	yes	—
110	#322	C₄×C₅₃	1	2	yes	—
111	C₅₃	#8295	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
116	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
117	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
118	#108	—	2	4	no	—
119	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
126	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
127	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
128	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
129	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
131	C₄×C₈	D₄×C₅₃	1	2	yes	—
132	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
134	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
135	M₁₆	—	2	4	no	—
136	M₁₆	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
144	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
145	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
146	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
147	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
148	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
149	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
150	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
151	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
152	C₄²	#41×C₅₃	1	1	yes	characteristic
153	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
158	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
160	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
161	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
163	C₂×C₄	#131×C₅₃	1	2	yes	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	4	no	—
169	C₈	—	4	4	no	—
170	C₈	#97×C₅₃	1	2	yes	—
171	C₈	#97×C₅₃	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂²	C₅₃×#446	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
179	C₄	C₅₃×#405	1	1	yes	characteristic
180	C₄	C₅₃×#430	1	2	yes	—
181	C₄	C₅₃×#436	1	2	yes	—
182	C₄	C₅₃×#444	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	C₅₃×#1770	1	1	yes	characteristic
185	C₂	C₅₃×#1773	1	1	yes	characteristic
186	C₂	C₅₃×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
78. #8295	C₅₃	direct (N × H)
111. C₅₃	#8295	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#108×C₅₃	2
C₅₃×#1564	2
C₅₃×#1451	2
C₂×C₅₃×#322	2
C₅₃×#1278	2
#8295	53

Subgroups — C53×#8295 #27215807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8295 #27215807