Subgroups — C₅₃×#22249 #27229761

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1991	C₂	1	2	yes	maximal
4	C₅₃×#1921	C₂	1	2	yes	maximal
5	C₅₃×#1920	C₂	1	2	yes	maximal
6	C₅₃×#1919	C₂	1	2	yes	maximal
7	C₅₃×#1918	C₂	1	2	yes	maximal
8	C₅₃×#1835	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#499	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#389	C₂²	1	2	yes	—
18	C₅₃×#389	C₂²	1	2	yes	—
19	C₅₃×#387	—	2	2	no	—
20	C₅₃×#387	—	2	2	no	—
21	C₅₃×#382	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₄³×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#340	C₂²	1	2	yes	—
28	C₅₃×#340	C₂²	1	2	yes	—
29	C₅₃×#340	C₂²	1	2	yes	—
30	C₅₃×#340	C₂²	1	1	yes	characteristic
31	C₅₃×#340	C₂²	1	2	yes	—
32	C₅₃×#340	C₂²	1	1	yes	characteristic
33	C₅₃×#340	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	#125×C₅₃	—	2	4	no	—
40	#125×C₅₃	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	D₄	1	2	yes	—
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	#107×C₅₃	C₂×C₄	1	4	yes	—
57	#107×C₅₃	C₂×C₄	1	4	yes	—
58	#106×C₅₃	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₅₃	—	2	2	no	—
60	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
61	#95×C₅₃	—	2	2	no	—
62	#95×C₅₃	—	2	2	no	—
63	#95×C₅₃	—	2	2	no	—
64	#95×C₅₃	—	2	2	no	—
65	#95×C₅₃	—	2	2	no	—
66	#95×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₅₃	—	2	8	no	—
84	C₄⋊C₄×C₅₃	—	2	8	no	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	4	4	no	—
92	C₄⋊C₄×C₅₃	—	4	4	no	—
93	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
95	C₄²×C₅₃	—	2	4	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	4	1	no	—
99	C₄²×C₅₃	—	4	1	no	—
100	C₄²×C₅₃	—	4	4	no	—
101	C₄²×C₅₃	—	4	4	no	—
102	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₅₃	#41	1	2	yes	—
104	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₅₃	#135	1	4	yes	—
128	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
131	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
132	C₈×C₅₃	—	2	2	no	—
133	C₈×C₅₃	—	2	2	no	—
134	C₈×C₅₃	—	4	2	no	—
135	#22249	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₅₃	#536	1	2	yes	—
138	C₂²×C₅₃	C₂×#135	1	2	yes	—
139	C₂²×C₅₃	#575	1	1	yes	characteristic
140	C₂²×C₅₃	#575	1	1	yes	characteristic
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	2	4	no	—
144	C₄×C₅₃	—	4	4	no	—
145	C₄×C₅₃	—	4	4	no	—
146	C₄×C₅₃	—	4	2	no	—
147	C₄×C₅₃	—	8	2	no	—
148	C₄×C₅₃	—	8	2	no	—
149	C₄×C₅₃	#519	1	1	yes	characteristic
150	C₄×C₅₃	#519	1	1	yes	characteristic
151	C₄×C₅₃	#541	1	2	yes	—
152	C₂×#499	C₂×C₅₃	1	1	yes	characteristic
153	C₄×#125	C₂×C₅₃	1	2	yes	—
154	#1991	C₂×C₅₃	1	2	yes	—
155	#1921	C₂×C₅₃	1	2	yes	—
156	#1920	C₂×C₅₃	1	2	yes	—
157	#1919	C₂×C₅₃	1	2	yes	—
158	#1918	C₂×C₅₃	1	2	yes	—
159	#1835	C₂×C₅₃	1	1	yes	characteristic
160	#1744	C₂×C₅₃	1	1	yes	characteristic
161	C₂×C₅₃	#2327	1	1	yes	characteristic
162	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
163	C₂×C₅₃	#2943	1	1	yes	characteristic
164	C₂×C₅₃	#3232	1	2	yes	—
165	C₂×C₅₃	#3233	1	2	yes	—
166	C₂×#125	C₂²×C₅₃	1	2	yes	—
167	#499	C₄×C₅₃	1	8	yes	complemented
168	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
170	C₂×#106	C₂²×C₅₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
173	#389	C₂²×C₅₃	1	2	yes	—
174	#389	C₂²×C₅₃	1	2	yes	—
175	#387	—	2	2	no	—
176	#387	—	2	2	no	—
177	#382	C₂²×C₅₃	1	2	yes	—
178	C₄×C₄⋊C₄	—	2	2	no	—
179	C₄×C₄⋊C₄	—	2	2	no	—
180	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
182	C₄³	C₂²×C₅₃	1	1	yes	characteristic
183	#340	C₂²×C₅₃	1	2	yes	—
184	#340	C₂²×C₅₃	1	1	yes	characteristic
185	#340	C₂²×C₅₃	1	1	yes	characteristic
186	#340	C₂²×C₅₃	1	2	yes	—
187	#340	C₂²×C₅₃	1	2	yes	—
188	#340	C₂²×C₅₃	1	2	yes	—
189	#340	C₂²×C₅₃	1	2	yes	—
190	#336	C₂²×C₅₃	1	2	yes	—
191	#336	C₂²×C₅₃	1	2	yes	—
192	C₅₃	#22249	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
194	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
196	#125	—	2	4	no	—
197	#125	C₂×C₄×C₅₃	1	8	yes	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	D₄×C₅₃	1	2	yes	—
209	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
212	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
213	#107	C₂×C₄×C₅₃	1	4	yes	—
214	#107	C₂×C₄×C₅₃	1	4	yes	—
215	#106	C₂×C₄×C₅₃	1	8	yes	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	8	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
251	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	4	no	—
258	C₄²	—	4	4	no	—
259	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
260	C₄²	#41×C₅₃	1	2	yes	—
261	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
284	C₂×C₄	#135×C₅₃	1	4	yes	—
285	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
287	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
288	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
293	C₂²	C₅₃×#536	1	2	yes	—
294	C₂²	C₂×#135×C₅₃	1	2	yes	—
295	C₂²	C₅₃×#575	1	1	yes	characteristic
296	C₂²	C₅₃×#575	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₅₃×#519	1	1	yes	characteristic
306	C₄	C₅₃×#519	1	1	yes	characteristic
307	C₄	C₅₃×#541	1	2	yes	—
308	C₂	C₅₃×#2327	1	1	yes	characteristic
309	C₂	C₄×#135×C₅₃	1	1	yes	characteristic
310	C₂	C₅₃×#2943	1	1	yes	characteristic
311	C₂	C₅₃×#3232	1	2	yes	—
312	C₂	C₅₃×#3233	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#499	C₄	semidirect
135. #22249	C₅₃	direct (N × H)
167. #499	C₄×C₅₃	semidirect
192. C₅₃	#22249	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×#499	2
C₄×#125×C₅₃	2
C₅₃×#1991	2
C₅₃×#1921	2
C₅₃×#1920	2
C₅₃×#1919	2
C₅₃×#1918	2
C₅₃×#1835	2
C₅₃×#1744	2
#22249	53

Subgroups — C53×#22249 #27229761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22249 #27229761