Subgroups — C₅₃×#11331 #27218843

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#428	C₂	1	2	yes	maximal
2	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#343	C₂	1	2	yes	maximal
4	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#428	—	2	4	no	—
12	C₅₃×#428	—	2	4	no	—
13	C₂×#107×C₅₃	—	2	2	no	—
14	C₂×#107×C₅₃	—	2	4	no	—
15	C₂×#107×C₅₃	—	2	2	no	—
16	C₂×#106×C₅₃	—	2	2	no	—
17	C₅₃×#343	—	2	4	no	—
18	C₅₃×#341	C₄	1	4	yes	complemented
19	C₅₃×#336	—	2	2	no	—
20	C₂³×C₄×C₅₃	—	2	2	no	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	2	1	no	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
30	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
31	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	2	no	—
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	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
50	C₂×C₄²×C₅₃	—	2	1	no	—
51	#107×C₅₃	—	2	8	no	—
52	#107×C₅₃	—	2	8	no	—
53	#106×C₅₃	—	2	8	no	—
54	C₂⁴×C₅₃	#41	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	8	4	no	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	M₁₆×C₅₃	—	2	4	no	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	4	2	no	—
76	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	4	1	no	—
86	C₂×C₈×C₅₃	—	4	1	no	—
87	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₂²⋊C₄×C₅₃	—	4	2	no	—
95	C₄²×C₅₃	—	4	1	no	—
96	C₄²×C₅₃	—	4	1	no	—
97	C₂³×C₅₃	—	2	2	no	—
98	C₂³×C₅₃	—	2	2	no	—
99	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₅₃	#121	1	2	yes	—
101	C₂³×C₅₃	#121	1	1	yes	characteristic
102	C₂³×C₅₃	#125	1	2	yes	—
103	C₂³×C₅₃	#125	1	1	yes	characteristic
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	8	4	no	—
110	C₂×C₄×C₅₃	—	8	4	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#120	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#123	1	2	yes	—
125	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
130	C₈×C₅₃	—	2	4	no	—
131	C₈×C₅₃	—	2	8	no	—
132	C₈×C₅₃	—	4	2	no	—
133	#11331	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	#389	1	2	yes	—
140	C₂²×C₅₃	#389	1	1	yes	characteristic
141	C₂²×C₅₃	#392	1	1	yes	characteristic
142	C₂²×C₅₃	#398	1	2	yes	—
143	C₂²×C₅₃	#398	1	1	yes	characteristic
144	C₂²×C₅₃	#472	1	2	yes	—
145	C₂²×C₅₃	#473	1	2	yes	—
146	C₄×C₅₃	—	8	1	no	—
147	C₄×C₅₃	—	8	4	no	—
148	C₄×C₅₃	—	8	1	no	—
149	C₄×C₅₃	#384	1	1	yes	characteristic
150	C₄×C₅₃	#385	1	1	yes	characteristic
151	C₄×C₅₃	#385	1	2	yes	—
152	C₄×C₅₃	#395	1	1	yes	characteristic
153	C₄×C₅₃	#395	1	2	yes	—
154	C₄×C₅₃	#399	1	1	yes	characteristic
155	C₂×#428	C₂×C₅₃	1	2	yes	—
156	C₂²×#107	C₂×C₅₃	1	1	yes	characteristic
157	C₂×#343	C₂×C₅₃	1	2	yes	—
158	C₂×#341	C₂×C₅₃	1	1	yes	characteristic
159	#1729	C₂×C₅₃	1	1	yes	characteristic
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	#1444	1	1	yes	characteristic
163	C₂×C₅₃	#1952	1	2	yes	—
164	C₂×C₅₃	#2018	1	1	yes	characteristic
165	C₂×C₅₃	#2020	1	1	yes	characteristic
166	C₂×C₅₃	#2089	1	2	yes	—
167	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
168	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
169	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
170	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
171	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
172	#428	—	2	4	no	—
173	#428	—	2	4	no	—
174	C₂×#107	—	2	2	no	—
175	C₂×#107	—	2	4	no	—
176	C₂×#107	—	2	2	no	—
177	C₂×#106	—	2	2	no	—
178	#343	—	2	4	no	—
179	#341	C₄×C₅₃	1	4	yes	complemented
180	#336	—	2	2	no	—
181	C₅₃	#11331	1	1	yes	characteristic, direct product factor, Sylow
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
191	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
192	C₂×M₁₆	Q₈×C₅₃	1	2	yes	—
193	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
199	C₂²×C₈	D₄×C₅₃	1	2	yes	—
200	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
209	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	C₂×C₄²	—	2	1	no	—
213	#107	—	2	8	no	—
214	#107	—	2	8	no	—
215	#106	—	2	8	no	—
216	C₂⁴	#41×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
228	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
230	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	#41×C₅₃	1	2	yes	—
232	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	C₄⋊C₄×C₅₃	1	8	yes	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
262	C₂³	#121×C₅₃	1	2	yes	—
263	C₂³	#121×C₅₃	1	1	yes	characteristic
264	C₂³	#125×C₅₃	1	1	yes	characteristic
265	C₂³	#125×C₅₃	1	2	yes	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
278	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
279	C₂×C₄	#120×C₅₃	1	2	yes	—
280	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
281	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
282	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
283	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
285	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
286	C₂×C₄	#123×C₅₃	1	2	yes	—
287	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
288	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
289	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
290	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
291	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	8	no	—
294	C₈	—	4	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	C₅₃×#389	1	2	yes	—
301	C₂²	C₅₃×#389	1	1	yes	characteristic
302	C₂²	C₅₃×#392	1	1	yes	characteristic
303	C₂²	C₅₃×#398	1	2	yes	—
304	C₂²	C₅₃×#398	1	1	yes	characteristic
305	C₂²	C₅₃×#472	1	2	yes	—
306	C₂²	C₅₃×#473	1	2	yes	—
307	C₄	—	8	1	no	—
308	C₄	—	8	4	no	—
309	C₄	—	8	1	no	—
310	C₄	C₅₃×#384	1	1	yes	characteristic
311	C₄	C₅₃×#385	1	1	yes	characteristic
312	C₄	C₅₃×#385	1	2	yes	—
313	C₄	C₅₃×#395	1	1	yes	characteristic
314	C₄	C₅₃×#395	1	2	yes	—
315	C₄	C₅₃×#399	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	C₅₃×#1444	1	1	yes	characteristic
319	C₂	C₅₃×#1952	1	2	yes	—
320	C₂	C₅₃×#2018	1	1	yes	characteristic
321	C₂	C₅₃×#2020	1	1	yes	characteristic
322	C₂	C₅₃×#2089	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
18. C₅₃×#341	C₄	semidirect
133. #11331	C₅₃	direct (N × H)
179. #341	C₄×C₅₃	semidirect
181. C₅₃	#11331	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×#428	2
C₂²×#107×C₅₃	2
C₂×C₅₃×#343	2
C₂×C₅₃×#341	2
C₅₃×#1729	2
#11331	53

Subgroups — C53×#11331 #27218843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11331 #27218843