Subgroups — C₅₃×#10916 #27218428

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2317	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#344	C₂	1	2	yes	maximal
4	C₅₃×#1742	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#343	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#405	—	2	4	no	—
12	C₄×M₁₆×C₅₃	—	2	2	no	—
13	C₅₃×#389	—	2	2	no	—
14	C₅₃×#385	C₄	1	4	yes	—
15	C₅₃×#384	—	2	1	no	—
16	C₅₃×#382	—	2	1	no	—
17	C₅₃×#344	C₄	1	8	yes	complemented
18	C₅₃×#343	—	2	4	no	—
19	C₅₃×#343	—	2	1	no	—
20	C₅₃×#343	—	2	1	no	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
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	2	no	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
34	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	#117×C₅₃	—	2	2	no	—
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
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	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	—	2	1	no	—
52	C₂×C₄²×C₅₃	D₄	1	2	yes	—
53	#97×C₅₃	—	2	4	no	—
54	C₄×C₈×C₅₃	—	2	4	no	—
55	#95×C₅₃	—	2	4	no	—
56	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	8	2	no	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	#41	1	2	yes	—
69	C₂²×C₄×C₅₃	#41	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
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	—	2	4	no	—
76	M₁₆×C₅₃	—	4	2	no	—
77	M₁₆×C₅₃	—	4	2	no	—
78	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	4	2	no	—
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	4	1	no	—
89	C₄⋊C₄×C₅₃	—	4	2	no	—
90	C₄⋊C₄×C₅₃	—	4	1	no	—
91	C₄⋊C₄×C₅₃	—	8	4	no	—
92	C₂²⋊C₄×C₅₃	—	2	2	no	—
93	C₂²⋊C₄×C₅₃	—	2	4	no	—
94	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
95	C₄²×C₅₃	—	2	4	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₂³×C₅₃	—	4	1	no	—
99	C₂³×C₅₃	—	4	1	no	—
100	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₅₃	#117	1	1	yes	characteristic
103	C₂³×C₅₃	C₄×D₄	1	2	yes	—
104	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	8	4	no	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₃	#117	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	2	yes	—
123	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₅₃	#123	1	2	yes	—
126	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#125	1	2	yes	—
129	C₂×C₄×C₅₃	#126	1	2	yes	—
130	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
131	C₈×C₅₃	—	2	8	no	—
132	C₈×C₅₃	—	4	4	no	—
133	#10916	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	—	4	1	no	—
140	C₂²×C₅₃	—	4	2	no	—
141	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
143	C₂²×C₅₃	#386	1	1	yes	characteristic
144	C₂²×C₅₃	#387	1	2	yes	—
145	C₂²×C₅₃	#391	1	1	yes	characteristic
146	C₄×C₅₃	—	4	4	no	—
147	C₄×C₅₃	—	4	2	no	—
148	C₄×C₅₃	—	4	2	no	—
149	C₄×C₅₃	—	8	2	no	—
150	C₄×C₅₃	#385	1	2	yes	—
151	C₄×C₅₃	#387	1	2	yes	—
152	C₄×C₅₃	#388	1	2	yes	—
153	C₄×C₅₃	#389	1	2	yes	—
154	#2873	C₂×C₅₃	1	1	yes	characteristic
155	#2317	C₂×C₅₃	1	1	yes	characteristic
156	C₂×#344	C₂×C₅₃	1	2	yes	—
157	#1742	C₂×C₅₃	1	1	yes	characteristic
158	C₂×#343	C₂×C₅₃	1	2	yes	—
159	C₂×C₅₃	—	2	1	no	—
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	#1436	1	1	yes	characteristic
164	C₂×C₅₃	#1914	1	1	yes	characteristic
165	C₂×C₅₃	#1915	1	1	yes	characteristic
166	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
167	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
168	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
171	#405	—	2	4	no	—
172	C₄×M₁₆	—	2	2	no	—
173	#389	—	2	2	no	—
174	#385	C₄×C₅₃	1	4	yes	—
175	#384	—	2	1	no	—
176	#382	—	2	1	no	—
177	#344	C₄×C₅₃	1	8	yes	complemented
178	#343	—	2	4	no	—
179	#343	—	2	1	no	—
180	#343	—	2	1	no	—
181	C₅₃	#10916	1	1	yes	characteristic, direct product factor, Sylow
182	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
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	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
193	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
194	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
195	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₂×C₄×C₅₃	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
211	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	D₄×C₅₃	1	2	yes	—
214	#97	—	2	4	no	—
215	C₄×C₈	—	2	4	no	—
216	#95	—	2	4	no	—
217	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	8	2	no	—
222	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
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	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
229	C₂²×C₄	#41×C₅₃	1	2	yes	—
230	C₂²×C₄	#41×C₅₃	1	2	yes	—
231	C₂²×C₄	#41×C₅₃	1	2	yes	—
232	M₁₆	—	2	4	no	—
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	—	4	2	no	—
239	M₁₆	C₄⋊C₄×C₅₃	1	8	yes	complemented
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	8	4	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	C₄²×C₅₃	1	4	yes	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
263	C₂³	#117×C₅₃	1	1	yes	characteristic
264	C₂³	C₄×D₄×C₅₃	1	2	yes	—
265	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
277	C₂×C₄	#117×C₅₃	1	2	yes	—
278	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
280	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
281	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
282	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
283	C₂×C₄	#121×C₅₃	1	2	yes	—
284	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
285	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
286	C₂×C₄	#123×C₅₃	1	2	yes	—
287	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
288	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
289	C₂×C₄	#125×C₅₃	1	2	yes	—
290	C₂×C₄	#126×C₅₃	1	2	yes	—
291	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
292	C₈	—	2	8	no	—
293	C₈	—	4	4	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
302	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
303	C₂²	C₅₃×#386	1	1	yes	characteristic
304	C₂²	C₅₃×#387	1	2	yes	—
305	C₂²	C₅₃×#391	1	1	yes	characteristic
306	C₄	—	4	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₅₃×#385	1	2	yes	—
311	C₄	C₅₃×#387	1	2	yes	—
312	C₄	C₅₃×#388	1	2	yes	—
313	C₄	C₅₃×#389	1	2	yes	—
314	C₂	—	2	1	no	—
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	C₅₃×#1436	1	1	yes	characteristic
319	C₂	C₅₃×#1914	1	1	yes	characteristic
320	C₂	C₅₃×#1915	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₅₃×#344	C₄	semidirect
78. M₁₆×C₅₃	C₄⋊C₄	semidirect
133. #10916	C₅₃	direct (N × H)
177. #344	C₄×C₅₃	semidirect
181. C₅₃	#10916	direct (N × H)
239. M₁₆	C₄⋊C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2873	2
C₅₃×#2317	2
C₂×C₅₃×#344	2
C₅₃×#1742	2
C₂×C₅₃×#343	2
#10916	53

Subgroups — C53×#10916 #27218428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10916 #27218428