Subgroups — C₅₃×#11357 #27218869

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#340	C₂	1	4	yes	maximal
3	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₅₃×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#384	—	2	2	no	—
12	C₅₃×#384	C₄	1	4	yes	complemented
13	C₂×#95×C₅₃	—	2	1	no	—
14	C₂×#95×C₅₃	—	2	1	no	—
15	C₅₃×#340	—	2	8	no	—
16	C₅₃×#336	—	2	2	no	—
17	C₂³×C₄×C₅₃	—	2	2	no	—
18	C₂³×C₄×C₅₃	—	2	2	no	—
19	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
36	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
40	C₂×C₄²×C₅₃	—	2	4	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	#95×C₅₃	—	4	2	no	—
44	#95×C₅₃	—	4	2	no	—
45	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	—	4	4	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	—	4	4	no	—
60	C₂²×C₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	—	4	2	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	D₈	1	2	yes	—
65	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₃	#41	1	4	yes	—
73	C₂²×C₄×C₅₃	#41	1	2	yes	—
74	C₂²×C₄×C₅₃	#41	1	2	yes	—
75	C₂×C₈×C₅₃	—	2	8	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	4	4	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	2	8	no	—
81	C₄⋊C₄×C₅₃	—	2	8	no	—
82	C₄⋊C₄×C₅₃	—	4	8	no	—
83	C₄⋊C₄×C₅₃	—	4	8	no	—
84	C₄⋊C₄×C₅₃	—	4	2	no	—
85	C₄⋊C₄×C₅₃	—	4	4	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₄²×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₅₃	#121	1	2	yes	—
91	C₂³×C₅₃	#123	1	2	yes	—
92	C₂³×C₅₃	#124	1	1	yes	characteristic
93	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
94	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
95	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
99	C₂³×C₅₃	#137	1	2	yes	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	—	8	4	no	—
115	C₂×C₄×C₅₃	—	8	4	no	—
116	C₂×C₄×C₅₃	—	8	2	no	—
117	C₂×C₄×C₅₃	—	8	2	no	—
118	C₂×C₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	—	8	2	no	—
120	C₂×C₄×C₅₃	#102	1	4	yes	—
121	C₂×C₄×C₅₃	#103	1	4	yes	—
122	C₂×C₄×C₅₃	#117	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
126	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₅₃	#123	1	4	yes	—
128	C₂×C₄×C₅₃	#123	1	2	yes	—
129	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#125	1	4	yes	—
131	C₂×C₄×C₅₃	#126	1	4	yes	—
132	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
133	C₈×C₅₃	—	4	4	no	—
134	#11357	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₅₃	#381	1	1	yes	characteristic
136	C₂²×C₅₃	#390	1	1	yes	characteristic
137	C₂²×C₅₃	#391	1	1	yes	characteristic
138	C₂²×C₅₃	#397	1	2	yes	—
139	C₂²×C₅₃	#398	1	2	yes	—
140	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
142	C₂²×C₅₃	#418	1	1	yes	characteristic
143	C₂²×C₅₃	#419	1	1	yes	characteristic
144	C₂²×C₅₃	#459	1	2	yes	—
145	C₂²×C₅₃	#460	1	2	yes	—
146	C₂²×C₅₃	#461	1	2	yes	—
147	C₂²×C₅₃	#462	1	2	yes	—
148	C₂²×C₅₃	#480	1	2	yes	—
149	C₂²×C₅₃	#481	1	2	yes	—
150	C₂²×C₅₃	#483	1	2	yes	—
151	C₂²×C₅₃	#484	1	2	yes	—
152	C₂²×C₅₃	#486	1	1	yes	characteristic
153	C₂²×C₅₃	#487	1	1	yes	characteristic
154	C₂²×C₅₃	#489	1	2	yes	—
155	C₂²×C₅₃	#498	1	1	yes	characteristic
156	C₂²×C₅₃	#500	1	1	yes	characteristic
157	C₂²×C₅₃	#501	1	2	yes	—
158	C₄×C₅₃	—	2	4	no	—
159	C₄×C₅₃	—	2	4	no	—
160	C₄×C₅₃	—	8	4	no	—
161	C₄×C₅₃	—	8	4	no	—
162	C₄×C₅₃	#387	1	4	yes	—
163	C₄×C₅₃	#399	1	2	yes	—
164	C₄×C₅₃	#400	1	2	yes	—
165	C₂×#384	C₂×C₅₃	1	1	yes	characteristic
166	C₂×#340	C₂×C₅₃	1	4	yes	—
167	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
168	C₄⋊#95	C₂×C₅₃	1	1	yes	characteristic
169	C₂×C₅₃	#1445	1	1	yes	characteristic
170	C₂×C₅₃	#1823	1	1	yes	characteristic
171	C₂×C₅₃	#1824	1	1	yes	characteristic
172	C₂×C₅₃	#1981	1	1	yes	characteristic
173	C₂×C₅₃	#1982	1	1	yes	characteristic
174	C₂×C₅₃	#1990	1	1	yes	characteristic
175	C₂×C₅₃	#1993	1	1	yes	characteristic
176	C₂×C₅₃	#2071	1	2	yes	—
177	C₂×C₅₃	#2072	1	2	yes	—
178	C₂×C₅₃	#2083	1	2	yes	—
179	C₂×C₅₃	#2084	1	2	yes	—
180	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
181	C₂²×C₄⋊C₄	—	2	1	no	—
182	C₂²×C₄⋊C₄	—	2	1	no	—
183	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
184	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
185	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
186	#384	—	2	2	no	—
187	#384	C₄×C₅₃	1	4	yes	complemented
188	C₂×#95	—	2	1	no	—
189	C₂×#95	—	2	1	no	—
190	#340	—	2	8	no	—
191	#336	—	2	2	no	—
192	C₅₃	#11357	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
196	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
197	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	4	4	no	—
212	C₂×C₄⋊C₄	—	4	2	no	—
213	C₂×C₄⋊C₄	—	4	2	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
215	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
218	C₂×C₄²	D₄×C₅₃	1	2	yes	—
219	#95	—	4	2	no	—
220	#95	—	4	2	no	—
221	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
239	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
240	C₂²×C₄	D₈×C₅₃	1	2	yes	—
241	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
242	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
243	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
248	C₂²×C₄	#41×C₅₃	1	4	yes	—
249	C₂²×C₄	#41×C₅₃	1	2	yes	—
250	C₂²×C₄	#41×C₅₃	1	2	yes	—
251	C₂×C₈	—	2	8	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	4	8	no	—
259	C₄⋊C₄	—	4	8	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	4	no	—
265	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
266	C₂³	#121×C₅₃	1	2	yes	—
267	C₂³	#123×C₅₃	1	2	yes	—
268	C₂³	#124×C₅₃	1	1	yes	characteristic
269	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
270	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
271	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
272	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
273	C₂³	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
274	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
275	C₂³	#137×C₅₃	1	2	yes	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	8	4	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	#102×C₅₃	1	4	yes	—
297	C₂×C₄	#103×C₅₃	1	4	yes	—
298	C₂×C₄	#117×C₅₃	1	2	yes	—
299	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
300	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
301	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
302	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
303	C₂×C₄	#123×C₅₃	1	4	yes	—
304	C₂×C₄	#123×C₅₃	1	2	yes	—
305	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
306	C₂×C₄	#125×C₅₃	1	4	yes	—
307	C₂×C₄	#126×C₅₃	1	4	yes	—
308	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
309	C₈	—	4	4	no	—
310	C₂²	C₅₃×#381	1	1	yes	characteristic
311	C₂²	C₅₃×#390	1	1	yes	characteristic
312	C₂²	C₅₃×#391	1	1	yes	characteristic
313	C₂²	C₅₃×#397	1	2	yes	—
314	C₂²	C₅₃×#398	1	2	yes	—
315	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
316	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
317	C₂²	C₅₃×#418	1	1	yes	characteristic
318	C₂²	C₅₃×#419	1	1	yes	characteristic
319	C₂²	C₅₃×#459	1	2	yes	—
320	C₂²	C₅₃×#460	1	2	yes	—
321	C₂²	C₅₃×#461	1	2	yes	—
322	C₂²	C₅₃×#462	1	2	yes	—
323	C₂²	C₅₃×#480	1	2	yes	—
324	C₂²	C₅₃×#481	1	2	yes	—
325	C₂²	C₅₃×#483	1	2	yes	—
326	C₂²	C₅₃×#484	1	2	yes	—
327	C₂²	C₅₃×#486	1	1	yes	characteristic
328	C₂²	C₅₃×#487	1	1	yes	characteristic
329	C₂²	C₅₃×#489	1	2	yes	—
330	C₂²	C₅₃×#498	1	1	yes	characteristic
331	C₂²	C₅₃×#500	1	1	yes	characteristic
332	C₂²	C₅₃×#501	1	2	yes	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	8	4	no	—
336	C₄	—	8	4	no	—
337	C₄	C₅₃×#387	1	4	yes	—
338	C₄	C₅₃×#399	1	2	yes	—
339	C₄	C₅₃×#400	1	2	yes	—
340	C₂	C₅₃×#1445	1	1	yes	characteristic
341	C₂	C₅₃×#1823	1	1	yes	characteristic
342	C₂	C₅₃×#1824	1	1	yes	characteristic
343	C₂	C₅₃×#1981	1	1	yes	characteristic
344	C₂	C₅₃×#1982	1	1	yes	characteristic
345	C₂	C₅₃×#1990	1	1	yes	characteristic
346	C₂	C₅₃×#1993	1	1	yes	characteristic
347	C₂	C₅₃×#2071	1	2	yes	—
348	C₂	C₅₃×#2072	1	2	yes	—
349	C₂	C₅₃×#2083	1	2	yes	—
350	C₂	C₅₃×#2084	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅₃×#384	C₄	semidirect
134. #11357	C₅₃	direct (N × H)
187. #384	C₄×C₅₃	semidirect
192. C₅₃	#11357	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×#384	2
C₂×C₅₃×#340	2
C₂×C₅₃×#336	2
C₅₃×C₄⋊#95	2
#11357	53

Subgroups — C53×#11357 #27218869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11357 #27218869