Subgroups — C₅₃×#7914 #27215426

Order 13568.

666 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1280	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#432	—	2	2	no	—
8	C₅₃×#432	C₄	1	4	yes	complemented
9	C₅₃×#431	—	2	2	no	—
10	C₅₃×#431	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
13	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	—	2	1	no	—
17	C₅₃×#382	—	2	1	no	—
18	C₅₃×#382	—	2	1	no	—
19	C₅₃×#382	—	2	1	no	—
20	C₅₃×#382	—	2	1	no	—
21	C₂×#95×C₅₃	C₄	1	2	yes	—
22	C₂×#95×C₅₃	C₂²	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
40	C₄⋊C₈×C₅₃	—	2	4	no	—
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₂²⋊C₈×C₅₃	—	2	2	no	—
44	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
46	#97×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	—	2	4	no	—
48	#95×C₅₃	—	2	2	no	—
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	2	no	—
51	#95×C₅₃	—	2	2	no	—
52	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	8	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	—	4	4	no	—
60	C₂²×C₄×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	4	4	no	—
78	C₂×C₈×C₅₃	—	4	4	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₄⋊C₄×C₅₃	—	4	4	no	—
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₂³×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	—	2	1	no	—
91	C₂³×C₅₃	—	2	1	no	—
92	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
93	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₅₃	#131	1	4	yes	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	—	4	1	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	—	8	8	no	—
114	C₂×C₄×C₅₃	#95	1	2	yes	—
115	C₂×C₄×C₅₃	#95	1	2	yes	—
116	C₂×C₄×C₅₃	#95	1	2	yes	—
117	C₂×C₄×C₅₃	#95	1	2	yes	—
118	C₂×C₄×C₅₃	#100	1	2	yes	—
119	C₂×C₄×C₅₃	#101	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₈×C₅₃	—	4	4	no	—
125	C₈×C₅₃	—	4	4	no	—
126	#7914	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
134	C₂²×C₅₃	#405	1	2	yes	—
135	C₂²×C₅₃	#408	1	2	yes	—
136	C₂²×C₅₃	#410	1	2	yes	—
137	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
139	C₂²×C₅₃	#424	1	2	yes	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	8	4	no	—
148	C₄×C₅₃	#343	1	4	yes	—
149	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
150	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
151	C₂×#382	C₂×C₅₃	1	1	yes	characteristic
152	#1280	C₂×C₅₃	1	4	yes	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	#1469	1	2	yes	—
156	C₂×C₅₃	#1470	1	2	yes	—
157	C₂×C₅₃	#1729	1	1	yes	characteristic
158	C₂×C₅₃	#1740	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#343	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
161	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
162	#432	—	2	2	no	—
163	#432	C₄×C₅₃	1	4	yes	complemented
164	#431	—	2	2	no	—
165	#431	C₄×C₅₃	1	4	yes	complemented
166	C₂×C₄⋊C₈	—	2	1	no	—
167	C₂×C₄⋊C₈	—	2	1	no	—
168	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
170	C₂×#97	—	2	1	no	—
171	C₂×C₄×C₈	—	2	1	no	—
172	#382	—	2	1	no	—
173	#382	—	2	1	no	—
174	#382	—	2	1	no	—
175	#382	—	2	1	no	—
176	C₂×#95	C₄×C₅₃	1	2	yes	—
177	C₂×#95	C₂²×C₅₃	1	2	yes	—
178	C₅₃	#7914	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
180	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
182	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	4	4	no	—
188	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
189	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
190	C₂×C₄²	D₄×C₅₃	1	2	yes	—
191	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
194	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
195	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
196	C₄⋊C₈	—	2	4	no	—
197	C₄⋊C₈	—	2	4	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
201	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
218	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
238	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
239	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
240	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
241	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
242	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
243	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
244	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
249	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
251	C₂³	#131×C₅₃	1	4	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	8	no	—
270	C₂×C₄	#95×C₅₃	1	2	yes	—
271	C₂×C₄	#95×C₅₃	1	2	yes	—
272	C₂×C₄	#95×C₅₃	1	2	yes	—
273	C₂×C₄	#95×C₅₃	1	2	yes	—
274	C₂×C₄	#100×C₅₃	1	2	yes	—
275	C₂×C₄	#101×C₅₃	1	2	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
280	C₈	—	4	4	no	—
281	C₈	—	4	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
289	C₂²	C₅₃×#405	1	2	yes	—
290	C₂²	C₅₃×#408	1	2	yes	—
291	C₂²	C₅₃×#410	1	2	yes	—
292	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
293	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
294	C₂²	C₅₃×#424	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	8	4	no	—
303	C₄	C₅₃×#343	1	4	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	C₅₃×#1469	1	2	yes	—
307	C₂	C₅₃×#1470	1	2	yes	—
308	C₂	C₅₃×#1729	1	1	yes	characteristic
309	C₂	C₅₃×#1740	1	1	yes	characteristic
310	C₂	C₂×C₅₃×#343	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
8. C₅₃×#432	C₄	semidirect
10. C₅₃×#431	C₄	semidirect
126. #7914	C₅₃	direct (N × H)
163. #432	C₄×C₅₃	semidirect
165. #431	C₄×C₅₃	semidirect
178. C₅₃	#7914	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×#432	2
C₂×C₅₃×#431	2
C₂×C₅₃×#382	2
C₅₃×#1280	2
#7914	53

Subgroups — C53×#7914 #27215426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7914 #27215426