Subgroups — C₅₃×#8017 #27215529

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 320 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	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1294	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#432	—	2	2	no	—
8	C₅₃×#432	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×#97×C₅₃	—	2	1	no	—
12	C₅₃×#383	—	2	1	no	—
13	C₅₃×#382	—	2	2	no	—
14	C₅₃×#382	C₄	1	2	yes	complemented
15	C₅₃×#382	C₄	1	4	yes	complemented
16	C₅₃×#381	—	2	1	no	—
17	C₅₃×#380	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	complemented
19	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
20	C₅₃×#376	—	2	1	no	—
21	C₂×#95×C₅₃	—	2	1	no	—
22	C₂×#95×C₅₃	C₂²	1	2	yes	—
23	C₂³×C₄×C₅₃	—	2	2	no	—
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
44	#97×C₅₃	—	2	4	no	—
45	#95×C₅₃	—	2	4	no	—
46	#95×C₅₃	—	2	2	no	—
47	#95×C₅₃	—	2	2	no	—
48	#95×C₅₃	—	2	2	no	—
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	2	no	—
51	#95×C₅₃	C₂×C₄	1	4	yes	—
52	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	4	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	4	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	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	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	4	4	no	—
78	C₄⋊C₄×C₅₃	—	4	8	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₂²⋊C₄×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	—	4	4	no	—
83	C₄²×C₅₃	C₄²	1	2	yes	—
84	C₄²×C₅₃	C₄²	1	2	yes	—
85	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
97	C₂³×C₅₃	#137	1	2	yes	—
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	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	—	4	1	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	—	4	4	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	—	4	4	no	—
119	C₂×C₄×C₅₃	—	8	8	no	—
120	C₂×C₄×C₅₃	#95	1	2	yes	—
121	C₂×C₄×C₅₃	#95	1	2	yes	—
122	C₂×C₄×C₅₃	#95	1	2	yes	—
123	C₂×C₄×C₅₃	#95	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₅₃	—	4	4	no	—
129	#8017	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
137	C₂²×C₅₃	#410	1	2	yes	—
138	C₂²×C₅₃	#417	1	2	yes	—
139	C₂²×C₅₃	#418	1	1	yes	characteristic
140	C₂²×C₅₃	#419	1	1	yes	characteristic
141	C₂²×C₅₃	#421	1	2	yes	—
142	C₂²×C₅₃	#428	1	2	yes	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	8	4	no	—
152	C₄×C₅₃	—	8	4	no	—
153	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
154	#2283	C₂×C₅₃	1	1	yes	characteristic
155	C₂×#382	C₂×C₅₃	1	1	yes	characteristic
156	#1294	C₂×C₅₃	1	4	yes	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	#1512	1	4	yes	—
161	C₂×C₅₃	#1733	1	1	yes	characteristic
162	C₂×C₅₃	#1736	1	1	yes	characteristic
163	C₂×C₅₃	#1740	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	2	1	no	—
165	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
166	#432	—	2	2	no	—
167	#432	C₄×C₅₃	1	4	yes	complemented
168	C₂×C₄⋊C₈	—	2	1	no	—
169	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
170	C₂×#97	—	2	1	no	—
171	#383	—	2	1	no	—
172	#382	—	2	2	no	—
173	#382	C₄×C₅₃	1	2	yes	complemented
174	#382	C₄×C₅₃	1	4	yes	complemented
175	#381	—	2	1	no	—
176	#380	C₂²×C₅₃	1	2	yes	—
177	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	complemented
178	C₄×C₂²⋊C₄	—	2	1	no	—
179	#376	—	2	1	no	—
180	C₂×#95	—	2	1	no	—
181	C₂×#95	C₂²×C₅₃	1	2	yes	—
182	C₅₃	#8017	1	1	yes	characteristic, direct product factor, Sylow
183	C₂³×C₄	—	2	2	no	—
184	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
185	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
187	C₂²×C₈	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	4	4	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
194	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
197	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₅₃	1	2	yes	—
199	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
200	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
201	C₄⋊C₈	—	2	4	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
204	#97	—	2	4	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	#95	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	C₂×C₄×C₅₃	1	4	yes	—
212	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
227	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
228	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
229	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	8	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	4	4	no	—
243	C₄²	C₄²×C₅₃	1	2	yes	—
244	C₄²	C₄²×C₅₃	1	2	yes	—
245	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
246	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
247	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
248	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
249	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
250	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
254	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
255	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
256	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
257	C₂³	#137×C₅₃	1	2	yes	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	#95×C₅₃	1	2	yes	—
281	C₂×C₄	#95×C₅₃	1	2	yes	—
282	C₂×C₄	#95×C₅₃	1	2	yes	—
283	C₂×C₄	#95×C₅₃	1	2	yes	—
284	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
288	C₈	—	4	4	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
296	C₂²	C₅₃×#410	1	2	yes	—
297	C₂²	C₅₃×#417	1	2	yes	—
298	C₂²	C₅₃×#418	1	1	yes	characteristic
299	C₂²	C₅₃×#419	1	1	yes	characteristic
300	C₂²	C₅₃×#421	1	2	yes	—
301	C₂²	C₅₃×#428	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	4	no	—
311	C₄	—	8	4	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	C₅₃×#1512	1	4	yes	—
316	C₂	C₅₃×#1733	1	1	yes	characteristic
317	C₂	C₅₃×#1736	1	1	yes	characteristic
318	C₂	C₅₃×#1740	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
14. C₅₃×#382	C₄	semidirect
15. C₅₃×#382	C₄	semidirect
18. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
129. #8017	C₅₃	direct (N × H)
167. #432	C₄×C₅₃	semidirect
173. #382	C₄×C₅₃	semidirect
174. #382	C₄×C₅₃	semidirect
177. C₄×C₄⋊C₄	C₄×C₅₃	semidirect
182. C₅₃	#8017	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
C₅₃×#2283	2
C₂×C₅₃×#382	2
C₅₃×#1294	2
#8017	53

Subgroups — C53×#8017 #27215529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8017 #27215529