Subgroups — C₅₃×#8432 #27215944

Order 13568.

740 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3501	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1804	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1518	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1481	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1478	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#554	—	2	1	no	—
9	C₅₃×#554	—	2	2	no	—
10	C₅₃×#554	—	2	2	no	—
11	C₅₃×#554	C₄	1	2	yes	complemented
12	D₄×Q₈×C₅₃	—	2	1	no	—
13	D₄×Q₈×C₅₃	C₄	1	2	yes	complemented
14	C₅₃×#544	—	2	1	no	—
15	C₅₃×#543	—	2	2	no	—
16	C₅₃×#533	C₂²	1	1	yes	characteristic
17	C₅₃×#533	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₅₃×#518	—	2	1	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#102×C₅₃	—	2	1	no	—
25	C₅₃×#382	C₂²	1	1	yes	characteristic
26	C₅₃×#340	—	2	1	no	—
27	C₅₃×#333	—	2	1	no	—
28	C₅₃×#332	—	2	1	no	—
29	C₅₃×#328	—	2	2	no	—
30	C₅₃×#328	—	2	2	no	—
31	C₅₃×Q₈⋊C₈	—	2	2	no	—
32	C₅₃×D₄⋊C₈	—	2	2	no	—
33	C₂²×Q₈×C₅₃	—	2	2	no	—
34	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	—	2	1	no	—
37	C₄⋊Q₈×C₅₃	—	2	2	no	—
38	C₄⋊Q₈×C₅₃	—	2	2	no	—
39	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
43	#125×C₅₃	—	2	2	no	—
44	#125×C₅₃	—	2	2	no	—
45	#125×C₅₃	—	2	1	no	—
46	#125×C₅₃	—	2	1	no	—
47	C₂²⋊Q₈×C₅₃	—	2	4	no	—
48	C₂²⋊Q₈×C₅₃	—	2	4	no	—
49	C₂²⋊Q₈×C₅₃	—	2	4	no	—
50	C₂²⋊Q₈×C₅₃	—	2	4	no	—
51	C₂²⋊Q₈×C₅₃	—	2	2	no	—
52	C₂²⋊Q₈×C₅₃	—	2	4	no	—
53	C₂²⋊Q₈×C₅₃	—	2	2	no	—
54	C₂²⋊Q₈×C₅₃	—	2	4	no	—
55	C₂²⋊Q₈×C₅₃	—	2	1	no	—
56	C₂²⋊Q₈×C₅₃	—	2	1	no	—
57	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
58	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
59	C₄×Q₈×C₅₃	—	2	1	no	—
60	C₄×Q₈×C₅₃	—	2	1	no	—
61	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×Q₈×C₅₃	D₄	1	2	yes	—
63	C₄×D₄×C₅₃	—	2	2	no	—
64	C₄×D₄×C₅₃	—	2	2	no	—
65	C₄×D₄×C₅₃	—	2	1	no	—
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	—	2	2	no	—
68	C₄×D₄×C₅₃	—	2	2	no	—
69	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×D₄×C₅₃	D₄	1	2	yes	—
71	#117×C₅₃	—	2	1	no	—
72	#117×C₅₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₅₃	—	2	1	no	—
81	C₄⋊C₈×C₅₃	—	2	1	no	—
82	C₄⋊C₈×C₅₃	—	2	1	no	—
83	C₄⋊C₈×C₅₃	—	2	1	no	—
84	#102×C₅₃	—	4	2	no	—
85	C₂²⋊C₈×C₅₃	—	2	1	no	—
86	C₂²⋊C₈×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	—	2	1	no	—
89	#95×C₅₃	—	2	1	no	—
90	#95×C₅₃	—	2	1	no	—
91	C₂⁴×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	4	no	—
93	C₂×Q₈×C₅₃	—	2	1	no	—
94	C₂×Q₈×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	—	8	4	no	—
96	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
97	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
102	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
103	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
104	C₂²×C₄×C₅₃	—	2	4	no	—
105	C₂²×C₄×C₅₃	—	2	4	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	4	1	no	—
110	C₂²×C₄×C₅₃	—	8	2	no	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	—	4	1	no	—
115	C₂×C₈×C₅₃	—	4	1	no	—
116	C₂×C₈×C₅₃	—	4	1	no	—
117	C₂×C₈×C₅₃	—	4	1	no	—
118	C₂×C₈×C₅₃	—	4	1	no	—
119	C₄⋊C₄×C₅₃	—	2	4	no	—
120	C₄⋊C₄×C₅₃	—	2	4	no	—
121	C₄⋊C₄×C₅₃	—	2	4	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	4	1	no	—
134	C₄⋊C₄×C₅₃	—	4	1	no	—
135	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
136	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
137	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
138	C₂²⋊C₄×C₅₃	—	2	4	no	—
139	C₂²⋊C₄×C₅₃	—	2	4	no	—
140	C₂²⋊C₄×C₅₃	—	2	2	no	—
141	C₂²⋊C₄×C₅₃	—	2	2	no	—
142	C₂²⋊C₄×C₅₃	—	2	2	no	—
143	C₂²⋊C₄×C₅₃	—	2	2	no	—
144	C₄²×C₅₃	—	2	2	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₂³×C₅₃	—	2	2	no	—
151	C₂³×C₅₃	—	2	2	no	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	—	8	1	no	—
154	C₂³×C₅₃	—	8	1	no	—
155	C₂³×C₅₃	#120	1	1	yes	characteristic
156	Q₈×C₅₃	—	4	2	no	—
157	Q₈×C₅₃	—	4	2	no	—
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	8	1	no	—
160	Q₈×C₅₃	—	8	4	no	—
161	D₄×C₅₃	—	2	2	no	—
162	D₄×C₅₃	—	2	2	no	—
163	D₄×C₅₃	—	2	2	no	—
164	D₄×C₅₃	#103	1	4	yes	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	4	no	—
173	C₂×C₄×C₅₃	—	8	2	no	—
174	C₂×C₄×C₅₃	—	8	1	no	—
175	C₂×C₄×C₅₃	—	8	4	no	—
176	C₂×C₄×C₅₃	—	8	4	no	—
177	C₂×C₄×C₅₃	—	8	1	no	—
178	C₂×C₄×C₅₃	—	8	1	no	—
179	C₂×C₄×C₅₃	—	8	1	no	—
180	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
184	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
185	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
188	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
189	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
190	C₈×C₅₃	—	8	1	no	—
191	C₈×C₅₃	—	8	1	no	—
192	C₈×C₅₃	—	8	1	no	—
193	#8432	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	2	4	no	—
197	C₂²×C₅₃	—	4	2	no	—
198	C₂²×C₅₃	—	4	2	no	—
199	C₂²×C₅₃	—	8	2	no	—
200	C₂²×C₅₃	—	8	2	no	—
201	C₂²×C₅₃	#379	1	1	yes	characteristic
202	C₂²×C₅₃	#448	1	1	yes	characteristic
203	C₂²×C₅₃	#451	1	1	yes	characteristic
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	4	2	no	—
209	C₄×C₅₃	—	4	2	no	—
210	C₄×C₅₃	—	4	2	no	—
211	C₄×C₅₃	—	4	2	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	16	1	no	—
214	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
215	C₄×C₅₃	#419	1	1	yes	characteristic
216	C₄×C₅₃	#448	1	1	yes	characteristic
217	C₄×C₅₃	#451	1	1	yes	characteristic
218	#3501	C₂×C₅₃	1	1	yes	characteristic
219	#1804	C₂×C₅₃	1	1	yes	characteristic
220	#1542	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#1518	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1518	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#1481	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#1478	C₂×C₅₃	1	1	yes	characteristic
225	C₂×C₅₃	—	2	1	no	—
226	C₂×C₅₃	—	2	1	no	—
227	C₂×C₅₃	—	4	4	no	—
228	C₂×C₅₃	#1787	1	1	yes	characteristic
229	C₂×C₅₃	#1794	1	1	yes	characteristic
230	C₂×C₅₃	#1795	1	1	yes	characteristic
231	#554	—	2	1	no	—
232	#554	—	2	2	no	—
233	#554	—	2	2	no	—
234	#554	C₄×C₅₃	1	2	yes	complemented
235	D₄×Q₈	—	2	1	no	—
236	D₄×Q₈	C₄×C₅₃	1	2	yes	complemented
237	#544	—	2	1	no	—
238	#543	—	2	2	no	—
239	#533	C₂²×C₅₃	1	1	yes	characteristic
240	#533	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×C₂²⋊Q₈	—	2	2	no	—
242	#518	—	2	1	no	—
243	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
244	#431	C₂²×C₅₃	1	1	yes	characteristic
245	#431	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×#102	—	2	1	no	—
248	#382	C₂²×C₅₃	1	1	yes	characteristic
249	#340	—	2	1	no	—
250	#333	—	2	1	no	—
251	#332	—	2	1	no	—
252	#328	—	2	2	no	—
253	#328	—	2	2	no	—
254	Q₈⋊C₈	—	2	2	no	—
255	D₄⋊C₈	—	2	2	no	—
256	C₅₃	#8432	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²×Q₈	—	2	2	no	—
258	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
259	C₂³×C₄	—	2	1	no	—
260	C₂²×C₈	—	2	1	no	—
261	C₄⋊Q₈	—	2	2	no	—
262	C₄⋊Q₈	—	2	2	no	—
263	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
264	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
265	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
266	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
267	#125	—	2	2	no	—
268	#125	—	2	2	no	—
269	#125	—	2	1	no	—
270	#125	—	2	1	no	—
271	C₂²⋊Q₈	—	2	4	no	—
272	C₂²⋊Q₈	—	2	4	no	—
273	C₂²⋊Q₈	—	2	4	no	—
274	C₂²⋊Q₈	—	2	4	no	—
275	C₂²⋊Q₈	—	2	2	no	—
276	C₂²⋊Q₈	—	2	4	no	—
277	C₂²⋊Q₈	—	2	1	no	—
278	C₂²⋊Q₈	—	2	1	no	—
279	C₂²⋊Q₈	—	2	2	no	—
280	C₂²⋊Q₈	—	2	4	no	—
281	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
282	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
283	C₄×Q₈	—	2	1	no	—
284	C₄×Q₈	—	2	1	no	—
285	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
286	C₄×Q₈	D₄×C₅₃	1	2	yes	—
287	C₄×D₄	—	2	2	no	—
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	—	2	2	no	—
290	C₄×D₄	—	2	2	no	—
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	—	2	1	no	—
293	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
294	C₄×D₄	D₄×C₅₃	1	2	yes	—
295	#117	—	2	1	no	—
296	#117	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	4	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	4	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	—	2	1	no	—
306	C₄⋊C₈	—	2	1	no	—
307	C₄⋊C₈	—	2	1	no	—
308	#102	—	4	2	no	—
309	C₂²⋊C₈	—	2	1	no	—
310	C₂²⋊C₈	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	#95	—	2	1	no	—
314	#95	—	2	1	no	—
315	C₂⁴	—	2	1	no	—
316	C₂×Q₈	—	2	4	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	8	4	no	—
320	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
321	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
326	C₂×D₄	SD₁₆×C₅₃	1	2	yes	—
327	C₂×D₄	Q₁₆×C₅₃	1	2	yes	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	—	8	2	no	—
335	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₂×C₈	—	4	1	no	—
339	C₂×C₈	—	4	1	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	1	no	—
342	C₂×C₈	—	4	1	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
360	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
361	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	2	4	no	—
364	C₂²⋊C₄	—	2	2	no	—
365	C₂²⋊C₄	—	2	2	no	—
366	C₂²⋊C₄	—	2	2	no	—
367	C₂²⋊C₄	—	2	2	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
371	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
373	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
374	C₂³	—	2	2	no	—
375	C₂³	—	2	2	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	8	1	no	—
378	C₂³	—	8	1	no	—
379	C₂³	#120×C₅₃	1	1	yes	characteristic
380	Q₈	—	4	2	no	—
381	Q₈	—	4	2	no	—
382	Q₈	—	4	2	no	—
383	Q₈	—	8	1	no	—
384	Q₈	—	8	4	no	—
385	D₄	—	2	2	no	—
386	D₄	—	2	2	no	—
387	D₄	—	2	2	no	—
388	D₄	#103×C₅₃	1	4	yes	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	4	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	—	8	1	no	—
399	C₂×C₄	—	8	4	no	—
400	C₂×C₄	—	8	4	no	—
401	C₂×C₄	—	8	1	no	—
402	C₂×C₄	—	8	1	no	—
403	C₂×C₄	—	8	1	no	—
404	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
405	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
406	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
407	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
408	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
409	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
410	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
411	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
412	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
413	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
414	C₈	—	8	1	no	—
415	C₈	—	8	1	no	—
416	C₈	—	8	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	8	2	no	—
423	C₂²	—	8	2	no	—
424	C₂²	C₅₃×#379	1	1	yes	characteristic
425	C₂²	C₅₃×#448	1	1	yes	characteristic
426	C₂²	C₅₃×#451	1	1	yes	characteristic
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	16	1	no	—
437	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
438	C₄	C₅₃×#419	1	1	yes	characteristic
439	C₄	C₅₃×#448	1	1	yes	characteristic
440	C₄	C₅₃×#451	1	1	yes	characteristic
441	C₂	—	2	1	no	—
442	C₂	—	2	1	no	—
443	C₂	—	4	4	no	—
444	C₂	C₅₃×#1787	1	1	yes	characteristic
445	C₂	C₅₃×#1794	1	1	yes	characteristic
446	C₂	C₅₃×#1795	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#379
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1542	C₂	semidirect
4. C₅₃×#1518	C₂	semidirect
5. C₅₃×#1518	C₂	semidirect
6. C₅₃×#1481	C₂	semidirect
11. C₅₃×#554	C₄	semidirect
13. D₄×Q₈×C₅₃	C₄	semidirect
193. #8432	C₅₃	direct (N × H)
220. #1542	C₂×C₅₃	semidirect
221. #1518	C₂×C₅₃	semidirect
222. #1518	C₂×C₅₃	semidirect
223. #1481	C₂×C₅₃	semidirect
234. #554	C₄×C₅₃	semidirect
236. D₄×Q₈	C₄×C₅₃	semidirect
256. C₅₃	#8432	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3501	2
C₅₃×#1804	2
C₅₃×#1542	2
C₅₃×#1518	2
C₅₃×#1481	2
C₅₃×#1478	2
#8432	53

Subgroups — C53×#8432 #27215944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8432 #27215944