Subgroups — C₅₃×#18749 #27226261

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#554	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3077	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1644	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1632	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1478	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#554	—	2	2	no	—
11	C₅₃×#554	—	2	2	no	—
12	C₅₃×#554	—	2	2	no	—
13	C₅₃×#554	C₂²	1	4	yes	—
14	C₅₃×#533	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#125×C₅₃	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	—	2	1	no	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₅₃×#494	C₂²	1	2	yes	complemented
24	C₅₃×#487	C₂²	1	2	yes	complemented
25	C₅₃×#484	—	2	1	no	—
26	C₅₃×#478	—	2	2	no	—
27	C₅₃×#477	C₂²	1	2	yes	complemented
28	C₅₃×#476	C₂²	1	2	yes	—
29	C₅₃×#474	C₂²	1	2	yes	—
30	C₅₃×#462	—	2	2	no	—
31	C₅₃×#462	C₂²	1	2	yes	complemented
32	C₅₃×#451	—	2	2	no	—
33	C₅₃×#451	—	2	1	no	—
34	C₄×Q₁₆×C₅₃	—	2	2	no	—
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#428	—	2	1	no	—
38	C₅₃×#423	C₂²	1	1	yes	characteristic
39	C₅₃×#418	—	2	1	no	—
40	C₅₃×#333	C₂²	1	4	yes	complemented
41	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
45	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	—	2	2	no	—
47	C₂³×C₄×C₅₃	—	2	1	no	—
48	C₂×Q₁₆×C₅₃	—	2	2	no	—
49	C₂×Q₁₆×C₅₃	—	2	4	no	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
54	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
57	#125×C₅₃	—	2	4	no	—
58	#125×C₅₃	—	2	4	no	—
59	#125×C₅₃	—	2	1	no	—
60	C₂²⋊Q₈×C₅₃	—	2	8	no	—
61	C₂²⋊Q₈×C₅₃	—	2	4	no	—
62	C₂²⋊Q₈×C₅₃	—	2	8	no	—
63	C₂²⋊Q₈×C₅₃	—	2	4	no	—
64	C₂²⋊Q₈×C₅₃	—	2	1	no	—
65	C₂²⋊Q₈×C₅₃	—	2	1	no	—
66	C₄×Q₈×C₅₃	—	2	1	no	—
67	C₄×Q₈×C₅₃	—	2	1	no	—
68	C₄×Q₈×C₅₃	—	2	1	no	—
69	C₄×Q₈×C₅₃	—	2	1	no	—
70	C₄×Q₈×C₅₃	D₄	1	2	yes	—
71	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
72	C₄×D₄×C₅₃	—	2	4	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	2	1	no	—
75	C₄×D₄×C₅₃	—	2	4	no	—
76	C₄×D₄×C₅₃	D₄	1	2	yes	—
77	C₄×D₄×C₅₃	C₂³	1	2	yes	—
78	#117×C₅₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
90	C₂×C₄²×C₅₃	—	2	1	no	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	#107×C₅₃	—	2	2	no	—
93	#107×C₅₃	—	2	2	no	—
94	#106×C₅₃	—	2	2	no	—
95	C₄⋊C₈×C₅₃	—	2	1	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	#103×C₅₃	—	2	2	no	—
100	#103×C₅₃	—	2	4	no	—
101	#103×C₅₃	—	2	2	no	—
102	#102×C₅₃	—	2	2	no	—
103	#102×C₅₃	—	2	2	no	—
104	C₂²⋊C₈×C₅₃	—	2	1	no	—
105	C₂²⋊C₈×C₅₃	—	2	1	no	—
106	C₄×C₈×C₅₃	—	2	1	no	—
107	C₄×C₈×C₅₃	C₂³	1	2	yes	—
108	C₂⁴×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	—	2	2	no	—
111	C₂×Q₈×C₅₃	—	2	1	no	—
112	C₂×Q₈×C₅₃	—	2	1	no	—
113	C₂×Q₈×C₅₃	—	4	2	no	—
114	C₂×Q₈×C₅₃	—	8	2	no	—
115	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
116	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	2	2	no	—
119	C₂×D₄×C₅₃	Q₁₆	1	4	yes	complemented
120	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	8	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	4	4	no	—
129	C₂²×C₄×C₅₃	—	8	4	no	—
130	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	Q₁₆×C₅₃	—	4	4	no	—
134	Q₁₆×C₅₃	—	8	4	no	—
135	M₁₆×C₅₃	—	4	2	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	2	8	no	—
140	C₄⋊C₄×C₅₃	—	2	8	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	—	2	1	no	—
152	C₄⋊C₄×C₅₃	—	2	1	no	—
153	C₄⋊C₄×C₅₃	—	4	1	no	—
154	C₄⋊C₄×C₅₃	—	4	1	no	—
155	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₅₃	—	2	8	no	—
158	C₂²⋊C₄×C₅₃	—	2	2	no	—
159	C₂²⋊C₄×C₅₃	—	2	2	no	—
160	C₂²⋊C₄×C₅₃	—	4	1	no	—
161	C₄²×C₅₃	—	2	2	no	—
162	C₄²×C₅₃	—	2	2	no	—
163	C₄²×C₅₃	—	4	1	no	—
164	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
168	C₂³×C₅₃	—	2	2	no	—
169	C₂³×C₅₃	—	2	4	no	—
170	C₂³×C₅₃	—	4	2	no	—
171	C₂³×C₅₃	—	4	2	no	—
172	C₂³×C₅₃	#120	1	1	yes	characteristic
173	Q₈×C₅₃	—	2	2	no	—
174	Q₈×C₅₃	—	2	2	no	—
175	Q₈×C₅₃	—	4	2	no	—
176	Q₈×C₅₃	—	4	4	no	—
177	Q₈×C₅₃	—	4	2	no	—
178	Q₈×C₅₃	—	4	2	no	—
179	Q₈×C₅₃	—	8	2	no	—
180	D₄×C₅₃	—	2	2	no	—
181	D₄×C₅₃	—	2	4	no	—
182	D₄×C₅₃	C₂×Q₁₆	1	4	yes	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	4	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	4	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	4	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	4	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	8	8	no	—
198	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
199	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
200	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
202	C₂×C₄×C₅₃	#137	1	2	yes	—
203	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
206	C₈×C₅₃	—	4	2	no	—
207	C₈×C₅₃	—	4	2	no	—
208	C₈×C₅₃	—	4	2	no	—
209	#18749	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂²×C₅₃	—	2	1	no	—
211	C₂²×C₅₃	—	2	1	no	—
212	C₂²×C₅₃	—	2	4	no	—
213	C₂²×C₅₃	—	4	8	no	—
214	C₂²×C₅₃	—	4	2	no	—
215	C₂²×C₅₃	—	4	2	no	—
216	C₂²×C₅₃	#451	1	2	yes	—
217	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	4	2	no	—
222	C₄×C₅₃	—	4	2	no	—
223	C₄×C₅₃	—	4	2	no	—
224	C₄×C₅₃	—	4	4	no	—
225	C₄×C₅₃	—	8	1	no	—
226	C₄×C₅₃	—	8	1	no	—
227	C₄×C₅₃	#451	1	2	yes	—
228	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
229	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
230	C₄×C₅₃	#575	1	2	yes	—
231	C₂×#554	C₂×C₅₃	1	1	yes	characteristic
232	#3209	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#3104	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#3077	C₂×C₅₃	1	1	yes	characteristic
235	#1644	C₂×C₅₃	1	4	yes	complemented
236	#1632	C₂×C₅₃	1	4	yes	complemented
237	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
238	#1487	C₂×C₅₃	1	1	yes	characteristic, complemented
239	#1478	C₂×C₅₃	1	1	yes	characteristic
240	C₂×C₅₃	—	2	2	no	—
241	C₂×C₅₃	—	4	4	no	—
242	C₂×C₅₃	C₂×#451	1	1	yes	characteristic
243	C₂×C₅₃	#3005	1	1	yes	characteristic
244	C₂×C₅₃	#3019	1	1	yes	characteristic
245	#554	—	2	2	no	—
246	#554	—	2	2	no	—
247	#554	—	2	2	no	—
248	#554	C₂²×C₅₃	1	4	yes	—
249	#533	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
251	C₂×#125	—	2	1	no	—
252	C₂×C₂²⋊Q₈	—	2	1	no	—
253	C₂×C₂²⋊Q₈	—	2	2	no	—
254	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
255	C₂×C₄×D₄	—	2	1	no	—
256	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
257	C₂²×C₄⋊C₄	—	2	1	no	—
258	#494	C₂²×C₅₃	1	2	yes	complemented
259	#487	C₂²×C₅₃	1	2	yes	complemented
260	#484	—	2	1	no	—
261	#478	—	2	2	no	—
262	#477	C₂²×C₅₃	1	2	yes	complemented
263	#476	C₂²×C₅₃	1	2	yes	—
264	#474	C₂²×C₅₃	1	2	yes	—
265	#462	—	2	2	no	—
266	#462	C₂²×C₅₃	1	2	yes	complemented
267	#451	—	2	1	no	—
268	#451	—	2	2	no	—
269	C₄×Q₁₆	—	2	2	no	—
270	#431	C₂²×C₅₃	1	1	yes	characteristic
271	#431	C₂²×C₅₃	1	1	yes	characteristic
272	#428	—	2	1	no	—
273	#423	C₂²×C₅₃	1	1	yes	characteristic
274	#418	—	2	1	no	—
275	#333	C₂²×C₅₃	1	4	yes	complemented
276	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	complemented
277	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
278	C₅₃	#18749	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
280	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
281	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
282	C₂³×C₄	—	2	2	no	—
283	C₂³×C₄	—	2	1	no	—
284	C₂×Q₁₆	—	2	2	no	—
285	C₂×Q₁₆	—	2	4	no	—
286	C₂×M₁₆	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	—	2	1	no	—
289	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
290	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
291	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
292	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
293	#125	—	2	4	no	—
294	#125	—	2	4	no	—
295	#125	—	2	1	no	—
296	C₂²⋊Q₈	—	2	4	no	—
297	C₂²⋊Q₈	—	2	8	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	C₂²⋊Q₈	—	2	8	no	—
300	C₂²⋊Q₈	—	2	1	no	—
301	C₂²⋊Q₈	—	2	1	no	—
302	C₄×Q₈	—	2	1	no	—
303	C₄×Q₈	—	2	1	no	—
304	C₄×Q₈	—	2	1	no	—
305	C₄×Q₈	—	2	1	no	—
306	C₄×Q₈	D₄×C₅₃	1	2	yes	—
307	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	2	4	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	D₄×C₅₃	1	2	yes	—
313	C₄×D₄	C₂³×C₅₃	1	2	yes	—
314	#117	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	4	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	4	no	—
322	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
323	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
328	#107	—	2	2	no	—
329	#107	—	2	2	no	—
330	#106	—	2	2	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
334	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
335	#103	—	2	2	no	—
336	#103	—	2	4	no	—
337	#103	—	2	2	no	—
338	#102	—	2	2	no	—
339	#102	—	2	2	no	—
340	C₂²⋊C₈	—	2	1	no	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₄×C₈	—	2	1	no	—
343	C₄×C₈	C₂³×C₅₃	1	2	yes	—
344	C₂⁴	—	2	1	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	4	2	no	—
350	C₂×Q₈	—	8	2	no	—
351	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
352	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	Q₁₆×C₅₃	1	4	yes	complemented
356	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	4	4	no	—
365	C₂²×C₄	—	8	4	no	—
366	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
369	Q₁₆	—	4	4	no	—
370	Q₁₆	—	8	4	no	—
371	M₁₆	—	4	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₄⋊C₄	—	2	8	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	—	4	1	no	—
391	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
392	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	2	no	—
396	C₂²⋊C₄	—	4	1	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	4	1	no	—
400	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
401	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
402	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
403	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
404	C₂³	—	2	2	no	—
405	C₂³	—	2	4	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	#120×C₅₃	1	1	yes	characteristic
409	Q₈	—	2	2	no	—
410	Q₈	—	2	2	no	—
411	Q₈	—	4	4	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	2	no	—
415	Q₈	—	8	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	2	4	no	—
418	D₄	C₂×Q₁₆×C₅₃	1	4	yes	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	8	8	no	—
434	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
435	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
436	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
437	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
438	C₂×C₄	#137×C₅₃	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
441	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
442	C₈	—	4	2	no	—
443	C₈	—	4	2	no	—
444	C₈	—	4	2	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	1	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	4	8	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	C₅₃×#451	1	2	yes	—
452	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
453	C₄	—	2	1	no	—
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	4	no	—
460	C₄	—	8	1	no	—
461	C₄	—	8	1	no	—
462	C₄	C₅₃×#451	1	2	yes	—
463	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
464	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
465	C₄	C₅₃×#575	1	2	yes	—
466	C₂	—	2	2	no	—
467	C₂	—	4	4	no	—
468	C₂	C₂×C₅₃×#451	1	1	yes	characteristic
469	C₂	C₅₃×#3005	1	1	yes	characteristic
470	C₂	C₅₃×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×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
2. C₅₃×#3209	C₂	semidirect
3. C₅₃×#3104	C₂	semidirect
5. C₅₃×#1644	C₂	semidirect
6. C₅₃×#1632	C₂	semidirect
7. C₅₃×#1548	C₂	semidirect
8. C₅₃×#1487	C₂	semidirect
23. C₅₃×#494	C₂²	semidirect
24. C₅₃×#487	C₂²	semidirect
27. C₅₃×#477	C₂²	semidirect
31. C₅₃×#462	C₂²	semidirect
40. C₅₃×#333	C₂²	semidirect
41. C₅₃×Q₈⋊C₈	C₂²	semidirect
119. C₂×D₄×C₅₃	Q₁₆	semidirect
209. #18749	C₅₃	direct (N × H)
232. #3209	C₂×C₅₃	semidirect
233. #3104	C₂×C₅₃	semidirect
235. #1644	C₂×C₅₃	semidirect
236. #1632	C₂×C₅₃	semidirect
237. #1548	C₂×C₅₃	semidirect
238. #1487	C₂×C₅₃	semidirect
258. #494	C₂²×C₅₃	semidirect
259. #487	C₂²×C₅₃	semidirect
262. #477	C₂²×C₅₃	semidirect
266. #462	C₂²×C₅₃	semidirect
275. #333	C₂²×C₅₃	semidirect
276. Q₈⋊C₈	C₂²×C₅₃	semidirect
278. C₅₃	#18749	direct (N × H)
355. C₂×D₄	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#554	2
C₅₃×#3209	2
C₅₃×#3104	2
C₅₃×#3077	2
C₅₃×#1644	2
C₅₃×#1632	2
C₅₃×#1548	2
C₅₃×#1487	2
C₅₃×#1478	2
#18749	53

Subgroups — C53×#18749 #27226261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18749 #27226261