Subgroups — C₅₃×#19627 #27227139

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1720	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1717	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1711	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1709	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#498	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#494	C₂²	1	2	yes	complemented
18	C₅₃×#493	C₂²	1	2	yes	—
19	C₅₃×#492	C₂²	1	1	yes	characteristic, complemented
20	C₅₃×#490	—	2	2	no	—
21	C₅₃×#489	—	2	2	no	—
22	C₅₃×#487	C₂²	1	2	yes	complemented
23	C₅₃×#487	C₂²	1	1	yes	characteristic, complemented
24	C₅₃×#487	C₂²	1	2	yes	complemented
25	C₅₃×#486	—	2	2	no	—
26	C₅₃×#486	C₂²	1	2	yes	—
27	C₅₃×#486	C₂²	1	2	yes	—
28	C₅₃×#486	C₂²	1	2	yes	—
29	C₅₃×#486	C₂²	1	2	yes	—
30	C₅₃×#486	C₂²	1	2	yes	—
31	C₅₃×#486	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#486	C₂²	1	2	yes	complemented
33	C₅₃×#465	—	2	2	no	—
34	C₅₃×#462	—	2	2	no	—
35	C₅₃×#459	—	2	2	no	—
36	C₅₃×#451	—	2	2	no	—
37	C₅₃×#447	—	2	2	no	—
38	C₄×SD₁₆×C₅₃	—	2	2	no	—
39	C₅₃×#431	C₂²	1	1	yes	characteristic
40	C₅₃×#418	—	2	2	no	—
41	C₅₃×#417	—	2	2	no	—
42	C₂×#102×C₅₃	—	2	2	no	—
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₅₃	C₂²	1	2	yes	complemented
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	C₂²×D₄×C₅₃	—	2	1	no	—
52	C₂×Q₁₆×C₅₃	—	2	4	no	—
53	C₂×SD₁₆×C₅₃	—	2	2	no	—
54	C₂×D₈×C₅₃	—	2	4	no	—
55	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
56	C₂²×C₈×C₅₃	D₄	1	2	yes	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
64	#127×C₅₃	—	2	1	no	—
65	#127×C₅₃	—	2	1	no	—
66	#127×C₅₃	—	2	1	no	—
67	#127×C₅₃	C₂³	1	2	yes	—
68	#127×C₅₃	C₂³	1	1	yes	characteristic
69	#125×C₅₃	—	2	1	no	—
70	#124×C₅₃	—	2	1	no	—
71	C₂²⋊Q₈×C₅₃	—	2	1	no	—
72	C₂²⋊Q₈×C₅₃	—	2	1	no	—
73	#121×C₅₃	—	2	1	no	—
74	#121×C₅₃	—	2	1	no	—
75	C₄×Q₈×C₅₃	—	2	1	no	—
76	C₄×Q₈×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	—	2	1	no	—
78	C₄×D₄×C₅₃	—	2	1	no	—
79	#117×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	#106×C₅₃	—	2	2	no	—
84	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
85	#103×C₅₃	—	2	4	no	—
86	#103×C₅₃	—	2	2	no	—
87	#103×C₅₃	—	2	4	no	—
88	#102×C₅₃	—	2	4	no	—
89	#102×C₅₃	—	2	4	no	—
90	#102×C₅₃	—	2	4	no	—
91	#102×C₅₃	—	2	4	no	—
92	#102×C₅₃	—	2	4	no	—
93	#102×C₅₃	—	2	4	no	—
94	#102×C₅₃	—	2	2	no	—
95	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
96	#97×C₅₃	D₄	1	2	yes	—
97	C₄×C₈×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	2	yes	—
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂⁴×C₅₃	—	4	1	no	—
103	#41×C₅₃	—	4	4	no	—
104	C₂×Q₈×C₅₃	—	2	1	no	—
105	C₂×Q₈×C₅₃	—	2	1	no	—
106	C₂×Q₈×C₅₃	—	2	2	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	4	1	no	—
109	C₂×Q₈×C₅₃	—	8	2	no	—
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	4	4	no	—
117	C₂×D₄×C₅₃	—	4	1	no	—
118	C₂×D₄×C₅₃	—	4	1	no	—
119	C₂×D₄×C₅₃	—	4	1	no	—
120	C₂×D₄×C₅₃	—	4	1	no	—
121	C₂×D₄×C₅₃	—	4	4	no	—
122	C₂×D₄×C₅₃	—	8	2	no	—
123	C₂²×C₄×C₅₃	—	4	1	no	—
124	C₂²×C₄×C₅₃	—	4	1	no	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	Q₁₆×C₅₃	—	8	4	no	—
129	SD₁₆×C₅₃	—	4	4	no	—
130	D₈×C₅₃	—	8	4	no	—
131	M₁₆×C₅₃	—	2	4	no	—
132	C₂×C₈×C₅₃	—	4	2	no	—
133	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
136	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	2	2	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	4	1	no	—
146	C₄⋊C₄×C₅₃	—	4	1	no	—
147	C₄⋊C₄×C₅₃	—	4	1	no	—
148	C₄⋊C₄×C₅₃	—	4	1	no	—
149	C₂²⋊C₄×C₅₃	—	4	1	no	—
150	C₂²⋊C₄×C₅₃	—	4	1	no	—
151	C₂²⋊C₄×C₅₃	—	4	1	no	—
152	C₄²×C₅₃	—	4	1	no	—
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂³×C₅₃	—	4	1	no	—
158	C₂³×C₅₃	—	4	2	no	—
159	C₂³×C₅₃	—	8	2	no	—
160	C₂³×C₅₃	—	8	2	no	—
161	C₂³×C₅₃	—	8	2	no	—
162	C₂³×C₅₃	#127	1	1	yes	characteristic
163	Q₈×C₅₃	—	4	2	no	—
164	Q₈×C₅₃	—	4	2	no	—
165	Q₈×C₅₃	—	4	4	no	—
166	Q₈×C₅₃	—	4	2	no	—
167	Q₈×C₅₃	—	8	2	no	—
168	D₄×C₅₃	—	4	4	no	—
169	D₄×C₅₃	—	4	2	no	—
170	D₄×C₅₃	—	4	4	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	4	4	no	—
173	D₄×C₅₃	—	4	2	no	—
174	D₄×C₅₃	—	8	2	no	—
175	D₄×C₅₃	—	8	2	no	—
176	D₄×C₅₃	—	8	2	no	—
177	D₄×C₅₃	—	8	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	8	2	no	—
185	C₂×C₄×C₅₃	—	8	2	no	—
186	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
187	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
188	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
189	C₂×C₄×C₅₃	#135	1	4	yes	—
190	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
193	C₈×C₅₃	—	2	4	no	—
194	C₈×C₅₃	—	2	2	no	—
195	C₈×C₅₃	—	2	2	no	—
196	C₈×C₅₃	—	2	4	no	—
197	#19627	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂²×C₅₃	—	2	1	no	—
199	C₂²×C₅₃	—	2	1	no	—
200	C₂²×C₅₃	—	8	1	no	—
201	C₂²×C₅₃	—	8	2	no	—
202	C₂²×C₅₃	—	8	2	no	—
203	C₂²×C₅₃	—	8	1	no	—
204	C₂²×C₅₃	—	8	2	no	—
205	C₂²×C₅₃	—	8	1	no	—
206	C₂²×C₅₃	—	16	4	no	—
207	C₂²×C₅₃	#495	1	2	yes	—
208	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	2	1	no	—
211	C₄×C₅₃	—	8	1	no	—
212	C₄×C₅₃	—	8	1	no	—
213	C₄×C₅₃	—	8	2	no	—
214	C₄×C₅₃	—	8	1	no	—
215	C₄×C₅₃	#495	1	2	yes	—
216	C₄×C₅₃	C₂×#135	1	2	yes	—
217	C₄×C₅₃	#575	1	2	yes	—
218	C₄×C₅₃	#576	1	2	yes	—
219	#3209	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#3205	C₂×C₅₃	1	1	yes	characteristic
221	#3167	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#3138	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#1720	C₂×C₅₃	1	2	yes	complemented
226	#1717	C₂×C₅₃	1	2	yes	complemented
227	#1711	C₂×C₅₃	1	2	yes	complemented
228	#1709	C₂×C₅₃	1	2	yes	complemented
229	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
230	C₂×C₅₃	—	2	2	no	—
231	C₂×C₅₃	—	16	1	no	—
232	C₂×C₅₃	—	16	2	no	—
233	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
234	C₂×C₅₃	#3153	1	1	yes	characteristic
235	C₂×C₅₃	#3155	1	1	yes	characteristic
236	#533	C₂²×C₅₃	1	1	yes	characteristic
237	#532	C₂²×C₅₃	1	1	yes	characteristic
238	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
240	#498	C₂²×C₅₃	1	1	yes	characteristic, complemented
241	#494	C₂²×C₅₃	1	2	yes	complemented
242	#493	C₂²×C₅₃	1	2	yes	—
243	#492	C₂²×C₅₃	1	1	yes	characteristic, complemented
244	#490	—	2	2	no	—
245	#489	—	2	2	no	—
246	#487	C₂²×C₅₃	1	2	yes	complemented
247	#487	C₂²×C₅₃	1	1	yes	characteristic, complemented
248	#487	C₂²×C₅₃	1	2	yes	complemented
249	#486	—	2	2	no	—
250	#486	C₂²×C₅₃	1	2	yes	—
251	#486	C₂²×C₅₃	1	2	yes	—
252	#486	C₂²×C₅₃	1	2	yes	—
253	#486	C₂²×C₅₃	1	2	yes	—
254	#486	C₂²×C₅₃	1	2	yes	—
255	#486	C₂²×C₅₃	1	2	yes	complemented
256	#486	C₂²×C₅₃	1	1	yes	characteristic, complemented
257	#465	—	2	2	no	—
258	#462	—	2	2	no	—
259	#459	—	2	2	no	—
260	#451	—	2	2	no	—
261	#447	—	2	2	no	—
262	C₄×SD₁₆	—	2	2	no	—
263	#431	C₂²×C₅₃	1	1	yes	characteristic
264	#418	—	2	2	no	—
265	#417	—	2	2	no	—
266	C₂×#102	—	2	2	no	—
267	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
269	#322	C₂²×C₅₃	1	2	yes	complemented
270	C₈²	C₂²×C₅₃	1	2	yes	complemented
271	C₅₃	#19627	1	1	yes	characteristic, direct product factor, Sylow
272	C₂×#41	—	2	1	no	—
273	C₂²×Q₈	—	2	1	no	—
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	—	2	1	no	—
277	C₂×Q₁₆	—	2	4	no	—
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×D₈	—	2	4	no	—
280	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
281	C₂²×C₈	D₄×C₅₃	1	2	yes	—
282	C₄⋊Q₈	—	2	1	no	—
283	C₄⋊Q₈	—	2	1	no	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
286	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
287	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
288	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
289	#127	—	2	1	no	—
290	#127	—	2	1	no	—
291	#127	—	2	1	no	—
292	#127	C₂³×C₅₃	1	2	yes	—
293	#127	C₂³×C₅₃	1	1	yes	characteristic
294	#125	—	2	1	no	—
295	#124	—	2	1	no	—
296	C₂²⋊Q₈	—	2	1	no	—
297	C₂²⋊Q₈	—	2	1	no	—
298	#121	—	2	1	no	—
299	#121	—	2	1	no	—
300	C₄×Q₈	—	2	1	no	—
301	C₄×Q₈	—	2	1	no	—
302	C₄×D₄	—	2	1	no	—
303	C₄×D₄	—	2	1	no	—
304	#117	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
308	#106	—	2	2	no	—
309	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
310	#103	—	2	4	no	—
311	#103	—	2	2	no	—
312	#103	—	2	4	no	—
313	#102	—	2	4	no	—
314	#102	—	2	4	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	#102	—	2	4	no	—
318	#102	—	2	2	no	—
319	#102	—	2	4	no	—
320	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
321	#97	D₄×C₅₃	1	2	yes	—
322	C₄×C₈	D₄×C₅₃	1	2	yes	—
323	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
324	C₄×C₈	C₂³×C₅₃	1	2	yes	—
325	C₄×C₈	C₂³×C₅₃	1	2	yes	—
326	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
327	C₂⁴	—	4	1	no	—
328	#41	—	4	4	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	4	1	no	—
334	C₂×Q₈	—	8	2	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	4	4	no	—
342	C₂×D₄	—	4	1	no	—
343	C₂×D₄	—	4	1	no	—
344	C₂×D₄	—	4	1	no	—
345	C₂×D₄	—	4	1	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂×D₄	—	8	2	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	—	4	1	no	—
350	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
353	Q₁₆	—	8	4	no	—
354	SD₁₆	—	4	4	no	—
355	D₈	—	8	4	no	—
356	M₁₆	—	2	4	no	—
357	C₂×C₈	—	4	2	no	—
358	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
359	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
361	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	4	1	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₄⋊C₄	—	4	1	no	—
373	C₄⋊C₄	—	4	1	no	—
374	C₂²⋊C₄	—	4	1	no	—
375	C₂²⋊C₄	—	4	1	no	—
376	C₂²⋊C₄	—	4	1	no	—
377	C₄²	—	4	1	no	—
378	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
379	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
380	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
381	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
382	C₂³	—	4	1	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	8	2	no	—
385	C₂³	—	8	2	no	—
386	C₂³	—	8	2	no	—
387	C₂³	#127×C₅₃	1	1	yes	characteristic
388	Q₈	—	4	2	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	4	4	no	—
392	Q₈	—	8	2	no	—
393	D₄	—	4	4	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	4	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	4	no	—
399	D₄	—	8	2	no	—
400	D₄	—	8	2	no	—
401	D₄	—	8	2	no	—
402	D₄	—	8	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	8	2	no	—
411	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
412	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
413	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
414	C₂×C₄	#135×C₅₃	1	4	yes	—
415	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
418	C₈	—	2	4	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	—	2	4	no	—
422	C₂²	—	2	1	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	1	no	—
430	C₂²	—	16	4	no	—
431	C₂²	C₅₃×#495	1	2	yes	—
432	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	1	no	—
437	C₄	—	8	2	no	—
438	C₄	—	8	1	no	—
439	C₄	C₅₃×#495	1	2	yes	—
440	C₄	C₂×#135×C₅₃	1	2	yes	—
441	C₄	C₅₃×#575	1	2	yes	—
442	C₄	C₅₃×#576	1	2	yes	—
443	C₂	—	2	2	no	—
444	C₂	—	16	2	no	—
445	C₂	—	16	1	no	—
446	C₂	C₂×C₅₃×#495	1	1	yes	characteristic
447	C₂	C₅₃×#3153	1	1	yes	characteristic
448	C₂	C₅₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3205	C₂
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
1. C₅₃×#3209	C₂	semidirect
3. C₅₃×#3167	C₂	semidirect
4. C₅₃×#3141	C₂	semidirect
5. C₅₃×#3138	C₂	semidirect
6. C₂×C₅₃×#486	C₂	semidirect
7. C₅₃×#1720	C₂	semidirect
8. C₅₃×#1717	C₂	semidirect
9. C₅₃×#1711	C₂	semidirect
10. C₅₃×#1709	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
16. C₅₃×#498	C₂²	semidirect
17. C₅₃×#494	C₂²	semidirect
19. C₅₃×#492	C₂²	semidirect
22. C₅₃×#487	C₂²	semidirect
23. C₅₃×#487	C₂²	semidirect
24. C₅₃×#487	C₂²	semidirect
31. C₅₃×#486	C₂²	semidirect
32. C₅₃×#486	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₈²×C₅₃	C₂²	semidirect
197. #19627	C₅₃	direct (N × H)
219. #3209	C₂×C₅₃	semidirect
221. #3167	C₂×C₅₃	semidirect
222. #3141	C₂×C₅₃	semidirect
223. #3138	C₂×C₅₃	semidirect
224. C₂×#486	C₂×C₅₃	semidirect
225. #1720	C₂×C₅₃	semidirect
226. #1717	C₂×C₅₃	semidirect
227. #1711	C₂×C₅₃	semidirect
228. #1709	C₂×C₅₃	semidirect
229. C₈×M₁₆	C₂×C₅₃	semidirect
240. #498	C₂²×C₅₃	semidirect
241. #494	C₂²×C₅₃	semidirect
243. #492	C₂²×C₅₃	semidirect
246. #487	C₂²×C₅₃	semidirect
247. #487	C₂²×C₅₃	semidirect
248. #487	C₂²×C₅₃	semidirect
255. #486	C₂²×C₅₃	semidirect
256. #486	C₂²×C₅₃	semidirect
269. #322	C₂²×C₅₃	semidirect
270. C₈²	C₂²×C₅₃	semidirect
271. C₅₃	#19627	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×#3209	2
C₅₃×#3205	2
C₅₃×#3167	2
C₅₃×#3141	2
C₅₃×#3138	2
C₂×C₅₃×#486	2
C₅₃×#1720	2
C₅₃×#1717	2
C₅₃×#1711	2
C₅₃×#1709	2
C₈×M₁₆×C₅₃	2
#19627	53

Subgroups — C53×#19627 #27227139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19627 #27227139