Subgroups — C₅₃×#19673 #27227185

Order 13568.

862 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	C₅₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3151	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1721	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1719	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1717	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1717	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
18	C₅₃×#496	—	2	2	no	—
19	C₅₃×#493	C₂²	1	2	yes	—
20	C₅₃×#493	C₂²	1	2	yes	—
21	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
22	C₅₃×#492	C₂²	1	2	yes	—
23	C₅₃×#492	C₂²	1	2	yes	complemented
24	C₅₃×#489	—	2	2	no	—
25	C₅₃×#487	C₂²	1	2	yes	complemented
26	C₅₃×#487	C₂²	1	2	yes	complemented
27	C₅₃×#486	—	2	2	no	—
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	1	yes	characteristic, complemented
33	C₅₃×#486	C₂²	1	2	yes	complemented
34	C₅₃×#466	—	2	2	no	—
35	C₅₃×#461	—	2	2	no	—
36	C₅₃×#459	—	2	2	no	—
37	C₅₃×#448	—	2	2	no	—
38	C₅₃×#447	—	2	2	no	—
39	C₄×D₈×C₅₃	—	2	2	no	—
40	C₅₃×#431	C₂²	1	1	yes	characteristic
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	C₅₃×#322	C₂²	1	2	yes	complemented
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂×#41×C₅₃	—	2	1	no	—
49	C₂²×Q₈×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	C₂²×D₄×C₅₃	—	2	1	no	—
52	C₂²×D₄×C₅₃	—	2	1	no	—
53	AGL(1,ℤ/8)×C₅₃	—	4	8	no	—
54	C₂×SD₁₆×C₅₃	—	2	4	no	—
55	C₂×SD₁₆×C₅₃	—	2	4	no	—
56	C₂×D₈×C₅₃	—	2	4	no	—
57	C₂×D₈×C₅₃	—	2	2	no	—
58	C₂×D₈×C₅₃	—	2	4	no	—
59	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
60	C₂²×C₈×C₅₃	D₄	1	2	yes	—
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	—	2	1	no	—
64	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
67	#127×C₅₃	—	2	1	no	—
68	#127×C₅₃	—	2	1	no	—
69	#127×C₅₃	—	2	1	no	—
70	#127×C₅₃	C₂³	1	2	yes	—
71	#127×C₅₃	C₂³	1	1	yes	characteristic
72	#127×C₅₃	C₂³	1	1	yes	characteristic
73	#124×C₅₃	—	2	1	no	—
74	#124×C₅₃	—	2	1	no	—
75	#121×C₅₃	—	2	1	no	—
76	#121×C₅₃	—	2	1	no	—
77	#121×C₅₃	—	2	1	no	—
78	#121×C₅₃	—	2	1	no	—
79	C₄×D₄×C₅₃	—	2	1	no	—
80	C₄×D₄×C₅₃	—	2	1	no	—
81	C₄×D₄×C₅₃	—	2	1	no	—
82	C₄×D₄×C₅₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	#107×C₅₃	—	2	2	no	—
87	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
88	#103×C₅₃	—	2	4	no	—
89	#103×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	#102×C₅₃	—	2	2	no	—
96	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
97	#97×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	D₄	1	2	yes	—
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₄×C₈×C₅₃	C₂³	1	2	yes	—
103	C₂⁴×C₅₃	—	4	1	no	—
104	#41×C₅₃	—	4	4	no	—
105	#41×C₅₃	—	4	4	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	2	no	—
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	4	1	no	—
110	C₂×Q₈×C₅₃	—	8	2	no	—
111	C₂×D₄×C₅₃	—	2	1	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	2	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	2	1	no	—
119	C₂×D₄×C₅₃	—	2	2	no	—
120	C₂×D₄×C₅₃	—	4	4	no	—
121	C₂×D₄×C₅₃	—	4	1	no	—
122	C₂×D₄×C₅₃	—	4	1	no	—
123	C₂×D₄×C₅₃	—	4	1	no	—
124	C₂×D₄×C₅₃	—	4	1	no	—
125	C₂×D₄×C₅₃	—	4	1	no	—
126	C₂×D₄×C₅₃	—	4	4	no	—
127	C₂×D₄×C₅₃	—	8	2	no	—
128	C₂²×C₄×C₅₃	—	4	1	no	—
129	C₂²×C₄×C₅₃	—	4	1	no	—
130	C₂²×C₄×C₅₃	—	4	1	no	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
134	SD₁₆×C₅₃	—	4	8	no	—
135	SD₁₆×C₅₃	—	8	4	no	—
136	D₈×C₅₃	—	4	4	no	—
137	D₈×C₅₃	—	4	8	no	—
138	D₈×C₅₃	—	8	4	no	—
139	M₁₆×C₅₃	—	2	4	no	—
140	C₂×C₈×C₅₃	—	4	2	no	—
141	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	1	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	—	4	1	no	—
152	C₄⋊C₄×C₅₃	—	4	1	no	—
153	C₂²⋊C₄×C₅₃	—	4	1	no	—
154	C₂²⋊C₄×C₅₃	—	4	1	no	—
155	C₂²⋊C₄×C₅₃	—	4	1	no	—
156	C₂²⋊C₄×C₅₃	—	4	1	no	—
157	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
161	C₂³×C₅₃	—	4	1	no	—
162	C₂³×C₅₃	—	4	1	no	—
163	C₂³×C₅₃	—	4	2	no	—
164	C₂³×C₅₃	—	8	2	no	—
165	C₂³×C₅₃	—	8	2	no	—
166	C₂³×C₅₃	—	8	2	no	—
167	C₂³×C₅₃	#127	1	1	yes	characteristic
168	Q₈×C₅₃	—	4	2	no	—
169	Q₈×C₅₃	—	4	2	no	—
170	Q₈×C₅₃	—	4	4	no	—
171	Q₈×C₅₃	—	8	2	no	—
172	D₄×C₅₃	—	4	4	no	—
173	D₄×C₅₃	—	4	2	no	—
174	D₄×C₅₃	—	4	2	no	—
175	D₄×C₅₃	—	4	4	no	—
176	D₄×C₅₃	—	4	2	no	—
177	D₄×C₅₃	—	4	2	no	—
178	D₄×C₅₃	—	4	4	no	—
179	D₄×C₅₃	—	4	2	no	—
180	D₄×C₅₃	—	4	2	no	—
181	D₄×C₅₃	—	8	2	no	—
182	D₄×C₅₃	—	8	2	no	—
183	D₄×C₅₃	—	8	2	no	—
184	D₄×C₅₃	—	8	2	no	—
185	D₄×C₅₃	—	8	2	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	1	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	8	2	no	—
192	C₂×C₄×C₅₃	—	8	2	no	—
193	C₂×C₄×C₅₃	—	8	2	no	—
194	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
195	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
196	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
197	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
198	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
201	C₈×C₅₃	—	2	4	no	—
202	C₈×C₅₃	—	2	2	no	—
203	C₈×C₅₃	—	2	2	no	—
204	C₈×C₅₃	—	2	4	no	—
205	#19673	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	8	1	no	—
209	C₂²×C₅₃	—	8	1	no	—
210	C₂²×C₅₃	—	8	2	no	—
211	C₂²×C₅₃	—	8	2	no	—
212	C₂²×C₅₃	—	8	1	no	—
213	C₂²×C₅₃	—	8	1	no	—
214	C₂²×C₅₃	—	8	2	no	—
215	C₂²×C₅₃	—	8	1	no	—
216	C₂²×C₅₃	—	8	1	no	—
217	C₂²×C₅₃	—	16	4	no	—
218	C₂²×C₅₃	#496	1	2	yes	—
219	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	2	1	no	—
222	C₄×C₅₃	—	8	1	no	—
223	C₄×C₅₃	—	8	1	no	—
224	C₄×C₅₃	—	8	2	no	—
225	C₄×C₅₃	#496	1	2	yes	—
226	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
227	C₄×C₅₃	#575	1	2	yes	—
228	C₄×C₅₃	#576	1	2	yes	—
229	#3206	C₂×C₅₃	1	1	yes	characteristic, complemented
230	#3205	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#3169	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#3151	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
235	#1721	C₂×C₅₃	1	2	yes	complemented
236	#1719	C₂×C₅₃	1	2	yes	complemented
237	#1717	C₂×C₅₃	1	2	yes	complemented
238	#1717	C₂×C₅₃	1	2	yes	complemented
239	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
240	C₂×C₅₃	—	2	2	no	—
241	C₂×C₅₃	—	16	1	no	—
242	C₂×C₅₃	—	16	1	no	—
243	C₂×C₅₃	—	16	2	no	—
244	C₂×C₅₃	C₂×#496	1	1	yes	characteristic
245	C₂×C₅₃	#3153	1	1	yes	characteristic
246	C₂×C₅₃	#3155	1	1	yes	characteristic
247	C₂×AGL(1,ℤ/8)	—	2	2	no	—
248	#532	C₂²×C₅₃	1	1	yes	characteristic
249	#532	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
251	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
252	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
253	#496	—	2	2	no	—
254	#493	C₂²×C₅₃	1	2	yes	—
255	#493	C₂²×C₅₃	1	2	yes	—
256	#493	C₂²×C₅₃	1	1	yes	characteristic, complemented
257	#492	C₂²×C₅₃	1	2	yes	—
258	#492	C₂²×C₅₃	1	2	yes	complemented
259	#489	—	2	2	no	—
260	#487	C₂²×C₅₃	1	2	yes	complemented
261	#487	C₂²×C₅₃	1	2	yes	complemented
262	#486	—	2	2	no	—
263	#486	C₂²×C₅₃	1	2	yes	—
264	#486	C₂²×C₅₃	1	2	yes	—
265	#486	C₂²×C₅₃	1	2	yes	—
266	#486	C₂²×C₅₃	1	2	yes	complemented
267	#486	C₂²×C₅₃	1	1	yes	characteristic, complemented
268	#486	C₂²×C₅₃	1	1	yes	characteristic, complemented
269	#466	—	2	2	no	—
270	#461	—	2	2	no	—
271	#459	—	2	2	no	—
272	#448	—	2	2	no	—
273	#447	—	2	2	no	—
274	C₄×D₈	—	2	2	no	—
275	#431	C₂²×C₅₃	1	1	yes	characteristic
276	#417	—	2	2	no	—
277	C₂×#102	—	2	2	no	—
278	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
280	#322	C₂²×C₅₃	1	2	yes	complemented
281	#322	C₂²×C₅₃	1	2	yes	complemented
282	C₅₃	#19673	1	1	yes	characteristic, direct product factor, Sylow
283	C₂×#41	—	2	1	no	—
284	C₂×#41	—	2	1	no	—
285	C₂²×Q₈	—	2	1	no	—
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	—	2	1	no	—
288	C₂²×D₄	—	2	1	no	—
289	AGL(1,ℤ/8)	—	4	8	no	—
290	C₂×SD₁₆	—	2	4	no	—
291	C₂×SD₁₆	—	2	4	no	—
292	C₂×D₈	—	2	4	no	—
293	C₂×D₈	—	2	2	no	—
294	C₂×D₈	—	2	4	no	—
295	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
296	C₂²×C₈	D₄×C₅₃	1	2	yes	—
297	C₄⋊Q₈	—	2	1	no	—
298	C₄⋊Q₈	—	2	1	no	—
299	C₄⋊Q₈	—	2	1	no	—
300	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
301	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
302	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
303	#127	—	2	1	no	—
304	#127	—	2	1	no	—
305	#127	—	2	1	no	—
306	#127	C₂³×C₅₃	1	2	yes	—
307	#127	C₂³×C₅₃	1	1	yes	characteristic
308	#127	C₂³×C₅₃	1	1	yes	characteristic
309	#124	—	2	1	no	—
310	#124	—	2	1	no	—
311	#121	—	2	1	no	—
312	#121	—	2	1	no	—
313	#121	—	2	1	no	—
314	#121	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
322	#107	—	2	2	no	—
323	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
324	#103	—	2	4	no	—
325	#103	—	2	4	no	—
326	#102	—	2	4	no	—
327	#102	—	2	4	no	—
328	#102	—	2	4	no	—
329	#102	—	2	2	no	—
330	#102	—	2	2	no	—
331	#102	—	2	4	no	—
332	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
333	#97	D₄×C₅₃	1	2	yes	—
334	C₄×C₈	D₄×C₅₃	1	2	yes	—
335	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
336	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
337	C₄×C₈	C₂³×C₅₃	1	2	yes	—
338	C₄×C₈	C₂³×C₅₃	1	2	yes	—
339	C₂⁴	—	4	1	no	—
340	#41	—	4	4	no	—
341	#41	—	4	4	no	—
342	C₂×Q₈	—	2	1	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	4	1	no	—
346	C₂×Q₈	—	8	2	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	4	4	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	—	4	1	no	—
359	C₂×D₄	—	4	1	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	—	4	1	no	—
362	C₂×D₄	—	4	4	no	—
363	C₂×D₄	—	8	2	no	—
364	C₂²×C₄	—	4	1	no	—
365	C₂²×C₄	—	4	1	no	—
366	C₂²×C₄	—	4	1	no	—
367	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
370	SD₁₆	—	4	8	no	—
371	SD₁₆	—	8	4	no	—
372	D₈	—	4	8	no	—
373	D₈	—	4	4	no	—
374	D₈	—	8	4	no	—
375	M₁₆	—	2	4	no	—
376	C₂×C₈	—	4	2	no	—
377	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
378	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
379	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
380	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	4	1	no	—
388	C₄⋊C₄	—	4	1	no	—
389	C₂²⋊C₄	—	4	1	no	—
390	C₂²⋊C₄	—	4	1	no	—
391	C₂²⋊C₄	—	4	1	no	—
392	C₂²⋊C₄	—	4	1	no	—
393	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
394	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
395	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	8	2	no	—
401	C₂³	—	8	2	no	—
402	C₂³	—	8	2	no	—
403	C₂³	#127×C₅₃	1	1	yes	characteristic
404	Q₈	—	4	2	no	—
405	Q₈	—	4	2	no	—
406	Q₈	—	4	4	no	—
407	Q₈	—	8	2	no	—
408	D₄	—	4	2	no	—
409	D₄	—	4	4	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	4	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	4	no	—
416	D₄	—	4	2	no	—
417	D₄	—	8	2	no	—
418	D₄	—	8	2	no	—
419	D₄	—	8	2	no	—
420	D₄	—	8	2	no	—
421	D₄	—	8	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	8	2	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	—	8	2	no	—
430	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
431	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
432	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
433	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
434	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
435	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
436	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
437	C₈	—	2	4	no	—
438	C₈	—	2	2	no	—
439	C₈	—	2	2	no	—
440	C₈	—	2	4	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	8	1	no	—
444	C₂²	—	8	1	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	1	no	—
448	C₂²	—	8	1	no	—
449	C₂²	—	8	2	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	16	4	no	—
453	C₂²	C₅₃×#496	1	2	yes	—
454	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	8	1	no	—
458	C₄	—	8	1	no	—
459	C₄	—	8	2	no	—
460	C₄	C₅₃×#496	1	2	yes	—
461	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
462	C₄	C₅₃×#575	1	2	yes	—
463	C₄	C₅₃×#576	1	2	yes	—
464	C₂	—	2	2	no	—
465	C₂	—	16	2	no	—
466	C₂	—	16	1	no	—
467	C₂	—	16	1	no	—
468	C₂	C₂×C₅₃×#496	1	1	yes	characteristic
469	C₂	C₅₃×#3153	1	1	yes	characteristic
470	C₂	C₅₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
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×#3206	C₂	semidirect
2. C₅₃×#3205	C₂	semidirect
3. C₅₃×#3169	C₂	semidirect
4. C₅₃×#3151	C₂	semidirect
5. C₅₃×#3141	C₂	semidirect
6. C₂×C₅₃×#486	C₂	semidirect
7. C₅₃×#1721	C₂	semidirect
8. C₅₃×#1719	C₂	semidirect
9. C₅₃×#1717	C₂	semidirect
10. C₅₃×#1717	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
17. C₅₃×#500	C₂²	semidirect
21. C₅₃×#493	C₂²	semidirect
23. C₅₃×#492	C₂²	semidirect
25. C₅₃×#487	C₂²	semidirect
26. C₅₃×#487	C₂²	semidirect
31. C₅₃×#486	C₂²	semidirect
32. C₅₃×#486	C₂²	semidirect
33. C₅₃×#486	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₅₃×#322	C₂²	semidirect
205. #19673	C₅₃	direct (N × H)
229. #3206	C₂×C₅₃	semidirect
230. #3205	C₂×C₅₃	semidirect
231. #3169	C₂×C₅₃	semidirect
232. #3151	C₂×C₅₃	semidirect
233. #3141	C₂×C₅₃	semidirect
234. C₂×#486	C₂×C₅₃	semidirect
235. #1721	C₂×C₅₃	semidirect
236. #1719	C₂×C₅₃	semidirect
237. #1717	C₂×C₅₃	semidirect
238. #1717	C₂×C₅₃	semidirect
239. #1451	C₂×C₅₃	semidirect
252. #500	C₂²×C₅₃	semidirect
256. #493	C₂²×C₅₃	semidirect
258. #492	C₂²×C₅₃	semidirect
260. #487	C₂²×C₅₃	semidirect
261. #487	C₂²×C₅₃	semidirect
266. #486	C₂²×C₅₃	semidirect
267. #486	C₂²×C₅₃	semidirect
268. #486	C₂²×C₅₃	semidirect
280. #322	C₂²×C₅₃	semidirect
281. #322	C₂²×C₅₃	semidirect
282. C₅₃	#19673	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×#3206	2
C₅₃×#3205	2
C₅₃×#3169	2
C₅₃×#3151	2
C₅₃×#3141	2
C₂×C₅₃×#486	2
C₅₃×#1721	2
C₅₃×#1719	2
C₅₃×#1717	2
C₅₃×#1451	2
#19673	53

Subgroups — C53×#19673 #27227185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19673 #27227185