Subgroups — C₅₃×#19198 #27226710

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 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
3	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1701	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1681	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₅₃	—	2	1	no	—
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#493	—	2	1	no	—
15	C₅₃×#492	—	2	1	no	—
16	C₅₃×#487	—	2	1	no	—
17	C₅₃×#486	—	2	1	no	—
18	C₅₃×#486	—	2	1	no	—
19	C₅₃×#486	—	2	1	no	—
20	C₅₃×#486	C₂²	1	2	yes	—
21	C₅₃×#486	C₂²	1	2	yes	—
22	C₅₃×#461	C₂²	1	4	yes	complemented
23	C₅₃×#459	C₂²	1	4	yes	—
24	C₅₃×#448	—	2	2	no	—
25	C₅₃×#447	—	2	2	no	—
26	C₅₃×#436	—	2	1	no	—
27	C₅₃×#436	C₂²	1	2	yes	complemented
28	C₈×D₄×C₅₃	—	2	1	no	—
29	C₈×D₄×C₅₃	C₂²	1	2	yes	complemented
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#423	C₂²	1	1	yes	characteristic
33	C₂×#102×C₅₃	—	2	1	no	—
34	C₂×#102×C₅₃	—	2	1	no	—
35	C₅₃×#408	—	2	1	no	—
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#335	C₂²	1	2	yes	complemented
38	C₅₃×#334	C₂²	1	2	yes	complemented
39	C₅₃×#332	C₂²	1	4	yes	complemented
40	C₅₃×#331	C₂²	1	4	yes	—
41	C₂×#41×C₅₃	—	2	1	no	—
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂²×D₄×C₅₃	—	2	1	no	—
44	C₂²×D₄×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	—	2	1	no	—
46	C₂×SD₁₆×C₅₃	—	2	4	no	—
47	C₂×D₈×C₅₃	—	2	4	no	—
48	#131×C₅₃	—	4	4	no	—
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
58	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
59	#127×C₅₃	—	2	1	no	—
60	#127×C₅₃	—	2	1	no	—
61	#127×C₅₃	—	2	1	no	—
62	#127×C₅₃	D₄	1	1	yes	characteristic
63	#127×C₅₃	C₂³	1	2	yes	—
64	#124×C₅₃	—	2	1	no	—
65	#121×C₅₃	—	2	1	no	—
66	#121×C₅₃	D₄	1	2	yes	—
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	—	2	1	no	—
73	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
74	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
75	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	4	no	—
77	#102×C₅₃	—	2	2	no	—
78	#102×C₅₃	—	2	2	no	—
79	#102×C₅₃	—	2	2	no	—
80	#102×C₅₃	—	2	2	no	—
81	#102×C₅₃	—	2	4	no	—
82	C₂²⋊C₈×C₅₃	—	2	2	no	—
83	C₂²⋊C₈×C₅₃	—	2	2	no	—
84	C₂²⋊C₈×C₅₃	—	2	1	no	—
85	C₂²⋊C₈×C₅₃	—	2	1	no	—
86	C₄×C₈×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂⁴×C₅₃	—	4	1	no	—
93	#41×C₅₃	—	4	2	no	—
94	#41×C₅₃	—	4	2	no	—
95	C₂×Q₈×C₅₃	—	2	1	no	—
96	C₂×Q₈×C₅₃	—	2	2	no	—
97	C₂×Q₈×C₅₃	—	4	1	no	—
98	C₂×Q₈×C₅₃	—	8	2	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	—	2	1	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	2	no	—
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	4	1	no	—
106	C₂×D₄×C₅₃	—	4	1	no	—
107	C₂×D₄×C₅₃	—	4	1	no	—
108	C₂×D₄×C₅₃	—	4	2	no	—
109	C₂×D₄×C₅₃	—	4	2	no	—
110	C₂×D₄×C₅₃	—	8	2	no	—
111	C₂×D₄×C₅₃	—	8	2	no	—
112	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	4	1	no	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	SD₁₆×C₅₃	—	8	4	no	—
119	D₈×C₅₃	—	8	4	no	—
120	M₁₆×C₅₃	—	4	4	no	—
121	M₁₆×C₅₃	—	4	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	4	4	no	—
128	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	4	1	no	—
136	C₄⋊C₄×C₅₃	—	4	1	no	—
137	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₅₃	—	2	2	no	—
139	C₂²⋊C₄×C₅₃	—	2	2	no	—
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅₃	#41	1	1	yes	characteristic
143	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
144	C₂³×C₅₃	—	2	2	no	—
145	C₂³×C₅₃	—	4	2	no	—
146	C₂³×C₅₃	—	8	2	no	—
147	C₂³×C₅₃	—	8	2	no	—
148	C₂³×C₅₃	—	8	2	no	—
149	C₂³×C₅₃	#121	1	1	yes	characteristic
150	Q₈×C₅₃	—	4	2	no	—
151	Q₈×C₅₃	—	4	4	no	—
152	Q₈×C₅₃	—	8	2	no	—
153	D₄×C₅₃	—	2	4	no	—
154	D₄×C₅₃	—	4	4	no	—
155	D₄×C₅₃	—	4	2	no	—
156	D₄×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	4	4	no	—
158	D₄×C₅₃	—	4	2	no	—
159	D₄×C₅₃	—	4	2	no	—
160	D₄×C₅₃	—	8	2	no	—
161	D₄×C₅₃	—	8	2	no	—
162	D₄×C₅₃	—	8	2	no	—
163	D₄×C₅₃	—	8	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	8	2	no	—
171	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
172	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
173	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
174	C₂×C₄×C₅₃	#135	1	2	yes	—
175	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₈×C₅₃	—	2	1	no	—
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	—	2	1	no	—
182	C₈×C₅₃	—	2	1	no	—
183	C₈×C₅₃	—	4	2	no	—
184	C₈×C₅₃	—	4	2	no	—
185	C₈×C₅₃	—	4	2	no	—
186	#19198	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	4	2	no	—
192	C₂²×C₅₃	—	8	2	no	—
193	C₂²×C₅₃	—	8	2	no	—
194	C₂²×C₅₃	—	8	2	no	—
195	C₂²×C₅₃	—	16	4	no	—
196	C₂²×C₅₃	#463	1	2	yes	—
197	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	4	2	no	—
201	C₄×C₅₃	—	8	2	no	—
202	C₄×C₅₃	#463	1	2	yes	—
203	C₄×C₅₃	#471	1	2	yes	—
204	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
205	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₄×C₅₃	#576	1	2	yes	—
207	#3206	C₂×C₅₃	1	1	yes	characteristic, complemented
208	#3205	C₂×C₅₃	1	1	yes	characteristic
209	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#2931	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#1701	C₂×C₅₃	1	4	yes	complemented
212	#1681	C₂×C₅₃	1	4	yes	complemented
213	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#1549	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#1545	C₂×C₅₃	1	1	yes	characteristic, complemented
216	C₂×C₅₃	—	2	2	no	—
217	C₂×C₅₃	—	8	2	no	—
218	C₂×C₅₃	—	16	2	no	—
219	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
220	C₂×C₅₃	#3058	1	1	yes	characteristic
221	C₂×C₅₃	C₂×#471	1	1	yes	characteristic
222	C₂×#131	—	2	1	no	—
223	#532	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
226	#493	—	2	1	no	—
227	#492	—	2	1	no	—
228	#487	—	2	1	no	—
229	#486	—	2	1	no	—
230	#486	—	2	1	no	—
231	#486	—	2	1	no	—
232	#486	C₂²×C₅₃	1	2	yes	—
233	#486	C₂²×C₅₃	1	2	yes	—
234	#461	C₂²×C₅₃	1	4	yes	complemented
235	#459	C₂²×C₅₃	1	4	yes	—
236	#448	—	2	2	no	—
237	#447	—	2	2	no	—
238	#436	—	2	1	no	—
239	#436	C₂²×C₅₃	1	2	yes	complemented
240	C₈×D₄	—	2	1	no	—
241	C₈×D₄	C₂²×C₅₃	1	2	yes	complemented
242	#431	C₂²×C₅₃	1	1	yes	characteristic
243	#431	C₂²×C₅₃	1	1	yes	characteristic
244	#423	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×#102	—	2	1	no	—
246	C₂×#102	—	2	1	no	—
247	#408	—	2	1	no	—
248	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
249	#335	C₂²×C₅₃	1	2	yes	complemented
250	#334	C₂²×C₅₃	1	2	yes	complemented
251	#332	C₂²×C₅₃	1	4	yes	complemented
252	#331	C₂²×C₅₃	1	4	yes	—
253	C₅₃	#19198	1	1	yes	characteristic, direct product factor, Sylow
254	C₂×#41	—	2	1	no	—
255	C₂²×Q₈	—	2	1	no	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	—	2	1	no	—
258	C₂²×D₄	—	2	1	no	—
259	C₂×SD₁₆	—	2	4	no	—
260	C₂×D₈	—	2	4	no	—
261	#131	—	4	4	no	—
262	C₂×M₁₆	—	2	1	no	—
263	C₂×M₁₆	—	2	2	no	—
264	C₂²×C₈	—	2	2	no	—
265	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
266	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	—	2	1	no	—
270	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
271	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
272	#127	—	2	1	no	—
273	#127	—	2	1	no	—
274	#127	—	2	1	no	—
275	#127	D₄×C₅₃	1	1	yes	characteristic
276	#127	C₂³×C₅₃	1	2	yes	—
277	#124	—	2	1	no	—
278	#121	—	2	1	no	—
279	#121	D₄×C₅₃	1	2	yes	—
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	C₂³×C₅₃	1	2	yes	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
287	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
288	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
289	#103	—	2	4	no	—
290	#102	—	2	2	no	—
291	#102	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	#102	—	2	4	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
302	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
305	C₂⁴	—	4	1	no	—
306	#41	—	4	2	no	—
307	#41	—	4	2	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	4	1	no	—
311	C₂×Q₈	—	8	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	1	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂×D₄	—	8	2	no	—
324	C₂×D₄	—	8	2	no	—
325	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	4	1	no	—
328	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
330	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
331	SD₁₆	—	8	4	no	—
332	D₈	—	8	4	no	—
333	M₁₆	—	4	4	no	—
334	M₁₆	—	4	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	4	no	—
341	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
355	C₄²	#41×C₅₃	1	1	yes	characteristic
356	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
357	C₂³	—	2	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	8	2	no	—
360	C₂³	—	8	2	no	—
361	C₂³	—	8	2	no	—
362	C₂³	#121×C₅₃	1	1	yes	characteristic
363	Q₈	—	4	2	no	—
364	Q₈	—	4	4	no	—
365	Q₈	—	8	2	no	—
366	D₄	—	2	4	no	—
367	D₄	—	4	4	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	4	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	2	no	—
375	D₄	—	8	2	no	—
376	D₄	—	8	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
385	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
386	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
387	C₂×C₄	#135×C₅₃	1	2	yes	—
388	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
389	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₈	—	4	2	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	1	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	16	4	no	—
408	C₂²	C₅₃×#463	1	2	yes	—
409	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
410	C₄	—	2	1	no	—
411	C₄	—	2	1	no	—
412	C₄	—	4	2	no	—
413	C₄	—	8	2	no	—
414	C₄	C₅₃×#463	1	2	yes	—
415	C₄	C₅₃×#471	1	2	yes	—
416	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
417	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
418	C₄	C₅₃×#576	1	2	yes	—
419	C₂	—	2	2	no	—
420	C₂	—	8	2	no	—
421	C₂	—	16	2	no	—
422	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
423	C₂	C₅₃×#3058	1	1	yes	characteristic
424	C₂	C₂×C₅₃×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
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×#3206	C₂	semidirect
3. C₂×C₅₃×#486	C₂	semidirect
4. C₅₃×#2931	C₂	semidirect
5. C₅₃×#1701	C₂	semidirect
6. C₅₃×#1681	C₂	semidirect
7. C₅₃×#1564	C₂	semidirect
8. C₅₃×#1549	C₂	semidirect
9. C₅₃×#1545	C₂	semidirect
22. C₅₃×#461	C₂²	semidirect
27. C₅₃×#436	C₂²	semidirect
29. C₈×D₄×C₅₃	C₂²	semidirect
37. C₅₃×#335	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
39. C₅₃×#332	C₂²	semidirect
186. #19198	C₅₃	direct (N × H)
207. #3206	C₂×C₅₃	semidirect
209. C₂×#486	C₂×C₅₃	semidirect
210. #2931	C₂×C₅₃	semidirect
211. #1701	C₂×C₅₃	semidirect
212. #1681	C₂×C₅₃	semidirect
213. #1564	C₂×C₅₃	semidirect
214. #1549	C₂×C₅₃	semidirect
215. #1545	C₂×C₅₃	semidirect
234. #461	C₂²×C₅₃	semidirect
239. #436	C₂²×C₅₃	semidirect
241. C₈×D₄	C₂²×C₅₃	semidirect
249. #335	C₂²×C₅₃	semidirect
250. #334	C₂²×C₅₃	semidirect
251. #332	C₂²×C₅₃	semidirect
253. C₅₃	#19198	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
C₂×C₅₃×#486	2
C₅₃×#2931	2
C₅₃×#1701	2
C₅₃×#1681	2
C₅₃×#1564	2
C₅₃×#1549	2
C₅₃×#1545	2
#19198	53

Subgroups — C53×#19198 #27226710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19198 #27226710