Subgroups — C₅₃×#19235 #27226747

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3246	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1703	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1699	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1683	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1679	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₅₃	—	2	1	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#500	—	2	1	no	—
17	C₅₃×#491	—	2	2	no	—
18	C₅₃×#487	C₂²	1	2	yes	—
19	C₅₃×#487	C₂²	1	2	yes	—
20	C₅₃×#486	—	2	1	no	—
21	C₅₃×#480	—	2	2	no	—
22	C₅₃×#476	C₂²	1	2	yes	—
23	C₅₃×#475	—	2	1	no	—
24	C₅₃×#474	C₂²	1	2	yes	—
25	C₅₃×#474	C₂²	1	2	yes	—
26	C₅₃×#465	—	2	2	no	—
27	C₅₃×#461	C₂²	1	2	yes	—
28	C₅₃×#460	—	2	1	no	—
29	C₅₃×#436	C₂²	1	2	yes	—
30	C₅₃×#435	—	2	2	no	—
31	C₈×D₄×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#419	—	2	1	no	—
36	C₅₃×#419	—	2	1	no	—
37	C₅₃×#408	—	2	1	no	—
38	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#335	C₂²	1	2	yes	—
40	C₅₃×#334	C₂²	1	2	yes	—
41	C₅₃×#333	C₂²	1	1	yes	characteristic
42	C₅₃×#333	C₂²	1	1	yes	characteristic
43	C₅₃×#332	C₂²	1	1	yes	characteristic
44	C₅₃×#332	C₂²	1	2	yes	—
45	C₅₃×#332	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	#131×C₅₃	—	4	4	no	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	—	2	2	no	—
55	C₂²×C₈×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
64	#127×C₅₃	D₄	1	1	yes	characteristic
65	#125×C₅₃	—	2	1	no	—
66	#124×C₅₃	—	2	1	no	—
67	C₂²⋊Q₈×C₅₃	—	2	1	no	—
68	C₂²⋊Q₈×C₅₃	—	2	1	no	—
69	#121×C₅₃	—	2	1	no	—
70	#121×C₅₃	D₄	1	2	yes	—
71	C₄×Q₈×C₅₃	—	2	1	no	—
72	C₄×Q₈×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	C₂³	1	2	yes	—
75	#117×C₅₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	#107×C₅₃	—	2	2	no	—
80	#107×C₅₃	—	2	2	no	—
81	#106×C₅₃	—	2	2	no	—
82	C₄⋊C₈×C₅₃	—	2	1	no	—
83	C₄⋊C₈×C₅₃	—	2	1	no	—
84	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
87	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
89	#103×C₅₃	—	2	2	no	—
90	#103×C₅₃	—	2	2	no	—
91	#103×C₅₃	—	2	2	no	—
92	#103×C₅₃	—	2	2	no	—
93	#103×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	2	2	no	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	—	2	2	no	—
97	C₂²⋊C₈×C₅₃	—	2	2	no	—
98	C₂²⋊C₈×C₅₃	—	2	2	no	—
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	#97×C₅₃	D₄	1	2	yes	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	C₂³	1	2	yes	—
103	#41×C₅₃	—	4	2	no	—
104	#41×C₅₃	—	4	2	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	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	4	1	no	—
110	C₂×Q₈×C₅₃	—	4	2	no	—
111	C₂×Q₈×C₅₃	—	4	2	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	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	—	4	1	no	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
121	SD₁₆×C₅₃	—	8	2	no	—
122	M₁₆×C₅₃	—	2	2	no	—
123	M₁₆×C₅₃	—	2	2	no	—
124	M₁₆×C₅₃	—	4	4	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	—	2	1	no	—
128	C₂×C₈×C₅₃	—	2	1	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	—	4	4	no	—
131	C₂×C₈×C₅₃	—	4	2	no	—
132	C₂×C₈×C₅₃	—	8	1	no	—
133	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
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	2	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	4	1	no	—
143	C₄⋊C₄×C₅₃	—	4	1	no	—
144	C₄⋊C₄×C₅₃	—	4	1	no	—
145	C₄⋊C₄×C₅₃	—	4	1	no	—
146	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
147	C₂²⋊C₄×C₅₃	—	2	2	no	—
148	C₂²⋊C₄×C₅₃	—	2	2	no	—
149	C₂²⋊C₄×C₅₃	—	4	1	no	—
150	C₄²×C₅₃	—	4	1	no	—
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	#41	1	1	yes	characteristic
154	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
155	C₂³×C₅₃	—	2	2	no	—
156	C₂³×C₅₃	#121	1	1	yes	characteristic
157	Q₈×C₅₃	—	4	2	no	—
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	4	2	no	—
160	Q₈×C₅₃	—	4	1	no	—
161	Q₈×C₅₃	—	4	1	no	—
162	Q₈×C₅₃	—	8	1	no	—
163	Q₈×C₅₃	—	8	2	no	—
164	D₄×C₅₃	—	2	2	no	—
165	D₄×C₅₃	—	2	2	no	—
166	D₄×C₅₃	—	4	4	no	—
167	D₄×C₅₃	—	4	2	no	—
168	D₄×C₅₃	—	8	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	4	1	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	8	2	no	—
179	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
180	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
181	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
182	C₂×C₄×C₅₃	#135	1	2	yes	—
183	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
187	C₈×C₅₃	—	2	2	no	—
188	C₈×C₅₃	—	2	2	no	—
189	C₈×C₅₃	—	4	1	no	—
190	C₈×C₅₃	—	4	1	no	—
191	C₈×C₅₃	—	4	1	no	—
192	C₈×C₅₃	—	4	2	no	—
193	C₈×C₅₃	—	4	1	no	—
194	#19235	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	2	1	no	—
197	C₂²×C₅₃	—	4	2	no	—
198	C₂²×C₅₃	—	4	2	no	—
199	C₂²×C₅₃	—	4	2	no	—
200	C₂²×C₅₃	#463	1	2	yes	—
201	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	4	2	no	—
206	C₄×C₅₃	—	8	2	no	—
207	C₄×C₅₃	—	8	1	no	—
208	C₄×C₅₃	—	8	1	no	—
209	C₄×C₅₃	#463	1	2	yes	—
210	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
211	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
212	C₄×C₅₃	#577	1	1	yes	characteristic
213	C₄×C₅₃	#578	1	1	yes	characteristic
214	#3246	C₂×C₅₃	1	1	yes	characteristic
215	#3226	C₂×C₅₃	1	1	yes	characteristic
216	#3139	C₂×C₅₃	1	1	yes	characteristic, complemented
217	#2935	C₂×C₅₃	1	1	yes	characteristic
218	#1703	C₂×C₅₃	1	2	yes	complemented
219	#1699	C₂×C₅₃	1	2	yes	complemented
220	#1683	C₂×C₅₃	1	2	yes	complemented
221	#1679	C₂×C₅₃	1	2	yes	complemented
222	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#1542	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₂×#332	C₂×C₅₃	1	1	yes	characteristic, complemented
225	C₂×C₅₃	—	2	2	no	—
226	C₂×C₅₃	—	8	2	no	—
227	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
228	C₂×C₅₃	#3063	1	1	yes	characteristic
229	C₂×C₅₃	#3069	1	1	yes	characteristic
230	C₂×#131	—	2	1	no	—
231	#533	C₂²×C₅₃	1	1	yes	characteristic
232	#532	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
234	#500	—	2	1	no	—
235	#491	—	2	2	no	—
236	#487	C₂²×C₅₃	1	2	yes	—
237	#487	C₂²×C₅₃	1	2	yes	—
238	#486	—	2	1	no	—
239	#480	—	2	2	no	—
240	#476	C₂²×C₅₃	1	2	yes	—
241	#475	—	2	1	no	—
242	#474	C₂²×C₅₃	1	2	yes	—
243	#474	C₂²×C₅₃	1	2	yes	—
244	#465	—	2	2	no	—
245	#461	C₂²×C₅₃	1	2	yes	—
246	#460	—	2	1	no	—
247	#436	C₂²×C₅₃	1	2	yes	—
248	#435	—	2	2	no	—
249	C₈×D₄	C₂²×C₅₃	1	2	yes	—
250	#431	C₂²×C₅₃	1	1	yes	characteristic
251	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
253	#419	—	2	1	no	—
254	#419	—	2	1	no	—
255	#408	—	2	1	no	—
256	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
257	#335	C₂²×C₅₃	1	2	yes	—
258	#334	C₂²×C₅₃	1	2	yes	—
259	#333	C₂²×C₅₃	1	1	yes	characteristic
260	#333	C₂²×C₅₃	1	1	yes	characteristic
261	#332	C₂²×C₅₃	1	2	yes	—
262	#332	C₂²×C₅₃	1	2	yes	—
263	#332	C₂²×C₅₃	1	1	yes	characteristic
264	#332	C₂²×C₅₃	1	1	yes	characteristic
265	C₅₃	#19235	1	1	yes	characteristic, direct product factor, Sylow
266	C₂×#41	—	2	1	no	—
267	C₂²×Q₈	—	2	1	no	—
268	C₂×SD₁₆	—	2	2	no	—
269	#131	—	4	4	no	—
270	C₂×M₁₆	—	2	2	no	—
271	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
272	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
273	C₂²×C₈	—	2	1	no	—
274	C₂²×C₈	—	2	1	no	—
275	C₂²×C₈	—	2	2	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄⋊Q₈	—	2	1	no	—
279	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
280	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
281	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
282	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
283	#127	D₄×C₅₃	1	1	yes	characteristic
284	#125	—	2	1	no	—
285	#124	—	2	1	no	—
286	C₂²⋊Q₈	—	2	1	no	—
287	C₂²⋊Q₈	—	2	1	no	—
288	#121	—	2	1	no	—
289	#121	D₄×C₅₃	1	2	yes	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	—	2	1	no	—
292	C₄×D₄	—	2	1	no	—
293	C₄×D₄	C₂³×C₅₃	1	2	yes	—
294	#117	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
298	#107	—	2	2	no	—
299	#107	—	2	2	no	—
300	#106	—	2	2	no	—
301	C₄⋊C₈	—	2	1	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
305	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
306	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
307	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
308	#103	—	2	2	no	—
309	#103	—	2	2	no	—
310	#103	—	2	2	no	—
311	#103	—	2	2	no	—
312	#103	—	2	2	no	—
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	#102	—	2	2	no	—
316	C₂²⋊C₈	—	2	1	no	—
317	C₂²⋊C₈	—	2	2	no	—
318	C₂²⋊C₈	—	2	2	no	—
319	#97	D₄×C₅₃	1	2	yes	—
320	C₄×C₈	—	2	1	no	—
321	C₄×C₈	C₂³×C₅₃	1	2	yes	—
322	#41	—	4	2	no	—
323	#41	—	4	2	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	1	no	—
328	C₂×Q₈	—	4	1	no	—
329	C₂×Q₈	—	4	2	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	4	1	no	—
337	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
340	SD₁₆	—	8	2	no	—
341	M₁₆	—	2	2	no	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	4	4	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	4	4	no	—
350	C₂×C₈	—	4	2	no	—
351	C₂×C₈	—	8	1	no	—
352	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
353	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₄⋊C₄	—	4	1	no	—
364	C₄⋊C₄	—	4	1	no	—
365	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
366	C₂²⋊C₄	—	2	2	no	—
367	C₂²⋊C₄	—	2	2	no	—
368	C₂²⋊C₄	—	4	1	no	—
369	C₄²	—	4	1	no	—
370	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
371	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₄²	#41×C₅₃	1	1	yes	characteristic
373	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
374	C₂³	—	2	2	no	—
375	C₂³	#121×C₅₃	1	1	yes	characteristic
376	Q₈	—	4	2	no	—
377	Q₈	—	4	2	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	4	1	no	—
380	Q₈	—	4	1	no	—
381	Q₈	—	8	1	no	—
382	Q₈	—	8	2	no	—
383	D₄	—	2	2	no	—
384	D₄	—	2	2	no	—
385	D₄	—	4	4	no	—
386	D₄	—	4	2	no	—
387	D₄	—	8	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
399	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
400	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
401	C₂×C₄	#135×C₅₃	1	2	yes	—
402	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
403	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	1	no	—
409	C₈	—	4	1	no	—
410	C₈	—	4	1	no	—
411	C₈	—	4	2	no	—
412	C₈	—	4	1	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	C₅₃×#463	1	2	yes	—
419	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	4	2	no	—
424	C₄	—	8	2	no	—
425	C₄	—	8	1	no	—
426	C₄	—	8	1	no	—
427	C₄	C₅₃×#463	1	2	yes	—
428	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
429	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
430	C₄	C₅₃×#577	1	1	yes	characteristic
431	C₄	C₅₃×#578	1	1	yes	characteristic
432	C₂	—	2	2	no	—
433	C₂	—	8	2	no	—
434	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
435	C₂	C₅₃×#3063	1	1	yes	characteristic
436	C₂	C₅₃×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#532	C₂²
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×#3139	C₂	semidirect
5. C₅₃×#1703	C₂	semidirect
6. C₅₃×#1699	C₂	semidirect
7. C₅₃×#1683	C₂	semidirect
8. C₅₃×#1679	C₂	semidirect
9. C₅₃×#1565	C₂	semidirect
10. C₅₃×#1542	C₂	semidirect
11. C₂×C₅₃×#332	C₂	semidirect
194. #19235	C₅₃	direct (N × H)
216. #3139	C₂×C₅₃	semidirect
218. #1703	C₂×C₅₃	semidirect
219. #1699	C₂×C₅₃	semidirect
220. #1683	C₂×C₅₃	semidirect
221. #1679	C₂×C₅₃	semidirect
222. #1565	C₂×C₅₃	semidirect
223. #1542	C₂×C₅₃	semidirect
224. C₂×#332	C₂×C₅₃	semidirect
265. C₅₃	#19235	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×#3246	2
C₅₃×#3226	2
C₅₃×#3139	2
C₅₃×#2935	2
C₅₃×#1703	2
C₅₃×#1699	2
C₅₃×#1683	2
C₅₃×#1679	2
C₅₃×#1565	2
C₅₃×#1542	2
C₂×C₅₃×#332	2
#19235	53

Subgroups — C53×#19235 #27226747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19235 #27226747