Subgroups — C₅₃×#19196 #27226708

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3227	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×#1697	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1677	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1548	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×#498	—	2	1	no	—
15	C₅₃×#493	—	2	1	no	—
16	C₅₃×#486	—	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×#481	—	2	2	no	—
23	C₅₃×#476	C₂²	1	4	yes	complemented
24	C₅₃×#459	C₂²	1	4	yes	—
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×#333	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×D₈×C₅₃	—	2	4	no	—
47	#131×C₅₃	—	4	4	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
58	#127×C₅₃	—	2	1	no	—
59	#127×C₅₃	—	2	1	no	—
60	#127×C₅₃	—	2	1	no	—
61	#127×C₅₃	D₄	1	1	yes	characteristic
62	#127×C₅₃	C₂³	1	2	yes	—
63	#124×C₅₃	—	2	1	no	—
64	#121×C₅₃	—	2	1	no	—
65	#121×C₅₃	D₄	1	2	yes	—
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#106×C₅₃	—	2	4	no	—
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	#102×C₅₃	—	2	2	no	—
77	#102×C₅₃	—	2	2	no	—
78	#102×C₅₃	—	2	2	no	—
79	#102×C₅₃	—	2	2	no	—
80	#102×C₅₃	—	2	4	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	D₈×C₅₃	—	8	4	no	—
119	M₁₆×C₅₃	—	4	4	no	—
120	M₁₆×C₅₃	—	4	2	no	—
121	C₂×C₈×C₅₃	—	2	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	—	4	4	no	—
127	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	4	1	no	—
135	C₄⋊C₄×C₅₃	—	4	1	no	—
136	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
137	C₂²⋊C₄×C₅₃	—	2	2	no	—
138	C₂²⋊C₄×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅₃	#41	1	1	yes	characteristic
142	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	4	2	no	—
145	C₂³×C₅₃	—	8	2	no	—
146	C₂³×C₅₃	—	8	2	no	—
147	C₂³×C₅₃	—	8	2	no	—
148	C₂³×C₅₃	#121	1	1	yes	characteristic
149	Q₈×C₅₃	—	4	2	no	—
150	Q₈×C₅₃	—	8	2	no	—
151	Q₈×C₅₃	—	8	2	no	—
152	D₄×C₅₃	—	2	4	no	—
153	D₄×C₅₃	—	4	4	no	—
154	D₄×C₅₃	—	4	2	no	—
155	D₄×C₅₃	—	4	2	no	—
156	D₄×C₅₃	—	4	4	no	—
157	D₄×C₅₃	—	4	2	no	—
158	D₄×C₅₃	—	4	2	no	—
159	D₄×C₅₃	—	8	2	no	—
160	D₄×C₅₃	—	8	2	no	—
161	D₄×C₅₃	—	8	2	no	—
162	D₄×C₅₃	—	8	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	8	2	no	—
170	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
171	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
172	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
173	C₂×C₄×C₅₃	#135	1	2	yes	—
174	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
178	C₈×C₅₃	—	2	1	no	—
179	C₈×C₅₃	—	2	1	no	—
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	—	2	1	no	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	—	4	2	no	—
184	C₈×C₅₃	—	4	2	no	—
185	#19196	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	—	4	2	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	8	2	no	—
192	C₂²×C₅₃	—	8	2	no	—
193	C₂²×C₅₃	—	8	2	no	—
194	C₂²×C₅₃	—	16	4	no	—
195	C₂²×C₅₃	#463	1	2	yes	—
196	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	#463	1	2	yes	—
202	C₄×C₅₃	#472	1	2	yes	—
203	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
204	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₄×C₅₃	#577	1	2	yes	—
206	#3227	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#3205	C₂×C₅₃	1	1	yes	characteristic
208	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
209	#2931	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1697	C₂×C₅₃	1	4	yes	complemented
211	#1677	C₂×C₅₃	1	4	yes	complemented
212	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#1548	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#1545	C₂×C₅₃	1	1	yes	characteristic, complemented
215	C₂×C₅₃	—	2	2	no	—
216	C₂×C₅₃	—	8	2	no	—
217	C₂×C₅₃	—	16	2	no	—
218	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
219	C₂×C₅₃	#3057	1	1	yes	characteristic
220	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
221	C₂×#131	—	2	1	no	—
222	#532	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
225	#498	—	2	1	no	—
226	#493	—	2	1	no	—
227	#486	—	2	1	no	—
228	#486	—	2	1	no	—
229	#486	—	2	1	no	—
230	#486	—	2	1	no	—
231	#486	C₂²×C₅₃	1	2	yes	—
232	#486	C₂²×C₅₃	1	2	yes	—
233	#481	—	2	2	no	—
234	#476	C₂²×C₅₃	1	4	yes	complemented
235	#459	C₂²×C₅₃	1	4	yes	—
236	#447	—	2	2	no	—
237	#436	—	2	1	no	—
238	#436	C₂²×C₅₃	1	2	yes	complemented
239	C₈×D₄	—	2	1	no	—
240	C₈×D₄	C₂²×C₅₃	1	2	yes	complemented
241	#431	C₂²×C₅₃	1	1	yes	characteristic
242	#431	C₂²×C₅₃	1	1	yes	characteristic
243	#423	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#102	—	2	1	no	—
245	C₂×#102	—	2	1	no	—
246	#408	—	2	1	no	—
247	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
248	#335	C₂²×C₅₃	1	2	yes	complemented
249	#334	C₂²×C₅₃	1	2	yes	complemented
250	#333	C₂²×C₅₃	1	4	yes	complemented
251	#331	C₂²×C₅₃	1	4	yes	—
252	C₅₃	#19196	1	1	yes	characteristic, direct product factor, Sylow
253	C₂×#41	—	2	1	no	—
254	C₂²×Q₈	—	2	1	no	—
255	C₂²×D₄	—	2	1	no	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	—	2	1	no	—
258	C₂×D₈	—	2	4	no	—
259	#131	—	4	4	no	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
264	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	—	2	1	no	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
269	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
270	#127	—	2	1	no	—
271	#127	—	2	1	no	—
272	#127	—	2	1	no	—
273	#127	D₄×C₅₃	1	1	yes	characteristic
274	#127	C₂³×C₅₃	1	2	yes	—
275	#124	—	2	1	no	—
276	#121	—	2	1	no	—
277	#121	D₄×C₅₃	1	2	yes	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	C₂³×C₅₃	1	2	yes	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
283	#106	—	2	4	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
286	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
287	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
288	#102	—	2	4	no	—
289	#102	—	2	2	no	—
290	#102	—	2	2	no	—
291	#102	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	4	no	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
301	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₂⁴	—	4	1	no	—
305	#41	—	4	2	no	—
306	#41	—	4	2	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	4	1	no	—
310	C₂×Q₈	—	8	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	4	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	2	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	8	2	no	—
323	C₂×D₄	—	8	2	no	—
324	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
329	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
330	D₈	—	8	4	no	—
331	M₁₆	—	4	4	no	—
332	M₁₆	—	4	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	4	4	no	—
339	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
341	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	4	1	no	—
347	C₄⋊C₄	—	4	1	no	—
348	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
352	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
353	C₄²	#41×C₅₃	1	1	yes	characteristic
354	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
355	C₂³	—	2	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	8	2	no	—
358	C₂³	—	8	2	no	—
359	C₂³	—	8	2	no	—
360	C₂³	#121×C₅₃	1	1	yes	characteristic
361	Q₈	—	4	2	no	—
362	Q₈	—	8	2	no	—
363	Q₈	—	8	2	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	4	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	2	no	—
368	D₄	—	4	4	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	8	2	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	2	no	—
382	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
383	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
384	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
385	C₂×C₄	#135×C₅₃	1	2	yes	—
386	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
387	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
388	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
390	C₈	—	2	1	no	—
391	C₈	—	2	1	no	—
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	2	no	—
396	C₈	—	4	2	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	16	4	no	—
406	C₂²	C₅₃×#463	1	2	yes	—
407	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
408	C₄	—	2	1	no	—
409	C₄	—	2	1	no	—
410	C₄	—	4	2	no	—
411	C₄	—	8	2	no	—
412	C₄	C₅₃×#463	1	2	yes	—
413	C₄	C₅₃×#472	1	2	yes	—
414	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
415	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
416	C₄	C₅₃×#577	1	2	yes	—
417	C₂	—	2	2	no	—
418	C₂	—	8	2	no	—
419	C₂	—	16	2	no	—
420	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
421	C₂	C₅₃×#3057	1	1	yes	characteristic
422	C₂	C₂×C₅₃×#472	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×#3227	C₂	semidirect
3. C₂×C₅₃×#486	C₂	semidirect
4. C₅₃×#2931	C₂	semidirect
5. C₅₃×#1697	C₂	semidirect
6. C₅₃×#1677	C₂	semidirect
7. C₅₃×#1564	C₂	semidirect
8. C₅₃×#1548	C₂	semidirect
9. C₅₃×#1545	C₂	semidirect
23. C₅₃×#476	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×#333	C₂²	semidirect
185. #19196	C₅₃	direct (N × H)
206. #3227	C₂×C₅₃	semidirect
208. C₂×#486	C₂×C₅₃	semidirect
209. #2931	C₂×C₅₃	semidirect
210. #1697	C₂×C₅₃	semidirect
211. #1677	C₂×C₅₃	semidirect
212. #1564	C₂×C₅₃	semidirect
213. #1548	C₂×C₅₃	semidirect
214. #1545	C₂×C₅₃	semidirect
234. #476	C₂²×C₅₃	semidirect
238. #436	C₂²×C₅₃	semidirect
240. C₈×D₄	C₂²×C₅₃	semidirect
248. #335	C₂²×C₅₃	semidirect
249. #334	C₂²×C₅₃	semidirect
250. #333	C₂²×C₅₃	semidirect
252. C₅₃	#19196	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×#3227	2
C₅₃×#3205	2
C₂×C₅₃×#486	2
C₅₃×#2931	2
C₅₃×#1697	2
C₅₃×#1677	2
C₅₃×#1564	2
C₅₃×#1548	2
C₅₃×#1545	2
#19196	53

Subgroups — C53×#19196 #27226708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19196 #27226708