Subgroups — C₅₃×#19422 #27226934

Order 13568.

768 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3036	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1701	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1687	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
11	C₂²×D₈×C₅₃	—	2	1	no	—
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#493	C₂²	1	2	yes	complemented
16	C₅₃×#486	C₂²	1	2	yes	complemented
17	C₅₃×#463	—	2	2	no	—
18	C₅₃×#461	C₂²	1	2	yes	complemented
19	C₅₃×#459	—	2	2	no	—
20	C₅₃×#459	—	2	2	no	—
21	C₅₃×#459	C₂²	1	4	yes	—
22	C₅₃×#459	C₂²	1	2	yes	complemented
23	C₅₃×#447	—	2	2	no	—
24	C₅₃×#436	—	2	2	no	—
25	C₅₃×#436	C₂²	1	4	yes	complemented
26	C₅₃×#431	C₂²	1	1	yes	characteristic
27	C₅₃×#431	C₂²	1	1	yes	characteristic
28	C₅₃×#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#417	—	2	1	no	—
31	C₂×#102×C₅₃	—	2	1	no	—
32	C₅₃×#407	—	2	2	no	—
33	C₅₃×#334	C₂²	1	4	yes	complemented
34	C₅₃×#332	C₂²	1	2	yes	complemented
35	C₅₃×#331	C₂²	1	2	yes	complemented
36	C₅₃×#331	C₂²	1	4	yes	complemented
37	C₂×#41×C₅₃	—	2	1	no	—
38	C₂²×D₄×C₅₃	—	2	1	no	—
39	C₂²×D₄×C₅₃	—	2	1	no	—
40	C₂²×D₄×C₅₃	—	2	1	no	—
41	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
44	C₂×SD₁₆×C₅₃	—	2	2	no	—
45	C₂×D₈×C₅₃	—	2	4	no	—
46	C₂×D₈×C₅₃	—	2	4	no	—
47	C₂×D₈×C₅₃	—	2	2	no	—
48	C₂×D₈×C₅₃	—	4	4	no	—
49	C₂×M₁₆×C₅₃	—	2	4	no	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂²×C₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
53	#127×C₅₃	—	2	1	no	—
54	#127×C₅₃	—	2	1	no	—
55	#127×C₅₃	—	2	1	no	—
56	#127×C₅₃	C₂³	1	2	yes	—
57	#127×C₅₃	C₂³	1	1	yes	characteristic
58	#124×C₅₃	—	2	1	no	—
59	#121×C₅₃	—	2	1	no	—
60	#121×C₅₃	—	2	1	no	—
61	C₄×D₄×C₅₃	—	2	1	no	—
62	C₄×D₄×C₅₃	—	2	1	no	—
63	C₄×D₄×C₅₃	—	2	1	no	—
64	C₄×D₄×C₅₃	—	2	1	no	—
65	C₄×D₄×C₅₃	D₄	1	2	yes	—
66	C₄×D₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₅₃	—	2	1	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
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	2	no	—
77	#102×C₅₃	—	2	4	no	—
78	#102×C₅₃	—	2	4	no	—
79	#102×C₅₃	—	2	2	no	—
80	C₂²⋊C₈×C₅₃	—	2	4	no	—
81	C₂²⋊C₈×C₅₃	—	2	1	no	—
82	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
83	C₄×C₈×C₅₃	—	2	1	no	—
84	C₄×C₈×C₅₃	C₂³	1	2	yes	—
85	C₂⁴×C₅₃	—	2	1	no	—
86	C₂⁴×C₅₃	—	4	1	no	—
87	#41×C₅₃	—	4	2	no	—
88	#41×C₅₃	—	4	2	no	—
89	C₂×Q₈×C₅₃	—	2	1	no	—
90	C₂×D₄×C₅₃	—	2	1	no	—
91	C₂×D₄×C₅₃	—	2	2	no	—
92	C₂×D₄×C₅₃	—	2	2	no	—
93	C₂×D₄×C₅₃	—	2	1	no	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	2	no	—
96	C₂×D₄×C₅₃	—	2	1	no	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	4	1	no	—
100	C₂×D₄×C₅₃	—	4	1	no	—
101	C₂×D₄×C₅₃	—	4	1	no	—
102	C₂×D₄×C₅₃	—	4	1	no	—
103	C₂×D₄×C₅₃	—	4	2	no	—
104	C₂×D₄×C₅₃	—	4	2	no	—
105	C₂×D₄×C₅₃	—	8	2	no	—
106	C₂×D₄×C₅₃	—	8	2	no	—
107	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	4	1	no	—
112	C₂²×C₄×C₅₃	—	8	2	no	—
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
116	SD₁₆×C₅₃	—	4	4	no	—
117	D₈×C₅₃	—	4	8	no	—
118	D₈×C₅₃	—	4	4	no	—
119	D₈×C₅₃	—	8	4	no	—
120	M₁₆×C₅₃	—	4	2	no	—
121	M₁₆×C₅₃	—	8	4	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	—	4	2	no	—
126	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
127	C₄⋊C₄×C₅₃	—	2	1	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
131	C₂²⋊C₄×C₅₃	—	2	2	no	—
132	C₂²⋊C₄×C₅₃	—	2	2	no	—
133	C₂²⋊C₄×C₅₃	—	4	1	no	—
134	C₂²⋊C₄×C₅₃	—	4	1	no	—
135	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₅₃	#41	1	1	yes	characteristic
138	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	—	4	2	no	—
141	C₂³×C₅₃	—	4	1	no	—
142	C₂³×C₅₃	—	4	2	no	—
143	C₂³×C₅₃	—	8	1	no	—
144	C₂³×C₅₃	—	8	1	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	1	no	—
150	Q₈×C₅₃	—	4	1	no	—
151	D₄×C₅₃	—	2	2	no	—
152	D₄×C₅₃	—	2	2	no	—
153	D₄×C₅₃	—	4	2	no	—
154	D₄×C₅₃	—	4	2	no	—
155	D₄×C₅₃	—	4	1	no	—
156	D₄×C₅₃	—	4	1	no	—
157	D₄×C₅₃	—	4	4	no	—
158	D₄×C₅₃	—	4	2	no	—
159	D₄×C₅₃	—	4	2	no	—
160	D₄×C₅₃	—	8	1	no	—
161	D₄×C₅₃	—	8	2	no	—
162	D₄×C₅₃	—	8	2	no	—
163	D₄×C₅₃	—	8	2	no	—
164	D₄×C₅₃	—	8	2	no	—
165	D₄×C₅₃	—	8	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	—	4	4	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	8	2	no	—
172	C₂×C₄×C₅₃	—	8	1	no	—
173	C₂×C₄×C₅₃	—	8	2	no	—
174	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
175	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
176	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
177	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₈×C₅₃	—	2	4	no	—
183	C₈×C₅₃	—	4	2	no	—
184	C₈×C₅₃	—	4	2	no	—
185	C₈×C₅₃	—	4	2	no	—
186	#19422	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	4	no	—
193	C₂²×C₅₃	—	8	2	no	—
194	C₂²×C₅₃	—	8	1	no	—
195	C₂²×C₅₃	—	8	1	no	—
196	C₂²×C₅₃	—	8	2	no	—
197	C₂²×C₅₃	—	8	2	no	—
198	C₂²×C₅₃	—	8	1	no	—
199	C₂²×C₅₃	—	16	4	no	—
200	C₂²×C₅₃	#469	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	1	no	—
207	C₄×C₅₃	#469	1	2	yes	—
208	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₄×C₅₃	#576	1	2	yes	—
211	#3205	C₂×C₅₃	1	1	yes	characteristic, complemented
212	#3036	C₂×C₅₃	1	1	yes	characteristic, complemented
213	C₂×#459	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#2975	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#1701	C₂×C₅₃	1	4	yes	complemented
216	#1687	C₂×C₅₃	1	4	yes	complemented
217	#1570	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#1545	C₂×C₅₃	1	1	yes	characteristic, complemented
219	#1540	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×C₅₃	—	2	2	no	—
221	C₂×C₅₃	—	8	2	no	—
222	C₂×C₅₃	—	16	1	no	—
223	C₂×C₅₃	—	16	2	no	—
224	C₂×C₅₃	#3041	1	1	yes	characteristic
225	C₂×C₅₃	C₂×#469	1	1	yes	characteristic
226	C₂×C₅₃	#3068	1	1	yes	characteristic
227	C₂×AGL(1,ℤ/8)	—	2	1	no	—
228	C₂²×D₈	—	2	1	no	—
229	#532	C₂²×C₅₃	1	1	yes	characteristic
230	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
232	#493	C₂²×C₅₃	1	2	yes	complemented
233	#486	C₂²×C₅₃	1	2	yes	complemented
234	#463	—	2	2	no	—
235	#461	C₂²×C₅₃	1	2	yes	complemented
236	#459	—	2	2	no	—
237	#459	—	2	2	no	—
238	#459	C₂²×C₅₃	1	4	yes	—
239	#459	C₂²×C₅₃	1	2	yes	complemented
240	#447	—	2	2	no	—
241	#436	—	2	2	no	—
242	#436	C₂²×C₅₃	1	4	yes	complemented
243	#431	C₂²×C₅₃	1	1	yes	characteristic
244	#431	C₂²×C₅₃	1	1	yes	characteristic
245	#423	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
247	#417	—	2	1	no	—
248	C₂×#102	—	2	1	no	—
249	#407	—	2	2	no	—
250	#334	C₂²×C₅₃	1	4	yes	complemented
251	#332	C₂²×C₅₃	1	2	yes	complemented
252	#331	C₂²×C₅₃	1	2	yes	complemented
253	#331	C₂²×C₅₃	1	4	yes	complemented
254	C₅₃	#19422	1	1	yes	characteristic, direct product factor, Sylow
255	C₂×#41	—	2	1	no	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	—	2	1	no	—
258	C₂²×D₄	—	2	1	no	—
259	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
260	C₂³×C₄	—	2	1	no	—
261	AGL(1,ℤ/8)	—	4	4	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	C₂×D₈	—	2	4	no	—
264	C₂×D₈	—	2	2	no	—
265	C₂×D₈	—	2	4	no	—
266	C₂×D₈	—	4	4	no	—
267	C₂×M₁₆	—	2	1	no	—
268	C₂×M₁₆	—	2	4	no	—
269	C₂²×C₈	—	2	1	no	—
270	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
271	#127	—	2	1	no	—
272	#127	—	2	1	no	—
273	#127	—	2	1	no	—
274	#127	C₂³×C₅₃	1	2	yes	—
275	#127	C₂³×C₅₃	1	1	yes	characteristic
276	#124	—	2	1	no	—
277	#121	—	2	1	no	—
278	#121	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	—	2	1	no	—
283	C₄×D₄	D₄×C₅₃	1	2	yes	—
284	C₄×D₄	C₂³×C₅₃	1	2	yes	—
285	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
286	C₂×C₂²⋊C₄	—	2	1	no	—
287	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
291	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
292	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
293	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
294	#103	—	2	2	no	—
295	#102	—	2	4	no	—
296	#102	—	2	4	no	—
297	#102	—	2	2	no	—
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	4	no	—
300	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
301	C₄×C₈	—	2	1	no	—
302	C₄×C₈	C₂³×C₅₃	1	2	yes	—
303	C₂⁴	—	2	1	no	—
304	C₂⁴	—	4	1	no	—
305	#41	—	4	2	no	—
306	#41	—	4	2	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	1	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	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	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	8	2	no	—
331	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
334	SD₁₆	—	4	4	no	—
335	D₈	—	4	8	no	—
336	D₈	—	4	4	no	—
337	D₈	—	8	4	no	—
338	M₁₆	—	4	2	no	—
339	M₁₆	—	8	4	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	4	2	no	—
344	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	4	1	no	—
352	C₂²⋊C₄	—	4	1	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	—	4	1	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	8	1	no	—
362	C₂³	—	8	1	no	—
363	C₂³	—	8	2	no	—
364	C₂³	—	8	2	no	—
365	C₂³	—	8	2	no	—
366	C₂³	#121×C₅₃	1	1	yes	characteristic
367	Q₈	—	4	1	no	—
368	Q₈	—	4	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	2	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	1	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	1	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	4	no	—
377	D₄	—	4	2	no	—
378	D₄	—	8	1	no	—
379	D₄	—	8	2	no	—
380	D₄	—	8	2	no	—
381	D₄	—	8	2	no	—
382	D₄	—	8	2	no	—
383	D₄	—	8	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	8	1	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
393	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
394	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
395	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
396	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
397	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
400	C₈	—	2	4	no	—
401	C₈	—	4	2	no	—
402	C₈	—	4	2	no	—
403	C₈	—	4	2	no	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	8	4	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	1	no	—
412	C₂²	—	8	1	no	—
413	C₂²	—	8	2	no	—
414	C₂²	—	8	2	no	—
415	C₂²	—	8	1	no	—
416	C₂²	—	16	4	no	—
417	C₂²	C₅₃×#469	1	2	yes	—
418	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
419	C₄	—	2	1	no	—
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	4	2	no	—
423	C₄	—	8	1	no	—
424	C₄	C₅₃×#469	1	2	yes	—
425	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
426	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
427	C₄	C₅₃×#576	1	2	yes	—
428	C₂	—	2	2	no	—
429	C₂	—	8	2	no	—
430	C₂	—	16	2	no	—
431	C₂	—	16	1	no	—
432	C₂	C₅₃×#3041	1	1	yes	characteristic
433	C₂	C₂×C₅₃×#469	1	1	yes	characteristic
434	C₂	C₅₃×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
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×#3205	C₂	semidirect
2. C₅₃×#3036	C₂	semidirect
3. C₂×C₅₃×#459	C₂	semidirect
4. C₅₃×#2975	C₂	semidirect
5. C₅₃×#1701	C₂	semidirect
6. C₅₃×#1687	C₂	semidirect
7. C₅₃×#1570	C₂	semidirect
8. C₅₃×#1545	C₂	semidirect
9. C₅₃×#1540	C₂	semidirect
15. C₅₃×#493	C₂²	semidirect
16. C₅₃×#486	C₂²	semidirect
18. C₅₃×#461	C₂²	semidirect
22. C₅₃×#459	C₂²	semidirect
25. C₅₃×#436	C₂²	semidirect
33. C₅₃×#334	C₂²	semidirect
34. C₅₃×#332	C₂²	semidirect
35. C₅₃×#331	C₂²	semidirect
36. C₅₃×#331	C₂²	semidirect
186. #19422	C₅₃	direct (N × H)
211. #3205	C₂×C₅₃	semidirect
212. #3036	C₂×C₅₃	semidirect
213. C₂×#459	C₂×C₅₃	semidirect
214. #2975	C₂×C₅₃	semidirect
215. #1701	C₂×C₅₃	semidirect
216. #1687	C₂×C₅₃	semidirect
217. #1570	C₂×C₅₃	semidirect
218. #1545	C₂×C₅₃	semidirect
219. #1540	C₂×C₅₃	semidirect
232. #493	C₂²×C₅₃	semidirect
233. #486	C₂²×C₅₃	semidirect
235. #461	C₂²×C₅₃	semidirect
239. #459	C₂²×C₅₃	semidirect
242. #436	C₂²×C₅₃	semidirect
250. #334	C₂²×C₅₃	semidirect
251. #332	C₂²×C₅₃	semidirect
252. #331	C₂²×C₅₃	semidirect
253. #331	C₂²×C₅₃	semidirect
254. C₅₃	#19422	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×#3205	2
C₅₃×#3036	2
C₂×C₅₃×#459	2
C₅₃×#2975	2
C₅₃×#1701	2
C₅₃×#1687	2
C₅₃×#1570	2
C₅₃×#1545	2
C₅₃×#1540	2
#19422	53

Subgroups — C53×#19422 #27226934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19422 #27226934