Subgroups — C₅₃×#19186 #27226698

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1685	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1665	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₅₃	—	2	1	no	—
11	C₂²×SD₁₆×C₅₃	—	2	1	no	—
12	C₂×#131×C₅₃	—	2	1	no	—
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#500	—	2	1	no	—
17	C₅₃×#493	—	2	1	no	—
18	C₅₃×#492	—	2	1	no	—
19	C₅₃×#492	—	2	1	no	—
20	C₅₃×#492	C₂²	1	2	yes	—
21	C₅₃×#492	C₂²	1	2	yes	—
22	C₅₃×#486	—	2	1	no	—
23	C₅₃×#486	—	2	1	no	—
24	C₅₃×#480	—	2	2	no	—
25	C₅₃×#474	C₂²	1	4	yes	complemented
26	C₅₃×#459	C₂²	1	4	yes	—
27	C₅₃×#447	—	2	2	no	—
28	C₅₃×#436	—	2	1	no	—
29	C₅₃×#436	C₂²	1	2	yes	complemented
30	C₈×D₄×C₅₃	—	2	1	no	—
31	C₈×D₄×C₅₃	C₂²	1	2	yes	complemented
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₅₃×#431	C₂²	1	1	yes	characteristic
34	C₅₃×#423	C₂²	1	1	yes	characteristic
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	2	no	—
47	C₂×SD₁₆×C₅₃	—	2	2	no	—
48	C₂×SD₁₆×C₅₃	—	2	2	no	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	C₂×SD₁₆×C₅₃	—	4	4	no	—
51	C₂×SD₁₆×C₅₃	—	4	4	no	—
52	C₂×D₈×C₅₃	—	2	4	no	—
53	#131×C₅₃	—	4	4	no	—
54	C₂×M₁₆×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	—	2	2	no	—
57	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
64	#127×C₅₃	—	2	1	no	—
65	#127×C₅₃	—	2	1	no	—
66	#127×C₅₃	—	2	1	no	—
67	#127×C₅₃	D₄	1	1	yes	characteristic
68	#127×C₅₃	C₂³	1	2	yes	—
69	#124×C₅₃	—	2	1	no	—
70	#121×C₅₃	—	2	1	no	—
71	#121×C₅₃	D₄	1	2	yes	—
72	C₄×D₄×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	#107×C₅₃	—	2	4	no	—
78	C₄⋊C₈×C₅₃	—	2	1	no	—
79	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
80	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
81	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
82	#102×C₅₃	—	2	4	no	—
83	#102×C₅₃	—	2	4	no	—
84	C₂²⋊C₈×C₅₃	—	2	2	no	—
85	C₂²⋊C₈×C₅₃	—	2	2	no	—
86	C₂²⋊C₈×C₅₃	—	2	1	no	—
87	C₂²⋊C₈×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	—	2	1	no	—
89	C₄×C₈×C₅₃	—	2	1	no	—
90	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
91	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂⁴×C₅₃	—	4	1	no	—
95	#41×C₅₃	—	4	2	no	—
96	#41×C₅₃	—	4	2	no	—
97	C₂×Q₈×C₅₃	—	2	1	no	—
98	C₂×Q₈×C₅₃	—	2	2	no	—
99	C₂×Q₈×C₅₃	—	4	1	no	—
100	C₂×Q₈×C₅₃	—	4	2	no	—
101	C₂×Q₈×C₅₃	—	4	2	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	—	2	1	no	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	2	2	no	—
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	4	1	no	—
109	C₂×D₄×C₅₃	—	4	1	no	—
110	C₂×D₄×C₅₃	—	4	1	no	—
111	C₂×D₄×C₅₃	—	4	2	no	—
112	C₂×D₄×C₅₃	—	4	2	no	—
113	C₂×D₄×C₅₃	—	8	2	no	—
114	C₂×D₄×C₅₃	—	8	2	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	—	4	4	no	—
122	SD₁₆×C₅₃	—	4	4	no	—
123	SD₁₆×C₅₃	—	4	4	no	—
124	SD₁₆×C₅₃	—	4	4	no	—
125	D₈×C₅₃	—	8	4	no	—
126	M₁₆×C₅₃	—	4	2	no	—
127	M₁₆×C₅₃	—	4	4	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	4	4	no	—
133	C₂×C₈×C₅₃	D₈	1	4	yes	complemented
134	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	4	1	no	—
142	C₄⋊C₄×C₅₃	—	4	1	no	—
143	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
144	C₂²⋊C₄×C₅₃	—	2	2	no	—
145	C₂²⋊C₄×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₅₃	#41	1	1	yes	characteristic
149	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
150	C₂³×C₅₃	—	2	2	no	—
151	C₂³×C₅₃	—	4	2	no	—
152	C₂³×C₅₃	—	8	2	no	—
153	C₂³×C₅₃	—	8	2	no	—
154	C₂³×C₅₃	—	8	2	no	—
155	C₂³×C₅₃	#121	1	1	yes	characteristic
156	Q₈×C₅₃	—	4	2	no	—
157	Q₈×C₅₃	—	4	2	no	—
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	8	2	no	—
160	D₄×C₅₃	—	2	4	no	—
161	D₄×C₅₃	—	4	4	no	—
162	D₄×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	4	2	no	—
164	D₄×C₅₃	—	4	4	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	4	2	no	—
167	D₄×C₅₃	—	8	2	no	—
168	D₄×C₅₃	—	8	2	no	—
169	D₄×C₅₃	—	8	2	no	—
170	D₄×C₅₃	—	8	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	8	2	no	—
178	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
180	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
182	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
186	C₈×C₅₃	—	2	1	no	—
187	C₈×C₅₃	—	2	1	no	—
188	C₈×C₅₃	—	4	2	no	—
189	C₈×C₅₃	—	4	2	no	—
190	C₈×C₅₃	—	4	2	no	—
191	C₈×C₅₃	C₂×D₈	1	2	yes	—
192	C₈×C₅₃	C₂×D₈	1	2	yes	—
193	#19186	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	4	2	no	—
197	C₂²×C₅₃	—	4	2	no	—
198	C₂²×C₅₃	—	4	2	no	—
199	C₂²×C₅₃	—	8	2	no	—
200	C₂²×C₅₃	—	8	2	no	—
201	C₂²×C₅₃	—	8	2	no	—
202	C₂²×C₅₃	—	16	4	no	—
203	C₂²×C₅₃	#459	1	2	yes	—
204	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	4	2	no	—
208	C₄×C₅₃	—	8	2	no	—
209	C₄×C₅₃	#459	1	2	yes	—
210	C₄×C₅₃	#471	1	2	yes	—
211	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
212	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₄×C₅₃	#577	1	2	yes	—
214	#3226	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#3205	C₂×C₅₃	1	1	yes	characteristic
216	C₂×#492	C₂×C₅₃	1	1	yes	characteristic, complemented
217	#2931	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#1685	C₂×C₅₃	1	4	yes	complemented
219	#1665	C₂×C₅₃	1	4	yes	complemented
220	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1545	C₂×C₅₃	1	1	yes	characteristic, complemented
223	C₂×C₅₃	—	2	2	no	—
224	C₂×C₅₃	—	8	2	no	—
225	C₂×C₅₃	—	16	2	no	—
226	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
227	C₂×C₅₃	#3057	1	1	yes	characteristic
228	C₂×C₅₃	C₂×#471	1	1	yes	characteristic
229	C₂²×SD₁₆	—	2	1	no	—
230	C₂²×SD₁₆	—	2	1	no	—
231	C₂×#131	—	2	1	no	—
232	#532	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
235	#500	—	2	1	no	—
236	#493	—	2	1	no	—
237	#492	—	2	1	no	—
238	#492	—	2	1	no	—
239	#492	C₂²×C₅₃	1	2	yes	—
240	#492	C₂²×C₅₃	1	2	yes	—
241	#486	—	2	1	no	—
242	#486	—	2	1	no	—
243	#480	—	2	2	no	—
244	#474	C₂²×C₅₃	1	4	yes	complemented
245	#459	C₂²×C₅₃	1	4	yes	—
246	#447	—	2	2	no	—
247	#436	—	2	1	no	—
248	#436	C₂²×C₅₃	1	2	yes	complemented
249	C₈×D₄	—	2	1	no	—
250	C₈×D₄	C₂²×C₅₃	1	2	yes	complemented
251	#431	C₂²×C₅₃	1	1	yes	characteristic
252	#431	C₂²×C₅₃	1	1	yes	characteristic
253	#423	C₂²×C₅₃	1	1	yes	characteristic
254	#408	—	2	1	no	—
255	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
256	#335	C₂²×C₅₃	1	2	yes	complemented
257	#334	C₂²×C₅₃	1	2	yes	complemented
258	#332	C₂²×C₅₃	1	4	yes	complemented
259	#331	C₂²×C₅₃	1	4	yes	—
260	C₅₃	#19186	1	1	yes	characteristic, direct product factor, Sylow
261	C₂×#41	—	2	1	no	—
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×SD₁₆	—	2	2	no	—
269	C₂×SD₁₆	—	2	2	no	—
270	C₂×SD₁₆	—	4	4	no	—
271	C₂×SD₁₆	—	4	4	no	—
272	C₂×D₈	—	2	4	no	—
273	#131	—	4	4	no	—
274	C₂×M₁₆	—	2	1	no	—
275	C₂×M₁₆	—	2	2	no	—
276	C₂²×C₈	—	2	2	no	—
277	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
279	C₄⋊Q₈	—	2	1	no	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
283	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
284	#127	—	2	1	no	—
285	#127	—	2	1	no	—
286	#127	—	2	1	no	—
287	#127	D₄×C₅₃	1	1	yes	characteristic
288	#127	C₂³×C₅₃	1	2	yes	—
289	#124	—	2	1	no	—
290	#121	—	2	1	no	—
291	#121	D₄×C₅₃	1	2	yes	—
292	C₄×D₄	—	2	1	no	—
293	C₄×D₄	C₂³×C₅₃	1	2	yes	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
297	#107	—	2	4	no	—
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
300	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
301	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
302	#102	—	2	4	no	—
303	#102	—	2	4	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	2	1	no	—
306	C₂²⋊C₈	—	2	2	no	—
307	C₂²⋊C₈	—	2	2	no	—
308	C₄×C₈	—	2	1	no	—
309	C₄×C₈	—	2	1	no	—
310	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
311	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
313	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
314	C₂⁴	—	4	1	no	—
315	#41	—	4	2	no	—
316	#41	—	4	2	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	4	1	no	—
320	C₂×Q₈	—	4	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂×D₄	—	4	1	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	1	no	—
338	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
340	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
341	SD₁₆	—	4	4	no	—
342	SD₁₆	—	4	4	no	—
343	SD₁₆	—	4	4	no	—
344	SD₁₆	—	4	4	no	—
345	D₈	—	8	4	no	—
346	M₁₆	—	4	2	no	—
347	M₁₆	—	4	4	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	4	4	no	—
353	C₂×C₈	D₈×C₅₃	1	4	yes	complemented
354	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
355	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
356	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
357	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
364	C₂²⋊C₄	—	2	2	no	—
365	C₂²⋊C₄	—	2	2	no	—
366	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
367	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₄²	#41×C₅₃	1	1	yes	characteristic
369	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
370	C₂³	—	2	2	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	8	2	no	—
373	C₂³	—	8	2	no	—
374	C₂³	—	8	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	—	8	2	no	—
380	D₄	—	2	4	no	—
381	D₄	—	4	4	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	4	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	2	no	—
387	D₄	—	8	2	no	—
388	D₄	—	8	2	no	—
389	D₄	—	8	2	no	—
390	D₄	—	8	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	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	C₂×D₈×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	1	no	—
407	C₈	—	2	1	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	2	no	—
410	C₈	—	4	2	no	—
411	C₈	C₂×D₈×C₅₃	1	2	yes	—
412	C₈	C₂×D₈×C₅₃	1	2	yes	—
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	—	8	2	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	16	4	no	—
422	C₂²	C₅₃×#459	1	2	yes	—
423	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	8	2	no	—
428	C₄	C₅₃×#459	1	2	yes	—
429	C₄	C₅₃×#471	1	2	yes	—
430	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
431	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
432	C₄	C₅₃×#577	1	2	yes	—
433	C₂	—	2	2	no	—
434	C₂	—	8	2	no	—
435	C₂	—	16	2	no	—
436	C₂	C₂×C₅₃×#459	1	1	yes	characteristic
437	C₂	C₅₃×#3057	1	1	yes	characteristic
438	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×#3226	C₂	semidirect
3. C₂×C₅₃×#492	C₂	semidirect
4. C₅₃×#2931	C₂	semidirect
5. C₅₃×#1685	C₂	semidirect
6. C₅₃×#1665	C₂	semidirect
7. C₅₃×#1564	C₂	semidirect
8. C₅₃×#1546	C₂	semidirect
9. C₅₃×#1545	C₂	semidirect
25. C₅₃×#474	C₂²	semidirect
29. C₅₃×#436	C₂²	semidirect
31. C₈×D₄×C₅₃	C₂²	semidirect
37. C₅₃×#335	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
39. C₅₃×#332	C₂²	semidirect
133. C₂×C₈×C₅₃	D₈	semidirect
193. #19186	C₅₃	direct (N × H)
214. #3226	C₂×C₅₃	semidirect
216. C₂×#492	C₂×C₅₃	semidirect
217. #2931	C₂×C₅₃	semidirect
218. #1685	C₂×C₅₃	semidirect
219. #1665	C₂×C₅₃	semidirect
220. #1564	C₂×C₅₃	semidirect
221. #1546	C₂×C₅₃	semidirect
222. #1545	C₂×C₅₃	semidirect
244. #474	C₂²×C₅₃	semidirect
248. #436	C₂²×C₅₃	semidirect
250. C₈×D₄	C₂²×C₅₃	semidirect
256. #335	C₂²×C₅₃	semidirect
257. #334	C₂²×C₅₃	semidirect
258. #332	C₂²×C₅₃	semidirect
260. C₅₃	#19186	direct (N × H)
353. C₂×C₈	D₈×C₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#3226	2
C₅₃×#3205	2
C₂×C₅₃×#492	2
C₅₃×#2931	2
C₅₃×#1685	2
C₅₃×#1665	2
C₅₃×#1564	2
C₅₃×#1546	2
C₅₃×#1545	2
#19186	53

Subgroups — C53×#19186 #27226698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19186 #27226698