Subgroups — C₅₃×#18089 #27225601

Order 13568.

604 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2979	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1592	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1583	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1582	C₂	1	2	yes	maximal, complemented
7	C₅₃×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1484	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₅₃×#498	C₄	1	1	yes	characteristic
15	C₅₃×#492	C₄	1	1	yes	characteristic
16	C₅₃×#487	C₄	1	1	yes	characteristic
17	C₅₃×#486	C₄	1	1	yes	characteristic
18	C₅₃×#465	C₄	1	2	yes	—
19	C₅₃×#465	C₄	1	2	yes	—
20	C₅₃×#446	—	2	1	no	—
21	C₅₃×#446	C₂²	1	2	yes	—
22	C₈×Q₈×C₅₃	—	2	1	no	—
23	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#436	—	2	1	no	—
27	C₅₃×#436	C₂²	1	2	yes	—
28	C₈×D₄×C₅₃	—	2	1	no	—
29	C₈×D₄×C₅₃	C₂²	1	2	yes	—
30	C₅₃×#433	—	2	2	no	—
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₅₃×#431	C₂²	1	1	yes	characteristic
34	C₅₃×#408	—	2	2	no	—
35	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#335	C₂²	1	1	yes	characteristic
38	C₅₃×#335	C₂²	1	1	yes	characteristic
39	C₅₃×#334	C₂²	1	1	yes	characteristic
40	C₅₃×#334	C₂²	1	1	yes	characteristic
41	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
44	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
45	C₅₃×#322	C₂²	1	2	yes	—
46	C₅₃×#322	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	2	1	no	—
52	C₂²×C₈×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#125×C₅₃	—	2	1	no	—
60	#124×C₅₃	—	2	1	no	—
61	C₂²⋊Q₈×C₅₃	—	2	1	no	—
62	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
63	#121×C₅₃	—	2	1	no	—
64	#121×C₅₃	C₂×C₄	1	2	yes	—
65	C₄×Q₈×C₅₃	—	2	1	no	—
66	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	C₂³	1	2	yes	—
69	#117×C₅₃	—	2	1	no	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#106×C₅₃	C₂×C₄	1	2	yes	—
72	#106×C₅₃	C₂×C₄	1	2	yes	—
73	C₄⋊C₈×C₅₃	—	2	2	no	—
74	C₄⋊C₈×C₅₃	—	2	2	no	—
75	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
81	#103×C₅₃	C₂×C₄	1	2	yes	—
82	#103×C₅₃	C₂×C₄	1	2	yes	—
83	#102×C₅₃	C₂×C₄	1	2	yes	—
84	#102×C₅₃	C₂×C₄	1	2	yes	—
85	C₂²⋊C₈×C₅₃	—	2	2	no	—
86	C₂²⋊C₈×C₅₃	—	2	2	no	—
87	C₂²⋊C₈×C₅₃	—	2	1	no	—
88	C₂²⋊C₈×C₅₃	—	2	1	no	—
89	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
90	#97×C₅₃	—	2	2	no	—
91	#97×C₅₃	—	2	1	no	—
92	C₄×C₈×C₅₃	—	2	2	no	—
93	C₄×C₈×C₅₃	—	2	1	no	—
94	C₄×C₈×C₅₃	—	2	1	no	—
95	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₅₃	C₂³	1	2	yes	—
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	#41×C₅₃	—	8	1	no	—
102	#41×C₅₃	—	8	1	no	—
103	C₂×Q₈×C₅₃	—	2	1	no	—
104	C₂×Q₈×C₅₃	—	2	1	no	—
105	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₅₃	—	2	2	no	—
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	—	2	1	no	—
109	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
114	SD₁₆×C₅₃	—	2	4	no	—
115	SD₁₆×C₅₃	—	4	2	no	—
116	M₁₆×C₅₃	—	4	2	no	—
117	M₁₆×C₅₃	—	8	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	4	1	no	—
125	C₂×C₈×C₅₃	—	8	2	no	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
140	C₂²⋊C₄×C₅₃	—	2	2	no	—
141	C₂²⋊C₄×C₅₃	—	2	2	no	—
142	C₂²⋊C₄×C₅₃	—	2	2	no	—
143	C₄²×C₅₃	—	2	2	no	—
144	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₅₃	#41	1	1	yes	characteristic
147	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	2	no	—
151	Q₈×C₅₃	—	2	2	no	—
152	Q₈×C₅₃	—	4	2	no	—
153	Q₈×C₅₃	—	8	1	no	—
154	D₄×C₅₃	—	2	2	no	—
155	D₄×C₅₃	—	2	2	no	—
156	D₄×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	8	1	no	—
158	D₄×C₅₃	—	8	2	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
175	C₈×C₅₃	—	2	1	no	—
176	C₈×C₅₃	—	2	1	no	—
177	C₈×C₅₃	—	2	1	no	—
178	C₈×C₅₃	—	2	2	no	—
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	4	2	no	—
181	C₈×C₅₃	—	4	1	no	—
182	C₈×C₅₃	—	4	1	no	—
183	C₈×C₅₃	—	4	1	no	—
184	C₈×C₅₃	—	4	1	no	—
185	C₈×C₅₃	#131	1	2	yes	—
186	#18089	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	—	2	1	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	4	2	no	—
192	C₂²×C₅₃	—	4	2	no	—
193	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	4	2	no	—
198	C₄×C₅₃	—	4	2	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	#444	1	2	yes	—
201	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
202	C₄×C₅₃	#568	1	1	yes	characteristic
203	C₄×C₅₃	#577	1	1	yes	characteristic
204	C₄×C₅₃	#577	1	1	yes	characteristic
205	#3167	C₂×C₅₃	1	1	yes	characteristic
206	#2986	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#2979	C₂×C₅₃	1	1	yes	characteristic
208	#1592	C₂×C₅₃	1	2	yes	complemented
209	#1583	C₂×C₅₃	1	2	yes	complemented
210	#1582	C₂×C₅₃	1	2	yes	complemented
211	SD₁₆⋊C₈	C₂×C₅₃	1	2	yes	complemented
212	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#1485	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#1484	C₂×C₅₃	1	1	yes	characteristic
215	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
216	C₂×C₅₃	—	4	1	no	—
217	C₂×C₅₃	—	8	2	no	—
218	C₂×C₅₃	#2935	1	1	yes	characteristic
219	C₂×C₅₃	#2949	1	1	yes	characteristic
220	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
221	#533	C₂²×C₅₃	1	1	yes	characteristic
222	#532	C₂²×C₅₃	1	1	yes	characteristic
223	#498	C₄×C₅₃	1	1	yes	characteristic
224	#492	C₄×C₅₃	1	1	yes	characteristic
225	#487	C₄×C₅₃	1	1	yes	characteristic
226	#486	C₄×C₅₃	1	1	yes	characteristic
227	#465	C₄×C₅₃	1	2	yes	—
228	#465	C₄×C₅₃	1	2	yes	—
229	#446	—	2	1	no	—
230	#446	C₂²×C₅₃	1	2	yes	—
231	C₈×Q₈	—	2	1	no	—
232	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
233	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
234	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
235	#436	—	2	1	no	—
236	#436	C₂²×C₅₃	1	2	yes	—
237	C₈×D₄	—	2	1	no	—
238	C₈×D₄	C₂²×C₅₃	1	2	yes	—
239	#433	—	2	2	no	—
240	#431	C₂²×C₅₃	1	1	yes	characteristic
241	#431	C₂²×C₅₃	1	1	yes	characteristic
242	#431	C₂²×C₅₃	1	1	yes	characteristic
243	#408	—	2	2	no	—
244	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
246	#335	C₂²×C₅₃	1	1	yes	characteristic
247	#335	C₂²×C₅₃	1	1	yes	characteristic
248	#334	C₂²×C₅₃	1	1	yes	characteristic
249	#334	C₂²×C₅₃	1	1	yes	characteristic
250	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
251	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
252	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
253	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
254	#322	C₂²×C₅₃	1	2	yes	—
255	#322	C₂²×C₅₃	1	2	yes	—
256	C₅₃	#18089	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×#41	—	2	1	no	—
258	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
259	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
264	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
265	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
266	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
267	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
268	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
269	#125	—	2	1	no	—
270	#124	—	2	1	no	—
271	C₂²⋊Q₈	—	2	1	no	—
272	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
273	#121	—	2	1	no	—
274	#121	C₂×C₄×C₅₃	1	2	yes	—
275	C₄×Q₈	—	2	1	no	—
276	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	C₂³×C₅₃	1	2	yes	—
279	#117	—	2	1	no	—
280	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
281	#106	C₂×C₄×C₅₃	1	2	yes	—
282	#106	C₂×C₄×C₅₃	1	2	yes	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	—	2	2	no	—
285	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
286	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
287	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
288	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
289	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
290	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
291	#103	C₂×C₄×C₅₃	1	2	yes	—
292	#103	C₂×C₄×C₅₃	1	2	yes	—
293	#102	C₂×C₄×C₅₃	1	2	yes	—
294	#102	C₂×C₄×C₅₃	1	2	yes	—
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	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
300	#97	—	2	2	no	—
301	#97	—	2	1	no	—
302	C₄×C₈	—	2	2	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
306	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
307	C₄×C₈	C₂³×C₅₃	1	2	yes	—
308	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
309	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₅₃	1	2	yes	—
311	#41	—	8	1	no	—
312	#41	—	8	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
323	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
324	SD₁₆	—	2	4	no	—
325	SD₁₆	—	4	2	no	—
326	M₁₆	—	4	2	no	—
327	M₁₆	—	8	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	8	2	no	—
336	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
337	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
338	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
339	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
349	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
356	C₄²	#41×C₅₃	1	1	yes	characteristic
357	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
358	C₂³	—	2	2	no	—
359	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	Q₈	—	2	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	8	1	no	—
364	D₄	—	2	2	no	—
365	D₄	—	2	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	1	no	—
368	D₄	—	8	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
385	C₈	—	2	1	no	—
386	C₈	—	2	1	no	—
387	C₈	—	2	1	no	—
388	C₈	—	2	2	no	—
389	C₈	—	2	2	no	—
390	C₈	—	4	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	1	no	—
395	C₈	#131×C₅₃	1	2	yes	—
396	C₂²	—	2	1	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	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	C₅₃×#444	1	2	yes	—
410	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
411	C₄	C₅₃×#568	1	1	yes	characteristic
412	C₄	C₅₃×#577	1	1	yes	characteristic
413	C₄	C₅₃×#577	1	1	yes	characteristic
414	C₂	—	4	1	no	—
415	C₂	—	8	2	no	—
416	C₂	C₅₃×#2935	1	1	yes	characteristic
417	C₂	C₅₃×#2949	1	1	yes	characteristic
418	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₅₃×#2986	C₂	semidirect
4. C₅₃×#1592	C₂	semidirect
5. C₅₃×#1583	C₂	semidirect
6. C₅₃×#1582	C₂	semidirect
7. C₅₃×SD₁₆⋊C₈	C₂	semidirect
8. C₅₃×#1564	C₂	semidirect
9. C₅₃×#1485	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
186. #18089	C₅₃	direct (N × H)
206. #2986	C₂×C₅₃	semidirect
208. #1592	C₂×C₅₃	semidirect
209. #1583	C₂×C₅₃	semidirect
210. #1582	C₂×C₅₃	semidirect
211. SD₁₆⋊C₈	C₂×C₅₃	semidirect
212. #1564	C₂×C₅₃	semidirect
213. #1485	C₂×C₅₃	semidirect
215. #1451	C₂×C₅₃	semidirect
256. C₅₃	#18089	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×#3167	2
C₅₃×#2986	2
C₅₃×#2979	2
C₅₃×#1592	2
C₅₃×#1583	2
C₅₃×#1582	2
C₅₃×SD₁₆⋊C₈	2
C₅₃×#1564	2
C₅₃×#1485	2
C₅₃×#1484	2
C₅₃×#1451	2
#18089	53

Subgroups — C53×#18089 #27225601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18089 #27225601