Subgroups — C₅₃×#18069 #27225581

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2969	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1583	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1582	C₂	1	2	yes	maximal, complemented
7	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1482	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#529	C₂²	1	2	yes	—
13	C₅₃×#499	C₄	1	1	yes	characteristic
14	C₅₃×#495	C₄	1	1	yes	characteristic
15	C₅₃×#488	C₄	1	2	yes	—
16	C₅₃×#467	C₄	1	1	yes	characteristic
17	C₅₃×#466	C₄	1	1	yes	characteristic
18	C₅₃×#465	C₄	1	2	yes	—
19	C₅₃×#446	C₂²	1	2	yes	—
20	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
23	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#436	—	2	2	no	—
25	C₅₃×#436	C₂²	1	2	yes	—
26	C₅₃×#435	—	2	2	no	—
27	C₈×D₄×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#433	—	2	2	no	—
29	C₅₃×#431	C₂²	1	2	yes	—
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#405	—	2	2	no	—
32	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#335	C₂²	1	1	yes	characteristic
35	C₅₃×#335	C₂²	1	1	yes	characteristic
36	C₅₃×#334	C₂²	1	2	yes	—
37	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
39	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
40	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
41	C₅₃×#322	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	1	yes	characteristic
43	C₈²×C₅₃	C₂²	1	1	yes	characteristic
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#126×C₅₃	—	2	2	no	—
55	#125×C₅₃	C₂×C₄	1	2	yes	—
56	#124×C₅₃	C₂×C₄	1	2	yes	—
57	#123×C₅₃	—	2	2	no	—
58	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
59	#121×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
61	C₄×D₄×C₅₃	—	2	2	no	—
62	C₄×D₄×C₅₃	C₂³	1	2	yes	—
63	#117×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#106×C₅₃	C₂×C₄	1	2	yes	—
68	C₄⋊C₈×C₅₃	—	2	2	no	—
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	1	no	—
71	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
72	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
73	#103×C₅₃	C₂×C₄	1	2	yes	—
74	#103×C₅₃	C₂×C₄	1	2	yes	—
75	#102×C₅₃	C₂×C₄	1	2	yes	—
76	#102×C₅₃	C₂×C₄	1	2	yes	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	C₂²⋊C₈×C₅₃	—	2	2	no	—
80	C₂²⋊C₈×C₅₃	—	2	1	no	—
81	#97×C₅₃	—	2	2	no	—
82	#97×C₅₃	D₄	1	1	yes	characteristic
83	#97×C₅₃	D₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	—	2	2	no	—
85	C₄×C₈×C₅₃	—	2	2	no	—
86	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×D₄×C₅₃	—	2	2	no	—
96	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
101	Q₁₆×C₅₃	—	2	2	no	—
102	SD₁₆×C₅₃	—	4	2	no	—
103	D₈×C₅₃	—	2	2	no	—
104	M₁₆×C₅₃	—	2	2	no	—
105	M₁₆×C₅₃	—	2	2	no	—
106	M₁₆×C₅₃	—	4	4	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	4	1	no	—
113	C₂×C₈×C₅₃	—	4	4	no	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₂²⋊C₄×C₅₃	—	2	2	no	—
130	C₂²⋊C₄×C₅₃	—	2	2	no	—
131	C₂²⋊C₄×C₅₃	—	2	2	no	—
132	C₂²⋊C₄×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅₃	#41	1	1	yes	characteristic
137	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
140	Q₈×C₅₃	—	2	2	no	—
141	Q₈×C₅₃	—	2	2	no	—
142	D₄×C₅₃	—	2	2	no	—
143	D₄×C₅₃	—	2	2	no	—
144	D₄×C₅₃	—	8	2	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	—	2	1	no	—
166	C₈×C₅₃	—	4	1	no	—
167	C₈×C₅₃	—	4	1	no	—
168	C₈×C₅₃	—	4	2	no	—
169	C₈×C₅₃	—	4	2	no	—
170	C₈×C₅₃	#131	1	2	yes	—
171	C₈×C₅₃	#135	1	2	yes	—
172	C₈×C₅₃	#135	1	2	yes	—
173	#18069	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	2	1	no	—
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	4	2	no	—
178	C₂²×C₅₃	—	4	2	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	—	4	2	no	—
186	C₄×C₅₃	—	4	2	no	—
187	C₄×C₅₃	—	4	2	no	—
188	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
189	C₄×C₅₃	#568	1	1	yes	characteristic
190	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
191	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
192	#3168	C₂×C₅₃	1	1	yes	characteristic
193	#2969	C₂×C₅₃	1	2	yes	complemented
194	#1591	C₂×C₅₃	1	1	yes	characteristic, complemented
195	#1589	C₂×C₅₃	1	1	yes	characteristic, complemented
196	#1583	C₂×C₅₃	1	2	yes	complemented
197	#1582	C₂×C₅₃	1	2	yes	complemented
198	C₈×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
199	C₈×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
201	#1482	C₂×C₅₃	1	2	yes	complemented
202	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
203	C₂×C₅₃	—	4	1	no	—
204	C₂×C₅₃	—	8	2	no	—
205	C₂×C₅₃	#2935	1	1	yes	characteristic
206	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
207	C₂×C₅₃	#2957	1	1	yes	characteristic
208	#529	C₂²×C₅₃	1	2	yes	—
209	#499	C₄×C₅₃	1	1	yes	characteristic
210	#495	C₄×C₅₃	1	1	yes	characteristic
211	#488	C₄×C₅₃	1	2	yes	—
212	#467	C₄×C₅₃	1	1	yes	characteristic
213	#466	C₄×C₅₃	1	1	yes	characteristic
214	#465	C₄×C₅₃	1	2	yes	—
215	#446	C₂²×C₅₃	1	2	yes	—
216	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
217	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
218	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
219	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
220	#436	—	2	2	no	—
221	#436	C₂²×C₅₃	1	2	yes	—
222	#435	—	2	2	no	—
223	C₈×D₄	C₂²×C₅₃	1	2	yes	—
224	#433	—	2	2	no	—
225	#431	C₂²×C₅₃	1	2	yes	—
226	#431	C₂²×C₅₃	1	1	yes	characteristic
227	#405	—	2	2	no	—
228	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
230	#335	C₂²×C₅₃	1	1	yes	characteristic
231	#335	C₂²×C₅₃	1	1	yes	characteristic
232	#334	C₂²×C₅₃	1	2	yes	—
233	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
234	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
235	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
236	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
237	#322	C₂²×C₅₃	1	1	yes	characteristic
238	#322	C₂²×C₅₃	1	1	yes	characteristic
239	C₈²	C₂²×C₅₃	1	1	yes	characteristic
240	C₈²	C₂²×C₅₃	1	1	yes	characteristic
241	C₅₃	#18069	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
243	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
244	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
247	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
251	#126	—	2	2	no	—
252	#125	C₂×C₄×C₅₃	1	2	yes	—
253	#124	C₂×C₄×C₅₃	1	2	yes	—
254	#123	—	2	2	no	—
255	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
256	#121	C₂×C₄×C₅₃	1	2	yes	—
257	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
258	C₄×D₄	—	2	2	no	—
259	C₄×D₄	C₂³×C₅₃	1	2	yes	—
260	#117	—	2	2	no	—
261	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
262	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
263	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
264	#106	C₂×C₄×C₅₃	1	2	yes	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
269	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
270	#103	C₂×C₄×C₅₃	1	2	yes	—
271	#103	C₂×C₄×C₅₃	1	2	yes	—
272	#102	C₂×C₄×C₅₃	1	2	yes	—
273	#102	C₂×C₄×C₅₃	1	2	yes	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	2	1	no	—
276	C₂²⋊C₈	—	2	2	no	—
277	C₂²⋊C₈	—	2	2	no	—
278	#97	—	2	2	no	—
279	#97	D₄×C₅₃	1	1	yes	characteristic
280	#97	D₄×C₅₃	1	1	yes	characteristic
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	—	2	2	no	—
283	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
284	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
286	C₄×C₈	C₂³×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	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
297	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	4	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	2	no	—
303	M₁₆	—	4	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	4	4	no	—
311	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
312	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
313	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
314	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
315	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
316	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
324	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
332	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
333	C₄²	#41×C₅₃	1	1	yes	characteristic
334	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	8	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
353	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	4	1	no	—
365	C₈	—	4	2	no	—
366	C₈	—	4	2	no	—
367	C₈	#131×C₅₃	1	2	yes	—
368	C₈	#135×C₅₃	1	2	yes	—
369	C₈	#135×C₅₃	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
385	C₄	C₅₃×#568	1	1	yes	characteristic
386	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
387	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	8	2	no	—
390	C₂	C₅₃×#2935	1	1	yes	characteristic
391	C₂	C₄×#135×C₅₃	1	1	yes	characteristic
392	C₂	C₅₃×#2957	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×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2969	C₂	semidirect
3. C₅₃×#1591	C₂	semidirect
4. C₅₃×#1589	C₂	semidirect
5. C₅₃×#1583	C₂	semidirect
6. C₅₃×#1582	C₂	semidirect
7. C₈×Q₁₆×C₅₃	C₂	semidirect
8. C₈×D₈×C₅₃	C₂	semidirect
9. C₅₃×#1564	C₂	semidirect
10. C₅₃×#1482	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
173. #18069	C₅₃	direct (N × H)
193. #2969	C₂×C₅₃	semidirect
194. #1591	C₂×C₅₃	semidirect
195. #1589	C₂×C₅₃	semidirect
196. #1583	C₂×C₅₃	semidirect
197. #1582	C₂×C₅₃	semidirect
198. C₈×Q₁₆	C₂×C₅₃	semidirect
199. C₈×D₈	C₂×C₅₃	semidirect
200. #1564	C₂×C₅₃	semidirect
201. #1482	C₂×C₅₃	semidirect
202. C₈×M₁₆	C₂×C₅₃	semidirect
241. C₅₃	#18069	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×#3168	2
C₅₃×#2969	2
C₅₃×#1591	2
C₅₃×#1589	2
C₅₃×#1583	2
C₅₃×#1582	2
C₈×Q₁₆×C₅₃	2
C₈×D₈×C₅₃	2
C₅₃×#1564	2
C₅₃×#1482	2
C₈×M₁₆×C₅₃	2
#18069	53

Subgroups — C53×#18069 #27225581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18069 #27225581