Subgroups — C₅₃×#22278 #27229790

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#497	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2327	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2326	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1958	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1880	C₂	1	2	yes	maximal
7	C₅₃×#1878	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1878	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1877	C₂	1	2	yes	maximal
10	C₅₃×#1836	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#497	C₄	1	8	yes	complemented
20	C₂×#103×C₅₃	C₂²	1	2	yes	—
21	C₂×#103×C₅₃	C₂²	1	2	yes	—
22	C₂×#103×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	2	yes	—
24	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#391	C₂²	1	1	yes	characteristic
27	C₅₃×#391	C₂²	1	2	yes	—
28	C₅₃×#390	C₂²	1	1	yes	characteristic
29	C₅₃×#388	—	2	2	no	—
30	C₅₃×#387	—	2	2	no	—
31	C₅₃×#387	—	2	2	no	—
32	C₅₃×#385	—	2	2	no	—
33	C₅₃×#384	C₂²	1	1	yes	characteristic
34	C₅₃×#382	C₂²	1	1	yes	characteristic
35	C₅₃×#376	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	2	yes	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	—	2	1	no	—
44	C₂×Q₁₆×C₅₃	—	4	2	no	—
45	C₂×Q₁₆×C₅₃	—	4	2	no	—
46	C₂×Q₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
47	C₂×SD₁₆×C₅₃	—	4	2	no	—
48	C₂×SD₁₆×C₅₃	—	4	2	no	—
49	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₅₃	—	2	2	no	—
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
54	#124×C₅₃	—	2	2	no	—
55	#124×C₅₃	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₅₃	—	4	2	no	—
57	C₄×Q₈×C₅₃	—	4	2	no	—
58	C₄×Q₈×C₅₃	—	4	2	no	—
59	C₄×D₄×C₅₃	—	4	2	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	—	2	1	no	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	#103×C₅₃	—	4	2	no	—
76	#103×C₅₃	—	4	2	no	—
77	#103×C₅₃	—	4	2	no	—
78	#102×C₅₃	—	4	2	no	—
79	#97×C₅₃	C₂×C₄	1	4	yes	complemented
80	C₄×C₈×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	2	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	D₄	1	2	yes	—
86	#95×C₅₃	D₄	1	2	yes	—
87	C₂⁴×C₅₃	—	2	1	no	—
88	C₂×Q₈×C₅₃	—	2	1	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	—	2	2	no	—
91	C₂×Q₈×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	—	2	2	no	—
94	C₂×Q₈×C₅₃	—	2	2	no	—
95	C₂×Q₈×C₅₃	—	2	2	no	—
96	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	4	2	no	—
108	C₂²×C₄×C₅₃	—	8	2	no	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₅₃	—	4	8	no	—
115	SD₁₆×C₅₃	—	4	8	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
122	C₄⋊C₄×C₅₃	—	2	8	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	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	4	2	no	—
128	C₄⋊C₄×C₅₃	—	4	4	no	—
129	C₄⋊C₄×C₅₃	—	8	2	no	—
130	C₂²⋊C₄×C₅₃	—	2	8	no	—
131	C₂²⋊C₄×C₅₃	—	4	2	no	—
132	C₂²⋊C₄×C₅₃	—	8	2	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	4	2	no	—
136	C₄²×C₅₃	—	4	4	no	—
137	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
138	C₄²×C₅₃	#41	1	2	yes	—
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	—	4	1	no	—
141	C₂³×C₅₃	—	4	1	no	—
142	C₂³×C₅₃	—	4	2	no	—
143	C₂³×C₅₃	—	4	2	no	—
144	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₅₃	—	2	4	no	—
146	Q₈×C₅₃	—	2	4	no	—
147	Q₈×C₅₃	—	2	4	no	—
148	Q₈×C₅₃	—	4	2	no	—
149	Q₈×C₅₃	—	4	2	no	—
150	Q₈×C₅₃	—	4	2	no	—
151	D₄×C₅₃	—	2	4	no	—
152	D₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	4	1	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	—	4	1	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	8	4	no	—
176	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
177	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
180	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
181	C₈×C₅₃	—	2	4	no	—
182	C₈×C₅₃	—	4	2	no	—
183	#22278	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂²×C₅₃	—	4	2	no	—
185	C₂²×C₅₃	—	4	4	no	—
186	C₂²×C₅₃	—	8	1	no	—
187	C₂²×C₅₃	—	8	1	no	—
188	C₂²×C₅₃	—	8	2	no	—
189	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
190	C₂²×C₅₃	#535	1	1	yes	characteristic
191	C₂²×C₅₃	#535	1	1	yes	characteristic
192	C₂²×C₅₃	#577	1	2	yes	—
193	C₂²×C₅₃	#578	1	2	yes	—
194	C₄×C₅₃	—	2	4	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	4	4	no	—
197	C₄×C₅₃	—	4	1	no	—
198	C₄×C₅₃	—	4	1	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	8	1	no	—
201	C₄×C₅₃	—	8	1	no	—
202	C₄×C₅₃	—	8	2	no	—
203	C₄×C₅₃	#518	1	2	yes	—
204	C₄×C₅₃	#540	1	2	yes	—
205	C₂×#497	C₂×C₅₃	1	1	yes	characteristic
206	#2327	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#2326	C₂×C₅₃	1	1	yes	characteristic
208	#1958	C₂×C₅₃	1	1	yes	characteristic
209	#1956	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1880	C₂×C₅₃	1	2	yes	—
211	#1878	C₂×C₅₃	1	2	yes	complemented
212	#1878	C₂×C₅₃	1	2	yes	complemented
213	#1877	C₂×C₅₃	1	2	yes	—
214	#1836	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
216	C₂×C₅₃	—	8	2	no	—
217	C₂×C₅₃	#2328	1	1	yes	characteristic
218	C₂×C₅₃	#2947	1	2	yes	—
219	C₂×C₅₃	#3222	1	1	yes	characteristic
220	C₂×C₅₃	#3223	1	1	yes	characteristic
221	C₂×C₅₃	#3225	1	2	yes	—
222	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
223	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
224	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
227	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
228	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
229	#497	C₄×C₅₃	1	8	yes	complemented
230	C₂×#103	C₂²×C₅₃	1	2	yes	—
231	C₂×#103	C₂²×C₅₃	1	2	yes	—
232	C₂×#103	C₂²×C₅₃	1	2	yes	—
233	C₂×#102	C₂²×C₅₃	1	2	yes	—
234	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
236	#391	C₂²×C₅₃	1	1	yes	characteristic
237	#391	C₂²×C₅₃	1	2	yes	—
238	#390	C₂²×C₅₃	1	1	yes	characteristic
239	#388	—	2	2	no	—
240	#387	—	2	2	no	—
241	#387	—	2	2	no	—
242	#385	—	2	2	no	—
243	#384	C₂²×C₅₃	1	1	yes	characteristic
244	#382	C₂²×C₅₃	1	1	yes	characteristic
245	#376	C₂²×C₅₃	1	1	yes	characteristic
246	#340	C₂²×C₅₃	1	2	yes	—
247	#340	C₂²×C₅₃	1	2	yes	—
248	#336	C₂²×C₅₃	1	2	yes	—
249	#336	C₂²×C₅₃	1	2	yes	—
250	C₅₃	#22278	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
252	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
253	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
254	C₂³×C₄	—	2	1	no	—
255	C₂×Q₁₆	—	4	2	no	—
256	C₂×Q₁₆	—	4	2	no	—
257	C₂×Q₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
258	C₂×SD₁₆	—	4	2	no	—
259	C₂×SD₁₆	—	4	2	no	—
260	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	—
261	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
262	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
263	C₄⋊Q₈	—	2	2	no	—
264	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
265	#124	—	2	2	no	—
266	#124	C₂×C₄×C₅₃	1	4	yes	—
267	C₄×Q₈	—	4	2	no	—
268	C₄×Q₈	—	4	2	no	—
269	C₄×Q₈	—	4	2	no	—
270	C₄×D₄	—	4	2	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
278	C₂×C₂²⋊C₄	—	2	2	no	—
279	C₂×C₂²⋊C₄	—	2	1	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₄²	—	2	2	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
286	#103	—	4	2	no	—
287	#103	—	4	2	no	—
288	#103	—	4	2	no	—
289	#102	—	4	2	no	—
290	#97	C₂×C₄×C₅₃	1	4	yes	complemented
291	C₄×C₈	—	2	2	no	—
292	#95	—	2	2	no	—
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	D₄×C₅₃	1	2	yes	—
297	#95	D₄×C₅₃	1	2	yes	—
298	C₂⁴	—	2	1	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
308	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
309	C₂×D₄	—	2	1	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	—	8	2	no	—
320	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
325	Q₁₆	—	4	8	no	—
326	SD₁₆	—	4	8	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
333	C₄⋊C₄	—	2	8	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	4	4	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	8	2	no	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₂²⋊C₄	—	8	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	4	2	no	—
347	C₄²	—	4	4	no	—
348	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
349	C₄²	#41×C₅₃	1	2	yes	—
350	C₂³	—	2	2	no	—
351	C₂³	—	4	1	no	—
352	C₂³	—	4	1	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
356	Q₈	—	2	4	no	—
357	Q₈	—	2	4	no	—
358	Q₈	—	2	4	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	4	2	no	—
361	Q₈	—	4	2	no	—
362	D₄	—	2	4	no	—
363	D₄	—	4	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	8	4	no	—
387	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
388	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
390	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
391	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
392	C₈	—	2	4	no	—
393	C₈	—	4	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	8	1	no	—
397	C₂²	—	8	1	no	—
398	C₂²	—	8	2	no	—
399	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
400	C₂²	C₅₃×#535	1	1	yes	characteristic
401	C₂²	C₅₃×#535	1	1	yes	characteristic
402	C₂²	C₅₃×#577	1	2	yes	—
403	C₂²	C₅₃×#578	1	2	yes	—
404	C₄	—	2	4	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	1	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	4	no	—
410	C₄	—	8	1	no	—
411	C₄	—	8	1	no	—
412	C₄	—	8	2	no	—
413	C₄	C₅₃×#518	1	2	yes	—
414	C₄	C₅₃×#540	1	2	yes	—
415	C₂	—	8	2	no	—
416	C₂	C₅₃×#2328	1	1	yes	characteristic
417	C₂	C₅₃×#2947	1	2	yes	—
418	C₂	C₅₃×#3222	1	1	yes	characteristic
419	C₂	C₅₃×#3223	1	1	yes	characteristic
420	C₂	C₅₃×#3225	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2327	C₂	semidirect
5. C₅₃×#1956	C₂	semidirect
7. C₅₃×#1878	C₂	semidirect
8. C₅₃×#1878	C₂	semidirect
10. C₅₃×#1836	C₂	semidirect
11. C₅₃×#1746	C₂	semidirect
19. C₅₃×#497	C₄	semidirect
46. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
53. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
79. #97×C₅₃	C₂×C₄	semidirect
183. #22278	C₅₃	direct (N × H)
206. #2327	C₂×C₅₃	semidirect
209. #1956	C₂×C₅₃	semidirect
211. #1878	C₂×C₅₃	semidirect
212. #1878	C₂×C₅₃	semidirect
214. #1836	C₂×C₅₃	semidirect
215. #1746	C₂×C₅₃	semidirect
229. #497	C₄×C₅₃	semidirect
250. C₅₃	#22278	direct (N × H)
257. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
264. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
290. #97	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#497	2
C₅₃×#2327	2
C₅₃×#2326	2
C₅₃×#1958	2
C₅₃×#1956	2
C₅₃×#1880	2
C₅₃×#1878	2
C₅₃×#1877	2
C₅₃×#1836	2
C₅₃×#1746	2
#22278	53

Subgroups — C53×#22278 #27229790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22278 #27229790