Subgroups — C₅₃×#21088 #27228600

Order 13568.

1074 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	C₂×C₅₃×#419	C₂	1	1	yes	maximal, characteristic
2	C₂²×#103×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2316	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2310	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1935	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1933	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1736	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1735	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#419	—	2	4	no	—
18	C₂×#103×C₅₃	C₄	1	8	yes	complemented
19	C₂×#103×C₅₃	C₂²	1	4	yes	—
20	C₂×#103×C₅₃	C₂²	1	2	yes	—
21	C₂×#103×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#405	—	2	4	no	—
23	C₂×#97×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#391	—	2	2	no	—
26	C₅₃×#391	C₂²	1	4	yes	—
27	C₅₃×#389	—	2	2	no	—
28	C₅₃×#389	C₂²	1	4	yes	—
29	C₅₃×#381	—	2	2	no	—
30	C₅₃×#380	—	2	2	no	—
31	C₅₃×#340	C₂²	1	2	yes	—
32	C₅₃×#340	C₂²	1	4	yes	—
33	C₅₃×#340	C₂²	1	2	yes	—
34	C₂²×Q₈×C₅₃	—	2	1	no	—
35	C₂²×Q₈×C₅₃	—	2	2	no	—
36	C₂²×Q₈×C₅₃	—	2	2	no	—
37	C₂²×Q₈×C₅₃	—	2	2	no	—
38	C₂²×Q₈×C₅₃	—	2	2	no	—
39	C₂²×Q₈×C₅₃	—	2	2	no	—
40	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	C₂³×C₄×C₅₃	—	2	1	no	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	#117×C₅₃	—	2	2	no	—
55	#117×C₅₃	—	2	2	no	—
56	#117×C₅₃	—	2	2	no	—
57	#117×C₅₃	—	2	2	no	—
58	#117×C₅₃	—	2	2	no	—
59	#117×C₅₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
71	C₂×C₄²×C₅₃	—	2	1	no	—
72	C₂×C₄²×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
74	#103×C₅₃	—	2	8	no	—
75	#103×C₅₃	C₂×C₄	1	8	yes	—
76	#103×C₅₃	C₂×C₄	1	8	yes	—
77	#97×C₅₃	—	2	4	no	—
78	C₄×C₈×C₅₃	—	2	4	no	—
79	#95×C₅₃	—	2	4	no	—
80	#95×C₅₃	—	2	8	no	—
81	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	4	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	2	8	no	—
85	C₂×Q₈×C₅₃	—	2	8	no	—
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	—	4	2	no	—
91	C₂×Q₈×C₅₃	—	4	4	no	—
92	C₂×Q₈×C₅₃	—	4	2	no	—
93	C₂×Q₈×C₅₃	—	4	4	no	—
94	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₅₃	—	2	4	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	2	4	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	4	2	no	—
102	C₂²×C₄×C₅₃	—	4	2	no	—
103	C₂²×C₄×C₅₃	—	8	2	no	—
104	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₅₃	#41	1	2	yes	—
109	C₂²×C₄×C₅₃	#41	1	2	yes	—
110	C₂²×C₄×C₅₃	#41	1	2	yes	—
111	C₂²×C₄×C₅₃	#41	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₅₃	—	2	8	no	—
114	C₂×C₈×C₅₃	—	2	8	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
124	C₄⋊C₄×C₅₃	—	2	4	no	—
125	C₄⋊C₄×C₅₃	—	2	4	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	—	8	4	no	—
132	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₂²⋊C₄×C₅₃	—	4	2	no	—
135	C₂²⋊C₄×C₅₃	—	4	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	4	2	no	—
139	C₄²×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	C₄×D₄	1	2	yes	—
143	C₂³×C₅₃	#121	1	2	yes	—
144	C₂³×C₅₃	#124	1	2	yes	—
145	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
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	8	no	—
151	C₂×C₄×C₅₃	—	2	4	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	2	4	no	—
154	C₂×C₄×C₅₃	—	2	4	no	—
155	C₂×C₄×C₅₃	—	2	4	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	4	no	—
159	C₂×C₄×C₅₃	—	4	8	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	8	4	no	—
167	C₂×C₄×C₅₃	#117	1	4	yes	—
168	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
169	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₅₃	#121	1	2	yes	—
171	C₂×C₄×C₅₃	#123	1	4	yes	—
172	C₂×C₄×C₅₃	#124	1	2	yes	—
173	C₂×C₄×C₅₃	#126	1	4	yes	—
174	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	2	2	no	—
182	C₈×C₅₃	—	2	4	no	—
183	#21088	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂²×C₅₃	—	2	4	no	—
185	C₂²×C₅₃	—	2	2	no	—
186	C₂²×C₅₃	—	2	2	no	—
187	C₂²×C₅₃	—	2	4	no	—
188	C₂²×C₅₃	#388	1	4	yes	—
189	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
192	C₂²×C₅₃	#577	1	2	yes	—
193	C₂²×C₅₃	#578	1	2	yes	—
194	C₄×C₅₃	—	4	4	no	—
195	C₄×C₅₃	—	4	4	no	—
196	C₄×C₅₃	—	4	4	no	—
197	C₄×C₅₃	—	8	2	no	—
198	C₄×C₅₃	—	8	2	no	—
199	C₄×C₅₃	#388	1	4	yes	—
200	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
201	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
202	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
203	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
204	C₂×#419	C₂×C₅₃	1	1	yes	characteristic
205	C₂²×#103	C₂×C₅₃	1	1	yes	characteristic
206	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
207	#2316	C₂×C₅₃	1	1	yes	characteristic
208	#2310	C₂×C₅₃	1	1	yes	characteristic
209	#1935	C₂×C₅₃	1	4	yes	complemented
210	#1933	C₂×C₅₃	1	4	yes	complemented
211	#1736	C₂×C₅₃	1	1	yes	characteristic
212	#1735	C₂×C₅₃	1	1	yes	characteristic
213	C₂×C₅₃	—	2	4	no	—
214	C₂×C₅₃	C₂×#388	1	1	yes	characteristic
215	C₂×C₅₃	#2949	1	1	yes	characteristic
216	C₂×C₅₃	#2950	1	1	yes	characteristic
217	C₂×C₅₃	#3063	1	2	yes	—
218	C₂×C₅₃	#3137	1	2	yes	—
219	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
220	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
221	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
222	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
225	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
226	#419	—	2	4	no	—
227	C₂×#103	C₄×C₅₃	1	8	yes	complemented
228	C₂×#103	C₂²×C₅₃	1	4	yes	—
229	C₂×#103	C₂²×C₅₃	1	2	yes	—
230	C₂×#103	C₂²×C₅₃	1	2	yes	—
231	#405	—	2	4	no	—
232	C₂×#97	C₂²×C₅₃	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
234	#391	—	2	2	no	—
235	#391	C₂²×C₅₃	1	4	yes	—
236	#389	—	2	2	no	—
237	#389	C₂²×C₅₃	1	4	yes	—
238	#381	—	2	2	no	—
239	#380	—	2	2	no	—
240	#340	C₂²×C₅₃	1	2	yes	—
241	#340	C₂²×C₅₃	1	4	yes	—
242	#340	C₂²×C₅₃	1	2	yes	—
243	C₅₃	#21088	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²×Q₈	—	2	1	no	—
245	C₂²×Q₈	—	2	2	no	—
246	C₂²×Q₈	—	2	2	no	—
247	C₂²×Q₈	—	2	2	no	—
248	C₂²×Q₈	—	2	2	no	—
249	C₂²×Q₈	—	2	2	no	—
250	C₂²×Q₈	C₂×C₄×C₅₃	1	4	yes	—
251	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
252	C₂³×C₄	—	2	1	no	—
253	C₂³×C₄	—	2	1	no	—
254	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
258	C₂²×C₈	—	2	2	no	—
259	C₂²×C₈	—	2	2	no	—
260	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
261	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
264	#117	—	2	2	no	—
265	#117	—	2	2	no	—
266	#117	—	2	2	no	—
267	#117	—	2	2	no	—
268	#117	—	2	2	no	—
269	#117	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	4	no	—
273	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
274	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
278	C₂×C₂²⋊C₄	—	2	1	no	—
279	C₂×C₂²⋊C₄	—	2	1	no	—
280	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
284	#103	—	2	8	no	—
285	#103	C₂×C₄×C₅₃	1	8	yes	—
286	#103	C₂×C₄×C₅₃	1	8	yes	—
287	#97	—	2	4	no	—
288	C₄×C₈	—	2	4	no	—
289	#95	—	2	4	no	—
290	#95	—	2	8	no	—
291	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
292	C₂×Q₈	—	2	4	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	8	no	—
295	C₂×Q₈	—	2	8	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	4	2	no	—
301	C₂×Q₈	—	4	4	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×Q₈	—	4	4	no	—
304	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	2	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	—	8	2	no	—
314	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
318	C₂²×C₄	#41×C₅₃	1	2	yes	—
319	C₂²×C₄	#41×C₅₃	1	2	yes	—
320	C₂²×C₄	#41×C₅₃	1	2	yes	—
321	C₂²×C₄	#41×C₅₃	1	2	yes	—
322	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
323	M₁₆	—	2	8	no	—
324	C₂×C₈	—	2	8	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
332	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
333	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	8	4	no	—
342	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	4	2	no	—
345	C₂²⋊C₄	—	4	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	4	2	no	—
349	C₄²	—	4	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	2	no	—
352	C₂³	C₄×D₄×C₅₃	1	2	yes	—
353	C₂³	#121×C₅₃	1	2	yes	—
354	C₂³	#124×C₅₃	1	2	yes	—
355	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
356	Q₈	—	2	4	no	—
357	Q₈	—	2	4	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	4	8	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	—	4	8	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	8	4	no	—
377	C₂×C₄	#117×C₅₃	1	4	yes	—
378	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
379	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
380	C₂×C₄	#121×C₅₃	1	2	yes	—
381	C₂×C₄	#123×C₅₃	1	4	yes	—
382	C₂×C₄	#124×C₅₃	1	2	yes	—
383	C₂×C₄	#126×C₅₃	1	4	yes	—
384	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
386	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	2	2	no	—
392	C₈	—	2	4	no	—
393	C₂²	—	2	2	no	—
394	C₂²	—	2	2	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	4	no	—
397	C₂²	C₅₃×#388	1	4	yes	—
398	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
399	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
400	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
401	C₂²	C₅₃×#577	1	2	yes	—
402	C₂²	C₅₃×#578	1	2	yes	—
403	C₄	—	4	4	no	—
404	C₄	—	4	4	no	—
405	C₄	—	4	4	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	2	no	—
408	C₄	C₅₃×#388	1	4	yes	—
409	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
410	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
411	C₄	C₂×#123×C₅₃	1	1	yes	characteristic
412	C₄	C₂×#126×C₅₃	1	1	yes	characteristic
413	C₂	—	2	4	no	—
414	C₂	C₂×C₅₃×#388	1	1	yes	characteristic
415	C₂	C₅₃×#2949	1	1	yes	characteristic
416	C₂	C₅₃×#2950	1	1	yes	characteristic
417	C₂	C₅₃×#3063	1	2	yes	—
418	C₂	C₅₃×#3137	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	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅₃×#1935	C₂	semidirect
7. C₅₃×#1933	C₂	semidirect
18. C₂×#103×C₅₃	C₄	semidirect
183. #21088	C₅₃	direct (N × H)
209. #1935	C₂×C₅₃	semidirect
210. #1933	C₂×C₅₃	semidirect
227. C₂×#103	C₄×C₅₃	semidirect
243. C₅₃	#21088	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#419	2
C₂²×#103×C₅₃	2
C₂×C₅₃×#405	2
C₅₃×#2316	2
C₅₃×#2310	2
C₅₃×#1935	2
C₅₃×#1933	2
C₅₃×#1736	2
C₅₃×#1735	2
#21088	53

Subgroups — C53×#21088 #27228600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21088 #27228600