Subgroups — C₅₃×#10743 #27218255

Order 13568.

1208 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3428	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#410	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2879	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1733	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#520	—	2	4	no	—
10	C₅₃×#519	C₄	1	2	yes	complemented
11	C₅₃×#518	—	2	2	no	—
12	C₅₃×#518	C₄	1	2	yes	complemented
13	C₄×#41×C₅₃	—	2	4	no	—
14	C₂×#117×C₅₃	—	2	2	no	—
15	C₂×#117×C₅₃	C₂²	1	2	yes	—
16	C₂×#117×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#410	—	2	2	no	—
18	C₅₃×#410	C₄	1	4	yes	complemented
19	C₂×#99×C₅₃	—	2	2	no	—
20	C₅₃×#408	—	2	2	no	—
21	C₅₃×#408	C₄	1	4	yes	complemented
22	C₅₃×#407	—	2	2	no	—
23	C₅₃×#339	—	2	4	no	—
24	C₅₃×H(3,ℤ/4)	—	2	4	no	—
25	C₂×#41×C₅₃	—	2	2	no	—
26	C₂×#41×C₅₃	—	2	2	no	—
27	C₂×#41×C₅₃	—	2	2	no	—
28	C₂×#41×C₅₃	—	2	2	no	—
29	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×#41×C₅₃	D₄	1	2	yes	—
32	C₂×#41×C₅₃	Q₈	1	2	yes	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₅₃	D₄	1	2	yes	—
36	C₂³×C₄×C₅₃	D₄	1	2	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	1	no	—
40	C₂×M₁₆×C₅₃	—	2	1	no	—
41	C₂×M₁₆×C₅₃	—	2	1	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
46	C₂²×C₈×C₅₃	—	2	1	no	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
48	C₄×Q₈×C₅₃	—	2	8	no	—
49	C₄×D₄×C₅₃	—	2	16	no	—
50	C₄×D₄×C₅₃	—	2	8	no	—
51	#117×C₅₃	—	2	4	no	—
52	#117×C₅₃	—	2	4	no	—
53	#117×C₅₃	—	2	4	no	—
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	C₂×C₄	1	4	yes	—
60	#117×C₅₃	C₂×C₄	1	4	yes	complemented
61	#117×C₅₃	D₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂×C₄²×C₅₃	—	2	4	no	—
69	C₂×C₄²×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	D₄	1	4	yes	—
71	#99×C₅₃	—	2	8	no	—
72	C₂²⋊C₈×C₅₃	—	2	8	no	—
73	C₂⁴×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
75	#41×C₅₃	—	2	4	no	—
76	#41×C₅₃	—	2	4	no	—
77	#41×C₅₃	—	2	4	no	—
78	#41×C₅₃	—	2	4	no	—
79	#41×C₅₃	—	2	4	no	—
80	#41×C₅₃	—	2	4	no	—
81	#41×C₅₃	—	2	4	no	—
82	#41×C₅₃	—	2	4	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
86	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₂×D₄×C₅₃	—	2	4	no	—
88	C₂×D₄×C₅₃	—	2	4	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	C₄²	1	2	yes	—
92	C₂×D₄×C₅₃	C₂²⋊C₄	1	4	yes	—
93	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
94	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	2	8	no	—
100	C₂²×C₄×C₅₃	—	2	4	no	—
101	C₂²×C₄×C₅₃	—	4	2	no	—
102	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
104	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	2	yes	—
114	C₂²×C₄×C₅₃	#41	1	2	yes	—
115	M₁₆×C₅₃	—	2	4	no	—
116	M₁₆×C₅₃	—	4	1	no	—
117	M₁₆×C₅₃	—	4	1	no	—
118	M₁₆×C₅₃	—	4	2	no	—
119	M₁₆×C₅₃	—	4	1	no	—
120	M₁₆×C₅₃	—	4	1	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	4	no	—
124	C₂×C₈×C₅₃	—	4	2	no	—
125	C₄⋊C₄×C₅₃	—	2	8	no	—
126	C₄⋊C₄×C₅₃	—	2	4	no	—
127	C₄⋊C₄×C₅₃	—	2	4	no	—
128	C₄⋊C₄×C₅₃	—	2	4	no	—
129	C₄⋊C₄×C₅₃	—	4	2	no	—
130	C₂²⋊C₄×C₅₃	—	2	8	no	—
131	C₂²⋊C₄×C₅₃	—	2	4	no	—
132	C₂²⋊C₄×C₅₃	—	2	4	no	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₂²⋊C₄×C₅₃	—	4	2	no	—
135	C₄²×C₅₃	—	2	8	no	—
136	C₄²×C₅₃	—	2	4	no	—
137	C₄²×C₅₃	—	2	4	no	—
138	C₄²×C₅₃	—	4	2	no	—
139	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	—	8	1	no	—
143	C₂³×C₅₃	—	8	2	no	—
144	C₂³×C₅₃	#95	1	4	yes	—
145	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂³×C₅₃	#120	1	2	yes	—
149	C₂³×C₅₃	#123	1	2	yes	—
150	Q₈×C₅₃	—	2	4	no	—
151	Q₈×C₅₃	—	2	4	no	—
152	D₄×C₅₃	—	2	8	no	—
153	D₄×C₅₃	—	2	8	no	—
154	D₄×C₅₃	—	2	4	no	—
155	D₄×C₅₃	—	2	4	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	—	2	4	no	—
160	C₂×C₄×C₅₃	—	2	4	no	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	2	8	no	—
163	C₂×C₄×C₅₃	—	4	4	no	—
164	C₂×C₄×C₅₃	—	8	4	no	—
165	C₂×C₄×C₅₃	—	8	8	no	—
166	C₂×C₄×C₅₃	—	8	4	no	—
167	C₂×C₄×C₅₃	#95	1	4	yes	—
168	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	#117	1	2	yes	—
174	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
180	C₂×C₄×C₅₃	#121	1	2	yes	—
181	C₂×C₄×C₅₃	#121	1	2	yes	—
182	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
184	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
185	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
186	C₈×C₅₃	—	4	2	no	—
187	C₈×C₅₃	—	4	2	no	—
188	#10743	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂²×C₅₃	—	2	4	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	4	2	no	—
192	C₂²×C₅₃	—	4	1	no	—
193	C₂²×C₅₃	—	4	1	no	—
194	C₂²×C₅₃	—	4	1	no	—
195	C₂²×C₅₃	—	4	4	no	—
196	C₂²×C₅₃	—	4	4	no	—
197	C₂²×C₅₃	—	8	2	no	—
198	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
199	C₂²×C₅₃	#379	1	1	yes	characteristic
200	C₂²×C₅₃	#381	1	1	yes	characteristic
201	C₂²×C₅₃	#385	1	2	yes	—
202	C₂²×C₅₃	#386	1	2	yes	—
203	C₄×C₅₃	—	2	4	no	—
204	C₄×C₅₃	—	4	8	no	—
205	C₄×C₅₃	—	8	4	no	—
206	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
207	C₄×C₅₃	#380	1	2	yes	—
208	C₄×C₅₃	#385	1	1	yes	characteristic
209	C₄×C₅₃	#385	1	1	yes	characteristic
210	C₄×C₅₃	#386	1	1	yes	characteristic
211	C₄×C₅₃	#386	1	1	yes	characteristic
212	#3428	C₂×C₅₃	1	1	yes	characteristic
213	C₂×#410	C₂×C₅₃	1	1	yes	characteristic
214	#2879	C₂×C₅₃	1	1	yes	characteristic
215	#1733	C₂×C₅₃	1	2	yes	complemented
216	C₂×H(3,ℤ/4)	C₂×C₅₃	1	2	yes	complemented
217	C₂×C₅₃	—	2	1	no	—
218	C₂×C₅₃	—	2	2	no	—
219	C₂×C₅₃	—	2	1	no	—
220	C₂×C₅₃	—	2	1	no	—
221	C₂×C₅₃	—	2	2	no	—
222	C₂×C₅₃	—	4	4	no	—
223	C₂×C₅₃	C₂²⋊#95	1	1	yes	characteristic
224	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
225	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
226	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
227	#520	—	2	4	no	—
228	#519	C₄×C₅₃	1	2	yes	complemented
229	#518	—	2	2	no	—
230	#518	C₄×C₅₃	1	2	yes	complemented
231	C₄×#41	—	2	4	no	—
232	C₂×#117	—	2	2	no	—
233	C₂×#117	C₂²×C₅₃	1	2	yes	—
234	C₂×#117	C₂²×C₅₃	1	2	yes	—
235	#410	—	2	2	no	—
236	#410	C₄×C₅₃	1	4	yes	complemented
237	C₂×#99	—	2	2	no	—
238	#408	—	2	2	no	—
239	#408	C₄×C₅₃	1	4	yes	complemented
240	#407	—	2	2	no	—
241	#339	—	2	4	no	—
242	H(3,ℤ/4)	—	2	4	no	—
243	C₅₃	#10743	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×#41	—	2	2	no	—
245	C₂×#41	—	2	2	no	—
246	C₂×#41	—	2	2	no	—
247	C₂×#41	—	2	2	no	—
248	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
249	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
250	C₂×#41	D₄×C₅₃	1	2	yes	—
251	C₂×#41	Q₈×C₅₃	1	2	yes	—
252	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
253	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
254	C₂²×D₄	D₄×C₅₃	1	2	yes	—
255	C₂³×C₄	D₄×C₅₃	1	2	yes	—
256	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	1	no	—
259	C₂×M₁₆	—	2	1	no	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	—	2	2	no	—
264	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	complemented
267	C₄×Q₈	—	2	8	no	—
268	C₄×D₄	—	2	16	no	—
269	C₄×D₄	—	2	8	no	—
270	#117	—	2	4	no	—
271	#117	—	2	4	no	—
272	#117	—	2	2	no	—
273	#117	—	2	2	no	—
274	#117	—	2	4	no	—
275	#117	—	2	2	no	—
276	#117	—	2	2	no	—
277	#117	—	2	2	no	—
278	#117	C₂×C₄×C₅₃	1	4	yes	—
279	#117	C₂×C₄×C₅₃	1	4	yes	complemented
280	#117	D₄×C₅₃	1	4	yes	—
281	C₂×C₄⋊C₄	—	2	4	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₂²⋊C₄	—	2	4	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₄²	—	2	4	no	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	D₄×C₅₃	1	4	yes	—
290	#99	—	2	8	no	—
291	C₂²⋊C₈	—	2	8	no	—
292	C₂⁴	C₂²⋊C₄×C₅₃	1	2	yes	—
293	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
294	#41	—	2	4	no	—
295	#41	—	2	4	no	—
296	#41	—	2	4	no	—
297	#41	—	2	4	no	—
298	#41	—	2	4	no	—
299	#41	—	2	4	no	—
300	#41	—	2	4	no	—
301	#41	—	2	4	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	C₄²×C₅₃	1	2	yes	—
305	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
306	C₂×D₄	—	2	4	no	—
307	C₂×D₄	—	2	4	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	C₄²×C₅₃	1	2	yes	—
311	C₂×D₄	C₂²⋊C₄×C₅₃	1	4	yes	—
312	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
313	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	4	2	no	—
321	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
322	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
323	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
324	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
331	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	#41×C₅₃	1	2	yes	—
333	C₂²×C₄	#41×C₅₃	1	2	yes	—
334	M₁₆	—	2	4	no	—
335	M₁₆	—	4	1	no	—
336	M₁₆	—	4	1	no	—
337	M₁₆	—	4	2	no	—
338	M₁₆	—	4	1	no	—
339	M₁₆	—	4	1	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	4	no	—
343	C₂×C₈	—	4	2	no	—
344	C₄⋊C₄	—	2	8	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₂²⋊C₄	—	2	8	no	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₂²⋊C₄	—	2	4	no	—
352	C₂²⋊C₄	—	2	4	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₄²	—	2	8	no	—
355	C₄²	—	2	4	no	—
356	C₄²	—	2	4	no	—
357	C₄²	—	4	2	no	—
358	C₄²	C₂²⋊C₄×C₅₃	1	8	yes	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	8	1	no	—
362	C₂³	—	8	2	no	—
363	C₂³	#95×C₅₃	1	4	yes	—
364	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
365	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
366	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
367	C₂³	#120×C₅₃	1	2	yes	—
368	C₂³	#123×C₅₃	1	2	yes	—
369	Q₈	—	2	4	no	—
370	Q₈	—	2	4	no	—
371	D₄	—	2	8	no	—
372	D₄	—	2	8	no	—
373	D₄	—	2	4	no	—
374	D₄	—	2	4	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	4	4	no	—
383	C₂×C₄	—	8	4	no	—
384	C₂×C₄	—	8	8	no	—
385	C₂×C₄	—	8	4	no	—
386	C₂×C₄	#95×C₅₃	1	4	yes	—
387	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
388	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
389	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
392	C₂×C₄	#117×C₅₃	1	2	yes	—
393	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
394	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
395	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
396	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
397	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
398	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
399	C₂×C₄	#121×C₅₃	1	2	yes	—
400	C₂×C₄	#121×C₅₃	1	2	yes	—
401	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
402	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
403	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
404	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	4	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	4	no	—
414	C₂²	—	4	4	no	—
415	C₂²	—	8	2	no	—
416	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
417	C₂²	C₅₃×#379	1	1	yes	characteristic
418	C₂²	C₅₃×#381	1	1	yes	characteristic
419	C₂²	C₅₃×#385	1	2	yes	—
420	C₂²	C₅₃×#386	1	2	yes	—
421	C₄	—	2	4	no	—
422	C₄	—	4	8	no	—
423	C₄	—	8	4	no	—
424	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
425	C₄	C₅₃×#380	1	2	yes	—
426	C₄	C₅₃×#385	1	1	yes	characteristic
427	C₄	C₅₃×#385	1	1	yes	characteristic
428	C₄	C₅₃×#386	1	1	yes	characteristic
429	C₄	C₅₃×#386	1	1	yes	characteristic
430	C₂	—	2	1	no	—
431	C₂	—	2	2	no	—
432	C₂	—	2	1	no	—
433	C₂	—	2	1	no	—
434	C₂	—	2	2	no	—
435	C₂	—	4	4	no	—
436	C₂	C₅₃×C₂²⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#386
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂×C₅₃	C₂²⋊#95
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1733	C₂	semidirect
5. C₂×C₅₃×H(3,ℤ/4)	C₂	semidirect
10. C₅₃×#519	C₄	semidirect
12. C₅₃×#518	C₄	semidirect
18. C₅₃×#410	C₄	semidirect
21. C₅₃×#408	C₄	semidirect
45. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
47. C₂²×C₈×C₅₃	C₂×C₄	semidirect
60. #117×C₅₃	C₂×C₄	semidirect
188. #10743	C₅₃	direct (N × H)
215. #1733	C₂×C₅₃	semidirect
216. C₂×H(3,ℤ/4)	C₂×C₅₃	semidirect
228. #519	C₄×C₅₃	semidirect
230. #518	C₄×C₅₃	semidirect
236. #410	C₄×C₅₃	semidirect
239. #408	C₄×C₅₃	semidirect
243. C₅₃	#10743	direct (N × H)
264. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
266. C₂²×C₈	C₂×C₄×C₅₃	semidirect
279. #117	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
C₅₃×#3428	2
C₂×C₅₃×#410	2
C₅₃×#2879	2
C₅₃×#1733	2
C₂×C₅₃×H(3,ℤ/4)	2
#10743	53

Subgroups — C53×#10743 #27218255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10743 #27218255