Subgroups — C₅₃×#8219 #27215731

Order 13568.

720 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#328	C₂	1	2	yes	maximal
4	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1294	C₂	1	2	yes	maximal
6	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂²×C₄²×C₅₃	—	2	1	no	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×#97×C₅₃	—	2	1	no	—
12	C₅₃×#391	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#389	C₄	1	1	yes	characteristic, complemented
14	C₅₃×#387	—	2	1	no	—
15	C₅₃×#385	—	2	1	no	—
16	C₅₃×#380	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
20	C₂×#95×C₅₃	—	2	1	no	—
21	C₂×#95×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#336	—	2	1	no	—
23	C₅₃×#328	C₄	1	8	yes	complemented
24	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	—	2	2	no	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²⋊Q₈×C₅₃	—	2	2	no	—
31	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₄×Q₈×C₅₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	—	2	4	no	—
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	—	4	2	no	—
45	C₂×C₄²×C₅₃	—	4	2	no	—
46	C₂×C₄²×C₅₃	—	4	2	no	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
53	#97×C₅₃	—	4	2	no	—
54	#95×C₅₃	—	2	4	no	—
55	#95×C₅₃	—	2	1	no	—
56	#95×C₅₃	—	2	1	no	—
57	#95×C₅₃	—	2	1	no	—
58	#95×C₅₃	—	2	1	no	—
59	#95×C₅₃	—	2	1	no	—
60	#95×C₅₃	—	2	2	no	—
61	#95×C₅₃	C₂×C₄	1	4	yes	—
62	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Q₈×C₅₃	—	2	1	no	—
64	C₂×Q₈×C₅₃	—	4	1	no	—
65	C₂×Q₈×C₅₃	—	4	1	no	—
66	C₂×Q₈×C₅₃	C₄²	1	2	yes	complemented
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	2	4	no	—
72	C₂²×C₄×C₅₃	—	2	4	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	—	2	1	no	—
75	C₂²×C₄×C₅₃	—	2	4	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	2	2	no	—
78	C₂²×C₄×C₅₃	—	2	1	no	—
79	C₂²×C₄×C₅₃	—	2	1	no	—
80	C₂²×C₄×C₅₃	—	4	1	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	—	4	2	no	—
84	C₂²×C₄×C₅₃	—	4	2	no	—
85	C₂²×C₄×C₅₃	—	4	2	no	—
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	#41	1	2	yes	—
95	C₂×C₈×C₅₃	—	2	4	no	—
96	C₂×C₈×C₅₃	—	4	2	no	—
97	C₂×C₈×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	2	4	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	C₄²	1	2	yes	complemented
105	C₂²⋊C₄×C₅₃	—	2	4	no	—
106	C₂²⋊C₄×C₅₃	—	4	1	no	—
107	C₂²⋊C₄×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	4	1	no	—
111	C₄²×C₅₃	—	4	1	no	—
112	C₄²×C₅₃	—	4	8	no	—
113	C₄²×C₅₃	—	4	2	no	—
114	C₄²×C₅₃	—	4	2	no	—
115	C₂³×C₅₃	—	2	1	no	—
116	C₂³×C₅₃	—	2	1	no	—
117	C₂³×C₅₃	—	2	1	no	—
118	C₂³×C₅₃	—	2	1	no	—
119	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₅₃	C₄×D₄	1	2	yes	—
121	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂³×C₅₃	#137	1	1	yes	characteristic
123	C₂³×C₅₃	#137	1	2	yes	—
124	Q₈×C₅₃	—	4	2	no	—
125	Q₈×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	2	4	no	—
127	C₂×C₄×C₅₃	—	2	4	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	8	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	4	no	—
147	C₂×C₄×C₅₃	—	4	4	no	—
148	C₂×C₄×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	—	4	4	no	—
150	C₂×C₄×C₅₃	—	4	4	no	—
151	C₂×C₄×C₅₃	—	4	4	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	4	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	4	1	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	—	4	1	no	—
163	C₂×C₄×C₅₃	#99	1	2	yes	—
164	C₂×C₄×C₅₃	#104	1	2	yes	—
165	C₂×C₄×C₅₃	#104	1	2	yes	—
166	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
167	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
169	C₈×C₅₃	—	4	4	no	—
170	#8219	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	2	2	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	2	2	no	—
176	C₂²×C₅₃	—	2	2	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₅₃	C₂×#99	1	1	yes	characteristic
181	C₂²×C₅₃	#410	1	1	yes	characteristic
182	C₂²×C₅₃	#417	1	1	yes	characteristic
183	C₂²×C₅₃	#419	1	1	yes	characteristic
184	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
185	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
186	C₂²×C₅₃	#440	1	2	yes	—
187	C₂²×C₅₃	#441	1	2	yes	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	2	no	—
192	C₄×C₅₃	—	2	2	no	—
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	4	8	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	4	2	no	—
197	C₄×C₅₃	—	4	1	no	—
198	C₄×C₅₃	—	8	1	no	—
199	C₄×C₅₃	—	8	1	no	—
200	C₄×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
201	#1749	C₂×C₅₃	1	1	yes	characteristic
202	C₂×#328	C₂×C₅₃	1	2	yes	—
203	C₄×#95	C₂×C₅₃	1	1	yes	characteristic
204	#1294	C₂×C₅₃	1	2	yes	—
205	C₂×C₅₃	—	2	2	no	—
206	C₂×C₅₃	—	2	2	no	—
207	C₂×C₅₃	#1504	1	1	yes	characteristic
208	C₂×C₅₃	#1506	1	1	yes	characteristic
209	C₂×C₅₃	#1508	1	1	yes	characteristic
210	C₂×C₅₃	#1509	1	1	yes	characteristic
211	C₂×C₅₃	C₄×#99	1	1	yes	characteristic
212	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
213	C₂×C₅₃	#1763	1	1	yes	characteristic
214	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
215	C₂×C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic, complemented
216	C₂²×C₄²	—	2	1	no	—
217	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
218	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
219	C₂×#97	—	2	1	no	—
220	#391	C₄×C₅₃	1	1	yes	characteristic, complemented
221	#389	C₄×C₅₃	1	1	yes	characteristic, complemented
222	#387	—	2	1	no	—
223	#385	—	2	1	no	—
224	#380	C₂²×C₅₃	1	1	yes	characteristic
225	C₄×C₄⋊C₄	—	2	1	no	—
226	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic, complemented
227	C₄×C₂²⋊C₄	—	2	1	no	—
228	C₂×#95	—	2	1	no	—
229	C₂×#95	C₂²×C₅₃	1	2	yes	—
230	#336	—	2	1	no	—
231	#328	C₄×C₅₃	1	8	yes	complemented
232	C₅₃	#8219	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
234	C₂³×C₄	—	2	2	no	—
235	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
236	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
237	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
238	C₂²×C₈	—	2	2	no	—
239	C₂²⋊Q₈	—	2	2	no	—
240	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
241	C₄×Q₈	—	4	2	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₂²⋊C₄	—	2	1	no	—
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₄²	—	2	4	no	—
251	C₂×C₄²	—	2	1	no	—
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	—	4	2	no	—
254	C₂×C₄²	—	4	2	no	—
255	C₂×C₄²	—	4	2	no	—
256	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
257	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
258	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
260	C₂²⋊C₈	—	2	2	no	—
261	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
262	#97	—	4	2	no	—
263	#95	—	2	4	no	—
264	#95	—	2	1	no	—
265	#95	—	2	2	no	—
266	#95	—	2	1	no	—
267	#95	—	2	1	no	—
268	#95	—	2	1	no	—
269	#95	—	2	1	no	—
270	#95	C₂×C₄×C₅₃	1	4	yes	—
271	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	C₄²×C₅₃	1	2	yes	complemented
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	#41×C₅₃	1	2	yes	—
304	C₂×C₈	—	2	4	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	—	4	2	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	C₄²×C₅₃	1	2	yes	complemented
314	C₂²⋊C₄	—	2	4	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	4	1	no	—
320	C₄²	—	4	1	no	—
321	C₄²	—	4	8	no	—
322	C₄²	—	4	2	no	—
323	C₄²	—	4	2	no	—
324	C₂³	—	2	1	no	—
325	C₂³	—	2	1	no	—
326	C₂³	—	2	1	no	—
327	C₂³	—	2	1	no	—
328	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
329	C₂³	C₄×D₄×C₅₃	1	2	yes	—
330	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
331	C₂³	#137×C₅₃	1	1	yes	characteristic
332	C₂³	#137×C₅₃	1	2	yes	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	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	1	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	#99×C₅₃	1	2	yes	—
373	C₂×C₄	#104×C₅₃	1	2	yes	—
374	C₂×C₄	#104×C₅₃	1	2	yes	—
375	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
376	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
378	C₈	—	4	4	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
388	C₂²	C₂×#99×C₅₃	1	1	yes	characteristic
389	C₂²	C₅₃×#410	1	1	yes	characteristic
390	C₂²	C₅₃×#417	1	1	yes	characteristic
391	C₂²	C₅₃×#419	1	1	yes	characteristic
392	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
393	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
394	C₂²	C₅₃×#440	1	2	yes	—
395	C₂²	C₅₃×#441	1	2	yes	—
396	C₄	—	2	2	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	2	no	—
401	C₄	—	2	2	no	—
402	C₄	—	4	8	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	1	no	—
406	C₄	—	8	1	no	—
407	C₄	—	8	1	no	—
408	C₂	—	2	2	no	—
409	C₂	—	2	2	no	—
410	C₂	C₅₃×#1504	1	1	yes	characteristic
411	C₂	C₅₃×#1506	1	1	yes	characteristic
412	C₂	C₅₃×#1508	1	1	yes	characteristic
413	C₂	C₅₃×#1509	1	1	yes	characteristic
414	C₂	C₄×#99×C₅₃	1	1	yes	characteristic
415	C₂	C₄×#104×C₅₃	1	1	yes	characteristic
416	C₂	C₅₃×#1763	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₂×C₄×Q₈×C₅₃	C₄	semidirect
12. C₅₃×#391	C₄	semidirect
13. C₅₃×#389	C₄	semidirect
18. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
23. C₅₃×#328	C₄	semidirect
66. C₂×Q₈×C₅₃	C₄²	semidirect
104. C₄⋊C₄×C₅₃	C₄²	semidirect
170. #8219	C₅₃	direct (N × H)
215. C₂×C₄×Q₈	C₄×C₅₃	semidirect
220. #391	C₄×C₅₃	semidirect
221. #389	C₄×C₅₃	semidirect
226. C₄×C₄⋊C₄	C₄×C₅₃	semidirect
231. #328	C₄×C₅₃	semidirect
232. C₅₃	#8219	direct (N × H)
275. C₂×Q₈	C₄²×C₅₃	semidirect
313. C₄⋊C₄	C₄²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#1749	2
C₂×C₅₃×#328	2
C₄×#95×C₅₃	2
C₅₃×#1294	2
#8219	53

Subgroups — C53×#8219 #27215731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8219 #27215731