Subgroups — C₅₃×#11327 #27218839

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#389	C₂	1	2	yes	maximal
3	C₂×C₅₃×#340	C₂	1	2	yes	maximal
4	C₂×C₅₃×#340	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#106×C₅₃	—	2	2	no	—
13	C₂×#106×C₅₃	—	2	4	no	—
14	C₅₃×#389	—	2	4	no	—
15	C₅₃×#389	—	2	4	no	—
16	C₂×#95×C₅₃	—	2	2	no	—
17	C₅₃×#340	—	2	4	no	—
18	C₅₃×#340	C₄	1	4	yes	complemented
19	C₅₃×#336	—	2	2	no	—
20	C₂³×C₄×C₅₃	—	2	2	no	—
21	C₂³×C₄×C₅₃	—	2	2	no	—
22	C₂³×C₄×C₅₃	D₄	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₅₃	Q₈	1	4	yes	—
49	C₂×C₄²×C₅₃	—	2	1	no	—
50	C₂×C₄²×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	#106×C₅₃	—	2	8	no	—
54	#95×C₅₃	—	2	8	no	—
55	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	4	2	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	—	4	2	no	—
74	C₂²×C₄×C₅₃	—	4	2	no	—
75	C₂²×C₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	#41	1	2	yes	—
85	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	4	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	—	4	2	no	—
95	C₂×C₈×C₅₃	—	4	1	no	—
96	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
97	C₄⋊C₄×C₅₃	—	2	4	no	—
98	C₄⋊C₄×C₅₃	—	2	4	no	—
99	C₄⋊C₄×C₅₃	—	2	4	no	—
100	C₄⋊C₄×C₅₃	—	2	4	no	—
101	C₄⋊C₄×C₅₃	—	2	4	no	—
102	C₄⋊C₄×C₅₃	—	4	16	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	—	4	2	no	—
105	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	complemented
106	C₄²×C₅₃	—	4	1	no	—
107	C₄²×C₅₃	—	4	2	no	—
108	C₄²×C₅₃	—	4	1	no	—
109	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₅₃	#120	1	2	yes	—
111	C₂³×C₅₃	#121	1	1	yes	characteristic
112	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
113	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
114	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
115	C₂³×C₅₃	#135	1	2	yes	—
116	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₂³×C₅₃	#137	1	2	yes	—
118	C₂×C₄×C₅₃	—	2	8	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	4	no	—
124	C₂×C₄×C₅₃	—	4	4	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	4	no	—
127	C₂×C₄×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	4	4	no	—
131	C₂×C₄×C₅₃	—	4	4	no	—
132	C₂×C₄×C₅₃	—	4	4	no	—
133	C₂×C₄×C₅₃	—	4	4	no	—
134	C₂×C₄×C₅₃	—	4	4	no	—
135	C₂×C₄×C₅₃	—	8	2	no	—
136	C₂×C₄×C₅₃	—	8	2	no	—
137	C₂×C₄×C₅₃	—	8	2	no	—
138	C₂×C₄×C₅₃	—	8	2	no	—
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#121	1	2	yes	—
145	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
148	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#123	1	2	yes	—
151	C₂×C₄×C₅₃	#125	1	2	yes	—
152	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
157	C₈×C₅₃	—	2	4	no	—
158	C₈×C₅₃	—	4	2	no	—
159	#11327	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	#385	1	2	yes	—
161	C₂²×C₅₃	#389	1	1	yes	characteristic
162	C₂²×C₅₃	#392	1	1	yes	characteristic
163	C₂²×C₅₃	#395	1	2	yes	—
164	C₂²×C₅₃	#398	1	1	yes	characteristic
165	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
166	C₂²×C₅₃	#441	1	1	yes	characteristic
167	C₂²×C₅₃	#442	1	1	yes	characteristic
168	C₂²×C₅₃	#448	1	2	yes	—
169	C₂²×C₅₃	#449	1	2	yes	—
170	C₂²×C₅₃	#450	1	2	yes	—
171	C₂²×C₅₃	#452	1	2	yes	—
172	C₂²×C₅₃	#465	1	2	yes	—
173	C₂²×C₅₃	#468	1	2	yes	—
174	C₂²×C₅₃	#475	1	2	yes	—
175	C₂²×C₅₃	#476	1	2	yes	—
176	C₂²×C₅₃	#478	1	2	yes	—
177	C₂²×C₅₃	#479	1	2	yes	—
178	C₂²×C₅₃	#498	1	1	yes	characteristic
179	C₂²×C₅₃	#499	1	1	yes	characteristic
180	C₂²×C₅₃	#501	1	2	yes	—
181	C₄×C₅₃	—	4	8	no	—
182	C₄×C₅₃	—	4	4	no	—
183	C₄×C₅₃	—	8	4	no	—
184	C₄×C₅₃	#384	1	1	yes	characteristic
185	C₄×C₅₃	#385	1	1	yes	characteristic
186	C₄×C₅₃	#389	1	2	yes	—
187	C₄×C₅₃	#395	1	1	yes	characteristic
188	C₄×C₅₃	#398	1	2	yes	—
189	C₄×C₅₃	#399	1	1	yes	characteristic
190	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic
191	C₂×#389	C₂×C₅₃	1	2	yes	—
192	C₂×#340	C₂×C₅₃	1	2	yes	—
193	C₂×#340	C₂×C₅₃	1	1	yes	characteristic
194	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
195	C₂×C₅₃	#1444	1	1	yes	characteristic
196	C₂×C₅₃	#1864	1	2	yes	—
197	C₂×C₅₃	#1865	1	2	yes	—
198	C₂×C₅₃	#1941	1	1	yes	characteristic
199	C₂×C₅₃	#1943	1	1	yes	characteristic
200	C₂×C₅₃	#2000	1	2	yes	—
201	C₂×C₅₃	#2058	1	2	yes	—
202	C₂×C₅₃	#2059	1	2	yes	—
203	C₂×C₅₃	#2088	1	2	yes	—
204	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
205	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
206	C₂²×C₄⋊C₄	—	2	2	no	—
207	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
208	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
209	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×#106	—	2	2	no	—
211	C₂×#106	—	2	4	no	—
212	#389	—	2	4	no	—
213	#389	—	2	4	no	—
214	C₂×#95	—	2	2	no	—
215	#340	—	2	4	no	—
216	#340	C₄×C₅₃	1	4	yes	complemented
217	#336	—	2	2	no	—
218	C₅₃	#11327	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³×C₄	—	2	2	no	—
220	C₂³×C₄	—	2	2	no	—
221	C₂³×C₄	D₄×C₅₃	1	2	yes	—
222	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
223	C₂²×C₈	—	2	2	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	2	no	—
227	C₂²×C₈	—	2	2	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
230	C₂²×C₈	D₄×C₅₃	1	2	yes	—
231	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
232	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	8	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	4	4	no	—
243	C₂×C₄⋊C₄	—	4	4	no	—
244	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
245	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
246	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
247	C₂×C₄⋊C₄	Q₈×C₅₃	1	4	yes	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	—	2	2	no	—
251	C₂×C₄²	—	2	1	no	—
252	#106	—	2	8	no	—
253	#95	—	2	8	no	—
254	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
276	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
279	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
280	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
282	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	#41×C₅₃	1	2	yes	—
284	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	complemented
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	4	16	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	complemented
305	C₄²	—	4	1	no	—
306	C₄²	—	4	2	no	—
307	C₄²	—	4	1	no	—
308	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
309	C₂³	#120×C₅₃	1	2	yes	—
310	C₂³	#121×C₅₃	1	1	yes	characteristic
311	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
312	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
313	C₂³	C₂×SD₁₆×C₅₃	1	2	yes	—
314	C₂³	#135×C₅₃	1	2	yes	—
315	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
316	C₂³	#137×C₅₃	1	2	yes	—
317	C₂×C₄	—	2	8	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
340	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
341	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
342	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
343	C₂×C₄	#121×C₅₃	1	2	yes	—
344	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
345	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
346	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
347	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
348	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
349	C₂×C₄	#123×C₅₃	1	2	yes	—
350	C₂×C₄	#125×C₅₃	1	2	yes	—
351	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
352	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
353	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
355	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
356	C₈	—	2	4	no	—
357	C₈	—	4	2	no	—
358	C₂²	C₅₃×#385	1	2	yes	—
359	C₂²	C₅₃×#389	1	1	yes	characteristic
360	C₂²	C₅₃×#392	1	1	yes	characteristic
361	C₂²	C₅₃×#395	1	2	yes	—
362	C₂²	C₅₃×#398	1	1	yes	characteristic
363	C₂²	C₄×SD₁₆×C₅₃	1	2	yes	—
364	C₂²	C₅₃×#441	1	1	yes	characteristic
365	C₂²	C₅₃×#442	1	1	yes	characteristic
366	C₂²	C₅₃×#448	1	2	yes	—
367	C₂²	C₅₃×#449	1	2	yes	—
368	C₂²	C₅₃×#450	1	2	yes	—
369	C₂²	C₅₃×#452	1	2	yes	—
370	C₂²	C₅₃×#465	1	2	yes	—
371	C₂²	C₅₃×#468	1	2	yes	—
372	C₂²	C₅₃×#475	1	2	yes	—
373	C₂²	C₅₃×#476	1	2	yes	—
374	C₂²	C₅₃×#478	1	2	yes	—
375	C₂²	C₅₃×#479	1	2	yes	—
376	C₂²	C₅₃×#498	1	1	yes	characteristic
377	C₂²	C₅₃×#499	1	1	yes	characteristic
378	C₂²	C₅₃×#501	1	2	yes	—
379	C₄	—	4	4	no	—
380	C₄	—	4	8	no	—
381	C₄	—	8	4	no	—
382	C₄	C₅₃×#384	1	1	yes	characteristic
383	C₄	C₅₃×#385	1	1	yes	characteristic
384	C₄	C₅₃×#389	1	2	yes	—
385	C₄	C₅₃×#395	1	1	yes	characteristic
386	C₄	C₅₃×#398	1	2	yes	—
387	C₄	C₅₃×#399	1	1	yes	characteristic
388	C₂	C₅₃×#1444	1	1	yes	characteristic
389	C₂	C₅₃×#1864	1	2	yes	—
390	C₂	C₅₃×#1865	1	2	yes	—
391	C₂	C₅₃×#1941	1	1	yes	characteristic
392	C₂	C₅₃×#1943	1	1	yes	characteristic
393	C₂	C₅₃×#2000	1	2	yes	—
394	C₂	C₅₃×#2058	1	2	yes	—
395	C₂	C₅₃×#2059	1	2	yes	—
396	C₂	C₅₃×#2088	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
18. C₅₃×#340	C₄	semidirect
96. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
105. C₄⋊C₄×C₅₃	C₄⋊C₄	semidirect
159. #11327	C₅₃	direct (N × H)
216. #340	C₄×C₅₃	semidirect
218. C₅₃	#11327	direct (N × H)
295. C₂×C₈	C₄⋊C₄×C₅₃	semidirect
304. C₄⋊C₄	C₄⋊C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#106×C₅₃	2
C₂×C₅₃×#389	2
C₂×C₅₃×#340	2
C₂×C₅₃×#336	2
#11327	53

Subgroups — C53×#11327 #27218839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11327 #27218839