Subgroups — C₅₃×#17831 #27225343

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2965	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2925	C₂	1	2	yes	maximal
3	C₅₃×#1574	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1573	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1572	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1569	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1565	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#446	C₄	1	1	yes	characteristic
12	C₅₃×#446	C₄	1	1	yes	characteristic
13	C₅₃×#436	C₄	1	1	yes	characteristic
14	C₅₃×#436	C₄	1	1	yes	characteristic
15	C₅₃×#435	C₄	1	2	yes	—
16	C₅₃×#435	C₄	1	2	yes	—
17	C₅₃×#433	—	2	2	no	—
18	C₅₃×#433	—	2	2	no	—
19	C₅₃×#432	C₂²	1	2	yes	—
20	C₅₃×#432	C₂²	1	2	yes	—
21	C₅₃×#431	C₂²	1	2	yes	—
22	C₅₃×#431	C₂²	1	2	yes	—
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₅₃×#431	C₂²	1	2	yes	—
25	C₅₃×#423	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#408	—	2	2	no	—
28	C₅₃×#405	—	2	2	no	—
29	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#335	C₂²	1	2	yes	—
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#335	C₂²	1	2	yes	—
34	C₅₃×#335	C₂²	1	2	yes	—
35	C₅₃×#334	C₂²	1	1	yes	characteristic
36	C₅₃×#334	C₂²	1	1	yes	characteristic
37	C₅₃×#334	C₂²	1	2	yes	—
38	C₅₃×#334	C₂²	1	1	yes	characteristic
39	C₅₃×#334	C₂²	1	1	yes	characteristic
40	C₅₃×#334	C₂²	1	2	yes	—
41	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
50	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄×C₅₃	Q₈	1	2	yes	—
53	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
54	#117×C₅₃	D₄	1	2	yes	—
55	#117×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
58	C₄⋊C₈×C₅₃	—	2	2	no	—
59	C₄⋊C₈×C₅₃	—	2	2	no	—
60	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
65	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
66	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
67	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
68	C₂²⋊C₈×C₅₃	—	2	2	no	—
69	C₂²⋊C₈×C₅₃	—	2	2	no	—
70	C₂²⋊C₈×C₅₃	—	2	2	no	—
71	C₂²⋊C₈×C₅₃	—	2	2	no	—
72	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
74	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
75	#97×C₅₃	—	2	2	no	—
76	#97×C₅₃	—	2	2	no	—
77	#97×C₅₃	C₂×C₄	1	2	yes	—
78	#97×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	—	2	2	no	—
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	#41×C₅₃	—	4	2	no	—
87	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
96	M₁₆×C₅₃	—	2	4	no	—
97	M₁₆×C₅₃	—	4	2	no	—
98	M₁₆×C₅₃	—	4	4	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	4	1	no	—
105	C₂×C₈×C₅₃	—	4	4	no	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
113	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
115	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
117	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
119	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
121	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅₃	C₂×D₄	1	2	yes	—
123	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
124	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
125	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	Q₈×C₅₃	—	4	1	no	—
129	D₄×C₅₃	—	4	1	no	—
130	D₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₅₃	—	2	1	no	—
150	C₈×C₅₃	—	2	2	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	4	2	no	—
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	—	4	2	no	—
157	C₈×C₅₃	—	4	2	no	—
158	C₈×C₅₃	#131	1	2	yes	—
159	#17831	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	2	no	—
166	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	2	no	—
174	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
175	C₄×C₅₃	#568	1	1	yes	characteristic
176	C₄×C₅₃	#575	1	2	yes	—
177	#2965	C₂×C₅₃	1	1	yes	characteristic
178	#2925	C₂×C₅₃	1	2	yes	—
179	#1574	C₂×C₅₃	1	2	yes	complemented
180	#1573	C₂×C₅₃	1	2	yes	complemented
181	#1572	C₂×C₅₃	1	2	yes	complemented
182	#1569	C₂×C₅₃	1	2	yes	complemented
183	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
184	#1565	C₂×C₅₃	1	2	yes	complemented
185	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
186	C₂×C₅₃	—	4	1	no	—
187	C₂×C₅₃	—	4	2	no	—
188	C₂×C₅₃	#2906	1	1	yes	characteristic
189	C₂×C₅₃	#2911	1	1	yes	characteristic
190	C₂×C₅₃	#2916	1	1	yes	characteristic
191	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
192	#446	C₄×C₅₃	1	1	yes	characteristic
193	#446	C₄×C₅₃	1	1	yes	characteristic
194	#436	C₄×C₅₃	1	1	yes	characteristic
195	#436	C₄×C₅₃	1	1	yes	characteristic
196	#435	C₄×C₅₃	1	2	yes	—
197	#435	C₄×C₅₃	1	2	yes	—
198	#433	—	2	2	no	—
199	#433	—	2	2	no	—
200	#432	C₂²×C₅₃	1	2	yes	—
201	#432	C₂²×C₅₃	1	2	yes	—
202	#431	C₂²×C₅₃	1	2	yes	—
203	#431	C₂²×C₅₃	1	2	yes	—
204	#431	C₂²×C₅₃	1	1	yes	characteristic
205	#431	C₂²×C₅₃	1	2	yes	—
206	#423	C₂²×C₅₃	1	2	yes	—
207	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
208	#408	—	2	2	no	—
209	#405	—	2	2	no	—
210	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
211	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
212	#335	C₂²×C₅₃	1	2	yes	—
213	#335	C₂²×C₅₃	1	2	yes	—
214	#335	C₂²×C₅₃	1	2	yes	—
215	#335	C₂²×C₅₃	1	2	yes	—
216	#334	C₂²×C₅₃	1	1	yes	characteristic
217	#334	C₂²×C₅₃	1	1	yes	characteristic
218	#334	C₂²×C₅₃	1	2	yes	—
219	#334	C₂²×C₅₃	1	1	yes	characteristic
220	#334	C₂²×C₅₃	1	1	yes	characteristic
221	#334	C₂²×C₅₃	1	2	yes	—
222	C₅₃	#17831	1	1	yes	characteristic, direct product factor, Sylow
223	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
226	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
227	C₂²×C₈	—	2	2	no	—
228	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₄×Q₈	Q₈×C₅₃	1	1	yes	characteristic
232	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
233	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
234	C₄×D₄	Q₈×C₅₃	1	1	yes	characteristic
235	C₄×D₄	Q₈×C₅₃	1	2	yes	—
236	#117	D₄×C₅₃	1	2	yes	—
237	#117	D₄×C₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
243	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
244	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
245	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
246	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
247	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
248	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
249	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
250	C₂²⋊C₈	—	2	2	no	—
251	C₂²⋊C₈	—	2	2	no	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
255	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
256	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
257	#97	—	2	2	no	—
258	#97	—	2	2	no	—
259	#97	C₂×C₄×C₅₃	1	2	yes	—
260	#97	C₂×C₄×C₅₃	1	2	yes	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
264	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
265	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
266	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂³×C₅₃	1	2	yes	—
268	#41	—	4	2	no	—
269	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
270	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
271	C₂×D₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
276	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	4	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	4	no	—
288	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
290	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
292	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
293	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
294	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
295	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
296	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
297	C₄⋊C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
298	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
299	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
300	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
301	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
302	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
303	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₅₃	1	2	yes	—
305	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
306	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
307	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
309	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
310	Q₈	—	4	1	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
326	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
328	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	#131×C₅₃	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	2	no	—
347	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
356	C₄	C₅₃×#568	1	1	yes	characteristic
357	C₄	C₅₃×#575	1	2	yes	—
358	C₂	—	4	1	no	—
359	C₂	—	4	2	no	—
360	C₂	C₅₃×#2906	1	1	yes	characteristic
361	C₂	C₅₃×#2911	1	1	yes	characteristic
362	C₂	C₅₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1574	C₂	semidirect
4. C₅₃×#1573	C₂	semidirect
5. C₅₃×#1572	C₂	semidirect
6. C₅₃×#1569	C₂	semidirect
7. C₅₃×#1568	C₂	semidirect
8. C₅₃×#1565	C₂	semidirect
9. C₅₃×#1564	C₂	semidirect
159. #17831	C₅₃	direct (N × H)
179. #1574	C₂×C₅₃	semidirect
180. #1573	C₂×C₅₃	semidirect
181. #1572	C₂×C₅₃	semidirect
182. #1569	C₂×C₅₃	semidirect
183. #1568	C₂×C₅₃	semidirect
184. #1565	C₂×C₅₃	semidirect
185. #1564	C₂×C₅₃	semidirect
222. C₅₃	#17831	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2965	2
C₅₃×#2925	2
C₅₃×#1574	2
C₅₃×#1573	2
C₅₃×#1572	2
C₅₃×#1569	2
C₅₃×#1568	2
C₅₃×#1565	2
C₅₃×#1564	2
#17831	53

Subgroups — C53×#17831 #27225343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17831 #27225343