Subgroups — C₅₃×#17811 #27225323

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2989	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1574	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1573	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1572	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1570	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#446	—	2	1	no	—
14	C₈×Q₈×C₅₃	—	2	1	no	—
15	C₅₃×#436	—	2	1	no	—
16	C₅₃×#435	—	2	2	no	—
17	C₈×D₄×C₅₃	—	2	2	no	—
18	C₈×D₄×C₅₃	—	2	1	no	—
19	C₅₃×#433	C₄	1	4	yes	—
20	C₅₃×#432	C₂²	1	2	yes	—
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₅₃×#431	C₂²	1	2	yes	—
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₅₃×#431	C₂²	1	2	yes	—
25	C₅₃×#431	C₂²	1	2	yes	—
26	C₅₃×#424	C₄	1	2	yes	—
27	C₅₃×#423	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#408	C₄	1	2	yes	—
32	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#335	C₂²	1	1	yes	characteristic
35	C₅₃×#335	C₂²	1	1	yes	characteristic
36	C₅₃×#335	C₂²	1	2	yes	—
37	C₅₃×#335	C₂²	1	2	yes	—
38	C₅₃×#334	C₂²	1	2	yes	—
39	C₅₃×#334	C₂²	1	2	yes	—
40	C₅₃×#334	C₂²	1	1	yes	characteristic
41	C₅₃×#334	C₂²	1	1	yes	characteristic
42	C₅₃×#334	C₂²	1	2	yes	—
43	C₅₃×#334	C₂²	1	2	yes	—
44	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
53	C₄×D₄×C₅₃	D₄	1	2	yes	—
54	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
55	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
56	C₄×D₄×C₅₃	Q₈	1	2	yes	—
57	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
58	#117×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	—	2	2	no	—
63	C₄⋊C₈×C₅₃	—	2	1	no	—
64	C₄⋊C₈×C₅₃	—	2	2	no	—
65	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
66	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
67	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
68	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
70	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²⋊C₈×C₅₃	—	2	1	no	—
74	C₂²⋊C₈×C₅₃	—	2	2	no	—
75	C₂²⋊C₈×C₅₃	—	2	1	no	—
76	C₂²⋊C₈×C₅₃	—	2	2	no	—
77	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
78	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
79	#97×C₅₃	—	2	2	no	—
80	#97×C₅₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₅₃	—	2	1	no	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	—	2	1	no	—
84	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	2	yes	—
88	#41×C₅₃	—	2	4	no	—
89	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
98	M₁₆×C₅₃	—	2	4	no	—
99	M₁₆×C₅₃	—	8	2	no	—
100	C₂×C₈×C₅₃	—	2	4	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	8	1	no	—
108	C₂×C₈×C₅₃	—	8	2	no	—
109	C₂×C₈×C₅₃	—	8	1	no	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₂×D₄	1	2	yes	—
124	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
125	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
126	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈×C₅₃	—	2	2	no	—
130	D₄×C₅₃	—	2	4	no	—
131	D₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	2	no	—
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
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	#131	1	4	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	2	no	—
150	C₈×C₅₃	—	2	2	no	—
151	C₈×C₅₃	—	2	4	no	—
152	C₈×C₅₃	—	4	1	no	—
153	C₈×C₅₃	—	4	1	no	—
154	C₈×C₅₃	—	4	1	no	—
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	—	4	2	no	—
157	C₈×C₅₃	—	4	1	no	—
158	#17811	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	2	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	#424	1	2	yes	—
165	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	#424	1	2	yes	—
173	C₄×C₅₃	C₂×#131	1	2	yes	—
174	C₄×C₅₃	#577	1	1	yes	characteristic
175	C₄×C₅₃	#577	1	1	yes	characteristic
176	#2989	C₂×C₅₃	1	1	yes	characteristic
177	#2988	C₂×C₅₃	1	1	yes	characteristic
178	#2922	C₂×C₅₃	1	1	yes	characteristic
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	#1570	C₂×C₅₃	1	2	yes	complemented
183	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
184	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
185	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
186	C₂×#334	C₂×C₅₃	1	1	yes	characteristic, complemented
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	—	4	2	no	—
189	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
190	C₂×C₅₃	#2912	1	1	yes	characteristic
191	C₂×C₅₃	#2916	1	1	yes	characteristic
192	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
193	#446	—	2	1	no	—
194	C₈×Q₈	—	2	1	no	—
195	#436	—	2	1	no	—
196	#435	—	2	2	no	—
197	C₈×D₄	—	2	2	no	—
198	C₈×D₄	—	2	1	no	—
199	#433	C₄×C₅₃	1	4	yes	—
200	#432	C₂²×C₅₃	1	2	yes	—
201	#431	C₂²×C₅₃	1	1	yes	characteristic
202	#431	C₂²×C₅₃	1	1	yes	characteristic
203	#431	C₂²×C₅₃	1	2	yes	—
204	#431	C₂²×C₅₃	1	2	yes	—
205	#431	C₂²×C₅₃	1	2	yes	—
206	#424	C₄×C₅₃	1	2	yes	—
207	#423	C₂²×C₅₃	1	2	yes	—
208	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
209	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
211	#408	C₄×C₅₃	1	2	yes	—
212	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
214	#335	C₂²×C₅₃	1	1	yes	characteristic
215	#335	C₂²×C₅₃	1	1	yes	characteristic
216	#335	C₂²×C₅₃	1	2	yes	—
217	#335	C₂²×C₅₃	1	2	yes	—
218	#334	C₂²×C₅₃	1	2	yes	—
219	#334	C₂²×C₅₃	1	2	yes	—
220	#334	C₂²×C₅₃	1	1	yes	characteristic
221	#334	C₂²×C₅₃	1	1	yes	characteristic
222	#334	C₂²×C₅₃	1	2	yes	—
223	#334	C₂²×C₅₃	1	2	yes	—
224	C₅₃	#17811	1	1	yes	characteristic, direct product factor, Sylow
225	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
232	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
233	C₄×Q₈	Q₈×C₅₃	1	1	yes	characteristic
234	C₄×D₄	D₄×C₅₃	1	2	yes	—
235	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
236	C₄×D₄	Q₈×C₅₃	1	2	yes	—
237	C₄×D₄	Q₈×C₅₃	1	1	yes	characteristic
238	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
239	#117	C₂×C₄×C₅₃	1	2	yes	—
240	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
247	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
248	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
249	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
250	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
251	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
252	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
253	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	2	1	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
259	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
260	#97	—	2	2	no	—
261	#97	C₂×C₄×C₅₃	1	2	yes	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	2	1	no	—
265	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
266	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
268	C₄×C₈	C₂³×C₅₃	1	2	yes	—
269	#41	—	2	4	no	—
270	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
271	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
272	C₂×D₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
274	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
277	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
278	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
279	M₁₆	—	2	4	no	—
280	M₁₆	—	8	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	8	1	no	—
289	C₂×C₈	—	8	2	no	—
290	C₂×C₈	—	8	1	no	—
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	4	yes	—
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	2	yes	—
306	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
307	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
309	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
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	2	yes	—
325	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
326	C₂×C₄	#131×C₅₃	1	4	yes	—
327	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	4	no	—
333	C₈	—	4	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	C₅₃×#424	1	2	yes	—
345	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	C₅₃×#424	1	2	yes	—
353	C₄	C₂×#131×C₅₃	1	2	yes	—
354	C₄	C₅₃×#577	1	1	yes	characteristic
355	C₄	C₅₃×#577	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×C₅₃×#424	1	1	yes	characteristic
359	C₂	C₅₃×#2912	1	1	yes	characteristic
360	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	C₂×C₄
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
4. C₅₃×#1574	C₂	semidirect
5. C₅₃×#1573	C₂	semidirect
6. C₅₃×#1572	C₂	semidirect
7. C₅₃×#1570	C₂	semidirect
8. C₅₃×#1565	C₂	semidirect
9. C₅₃×#1565	C₂	semidirect
10. C₅₃×#1564	C₂	semidirect
11. C₂×C₅₃×#334	C₂	semidirect
158. #17811	C₅₃	direct (N × H)
179. #1574	C₂×C₅₃	semidirect
180. #1573	C₂×C₅₃	semidirect
181. #1572	C₂×C₅₃	semidirect
182. #1570	C₂×C₅₃	semidirect
183. #1565	C₂×C₅₃	semidirect
184. #1565	C₂×C₅₃	semidirect
185. #1564	C₂×C₅₃	semidirect
186. C₂×#334	C₂×C₅₃	semidirect
224. C₅₃	#17811	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×#2989	2
C₅₃×#2988	2
C₅₃×#2922	2
C₅₃×#1574	2
C₅₃×#1573	2
C₅₃×#1572	2
C₅₃×#1570	2
C₅₃×#1565	2
C₅₃×#1564	2
C₂×C₅₃×#334	2
#17811	53

Subgroups — C53×#17811 #27225323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17811 #27225323