Subgroups — C₅₃×#7887 #27215399

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1278	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	—	2	1	no	—
6	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#432	—	2	1	no	—
9	C₅₃×#432	C₄	1	2	yes	—
10	C₅₃×#431	—	2	1	no	—
11	C₅₃×#431	C₂²	1	2	yes	—
12	C₅₃×#423	C₄	1	2	yes	—
13	C₅₃×#407	—	2	1	no	—
14	C₄×M₁₆×C₅₃	—	2	1	no	—
15	C₄×M₁₆×C₅₃	—	2	1	no	—
16	C₄×M₁₆×C₅₃	—	2	1	no	—
17	C₄×M₁₆×C₅₃	—	2	1	no	—
18	C₄×M₁₆×C₅₃	—	2	1	no	—
19	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
20	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
21	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
22	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	—	2	1	no	—
24	C₂×#97×C₅₃	—	2	1	no	—
25	C₂×C₄×C₈×C₅₃	—	2	1	no	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₅₃	—	2	8	no	—
31	C₂×M₁₆×C₅₃	—	2	8	no	—
32	C₂×M₁₆×C₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	—	2	2	no	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
53	C₂²⋊C₈×C₅₃	—	2	2	no	—
54	C₂²⋊C₈×C₅₃	—	2	2	no	—
55	#97×C₅₃	—	2	2	no	—
56	#97×C₅₃	—	2	2	no	—
57	#97×C₅₃	—	2	2	no	—
58	#97×C₅₃	—	2	2	no	—
59	#97×C₅₃	—	2	2	no	—
60	C₄×C₈×C₅₃	—	2	2	no	—
61	C₄×C₈×C₅₃	—	2	2	no	—
62	C₄×C₈×C₅₃	—	2	2	no	—
63	C₄×C₈×C₅₃	—	2	2	no	—
64	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
65	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	M₁₆×C₅₃	—	2	8	no	—
88	M₁₆×C₅₃	—	2	8	no	—
89	M₁₆×C₅₃	—	2	8	no	—
90	M₁₆×C₅₃	—	2	4	no	—
91	M₁₆×C₅₃	—	4	4	no	—
92	M₁₆×C₅₃	M₁₆	1	8	yes	complemented
93	C₂×C₈×C₅₃	—	2	8	no	—
94	C₂×C₈×C₅₃	—	2	8	no	—
95	C₂×C₈×C₅₃	—	2	4	no	—
96	C₂×C₈×C₅₃	—	2	4	no	—
97	C₂×C₈×C₅₃	—	2	4	no	—
98	C₄²×C₅₃	C₄²	1	1	yes	characteristic
99	C₄²×C₅₃	C₄²	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₅₃	—	2	2	no	—
115	C₂³×C₅₃	—	2	2	no	—
116	C₂³×C₅₃	—	4	1	no	—
117	C₂³×C₅₃	#95	1	1	yes	characteristic
118	C₂³×C₅₃	#95	1	1	yes	characteristic
119	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#104	1	4	yes	—
143	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
150	C₈×C₅₃	—	2	8	no	—
151	C₈×C₅₃	—	2	8	no	—
152	C₈×C₅₃	—	4	4	no	—
153	#7887	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₅₃	—	2	1	no	—
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	4	4	no	—
162	C₂²×C₅₃	#339	1	2	yes	—
163	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
164	C₂²×C₅₃	#423	1	2	yes	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	#339	1	2	yes	—
173	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
174	C₄×C₅₃	#407	1	2	yes	—
175	C₄×C₅₃	C₂×#104	1	2	yes	—
176	C₄×C₅₃	#430	1	2	yes	—
177	#2920	C₂×C₅₃	1	1	yes	characteristic
178	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
179	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
180	#1278	C₂×C₅₃	1	4	yes	complemented
181	C₂×C₅₃	—	2	2	no	—
182	C₂×C₅₃	—	2	2	no	—
183	C₂×C₅₃	—	4	1	no	—
184	C₂×C₅₃	#1728	1	1	yes	characteristic
185	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
186	C₂×C₅₃	#1738	1	1	yes	characteristic
187	C₂²×M₁₆	—	2	1	no	—
188	C₂²×M₁₆	C₄×C₅₃	1	2	yes	—
189	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
190	#432	—	2	1	no	—
191	#432	C₄×C₅₃	1	2	yes	—
192	#431	—	2	1	no	—
193	#431	C₂²×C₅₃	1	2	yes	—
194	#423	C₄×C₅₃	1	2	yes	—
195	#407	—	2	1	no	—
196	C₄×M₁₆	—	2	1	no	—
197	C₄×M₁₆	—	2	1	no	—
198	C₄×M₁₆	—	2	1	no	—
199	C₄×M₁₆	—	2	1	no	—
200	C₄×M₁₆	—	2	1	no	—
201	C₄×M₁₆	C₄×C₅₃	1	2	yes	—
202	C₄×M₁₆	C₄×C₅₃	1	2	yes	—
203	C₄×M₁₆	C₄×C₅₃	1	2	yes	—
204	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
205	C₂×#97	—	2	1	no	—
206	C₂×#97	—	2	1	no	—
207	C₂×C₄×C₈	—	2	1	no	—
208	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
209	C₅₃	#7887	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
211	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
212	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
213	C₂×M₁₆	—	2	8	no	—
214	C₂×M₁₆	—	2	2	no	—
215	C₂×M₁₆	—	2	2	no	—
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	8	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	—	2	2	no	—
222	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	2	2	no	—
238	#97	—	2	2	no	—
239	#97	—	2	2	no	—
240	#97	—	2	2	no	—
241	#97	—	2	2	no	—
242	#97	—	2	2	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	—	2	2	no	—
247	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
248	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
270	M₁₆	—	2	8	no	—
271	M₁₆	—	2	8	no	—
272	M₁₆	—	2	8	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	4	4	no	—
275	M₁₆	M₁₆×C₅₃	1	8	yes	complemented
276	C₂×C₈	—	2	8	no	—
277	C₂×C₈	—	2	8	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₄²	C₄²×C₅₃	1	1	yes	characteristic
282	C₄²	C₄²×C₅₃	1	1	yes	characteristic
283	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
284	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
285	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
286	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
287	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
288	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
289	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
290	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
291	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
292	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
293	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
294	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
296	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	#95×C₅₃	1	1	yes	characteristic
301	C₂³	#95×C₅₃	1	1	yes	characteristic
302	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
320	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
321	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
322	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
323	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
324	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
325	C₂×C₄	#104×C₅₃	1	4	yes	—
326	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
333	C₈	—	2	8	no	—
334	C₈	—	2	8	no	—
335	C₈	—	4	4	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	4	no	—
344	C₂²	C₅₃×#339	1	2	yes	—
345	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
346	C₂²	C₅₃×#423	1	2	yes	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	C₅₃×#339	1	2	yes	—
355	C₄	C₄×M₁₆×C₅₃	1	2	yes	—
356	C₄	C₅₃×#407	1	2	yes	—
357	C₄	C₂×#104×C₅₃	1	2	yes	—
358	C₄	C₅₃×#430	1	2	yes	—
359	C₂	—	2	2	no	—
360	C₂	—	2	2	no	—
361	C₂	—	4	1	no	—
362	C₂	C₅₃×#1728	1	1	yes	characteristic
363	C₂	C₂×C₅₃×#339	1	1	yes	characteristic
364	C₂	C₅₃×#1738	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1278	C₂	semidirect
92. M₁₆×C₅₃	M₁₆	semidirect
153. #7887	C₅₃	direct (N × H)
180. #1278	C₂×C₅₃	semidirect
209. C₅₃	#7887	direct (N × H)
275. M₁₆	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2920	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1278	2
#7887	53

Subgroups — C53×#7887 #27215399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7887 #27215399