Subgroups — C₅₃×#17952 #27225464

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3101	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2975	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1592	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1590	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1588	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1583	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1572	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1490	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#483	C₄	1	2	yes	—
15	C₅₃×#476	C₄	1	2	yes	—
16	C₅₃×#461	C₄	1	2	yes	—
17	C₅₃×#450	C₄	1	2	yes	—
18	C₅₃×#446	—	2	2	no	—
19	C₅₃×#446	C₂²	1	2	yes	complemented
20	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
21	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
22	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#436	—	2	2	no	—
24	C₅₃×#436	C₂²	1	2	yes	—
25	C₅₃×#436	C₂²	1	2	yes	—
26	C₅₃×#433	—	2	2	no	—
27	C₅₃×#432	C₂²	1	1	yes	characteristic
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#407	—	2	2	no	—
32	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#335	C₂²	1	2	yes	complemented
34	C₅₃×#334	C₂²	1	2	yes	complemented
35	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
36	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
37	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
38	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
39	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	1	no	—
50	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#125×C₅₃	—	2	1	no	—
53	C₂²⋊Q₈×C₅₃	—	2	1	no	—
54	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₄×Q₈×C₅₃	—	2	1	no	—
56	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
57	C₄×D₄×C₅₃	—	2	1	no	—
58	C₄×D₄×C₅₃	—	2	1	no	—
59	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₅₃	C₂³	1	2	yes	—
61	#117×C₅₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#106×C₅₃	C₂×C₄	1	2	yes	—
66	#106×C₅₃	C₂×C₄	1	2	yes	—
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	—	2	2	no	—
69	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
71	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
72	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
73	#103×C₅₃	C₂×C₄	1	2	yes	—
74	#103×C₅₃	C₂×C₄	1	2	yes	—
75	#102×C₅₃	C₂×C₄	1	2	yes	—
76	#102×C₅₃	C₂×C₄	1	2	yes	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	C₂²⋊C₈×C₅₃	—	2	1	no	—
80	C₂²⋊C₈×C₅₃	—	2	1	no	—
81	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
82	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
83	#97×C₅₃	—	2	2	no	—
84	#97×C₅₃	—	2	1	no	—
85	#97×C₅₃	—	2	1	no	—
86	C₄×C₈×C₅₃	—	2	2	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂⁴×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	—	2	1	no	—
96	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	4	2	no	—
99	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅₃	—	2	1	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	8	2	no	—
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
107	SD₁₆×C₅₃	—	4	2	no	—
108	SD₁₆×C₅₃	—	4	2	no	—
109	M₁₆×C₅₃	—	4	2	no	—
110	M₁₆×C₅₃	—	8	2	no	—
111	M₁₆×C₅₃	—	8	2	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₂²⋊C₄×C₅₃	—	2	2	no	—
130	C₂²⋊C₄×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₅₃	#41	1	1	yes	characteristic
135	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	—	4	2	no	—
138	C₂³×C₅₃	—	8	1	no	—
139	C₂³×C₅₃	—	8	1	no	—
140	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₅₃	—	2	2	no	—
142	Q₈×C₅₃	—	2	2	no	—
143	Q₈×C₅₃	—	8	1	no	—
144	D₄×C₅₃	—	2	2	no	—
145	D₄×C₅₃	—	2	2	no	—
146	D₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	8	2	no	—
156	C₂×C₄×C₅₃	—	8	1	no	—
157	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₈×C₅₃	—	2	2	no	—
164	C₈×C₅₃	—	2	2	no	—
165	C₈×C₅₃	—	2	2	no	—
166	C₈×C₅₃	—	2	2	no	—
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	4	1	no	—
169	C₈×C₅₃	—	4	2	no	—
170	C₈×C₅₃	—	4	2	no	—
171	#17952	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂²×C₅₃	—	2	1	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	4	2	no	—
176	C₂²×C₅₃	—	4	2	no	—
177	C₂²×C₅₃	—	4	2	no	—
178	C₂²×C₅₃	—	8	4	no	—
179	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	4	2	no	—
185	C₄×C₅₃	—	4	2	no	—
186	C₄×C₅₃	—	4	2	no	—
187	C₄×C₅₃	#568	1	2	yes	—
188	C₄×C₅₃	#577	1	2	yes	—
189	#3101	C₂×C₅₃	1	1	yes	characteristic
190	#2985	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#2975	C₂×C₅₃	1	1	yes	characteristic
192	#1592	C₂×C₅₃	1	2	yes	complemented
193	#1590	C₂×C₅₃	1	2	yes	complemented
194	#1588	C₂×C₅₃	1	2	yes	complemented
195	#1583	C₂×C₅₃	1	2	yes	complemented
196	#1572	C₂×C₅₃	1	1	yes	characteristic, complemented
197	#1490	C₂×C₅₃	1	1	yes	characteristic
198	#1485	C₂×C₅₃	1	1	yes	characteristic, complemented
199	#1453	C₂×C₅₃	1	1	yes	characteristic, complemented
200	C₂×C₅₃	—	4	1	no	—
201	C₂×C₅₃	—	8	2	no	—
202	C₂×C₅₃	#2930	1	1	yes	characteristic
203	C₂×C₅₃	#2949	1	1	yes	characteristic
204	C₂×C₅₃	#2956	1	1	yes	characteristic
205	#533	C₂²×C₅₃	1	1	yes	characteristic
206	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
207	#483	C₄×C₅₃	1	2	yes	—
208	#476	C₄×C₅₃	1	2	yes	—
209	#461	C₄×C₅₃	1	2	yes	—
210	#450	C₄×C₅₃	1	2	yes	—
211	#446	—	2	2	no	—
212	#446	C₂²×C₅₃	1	2	yes	complemented
213	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
214	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
215	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
216	#436	—	2	2	no	—
217	#436	C₂²×C₅₃	1	2	yes	—
218	#436	C₂²×C₅₃	1	2	yes	—
219	#433	—	2	2	no	—
220	#432	C₂²×C₅₃	1	1	yes	characteristic
221	#431	C₂²×C₅₃	1	1	yes	characteristic
222	#431	C₂²×C₅₃	1	1	yes	characteristic
223	#431	C₂²×C₅₃	1	1	yes	characteristic
224	#407	—	2	2	no	—
225	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
226	#335	C₂²×C₅₃	1	2	yes	complemented
227	#334	C₂²×C₅₃	1	2	yes	complemented
228	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
229	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
230	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
231	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
232	#322	C₂²×C₅₃	1	1	yes	characteristic, complemented
233	#322	C₂²×C₅₃	1	1	yes	characteristic, complemented
234	#322	C₂²×C₅₃	1	1	yes	characteristic, complemented
235	C₈²	C₂²×C₅₃	1	1	yes	characteristic, complemented
236	C₅₃	#17952	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
238	C₂³×C₄	—	2	1	no	—
239	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
240	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
241	C₂×M₁₆	—	2	1	no	—
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
245	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
246	#125	—	2	1	no	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
254	C₄×D₄	C₂³×C₅₃	1	2	yes	—
255	#117	—	2	1	no	—
256	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
257	C₂×C₂²⋊C₄	—	2	1	no	—
258	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
259	#106	C₂×C₄×C₅₃	1	2	yes	—
260	#106	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	2	yes	—
264	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
265	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
266	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
267	#103	C₂×C₄×C₅₃	1	2	yes	—
268	#103	C₂×C₄×C₅₃	1	2	yes	—
269	#102	C₂×C₄×C₅₃	1	2	yes	—
270	#102	C₂×C₄×C₅₃	1	2	yes	—
271	C₂²⋊C₈	—	2	1	no	—
272	C₂²⋊C₈	—	2	1	no	—
273	C₂²⋊C₈	—	2	2	no	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
276	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
277	#97	—	2	2	no	—
278	#97	—	2	1	no	—
279	#97	—	2	1	no	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
283	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
284	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
286	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
287	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
288	C₂⁴	—	2	1	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	—	4	2	no	—
293	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	8	2	no	—
298	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
300	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
301	SD₁₆	—	4	2	no	—
302	SD₁₆	—	4	2	no	—
303	M₁₆	—	4	2	no	—
304	M₁₆	—	8	2	no	—
305	M₁₆	—	8	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
311	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
312	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
313	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
321	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
328	C₄²	#41×C₅₃	1	1	yes	characteristic
329	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
330	C₂³	—	2	2	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	8	1	no	—
333	C₂³	—	8	1	no	—
334	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	8	1	no	—
338	D₄	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	4	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
353	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	2	no	—
364	C₈	—	4	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	8	4	no	—
372	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	2	no	—
380	C₄	C₅₃×#568	1	2	yes	—
381	C₄	C₅₃×#577	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	8	2	no	—
384	C₂	C₅₃×#2930	1	1	yes	characteristic
385	C₂	C₅₃×#2949	1	1	yes	characteristic
386	C₂	C₅₃×#2956	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	C₂×C₄×D₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2985	C₂	semidirect
4. C₅₃×#1592	C₂	semidirect
5. C₅₃×#1590	C₂	semidirect
6. C₅₃×#1588	C₂	semidirect
7. C₅₃×#1583	C₂	semidirect
8. C₅₃×#1572	C₂	semidirect
10. C₅₃×#1485	C₂	semidirect
11. C₅₃×#1453	C₂	semidirect
19. C₅₃×#446	C₂²	semidirect
20. C₈×Q₈×C₅₃	C₂²	semidirect
33. C₅₃×#335	C₂²	semidirect
34. C₅₃×#334	C₂²	semidirect
35. C₅₃×Q₈⋊C₈	C₂²	semidirect
36. C₅₃×Q₈⋊C₈	C₂²	semidirect
39. C₅₃×#322	C₂²	semidirect
40. C₅₃×#322	C₂²	semidirect
41. C₅₃×#322	C₂²	semidirect
42. C₈²×C₅₃	C₂²	semidirect
171. #17952	C₅₃	direct (N × H)
190. #2985	C₂×C₅₃	semidirect
192. #1592	C₂×C₅₃	semidirect
193. #1590	C₂×C₅₃	semidirect
194. #1588	C₂×C₅₃	semidirect
195. #1583	C₂×C₅₃	semidirect
196. #1572	C₂×C₅₃	semidirect
198. #1485	C₂×C₅₃	semidirect
199. #1453	C₂×C₅₃	semidirect
212. #446	C₂²×C₅₃	semidirect
213. C₈×Q₈	C₂²×C₅₃	semidirect
226. #335	C₂²×C₅₃	semidirect
227. #334	C₂²×C₅₃	semidirect
228. Q₈⋊C₈	C₂²×C₅₃	semidirect
229. Q₈⋊C₈	C₂²×C₅₃	semidirect
232. #322	C₂²×C₅₃	semidirect
233. #322	C₂²×C₅₃	semidirect
234. #322	C₂²×C₅₃	semidirect
235. C₈²	C₂²×C₅₃	semidirect
236. C₅₃	#17952	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×#3101	2
C₅₃×#2985	2
C₅₃×#2975	2
C₅₃×#1592	2
C₅₃×#1590	2
C₅₃×#1588	2
C₅₃×#1583	2
C₅₃×#1572	2
C₅₃×#1490	2
C₅₃×#1485	2
C₅₃×#1453	2
#17952	53

Subgroups — C53×#17952 #27225464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17952 #27225464