Subgroups — C₅₃×#10794 #27218306

Order 13568.

1164 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#421	C₂	1	2	yes	maximal
3	C₅₃×#1742	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#568	—	2	4	no	—
8	C₅₃×#568	—	2	4	no	—
9	C₂×#131×C₅₃	—	2	2	no	—
10	C₂×#131×C₅₃	C₄	1	4	yes	complemented
11	C₂²×M₁₆×C₅₃	—	2	2	no	—
12	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#421	—	2	4	no	—
17	C₅₃×#421	—	2	4	no	—
18	C₂×#104×C₅₃	C₄	1	8	yes	complemented
19	C₅₃×#343	—	2	4	no	—
20	C₅₃×#338	—	2	2	no	—
21	C₅₃×H(3,ℤ/4)	—	2	2	no	—
22	C₂×#41×C₅₃	—	2	1	no	—
23	C₂×#41×C₅₃	—	2	2	no	—
24	C₂×#41×C₅₃	—	2	2	no	—
25	C₂×#41×C₅₃	—	2	1	no	—
26	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×#41×C₅₃	D₄	1	4	yes	—
29	C₂×#41×C₅₃	D₄	1	2	yes	—
30	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×D₄×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	—	2	1	no	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	#131×C₅₃	—	2	16	no	—
36	#131×C₅₃	—	2	8	no	—
37	#131×C₅₃	—	2	8	no	—
38	C₂×M₁₆×C₅₃	—	2	16	no	—
39	C₂×M₁₆×C₅₃	—	2	4	no	—
40	C₂×M₁₆×C₅₃	—	2	1	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
52	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
53	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
54	C₂²×C₈×C₅₃	—	2	4	no	—
55	C₂²×C₈×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	—	2	2	no	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	#117×C₅₃	—	2	2	no	—
59	#117×C₅₃	—	2	2	no	—
60	#117×C₅₃	—	2	2	no	—
61	#117×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
65	#104×C₅₃	—	2	16	no	—
66	C₂⁴×C₅₃	—	2	1	no	—
67	C₂⁴×C₅₃	#41	1	1	yes	characteristic
68	#41×C₅₃	—	2	2	no	—
69	#41×C₅₃	—	2	2	no	—
70	#41×C₅₃	—	2	4	no	—
71	#41×C₅₃	—	2	4	no	—
72	#41×C₅₃	—	2	4	no	—
73	#41×C₅₃	—	2	4	no	—
74	#41×C₅₃	—	2	4	no	—
75	#41×C₅₃	C₄²	1	4	yes	—
76	#41×C₅₃	C₂²⋊C₄	1	8	yes	—
77	#41×C₅₃	C₂²⋊C₄	1	4	yes	—
78	C₂×Q₈×C₅₃	—	2	1	no	—
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
82	C₂×D₄×C₅₃	—	2	2	no	—
83	C₂×D₄×C₅₃	—	2	4	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	4	no	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	1	no	—
88	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	4	no	—
93	C₂²×C₄×C₅₃	—	2	4	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	4	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	#41	1	2	yes	—
105	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
106	M₁₆×C₅₃	—	2	16	no	—
107	M₁₆×C₅₃	—	2	2	no	—
108	M₁₆×C₅₃	—	2	4	no	—
109	M₁₆×C₅₃	—	2	4	no	—
110	M₁₆×C₅₃	—	2	4	no	—
111	M₁₆×C₅₃	—	2	4	no	—
112	M₁₆×C₅₃	—	2	4	no	—
113	M₁₆×C₅₃	—	4	2	no	—
114	M₁₆×C₅₃	—	4	2	no	—
115	M₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
116	C₂×C₈×C₅₃	—	2	16	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	4	no	—
121	C₂×C₈×C₅₃	—	2	4	no	—
122	C₂×C₈×C₅₃	—	4	2	no	—
123	C₂×C₈×C₅₃	—	4	2	no	—
124	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
125	C₄⋊C₄×C₅₃	—	4	2	no	—
126	C₄⋊C₄×C₅₃	—	4	2	no	—
127	C₂²⋊C₄×C₅₃	—	4	2	no	—
128	C₂²⋊C₄×C₅₃	—	4	2	no	—
129	C₄²×C₅₃	—	2	4	no	—
130	C₄²×C₅₃	—	2	4	no	—
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	—	2	4	no	—
133	C₂³×C₅₃	—	8	1	no	—
134	C₂³×C₅₃	—	8	2	no	—
135	C₂³×C₅₃	#117	1	1	yes	characteristic
136	C₂³×C₅₃	C₄×D₄	1	2	yes	—
137	C₂³×C₅₃	#121	1	1	yes	characteristic
138	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂³×C₅₃	#123	1	2	yes	—
140	Q₈×C₅₃	—	2	2	no	—
141	Q₈×C₅₃	—	2	4	no	—
142	Q₈×C₅₃	#95	1	4	yes	—
143	D₄×C₅₃	—	2	4	no	—
144	D₄×C₅₃	—	2	4	no	—
145	D₄×C₅₃	—	2	8	no	—
146	D₄×C₅₃	—	2	2	no	—
147	D₄×C₅₃	—	2	4	no	—
148	D₄×C₅₃	#95	1	4	yes	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	4	no	—
151	C₂×C₄×C₅₃	—	2	4	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	2	8	no	—
154	C₂×C₄×C₅₃	—	4	4	no	—
155	C₂×C₄×C₅₃	—	8	4	no	—
156	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₅₃	#120	1	2	yes	—
168	C₂×C₄×C₅₃	#120	1	2	yes	—
169	C₂×C₄×C₅₃	#121	1	2	yes	—
170	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
172	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₄×C₅₃	#123	1	2	yes	—
174	C₈×C₅₃	—	2	4	no	—
175	C₈×C₅₃	—	2	4	no	—
176	C₈×C₅₃	—	4	4	no	—
177	#10794	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₅₃	—	2	4	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	4	1	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	—	4	2	no	—
184	C₂²×C₅₃	—	4	4	no	—
185	C₂²×C₅₃	—	8	4	no	—
186	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
187	C₂²×C₅₃	#380	1	1	yes	characteristic
188	C₂²×C₅₃	#381	1	1	yes	characteristic
189	C₂²×C₅₃	#385	1	2	yes	—
190	C₂²×C₅₃	#386	1	2	yes	—
191	C₄×C₅₃	—	2	4	no	—
192	C₄×C₅₃	—	8	4	no	—
193	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
194	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
195	C₄×C₅₃	#379	1	1	yes	characteristic
196	C₄×C₅₃	#380	1	1	yes	characteristic
197	C₄×C₅₃	#385	1	2	yes	—
198	C₄×C₅₃	#386	1	2	yes	—
199	C₂×#568	C₂×C₅₃	1	1	yes	characteristic
200	C₂×#421	C₂×C₅₃	1	2	yes	—
201	#1742	C₂×C₅₃	1	2	yes	complemented
202	C₂×#338	C₂×C₅₃	1	1	yes	characteristic, complemented
203	C₂×H(3,ℤ/4)	C₂×C₅₃	1	1	yes	characteristic, complemented
204	C₂×C₅₃	—	2	2	no	—
205	C₂×C₅₃	—	2	1	no	—
206	C₂×C₅₃	—	2	2	no	—
207	C₂×C₅₃	—	2	1	no	—
208	C₂×C₅₃	—	2	1	no	—
209	C₂×C₅₃	—	4	4	no	—
210	C₂×C₅₃	C₂²⋊#95	1	1	yes	characteristic
211	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
212	#568	—	2	4	no	—
213	#568	—	2	4	no	—
214	C₂×#131	—	2	2	no	—
215	C₂×#131	C₄×C₅₃	1	4	yes	complemented
216	C₂²×M₁₆	—	2	2	no	—
217	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
218	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
219	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
220	C₂×#117	C₂²×C₅₃	1	2	yes	—
221	#421	—	2	4	no	—
222	#421	—	2	4	no	—
223	C₂×#104	C₄×C₅₃	1	8	yes	complemented
224	#343	—	2	4	no	—
225	#338	—	2	2	no	—
226	H(3,ℤ/4)	—	2	2	no	—
227	C₅₃	#10794	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×#41	—	2	1	no	—
229	C₂×#41	—	2	2	no	—
230	C₂×#41	—	2	2	no	—
231	C₂×#41	—	2	1	no	—
232	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
233	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
234	C₂×#41	D₄×C₅₃	1	4	yes	—
235	C₂×#41	D₄×C₅₃	1	2	yes	—
236	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
237	C₂²×D₄	—	2	1	no	—
238	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
241	#131	—	2	16	no	—
242	#131	—	2	8	no	—
243	#131	—	2	8	no	—
244	C₂×M₁₆	—	2	16	no	—
245	C₂×M₁₆	—	2	1	no	—
246	C₂×M₁₆	—	2	2	no	—
247	C₂×M₁₆	—	2	2	no	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂×M₁₆	—	2	4	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	2	2	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
256	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
257	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
258	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
259	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	—	2	4	no	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
264	#117	—	2	2	no	—
265	#117	—	2	2	no	—
266	#117	—	2	2	no	—
267	#117	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
271	#104	—	2	16	no	—
272	C₂⁴	—	2	1	no	—
273	C₂⁴	#41×C₅₃	1	1	yes	characteristic
274	#41	—	2	2	no	—
275	#41	—	2	2	no	—
276	#41	—	2	4	no	—
277	#41	—	2	4	no	—
278	#41	—	2	4	no	—
279	#41	—	2	4	no	—
280	#41	—	2	4	no	—
281	#41	C₄²×C₅₃	1	4	yes	—
282	#41	C₂²⋊C₄×C₅₃	1	8	yes	—
283	#41	C₂²⋊C₄×C₅₃	1	4	yes	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
287	C₂×Q₈	C₄⋊C₄×C₅₃	1	4	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	4	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	4	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
295	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
305	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
309	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
310	C₂²×C₄	#41×C₅₃	1	2	yes	—
311	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
312	M₁₆	—	2	16	no	—
313	M₁₆	—	2	2	no	—
314	M₁₆	—	2	4	no	—
315	M₁₆	—	2	4	no	—
316	M₁₆	—	2	4	no	—
317	M₁₆	—	2	4	no	—
318	M₁₆	—	2	4	no	—
319	M₁₆	—	4	2	no	—
320	M₁₆	—	4	2	no	—
321	M₁₆	C₂²⋊C₄×C₅₃	1	4	yes	—
322	C₂×C₈	—	2	16	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	4	no	—
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	4	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₂²⋊C₄	—	4	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₄²	—	2	4	no	—
336	C₄²	—	2	4	no	—
337	C₂³	—	2	2	no	—
338	C₂³	—	2	4	no	—
339	C₂³	—	8	1	no	—
340	C₂³	—	8	2	no	—
341	C₂³	#117×C₅₃	1	1	yes	characteristic
342	C₂³	C₄×D₄×C₅₃	1	2	yes	—
343	C₂³	#121×C₅₃	1	1	yes	characteristic
344	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
345	C₂³	#123×C₅₃	1	2	yes	—
346	Q₈	—	2	2	no	—
347	Q₈	—	2	4	no	—
348	Q₈	#95×C₅₃	1	4	yes	—
349	D₄	—	2	4	no	—
350	D₄	—	2	4	no	—
351	D₄	—	2	8	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	4	no	—
354	D₄	#95×C₅₃	1	4	yes	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	8	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	—	8	4	no	—
362	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
364	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
365	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
367	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
369	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
370	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
371	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
372	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
373	C₂×C₄	#120×C₅₃	1	2	yes	—
374	C₂×C₄	#120×C₅₃	1	2	yes	—
375	C₂×C₄	#121×C₅₃	1	2	yes	—
376	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
378	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
379	C₂×C₄	#123×C₅₃	1	2	yes	—
380	C₈	—	2	4	no	—
381	C₈	—	2	4	no	—
382	C₈	—	4	4	no	—
383	C₂²	—	2	4	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	4	no	—
390	C₂²	—	8	4	no	—
391	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
392	C₂²	C₅₃×#380	1	1	yes	characteristic
393	C₂²	C₅₃×#381	1	1	yes	characteristic
394	C₂²	C₅₃×#385	1	2	yes	—
395	C₂²	C₅₃×#386	1	2	yes	—
396	C₄	—	2	4	no	—
397	C₄	—	8	4	no	—
398	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
399	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
400	C₄	C₅₃×#379	1	1	yes	characteristic
401	C₄	C₅₃×#380	1	1	yes	characteristic
402	C₄	C₅₃×#385	1	2	yes	—
403	C₄	C₅₃×#386	1	2	yes	—
404	C₂	—	2	2	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	2	no	—
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	—	4	4	no	—
410	C₂	C₅₃×C₂²⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂×C₅₃	C₂²⋊#95
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1742	C₂	semidirect
4. C₂×C₅₃×#338	C₂	semidirect
5. C₂×C₅₃×H(3,ℤ/4)	C₂	semidirect
10. C₂×#131×C₅₃	C₄	semidirect
18. C₂×#104×C₅₃	C₄	semidirect
177. #10794	C₅₃	direct (N × H)
201. #1742	C₂×C₅₃	semidirect
202. C₂×#338	C₂×C₅₃	semidirect
203. C₂×H(3,ℤ/4)	C₂×C₅₃	semidirect
215. C₂×#131	C₄×C₅₃	semidirect
223. C₂×#104	C₄×C₅₃	semidirect
227. C₅₃	#10794	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#568	2
C₂×C₅₃×#421	2
C₅₃×#1742	2
C₂×C₅₃×#338	2
C₂×C₅₃×H(3,ℤ/4)	2
#10794	53

Subgroups — C53×#10794 #27218306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10794 #27218306