Subgroups — C₅₃×#19644 #27227156

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3210	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3207	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3182	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1719	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1717	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1711	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1708	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#533	C₂²	1	1	yes	characteristic
17	C₅₃×#532	C₂²	1	1	yes	characteristic
18	C₅₃×#532	C₂²	1	1	yes	characteristic
19	C₅₃×#532	C₂²	1	1	yes	characteristic
20	C₅₃×#500	C₂²	1	1	yes	characteristic
21	C₅₃×#500	C₂²	1	1	yes	characteristic
22	C₅₃×#498	C₂²	1	1	yes	characteristic
23	C₅₃×#494	C₂²	1	1	yes	characteristic
24	C₅₃×#493	C₂²	1	1	yes	characteristic
25	C₅₃×#493	C₂²	1	1	yes	characteristic
26	C₅₃×#493	C₂²	1	1	yes	characteristic
27	C₅₃×#492	C₂²	1	1	yes	characteristic
28	C₅₃×#492	C₂²	1	1	yes	characteristic
29	C₅₃×#492	C₂²	1	1	yes	characteristic
30	C₅₃×#488	—	2	2	no	—
31	C₅₃×#487	C₂²	1	1	yes	characteristic
32	C₅₃×#487	C₂²	1	1	yes	characteristic
33	C₅₃×#487	C₂²	1	1	yes	characteristic
34	C₅₃×#487	C₂²	1	1	yes	characteristic
35	C₅₃×#487	C₂²	1	1	yes	characteristic
36	C₅₃×#486	C₂²	1	1	yes	characteristic
37	C₅₃×#486	C₂²	1	1	yes	characteristic
38	C₅₃×#486	C₂²	1	1	yes	characteristic
39	C₅₃×#486	C₂²	1	1	yes	characteristic
40	C₅₃×#486	C₂²	1	1	yes	characteristic
41	C₅₃×#486	C₂²	1	1	yes	characteristic
42	C₅₃×#486	C₂²	1	1	yes	characteristic
43	C₅₃×#486	C₂²	1	1	yes	characteristic
44	C₅₃×#486	C₂²	1	1	yes	characteristic
45	C₅₃×#469	—	2	2	no	—
46	C₅₃×#468	—	2	2	no	—
47	C₅₃×#466	—	2	2	no	—
48	C₅₃×#464	—	2	2	no	—
49	C₅₃×#463	—	2	2	no	—
50	C₅₃×#452	—	2	2	no	—
51	C₅₃×#449	—	2	2	no	—
52	C₅₃×#442	—	2	2	no	—
53	C₅₃×#440	—	2	2	no	—
54	C₄×D₈×C₅₃	—	2	2	no	—
55	C₅₃×#431	C₂²	1	1	yes	characteristic
56	C₅₃×#416	—	2	2	no	—
57	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₅₃×#322	C₂²	1	1	yes	characteristic
60	C₅₃×#322	C₂²	1	1	yes	characteristic
61	C₈²×C₅₃	C₂²	1	1	yes	characteristic
62	C₈²×C₅₃	C₂²	1	1	yes	characteristic
63	C₂×#41×C₅₃	—	2	1	no	—
64	C₂×#41×C₅₃	—	2	1	no	—
65	C₂×#41×C₅₃	—	2	1	no	—
66	C₂×Q₁₆×C₅₃	—	2	2	no	—
67	C₂×SD₁₆×C₅₃	—	2	2	no	—
68	C₂×SD₁₆×C₅₃	—	2	2	no	—
69	C₂×SD₁₆×C₅₃	—	2	2	no	—
70	C₂×D₈×C₅₃	—	2	2	no	—
71	C₂×D₈×C₅₃	—	2	2	no	—
72	C₂×D₈×C₅₃	—	2	2	no	—
73	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
74	C₂²×C₈×C₅₃	D₄	1	2	yes	—
75	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
80	#127×C₅₃	C₂³	1	1	yes	characteristic
81	#127×C₅₃	C₂³	1	1	yes	characteristic
82	#127×C₅₃	C₂³	1	1	yes	characteristic
83	#125×C₅₃	—	2	1	no	—
84	#124×C₅₃	—	2	1	no	—
85	#124×C₅₃	—	2	1	no	—
86	#124×C₅₃	—	2	1	no	—
87	C₂²⋊Q₈×C₅₃	—	2	1	no	—
88	C₂²⋊Q₈×C₅₃	—	2	1	no	—
89	#121×C₅₃	—	2	1	no	—
90	#121×C₅₃	—	2	1	no	—
91	#121×C₅₃	—	2	1	no	—
92	#121×C₅₃	—	2	1	no	—
93	#121×C₅₃	—	2	1	no	—
94	#121×C₅₃	—	2	1	no	—
95	C₄×Q₈×C₅₃	—	2	1	no	—
96	C₄×Q₈×C₅₃	—	2	1	no	—
97	C₄×D₄×C₅₃	—	2	1	no	—
98	C₄×D₄×C₅₃	—	2	1	no	—
99	C₄×D₄×C₅₃	—	2	1	no	—
100	C₄×D₄×C₅₃	—	2	1	no	—
101	C₄×D₄×C₅₃	—	2	1	no	—
102	C₄×D₄×C₅₃	—	2	1	no	—
103	#117×C₅₃	—	2	1	no	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	#107×C₅₃	—	2	2	no	—
106	#107×C₅₃	—	2	2	no	—
107	#106×C₅₃	—	2	2	no	—
108	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
109	#103×C₅₃	—	2	2	no	—
110	#103×C₅₃	—	2	2	no	—
111	#103×C₅₃	—	2	2	no	—
112	#103×C₅₃	—	2	2	no	—
113	#103×C₅₃	—	2	2	no	—
114	#102×C₅₃	—	2	2	no	—
115	#102×C₅₃	—	2	2	no	—
116	#102×C₅₃	—	2	2	no	—
117	#102×C₅₃	—	2	2	no	—
118	#102×C₅₃	—	2	2	no	—
119	#102×C₅₃	—	2	2	no	—
120	#102×C₅₃	—	2	2	no	—
121	#102×C₅₃	—	2	2	no	—
122	#102×C₅₃	—	2	2	no	—
123	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
124	#97×C₅₃	D₄	1	2	yes	—
125	C₄×C₈×C₅₃	D₄	1	2	yes	—
126	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
127	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
132	#41×C₅₃	—	4	4	no	—
133	#41×C₅₃	—	8	2	no	—
134	#41×C₅₃	—	8	2	no	—
135	C₂×Q₈×C₅₃	—	2	1	no	—
136	C₂×Q₈×C₅₃	—	2	1	no	—
137	C₂×Q₈×C₅₃	—	2	1	no	—
138	C₂×Q₈×C₅₃	—	2	1	no	—
139	C₂×Q₈×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	1	no	—
141	C₂×D₄×C₅₃	—	2	1	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	—	2	1	no	—
145	C₂×D₄×C₅₃	—	2	1	no	—
146	C₂×D₄×C₅₃	—	2	1	no	—
147	C₂×D₄×C₅₃	—	2	1	no	—
148	C₂×D₄×C₅₃	—	2	1	no	—
149	C₂×D₄×C₅₃	—	4	1	no	—
150	C₂×D₄×C₅₃	—	4	1	no	—
151	C₂×D₄×C₅₃	—	4	1	no	—
152	C₂²×C₄×C₅₃	—	4	1	no	—
153	C₂²×C₄×C₅₃	—	4	1	no	—
154	C₂²×C₄×C₅₃	—	4	1	no	—
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
158	Q₁₆×C₅₃	—	8	2	no	—
159	SD₁₆×C₅₃	—	4	4	no	—
160	SD₁₆×C₅₃	—	8	2	no	—
161	SD₁₆×C₅₃	—	8	2	no	—
162	D₈×C₅₃	—	4	4	no	—
163	D₈×C₅₃	—	4	4	no	—
164	D₈×C₅₃	—	8	2	no	—
165	M₁₆×C₅₃	—	2	4	no	—
166	C₂×C₈×C₅₃	—	4	2	no	—
167	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₅₃	—	2	1	no	—
174	C₄⋊C₄×C₅₃	—	2	1	no	—
175	C₄⋊C₄×C₅₃	—	2	1	no	—
176	C₄⋊C₄×C₅₃	—	2	1	no	—
177	C₄⋊C₄×C₅₃	—	2	1	no	—
178	C₄⋊C₄×C₅₃	—	2	1	no	—
179	C₄⋊C₄×C₅₃	—	2	1	no	—
180	C₄⋊C₄×C₅₃	—	2	1	no	—
181	C₄⋊C₄×C₅₃	—	2	1	no	—
182	C₄⋊C₄×C₅₃	—	2	1	no	—
183	C₄⋊C₄×C₅₃	—	4	1	no	—
184	C₄⋊C₄×C₅₃	—	4	1	no	—
185	C₂²⋊C₄×C₅₃	—	4	1	no	—
186	C₂²⋊C₄×C₅₃	—	4	1	no	—
187	C₂²⋊C₄×C₅₃	—	4	1	no	—
188	C₂²⋊C₄×C₅₃	—	4	1	no	—
189	C₂²⋊C₄×C₅₃	—	4	1	no	—
190	C₂²⋊C₄×C₅₃	—	4	1	no	—
191	C₂²⋊C₄×C₅₃	—	4	1	no	—
192	C₄²×C₅₃	—	4	1	no	—
193	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
197	C₂³×C₅₃	—	4	1	no	—
198	C₂³×C₅₃	—	4	1	no	—
199	C₂³×C₅₃	—	4	1	no	—
200	C₂³×C₅₃	#127	1	1	yes	characteristic
201	Q₈×C₅₃	—	4	2	no	—
202	Q₈×C₅₃	—	4	2	no	—
203	Q₈×C₅₃	—	4	2	no	—
204	Q₈×C₅₃	—	4	2	no	—
205	Q₈×C₅₃	—	4	2	no	—
206	D₄×C₅₃	—	4	2	no	—
207	D₄×C₅₃	—	4	2	no	—
208	D₄×C₅₃	—	4	2	no	—
209	D₄×C₅₃	—	4	2	no	—
210	D₄×C₅₃	—	4	2	no	—
211	D₄×C₅₃	—	4	2	no	—
212	D₄×C₅₃	—	4	2	no	—
213	D₄×C₅₃	—	4	2	no	—
214	D₄×C₅₃	—	4	2	no	—
215	D₄×C₅₃	—	8	2	no	—
216	D₄×C₅₃	—	8	2	no	—
217	D₄×C₅₃	—	8	2	no	—
218	C₂×C₄×C₅₃	—	2	2	no	—
219	C₂×C₄×C₅₃	—	2	2	no	—
220	C₂×C₄×C₅₃	—	2	2	no	—
221	C₂×C₄×C₅₃	—	4	1	no	—
222	C₂×C₄×C₅₃	—	4	1	no	—
223	C₂×C₄×C₅₃	—	4	1	no	—
224	C₂×C₄×C₅₃	—	4	1	no	—
225	C₂×C₄×C₅₃	—	4	1	no	—
226	C₂×C₄×C₅₃	—	8	2	no	—
227	C₂×C₄×C₅₃	—	8	2	no	—
228	C₂×C₄×C₅₃	—	8	2	no	—
229	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
230	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
231	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
232	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
235	C₈×C₅₃	—	2	2	no	—
236	C₈×C₅₃	—	2	2	no	—
237	C₈×C₅₃	—	2	2	no	—
238	C₈×C₅₃	—	2	2	no	—
239	C₈×C₅₃	—	2	2	no	—
240	C₈×C₅₃	#135	1	4	yes	—
241	#19644	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂²×C₅₃	—	2	1	no	—
243	C₂²×C₅₃	—	2	1	no	—
244	C₂²×C₅₃	—	2	1	no	—
245	C₂²×C₅₃	—	8	1	no	—
246	C₂²×C₅₃	—	8	1	no	—
247	C₂²×C₅₃	—	8	1	no	—
248	C₂²×C₅₃	—	8	1	no	—
249	C₂²×C₅₃	—	8	1	no	—
250	C₂²×C₅₃	—	8	1	no	—
251	C₂²×C₅₃	—	8	1	no	—
252	C₂²×C₅₃	—	8	1	no	—
253	C₂²×C₅₃	—	8	1	no	—
254	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
255	C₄×C₅₃	—	2	1	no	—
256	C₄×C₅₃	—	2	1	no	—
257	C₄×C₅₃	—	2	1	no	—
258	C₄×C₅₃	—	8	1	no	—
259	C₄×C₅₃	—	8	1	no	—
260	C₄×C₅₃	—	8	1	no	—
261	C₄×C₅₃	—	8	1	no	—
262	C₄×C₅₃	—	8	1	no	—
263	C₄×C₅₃	C₂×#135	1	2	yes	—
264	C₄×C₅₃	#575	1	2	yes	—
265	C₄×C₅₃	#576	1	2	yes	—
266	#3210	C₂×C₅₃	1	1	yes	characteristic, complemented
267	#3207	C₂×C₅₃	1	1	yes	characteristic, complemented
268	#3184	C₂×C₅₃	1	1	yes	characteristic, complemented
269	#3182	C₂×C₅₃	1	1	yes	characteristic, complemented
270	#3169	C₂×C₅₃	1	1	yes	characteristic, complemented
271	#3131	C₂×C₅₃	1	1	yes	characteristic, complemented
272	#1723	C₂×C₅₃	1	1	yes	characteristic, complemented
273	#1721	C₂×C₅₃	1	1	yes	characteristic, complemented
274	#1719	C₂×C₅₃	1	1	yes	characteristic, complemented
275	#1717	C₂×C₅₃	1	1	yes	characteristic, complemented
276	#1711	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#1709	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#1708	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#1706	C₂×C₅₃	1	1	yes	characteristic, complemented
280	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
281	C₂×C₅₃	—	4	1	no	—
282	C₂×C₅₃	—	16	1	no	—
283	C₂×C₅₃	—	16	1	no	—
284	C₂×C₅₃	—	16	1	no	—
285	C₂×C₅₃	#3147	1	1	yes	characteristic
286	C₂×C₅₃	#3153	1	1	yes	characteristic
287	C₂×C₅₃	#3155	1	1	yes	characteristic
288	#533	C₂²×C₅₃	1	1	yes	characteristic
289	#532	C₂²×C₅₃	1	1	yes	characteristic
290	#532	C₂²×C₅₃	1	1	yes	characteristic
291	#532	C₂²×C₅₃	1	1	yes	characteristic
292	#500	C₂²×C₅₃	1	1	yes	characteristic
293	#500	C₂²×C₅₃	1	1	yes	characteristic
294	#498	C₂²×C₅₃	1	1	yes	characteristic
295	#494	C₂²×C₅₃	1	1	yes	characteristic
296	#493	C₂²×C₅₃	1	1	yes	characteristic
297	#493	C₂²×C₅₃	1	1	yes	characteristic
298	#493	C₂²×C₅₃	1	1	yes	characteristic
299	#492	C₂²×C₅₃	1	1	yes	characteristic
300	#492	C₂²×C₅₃	1	1	yes	characteristic
301	#492	C₂²×C₅₃	1	1	yes	characteristic
302	#488	—	2	2	no	—
303	#487	C₂²×C₅₃	1	1	yes	characteristic
304	#487	C₂²×C₅₃	1	1	yes	characteristic
305	#487	C₂²×C₅₃	1	1	yes	characteristic
306	#487	C₂²×C₅₃	1	1	yes	characteristic
307	#487	C₂²×C₅₃	1	1	yes	characteristic
308	#486	C₂²×C₅₃	1	1	yes	characteristic
309	#486	C₂²×C₅₃	1	1	yes	characteristic
310	#486	C₂²×C₅₃	1	1	yes	characteristic
311	#486	C₂²×C₅₃	1	1	yes	characteristic
312	#486	C₂²×C₅₃	1	1	yes	characteristic
313	#486	C₂²×C₅₃	1	1	yes	characteristic
314	#486	C₂²×C₅₃	1	1	yes	characteristic
315	#486	C₂²×C₅₃	1	1	yes	characteristic
316	#486	C₂²×C₅₃	1	1	yes	characteristic
317	#469	—	2	2	no	—
318	#468	—	2	2	no	—
319	#466	—	2	2	no	—
320	#464	—	2	2	no	—
321	#463	—	2	2	no	—
322	#452	—	2	2	no	—
323	#449	—	2	2	no	—
324	#442	—	2	2	no	—
325	#440	—	2	2	no	—
326	C₄×D₈	—	2	2	no	—
327	#431	C₂²×C₅₃	1	1	yes	characteristic
328	#416	—	2	2	no	—
329	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
330	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
331	#322	C₂²×C₅₃	1	1	yes	characteristic
332	#322	C₂²×C₅₃	1	1	yes	characteristic
333	C₈²	C₂²×C₅₃	1	1	yes	characteristic
334	C₈²	C₂²×C₅₃	1	1	yes	characteristic
335	C₅₃	#19644	1	1	yes	characteristic, direct product factor, Sylow
336	C₂×#41	—	2	1	no	—
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×Q₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×SD₁₆	—	2	2	no	—
343	C₂×D₈	—	2	2	no	—
344	C₂×D₈	—	2	2	no	—
345	C₂×D₈	—	2	2	no	—
346	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
347	C₂²×C₈	D₄×C₅₃	1	2	yes	—
348	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
350	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
352	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
353	#127	C₂³×C₅₃	1	1	yes	characteristic
354	#127	C₂³×C₅₃	1	1	yes	characteristic
355	#127	C₂³×C₅₃	1	1	yes	characteristic
356	#125	—	2	1	no	—
357	#124	—	2	1	no	—
358	#124	—	2	1	no	—
359	#124	—	2	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	—	2	1	no	—
362	#121	—	2	1	no	—
363	#121	—	2	1	no	—
364	#121	—	2	1	no	—
365	#121	—	2	1	no	—
366	#121	—	2	1	no	—
367	#121	—	2	1	no	—
368	C₄×Q₈	—	2	1	no	—
369	C₄×Q₈	—	2	1	no	—
370	C₄×D₄	—	2	1	no	—
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	#117	—	2	1	no	—
377	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
378	#107	—	2	2	no	—
379	#107	—	2	2	no	—
380	#106	—	2	2	no	—
381	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
382	#103	—	2	2	no	—
383	#103	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#103	—	2	2	no	—
387	#102	—	2	2	no	—
388	#102	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	#102	—	2	2	no	—
396	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
397	#97	D₄×C₅₃	1	2	yes	—
398	C₄×C₈	D₄×C₅₃	1	2	yes	—
399	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
402	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
404	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
405	#41	—	4	4	no	—
406	#41	—	8	2	no	—
407	#41	—	8	2	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	4	1	no	—
423	C₂×D₄	—	4	1	no	—
424	C₂×D₄	—	4	1	no	—
425	C₂²×C₄	—	4	1	no	—
426	C₂²×C₄	—	4	1	no	—
427	C₂²×C₄	—	4	1	no	—
428	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
431	Q₁₆	—	8	2	no	—
432	SD₁₆	—	4	4	no	—
433	SD₁₆	—	8	2	no	—
434	SD₁₆	—	8	2	no	—
435	D₈	—	4	4	no	—
436	D₈	—	4	4	no	—
437	D₈	—	8	2	no	—
438	M₁₆	—	2	4	no	—
439	C₂×C₈	—	4	2	no	—
440	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
441	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
442	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
443	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
444	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
445	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	4	1	no	—
457	C₄⋊C₄	—	4	1	no	—
458	C₂²⋊C₄	—	4	1	no	—
459	C₂²⋊C₄	—	4	1	no	—
460	C₂²⋊C₄	—	4	1	no	—
461	C₂²⋊C₄	—	4	1	no	—
462	C₂²⋊C₄	—	4	1	no	—
463	C₂²⋊C₄	—	4	1	no	—
464	C₂²⋊C₄	—	4	1	no	—
465	C₄²	—	4	1	no	—
466	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
467	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
469	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	1	no	—
473	C₂³	#127×C₅₃	1	1	yes	characteristic
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	2	no	—
477	Q₈	—	4	2	no	—
478	Q₈	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	2	no	—
483	D₄	—	4	2	no	—
484	D₄	—	4	2	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	8	2	no	—
489	D₄	—	8	2	no	—
490	D₄	—	8	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	8	2	no	—
500	C₂×C₄	—	8	2	no	—
501	C₂×C₄	—	8	2	no	—
502	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
503	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
504	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
508	C₈	—	2	2	no	—
509	C₈	—	2	2	no	—
510	C₈	—	2	2	no	—
511	C₈	—	2	2	no	—
512	C₈	—	2	2	no	—
513	C₈	#135×C₅₃	1	4	yes	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	1	no	—
519	C₂²	—	8	1	no	—
520	C₂²	—	8	1	no	—
521	C₂²	—	8	1	no	—
522	C₂²	—	8	1	no	—
523	C₂²	—	8	1	no	—
524	C₂²	—	8	1	no	—
525	C₂²	—	8	1	no	—
526	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
527	C₄	—	2	1	no	—
528	C₄	—	2	1	no	—
529	C₄	—	2	1	no	—
530	C₄	—	8	1	no	—
531	C₄	—	8	1	no	—
532	C₄	—	8	1	no	—
533	C₄	—	8	1	no	—
534	C₄	—	8	1	no	—
535	C₄	C₂×#135×C₅₃	1	2	yes	—
536	C₄	C₅₃×#575	1	2	yes	—
537	C₄	C₅₃×#576	1	2	yes	—
538	C₂	—	4	1	no	—
539	C₂	—	16	1	no	—
540	C₂	—	16	1	no	—
541	C₂	—	16	1	no	—
542	C₂	C₅₃×#3147	1	1	yes	characteristic
543	C₂	C₅₃×#3153	1	1	yes	characteristic
544	C₂	C₅₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3210	C₂	semidirect
2. C₅₃×#3207	C₂	semidirect
3. C₅₃×#3184	C₂	semidirect
4. C₅₃×#3182	C₂	semidirect
5. C₅₃×#3169	C₂	semidirect
6. C₅₃×#3131	C₂	semidirect
7. C₅₃×#1723	C₂	semidirect
8. C₅₃×#1721	C₂	semidirect
9. C₅₃×#1719	C₂	semidirect
10. C₅₃×#1717	C₂	semidirect
11. C₅₃×#1711	C₂	semidirect
12. C₅₃×#1709	C₂	semidirect
13. C₅₃×#1708	C₂	semidirect
14. C₅₃×#1706	C₂	semidirect
15. C₈×M₁₆×C₅₃	C₂	semidirect
241. #19644	C₅₃	direct (N × H)
266. #3210	C₂×C₅₃	semidirect
267. #3207	C₂×C₅₃	semidirect
268. #3184	C₂×C₅₃	semidirect
269. #3182	C₂×C₅₃	semidirect
270. #3169	C₂×C₅₃	semidirect
271. #3131	C₂×C₅₃	semidirect
272. #1723	C₂×C₅₃	semidirect
273. #1721	C₂×C₅₃	semidirect
274. #1719	C₂×C₅₃	semidirect
275. #1717	C₂×C₅₃	semidirect
276. #1711	C₂×C₅₃	semidirect
277. #1709	C₂×C₅₃	semidirect
278. #1708	C₂×C₅₃	semidirect
279. #1706	C₂×C₅₃	semidirect
280. C₈×M₁₆	C₂×C₅₃	semidirect
335. C₅₃	#19644	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×#3210	2
C₅₃×#3207	2
C₅₃×#3184	2
C₅₃×#3182	2
C₅₃×#3169	2
C₅₃×#3131	2
C₅₃×#1723	2
C₅₃×#1721	2
C₅₃×#1719	2
C₅₃×#1717	2
C₅₃×#1711	2
C₅₃×#1709	2
C₅₃×#1708	2
C₅₃×#1706	2
C₈×M₁₆×C₅₃	2
#19644	53

Subgroups — C53×#19644 #27227156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19644 #27227156