Subgroups — C₅₃×#18819 #27226331

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3508	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3116	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1642	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1640	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1631	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1630	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1555	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1488	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#559	—	2	1	no	—
13	C₅₃×#559	C₂²	1	2	yes	—
14	C₅₃×#550	—	2	1	no	—
15	C₅₃×#550	C₂²	1	2	yes	—
16	C₅₃×#546	—	2	2	no	—
17	C₅₃×#546	—	2	2	no	—
18	C₅₃×#540	—	2	2	no	—
19	C₅₃×#535	—	2	1	no	—
20	C₅₃×#533	C₂²	1	1	yes	characteristic
21	C₅₃×#533	C₂²	1	1	yes	characteristic
22	C₅₃×#532	—	2	1	no	—
23	C₅₃×#532	—	2	1	no	—
24	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#518	—	2	1	no	—
26	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#501	—	2	1	no	—
28	C₅₃×#492	C₂²	1	2	yes	—
29	C₅₃×#491	—	2	1	no	—
30	C₅₃×#486	C₂²	1	2	yes	—
31	C₅₃×#485	—	2	2	no	—
32	C₅₃×#485	—	2	1	no	—
33	C₅₃×#477	C₂²	1	2	yes	complemented
34	C₅₃×#477	C₂²	1	1	yes	characteristic, complemented
35	C₅₃×#476	C₂²	1	1	yes	characteristic, complemented
36	C₅₃×#475	C₂²	1	2	yes	complemented
37	C₅₃×#462	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#461	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#460	—	2	2	no	—
40	C₅₃×#452	—	2	1	no	—
41	C₅₃×#450	—	2	2	no	—
42	C₅₃×#441	—	2	1	no	—
43	C₅₃×#440	—	2	1	no	—
44	C₅₃×#432	C₂²	1	1	yes	characteristic
45	C₅₃×#432	C₂²	1	1	yes	characteristic
46	C₅₃×#431	C₂²	1	1	yes	characteristic
47	C₅₃×#331	C₂²	1	2	yes	—
48	C₅₃×#331	C₂²	1	2	yes	—
49	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
50	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
51	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
52	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
53	C₂×#41×C₅₃	—	2	2	no	—
54	C₂×#41×C₅₃	—	2	2	no	—
55	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₅₃	—	2	1	no	—
57	C₂²×D₄×C₅₃	—	2	1	no	—
58	C₂²×D₄×C₅₃	D₄	1	2	yes	—
59	C₂×Q₁₆×C₅₃	—	2	1	no	—
60	C₂×SD₁₆×C₅₃	—	2	1	no	—
61	C₂×SD₁₆×C₅₃	—	2	4	no	—
62	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
66	#127×C₅₃	—	2	1	no	—
67	#127×C₅₃	—	2	1	no	—
68	#127×C₅₃	—	2	2	no	—
69	#127×C₅₃	D₄	1	2	yes	—
70	#127×C₅₃	C₂³	1	2	yes	—
71	#125×C₅₃	—	2	1	no	—
72	#125×C₅₃	—	2	1	no	—
73	#124×C₅₃	—	2	2	no	—
74	#124×C₅₃	—	2	2	no	—
75	#124×C₅₃	—	2	2	no	—
76	#123×C₅₃	—	2	4	no	—
77	#123×C₅₃	—	2	4	no	—
78	C₂²⋊Q₈×C₅₃	—	2	1	no	—
79	C₂²⋊Q₈×C₅₃	—	2	1	no	—
80	C₂²⋊Q₈×C₅₃	—	2	1	no	—
81	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
82	#121×C₅₃	—	2	2	no	—
83	#121×C₅₃	—	2	2	no	—
84	#121×C₅₃	—	2	2	no	—
85	#121×C₅₃	—	2	4	no	—
86	#121×C₅₃	—	2	2	no	—
87	#121×C₅₃	—	2	4	no	—
88	#120×C₅₃	—	2	4	no	—
89	C₄×Q₈×C₅₃	—	2	1	no	—
90	C₄×Q₈×C₅₃	—	2	1	no	—
91	C₄×Q₈×C₅₃	—	2	1	no	—
92	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
93	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
95	C₄×D₄×C₅₃	—	2	2	no	—
96	C₄×D₄×C₅₃	—	2	2	no	—
97	C₄×D₄×C₅₃	—	2	2	no	—
98	C₄×D₄×C₅₃	—	2	2	no	—
99	C₄×D₄×C₅₃	—	2	2	no	—
100	C₄×D₄×C₅₃	—	2	2	no	—
101	C₄×D₄×C₅₃	—	2	1	no	—
102	C₄×D₄×C₅₃	—	2	1	no	—
103	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
104	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
105	#117×C₅₃	—	2	1	no	—
106	#117×C₅₃	—	2	1	no	—
107	#117×C₅₃	—	2	1	no	—
108	#117×C₅₃	D₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
110	C₂×C₄²×C₅₃	—	2	1	no	—
111	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
112	#107×C₅₃	—	2	2	no	—
113	#107×C₅₃	—	2	1	no	—
114	#106×C₅₃	—	2	2	no	—
115	#106×C₅₃	—	2	1	no	—
116	C₄⋊C₈×C₅₃	—	2	1	no	—
117	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
118	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
120	#103×C₅₃	—	2	1	no	—
121	#103×C₅₃	—	2	1	no	—
122	#103×C₅₃	—	2	2	no	—
123	#103×C₅₃	—	2	2	no	—
124	#103×C₅₃	—	2	1	no	—
125	#102×C₅₃	—	2	4	no	—
126	#102×C₅₃	—	2	1	no	—
127	C₂²⋊C₈×C₅₃	—	2	1	no	—
128	C₂²⋊C₈×C₅₃	—	2	1	no	—
129	C₂²⋊C₈×C₅₃	—	2	1	no	—
130	#97×C₅₃	—	2	1	no	—
131	#97×C₅₃	—	2	1	no	—
132	C₄×C₈×C₅₃	C₂³	1	2	yes	—
133	C₂⁴×C₅₃	—	2	2	no	—
134	#41×C₅₃	—	4	1	no	—
135	#41×C₅₃	—	4	1	no	—
136	#41×C₅₃	—	8	4	no	—
137	#41×C₅₃	—	8	4	no	—
138	C₂×Q₈×C₅₃	—	2	1	no	—
139	C₂×Q₈×C₅₃	—	2	1	no	—
140	C₂×Q₈×C₅₃	—	2	1	no	—
141	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
142	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	4	no	—
146	C₂×D₄×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	—	2	2	no	—
148	C₂×D₄×C₅₃	—	2	2	no	—
149	C₂×D₄×C₅₃	—	2	2	no	—
150	C₂×D₄×C₅₃	—	2	2	no	—
151	C₂×D₄×C₅₃	—	4	2	no	—
152	C₂×D₄×C₅₃	—	4	2	no	—
153	C₂×D₄×C₅₃	—	8	2	no	—
154	C₂×D₄×C₅₃	—	8	2	no	—
155	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₅₃	—	2	4	no	—
158	C₂²×C₄×C₅₃	—	2	4	no	—
159	C₂²×C₄×C₅₃	—	2	1	no	—
160	C₂²×C₄×C₅₃	—	2	1	no	—
161	C₂²×C₄×C₅₃	—	2	1	no	—
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
165	Q₁₆×C₅₃	—	4	2	no	—
166	SD₁₆×C₅₃	—	4	2	no	—
167	SD₁₆×C₅₃	—	8	4	no	—
168	C₂×C₈×C₅₃	—	2	2	no	—
169	C₂×C₈×C₅₃	—	2	1	no	—
170	C₂×C₈×C₅₃	—	2	2	no	—
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₄⋊C₄×C₅₃	—	2	1	no	—
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	2	no	—
177	C₄⋊C₄×C₅₃	—	2	1	no	—
178	C₄⋊C₄×C₅₃	—	2	2	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	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
186	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
187	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
188	C₂²⋊C₄×C₅₃	—	2	4	no	—
189	C₂²⋊C₄×C₅₃	—	2	4	no	—
190	C₂²⋊C₄×C₅₃	—	2	4	no	—
191	C₂²⋊C₄×C₅₃	—	2	4	no	—
192	C₂²⋊C₄×C₅₃	—	2	1	no	—
193	C₂²⋊C₄×C₅₃	—	2	2	no	—
194	C₂²⋊C₄×C₅₃	—	2	1	no	—
195	C₂²⋊C₄×C₅₃	—	2	1	no	—
196	C₂²⋊C₄×C₅₃	—	4	1	no	—
197	C₄²×C₅₃	—	2	1	no	—
198	C₄²×C₅₃	—	2	1	no	—
199	C₄²×C₅₃	—	2	1	no	—
200	C₄²×C₅₃	—	2	2	no	—
201	C₄²×C₅₃	—	4	1	no	—
202	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
206	C₂³×C₅₃	—	2	1	no	—
207	C₂³×C₅₃	—	2	4	no	—
208	C₂³×C₅₃	—	4	4	no	—
209	C₂³×C₅₃	—	8	2	no	—
210	C₂³×C₅₃	—	8	2	no	—
211	C₂³×C₅₃	#120	1	1	yes	characteristic
212	Q₈×C₅₃	—	2	1	no	—
213	Q₈×C₅₃	—	2	1	no	—
214	Q₈×C₅₃	—	2	4	no	—
215	Q₈×C₅₃	—	4	2	no	—
216	Q₈×C₅₃	—	4	2	no	—
217	Q₈×C₅₃	—	4	1	no	—
218	Q₈×C₅₃	—	4	1	no	—
219	D₄×C₅₃	—	2	1	no	—
220	D₄×C₅₃	—	2	1	no	—
221	D₄×C₅₃	—	4	4	no	—
222	D₄×C₅₃	—	4	4	no	—
223	D₄×C₅₃	—	4	4	no	—
224	D₄×C₅₃	—	4	2	no	—
225	D₄×C₅₃	—	4	2	no	—
226	D₄×C₅₃	—	4	2	no	—
227	D₄×C₅₃	—	4	4	no	—
228	D₄×C₅₃	—	8	2	no	—
229	D₄×C₅₃	—	8	4	no	—
230	D₄×C₅₃	—	8	4	no	—
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	4	1	no	—
240	C₂×C₄×C₅₃	—	4	2	no	—
241	C₂×C₄×C₅₃	—	4	2	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	—	4	1	no	—
244	C₂×C₄×C₅₃	—	4	1	no	—
245	C₂×C₄×C₅₃	—	8	4	no	—
246	C₂×C₄×C₅₃	—	8	4	no	—
247	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
248	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
249	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
253	C₈×C₅₃	—	4	2	no	—
254	C₈×C₅₃	—	4	1	no	—
255	C₈×C₅₃	—	4	1	no	—
256	C₈×C₅₃	—	4	2	no	—
257	#18819	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₅₃	—	2	1	no	—
259	C₂²×C₅₃	—	2	1	no	—
260	C₂²×C₅₃	—	2	1	no	—
261	C₂²×C₅₃	—	4	1	no	—
262	C₂²×C₅₃	—	4	4	no	—
263	C₂²×C₅₃	—	4	1	no	—
264	C₂²×C₅₃	—	4	4	no	—
265	C₂²×C₅₃	—	4	1	no	—
266	C₂²×C₅₃	—	4	4	no	—
267	C₂²×C₅₃	—	16	4	no	—
268	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
269	C₄×C₅₃	—	2	1	no	—
270	C₄×C₅₃	—	2	1	no	—
271	C₄×C₅₃	—	2	1	no	—
272	C₄×C₅₃	—	2	1	no	—
273	C₄×C₅₃	—	4	1	no	—
274	C₄×C₅₃	—	4	1	no	—
275	C₄×C₅₃	—	4	2	no	—
276	C₄×C₅₃	—	4	1	no	—
277	C₄×C₅₃	—	4	2	no	—
278	C₄×C₅₃	—	8	1	no	—
279	C₄×C₅₃	—	8	1	no	—
280	C₄×C₅₃	#575	1	1	yes	characteristic
281	C₄×C₅₃	#575	1	1	yes	characteristic
282	C₄×C₅₃	#577	1	2	yes	—
283	#3508	C₂×C₅₃	1	1	yes	characteristic
284	#3237	C₂×C₅₃	1	1	yes	characteristic, complemented
285	#3208	C₂×C₅₃	1	1	yes	characteristic, complemented
286	#3116	C₂×C₅₃	1	1	yes	characteristic, complemented
287	#1642	C₂×C₅₃	1	2	yes	complemented
288	#1640	C₂×C₅₃	1	2	yes	complemented
289	#1631	C₂×C₅₃	1	2	yes	complemented
290	#1630	C₂×C₅₃	1	2	yes	complemented
291	#1555	C₂×C₅₃	1	1	yes	characteristic, complemented
292	#1497	C₂×C₅₃	1	1	yes	characteristic, complemented
293	#1488	C₂×C₅₃	1	1	yes	characteristic, complemented
294	C₂×C₅₃	—	4	1	no	—
295	C₂×C₅₃	—	8	1	no	—
296	C₂×C₅₃	—	8	4	no	—
297	C₂×C₅₃	#3005	1	1	yes	characteristic
298	C₂×C₅₃	#3009	1	1	yes	characteristic
299	C₂×C₅₃	#3019	1	1	yes	characteristic
300	#559	—	2	1	no	—
301	#559	C₂²×C₅₃	1	2	yes	—
302	#550	—	2	1	no	—
303	#550	C₂²×C₅₃	1	2	yes	—
304	#546	—	2	2	no	—
305	#546	—	2	2	no	—
306	#540	—	2	2	no	—
307	#535	—	2	1	no	—
308	#533	C₂²×C₅₃	1	1	yes	characteristic
309	#533	C₂²×C₅₃	1	1	yes	characteristic
310	#532	—	2	1	no	—
311	#532	—	2	1	no	—
312	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
313	#518	—	2	1	no	—
314	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
315	#501	—	2	1	no	—
316	#492	C₂²×C₅₃	1	2	yes	—
317	#491	—	2	1	no	—
318	#486	C₂²×C₅₃	1	2	yes	—
319	#485	—	2	2	no	—
320	#485	—	2	1	no	—
321	#477	C₂²×C₅₃	1	1	yes	characteristic, complemented
322	#477	C₂²×C₅₃	1	2	yes	complemented
323	#476	C₂²×C₅₃	1	1	yes	characteristic, complemented
324	#475	C₂²×C₅₃	1	2	yes	complemented
325	#462	C₂²×C₅₃	1	1	yes	characteristic, complemented
326	#461	C₂²×C₅₃	1	1	yes	characteristic, complemented
327	#460	—	2	2	no	—
328	#452	—	2	1	no	—
329	#450	—	2	2	no	—
330	#441	—	2	1	no	—
331	#440	—	2	1	no	—
332	#432	C₂²×C₅₃	1	1	yes	characteristic
333	#432	C₂²×C₅₃	1	1	yes	characteristic
334	#431	C₂²×C₅₃	1	1	yes	characteristic
335	#331	C₂²×C₅₃	1	2	yes	—
336	#331	C₂²×C₅₃	1	2	yes	—
337	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
338	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
339	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
340	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
341	C₅₃	#18819	1	1	yes	characteristic, direct product factor, Sylow
342	C₂×#41	—	2	2	no	—
343	C₂×#41	—	2	2	no	—
344	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
345	C₂²×D₄	—	2	1	no	—
346	C₂²×D₄	—	2	1	no	—
347	C₂²×D₄	D₄×C₅₃	1	2	yes	—
348	C₂×Q₁₆	—	2	1	no	—
349	C₂×SD₁₆	—	2	1	no	—
350	C₂×SD₁₆	—	2	4	no	—
351	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
352	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
354	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
355	#127	—	2	2	no	—
356	#127	—	2	1	no	—
357	#127	—	2	1	no	—
358	#127	D₄×C₅₃	1	2	yes	—
359	#127	C₂³×C₅₃	1	2	yes	—
360	#125	—	2	1	no	—
361	#125	—	2	1	no	—
362	#124	—	2	2	no	—
363	#124	—	2	2	no	—
364	#124	—	2	2	no	—
365	#123	—	2	4	no	—
366	#123	—	2	4	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	C₂²⋊Q₈	—	2	1	no	—
369	C₂²⋊Q₈	—	2	1	no	—
370	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
371	#121	—	2	2	no	—
372	#121	—	2	2	no	—
373	#121	—	2	2	no	—
374	#121	—	2	4	no	—
375	#121	—	2	4	no	—
376	#121	—	2	2	no	—
377	#120	—	2	4	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
382	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
383	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
393	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	#117	—	2	1	no	—
397	#117	D₄×C₅₃	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	—	2	4	no	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
401	#107	—	2	1	no	—
402	#107	—	2	2	no	—
403	#106	—	2	1	no	—
404	#106	—	2	2	no	—
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
407	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
408	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
409	#103	—	2	1	no	—
410	#103	—	2	2	no	—
411	#103	—	2	2	no	—
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#102	—	2	1	no	—
415	#102	—	2	4	no	—
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	—	2	1	no	—
418	C₂²⋊C₈	—	2	1	no	—
419	#97	—	2	1	no	—
420	#97	—	2	1	no	—
421	C₄×C₈	C₂³×C₅₃	1	2	yes	—
422	C₂⁴	—	2	2	no	—
423	#41	—	4	1	no	—
424	#41	—	4	1	no	—
425	#41	—	8	4	no	—
426	#41	—	8	4	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	—	4	2	no	—
442	C₂×D₄	—	8	2	no	—
443	C₂×D₄	—	8	2	no	—
444	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
445	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
454	Q₁₆	—	4	2	no	—
455	SD₁₆	—	4	2	no	—
456	SD₁₆	—	8	4	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₄⋊C₄	—	4	1	no	—
474	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
475	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
476	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	2	1	no	—
484	C₂²⋊C₄	—	2	1	no	—
485	C₂²⋊C₄	—	4	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	2	no	—
490	C₄²	—	4	1	no	—
491	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
492	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
493	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
494	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
495	C₂³	—	2	4	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	4	4	no	—
498	C₂³	—	8	2	no	—
499	C₂³	—	8	2	no	—
500	C₂³	#120×C₅₃	1	1	yes	characteristic
501	Q₈	—	2	1	no	—
502	Q₈	—	2	1	no	—
503	Q₈	—	2	4	no	—
504	Q₈	—	4	2	no	—
505	Q₈	—	4	2	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	D₄	—	2	1	no	—
509	D₄	—	2	1	no	—
510	D₄	—	4	4	no	—
511	D₄	—	4	4	no	—
512	D₄	—	4	4	no	—
513	D₄	—	4	2	no	—
514	D₄	—	4	2	no	—
515	D₄	—	4	2	no	—
516	D₄	—	4	4	no	—
517	D₄	—	8	2	no	—
518	D₄	—	8	4	no	—
519	D₄	—	8	4	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	8	4	no	—
535	C₂×C₄	—	8	4	no	—
536	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
537	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
538	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
539	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
540	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
542	C₈	—	4	2	no	—
543	C₈	—	4	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	2	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	2	1	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	16	4	no	—
556	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
557	C₄	—	2	1	no	—
558	C₄	—	2	1	no	—
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	4	1	no	—
562	C₄	—	4	1	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	2	no	—
566	C₄	—	8	1	no	—
567	C₄	—	8	1	no	—
568	C₄	C₅₃×#575	1	1	yes	characteristic
569	C₄	C₅₃×#575	1	1	yes	characteristic
570	C₄	C₅₃×#577	1	2	yes	—
571	C₂	—	4	1	no	—
572	C₂	—	8	4	no	—
573	C₂	—	8	1	no	—
574	C₂	C₅₃×#3005	1	1	yes	characteristic
575	C₂	C₅₃×#3009	1	1	yes	characteristic
576	C₂	C₅₃×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3508	C₂
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
2. C₅₃×#3237	C₂	semidirect
3. C₅₃×#3208	C₂	semidirect
4. C₅₃×#3116	C₂	semidirect
5. C₅₃×#1642	C₂	semidirect
6. C₅₃×#1640	C₂	semidirect
7. C₅₃×#1631	C₂	semidirect
8. C₅₃×#1630	C₂	semidirect
9. C₅₃×#1555	C₂	semidirect
10. C₅₃×#1497	C₂	semidirect
11. C₅₃×#1488	C₂	semidirect
33. C₅₃×#477	C₂²	semidirect
34. C₅₃×#477	C₂²	semidirect
35. C₅₃×#476	C₂²	semidirect
36. C₅₃×#475	C₂²	semidirect
37. C₅₃×#462	C₂²	semidirect
38. C₅₃×#461	C₂²	semidirect
49. C₅₃×Q₈⋊C₈	C₂²	semidirect
50. C₅₃×Q₈⋊C₈	C₂²	semidirect
51. C₅₃×Q₈⋊C₈	C₂²	semidirect
52. C₅₃×D₄⋊C₈	C₂²	semidirect
257. #18819	C₅₃	direct (N × H)
284. #3237	C₂×C₅₃	semidirect
285. #3208	C₂×C₅₃	semidirect
286. #3116	C₂×C₅₃	semidirect
287. #1642	C₂×C₅₃	semidirect
288. #1640	C₂×C₅₃	semidirect
289. #1631	C₂×C₅₃	semidirect
290. #1630	C₂×C₅₃	semidirect
291. #1555	C₂×C₅₃	semidirect
292. #1497	C₂×C₅₃	semidirect
293. #1488	C₂×C₅₃	semidirect
321. #477	C₂²×C₅₃	semidirect
322. #477	C₂²×C₅₃	semidirect
323. #476	C₂²×C₅₃	semidirect
324. #475	C₂²×C₅₃	semidirect
325. #462	C₂²×C₅₃	semidirect
326. #461	C₂²×C₅₃	semidirect
337. Q₈⋊C₈	C₂²×C₅₃	semidirect
338. Q₈⋊C₈	C₂²×C₅₃	semidirect
339. Q₈⋊C₈	C₂²×C₅₃	semidirect
340. D₄⋊C₈	C₂²×C₅₃	semidirect
341. C₅₃	#18819	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×#3508	2
C₅₃×#3237	2
C₅₃×#3208	2
C₅₃×#3116	2
C₅₃×#1642	2
C₅₃×#1640	2
C₅₃×#1631	2
C₅₃×#1630	2
C₅₃×#1555	2
C₅₃×#1497	2
C₅₃×#1488	2
#18819	53

Subgroups — C53×#18819 #27226331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18819 #27226331