Subgroups — #27351249

Order 13568.

1150 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810469	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273227	C₂	1	2	yes	maximal, complemented
4	#20273223	C₂	1	2	yes	maximal, complemented
5	#20273219	C₂	1	4	yes	maximal, complemented
6	#20273201	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3444	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×Dic₅₃	—	2	1	no	—
9	#16810680	—	2	1	no	—
10	#16810680	C₂²	1	2	yes	—
11	#16810676	—	2	1	no	—
12	#16810676	C₂²	1	2	yes	—
13	#16810647	—	2	1	no	—
14	#16810587	C₂²	1	2	yes	—
15	C₄×#11719917	—	2	1	no	—
16	#16810469	—	2	1	no	—
17	#16810469	—	2	1	no	—
18	#16810469	C₂²	1	2	yes	—
19	#16810469	C₂²	1	2	yes	—
20	C₂×#13133124	—	2	1	no	—
21	C₂×#13133124	C₂²	1	2	yes	—
22	#16810360	C₂²	1	8	yes	—
23	#16810357	C₂²	1	8	yes	—
24	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
25	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
26	C₅₃×#533	C₂²	1	1	yes	characteristic
27	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
28	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
29	C₅₃×#532	C₂²	1	1	yes	characteristic
30	C₅₃×#529	—	2	1	no	—
31	C₅₃×#529	—	2	1	no	—
32	C₅₃×#529	—	2	1	no	—
33	C₅₃×#529	—	2	1	no	—
34	C₅₃×#529	C₄	1	2	yes	complemented
35	C₅₃×#529	C₂²	1	2	yes	—
36	C₅₃×#525	—	2	1	no	—
37	C₅₃×#525	—	2	1	no	—
38	C₅₃×#525	C₂²	1	2	yes	—
39	C₄×#41×C₅₃	—	2	1	no	—
40	C₄×#41×C₅₃	—	2	1	no	—
41	C₄×#41×C₅₃	C₄	1	2	yes	complemented
42	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂³×Dic₅₃	—	2	2	no	—
44	C₂×#11719917	—	2	2	no	—
45	C₂×C₄×Dic₅₃	—	2	8	no	—
46	C₂×C₄×Dic₅₃	—	2	2	no	—
47	C₂×C₄×Dic₅₃	—	2	2	no	—
48	C₄⋊#88322	—	2	2	no	—
49	C₄⋊#88322	—	2	2	no	—
50	C₂²⋊#88322	—	2	2	no	—
51	C₂²⋊#88322	C₂³	1	4	yes	—
52	#13133128	—	2	2	no	—
53	C₄×#88322	—	2	2	no	—
54	#13133124	—	2	2	no	—
55	#13133124	—	2	2	no	—
56	#13133124	—	2	2	no	—
57	#13133124	C₂³	1	4	yes	—
58	C₂×#41×C₅₃	—	2	1	no	—
59	C₂×#41×C₅₃	—	2	1	no	—
60	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
61	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
65	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
66	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
67	#127×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
68	#126×C₅₃	—	2	2	no	—
69	#126×C₅₃	—	2	2	no	—
70	#125×C₅₃	—	2	1	no	—
71	#125×C₅₃	—	2	1	no	—
72	#125×C₅₃	C₂×C₄	1	2	yes	complemented
73	#124×C₅₃	—	2	1	no	—
74	#124×C₅₃	—	2	1	no	—
75	#124×C₅₃	C₂×C₄	1	2	yes	complemented
76	#123×C₅₃	—	2	2	no	—
77	#123×C₅₃	—	2	2	no	—
78	#123×C₅₃	—	2	2	no	—
79	C₂²⋊Q₈×C₅₃	—	2	1	no	—
80	C₂²⋊Q₈×C₅₃	—	2	1	no	—
81	C₂²⋊Q₈×C₅₃	—	2	1	no	—
82	C₂²⋊Q₈×C₅₃	—	2	1	no	—
83	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
84	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
85	#121×C₅₃	—	2	1	no	—
86	#121×C₅₃	—	2	1	no	—
87	#121×C₅₃	—	2	1	no	—
88	#121×C₅₃	—	2	1	no	—
89	#121×C₅₃	C₂×C₄	1	2	yes	—
90	#121×C₅₃	C₂³	1	2	yes	—
91	#120×C₅₃	—	2	2	no	—
92	C₄×Q₈×C₅₃	—	2	1	no	—
93	C₄×Q₈×C₅₃	—	2	1	no	—
94	C₄×Q₈×C₅₃	—	2	1	no	—
95	C₄×Q₈×C₅₃	—	2	1	no	—
96	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
97	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
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	2	no	—
103	C₄×D₄×C₅₃	—	2	2	no	—
104	C₄×D₄×C₅₃	—	2	2	no	—
105	C₄×D₄×C₅₃	—	2	2	no	—
106	C₄×D₄×C₅₃	—	2	2	no	—
107	C₄×D₄×C₅₃	—	2	2	no	—
108	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
109	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
110	#117×C₅₃	—	2	2	no	—
111	#117×C₅₃	—	2	2	no	—
112	#117×C₅₃	—	2	2	no	—
113	#117×C₅₃	—	2	1	no	—
114	#117×C₅₃	—	2	1	no	—
115	#117×C₅₃	C₂×C₄	1	2	yes	—
116	C₂×C₄²×C₅₃	—	2	2	no	—
117	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
122	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
129	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
130	#41×C₅₃	—	4	4	no	—
131	#41×C₅₃	—	4	4	no	—
132	#41×C₅₃	—	4	4	no	—
133	C₂×Q₈×C₅₃	—	2	1	no	—
134	C₂×Q₈×C₅₃	—	2	1	no	—
135	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₅₃	—	2	2	no	—
137	C₂×D₄×C₅₃	—	2	2	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	1	no	—
141	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₅₃	—	2	4	no	—
143	C₂²×C₄×C₅₃	—	2	2	no	—
144	C₂²×C₄×C₅₃	—	2	2	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	4	2	no	—
147	C₂²×C₄×C₅₃	—	4	2	no	—
148	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
166	C₂²×Dic₅₃	—	2	8	no	—
167	C₂²×Dic₅₃	—	2	4	no	—
168	C₂²×Dic₅₃	—	4	4	no	—
169	C₂²×Dic₅₃	—	4	4	no	—
170	C₄⋊C₄×C₅₃	—	2	1	no	—
171	C₄⋊C₄×C₅₃	—	2	1	no	—
172	C₄⋊C₄×C₅₃	—	2	1	no	—
173	C₄⋊C₄×C₅₃	—	2	1	no	—
174	C₄⋊C₄×C₅₃	—	2	2	no	—
175	C₄⋊C₄×C₅₃	—	2	2	no	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	—	2	2	no	—
178	C₄⋊C₄×C₅₃	—	2	2	no	—
179	C₄⋊C₄×C₅₃	—	2	2	no	—
180	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
181	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
182	C₂²⋊C₄×C₅₃	—	2	2	no	—
183	C₂²⋊C₄×C₅₃	—	2	2	no	—
184	C₂²⋊C₄×C₅₃	—	2	2	no	—
185	C₂²⋊C₄×C₅₃	—	2	2	no	—
186	C₂²⋊C₄×C₅₃	—	2	2	no	—
187	C₂²⋊C₄×C₅₃	—	2	2	no	—
188	C₂²⋊C₄×C₅₃	—	2	2	no	—
189	C₂²⋊C₄×C₅₃	—	2	2	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₄²×C₅₃	—	2	2	no	—
192	C₄²×C₅₃	—	2	2	no	—
193	C₄²×C₅₃	—	2	2	no	—
194	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
195	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
196	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
197	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
198	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
203	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
210	C₄×Dic₅₃	—	2	8	no	—
211	C₄×Dic₅₃	—	2	8	no	—
212	#11719917	—	4	4	no	—
213	C₂×#88322	—	2	4	no	—
214	C₂³×C₅₃	—	2	2	no	—
215	C₂³×C₅₃	—	2	2	no	—
216	C₂³×C₅₃	—	2	2	no	—
217	C₂³×C₅₃	—	2	2	no	—
218	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₅₃	—	4	2	no	—
222	Q₈×C₅₃	—	4	2	no	—
223	Q₈×C₅₃	—	4	2	no	—
224	D₄×C₅₃	—	4	2	no	—
225	D₄×C₅₃	—	4	2	no	—
226	D₄×C₅₃	—	4	4	no	—
227	D₄×C₅₃	—	4	4	no	—
228	D₄×C₅₃	—	4	4	no	—
229	D₄×C₅₃	—	4	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	4	1	no	—
243	C₂×C₄×C₅₃	—	4	1	no	—
244	C₂×C₄×C₅₃	—	4	1	no	—
245	C₂×C₄×C₅₃	—	4	1	no	—
246	C₂×C₄×C₅₃	—	4	1	no	—
247	C₂×C₄×C₅₃	—	4	1	no	—
248	C₂×C₄×C₅₃	—	4	1	no	—
249	C₂×C₄×C₅₃	—	4	1	no	—
250	C₂×C₄×C₅₃	—	4	4	no	—
251	C₂×C₄×C₅₃	—	4	4	no	—
252	C₂×C₄×C₅₃	—	4	4	no	—
253	C₂×C₄×C₅₃	#104	1	4	yes	—
254	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
265	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
266	C₂×Dic₅₃	—	2	8	no	—
267	C₂×Dic₅₃	—	4	16	no	—
268	C₂×Dic₅₃	—	4	4	no	—
269	C₂×Dic₅₃	—	4	4	no	—
270	#88322	—	4	4	no	—
271	#17541	—	53	1	no	maximal, Sylow
272	C₂²×C₅₃	—	2	8	no	—
273	C₂²×C₅₃	—	2	1	no	—
274	C₂²×C₅₃	—	2	1	no	—
275	C₂²×C₅₃	—	2	1	no	—
276	C₂²×C₅₃	—	2	1	no	—
277	C₂²×C₅₃	—	2	1	no	—
278	C₂²×C₅₃	—	2	1	no	—
279	C₂²×C₅₃	—	4	2	no	—
280	C₂²×C₅₃	—	4	2	no	—
281	C₂²×C₅₃	—	4	2	no	—
282	C₂²×C₅₃	#410	1	2	yes	—
283	C₂²×C₅₃	C₂×#104	1	2	yes	—
284	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
285	C₂²×C₅₃	#575	1	2	yes	—
286	C₄×C₅₃	—	2	1	no	—
287	C₄×C₅₃	—	2	1	no	—
288	C₄×C₅₃	—	2	1	no	—
289	C₄×C₅₃	—	2	1	no	—
290	C₄×C₅₃	—	2	1	no	—
291	C₄×C₅₃	—	2	1	no	—
292	C₄×C₅₃	—	2	1	no	—
293	C₄×C₅₃	—	2	1	no	—
294	C₄×C₅₃	—	2	1	no	—
295	C₄×C₅₃	—	2	1	no	—
296	C₄×C₅₃	—	4	2	no	—
297	C₄×C₅₃	—	4	2	no	—
298	C₄×C₅₃	—	4	2	no	—
299	C₄×C₅₃	C₂×#104	1	2	yes	—
300	C₄×C₅₃	C₂×#104	1	2	yes	—
301	Dic₅₃	—	4	8	no	—
302	#3444	D₅₃	1	1	yes	characteristic
303	#2920	—	53	1	no	—
304	C₂×#376	—	53	1	no	—
305	#1506	—	53	2	no	—
306	#1505	—	53	2	no	—
307	#1504	—	53	4	no	—
308	#1500	—	53	4	no	—
309	C₂×C₅₃	—	2	2	no	—
310	C₂×C₅₃	—	2	2	no	—
311	C₂×C₅₃	—	2	2	no	—
312	C₂×C₅₃	—	8	2	no	—
313	C₂×C₅₃	C₂×#410	1	1	yes	characteristic
314	C₂×C₅₃	#2896	1	1	yes	characteristic
315	C₂×C₅₃	C₂²×#104	1	1	yes	characteristic
316	#533	Dic₅₃	1	1	yes	characteristic, complemented
317	#533	Dic₅₃	1	1	yes	characteristic, complemented
318	#533	C₂×D₅₃	1	1	yes	characteristic
319	#532	Dic₅₃	1	1	yes	characteristic, complemented
320	#532	Dic₅₃	1	1	yes	characteristic, complemented
321	#532	C₂×D₅₃	1	1	yes	characteristic
322	#529	—	2	1	no	—
323	#529	—	2	1	no	—
324	#529	—	2	1	no	—
325	#529	—	2	1	no	—
326	#529	Dic₅₃	1	2	yes	complemented
327	#529	C₂×D₅₃	1	2	yes	—
328	#525	—	2	1	no	—
329	#525	—	2	1	no	—
330	#525	C₂×D₅₃	1	2	yes	—
331	C₄×#41	—	2	1	no	—
332	C₄×#41	—	2	1	no	—
333	C₄×#41	Dic₅₃	1	2	yes	complemented
334	C₂²×C₄²	—	106	1	no	—
335	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
336	#432	—	53	2	no	—
337	#432	—	106	1	no	—
338	#431	—	53	2	no	—
339	#431	—	106	1	no	—
340	#423	—	106	1	no	—
341	#407	—	53	2	no	—
342	C₄×M₁₆	—	106	1	no	—
343	#376	—	53	2	no	—
344	#376	—	53	2	no	—
345	#376	—	106	1	no	—
346	#376	—	106	1	no	—
347	C₂×#95	—	53	2	no	—
348	C₂×#95	—	106	1	no	—
349	#328	—	53	8	no	—
350	#327	—	53	8	no	—
351	C₅₃	#17541	1	1	yes	characteristic, complemented, Sylow
352	C₂×#41	—	2	1	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	C₂×Dic₅₃	1	2	yes	—
355	C₂³×C₄	—	106	2	no	—
356	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
357	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
358	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
359	C₂×M₁₆	—	106	2	no	—
360	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
361	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
362	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
363	#127	C₂×Dic₅₃	1	1	yes	characteristic, complemented
364	#126	—	2	2	no	—
365	#126	—	2	2	no	—
366	#125	—	2	1	no	—
367	#125	—	2	1	no	—
368	#125	C₂×Dic₅₃	1	2	yes	complemented
369	#124	—	2	1	no	—
370	#124	—	2	1	no	—
371	#124	C₂×Dic₅₃	1	2	yes	complemented
372	#123	—	2	2	no	—
373	#123	—	2	2	no	—
374	#123	—	2	2	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	C₂×Dic₅₃	1	2	yes	—
380	C₂²⋊Q₈	C₂²×D₅₃	1	2	yes	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	#121	—	2	1	no	—
384	#121	—	2	1	no	—
385	#121	C₂×Dic₅₃	1	2	yes	—
386	#121	C₂²×D₅₃	1	2	yes	—
387	#120	—	2	2	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	—	2	1	no	—
392	C₄×Q₈	C₂×Dic₅₃	1	2	yes	complemented
393	C₄×Q₈	C₂×Dic₅₃	1	2	yes	complemented
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	C₂×Dic₅₃	1	2	yes	complemented
405	C₄×D₄	C₂×Dic₅₃	1	2	yes	complemented
406	#117	—	2	1	no	—
407	#117	—	2	1	no	—
408	#117	—	2	2	no	—
409	#117	—	2	2	no	—
410	#117	—	2	2	no	—
411	#117	C₂×Dic₅₃	1	2	yes	—
412	C₂×C₄²	—	2	2	no	—
413	C₂×C₄²	—	106	8	no	—
414	C₂×C₄²	—	106	2	no	—
415	C₂×C₄²	—	106	2	no	—
416	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
417	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
418	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
419	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
420	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
421	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
422	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
423	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
424	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
425	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
426	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
427	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
428	C₄⋊C₈	—	106	2	no	—
429	C₄⋊C₈	—	106	2	no	—
430	C₂²⋊C₈	—	53	4	no	—
431	C₂²⋊C₈	—	106	2	no	—
432	#97	—	106	2	no	—
433	C₄×C₈	—	106	2	no	—
434	#95	—	53	4	no	—
435	#95	—	106	2	no	—
436	#95	—	106	2	no	—
437	#95	—	106	2	no	—
438	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
439	#41	—	4	4	no	—
440	#41	—	4	4	no	—
441	#41	—	4	4	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	4	2	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	106	8	no	—
458	C₂²×C₄	—	106	4	no	—
459	C₂²×C₄	—	212	4	no	—
460	C₂²×C₄	—	212	4	no	—
461	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
465	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
467	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
468	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
469	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
470	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
472	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
473	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
474	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
475	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
476	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
478	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
479	M₁₆	—	212	4	no	—
480	C₂×C₈	—	106	4	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
492	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₄²	—	2	2	no	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	106	8	no	—
506	C₄²	—	106	8	no	—
507	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
508	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
509	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
510	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
511	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
512	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
513	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
514	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
515	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
516	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
517	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
518	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
519	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
520	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
521	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
522	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
528	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
529	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
530	Q₈	—	4	2	no	—
531	Q₈	—	4	2	no	—
532	Q₈	—	4	2	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	2	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	D₄	—	4	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	106	8	no	—
563	C₂×C₄	—	212	16	no	—
564	C₂×C₄	—	212	4	no	—
565	C₂×C₄	—	212	4	no	—
566	C₂×C₄	#13133144	1	4	yes	—
567	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
569	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
570	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
571	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
572	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
573	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
574	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
575	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
577	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
578	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
579	C₈	—	212	4	no	—
580	C₂²	—	2	8	no	—
581	C₂²	—	2	1	no	—
582	C₂²	—	2	1	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	#16810597	1	2	yes	—
591	C₂²	C₂×#13133144	1	2	yes	—
592	C₂²	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
593	C₂²	#16811421	1	2	yes	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	1	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	2	1	no	—
603	C₄	—	2	1	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	212	8	no	—
608	C₄	C₂×#13133144	1	2	yes	—
609	C₄	C₂×#13133144	1	2	yes	—
610	C₂	—	2	2	no	—
611	C₂	—	2	2	no	—
612	C₂	—	2	2	no	—
613	C₂	—	8	2	no	—
614	C₂	C₂×#16810597	1	1	yes	characteristic
615	C₂	#20283234	1	1	yes	characteristic
616	C₂	C₂²×#13133144	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3444	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3444	D₅₃
hypocenter	C₅₃	#17541
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #3444

Complements

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

N	H ≅ G/N	type
1. #20283348	C₂	semidirect
2. C₂×#16810469	C₂	semidirect
3. #20273227	C₂	semidirect
4. #20273223	C₂	semidirect
5. #20273219	C₂	semidirect
6. #20273201	C₂	semidirect
24. C₅₃×#533	C₄	semidirect
25. C₅₃×#533	C₄	semidirect
27. C₅₃×#532	C₄	semidirect
28. C₅₃×#532	C₄	semidirect
34. C₅₃×#529	C₄	semidirect
41. C₄×#41×C₅₃	C₄	semidirect
64. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
65. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
66. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
67. #127×C₅₃	C₂×C₄	semidirect
72. #125×C₅₃	C₂×C₄	semidirect
75. #124×C₅₃	C₂×C₄	semidirect
96. C₄×Q₈×C₅₃	C₂×C₄	semidirect
97. C₄×Q₈×C₅₃	C₂×C₄	semidirect
108. C₄×D₄×C₅₃	C₂×C₄	semidirect
109. C₄×D₄×C₅₃	C₂×C₄	semidirect
316. #533	Dic₅₃	semidirect
317. #533	Dic₅₃	semidirect
319. #532	Dic₅₃	semidirect
320. #532	Dic₅₃	semidirect
326. #529	Dic₅₃	semidirect
333. C₄×#41	Dic₅₃	semidirect
351. C₅₃	#17541	semidirect
360. C₄⋊Q₈	C₂×Dic₅₃	semidirect
361. C₄⋊Q₈	C₂×Dic₅₃	semidirect
362. C₄⋊Q₈	C₂×Dic₅₃	semidirect
363. #127	C₂×Dic₅₃	semidirect
368. #125	C₂×Dic₅₃	semidirect
371. #124	C₂×Dic₅₃	semidirect
392. C₄×Q₈	C₂×Dic₅₃	semidirect
393. C₄×Q₈	C₂×Dic₅₃	semidirect
404. C₄×D₄	C₂×Dic₅₃	semidirect
405. C₄×D₄	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×#16810469	2
#20273227	2
#20273223	2
#20273219	2
#20273201	2
C₅₃×#3444	2
#17541	53