Subgroups — #27353131

Order 13568.

666 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283834	C₂	1	2	yes	maximal
2	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
3	#20273483	C₂	1	2	yes	maximal, complemented
4	#20273481	C₂	1	2	yes	maximal, complemented
5	#20273475	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273471	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273469	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273466	C₂	1	2	yes	maximal, complemented
9	#20273460	C₂	1	1	yes	maximal, characteristic, complemented
10	#334×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	C₄	1	1	yes	characteristic
13	C₂×#13133223	C₄	1	1	yes	characteristic
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	#16810751	—	2	2	no	—
16	#16810698	—	2	2	no	—
17	#16810697	—	2	2	no	—
18	#16810696	—	2	2	no	—
19	#16810679	C₂²	1	2	yes	—
20	#16810675	C₂²	1	2	yes	—
21	#16810647	C₂²	1	2	yes	—
22	#16810645	C₂²	1	2	yes	—
23	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
24	#16810578	C₄	1	1	yes	characteristic
25	#16810578	C₄	1	1	yes	characteristic
26	#16810577	C₄	1	1	yes	characteristic
27	#16810577	C₄	1	1	yes	characteristic
28	#16810577	C₄	1	1	yes	characteristic
29	#16810577	C₄	1	1	yes	characteristic
30	C₄×#11719917	C₂²	1	1	yes	characteristic
31	C₄×#11719912	C₂²	1	1	yes	characteristic
32	C₄×#11719912	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	#16810375	C₂²	1	2	yes	—
36	#16810375	C₂²	1	2	yes	—
37	#16810374	C₂²	1	1	yes	characteristic
38	#16810374	C₂²	1	1	yes	characteristic
39	#16810374	C₂²	1	2	yes	—
40	#16810373	C₂²	1	2	yes	—
41	#16810373	C₂²	1	2	yes	—
42	C₅₃×#432	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#335	C₂²	1	2	yes	—
45	C₅₃×#334	C₂²	1	1	yes	characteristic
46	C₅₃×#334	C₂²	1	1	yes	characteristic
47	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
48	#13133223	—	2	4	no	—
49	#13133223	—	2	4	no	—
50	C₂×#11719917	—	2	2	no	—
51	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
52	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719912	—	2	2	no	—
54	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
56	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
57	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₅₃	—	2	2	no	—
59	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
64	C₄×Dic₁₀₆	Q₈	1	1	yes	characteristic
65	C₄×D₂₁₂	D₄	1	1	yes	characteristic
66	C₄×D₂₁₂	Q₈	1	1	yes	characteristic
67	C₄×C₂²⋊D₅₃	D₄	1	1	yes	characteristic
68	C₄×C₂²⋊D₅₃	D₄	1	1	yes	characteristic
69	C₄×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
70	C₄×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
71	#13133176	C₂×C₄	1	1	yes	characteristic
72	#13133174	C₂×C₄	1	1	yes	characteristic
73	#13133174	C₂×C₄	1	1	yes	characteristic
74	C₄²×D₅₃	C₂³	1	1	yes	characteristic
75	C₄²×D₅₃	C₂³	1	1	yes	characteristic
76	C₄⋊#88322	—	2	2	no	—
77	C₄⋊#88322	C₂³	1	2	yes	—
78	C₄⋊#88322	C₂³	1	2	yes	—
79	#13133146	—	2	2	no	—
80	#13133146	—	2	2	no	—
81	#13133129	—	2	2	no	—
82	#13133129	—	2	2	no	—
83	#13133128	C₂×C₄	1	1	yes	characteristic
84	#13133128	C₂×C₄	1	1	yes	characteristic
85	#13133127	—	2	2	no	—
86	#13133127	C₂×C₄	1	1	yes	characteristic
87	#13133127	C₂×C₄	1	1	yes	characteristic
88	#13133127	C₂×C₄	1	1	yes	characteristic
89	#13133127	C₂×C₄	1	1	yes	characteristic
90	C₄×#88322	C₂³	1	1	yes	characteristic
91	C₄×#88322	C₂³	1	1	yes	characteristic
92	C₈×Dic₅₃	—	2	2	no	—
93	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
95	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
96	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
101	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
102	C₂²⋊C₈×C₅₃	—	2	2	no	—
103	#97×C₅₃	—	2	2	no	—
104	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
113	#11719945	—	2	2	no	—
114	#11719945	—	2	2	no	—
115	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
122	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
134	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
135	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
136	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
137	#11719921	C₄⋊C₄	1	1	yes	characteristic
138	#11719921	C₄⋊C₄	1	1	yes	characteristic
139	#11719921	C₄⋊C₄	1	1	yes	characteristic
140	#11719921	C₄⋊C₄	1	1	yes	characteristic
141	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
142	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
143	#11719919	C₄⋊C₄	1	1	yes	characteristic
144	#11719919	C₄⋊C₄	1	1	yes	characteristic
145	#11719919	C₄⋊C₄	1	1	yes	characteristic
146	#11719919	C₄⋊C₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
153	#11719917	—	2	2	no	—
154	#11719917	—	2	2	no	—
155	#11719917	—	8	2	no	—
156	C₂×#88322	—	2	2	no	—
157	C₂×#88322	—	2	2	no	—
158	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
159	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
160	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
161	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
162	#11719912	—	2	2	no	—
163	#11719912	—	2	2	no	—
164	#11719912	—	2	2	no	—
165	#11719912	—	8	2	no	—
166	#11719912	M₁₆	1	4	yes	complemented
167	C₈×D₅₃	—	2	2	no	—
168	C₈×D₅₃	—	2	2	no	—
169	C₈×D₅₃	—	2	2	no	—
170	C₈×D₅₃	—	8	2	no	—
171	C₈×D₅₃	M₁₆	1	4	yes	complemented
172	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
185	C₂²⋊D₅₃	—	2	2	no	—
186	C₂²⋊D₅₃	—	2	2	no	—
187	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
193	D₂₁₂	—	2	2	no	—
194	C₄×D₅₃	—	2	1	no	—
195	C₄×D₅₃	—	2	1	no	—
196	C₄×D₅₃	—	2	2	no	—
197	C₄×D₅₃	—	2	2	no	—
198	C₄×D₅₃	—	4	1	no	—
199	C₄×D₅₃	—	4	1	no	—
200	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
201	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
202	Dic₁₀₆	—	2	2	no	—
203	C₈×C₅₃	—	2	1	no	—
204	C₈×C₅₃	—	2	1	no	—
205	C₈×C₅₃	—	4	2	no	—
206	C₈×C₅₃	—	4	2	no	—
207	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
208	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
209	#88322	—	2	1	no	—
210	#88322	—	2	1	no	—
211	#88322	—	4	2	no	—
212	#88322	—	4	2	no	—
213	#88322	C₂×M₁₆	1	2	yes	—
214	#88322	C₂×M₁₆	1	2	yes	—
215	#17798	—	53	1	no	maximal, Sylow
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
220	C₂×D₅₃	—	2	1	no	—
221	C₂×D₅₃	—	2	1	no	—
222	C₂×D₅₃	—	2	1	no	—
223	C₂×D₅₃	—	2	1	no	—
224	C₂×D₅₃	—	2	1	no	—
225	C₂×D₅₃	#423	1	2	yes	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	2	1	no	—
228	C₄×C₅₃	—	2	1	no	—
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
231	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
232	C₄×C₅₃	#577	1	2	yes	—
233	Dic₅₃	—	2	1	no	—
234	Dic₅₃	—	2	1	no	—
235	Dic₅₃	—	2	1	no	—
236	Dic₅₃	—	2	1	no	—
237	Dic₅₃	—	2	1	no	—
238	Dic₅₃	#423	1	2	yes	—
239	#2989	—	53	2	no	—
240	C₄×#131	—	53	1	no	—
241	#1569	—	53	2	no	—
242	#1569	—	53	2	no	—
243	#1568	—	53	1	no	—
244	#1568	D₅₃	1	1	yes	characteristic, complemented
245	#1566	—	53	1	no	—
246	#1566	—	53	1	no	—
247	#1565	—	53	2	no	—
248	#1564	—	53	1	no	—
249	C₂×#334	—	53	1	no	—
250	C₂×C₅₃	—	4	1	no	—
251	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
252	C₂×C₅₃	#2912	1	1	yes	characteristic
253	C₂×C₅₃	#2916	1	1	yes	characteristic
254	D₅₃	—	2	2	no	—
255	D₅₃	—	4	1	no	—
256	C₂×#131	—	53	1	no	—
257	C₂×#131	—	53	1	no	—
258	C₄×#41	—	53	1	no	—
259	#446	—	106	2	no	—
260	#436	—	106	2	no	—
261	#435	—	106	2	no	—
262	#435	—	106	2	no	—
263	#432	—	53	2	no	—
264	#432	C₂×D₅₃	1	2	yes	—
265	#431	—	53	2	no	—
266	#423	—	53	2	no	—
267	#423	—	53	2	no	—
268	C₂×C₄⋊C₈	—	53	2	no	—
269	#405	—	53	1	no	—
270	#405	—	53	1	no	—
271	#405	—	53	1	no	—
272	#405	—	53	1	no	—
273	#405	—	53	1	no	—
274	#405	—	53	1	no	—
275	C₄×M₁₆	—	53	1	no	—
276	C₄×M₁₆	—	53	1	no	—
277	C₄×M₁₆	—	53	1	no	—
278	C₂×C₄×C₈	—	53	1	no	—
279	C₂×C₄×C₈	—	53	1	no	—
280	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
281	#335	—	53	2	no	—
282	#335	—	53	2	no	—
283	#335	C₂×D₅₃	1	2	yes	—
284	#334	—	53	2	no	—
285	#334	—	53	2	no	—
286	#334	—	53	1	no	—
287	#334	—	53	1	no	—
288	#334	—	53	2	no	—
289	#334	C₂×D₅₃	1	1	yes	characteristic
290	#334	C₂×D₅₃	1	1	yes	characteristic
291	C₅₃	#17798	1	1	yes	characteristic, complemented, Sylow
292	C₂×#41	—	53	1	no	—
293	#131	—	106	4	no	—
294	#131	—	106	4	no	—
295	C₂×M₁₆	—	53	1	no	—
296	C₂×M₁₆	—	53	1	no	—
297	C₂×M₁₆	—	53	1	no	—
298	C₂×M₁₆	—	53	1	no	—
299	C₂×M₁₆	—	53	1	no	—
300	C₂×M₁₆	—	53	1	no	—
301	C₂×M₁₆	—	106	2	no	—
302	C₂×M₁₆	—	106	2	no	—
303	C₂²×C₈	—	53	1	no	—
304	C₂²×C₈	—	53	1	no	—
305	C₂²×C₈	—	53	1	no	—
306	C₂²×C₈	—	53	1	no	—
307	C₂²×C₈	—	106	2	no	—
308	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
309	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
310	C₄×Q₈	—	53	1	no	—
311	C₄×Q₈	—	53	1	no	—
312	C₄×D₄	—	53	1	no	—
313	C₄×D₄	—	53	1	no	—
314	C₄×D₄	—	53	1	no	—
315	C₄×D₄	—	53	1	no	—
316	C₄×D₄	—	53	1	no	—
317	C₄×D₄	—	53	1	no	—
318	#117	—	53	1	no	—
319	#117	—	53	1	no	—
320	#117	—	53	1	no	—
321	C₂×C₄²	—	53	1	no	—
322	C₂×C₄²	—	53	1	no	—
323	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
324	C₄⋊C₈	—	53	2	no	—
325	C₄⋊C₈	—	53	2	no	—
326	C₄⋊C₈	—	106	2	no	—
327	C₄⋊C₈	—	106	2	no	—
328	C₄⋊C₈	—	106	2	no	—
329	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
330	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
331	C₂²⋊C₈	—	2	2	no	—
332	C₂²⋊C₈	—	106	2	no	—
333	C₂²⋊C₈	—	106	2	no	—
334	#97	—	2	2	no	—
335	#97	—	53	1	no	—
336	#97	—	53	1	no	—
337	#97	—	53	1	no	—
338	#97	—	53	1	no	—
339	#97	—	53	1	no	—
340	#97	—	53	1	no	—
341	#97	—	106	2	no	—
342	C₄×C₈	—	53	1	no	—
343	C₄×C₈	—	53	1	no	—
344	C₄×C₈	—	53	1	no	—
345	C₄×C₈	—	53	1	no	—
346	C₄×C₈	—	53	1	no	—
347	C₄×C₈	—	53	1	no	—
348	C₄×C₈	—	106	2	no	—
349	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
350	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
351	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
352	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
353	#41	—	106	2	no	—
354	#41	—	106	2	no	—
355	C₂×Q₈	—	53	1	no	—
356	C₂×D₄	—	53	1	no	—
357	C₂×D₄	—	53	1	no	—
358	C₂×D₄	—	53	1	no	—
359	C₂²×C₄	—	53	1	no	—
360	C₂²×C₄	—	53	1	no	—
361	C₂²×C₄	—	53	1	no	—
362	C₂²×C₄	—	53	1	no	—
363	C₂²×C₄	—	53	1	no	—
364	C₂²×C₄	—	53	1	no	—
365	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
366	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
367	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
368	M₁₆	—	53	4	no	—
369	M₁₆	—	106	2	no	—
370	M₁₆	—	106	2	no	—
371	M₁₆	—	106	2	no	—
372	M₁₆	—	106	2	no	—
373	M₁₆	—	106	2	no	—
374	M₁₆	—	424	2	no	—
375	M₁₆	—	424	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	53	1	no	—
381	C₂×C₈	—	53	1	no	—
382	C₂×C₈	—	53	4	no	—
383	C₂×C₈	—	53	1	no	—
384	C₂×C₈	—	53	1	no	—
385	C₂×C₈	—	106	2	no	—
386	C₂×C₈	—	106	2	no	—
387	C₂×C₈	—	106	2	no	—
388	C₂×C₈	—	106	2	no	—
389	C₂×C₈	—	106	2	no	—
390	C₂×C₈	—	424	2	no	—
391	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
392	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
393	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
394	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
395	C₄⋊C₄	—	53	1	no	—
396	C₄⋊C₄	—	53	1	no	—
397	C₄⋊C₄	—	53	1	no	—
398	C₄⋊C₄	—	53	1	no	—
399	C₄⋊C₄	—	53	1	no	—
400	C₄⋊C₄	—	53	1	no	—
401	C₂²⋊C₄	—	53	1	no	—
402	C₂²⋊C₄	—	53	1	no	—
403	C₂²⋊C₄	—	53	1	no	—
404	C₂²⋊C₄	—	53	1	no	—
405	C₂²⋊C₄	—	53	1	no	—
406	C₂²⋊C₄	—	53	1	no	—
407	C₄²	—	53	1	no	—
408	C₄²	—	53	1	no	—
409	C₄²	—	53	1	no	—
410	C₄²	—	53	1	no	—
411	C₄²	—	53	1	no	—
412	C₄²	—	53	1	no	—
413	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
414	C₄²	D₄×D₅₃	1	1	yes	characteristic
415	C₄²	Q₈×D₅₃	1	1	yes	characteristic
416	C₄²	C₂³×D₅₃	1	1	yes	characteristic
417	C₂³	—	53	1	no	—
418	C₂³	—	53	1	no	—
419	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
420	Q₈	—	106	2	no	—
421	D₄	—	106	2	no	—
422	D₄	—	106	2	no	—
423	D₄	—	106	2	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	53	1	no	—
429	C₂×C₄	—	53	2	no	—
430	C₂×C₄	—	53	2	no	—
431	C₂×C₄	—	53	1	no	—
432	C₂×C₄	—	53	1	no	—
433	C₂×C₄	—	53	1	no	—
434	C₂×C₄	—	53	1	no	—
435	C₂×C₄	—	53	1	no	—
436	C₂×C₄	—	106	1	no	—
437	C₂×C₄	—	106	1	no	—
438	C₂×C₄	—	106	2	no	—
439	C₂×C₄	—	106	2	no	—
440	C₂×C₄	—	212	1	no	—
441	C₂×C₄	—	212	1	no	—
442	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
443	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
444	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
445	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
446	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
447	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
448	C₈	—	2	1	no	—
449	C₈	—	2	1	no	—
450	C₈	—	4	2	no	—
451	C₈	—	4	2	no	—
452	C₈	—	53	2	no	—
453	C₈	—	53	2	no	—
454	C₈	—	106	1	no	—
455	C₈	—	106	1	no	—
456	C₈	—	212	2	no	—
457	C₈	—	212	2	no	—
458	C₈	M₁₆×D₅₃	1	2	yes	—
459	C₈	M₁₆×D₅₃	1	2	yes	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	53	2	no	—
464	C₂²	—	106	1	no	—
465	C₂²	—	106	1	no	—
466	C₂²	—	106	1	no	—
467	C₂²	—	106	1	no	—
468	C₂²	—	106	1	no	—
469	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	53	2	no	—
475	C₄	—	106	1	no	—
476	C₄	—	106	1	no	—
477	C₄	—	106	1	no	—
478	C₄	—	106	1	no	—
479	C₄	—	106	1	no	—
480	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
481	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
482	C₄	#16811430	1	2	yes	—
483	C₂	—	4	1	no	—
484	C₂	—	106	2	no	—
485	C₂	—	212	1	no	—
486	C₂	#423×D₅₃	1	1	yes	characteristic
487	C₂	#20283307	1	1	yes	characteristic
488	C₂	#20283323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1568	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#17798
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #20273483	C₂	semidirect
4. #20273481	C₂	semidirect
5. #20273475	C₂	semidirect
6. #20273471	C₂	semidirect
7. #20273469	C₂	semidirect
8. #20273466	C₂	semidirect
9. #20273460	C₂	semidirect
10. #334×D₅₃	C₂	semidirect
11. C₅₃×#1568	C₂	semidirect
166. #11719912	M₁₆	semidirect
171. C₈×D₅₃	M₁₆	semidirect
244. #1568	D₅₃	semidirect
291. C₅₃	#17798	semidirect

Maximal subgroups

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

subgroup	index
#20283834	2
C₄×#13133223	2
#20273483	2
#20273481	2
#20273475	2
#20273471	2
#20273469	2
#20273466	2
#20273460	2
#334×D₅₃	2
C₅₃×#1568	2
#17798	53