Subgroups — #27387316

Order 13568.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133237	C₂	1	1	yes	maximal, characteristic
2	#20283357	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275363	C₂	1	2	yes	maximal, complemented
4	#20275314	C₂	1	2	yes	maximal, complemented
5	#20275042	C₂	1	2	yes	maximal
6	#20274990	C₂	1	2	yes	maximal, complemented
7	C₄×#13133152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13133143	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133144	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274460	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133237	C₂²	1	1	yes	characteristic
13	C₄×#11719945	C₂²	1	1	yes	characteristic
14	C₄×#11719925	C₄	1	1	yes	characteristic, complemented
15	C₄×#11719925	C₄	1	1	yes	characteristic, complemented
16	C₄×#11719924	C₄	1	1	yes	characteristic, complemented
17	C₄×#11719924	C₄	1	1	yes	characteristic, complemented
18	C₄×#11719923	C₄	1	1	yes	characteristic, complemented
19	C₄×#11719923	C₄	1	1	yes	characteristic, complemented
20	C₄×#11719922	C₄	1	1	yes	characteristic, complemented
21	C₄×#11719922	C₄	1	1	yes	characteristic, complemented
22	#16810682	—	2	2	no	—
23	#16810674	—	2	1	no	—
24	#16810674	—	2	1	no	—
25	C₂×#13133152	C₂²	1	2	yes	—
26	#16810659	C₂²	1	1	yes	characteristic
27	#16810649	C₂²	1	2	yes	—
28	C₂×#13133144	C₂²	1	2	yes	—
29	C₂×#13133143	C₂²	1	2	yes	—
30	#16810619	C₂²	1	1	yes	characteristic
31	#16810617	C₂²	1	1	yes	characteristic
32	#16810589	C₂²	1	2	yes	—
33	C₄×#11719917	C₂²	1	1	yes	characteristic
34	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
35	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
36	#16810387	C₂²	1	2	yes	—
37	#16810384	C₂²	1	2	yes	—
38	#16810382	C₂²	1	2	yes	—
39	#16810377	C₂²	1	2	yes	—
40	#16810377	C₂²	1	2	yes	—
41	C₂×#131×C₅₃	C₂²	1	2	yes	—
42	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#405	—	2	2	no	—
44	C₅₃×#405	C₂²	1	2	yes	—
45	C₄×M₁₆×C₅₃	—	2	1	no	—
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₅₃	—	2	1	no	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#11719945	C₂³	1	1	yes	characteristic
50	#13133237	—	2	4	no	—
51	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
52	C₂×#11719923	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719922	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719917	C₂³	1	2	yes	—
56	C₂²×#88322	C₂³	1	1	yes	characteristic
57	C₂²×#88322	C₂³	1	1	yes	characteristic
58	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
60	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
61	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
62	C₄×C₂²⋊D₅₃	—	2	1	no	—
63	C₄×C₂²⋊D₅₃	—	2	1	no	—
64	#13133176	C₂³	1	1	yes	characteristic
65	#13133174	—	2	1	no	—
66	C₂×C₄×Dic₅₃	—	2	1	no	—
67	C₂×C₄×Dic₅₃	—	2	1	no	—
68	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
69	C₄²×D₅₃	—	2	1	no	—
70	#13133152	—	2	4	no	—
71	#13133150	C₂×C₄	1	1	yes	characteristic
72	#13133150	C₂×C₄	1	1	yes	characteristic
73	#13133148	C₂×C₄	1	1	yes	characteristic
74	#13133148	C₂×C₄	1	1	yes	characteristic
75	#13133146	—	2	2	no	—
76	#13133146	—	2	2	no	—
77	#13133144	—	2	4	no	—
78	#13133143	—	2	4	no	—
79	#13133141	C₂×C₄	1	1	yes	characteristic
80	#13133141	C₂×C₄	1	1	yes	characteristic
81	#13133140	C₂×C₄	1	1	yes	characteristic
82	#13133140	C₂×C₄	1	1	yes	characteristic
83	#13133138	C₂×C₄	1	1	yes	characteristic
84	#13133138	C₂×C₄	1	1	yes	characteristic
85	#13133137	C₂×C₄	1	1	yes	characteristic
86	#13133137	C₂×C₄	1	1	yes	characteristic
87	#13133129	—	2	2	no	—
88	#13133129	—	2	2	no	—
89	#13133128	—	2	1	no	—
90	C₄×#88322	—	2	1	no	—
91	C₄×#88322	C₂×C₄	1	1	yes	characteristic
92	C₄×#88322	C₂×C₄	1	1	yes	characteristic
93	C₄×#88322	C₂×C₄	1	1	yes	characteristic
94	C₄×#88322	C₂×C₄	1	1	yes	characteristic
95	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
96	#131×C₅₃	—	2	4	no	—
97	#131×C₅₃	—	2	4	no	—
98	C₂×M₁₆×C₅₃	—	2	2	no	—
99	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
100	C₂²×C₈×C₅₃	—	2	2	no	—
101	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
102	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₅₃	—	2	1	no	—
105	C₄×D₄×C₅₃	—	2	1	no	—
106	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
108	#117×C₅₃	—	2	1	no	—
109	#117×C₅₃	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₅₃	—	2	1	no	—
111	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
112	#97×C₅₃	—	2	2	no	—
113	#97×C₅₃	—	2	1	no	—
114	C₄×C₈×C₅₃	—	2	2	no	—
115	C₄×C₈×C₅₃	—	2	1	no	—
116	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
117	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
118	C₄×C₈×C₅₃	D₄	1	2	yes	—
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	#41	1	4	yes	—
121	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄×C₅₃	—	2	1	no	—
123	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	—	2	1	no	—
127	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₂²⋊D₅₃	—	2	1	no	—
131	C₂²×Dic₅₃	—	2	2	no	—
132	#11719945	—	2	2	no	—
133	#11719945	—	2	2	no	—
134	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×D₅₃	—	2	1	no	—
136	C₂×C₄×D₅₃	—	2	1	no	—
137	C₂×C₄×D₅₃	—	2	1	no	—
138	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
139	M₁₆×C₅₃	—	2	8	no	—
140	M₁₆×C₅₃	—	2	2	no	—
141	M₁₆×C₅₃	#41	1	4	yes	—
142	C₂×C₈×C₅₃	—	2	8	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
148	C₄⋊C₄×C₅₃	—	2	1	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²⋊C₄×C₅₃	—	2	1	no	—
153	C₂²⋊C₄×C₅₃	—	2	1	no	—
154	C₂²⋊C₄×C₅₃	—	2	1	no	—
155	C₄²×C₅₃	—	2	1	no	—
156	C₄²×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	1	no	—
158	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²⋊Dic₅₃	—	2	1	no	—
163	#11719925	—	2	2	no	—
164	#11719924	—	2	2	no	—
165	#11719923	—	2	2	no	—
166	#11719922	—	2	2	no	—
167	#11719921	—	2	1	no	—
168	#11719921	—	2	1	no	—
169	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
171	#11719919	—	2	1	no	—
172	#11719919	—	2	1	no	—
173	C₄×Dic₅₃	—	2	4	no	—
174	C₄×Dic₅₃	—	2	1	no	—
175	C₄×Dic₅₃	—	2	1	no	—
176	C₄×Dic₅₃	—	2	1	no	—
177	C₄×Dic₅₃	—	2	2	no	—
178	C₄×Dic₅₃	—	2	2	no	—
179	C₄×Dic₅₃	—	4	2	no	—
180	#11719917	—	2	2	no	—
181	#11719917	—	2	2	no	—
182	C₂×#88322	—	2	2	no	—
183	C₂×#88322	—	2	2	no	—
184	C₂×#88322	—	2	2	no	—
185	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
186	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
187	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
188	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
189	C₂³×C₅₃	—	2	1	no	—
190	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
191	C₂²×D₅₃	—	2	1	no	—
192	Q₈×C₅₃	—	2	1	no	—
193	Q₈×C₅₃	#117	1	2	yes	—
194	D₄×C₅₃	—	2	4	no	—
195	D₄×C₅₃	—	2	1	no	—
196	D₄×C₅₃	#117	1	2	yes	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	2	no	—
205	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
208	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
212	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
213	C₂²⋊D₅₃	—	4	2	no	—
214	C₂×Dic₅₃	—	2	4	no	—
215	C₂×Dic₅₃	—	2	1	no	—
216	C₂×Dic₅₃	—	2	1	no	—
217	C₂×Dic₅₃	—	2	1	no	—
218	C₂×Dic₅₃	—	4	2	no	—
219	C₂×Dic₅₃	—	4	2	no	—
220	D₂₁₂	—	2	1	no	—
221	D₂₁₂	—	2	1	no	—
222	C₄×D₅₃	—	4	2	no	—
223	C₄×D₅₃	—	4	2	no	—
224	C₄×D₅₃	—	4	2	no	—
225	Dic₁₀₆	—	2	1	no	—
226	Dic₁₀₆	—	2	1	no	—
227	C₈×C₅₃	—	2	2	no	—
228	C₈×C₅₃	—	2	4	no	—
229	C₈×C₅₃	—	2	2	no	—
230	#88322	—	2	1	no	—
231	#88322	—	2	1	no	—
232	#88322	—	2	1	no	—
233	#88322	—	2	1	no	—
234	#88322	—	4	2	no	—
235	#21137	—	53	1	no	maximal, Sylow
236	C₂²×C₅₃	—	2	1	no	—
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	2	2	no	—
239	C₂²×C₅₃	—	4	1	no	—
240	C₂²×C₅₃	—	4	1	no	—
241	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
242	C₂²×C₅₃	#529	1	2	yes	—
243	C₂×D₅₃	—	4	1	no	—
244	C₂×D₅₃	—	4	1	no	—
245	C₂×D₅₃	—	4	1	no	—
246	C₄×C₅₃	—	2	1	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	4	1	no	—
250	C₄×C₅₃	—	4	1	no	—
251	C₄×C₅₃	#443	1	2	yes	—
252	C₄×C₅₃	#444	1	2	yes	—
253	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
254	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
255	C₄×C₅₃	#525	1	2	yes	—
256	Dic₅₃	—	4	4	no	—
257	Dic₅₃	—	4	1	no	—
258	Dic₅₃	—	4	1	no	—
259	Dic₅₃	—	4	1	no	—
260	C₄×#135	—	53	1	no	—
261	#2922	—	53	1	no	—
262	C₄×#131	D₅₃	1	1	yes	characteristic, complemented
263	#1929	—	53	2	no	—
264	#1922	—	53	2	no	—
265	#1876	—	53	2	no	—
266	#1866	—	53	2	no	—
267	C₄×#108	—	53	1	no	—
268	C₄×#104	—	53	1	no	—
269	C₄×#104	—	53	1	no	—
270	#1744	—	53	1	no	—
271	C₂×C₅₃	—	2	2	no	—
272	C₂×C₅₃	—	4	2	no	—
273	C₂×C₅₃	#2290	1	1	yes	characteristic
274	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
275	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
276	D₅₃	—	8	1	no	—
277	C₂×#135	—	53	1	no	—
278	C₂×#131	C₂×D₅₃	1	2	yes	—
279	C₄×#41	—	53	1	no	—
280	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
281	C₄×Q₁₆	—	53	1	no	—
282	C₄×Q₁₆	—	53	1	no	—
283	C₄×SD₁₆	—	53	1	no	—
284	C₄×SD₁₆	—	53	1	no	—
285	C₄×SD₁₆	—	53	1	no	—
286	C₄×SD₁₆	—	53	1	no	—
287	C₄×D₈	—	53	1	no	—
288	C₄×D₈	—	53	1	no	—
289	#433	—	106	2	no	—
290	#431	—	106	1	no	—
291	#431	—	106	1	no	—
292	C₂×#108	—	53	2	no	—
293	#427	—	53	1	no	—
294	#424	—	53	2	no	—
295	C₂×#104	—	53	2	no	—
296	C₂×#104	—	53	2	no	—
297	#416	—	53	1	no	—
298	#416	—	53	1	no	—
299	#408	—	53	2	no	—
300	#405	—	2	2	no	—
301	#405	C₂×D₅₃	1	2	yes	—
302	C₄×M₁₆	—	2	1	no	—
303	C₄×M₁₆	—	53	1	no	—
304	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
305	C₂×C₄×C₈	—	2	1	no	—
306	C₂×C₄×C₈	—	53	1	no	—
307	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
308	C₄³	—	53	1	no	—
309	#341	—	53	2	no	—
310	#339	—	53	2	no	—
311	#339	—	53	2	no	—
312	#336	—	53	2	no	—
313	#336	—	53	2	no	—
314	C₅₃	#21137	1	1	yes	characteristic, complemented, Sylow
315	C₂×#41	—	53	1	no	—
316	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
317	#135	—	106	4	no	—
318	C₂×Q₁₆	—	53	1	no	—
319	C₂×SD₁₆	—	53	1	no	—
320	C₂×SD₁₆	—	53	1	no	—
321	C₂×D₈	—	53	1	no	—
322	#131	—	2	4	no	—
323	#131	—	2	4	no	—
324	C₂×M₁₆	—	2	2	no	—
325	C₂×M₁₆	—	53	2	no	—
326	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
327	C₂²×C₈	—	2	2	no	—
328	C₂²×C₈	—	53	1	no	—
329	C₂²×C₈	—	53	1	no	—
330	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
331	C₄×Q₈	—	53	1	no	—
332	C₄×Q₈	—	53	1	no	—
333	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
334	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	53	1	no	—
338	C₄×D₄	—	53	1	no	—
339	C₄×D₄	—	106	1	no	—
340	C₄×D₄	—	106	1	no	—
341	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
342	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
343	#117	—	2	1	no	—
344	#117	—	53	1	no	—
345	#117	—	106	1	no	—
346	#117	C₂²×D₅₃	1	1	yes	characteristic
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	53	2	no	—
349	C₂×C₄²	—	106	1	no	—
350	C₂×C₄²	—	106	1	no	—
351	C₂×C₄²	—	106	1	no	—
352	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
353	#108	—	106	4	no	—
354	#107	—	53	1	no	—
355	#107	—	53	1	no	—
356	#106	—	53	1	no	—
357	#106	—	53	1	no	—
358	C₄⋊C₈	—	106	2	no	—
359	C₄⋊C₈	—	106	2	no	—
360	#104	—	106	4	no	—
361	#104	—	106	4	no	—
362	#103	—	53	1	no	—
363	#103	—	53	1	no	—
364	#103	—	53	1	no	—
365	#103	—	53	1	no	—
366	#102	—	53	1	no	—
367	#102	—	53	1	no	—
368	#102	—	53	1	no	—
369	#102	—	53	1	no	—
370	C₂²⋊C₈	—	106	2	no	—
371	C₂²⋊C₈	—	106	2	no	—
372	#97	—	2	2	no	—
373	#97	—	2	1	no	—
374	#97	—	106	1	no	—
375	C₄×C₈	—	2	2	no	—
376	C₄×C₈	—	2	1	no	—
377	C₄×C₈	—	53	1	no	—
378	C₄×C₈	—	53	1	no	—
379	C₄×C₈	—	53	1	no	—
380	C₄×C₈	—	53	1	no	—
381	C₄×C₈	—	106	1	no	—
382	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
383	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
384	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
385	#41	—	2	2	no	—
386	#41	—	106	2	no	—
387	#41	—	106	2	no	—
388	#41	#11719945	1	4	yes	—
389	C₂×Q₈	—	53	1	no	—
390	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	53	1	no	—
393	C₂×D₄	—	106	1	no	—
394	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	106	2	no	—
399	C₂²×C₄	—	106	1	no	—
400	C₂²×C₄	—	106	1	no	—
401	C₂²×C₄	—	106	1	no	—
402	C₂²×C₄	#11719945	1	1	yes	characteristic
403	C₂²×C₄	#11719945	1	1	yes	characteristic
404	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
405	Q₁₆	—	106	2	no	—
406	SD₁₆	—	106	2	no	—
407	SD₁₆	—	106	2	no	—
408	D₈	—	106	2	no	—
409	M₁₆	—	2	8	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	106	2	no	—
412	M₁₆	—	106	2	no	—
413	M₁₆	#11719945	1	4	yes	—
414	C₂×C₈	—	2	8	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	53	1	no	—
419	C₂×C₈	—	53	1	no	—
420	C₂×C₈	—	53	1	no	—
421	C₂×C₈	—	53	1	no	—
422	C₂×C₈	—	106	2	no	—
423	C₂×C₈	—	106	2	no	—
424	C₂×C₈	—	106	2	no	—
425	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
426	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	53	1	no	—
430	C₄⋊C₄	—	53	1	no	—
431	C₄⋊C₄	—	106	1	no	—
432	C₄⋊C₄	—	106	1	no	—
433	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
434	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	2	1	no	—
438	C₂²⋊C₄	—	106	1	no	—
439	C₂²⋊C₄	—	106	1	no	—
440	C₂²⋊C₄	—	106	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	106	4	no	—
445	C₄²	—	106	1	no	—
446	C₄²	—	106	1	no	—
447	C₄²	—	106	1	no	—
448	C₄²	—	106	2	no	—
449	C₄²	—	106	2	no	—
450	C₄²	—	212	2	no	—
451	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
452	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
453	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
454	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
455	C₂³	—	2	1	no	—
456	C₂³	—	106	1	no	—
457	C₂³	C₂×#11719945	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	—	106	1	no	—
460	Q₈	—	106	1	no	—
461	Q₈	#13133174	1	2	yes	—
462	D₄	—	2	4	no	—
463	D₄	—	2	1	no	—
464	D₄	—	106	1	no	—
465	D₄	—	106	1	no	—
466	D₄	—	212	2	no	—
467	D₄	#13133174	1	2	yes	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	106	4	no	—
477	C₂×C₄	—	106	1	no	—
478	C₂×C₄	—	106	1	no	—
479	C₂×C₄	—	106	1	no	—
480	C₂×C₄	—	212	2	no	—
481	C₂×C₄	—	212	2	no	—
482	C₂×C₄	—	212	2	no	—
483	C₂×C₄	—	212	2	no	—
484	C₂×C₄	—	212	2	no	—
485	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
486	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
487	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
488	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
489	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
491	C₂×C₄	C₂×#11719945	1	2	yes	—
492	C₂×C₄	C₂×#11719945	1	2	yes	—
493	C₈	—	2	2	no	—
494	C₈	—	2	4	no	—
495	C₈	—	2	2	no	—
496	C₈	—	106	1	no	—
497	C₈	—	106	1	no	—
498	C₈	—	106	1	no	—
499	C₈	—	106	1	no	—
500	C₈	—	212	2	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	212	1	no	—
507	C₂²	—	212	1	no	—
508	C₂²	—	212	1	no	—
509	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
510	C₂²	#16811157	1	2	yes	—
511	C₄	—	2	2	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	4	1	no	—
516	C₄	—	212	4	no	—
517	C₄	—	212	1	no	—
518	C₄	—	212	1	no	—
519	C₄	—	212	1	no	—
520	C₄	#16810738	1	2	yes	—
521	C₄	#16810743	1	2	yes	—
522	C₄	C₂×#13133174	1	1	yes	characteristic
523	C₄	C₄×#11719945	1	1	yes	characteristic
524	C₄	#16811134	1	2	yes	—
525	C₂	—	2	2	no	—
526	C₂	—	4	2	no	—
527	C₂	—	424	1	no	—
528	C₂	#20277158	1	1	yes	characteristic
529	C₂	C₂×#16810738	1	1	yes	characteristic
530	C₂	C₂×#16810743	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#21137
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20283357	C₂	semidirect
3. #20275363	C₂	semidirect
4. #20275314	C₂	semidirect
6. #20274990	C₂	semidirect
7. C₄×#13133152	C₂	semidirect
8. C₄×#13133143	C₂	semidirect
9. C₄×#13133144	C₂	semidirect
10. #20274460	C₂	semidirect
11. C₄×#131×C₅₃	C₂	semidirect
14. C₄×#11719925	C₄	semidirect
15. C₄×#11719925	C₄	semidirect
16. C₄×#11719924	C₄	semidirect
17. C₄×#11719924	C₄	semidirect
18. C₄×#11719923	C₄	semidirect
19. C₄×#11719923	C₄	semidirect
20. C₄×#11719922	C₄	semidirect
21. C₄×#11719922	C₄	semidirect
262. C₄×#131	D₅₃	semidirect
314. C₅₃	#21137	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133237	2
#20283357	2
#20275363	2
#20275314	2
#20275042	2
#20274990	2
C₄×#13133152	2
C₄×#13133143	2
C₄×#13133144	2
#20274460	2
C₄×#131×C₅₃	2
#21137	53