Subgroups — #27391031

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284435	C₂	1	1	yes	maximal, characteristic
2	#20283616	C₂	1	2	yes	maximal, complemented
3	C₂×#16810741	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275613	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275505	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275384	C₂	1	2	yes	maximal, complemented
7	#20275216	C₂	1	2	yes	maximal, complemented
8	#20274769	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274516	C₂	1	2	yes	maximal, complemented
11	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811421	C₂²	1	2	yes	—
13	C₂×#13133239	C₂²	1	1	yes	characteristic
14	C₂×#13133223	C₂²	1	2	yes	—
15	#16810898	C₄	1	2	yes	complemented
16	#16810893	C₄	1	2	yes	complemented
17	#16810886	C₄	1	4	yes	complemented
18	#16810743	—	2	4	no	—
19	#16810741	—	2	1	no	—
20	#16810741	—	2	1	no	—
21	#16810741	—	2	1	no	—
22	#16810741	—	2	1	no	—
23	#16810738	—	2	4	no	—
24	#16810671	C₂²	1	2	yes	—
25	C₂×#13133153	C₂²	1	1	yes	characteristic
26	C₂×#13133153	C₂²	1	1	yes	characteristic
27	#16810661	C₂²	1	1	yes	characteristic
28	#16810640	C₂²	1	2	yes	—
29	#16810639	C₂²	1	2	yes	—
30	C₂×#13133145	C₂²	1	2	yes	—
31	#16810605	C₂²	1	2	yes	—
32	#16810596	C₂²	1	2	yes	—
33	#16810578	C₂²	1	2	yes	—
34	#16810395	C₂²	1	2	yes	—
35	#16810393	C₂²	1	2	yes	—
36	#16810383	C₂²	1	2	yes	—
37	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#405	—	2	1	no	—
41	C₅₃×#405	—	2	1	no	—
42	C₅₃×#405	C₂²	1	1	yes	characteristic
43	C₅₃×#405	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₅₃	—	2	1	no	—
45	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	—	2	1	no	—
47	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
48	C₂×#11719945	C₂³	1	2	yes	—
49	#13133248	—	2	2	no	—
50	#13133247	C₂×C₄	1	4	yes	—
51	#13133243	—	2	2	no	—
52	#13133242	C₂×C₄	1	4	yes	—
53	#13133239	—	2	1	no	—
54	#13133239	—	2	1	no	—
55	#13133239	—	2	1	no	—
56	#13133239	—	2	1	no	—
57	#13133237	—	2	4	no	—
58	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic, complemented
59	C₂×#11719913	C₂×C₄	1	2	yes	complemented
60	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
61	#13133223	—	2	2	no	—
62	#13133223	—	2	2	no	—
63	#13133223	—	2	2	no	—
64	#13133223	—	2	2	no	—
65	C₂×#11719917	C₂³	1	2	yes	—
66	C₂×#11719917	C₂³	1	2	yes	—
67	C₂×#11719912	—	2	2	no	—
68	C₂×C₈×D₅₃	—	2	2	no	—
69	#13133176	C₂³	1	2	yes	—
70	#13133153	—	2	1	no	—
71	#13133153	—	2	1	no	—
72	#13133153	C₂×C₄	1	2	yes	complemented
73	#13133153	C₂×C₄	1	2	yes	complemented
74	#13133152	—	2	4	no	—
75	#13133151	—	2	1	no	—
76	#13133149	—	2	1	no	—
77	#13133145	—	2	2	no	—
78	#13133145	—	2	2	no	—
79	#13133144	—	2	4	no	—
80	#13133143	—	2	4	no	—
81	#13133135	C₂×C₄	1	4	yes	complemented
82	#13133133	C₂×C₄	1	4	yes	complemented
83	#13133131	—	2	4	no	—
84	#13133127	—	2	2	no	—
85	C₈×Dic₅₃	—	2	2	no	—
86	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
87	#131×C₅₃	—	2	1	no	—
88	#131×C₅₃	—	2	1	no	—
89	#131×C₅₃	—	2	1	no	—
90	#131×C₅₃	—	2	1	no	—
91	#131×C₅₃	C₂×C₄	1	2	yes	—
92	#131×C₅₃	C₂×C₄	1	2	yes	—
93	#131×C₅₃	D₄	1	2	yes	—
94	#131×C₅₃	D₄	1	2	yes	—
95	C₂×M₁₆×C₅₃	—	2	1	no	—
96	C₂×M₁₆×C₅₃	—	2	1	no	—
97	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₅₃	—	2	1	no	—
100	C₂²×C₈×C₅₃	—	2	1	no	—
101	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₄×Q₈×C₅₃	—	2	1	no	—
104	C₄×D₄×C₅₃	—	2	2	no	—
105	C₄×D₄×C₅₃	—	2	1	no	—
106	#117×C₅₃	—	2	1	no	—
107	#117×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₅₃	—	2	1	no	—
109	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
110	#97×C₅₃	—	2	1	no	—
111	#97×C₅₃	—	2	1	no	—
112	#97×C₅₃	—	2	1	no	—
113	C₄×C₈×C₅₃	—	2	1	no	—
114	C₄×C₈×C₅₃	—	2	1	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	1	yes	characteristic
119	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
120	#41×C₅₃	—	2	1	no	—
121	#41×C₅₃	—	2	1	no	—
122	#41×C₅₃	C₂²×C₄	1	2	yes	—
123	#41×C₅₃	C₂×D₄	1	2	yes	—
124	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
125	C₂×D₄×C₅₃	—	2	1	no	—
126	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	—	2	1	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₂×C₂²⋊D₅₃	—	2	2	no	—
132	#11719945	—	2	2	no	—
133	#11719945	—	2	2	no	—
134	#11719945	—	2	2	no	—
135	#11719945	—	2	2	no	—
136	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
137	C₂×C₄×D₅₃	—	2	2	no	—
138	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
139	M₁₆×C₅₃	—	2	1	no	—
140	M₁₆×C₅₃	—	2	1	no	—
141	M₁₆×C₅₃	—	2	2	no	—
142	M₁₆×C₅₃	—	2	2	no	—
143	M₁₆×C₅₃	—	2	2	no	—
144	M₁₆×C₅₃	—	2	2	no	—
145	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
146	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	1	no	—
149	C₂×C₈×C₅₃	—	2	2	no	—
150	C₂×C₈×C₅₃	—	2	2	no	—
151	C₂×C₈×C₅₃	—	2	2	no	—
152	C₂×C₈×C₅₃	—	2	2	no	—
153	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
162	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
163	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
164	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	—	2	2	no	—
166	C₄⋊C₄×C₅₃	—	2	1	no	—
167	C₂²⋊C₄×C₅₃	—	2	2	no	—
168	C₂²⋊C₄×C₅₃	#41	1	2	yes	—
169	C₄²×C₅₃	—	2	2	no	—
170	C₄²×C₅₃	—	2	1	no	—
171	C₄²×C₅₃	—	2	1	no	—
172	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
174	C₂²⋊Dic₅₃	—	2	2	no	—
175	#11719925	—	2	4	no	—
176	#11719924	—	2	4	no	—
177	#11719923	—	2	4	no	—
178	#11719922	—	2	4	no	—
179	C₄⋊Dic₅₃	—	2	2	no	—
180	C₄×Dic₅₃	—	2	2	no	—
181	#11719917	—	2	2	no	—
182	#11719917	—	2	2	no	—
183	#11719917	—	2	2	no	—
184	#11719917	C₂²×C₄	1	4	yes	—
185	C₂×#88322	—	2	2	no	—
186	C₂×#88322	—	2	2	no	—
187	Dic₂₁₂	—	2	1	no	—
188	Dic₂₁₂	—	2	1	no	—
189	D₄₂₄	—	2	1	no	—
190	D₄₂₄	—	2	1	no	—
191	#11719913	—	2	2	no	—
192	#11719913	—	2	2	no	—
193	#11719912	—	4	2	no	—
194	#11719912	—	4	2	no	—
195	C₈×D₅₃	—	4	2	no	—
196	C₈×D₅₃	—	4	2	no	—
197	C₂³×C₅₃	—	2	1	no	—
198	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₂²×D₅₃	—	2	2	no	—
200	Q₈×C₅₃	—	2	1	no	—
201	Q₈×C₅₃	C₄×D₄	1	2	yes	—
202	D₄×C₅₃	—	2	2	no	—
203	D₄×C₅₃	—	2	2	no	—
204	D₄×C₅₃	—	2	1	no	—
205	D₄×C₅₃	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	4	2	no	—
213	C₂×C₄×C₅₃	—	4	4	no	—
214	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
220	C₂²⋊D₅₃	—	4	2	no	—
221	C₂²⋊D₅₃	—	4	2	no	—
222	C₂×Dic₅₃	—	2	2	no	—
223	C₂×Dic₅₃	—	4	2	no	—
224	D₂₁₂	—	2	2	no	—
225	D₂₁₂	—	2	2	no	—
226	C₄×D₅₃	—	4	2	no	—
227	C₄×D₅₃	—	4	2	no	—
228	Dic₁₀₆	—	2	2	no	—
229	Dic₁₀₆	—	2	2	no	—
230	C₈×C₅₃	—	2	2	no	—
231	C₈×C₅₃	—	2	2	no	—
232	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
233	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
234	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
235	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
236	C₈×C₅₃	#121	1	1	yes	characteristic
237	C₈×C₅₃	#121	1	1	yes	characteristic
238	C₈×C₅₃	#121	1	1	yes	characteristic
239	C₈×C₅₃	#121	1	1	yes	characteristic
240	#88322	—	2	4	no	—
241	#88322	—	4	2	no	—
242	#21558	—	53	1	no	maximal, Sylow
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	4	1	no	—
245	C₂²×C₅₃	—	4	2	no	—
246	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
247	C₂²×C₅₃	#525	1	1	yes	characteristic
248	C₂²×C₅₃	#532	1	1	yes	characteristic
249	C₂×D₅₃	—	4	2	no	—
250	C₂×D₅₃	—	8	2	no	—
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	2	2	no	—
253	C₄×C₅₃	—	4	2	no	—
254	C₄×C₅₃	—	4	1	no	—
255	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
257	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
258	C₄×C₅₃	#529	1	1	yes	characteristic
259	Dic₅₃	—	4	2	no	—
260	Dic₅₃	—	8	2	no	—
261	#3067	—	53	1	no	—
262	#2955	—	53	2	no	—
263	C₂×#443	—	53	1	no	—
264	C₄×#131	D₅₃	1	1	yes	characteristic, complemented
265	#1976	—	53	1	no	—
266	#1953	—	53	1	no	—
267	#1932	—	53	2	no	—
268	#1906	—	53	2	no	—
269	#1814	—	53	1	no	—
270	#1772	—	53	1	no	—
271	#1757	—	53	2	no	—
272	C₂×C₅₃	—	2	1	no	—
273	C₂×C₅₃	—	2	1	no	—
274	C₂×C₅₃	—	2	1	no	—
275	C₂×C₅₃	—	4	2	no	—
276	C₂×C₅₃	C₄×#121	1	1	yes	characteristic
277	D₅₃	—	8	2	no	—
278	#575	—	53	2	no	—
279	C₂×#135	—	53	1	no	—
280	C₂×#131	—	53	2	no	—
281	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
282	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
283	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
284	#473	—	53	2	no	—
285	#472	—	53	2	no	—
286	#471	—	53	4	no	—
287	#444	—	106	4	no	—
288	#443	—	106	4	no	—
289	#443	—	106	1	no	—
290	#443	—	106	1	no	—
291	#443	—	106	1	no	—
292	#443	—	106	1	no	—
293	#430	—	53	2	no	—
294	C₂×#108	—	53	1	no	—
295	C₂×#108	—	53	1	no	—
296	#427	—	53	1	no	—
297	#421	—	53	2	no	—
298	#421	—	53	2	no	—
299	C₂×#104	—	53	2	no	—
300	#413	—	53	2	no	—
301	#410	—	53	2	no	—
302	#405	—	2	1	no	—
303	#405	—	2	1	no	—
304	#405	—	53	2	no	—
305	#405	C₂×D₅₃	1	1	yes	characteristic
306	#405	C₂×D₅₃	1	1	yes	characteristic
307	C₄×M₁₆	—	2	1	no	—
308	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
309	C₂×C₄×C₈	—	2	1	no	—
310	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
311	#344	—	53	2	no	—
312	#344	—	53	2	no	—
313	#339	—	53	2	no	—
314	C₅₃	#21558	1	1	yes	characteristic, complemented, Sylow
315	C₂×#41	—	53	2	no	—
316	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
317	#137	—	53	4	no	—
318	#137	—	106	2	no	—
319	AGL(1,ℤ/8)	—	53	4	no	—
320	AGL(1,ℤ/8)	—	106	2	no	—
321	#135	—	106	4	no	—
322	#135	—	106	1	no	—
323	#135	—	106	1	no	—
324	#135	—	106	1	no	—
325	#135	—	106	1	no	—
326	C₂×Q₁₆	—	53	1	no	—
327	C₂×SD₁₆	—	53	2	no	—
328	C₂×D₈	—	53	1	no	—
329	#131	—	2	1	no	—
330	#131	—	2	1	no	—
331	#131	—	2	1	no	—
332	#131	—	2	1	no	—
333	#131	—	106	2	no	—
334	#131	—	106	2	no	—
335	#131	—	106	2	no	—
336	#131	—	106	2	no	—
337	#131	C₄×D₅₃	1	2	yes	—
338	#131	C₄×D₅₃	1	2	yes	—
339	#131	D₂₁₂	1	2	yes	—
340	#131	D₂₁₂	1	2	yes	—
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂×M₁₆	—	53	2	no	—
344	C₂×M₁₆	—	53	2	no	—
345	C₂×M₁₆	—	106	2	no	—
346	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
347	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
348	C₂²×C₈	—	2	1	no	—
349	C₂²×C₈	—	2	1	no	—
350	C₂²×C₈	—	106	2	no	—
351	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
352	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
353	C₄×Q₈	—	2	1	no	—
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	—	2	2	no	—
356	#117	—	2	1	no	—
357	#117	—	53	2	no	—
358	#117	C₂²×D₅₃	1	1	yes	characteristic
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
361	#108	—	53	2	no	—
362	#108	—	53	2	no	—
363	#108	—	106	4	no	—
364	#108	—	106	1	no	—
365	#108	—	106	1	no	—
366	#107	—	106	1	no	—
367	#106	—	106	1	no	—
368	#104	—	106	4	no	—
369	#104	—	106	4	no	—
370	#104	—	106	2	no	—
371	#104	—	106	2	no	—
372	#101	—	53	4	no	—
373	#100	—	53	4	no	—
374	#99	—	106	4	no	—
375	#97	—	2	1	no	—
376	#97	—	2	1	no	—
377	#97	—	2	1	no	—
378	#97	—	106	2	no	—
379	C₄×C₈	—	2	1	no	—
380	C₄×C₈	—	2	1	no	—
381	C₄×C₈	—	2	1	no	—
382	C₄×C₈	—	106	2	no	—
383	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
384	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
385	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
386	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	106	2	no	—
390	#41	—	106	2	no	—
391	#41	—	106	2	no	—
392	#41	—	106	2	no	—
393	#41	C₂×C₄×D₅₃	1	2	yes	—
394	#41	C₂×D₂₁₂	1	2	yes	—
395	C₂×Q₈	—	53	2	no	—
396	C₂×Q₈	#11719945	1	1	yes	characteristic
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	53	2	no	—
399	C₂×D₄	—	106	2	no	—
400	C₂×D₄	#11719945	1	1	yes	characteristic
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	106	2	no	—
405	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
406	Q₁₆	—	106	4	no	—
407	Q₁₆	—	106	1	no	—
408	Q₁₆	—	106	1	no	—
409	SD₁₆	—	106	4	no	—
410	SD₁₆	—	106	4	no	—
411	SD₁₆	—	106	2	no	—
412	SD₁₆	—	106	2	no	—
413	D₈	—	106	4	no	—
414	D₈	—	106	1	no	—
415	D₈	—	106	1	no	—
416	M₁₆	—	2	1	no	—
417	M₁₆	—	2	1	no	—
418	M₁₆	—	2	2	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	2	no	—
421	M₁₆	—	2	2	no	—
422	M₁₆	—	53	4	no	—
423	M₁₆	—	106	2	no	—
424	M₁₆	—	106	2	no	—
425	M₁₆	—	106	2	no	—
426	M₁₆	—	212	2	no	—
427	M₁₆	—	212	2	no	—
428	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
429	M₁₆	C₂×D₂₁₂	1	2	yes	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	106	2	no	—
437	C₂×C₈	—	106	2	no	—
438	C₂×C₈	—	212	2	no	—
439	C₂×C₈	—	212	2	no	—
440	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
441	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
442	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
443	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
444	C₂×C₈	#11719945	1	1	yes	characteristic
445	C₂×C₈	#11719945	1	1	yes	characteristic
446	C₂×C₈	#11719945	1	1	yes	characteristic
447	C₂×C₈	#11719945	1	1	yes	characteristic
448	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
449	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
450	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
451	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	106	2	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	106	2	no	—
457	C₂²⋊C₄	#11719945	1	2	yes	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	1	no	—
460	C₄²	—	2	1	no	—
461	C₄²	—	106	2	no	—
462	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
463	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
464	C₂³	—	2	1	no	—
465	C₂³	—	106	2	no	—
466	C₂³	C₂×#11719945	1	1	yes	characteristic
467	Q₈	—	2	1	no	—
468	Q₈	—	106	2	no	—
469	Q₈	—	106	2	no	—
470	Q₈	C₄×D₂₁₂	1	2	yes	—
471	D₄	—	2	2	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	1	no	—
474	D₄	—	106	2	no	—
475	D₄	—	106	2	no	—
476	D₄	—	212	2	no	—
477	D₄	—	212	2	no	—
478	D₄	C₄×D₂₁₂	1	2	yes	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	106	2	no	—
488	C₂×C₄	—	212	2	no	—
489	C₂×C₄	—	212	2	no	—
490	C₂×C₄	—	212	2	no	—
491	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
492	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
494	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
495	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
496	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	2	2	no	—
499	C₈	—	106	4	no	—
500	C₈	—	212	2	no	—
501	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
502	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
503	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
504	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
505	C₈	#13133191	1	1	yes	characteristic
506	C₈	#13133191	1	1	yes	characteristic
507	C₈	#13133191	1	1	yes	characteristic
508	C₈	#13133191	1	1	yes	characteristic
509	C₂²	—	2	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	424	2	no	—
513	C₂²	—	212	2	no	—
514	C₂²	C₄×#11719945	1	1	yes	characteristic
515	C₂²	#16811134	1	1	yes	characteristic
516	C₂²	#16811173	1	1	yes	characteristic
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	1	no	—
521	C₄	—	424	2	no	—
522	C₄	—	212	2	no	—
523	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
524	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
525	C₄	C₂×#13133191	1	1	yes	characteristic
526	C₄	#16811157	1	1	yes	characteristic
527	C₂	—	2	1	no	—
528	C₂	—	2	1	no	—
529	C₂	—	2	1	no	—
530	C₂	—	4	2	no	—
531	C₂	—	424	2	no	—
532	C₂	C₄×#13133191	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133191
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂×C₅₃	C₄×#121
prime radical	#20284435	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#21558
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20283616	C₂	semidirect
3. C₂×#16810741	C₂	semidirect
4. #20275613	C₂	semidirect
5. #20275505	C₂	semidirect
6. #20275384	C₂	semidirect
7. #20275216	C₂	semidirect
8. #20274769	C₂	semidirect
9. #20274594	C₂	semidirect
10. #20274516	C₂	semidirect
11. C₄×#131×C₅₃	C₂	semidirect
15. #16810898	C₄	semidirect
16. #16810893	C₄	semidirect
17. #16810886	C₄	semidirect
58. C₂×Dic₂₁₂	C₂×C₄	semidirect
59. C₂×#11719913	C₂×C₄	semidirect
60. C₂×D₄₂₄	C₂×C₄	semidirect
72. #13133153	C₂×C₄	semidirect
73. #13133153	C₂×C₄	semidirect
81. #13133135	C₂×C₄	semidirect
82. #13133133	C₂×C₄	semidirect
264. C₄×#131	D₅₃	semidirect
314. C₅₃	#21558	semidirect

Maximal subgroups

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

subgroup	index
#20284435	2
#20283616	2
C₂×#16810741	2
#20275613	2
#20275505	2
#20275384	2
#20275216	2
#20274769	2
#20274594	2
#20274516	2
C₄×#131×C₅₃	2
#21558	53