Subgroups — #28318598

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#444	C₂	1	1	yes	maximal, characteristic
2	#20291469	C₂	1	1	yes	maximal, characteristic
3	#131×#6157	C₂	1	2	yes	maximal, complemented
4	#20290536	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290524	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290517	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290495	C₂	1	2	yes	maximal, complemented
8	#20290489	C₂	1	2	yes	maximal, complemented
9	#20290480	C₂	1	2	yes	maximal, complemented
10	#20290371	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×D₅₃	C₂	1	1	yes	maximal, characteristic
12	D₅₃⋊#131	C₂²	1	2	yes	—
13	#41×#6157	C₂²	1	2	yes	—
14	#16811670	C₄	1	2	yes	complemented
15	#16811669	C₄	1	2	yes	complemented
16	#16811668	C₄	1	4	yes	complemented
17	#16811664	C₄	1	2	yes	complemented
18	#16811663	C₄	1	2	yes	complemented
19	#16811662	C₄	1	4	yes	complemented
20	D₅₃⋊#108	C₂²	1	1	yes	characteristic
21	#16811633	C₂²	1	1	yes	characteristic
22	D₅₃⋊#104	C₂²	1	2	yes	—
23	#16811613	C₂²	1	2	yes	—
24	#16811589	—	2	2	no	—
25	#16811589	C₂²	1	2	yes	—
26	#16811586	—	2	2	no	—
27	#16811586	C₂²	1	2	yes	—
28	M₁₆×#6157	—	2	2	no	—
29	M₁₆×#6157	C₂²	1	2	yes	—
30	#16811582	C₂²	1	1	yes	characteristic
31	C₂×#13133271	C₂²	1	1	yes	characteristic
32	C₂×#13133271	C₂²	1	1	yes	characteristic
33	C₂×C₈×#6157	—	2	2	no	—
34	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
35	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
36	#16811559	C₂²	1	1	yes	characteristic
37	#16811498	C₂²	1	2	yes	—
38	#16811495	C₂²	1	2	yes	—
39	#16811494	C₂²	1	2	yes	—
40	#135×D₅₃	C₂²	1	1	yes	characteristic
41	#131×D₅₃	C₂²	1	2	yes	—
42	#16810741	C₄	1	1	yes	characteristic, complemented
43	#16810740	C₄	1	1	yes	characteristic, complemented
44	#16810739	C₄	1	1	yes	characteristic, complemented
45	#16810738	C₄	1	2	yes	complemented
46	#16810737	C₄	1	2	yes	complemented
47	#108×D₅₃	C₂²	1	1	yes	characteristic
48	#104×D₅₃	C₂²	1	2	yes	—
49	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
50	C₅₃×#443	C₄	1	1	yes	characteristic, complemented
51	#13133305	C₂×C₄	1	4	yes	complemented
52	#13133304	—	2	4	no	—
53	#13133304	C₂×C₄	1	4	yes	complemented
54	D₅₃⋊M₁₆	—	2	2	no	—
55	D₅₃⋊M₁₆	C₂³	1	2	yes	—
56	C₂×D₅₃⋊C₈	—	2	2	no	—
57	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
59	Q₈×#6157	C₂×C₄	1	4	yes	—
60	D₄×#6157	—	2	4	no	—
61	D₄×#6157	C₂×C₄	1	4	yes	—
62	#13133296	—	2	2	no	—
63	#13133296	C₂³	1	2	yes	—
64	C₂×C₄×#6157	—	2	2	no	—
65	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
66	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
67	#13133288	C₂×C₄	1	2	yes	complemented
68	#13133287	C₂×C₄	1	2	yes	complemented
69	#13133286	C₂×C₄	1	2	yes	—
70	#13133285	C₂×C₄	1	2	yes	—
71	#13133282	C₂×C₄	1	4	yes	complemented
72	#13133281	C₂×C₄	1	4	yes	complemented
73	#13133280	C₂×C₄	1	4	yes	—
74	#13133279	C₂×C₄	1	4	yes	—
75	#13133272	—	2	2	no	—
76	#13133271	—	2	2	no	—
77	#13133271	—	2	2	no	—
78	#13133271	—	2	2	no	—
79	#13133271	C₂×C₄	1	2	yes	—
80	#13133271	C₂×C₄	1	2	yes	—
81	#13133271	C₂×C₄	1	2	yes	complemented
82	#13133271	C₂×C₄	1	2	yes	complemented
83	C₈×#6157	—	2	2	no	—
84	C₈×#6157	—	2	2	no	—
85	C₈×#6157	—	2	1	no	—
86	C₈×#6157	—	2	1	no	—
87	C₈×#6157	—	2	2	no	—
88	C₈×#6157	D₄	1	2	yes	—
89	C₈×#6157	D₄	1	2	yes	—
90	(C₂×#6157)⋊C₄	—	2	2	no	—
91	#41×D₅₃	C₂³	1	2	yes	—
92	#13133240	C₂×C₄	1	1	yes	characteristic
93	#13133239	C₂×C₄	1	1	yes	characteristic
94	#13133238	C₂×C₄	1	2	yes	—
95	#13133237	C₂×C₄	1	2	yes	—
96	#13133236	C₂×C₄	1	1	yes	characteristic
97	Q₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
98	SD₁₆×D₅₃	C₂×C₄	1	2	yes	—
99	D₈×D₅₃	C₂×C₄	1	1	yes	characteristic
100	#13133225	—	2	2	no	—
101	#13133225	C₂×C₄	1	2	yes	complemented
102	#13133224	C₂×C₄	1	2	yes	complemented
103	#13133223	—	2	2	no	—
104	#13133223	C₂×C₄	1	2	yes	complemented
105	M₁₆×D₅₃	—	2	2	no	—
106	M₁₆×D₅₃	C₂³	1	2	yes	—
107	C₂×C₈×D₅₃	—	2	2	no	—
108	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
110	C₄²×D₅₃	C₂³	1	1	yes	characteristic
111	#13133153	C₂×C₄	1	1	yes	characteristic, complemented
112	#13133152	C₂×C₄	1	2	yes	complemented
113	#13133145	C₂×C₄	1	2	yes	complemented
114	#13133144	C₂×C₄	1	2	yes	complemented
115	#13133143	C₂×C₄	1	2	yes	complemented
116	C₄×#88322	C₂×C₄	1	1	yes	characteristic, complemented
117	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic, complemented
118	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic, complemented
119	#135×C₅₃	C₂×C₄	1	1	yes	characteristic
120	#131×C₅₃	C₂×C₄	1	2	yes	complemented
121	#108×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
122	#104×C₅₃	C₂×C₄	1	2	yes	complemented
123	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
124	C₂²×#6157	—	2	1	no	—
125	C₂²×#6157	—	2	4	no	—
126	C₂²×#6157	#41	1	2	yes	—
127	#41×C₅₃	C₂²×C₄	1	2	yes	—
128	#11719949	C₂²×C₄	1	2	yes	—
129	Q₈×D₅₃	C₂²×C₄	1	2	yes	—
130	#11719947	—	2	2	no	—
131	#11719947	C₂²×C₄	1	2	yes	—
132	D₄×D₅₃	—	2	2	no	—
133	D₄×D₅₃	C₂²×C₄	1	2	yes	—
134	#11719945	—	2	2	no	—
135	#11719945	C₂²×C₄	1	2	yes	—
136	C₂×C₄×D₅₃	—	2	2	no	—
137	C₂×C₄×D₅₃	—	2	1	no	—
138	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
139	C₂²⋊#6157	—	2	2	no	—
140	C₂²⋊#6157	—	2	4	no	—
141	#11719940	—	2	2	no	—
142	#11719940	—	2	4	no	—
143	C₂×#88324	—	2	1	no	—
144	C₂×#88324	—	2	4	no	—
145	C₂×#88324	#41	1	2	yes	—
146	C₄⋊#6157	—	2	4	no	—
147	C₄⋊#6157	C₂²×C₄	1	4	yes	—
148	C₄×#6157	—	2	4	no	—
149	C₄×#6157	—	2	2	no	—
150	C₄×#6157	—	2	1	no	—
151	C₄×#6157	C₂²×C₄	1	2	yes	—
152	C₄×#6157	C₂²×C₄	1	2	yes	—
153	C₄×#6157	C₂×D₄	1	2	yes	—
154	#11719936	—	2	4	no	—
155	#11719936	C₂²×C₄	1	4	yes	—
156	D₅₃⋊C₈	—	2	2	no	—
157	D₅₃⋊C₈	—	2	1	no	—
158	D₅₃⋊C₈	—	2	4	no	—
159	D₅₃⋊C₈	C₂²×C₄	1	2	yes	—
160	D₅₃⋊C₈	C₂²×C₄	1	2	yes	—
161	D₅₃⋊C₈	C₂×D₄	1	2	yes	—
162	Q₁₆×C₅₃	C₄²	1	1	yes	characteristic
163	SD₁₆×C₅₃	C₄²	1	2	yes	—
164	D₈×C₅₃	C₄²	1	1	yes	characteristic
165	M₁₆×C₅₃	—	2	2	no	—
166	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
167	C₂×C₈×C₅₃	—	2	2	no	—
168	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
171	#11719925	C₄²	1	2	yes	—
172	#11719924	C₄²	1	2	yes	—
173	#11719923	C₄²	1	2	yes	—
174	#11719922	C₄²	1	2	yes	—
175	C₄×Dic₅₃	—	2	1	no	—
176	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
177	#11719917	—	2	2	no	—
178	#11719917	C₂²×C₄	1	2	yes	—
179	C₂×#88322	—	2	2	no	—
180	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
181	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
182	Dic₂₁₂	C₄²	1	1	yes	characteristic
183	D₄₂₄	C₄²	1	1	yes	characteristic
184	#11719913	C₄²	1	2	yes	—
185	#11719912	—	2	2	no	—
186	#11719912	—	2	2	no	—
187	#11719912	—	2	2	no	—
188	C₈×D₅₃	—	2	2	no	—
189	C₈×D₅₃	—	2	2	no	—
190	C₈×D₅₃	—	2	2	no	—
191	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
192	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
195	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
196	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
197	C₈×D₅₃	#41	1	1	yes	characteristic
198	C₈×D₅₃	#41	1	1	yes	characteristic
199	C₂²×D₅₃	—	2	2	no	—
200	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
201	C₂×#6157	—	2	4	no	—
202	C₂×#6157	—	2	2	no	—
203	C₂×#6157	—	4	2	no	—
204	C₂×#6157	—	4	8	no	—
205	C₂×#6157	—	4	2	no	—
206	C₂×#6157	C₄×D₄	1	4	yes	—
207	Q₈×C₅₃	C₂×C₄²	1	2	yes	—
208	D₄×C₅₃	—	2	2	no	—
209	D₄×C₅₃	C₂×C₄²	1	2	yes	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
213	C₂²⋊D₅₃	—	2	2	no	—
214	C₂²⋊D₅₃	—	2	2	no	—
215	C₂²⋊D₅₃	—	2	2	no	—
216	C₂×Dic₅₃	—	2	2	no	—
217	C₂×Dic₅₃	—	2	1	no	—
218	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
219	D₂₁₂	—	2	2	no	—
220	D₂₁₂	C₂×C₄²	1	2	yes	—
221	C₄×D₅₃	—	2	2	no	—
222	C₄×D₅₃	—	2	2	no	—
223	C₄×D₅₃	—	2	2	no	—
224	C₄×D₅₃	—	4	2	no	—
225	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
226	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
227	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
228	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
229	Dic₁₀₆	—	2	2	no	—
230	Dic₁₀₆	C₂×C₄²	1	2	yes	—
231	#88324	—	2	4	no	—
232	#88324	—	2	2	no	—
233	#88324	C₄×D₄	1	4	yes	—
234	C₈×C₅₃	—	2	2	no	—
235	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
236	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
237	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
238	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
239	#88322	—	2	2	no	—
240	#88322	C₂×C₄²	1	1	yes	characteristic
241	#88322	C₂×C₄²	1	1	yes	characteristic
242	#88322	C₄×D₄	1	1	yes	characteristic
243	#88322	C₄×D₄	1	1	yes	characteristic
244	#20510	—	53	1	no	maximal, Sylow
245	C₂²×C₅₃	—	2	2	no	—
246	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
247	C₂×D₅₃	—	2	2	no	—
248	C₂×D₅₃	—	2	2	no	—
249	C₂×D₅₃	—	4	4	no	—
250	C₂×D₅₃	C₂×C₄×D₄	1	1	yes	characteristic
251	C₂×D₅₃	C₄×#41	1	1	yes	characteristic
252	#6157	—	2	4	no	—
253	#6157	—	4	2	no	—
254	#6157	—	4	4	no	—
255	C₄×C₅₃	—	2	2	no	—
256	C₄×C₅₃	—	4	1	no	—
257	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
258	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
259	Dic₅₃	—	2	2	no	—
260	Dic₅₃	—	4	1	no	—
261	Dic₅₃	C₂×C₄×D₄	1	1	yes	characteristic
262	Dic₅₃	C₄×#41	1	1	yes	characteristic
263	C₂×#444	—	53	1	no	—
264	C₂×#443	—	53	1	no	—
265	#2943	—	53	1	no	—
266	C₄×#131	—	53	2	no	—
267	#1778	—	53	1	no	—
268	#1773	—	53	1	no	—
269	#1772	—	53	1	no	—
270	#1765	—	53	2	no	—
271	#1764	—	53	2	no	—
272	#1759	—	53	2	no	—
273	#1726	—	53	1	no	—
274	C₂×C₅₃	—	2	1	no	—
275	C₂×C₅₃	—	4	2	no	—
276	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
277	D₅₃	—	2	1	no	—
278	D₅₃	—	4	2	no	—
279	D₅₃	#2955	1	2	yes	—
280	C₂×#135	—	53	1	no	—
281	C₂×#131	—	53	2	no	—
282	C₂×#131	—	53	2	no	—
283	C₄×#41	—	53	2	no	—
284	#444	—	53	4	no	—
285	#444	—	53	2	no	—
286	#444	—	53	2	no	—
287	#443	—	53	1	no	—
288	#443	—	53	2	no	—
289	#443	—	53	1	no	—
290	#443	—	53	2	no	—
291	#443	—	53	1	no	—
292	#443	#6157	1	1	yes	characteristic, complemented
293	#442	—	53	2	no	—
294	#441	—	53	2	no	—
295	#440	—	53	4	no	—
296	C₂×#108	—	53	1	no	—
297	C₂×#108	—	53	1	no	—
298	#427	—	53	1	no	—
299	C₂×#104	—	53	2	no	—
300	C₂×#104	—	53	2	no	—
301	#416	—	53	2	no	—
302	#405	—	53	2	no	—
303	#405	—	53	2	no	—
304	#405	—	106	2	no	—
305	#405	—	106	2	no	—
306	C₄×M₁₆	—	53	2	no	—
307	C₄×M₁₆	—	106	2	no	—
308	C₂×#97	—	53	1	no	—
309	C₂×#97	—	53	1	no	—
310	C₂×#97	—	53	1	no	—
311	C₂×C₄×C₈	—	53	1	no	—
312	C₂×C₄×C₈	—	53	1	no	—
313	C₂×C₄×C₈	—	53	1	no	—
314	C₂×C₄×C₈	—	106	2	no	—
315	#376	—	53	1	no	—
316	#341	—	53	2	no	—
317	#339	—	53	2	no	—
318	#339	—	53	2	no	—
319	C₅₃	#20510	1	1	yes	characteristic, complemented, Sylow
320	C₂×#41	—	53	2	no	—
321	#135	—	53	2	no	—
322	#135	—	53	1	no	—
323	#135	—	53	2	no	—
324	#135	—	53	1	no	—
325	#135	—	53	1	no	—
326	#135	C₂×#6157	1	1	yes	characteristic
327	C₂×Q₁₆	—	53	1	no	—
328	C₂×SD₁₆	—	53	2	no	—
329	C₂×D₈	—	53	1	no	—
330	#131	—	53	4	no	—
331	#131	—	53	4	no	—
332	#131	—	53	2	no	—
333	#131	—	53	2	no	—
334	#131	—	53	2	no	—
335	#131	—	106	2	no	—
336	#131	—	106	2	no	—
337	#131	—	106	4	no	—
338	#131	C₂×#6157	1	2	yes	complemented
339	C₂×M₁₆	—	53	2	no	—
340	C₂×M₁₆	—	53	2	no	—
341	C₂×M₁₆	—	106	2	no	—
342	C₂×M₁₆	—	106	2	no	—
343	C₂²×C₈	—	53	1	no	—
344	C₂²×C₈	—	53	1	no	—
345	C₂²×C₈	—	53	1	no	—
346	C₂²×C₈	—	53	1	no	—
347	C₂²×C₈	—	106	2	no	—
348	C₂²×C₈	—	106	2	no	—
349	C₄×Q₈	—	53	4	no	—
350	C₄×D₄	—	53	4	no	—
351	C₄×D₄	—	106	4	no	—
352	#117	—	53	2	no	—
353	#117	—	106	2	no	—
354	C₂×C₄²	—	53	1	no	—
355	C₂×C₄²	—	53	1	no	—
356	C₂×C₄²	—	53	1	no	—
357	C₂×C₄²	—	106	2	no	—
358	#108	—	53	2	no	—
359	#108	—	53	2	no	—
360	#108	—	53	2	no	—
361	#108	—	53	1	no	—
362	#108	C₂×#6157	1	1	yes	characteristic, complemented
363	#107	—	53	2	no	—
364	#106	—	53	2	no	—
365	#104	—	53	4	no	—
366	#104	—	53	4	no	—
367	#104	—	53	2	no	—
368	#104	—	53	2	no	—
369	#104	—	53	2	no	—
370	#104	C₂×#6157	1	2	yes	complemented
371	#103	—	53	4	no	—
372	#102	—	53	4	no	—
373	#97	—	53	2	no	—
374	#97	—	53	2	no	—
375	#97	—	53	2	no	—
376	#97	—	53	2	no	—
377	#97	—	106	2	no	—
378	#97	—	106	2	no	—
379	#97	—	106	2	no	—
380	#97	—	106	2	no	—
381	C₄×C₈	—	53	2	no	—
382	C₄×C₈	—	53	2	no	—
383	C₄×C₈	—	53	1	no	—
384	C₄×C₈	—	53	1	no	—
385	C₄×C₈	—	53	1	no	—
386	C₄×C₈	—	106	2	no	—
387	C₄×C₈	—	106	1	no	—
388	C₄×C₈	—	106	1	no	—
389	C₄×C₈	—	106	2	no	—
390	C₄×C₈	—	106	2	no	—
391	C₄×C₈	C₂×#6157	1	1	yes	characteristic, complemented
392	#95	—	106	2	no	—
393	#41	—	53	2	no	—
394	#41	—	53	2	no	—
395	#41	—	53	2	no	—
396	#41	—	106	2	no	—
397	#41	—	106	2	no	—
398	#41	C₂²×#6157	1	2	yes	—
399	C₂×Q₈	—	53	2	no	—
400	C₂×D₄	—	53	2	no	—
401	C₂×D₄	—	106	2	no	—
402	C₂²×C₄	—	53	1	no	—
403	C₂²×C₄	—	53	2	no	—
404	C₂²×C₄	—	106	2	no	—
405	C₂²×C₄	—	106	1	no	—
406	C₂²×C₄	—	106	4	no	—
407	C₂²×C₄	—	106	1	no	—
408	Q₁₆	—	53	2	no	—
409	Q₁₆	—	53	1	no	—
410	Q₁₆	C₄×#6157	1	1	yes	characteristic
411	SD₁₆	—	53	2	no	—
412	SD₁₆	—	53	2	no	—
413	SD₁₆	—	53	2	no	—
414	SD₁₆	C₄×#6157	1	2	yes	—
415	D₈	—	53	2	no	—
416	D₈	—	53	1	no	—
417	D₈	C₄×#6157	1	1	yes	characteristic
418	M₁₆	—	2	2	no	—
419	M₁₆	—	53	4	no	—
420	M₁₆	—	53	2	no	—
421	M₁₆	—	106	2	no	—
422	M₁₆	—	106	2	no	—
423	M₁₆	—	106	2	no	—
424	M₁₆	—	106	4	no	—
425	M₁₆	—	106	4	no	—
426	M₁₆	—	106	2	no	—
427	M₁₆	—	106	2	no	—
428	M₁₆	C₂²×#6157	1	2	yes	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	53	2	no	—
431	C₂×C₈	—	53	2	no	—
432	C₂×C₈	—	53	2	no	—
433	C₂×C₈	—	53	1	no	—
434	C₂×C₈	—	53	1	no	—
435	C₂×C₈	—	53	1	no	—
436	C₂×C₈	—	53	1	no	—
437	C₂×C₈	—	53	1	no	—
438	C₂×C₈	—	53	1	no	—
439	C₂×C₈	—	53	1	no	—
440	C₂×C₈	—	53	1	no	—
441	C₂×C₈	—	53	1	no	—
442	C₂×C₈	—	53	1	no	—
443	C₂×C₈	—	53	2	no	—
444	C₂×C₈	—	106	2	no	—
445	C₂×C₈	—	106	2	no	—
446	C₂×C₈	—	106	2	no	—
447	C₂×C₈	—	106	4	no	—
448	C₂×C₈	—	106	4	no	—
449	C₂×C₈	—	106	1	no	—
450	C₂×C₈	—	106	2	no	—
451	C₂×C₈	—	106	1	no	—
452	C₂×C₈	—	106	2	no	—
453	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
454	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
455	C₄⋊C₄	—	53	4	no	—
456	C₄⋊C₄	—	106	4	no	—
457	C₂²⋊C₄	—	106	4	no	—
458	C₂²⋊C₄	—	106	2	no	—
459	C₄²	—	53	2	no	—
460	C₄²	—	53	2	no	—
461	C₄²	—	53	2	no	—
462	C₄²	—	53	1	no	—
463	C₄²	—	106	1	no	—
464	C₄²	—	106	2	no	—
465	C₄²	—	106	1	no	—
466	C₄²	—	106	4	no	—
467	C₄²	C₂²×#6157	1	1	yes	characteristic
468	C₂³	—	53	1	no	—
469	C₂³	—	106	2	no	—
470	Q₈	—	53	2	no	—
471	Q₈	—	106	2	no	—
472	Q₈	C₂×C₄×#6157	1	2	yes	—
473	D₄	—	2	2	no	—
474	D₄	—	53	2	no	—
475	D₄	—	106	2	no	—
476	D₄	—	106	2	no	—
477	D₄	—	106	2	no	—
478	D₄	—	106	2	no	—
479	D₄	C₂×C₄×#6157	1	2	yes	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	53	1	no	—
483	C₂×C₄	—	53	1	no	—
484	C₂×C₄	—	53	1	no	—
485	C₂×C₄	—	53	1	no	—
486	C₂×C₄	—	53	1	no	—
487	C₂×C₄	—	53	4	no	—
488	C₂×C₄	—	106	2	no	—
489	C₂×C₄	—	106	2	no	—
490	C₂×C₄	—	106	2	no	—
491	C₂×C₄	—	106	2	no	—
492	C₂×C₄	—	106	2	no	—
493	C₂×C₄	—	106	4	no	—
494	C₂×C₄	—	106	1	no	—
495	C₂×C₄	—	212	2	no	—
496	C₂×C₄	—	212	2	no	—
497	C₂×C₄	—	212	8	no	—
498	C₂×C₄	—	212	2	no	—
499	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
500	C₈	—	2	2	no	—
501	C₈	—	53	4	no	—
502	C₈	—	53	1	no	—
503	C₈	—	53	1	no	—
504	C₈	—	53	1	no	—
505	C₈	—	53	1	no	—
506	C₈	—	106	2	no	—
507	C₈	—	106	4	no	—
508	C₈	—	106	2	no	—
509	C₈	C₂×C₄×#6157	1	1	yes	characteristic
510	C₈	C₂×C₄×#6157	1	1	yes	characteristic
511	C₈	D₄×#6157	1	1	yes	characteristic
512	C₈	D₄×#6157	1	1	yes	characteristic
513	C₂²	—	2	2	no	—
514	C₂²	—	53	1	no	—
515	C₂²	—	53	1	no	—
516	C₂²	—	106	2	no	—
517	C₂²	—	106	2	no	—
518	C₂²	—	212	4	no	—
519	C₂²	#41×#6157	1	1	yes	characteristic
520	C₄	—	2	2	no	—
521	C₄	—	4	1	no	—
522	C₄	—	53	1	no	—
523	C₄	—	53	1	no	—
524	C₄	—	106	4	no	—
525	C₄	—	106	2	no	—
526	C₄	—	212	1	no	—
527	C₄	—	212	4	no	—
528	C₄	—	212	2	no	—
529	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
530	C₄	C₂×D₄×#6157	1	1	yes	characteristic
531	C₂	—	2	1	no	—
532	C₂	—	4	2	no	—
533	C₂	—	53	2	no	—
534	C₂	—	106	1	no	—
535	C₂	—	212	2	no	—
536	C₂	C₄×D₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#443	C₄
socle	C₂×C₅₃	C₄²×D₄
prime radical	#135×D₅₃	C₂²
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#443	#6157
hypocenter	C₅₃	#20510
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
3. #131×#6157	C₂	semidirect
4. #20290536	C₂	semidirect
5. #20290524	C₂	semidirect
6. #20290517	C₂	semidirect
7. #20290495	C₂	semidirect
8. #20290489	C₂	semidirect
9. #20290480	C₂	semidirect
10. #20290371	C₂	semidirect
14. #16811670	C₄	semidirect
15. #16811669	C₄	semidirect
16. #16811668	C₄	semidirect
17. #16811664	C₄	semidirect
18. #16811663	C₄	semidirect
19. #16811662	C₄	semidirect
42. #16810741	C₄	semidirect
43. #16810740	C₄	semidirect
44. #16810739	C₄	semidirect
45. #16810738	C₄	semidirect
46. #16810737	C₄	semidirect
50. C₅₃×#443	C₄	semidirect
51. #13133305	C₂×C₄	semidirect
53. #13133304	C₂×C₄	semidirect
67. #13133288	C₂×C₄	semidirect
68. #13133287	C₂×C₄	semidirect
71. #13133282	C₂×C₄	semidirect
72. #13133281	C₂×C₄	semidirect
81. #13133271	C₂×C₄	semidirect
82. #13133271	C₂×C₄	semidirect
101. #13133225	C₂×C₄	semidirect
102. #13133224	C₂×C₄	semidirect
104. #13133223	C₂×C₄	semidirect
111. #13133153	C₂×C₄	semidirect
112. #13133152	C₂×C₄	semidirect
113. #13133145	C₂×C₄	semidirect
114. #13133144	C₂×C₄	semidirect
115. #13133143	C₂×C₄	semidirect
116. C₄×#88322	C₂×C₄	semidirect
117. C₈×Dic₅₃	C₂×C₄	semidirect
118. C₈×Dic₅₃	C₂×C₄	semidirect
120. #131×C₅₃	C₂×C₄	semidirect
121. #108×C₅₃	C₂×C₄	semidirect
122. #104×C₅₃	C₂×C₄	semidirect
123. C₄×C₈×C₅₃	C₂×C₄	semidirect
292. #443	#6157	semidirect
319. C₅₃	#20510	semidirect
338. #131	C₂×#6157	semidirect
362. #108	C₂×#6157	semidirect
370. #104	C₂×#6157	semidirect
391. C₄×C₈	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#444	2
#20291469	2
#131×#6157	2
#20290536	2
#20290524	2
#20290517	2
#20290495	2
#20290489	2
#20290480	2
#20290371	2
#443×D₅₃	2
#20510	53