Subgroups — #27402611

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283630	C₂	1	2	yes	maximal
2	C₂×#16810738	C₂	1	1	yes	maximal, characteristic
3	#20283323	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275502	C₂	1	2	yes	maximal, complemented
5	#20274955	C₂	1	2	yes	maximal, complemented
6	#20274952	C₂	1	2	yes	maximal, complemented
7	#20274885	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274770	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274769	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274597	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2916	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133237	C₂²	1	1	yes	characteristic
13	#16811389	C₂²	1	2	yes	—
14	C₂×#13133223	C₂²	1	1	yes	characteristic
15	#16810743	—	2	4	no	—
16	#16810738	—	2	4	no	—
17	#16810738	C₄	1	2	yes	complemented
18	#16810738	C₄	1	2	yes	complemented
19	#16810738	C₄	1	2	yes	complemented
20	#16810738	C₄	1	2	yes	complemented
21	#16810671	C₂²	1	2	yes	—
22	#16810670	—	2	2	no	—
23	C₂×#13133153	C₂²	1	1	yes	characteristic
24	C₂×#13133152	C₂²	1	2	yes	—
25	C₂×#13133152	C₂²	1	1	yes	characteristic
26	#108×D₅₃	—	2	2	no	—
27	#16810661	C₂²	1	1	yes	characteristic
28	#16810646	C₂²	1	2	yes	—
29	#16810640	C₂²	1	2	yes	—
30	#16810639	C₂²	1	2	yes	—
31	C₂×#13133144	C₂²	1	1	yes	characteristic
32	C₂×#13133143	C₂²	1	1	yes	characteristic
33	#16810581	C₂²	1	2	yes	—
34	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
35	#16810395	C₂²	1	2	yes	—
36	#16810394	C₂²	1	2	yes	—
37	#16810384	C₂²	1	2	yes	—
38	C₅₃×#568	C₂²	1	2	yes	—
39	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#430	—	2	2	no	—
41	C₅₃×#430	C₂²	1	2	yes	—
42	C₂×#108×C₅₃	—	2	2	no	—
43	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#11719945	C₂³	1	1	yes	characteristic
46	#13133237	C₂×C₄	1	2	yes	—
47	#13133237	C₂×C₄	1	2	yes	—
48	#13133237	D₄	1	2	yes	—
49	#13133237	D₄	1	2	yes	—
50	C₂×#11719925	Q₈	1	1	yes	characteristic
51	C₂×#11719923	Q₈	1	1	yes	characteristic
52	C₂×#11719924	Q₈	1	1	yes	characteristic
53	C₂×#11719922	Q₈	1	1	yes	characteristic
54	#13133225	—	4	2	no	—
55	#13133223	—	2	1	no	—
56	#13133223	—	2	1	no	—
57	#13133223	—	2	4	no	—
58	#13133223	C₂×C₄	1	2	yes	—
59	#13133223	C₂×C₄	1	2	yes	—
60	C₂×#11719917	C₂³	1	2	yes	—
61	C₂×#11719917	C₂³	1	1	yes	characteristic
62	C₂×#11719912	—	2	1	no	—
63	M₁₆×D₅₃	—	2	4	no	—
64	C₂²×#88322	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₅₃	—	2	1	no	—
66	#13133176	C₂³	1	2	yes	—
67	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	#13133153	—	2	1	no	—
69	#13133153	—	2	1	no	—
70	#13133152	—	2	4	no	—
71	#13133152	—	2	2	no	—
72	#13133152	—	2	2	no	—
73	#13133152	C₂×C₄	1	2	yes	—
74	#13133152	C₂×C₄	1	2	yes	—
75	#13133151	—	2	1	no	—
76	#13133149	—	2	1	no	—
77	#13133146	—	4	2	no	—
78	#13133144	—	2	4	no	—
79	#13133144	C₂×C₄	1	2	yes	—
80	#13133144	C₂×C₄	1	2	yes	—
81	#13133143	—	2	4	no	—
82	#13133143	C₂×C₄	1	2	yes	—
83	#13133143	C₂×C₄	1	2	yes	—
84	#13133127	—	2	2	no	—
85	C₈×Dic₅₃	—	2	2	no	—
86	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
91	#131×C₅₃	—	2	1	no	—
92	#131×C₅₃	—	2	1	no	—
93	#131×C₅₃	—	2	4	no	—
94	#131×C₅₃	—	4	2	no	—
95	#131×C₅₃	C₂×C₄	1	2	yes	—
96	#131×C₅₃	C₂×C₄	1	2	yes	—
97	C₂×M₁₆×C₅₃	—	2	1	no	—
98	C₂×M₁₆×C₅₃	—	2	4	no	—
99	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
100	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
102	C₂²×C₈×C₅₃	—	2	1	no	—
103	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	#108×C₅₃	—	2	4	no	—
105	#108×C₅₃	—	2	1	no	—
106	#108×C₅₃	—	2	1	no	—
107	#108×C₅₃	D₄	1	2	yes	—
108	#108×C₅₃	D₄	1	2	yes	—
109	#41×C₅₃	—	2	1	no	—
110	#41×C₅₃	—	2	1	no	—
111	#41×C₅₃	C₂²×C₄	1	2	yes	—
112	#41×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×Q₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₂²⋊D₅₃	—	2	1	no	—
119	C₂²×Dic₅₃	—	2	1	no	—
120	#11719945	—	2	1	no	—
121	#11719945	—	2	1	no	—
122	#11719945	C₂²×C₄	1	2	yes	—
123	#11719945	C₂×D₄	1	2	yes	—
124	C₂×D₂₁₂	C₂×Q₈	1	1	yes	characteristic
125	C₂×C₄×D₅₃	—	2	1	no	—
126	C₂×Dic₁₀₆	C₂×Q₈	1	1	yes	characteristic
127	M₁₆×C₅₃	—	2	1	no	—
128	M₁₆×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	2	2	no	—
130	M₁₆×C₅₃	—	2	2	no	—
131	M₁₆×C₅₃	—	2	2	no	—
132	M₁₆×C₅₃	—	2	2	no	—
133	M₁₆×C₅₃	—	4	4	no	—
134	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
135	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	4	4	no	—
142	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
147	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
148	C₂²⋊Dic₅₃	—	2	2	no	—
149	#11719925	C₄⋊C₄	1	2	yes	—
150	#11719925	C₄⋊C₄	1	2	yes	—
151	#11719924	C₄⋊C₄	1	2	yes	—
152	#11719924	C₄⋊C₄	1	2	yes	—
153	#11719923	C₄⋊C₄	1	2	yes	—
154	#11719923	C₄⋊C₄	1	2	yes	—
155	#11719922	C₄⋊C₄	1	2	yes	—
156	#11719922	C₄⋊C₄	1	2	yes	—
157	C₄⋊Dic₅₃	—	2	2	no	—
158	C₄×Dic₅₃	—	2	1	no	—
159	C₄×Dic₅₃	—	2	2	no	—
160	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
162	#11719917	—	2	1	no	—
163	#11719917	—	2	2	no	—
164	#11719917	—	2	2	no	—
165	#11719917	C₂²×C₄	1	2	yes	—
166	C₂×#88322	—	2	1	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
169	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
170	C₂×#88322	C₂×D₄	1	1	yes	characteristic
171	C₂×#88322	C₂×D₄	1	1	yes	characteristic
172	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
173	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
174	#11719912	—	2	2	no	—
175	#11719912	—	2	2	no	—
176	#11719912	—	4	4	no	—
177	C₈×D₅₃	—	2	2	no	—
178	C₈×D₅₃	—	2	2	no	—
179	C₈×D₅₃	—	4	4	no	—
180	C₂³×C₅₃	—	2	1	no	—
181	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
182	C₂²×D₅₃	—	2	1	no	—
183	Q₈×C₅₃	—	2	1	no	—
184	Q₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
185	D₄×C₅₃	—	2	2	no	—
186	D₄×C₅₃	—	2	2	no	—
187	D₄×C₅₃	—	2	1	no	—
188	D₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂²⋊D₅₃	—	2	2	no	—
199	C₂²⋊D₅₃	—	2	2	no	—
200	C₂×Dic₅₃	—	2	1	no	—
201	C₂×Dic₅₃	—	2	1	no	—
202	C₂×Dic₅₃	—	2	1	no	—
203	C₂×Dic₅₃	—	4	2	no	—
204	C₂×Dic₅₃	—	4	2	no	—
205	D₂₁₂	—	2	1	no	—
206	D₂₁₂	C₂×C₄⋊C₄	1	2	yes	—
207	C₄×D₅₃	—	2	2	no	—
208	C₄×D₅₃	—	2	2	no	—
209	Dic₁₀₆	—	2	1	no	—
210	Dic₁₀₆	C₂×C₄⋊C₄	1	2	yes	—
211	C₈×C₅₃	—	2	2	no	—
212	C₈×C₅₃	—	2	4	no	—
213	C₈×C₅₃	—	4	2	no	—
214	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
215	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
216	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
217	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
218	#88322	—	2	2	no	—
219	#88322	—	4	2	no	—
220	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
221	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
222	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
223	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
224	#22694	—	53	1	no	maximal, Sylow
225	C₂²×C₅₃	—	2	2	no	—
226	C₂²×C₅₃	—	4	1	no	—
227	C₂²×C₅₃	—	8	1	no	—
228	C₂²×C₅₃	#520	1	1	yes	characteristic
229	C₂²×C₅₃	#547	1	1	yes	characteristic
230	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
231	C₂×D₅₃	—	2	2	no	—
232	C₂×D₅₃	—	8	1	no	—
233	C₄×C₅₃	—	2	2	no	—
234	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
235	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
236	C₄×C₅₃	D₄²	1	1	yes	characteristic
237	C₄×C₅₃	#554	1	1	yes	characteristic
238	Dic₅₃	—	2	2	no	—
239	Dic₅₃	—	2	2	no	—
240	Dic₅₃	—	4	1	no	—
241	Dic₅₃	—	8	2	no	—
242	#2957	—	53	2	no	—
243	C₂×#443	—	53	1	no	—
244	#2916	—	53	1	no	—
245	#2916	D₅₃	1	1	yes	characteristic, complemented
246	#1953	—	53	2	no	—
247	#1861	—	53	2	no	—
248	#1861	—	53	2	no	—
249	#1848	—	53	1	no	—
250	#1814	—	53	1	no	—
251	#1814	—	53	1	no	—
252	#1772	—	53	1	no	—
253	C₂×C₅₃	—	2	1	no	—
254	C₂×C₅₃	—	2	1	no	—
255	C₂×C₅₃	—	2	1	no	—
256	C₂×C₅₃	—	4	2	no	—
257	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
258	D₅₃	—	4	2	no	—
259	C₂×#135	—	53	1	no	—
260	#568	—	53	2	no	—
261	#568	C₂×D₅₃	1	2	yes	—
262	C₂×#131	—	53	1	no	—
263	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
264	#444	—	106	4	no	—
265	#443	—	53	2	no	—
266	#443	—	53	2	no	—
267	#443	—	53	2	no	—
268	#443	—	53	2	no	—
269	#443	—	106	4	no	—
270	#430	—	2	2	no	—
271	#430	—	53	2	no	—
272	#430	—	106	2	no	—
273	#430	C₂×D₅₃	1	2	yes	—
274	C₂×#108	—	2	2	no	—
275	C₂×#108	—	53	1	no	—
276	C₂×#108	—	53	2	no	—
277	C₂×#108	—	53	1	no	—
278	C₂×#108	—	106	2	no	—
279	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
280	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
281	#427	—	53	1	no	—
282	#423	—	53	2	no	—
283	#421	—	53	2	no	—
284	#421	—	53	2	no	—
285	C₂×#104	—	53	1	no	—
286	C₂×#104	—	53	1	no	—
287	#405	—	53	2	no	—
288	C₂×C₄×C₈	—	53	1	no	—
289	#344	—	53	2	no	—
290	#344	—	53	2	no	—
291	#339	—	53	2	no	—
292	C₅₃	#22694	1	1	yes	characteristic, complemented, Sylow
293	C₂×#41	—	53	1	no	—
294	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
295	#135	—	53	2	no	—
296	#135	—	53	2	no	—
297	#135	—	53	2	no	—
298	#135	—	53	2	no	—
299	C₂×Q₁₆	—	53	1	no	—
300	C₂×SD₁₆	—	53	1	no	—
301	C₂×SD₁₆	—	53	1	no	—
302	C₂×D₈	—	53	1	no	—
303	#131	—	2	1	no	—
304	#131	—	2	1	no	—
305	#131	—	2	4	no	—
306	#131	—	4	2	no	—
307	#131	—	53	2	no	—
308	#131	—	53	2	no	—
309	#131	—	106	4	no	—
310	#131	—	106	1	no	—
311	#131	—	106	1	no	—
312	#131	—	212	2	no	—
313	#131	C₄×D₅₃	1	2	yes	—
314	#131	C₄×D₅₃	1	2	yes	—
315	C₂×M₁₆	—	2	1	no	—
316	C₂×M₁₆	—	2	4	no	—
317	C₂×M₁₆	—	53	1	no	—
318	C₂×M₁₆	—	53	2	no	—
319	C₂×M₁₆	—	106	1	no	—
320	C₂×M₁₆	—	106	4	no	—
321	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
322	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
323	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
324	C₂²×C₈	—	2	1	no	—
325	C₂²×C₈	—	53	1	no	—
326	C₂²×C₈	—	106	1	no	—
327	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
328	#117	—	53	2	no	—
329	C₂×C₄²	—	53	1	no	—
330	#108	—	2	4	no	—
331	#108	—	2	1	no	—
332	#108	—	2	1	no	—
333	#108	—	53	2	no	—
334	#108	—	53	2	no	—
335	#108	—	106	2	no	—
336	#108	—	106	2	no	—
337	#108	—	106	4	no	—
338	#108	—	106	1	no	—
339	#108	—	106	1	no	—
340	#108	C₂²⋊D₅₃	1	2	yes	—
341	#108	C₂²⋊D₅₃	1	2	yes	—
342	#107	—	106	1	no	—
343	#106	—	106	1	no	—
344	C₄⋊C₈	—	212	2	no	—
345	#104	—	53	2	no	—
346	#104	—	53	2	no	—
347	#104	—	53	2	no	—
348	#104	—	53	2	no	—
349	#104	—	106	4	no	—
350	#104	—	106	4	no	—
351	#97	—	106	2	no	—
352	C₄×C₈	—	53	1	no	—
353	C₄×C₈	—	53	1	no	—
354	C₄×C₈	—	53	1	no	—
355	C₄×C₈	—	53	1	no	—
356	C₄×C₈	—	106	2	no	—
357	#41	—	2	1	no	—
358	#41	—	2	1	no	—
359	#41	—	53	2	no	—
360	#41	—	53	2	no	—
361	#41	—	106	1	no	—
362	#41	—	106	1	no	—
363	#41	C₂×C₄×D₅₃	1	2	yes	—
364	#41	D₄×D₅₃	1	2	yes	—
365	C₂×Q₈	—	53	1	no	—
366	C₂×Q₈	Q₈×D₅₃	1	1	yes	characteristic
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	53	1	no	—
369	C₂×D₄	—	106	1	no	—
370	C₂×D₄	Q₈×D₅₃	1	1	yes	characteristic
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	106	1	no	—
373	C₂²×C₄	—	106	1	no	—
374	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
375	Q₁₆	—	53	2	no	—
376	Q₁₆	—	53	2	no	—
377	SD₁₆	—	53	2	no	—
378	SD₁₆	—	53	2	no	—
379	SD₁₆	—	53	2	no	—
380	SD₁₆	—	53	2	no	—
381	D₈	—	53	2	no	—
382	D₈	—	53	2	no	—
383	M₁₆	—	2	1	no	—
384	M₁₆	—	2	2	no	—
385	M₁₆	—	2	2	no	—
386	M₁₆	—	2	2	no	—
387	M₁₆	—	2	2	no	—
388	M₁₆	—	2	2	no	—
389	M₁₆	—	4	4	no	—
390	M₁₆	—	53	2	no	—
391	M₁₆	—	106	1	no	—
392	M₁₆	—	106	2	no	—
393	M₁₆	—	106	2	no	—
394	M₁₆	—	106	2	no	—
395	M₁₆	—	106	2	no	—
396	M₁₆	—	212	4	no	—
397	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
398	M₁₆	C₂×C₂²⋊D₅₃	1	4	yes	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	4	4	no	—
405	C₂×C₈	—	53	1	no	—
406	C₂×C₈	—	53	1	no	—
407	C₂×C₈	—	53	1	no	—
408	C₂×C₈	—	53	1	no	—
409	C₂×C₈	—	53	1	no	—
410	C₂×C₈	—	53	1	no	—
411	C₂×C₈	—	106	1	no	—
412	C₂×C₈	—	106	2	no	—
413	C₂×C₈	—	106	2	no	—
414	C₂×C₈	—	106	2	no	—
415	C₂×C₈	—	212	4	no	—
416	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
417	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
418	C₂×C₈	#11719945	1	1	yes	characteristic
419	C₂×C₈	#11719945	1	1	yes	characteristic
420	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
421	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
422	C₄⋊C₄	—	106	2	no	—
423	C₂²⋊C₄	—	106	2	no	—
424	C₄²	—	53	1	no	—
425	C₄²	—	53	1	no	—
426	C₄²	—	106	1	no	—
427	C₄²	—	106	2	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	106	1	no	—
430	C₂³	#13133265	1	1	yes	characteristic
431	Q₈	—	2	1	no	—
432	Q₈	—	53	2	no	—
433	Q₈	—	106	1	no	—
434	Q₈	C₄⋊C₄×D₅₃	1	2	yes	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	1	no	—
438	D₄	—	53	2	no	—
439	D₄	—	106	2	no	—
440	D₄	—	106	2	no	—
441	D₄	—	106	1	no	—
442	D₄	C₄⋊C₄×D₅₃	1	2	yes	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	106	1	no	—
447	C₂×C₄	—	106	1	no	—
448	C₂×C₄	—	106	2	no	—
449	C₂×C₄	—	106	2	no	—
450	C₂×C₄	—	106	1	no	—
451	C₂×C₄	—	212	2	no	—
452	C₂×C₄	—	212	2	no	—
453	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
454	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
455	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
456	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
457	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
458	C₂×C₄	#13133262	1	1	yes	characteristic
459	C₈	—	2	2	no	—
460	C₈	—	2	4	no	—
461	C₈	—	4	2	no	—
462	C₈	—	53	1	no	—
463	C₈	—	53	1	no	—
464	C₈	—	53	1	no	—
465	C₈	—	53	1	no	—
466	C₈	—	106	2	no	—
467	C₈	—	212	2	no	—
468	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
469	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
470	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
471	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
472	C₂²	—	2	2	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	8	1	no	—
475	C₂²	—	106	2	no	—
476	C₂²	—	424	1	no	—
477	C₂²	#16811104	1	1	yes	characteristic
478	C₂²	#16811274	1	1	yes	characteristic
479	C₂²	Q₈×C₂²⋊D₅₃	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	106	2	no	—
482	C₄	—	106	2	no	—
483	C₄	—	424	2	no	—
484	C₄	—	212	1	no	—
485	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
486	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
487	C₄	D₄×C₂²⋊D₅₃	1	1	yes	characteristic
488	C₄	#16811328	1	1	yes	characteristic
489	C₂	—	2	1	no	—
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	4	2	no	—
493	C₂	—	212	2	no	—
494	C₂	C₄⋊C₄×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2916	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2916	D₅₃
hypocenter	C₅₃	#22694
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2916

Complements

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

N	H ≅ G/N	type
3. #20283323	C₂	semidirect
4. #20275502	C₂	semidirect
5. #20274955	C₂	semidirect
6. #20274952	C₂	semidirect
7. #20274885	C₂	semidirect
8. #20274770	C₂	semidirect
9. #20274769	C₂	semidirect
10. #20274597	C₂	semidirect
11. C₅₃×#2916	C₂	semidirect
17. #16810738	C₄	semidirect
18. #16810738	C₄	semidirect
19. #16810738	C₄	semidirect
20. #16810738	C₄	semidirect
245. #2916	D₅₃	semidirect
292. C₅₃	#22694	semidirect

Maximal subgroups

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

subgroup	index
#20283630	2
C₂×#16810738	2
#20283323	2
#20275502	2
#20274955	2
#20274952	2
#20274885	2
#20274770	2
#20274769	2
#20274597	2
C₅₃×#2916	2
#22694	53