Subgroups — #27406170

Order 13568.

708 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	#20284503	C₂	1	1	yes	maximal, characteristic
2	#20283658	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275691	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275488	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275465	C₂	1	2	yes	maximal, complemented
6	#20275311	C₂	1	2	yes	maximal, complemented
7	#20274990	C₂	1	2	yes	maximal
8	#20274570	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274557	C₂	1	2	yes	maximal
10	C₄×#13133144	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2906	C₂	1	1	yes	maximal, characteristic
12	#16811180	C₂²	1	1	yes	characteristic
13	#16810930	C₄	1	2	yes	complemented
14	#16810923	C₄	1	2	yes	complemented
15	#16810918	C₄	1	1	yes	characteristic, complemented
16	#16810913	C₄	1	1	yes	characteristic, complemented
17	#16810908	C₄	1	1	yes	characteristic, complemented
18	#16810903	C₄	1	1	yes	characteristic, complemented
19	#16810752	—	2	2	no	—
20	#16810750	—	2	1	no	—
21	#16810747	—	2	1	no	—
22	C₂×#13133152	C₂²	1	2	yes	—
23	#16810659	C₂²	1	2	yes	—
24	#16810649	C₂²	1	2	yes	—
25	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×#13133144	C₂²	1	1	yes	characteristic
27	C₂×#13133144	C₂²	1	1	yes	characteristic
28	#16810619	C₂²	1	2	yes	—
29	#16810578	C₂²	1	2	yes	—
30	C₄×#11719917	C₂²	1	1	yes	characteristic
31	C₂×#13133127	C₂²	1	2	yes	—
32	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
33	#16810387	C₂²	1	2	yes	—
34	#16810384	C₂²	1	2	yes	—
35	#16810384	C₂²	1	2	yes	—
36	#16810382	C₂²	1	1	yes	characteristic
37	#16810382	C₂²	1	1	yes	characteristic
38	#16810377	C₂²	1	2	yes	—
39	C₂×#131×C₅₃	C₂²	1	2	yes	—
40	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#424	—	2	2	no	—
42	C₅₃×#424	C₂²	1	2	yes	—
43	C₅₃×#423	—	2	1	no	—
44	C₅₃×#423	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#11719917	C₂³	1	1	yes	characteristic
48	C₂×#11719917	C₂³	1	1	yes	characteristic
49	C₂²×#88322	C₂³	1	2	yes	—
50	#13133217	C₂×C₄	1	1	yes	characteristic
51	#13133217	C₂×C₄	1	1	yes	characteristic
52	#13133209	C₂×C₄	1	2	yes	—
53	#13133196	—	2	1	no	—
54	#13133196	—	2	1	no	—
55	C₄×Dic₁₀₆	—	2	1	no	—
56	C₄×Dic₁₀₆	—	2	1	no	—
57	#13133176	C₂³	1	2	yes	—
58	C₂×C₄×Dic₅₃	—	2	1	no	—
59	C₂×C₄×Dic₅₃	—	2	1	no	—
60	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
62	#13133152	—	4	2	no	—
63	#13133150	C₂×C₄	1	2	yes	—
64	#13133150	C₂×C₄	1	2	yes	—
65	#13133148	C₂×C₄	1	2	yes	—
66	#13133148	C₂×C₄	1	2	yes	—
67	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
68	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
69	C₄⋊#88322	C₂×C₄	1	2	yes	—
70	#13133146	—	2	2	no	—
71	#13133146	—	2	2	no	—
72	#13133144	—	2	2	no	—
73	#13133144	Q₈	1	4	yes	complemented
74	#13133141	C₂×C₄	1	2	yes	—
75	#13133141	C₂×C₄	1	2	yes	—
76	#13133138	C₂×C₄	1	2	yes	—
77	#13133138	C₂×C₄	1	2	yes	—
78	#13133128	—	2	1	no	—
79	#13133127	—	2	2	no	—
80	#13133127	—	4	2	no	—
81	C₄×#88322	—	2	1	no	—
82	C₈×Dic₅₃	—	2	2	no	—
83	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
84	#131×C₅₃	—	2	4	no	—
85	#131×C₅₃	—	4	2	no	—
86	C₂×M₁₆×C₅₃	—	2	2	no	—
87	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
88	C₂²×C₈×C₅₃	—	2	2	no	—
89	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
90	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₄×D₄×C₅₃	—	2	1	no	—
93	C₄×D₄×C₅₃	—	2	1	no	—
94	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
96	#117×C₅₃	—	2	1	no	—
97	#117×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	—	2	1	no	—
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₅₃	—	2	2	no	—
101	C₄⋊C₈×C₅₃	—	2	2	no	—
102	C₄⋊C₈×C₅₃	—	2	1	no	—
103	C₄⋊C₈×C₅₃	—	2	1	no	—
104	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
105	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
107	#41×C₅₃	—	2	1	no	—
108	#41×C₅₃	—	2	1	no	—
109	#41×C₅₃	C₂×Q₈	1	2	yes	—
110	#41×C₅₃	#41	1	2	yes	—
111	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂²×Dic₅₃	—	2	1	no	—
121	C₂²×Dic₅₃	—	2	1	no	—
122	C₂×Dic₁₀₆	—	2	1	no	—
123	C₂×Dic₁₀₆	—	2	1	no	—
124	M₁₆×C₅₃	—	2	2	no	—
125	M₁₆×C₅₃	—	2	2	no	—
126	M₁₆×C₅₃	—	4	4	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	—	4	4	no	—
131	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₅₃	—	2	1	no	—
138	C₂²⋊C₄×C₅₃	—	2	1	no	—
139	C₂²⋊C₄×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₂²⋊Dic₅₃	—	2	2	no	—
148	C₄⋊Dic₅₃	—	2	1	no	—
149	C₄⋊Dic₅₃	—	2	1	no	—
150	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
151	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
152	#11719919	—	2	2	no	—
153	#11719919	—	2	2	no	—
154	C₄×Dic₅₃	—	2	2	no	—
155	C₄×Dic₅₃	—	2	1	no	—
156	C₄×Dic₅₃	—	2	1	no	—
157	C₄×Dic₅₃	—	2	1	no	—
158	C₄×Dic₅₃	—	4	2	no	—
159	C₄×Dic₅₃	—	4	2	no	—
160	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
161	#11719917	—	2	1	no	—
162	#11719917	—	2	1	no	—
163	#11719917	—	2	1	no	—
164	#11719917	C₂×Q₈	1	2	yes	—
165	C₂×#88322	—	2	1	no	—
166	C₂×#88322	—	2	2	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	—	2	1	no	—
169	C₂×#88322	C₂²×C₄	1	2	yes	—
170	C₂×#88322	C₂²×C₄	1	2	yes	—
171	C₂³×C₅₃	—	2	1	no	—
172	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
173	Q₈×C₅₃	—	2	1	no	—
174	Q₈×C₅₃	C₄×Q₈	1	2	yes	—
175	D₄×C₅₃	—	2	2	no	—
176	D₄×C₅₃	—	2	2	no	—
177	D₄×C₅₃	—	2	1	no	—
178	D₄×C₅₃	C₄×Q₈	1	2	yes	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	8	1	no	—
188	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
198	C₂×Dic₅₃	—	2	2	no	—
199	C₂×Dic₅₃	—	2	2	no	—
200	C₂×Dic₅₃	—	2	2	no	—
201	C₂×Dic₅₃	—	2	2	no	—
202	C₂×Dic₅₃	—	4	1	no	—
203	C₂×Dic₅₃	—	4	1	no	—
204	C₂×Dic₅₃	—	4	1	no	—
205	C₂×Dic₅₃	—	4	1	no	—
206	Dic₁₀₆	—	4	2	no	—
207	Dic₁₀₆	—	8	1	no	—
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	—	2	2	no	—
210	C₈×C₅₃	—	4	2	no	—
211	#88322	—	2	2	no	—
212	#88322	—	2	2	no	—
213	#88322	—	2	2	no	—
214	#88322	—	4	1	no	—
215	#22983	—	53	1	no	maximal, Sylow
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	4	1	no	—
220	C₂²×C₅₃	—	4	1	no	—
221	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₅₃	#552	1	1	yes	characteristic
223	C₂²×C₅₃	#554	1	1	yes	characteristic
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	2	1	no	—
228	C₄×C₅₃	—	4	1	no	—
229	C₄×C₅₃	—	4	1	no	—
230	C₄×C₅₃	#444	1	2	yes	—
231	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
232	C₄×C₅₃	#520	1	1	yes	characteristic
233	C₄×C₅₃	#546	1	1	yes	characteristic
234	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
235	Dic₅₃	—	4	2	no	—
236	Dic₅₃	—	4	2	no	—
237	Dic₅₃	—	4	2	no	—
238	Dic₅₃	—	4	2	no	—
239	#3076	—	53	1	no	—
240	#2962	—	53	1	no	—
241	#2906	D₅₃	1	1	yes	characteristic
242	#1994	—	53	1	no	—
243	#1950	—	53	1	no	—
244	#1946	—	53	2	no	—
245	#1922	—	53	2	no	—
246	#1866	—	53	2	no	—
247	#1767	—	53	1	no	—
248	#1765	—	53	2	no	—
249	C₄×#104	—	53	1	no	—
250	C₂×C₅₃	—	2	1	no	—
251	C₂×C₅₃	—	2	1	no	—
252	C₂×C₅₃	—	4	2	no	—
253	C₂×C₅₃	#2360	1	1	yes	characteristic
254	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
255	C₂×C₅₃	#2956	1	1	yes	characteristic
256	C₂×#131	C₂×D₅₃	1	2	yes	—
257	#533	—	53	1	no	—
258	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
259	#479	—	53	2	no	—
260	#478	—	53	2	no	—
261	#477	—	53	1	no	—
262	#476	—	53	1	no	—
263	#475	—	53	1	no	—
264	#474	—	53	1	no	—
265	#446	—	106	2	no	—
266	#446	—	106	1	no	—
267	C₈×Q₈	—	106	1	no	—
268	C₂×#108	—	53	2	no	—
269	#427	—	53	2	no	—
270	#424	—	2	2	no	—
271	#424	—	53	2	no	—
272	#424	C₂×D₅₃	1	2	yes	—
273	#423	—	2	1	no	—
274	#423	C₂×D₅₃	1	1	yes	characteristic
275	C₂×C₄⋊C₈	—	2	1	no	—
276	C₂×C₄⋊C₈	—	53	1	no	—
277	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
278	C₂×#104	—	53	1	no	—
279	C₂×#104	—	53	1	no	—
280	#416	—	53	2	no	—
281	#405	—	53	2	no	—
282	C₄×M₁₆	—	53	1	no	—
283	C₂×#97	—	53	2	no	—
284	C₄³	—	53	1	no	—
285	#341	—	53	2	no	—
286	#339	—	53	2	no	—
287	#339	—	53	2	no	—
288	#339	—	53	1	no	—
289	#339	—	53	1	no	—
290	#336	—	53	2	no	—
291	C₅₃	#22983	1	1	yes	characteristic, complemented, Sylow
292	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
293	#131	—	2	4	no	—
294	#131	—	4	2	no	—
295	C₂×M₁₆	—	2	2	no	—
296	C₂×M₁₆	—	53	1	no	—
297	C₂×M₁₆	—	53	1	no	—
298	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	—	53	2	no	—
301	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
302	C₄⋊Q₈	—	53	1	no	—
303	C₄⋊Q₈	—	53	1	no	—
304	#125	—	53	2	no	—
305	C₂²⋊Q₈	—	106	1	no	—
306	C₂²⋊Q₈	—	106	1	no	—
307	C₄×Q₈	—	106	1	no	—
308	C₄×Q₈	—	106	1	no	—
309	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
310	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
314	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	—	53	2	no	—
317	#117	C₂²×D₅₃	1	1	yes	characteristic
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	53	1	no	—
320	C₂×C₄²	—	53	1	no	—
321	C₂×C₄²	—	106	1	no	—
322	C₂×C₄²	—	106	1	no	—
323	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
324	#108	—	212	2	no	—
325	#107	—	53	2	no	—
326	#107	—	53	2	no	—
327	#106	—	53	2	no	—
328	#106	—	53	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	—	53	1	no	—
334	C₄⋊C₈	—	53	1	no	—
335	C₄⋊C₈	—	53	2	no	—
336	C₄⋊C₈	—	106	2	no	—
337	C₄⋊C₈	—	106	2	no	—
338	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
339	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
340	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
341	#104	—	53	4	no	—
342	#104	—	106	2	no	—
343	#103	—	53	2	no	—
344	#103	—	53	2	no	—
345	#102	—	53	2	no	—
346	#102	—	53	2	no	—
347	#97	—	106	2	no	—
348	#97	—	106	1	no	—
349	#97	—	212	2	no	—
350	C₄×C₈	—	106	2	no	—
351	C₄×C₈	—	106	1	no	—
352	#41	—	2	1	no	—
353	#41	—	2	1	no	—
354	#41	#11719947	1	2	yes	—
355	#41	Q₈×D₅₃	1	2	yes	—
356	C₂×Q₈	—	106	1	no	—
357	C₂×Q₈	—	106	1	no	—
358	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	106	1	no	—
365	C₂²×C₄	—	106	1	no	—
366	C₂²×C₄	#11719945	1	1	yes	characteristic
367	C₂²×C₄	#11719945	1	1	yes	characteristic
368	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
369	M₁₆	—	2	2	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	4	4	no	—
372	M₁₆	—	53	2	no	—
373	M₁₆	—	106	1	no	—
374	M₁₆	—	106	1	no	—
375	M₁₆	—	106	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	4	4	no	—
380	C₂×C₈	—	53	2	no	—
381	C₂×C₈	—	53	2	no	—
382	C₂×C₈	—	106	1	no	—
383	C₂×C₈	—	106	2	no	—
384	C₂×C₈	—	106	2	no	—
385	C₂×C₈	—	106	1	no	—
386	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
387	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	53	2	no	—
391	C₄⋊C₄	—	53	2	no	—
392	C₄⋊C₄	—	106	2	no	—
393	C₄⋊C₄	—	106	2	no	—
394	C₄⋊C₄	—	106	1	no	—
395	C₄⋊C₄	—	106	1	no	—
396	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
397	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	1	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	106	2	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	53	2	no	—
406	C₄²	—	106	2	no	—
407	C₄²	—	106	1	no	—
408	C₄²	—	106	1	no	—
409	C₄²	—	106	1	no	—
410	C₄²	—	212	2	no	—
411	C₄²	—	212	2	no	—
412	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
413	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
414	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
415	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
416	C₂³	—	2	1	no	—
417	C₂³	C₂×#11719945	1	1	yes	characteristic
418	Q₈	—	2	1	no	—
419	Q₈	—	424	1	no	—
420	Q₈	—	212	2	no	—
421	Q₈	#13133182	1	2	yes	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	1	no	—
425	D₄	#13133182	1	2	yes	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	8	1	no	—
435	C₂×C₄	—	106	2	no	—
436	C₂×C₄	—	106	2	no	—
437	C₂×C₄	—	106	2	no	—
438	C₂×C₄	—	106	2	no	—
439	C₂×C₄	—	212	1	no	—
440	C₂×C₄	—	212	1	no	—
441	C₂×C₄	—	212	1	no	—
442	C₂×C₄	—	212	1	no	—
443	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
444	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
445	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
446	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
447	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
448	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
449	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
450	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
451	C₂×C₄	#13133262	1	1	yes	characteristic
452	C₂×C₄	#13133265	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	2	no	—
455	C₈	—	4	2	no	—
456	C₈	—	106	2	no	—
457	C₈	—	106	2	no	—
458	C₈	—	106	2	no	—
459	C₈	—	212	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
466	C₂²	#16811308	1	1	yes	characteristic
467	C₂²	#16811328	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	1	no	—
473	C₄	—	4	1	no	—
474	C₄	—	212	2	no	—
475	C₄	—	212	2	no	—
476	C₄	—	212	2	no	—
477	C₄	—	212	2	no	—
478	C₄	#16810743	1	2	yes	—
479	C₄	C₂×#13133182	1	1	yes	characteristic
480	C₄	#16811104	1	1	yes	characteristic
481	C₄	#16811267	1	1	yes	characteristic
482	C₄	Q₈×C₂²⋊D₅₃	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	#20277720	1	1	yes	characteristic
487	C₂	C₂×#16810743	1	1	yes	characteristic
488	C₂	#20283622	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2906	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#41×C₅₃	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2906	D₅₃
hypocenter	C₅₃	#22983
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #2906

Complements

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

N	H ≅ G/N	type
2. #20283658	C₂	semidirect
3. #20275691	C₂	semidirect
4. #20275488	C₂	semidirect
5. #20275465	C₂	semidirect
6. #20275311	C₂	semidirect
8. #20274570	C₂	semidirect
13. #16810930	C₄	semidirect
14. #16810923	C₄	semidirect
15. #16810918	C₄	semidirect
16. #16810913	C₄	semidirect
17. #16810908	C₄	semidirect
18. #16810903	C₄	semidirect
73. #13133144	Q₈	semidirect
291. C₅₃	#22983	semidirect

Maximal subgroups

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

subgroup	index
#20284503	2
#20283658	2
#20275691	2
#20275488	2
#20275465	2
#20275311	2
#20274990	2
#20274570	2
#20274557	2
C₄×#13133144	2
C₅₃×#2906	2
#22983	53