Subgroups — #27391933

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 506 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	#20283649	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275659	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275490	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275314	C₂	1	2	yes	maximal, complemented
6	#20274990	C₂	1	2	yes	maximal
7	#20274735	C₂	1	1	yes	maximal, characteristic
8	#20274600	C₂	1	2	yes	maximal, complemented
9	#20274582	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274557	C₂	1	2	yes	maximal
11	C₄×#131×C₅₃	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	#16810753	—	2	1	no	—
20	#16810753	—	2	1	no	—
21	C₈×Dic₁₀₆	—	2	1	no	—
22	C₈×Dic₁₀₆	—	2	1	no	—
23	C₂×#13133152	C₂²	1	2	yes	—
24	#16810659	C₂²	1	2	yes	—
25	#16810649	C₂²	1	2	yes	—
26	#16810647	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×#13133144	C₂²	1	2	yes	—
29	#16810619	C₂²	1	2	yes	—
30	#16810578	C₂²	1	2	yes	—
31	C₂×#13133127	C₂²	1	2	yes	—
32	C₄×C₄⋊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	2	yes	—
37	#16810377	C₂²	1	2	yes	—
38	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#405	—	2	1	no	—
42	C₅₃×#405	—	2	1	no	—
43	C₅₃×#405	C₂²	1	1	yes	characteristic
44	C₅₃×#405	C₂²	1	1	yes	characteristic
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×#11719917	C₂³	1	2	yes	—
50	C₂²×#88322	C₂³	1	2	yes	—
51	#13133217	C₂×C₄	1	1	yes	characteristic
52	#13133217	C₂×C₄	1	1	yes	characteristic
53	#13133209	C₂×C₄	1	2	yes	—
54	#13133196	—	2	1	no	—
55	#13133196	—	2	1	no	—
56	C₄×Dic₁₀₆	—	2	1	no	—
57	C₄×Dic₁₀₆	—	2	1	no	—
58	#13133176	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₅₃	—	2	1	no	—
60	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
61	#13133152	—	2	4	no	—
62	#13133150	C₂×C₄	1	2	yes	—
63	#13133150	C₂×C₄	1	2	yes	—
64	#13133148	C₂×C₄	1	2	yes	—
65	#13133148	C₂×C₄	1	2	yes	—
66	C₄⋊#88322	—	2	1	no	—
67	C₄⋊#88322	—	2	1	no	—
68	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
69	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
70	C₄⋊#88322	C₂×C₄	1	2	yes	—
71	#13133146	—	2	2	no	—
72	#13133146	—	2	2	no	—
73	#13133144	—	2	4	no	—
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	#13133127	—	2	2	no	—
79	#13133127	—	2	4	no	—
80	C₈×Dic₅₃	—	2	2	no	—
81	#13133124	—	2	1	no	—
82	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
83	#131×C₅₃	—	2	2	no	—
84	#131×C₅₃	—	2	2	no	—
85	#131×C₅₃	—	2	2	no	—
86	#131×C₅₃	Q₈	1	4	yes	complemented
87	C₂×M₁₆×C₅₃	—	2	1	no	—
88	C₂×M₁₆×C₅₃	—	2	1	no	—
89	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₅₃	—	2	1	no	—
92	C₂²×C₈×C₅₃	—	2	1	no	—
93	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₅₃	—	2	1	no	—
98	C₄×D₄×C₅₃	—	2	1	no	—
99	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
101	#117×C₅₃	—	2	1	no	—
102	#117×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	—	2	1	no	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	#97×C₅₃	—	2	1	no	—
106	#97×C₅₃	—	2	1	no	—
107	#97×C₅₃	—	2	1	no	—
108	C₄×C₈×C₅₃	—	2	1	no	—
109	C₄×C₈×C₅₃	—	2	1	no	—
110	C₄×C₈×C₅₃	—	2	1	no	—
111	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
112	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
113	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
114	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
115	#41×C₅₃	—	2	1	no	—
116	#41×C₅₃	—	2	1	no	—
117	#41×C₅₃	C₂×Q₈	1	2	yes	—
118	#41×C₅₃	#41	1	2	yes	—
119	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄×C₅₃	—	2	1	no	—
121	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂²×Dic₅₃	—	2	2	no	—
129	C₂×Dic₁₀₆	—	2	1	no	—
130	C₂×Dic₁₀₆	—	2	1	no	—
131	M₁₆×C₅₃	—	2	4	no	—
132	M₁₆×C₅₃	—	2	1	no	—
133	M₁₆×C₅₃	—	2	1	no	—
134	M₁₆×C₅₃	—	2	4	no	—
135	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
136	M₁₆×C₅₃	#41	1	2	yes	—
137	C₂×C₈×C₅₃	—	2	4	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	—	2	1	no	—
142	C₂×C₈×C₅₃	—	2	4	no	—
143	C₂×C₈×C₅₃	C₂×D₄	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	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
148	C₂×C₈×C₅₃	#41	1	2	yes	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²⋊C₄×C₅₃	—	2	1	no	—
154	C₂²⋊C₄×C₅₃	—	2	1	no	—
155	C₂²⋊C₄×C₅₃	—	2	1	no	—
156	C₄²×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	1	no	—
158	C₄²×C₅₃	—	2	1	no	—
159	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²⋊Dic₅₃	—	2	2	no	—
164	C₄⋊Dic₅₃	—	2	1	no	—
165	C₄⋊Dic₅₃	—	2	1	no	—
166	C₄⋊Dic₅₃	—	4	2	no	—
167	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
168	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
169	#11719919	—	2	2	no	—
170	#11719919	—	2	2	no	—
171	C₄×Dic₅₃	—	2	2	no	—
172	C₄×Dic₅₃	—	2	2	no	—
173	C₄×Dic₅₃	—	2	2	no	—
174	#11719917	—	2	2	no	—
175	#11719917	—	2	2	no	—
176	C₂×#88322	—	2	2	no	—
177	C₂×#88322	—	2	2	no	—
178	C₂×#88322	—	2	2	no	—
179	C₂×#88322	C₂²×C₄	1	2	yes	—
180	C₂×#88322	C₂²×C₄	1	2	yes	—
181	C₂³×C₅₃	—	2	1	no	—
182	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
183	Q₈×C₅₃	—	2	1	no	—
184	Q₈×C₅₃	C₄×Q₈	1	2	yes	—
185	D₄×C₅₃	—	2	2	no	—
186	D₄×C₅₃	—	2	2	no	—
187	D₄×C₅₃	—	2	1	no	—
188	D₄×C₅₃	C₄×Q₈	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	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	—	4	2	no	—
198	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
208	C₂×Dic₅₃	—	2	2	no	—
209	C₂×Dic₅₃	—	2	2	no	—
210	C₂×Dic₅₃	—	4	2	no	—
211	C₂×Dic₅₃	—	4	2	no	—
212	C₂×Dic₅₃	—	4	2	no	—
213	Dic₁₀₆	—	4	2	no	—
214	Dic₁₀₆	—	4	2	no	—
215	C₈×C₅₃	—	2	2	no	—
216	C₈×C₅₃	—	2	1	no	—
217	C₈×C₅₃	—	2	2	no	—
218	C₈×C₅₃	—	2	1	no	—
219	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
220	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
221	#88322	—	2	2	no	—
222	#88322	—	2	2	no	—
223	#88322	—	4	2	no	—
224	#21628	—	53	1	no	maximal, Sylow
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	2	1	no	—
227	C₂²×C₅₃	—	2	2	no	—
228	C₂²×C₅₃	—	4	1	no	—
229	C₂²×C₅₃	—	4	1	no	—
230	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂²×C₅₃	#529	1	1	yes	characteristic
232	C₂²×C₅₃	#533	1	1	yes	characteristic
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	2	2	no	—
237	C₄×C₅₃	—	4	1	no	—
238	C₄×C₅₃	—	4	1	no	—
239	C₄×C₅₃	#444	1	2	yes	—
240	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
241	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
242	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
243	C₄×C₅₃	#525	1	1	yes	characteristic
244	Dic₅₃	—	4	2	no	—
245	Dic₅₃	—	4	2	no	—
246	Dic₅₃	—	8	2	no	—
247	#3076	—	53	1	no	—
248	#2960	—	53	1	no	—
249	C₄×#131	D₅₃	1	1	yes	characteristic
250	#1986	—	53	1	no	—
251	#1950	—	53	1	no	—
252	#1922	—	53	2	no	—
253	#1866	—	53	2	no	—
254	#1806	—	53	1	no	—
255	#1773	—	53	2	no	—
256	#1770	—	53	1	no	—
257	#1765	—	53	2	no	—
258	C₂×C₅₃	—	2	1	no	—
259	C₂×C₅₃	—	2	1	no	—
260	C₂×C₅₃	—	4	2	no	—
261	C₂×C₅₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
262	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
263	C₂×C₅₃	#2955	1	1	yes	characteristic
264	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
265	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
266	#533	—	53	1	no	—
267	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
268	#479	—	53	2	no	—
269	#478	—	53	2	no	—
270	#477	—	53	1	no	—
271	#476	—	53	1	no	—
272	#475	—	53	1	no	—
273	#474	—	53	1	no	—
274	#446	—	106	1	no	—
275	#446	—	106	1	no	—
276	C₈×Q₈	—	106	1	no	—
277	C₈×Q₈	—	106	1	no	—
278	C₂×#108	—	53	2	no	—
279	#427	—	53	2	no	—
280	#424	—	53	2	no	—
281	#423	—	53	1	no	—
282	C₂×C₄⋊C₈	—	53	1	no	—
283	C₂×#104	—	53	2	no	—
284	#416	—	53	2	no	—
285	#405	—	2	1	no	—
286	#405	—	2	1	no	—
287	#405	—	53	2	no	—
288	#405	C₂×D₅₃	1	1	yes	characteristic
289	#405	C₂×D₅₃	1	1	yes	characteristic
290	C₄×M₁₆	—	2	1	no	—
291	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
292	C₂×#97	—	53	2	no	—
293	C₂×C₄×C₈	—	2	1	no	—
294	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
295	C₄×C₄⋊C₄	—	53	1	no	—
296	#341	—	53	2	no	—
297	#339	—	53	2	no	—
298	#339	—	53	2	no	—
299	#339	—	53	2	no	—
300	#336	—	53	2	no	—
301	C₅₃	#21628	1	1	yes	characteristic, complemented, Sylow
302	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
303	#131	—	2	2	no	—
304	#131	—	2	2	no	—
305	#131	—	2	2	no	—
306	#131	Dic₁₀₆	1	4	yes	complemented
307	C₂×M₁₆	—	2	1	no	—
308	C₂×M₁₆	—	2	1	no	—
309	C₂×M₁₆	—	53	2	no	—
310	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
311	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
312	C₂²×C₈	—	2	1	no	—
313	C₂²×C₈	—	2	1	no	—
314	C₂²×C₈	—	53	2	no	—
315	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
316	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
317	C₄⋊Q₈	—	53	1	no	—
318	C₄⋊Q₈	—	53	1	no	—
319	#125	—	53	2	no	—
320	C₂²⋊Q₈	—	106	1	no	—
321	C₂²⋊Q₈	—	106	1	no	—
322	C₄×Q₈	—	106	1	no	—
323	C₄×Q₈	—	106	1	no	—
324	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
325	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
329	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
330	#117	—	2	1	no	—
331	#117	—	53	2	no	—
332	#117	C₂²×D₅₃	1	1	yes	characteristic
333	C₂×C₄⋊C₄	—	106	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	53	2	no	—
336	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
337	#108	—	106	4	no	—
338	#107	—	53	2	no	—
339	#107	—	53	2	no	—
340	#106	—	53	2	no	—
341	#106	—	53	2	no	—
342	C₄⋊C₈	—	53	1	no	—
343	C₄⋊C₈	—	53	1	no	—
344	C₄⋊C₈	—	53	2	no	—
345	C₄⋊C₈	—	106	2	no	—
346	C₄⋊C₈	—	106	2	no	—
347	C₄⋊C₈	—	106	1	no	—
348	C₄⋊C₈	—	106	1	no	—
349	#104	—	106	4	no	—
350	#103	—	53	2	no	—
351	#103	—	53	2	no	—
352	#102	—	53	2	no	—
353	#102	—	53	2	no	—
354	#97	—	2	1	no	—
355	#97	—	2	1	no	—
356	#97	—	2	1	no	—
357	#97	—	106	4	no	—
358	#97	—	106	2	no	—
359	C₄×C₈	—	2	1	no	—
360	C₄×C₈	—	2	1	no	—
361	C₄×C₈	—	2	1	no	—
362	C₄×C₈	—	106	2	no	—
363	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
364	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
365	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
366	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
367	#95	—	106	1	no	—
368	#41	—	2	1	no	—
369	#41	—	2	1	no	—
370	#41	C₂×Dic₁₀₆	1	2	yes	—
371	#41	#11719945	1	2	yes	—
372	C₂×Q₈	—	106	1	no	—
373	C₂×Q₈	—	106	1	no	—
374	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	106	2	no	—
381	C₂²×C₄	#11719945	1	1	yes	characteristic
382	C₂²×C₄	#11719945	1	1	yes	characteristic
383	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
384	M₁₆	—	2	4	no	—
385	M₁₆	—	2	1	no	—
386	M₁₆	—	2	1	no	—
387	M₁₆	—	2	4	no	—
388	M₁₆	—	106	2	no	—
389	M₁₆	—	106	2	no	—
390	M₁₆	C₂×Dic₁₀₆	1	2	yes	—
391	M₁₆	#11719945	1	2	yes	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	4	no	—
398	C₂×C₈	—	53	2	no	—
399	C₂×C₈	—	53	2	no	—
400	C₂×C₈	—	106	2	no	—
401	C₂×C₈	—	106	2	no	—
402	C₂×C₈	—	106	2	no	—
403	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
404	C₂×C₈	#11719945	1	2	yes	—
405	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
406	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
407	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
408	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	53	2	no	—
412	C₄⋊C₄	—	53	2	no	—
413	C₄⋊C₄	—	106	2	no	—
414	C₄⋊C₄	—	106	2	no	—
415	C₄⋊C₄	—	106	1	no	—
416	C₄⋊C₄	—	106	1	no	—
417	C₄⋊C₄	—	212	2	no	—
418	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
419	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	106	2	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	106	2	no	—
428	C₄²	—	106	2	no	—
429	C₄²	—	106	2	no	—
430	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
431	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
432	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
433	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
434	C₂³	—	2	1	no	—
435	C₂³	C₂×#11719945	1	1	yes	characteristic
436	Q₈	—	2	1	no	—
437	Q₈	—	212	2	no	—
438	Q₈	—	212	2	no	—
439	Q₈	C₄×Dic₁₀₆	1	2	yes	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	2	1	no	—
443	D₄	C₄×Dic₁₀₆	1	2	yes	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	106	2	no	—
454	C₂×C₄	—	106	2	no	—
455	C₂×C₄	—	212	2	no	—
456	C₂×C₄	—	212	2	no	—
457	C₂×C₄	—	212	2	no	—
458	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
459	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
460	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
461	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
463	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
464	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
465	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
466	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
467	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	2	1	no	—
470	C₈	—	2	2	no	—
471	C₈	—	2	1	no	—
472	C₈	—	106	2	no	—
473	C₈	—	106	2	no	—
474	C₈	—	212	2	no	—
475	C₈	#13133196	1	2	yes	—
476	C₈	#13133196	1	2	yes	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
483	C₂²	#16811157	1	1	yes	characteristic
484	C₂²	#16811180	1	1	yes	characteristic
485	C₄	—	2	2	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	1	no	—
491	C₄	—	424	2	no	—
492	C₄	—	212	2	no	—
493	C₄	—	212	2	no	—
494	C₄	#16810743	1	2	yes	—
495	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
496	C₄	C₄×#11719945	1	1	yes	characteristic
497	C₄	C₂×#13133196	1	1	yes	characteristic
498	C₄	#16811134	1	1	yes	characteristic
499	C₂	—	2	1	no	—
500	C₂	—	2	1	no	—
501	C₂	—	4	2	no	—
502	C₂	C₄×#13133196	1	1	yes	characteristic
503	C₂	C₂×#16810743	1	1	yes	characteristic
504	C₂	#20283616	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	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#41×C₅₃	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#21628
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20283649	C₂	semidirect
3. #20275659	C₂	semidirect
4. #20275490	C₂	semidirect
5. #20275314	C₂	semidirect
8. #20274600	C₂	semidirect
9. #20274582	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
86. #131×C₅₃	Q₈	semidirect
301. C₅₃	#21628	semidirect
306. #131	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20284503	2
#20283649	2
#20275659	2
#20275490	2
#20275314	2
#20274990	2
#20274735	2
#20274600	2
#20274582	2
#20274557	2
C₄×#131×C₅₃	2
#21628	53