Subgroups — #27406613

Order 13568.

880 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810745	C₂	1	2	yes	maximal, complemented
2	#20283459	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283283	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275544	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275428	C₂	1	2	yes	maximal, complemented
7	#20275044	C₂	1	2	yes	maximal, complemented
8	#20274992	C₂	1	2	yes	maximal, complemented
9	#20274570	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133144	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3035	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	C₂²	1	2	yes	—
13	C₂×#13133224	C₂²	1	2	yes	—
14	#16810745	—	2	8	no	—
15	#16810701	—	2	1	no	—
16	#16810695	—	2	2	no	—
17	D₄×#88322	—	2	1	no	—
18	C₂×#13133153	C₂²	1	2	yes	—
19	#16810650	—	2	2	no	—
20	#16810650	C₂²	1	2	yes	—
21	#16810647	—	2	1	no	—
22	#16810647	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#88322	—	2	1	no	—
24	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×#13133144	C₂²	1	1	yes	characteristic
26	C₂×#13133144	C₂²	1	2	yes	—
27	C₂×#13133144	C₂²	1	2	yes	—
28	C₂×#13133144	C₂²	1	1	yes	characteristic
29	#16810590	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	#16810388	C₂²	1	2	yes	—
34	#16810382	C₂²	1	1	yes	characteristic
35	#16810382	C₂²	1	1	yes	characteristic
36	#16810377	C₂²	1	2	yes	—
37	C₂×#135×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#532	C₂²	1	1	yes	characteristic
39	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#464	C₄	1	2	yes	complemented
41	C₅₃×#463	C₄	1	2	yes	complemented
42	C₅₃×#462	C₄	1	1	yes	characteristic, complemented
43	C₅₃×#461	C₄	1	1	yes	characteristic, complemented
44	C₅₃×#460	C₄	1	1	yes	characteristic, complemented
45	C₅₃×#459	C₄	1	1	yes	characteristic, complemented
46	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#416	C₂²	1	2	yes	—
48	#13133224	—	2	4	no	—
49	#13133224	—	2	4	no	—
50	#13133224	—	4	2	no	—
51	#13133224	—	4	2	no	—
52	C₂×#11719917	—	2	2	no	—
53	C₂×#11719917	—	2	2	no	—
54	C₂×#11719917	C₂³	1	1	yes	characteristic
55	C₂×#11719917	C₂³	1	2	yes	—
56	C₂×#11719917	C₂³	1	1	yes	characteristic
57	C₂²×#88322	—	2	2	no	—
58	C₂²×#88322	—	2	2	no	—
59	C₂²×#88322	C₂³	1	2	yes	—
60	C₂×C₄×Dic₅₃	—	2	1	no	—
61	C₂×C₄×Dic₅₃	—	2	1	no	—
62	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
64	#13133153	—	2	4	no	—
65	C₄⋊#88322	—	2	1	no	—
66	C₄⋊#88322	—	2	1	no	—
67	C₄⋊#88322	—	2	2	no	—
68	C₄⋊#88322	—	2	2	no	—
69	C₄⋊#88322	D₄	1	1	yes	characteristic
70	C₄⋊#88322	D₄	1	1	yes	characteristic
71	C₄⋊#88322	D₄	1	2	yes	—
72	#13133144	—	2	4	no	—
73	#13133144	—	2	2	no	—
74	#13133144	—	4	2	no	—
75	#13133144	D₄	1	4	yes	complemented
76	C₂²⋊#88322	—	2	2	no	—
77	C₂²⋊#88322	—	2	2	no	—
78	#13133128	—	2	1	no	—
79	#13133127	—	2	4	no	—
80	C₄×#88322	—	2	1	no	—
81	C₂×#41×C₅₃	C₂³	1	2	yes	—
82	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
83	#135×C₅₃	—	2	4	no	—
84	#135×C₅₃	—	2	4	no	—
85	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
86	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
87	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
88	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
89	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
90	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#124×C₅₃	C₂×C₄	1	2	yes	—
93	#121×C₅₃	—	2	1	no	—
94	#121×C₅₃	—	2	1	no	—
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	—	2	1	no	—
100	C₄×D₄×C₅₃	—	2	1	no	—
101	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	#117×C₅₃	—	2	1	no	—
104	#117×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	—	2	1	no	—
106	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
110	#103×C₅₃	C₂×C₄	1	2	yes	—
111	#103×C₅₃	C₂×C₄	1	2	yes	—
112	#102×C₅₃	C₂×C₄	1	2	yes	—
113	#102×C₅₃	C₂×C₄	1	2	yes	—
114	#41×C₅₃	—	2	1	no	—
115	#41×C₅₃	—	2	1	no	—
116	#41×C₅₃	—	2	2	no	—
117	#41×C₅₃	—	2	2	no	—
118	#41×C₅₃	—	2	2	no	—
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	C₂×D₄	1	2	yes	—
121	#41×C₅₃	#41	1	2	yes	—
122	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×D₄×C₅₃	—	2	1	no	—
125	C₂×D₄×C₅₃	—	2	2	no	—
126	C₂×D₄×C₅₃	—	2	1	no	—
127	C₂×D₄×C₅₃	—	2	1	no	—
128	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	—	2	1	no	—
131	C₂²×C₄×C₅₃	—	2	1	no	—
132	C₂²×C₄×C₅₃	—	2	2	no	—
133	C₂²×C₄×C₅₃	—	2	1	no	—
134	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
137	C₂²×Dic₅₃	—	2	1	no	—
138	C₂²×Dic₅₃	—	2	1	no	—
139	Q₁₆×C₅₃	—	2	4	no	—
140	SD₁₆×C₅₃	—	2	4	no	—
141	SD₁₆×C₅₃	—	2	4	no	—
142	D₈×C₅₃	—	2	4	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
146	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	1	no	—
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	C₂²⋊C₄×C₅₃	—	2	2	no	—
157	C₂²⋊C₄×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	2	1	no	—
159	C₄²×C₅₃	—	2	1	no	—
160	C₄²×C₅₃	—	2	1	no	—
161	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
165	C₄×Dic₅₃	—	2	1	no	—
166	C₄×Dic₅₃	—	2	1	no	—
167	C₄×Dic₅₃	—	2	1	no	—
168	C₄×Dic₅₃	—	4	2	no	—
169	C₄×Dic₅₃	—	4	2	no	—
170	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
171	#11719917	—	2	1	no	—
172	#11719917	—	2	2	no	—
173	#11719917	—	2	1	no	—
174	#11719917	—	2	2	no	—
175	#11719917	—	2	1	no	—
176	#11719917	—	4	4	no	—
177	#11719917	—	4	4	no	—
178	#11719917	C₂×D₄	1	2	yes	—
179	C₂×#88322	—	2	1	no	—
180	C₂×#88322	—	2	2	no	—
181	C₂×#88322	—	2	2	no	—
182	C₂×#88322	—	2	1	no	—
183	C₂×#88322	—	2	2	no	—
184	C₂×#88322	—	4	4	no	—
185	C₂×#88322	—	4	4	no	—
186	C₂×#88322	C₂×D₄	1	2	yes	—
187	C₂×#88322	C₂×D₄	1	2	yes	—
188	C₂³×C₅₃	—	2	1	no	—
189	C₂³×C₅₃	—	2	2	no	—
190	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
191	Q₈×C₅₃	—	2	1	no	—
192	Q₈×C₅₃	—	2	2	no	—
193	Q₈×C₅₃	—	2	2	no	—
194	Q₈×C₅₃	C₄×D₄	1	2	yes	—
195	D₄×C₅₃	—	2	2	no	—
196	D₄×C₅₃	—	2	2	no	—
197	D₄×C₅₃	—	2	2	no	—
198	D₄×C₅₃	—	2	1	no	—
199	D₄×C₅₃	—	2	2	no	—
200	D₄×C₅₃	—	4	2	no	—
201	D₄×C₅₃	—	4	2	no	—
202	D₄×C₅₃	—	4	2	no	—
203	D₄×C₅₃	—	8	1	no	—
204	D₄×C₅₃	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	4	1	no	—
213	C₂×C₄×C₅₃	—	4	2	no	—
214	C₂×C₄×C₅₃	—	4	2	no	—
215	C₂×C₄×C₅₃	—	4	2	no	—
216	C₂×C₄×C₅₃	—	8	1	no	—
217	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
226	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
227	C₂×Dic₅₃	—	2	2	no	—
228	C₂×Dic₅₃	—	2	2	no	—
229	C₂×Dic₅₃	—	4	1	no	—
230	C₂×Dic₅₃	—	4	1	no	—
231	C₂×Dic₅₃	—	4	1	no	—
232	C₂×Dic₅₃	—	4	1	no	—
233	C₈×C₅₃	—	2	2	no	—
234	C₈×C₅₃	—	2	2	no	—
235	#88322	—	2	2	no	—
236	#88322	—	2	2	no	—
237	#88322	—	2	2	no	—
238	#88322	—	4	2	no	—
239	#88322	—	4	1	no	—
240	#23016	—	53	1	no	maximal, Sylow
241	C₂²×C₅₃	—	2	1	no	—
242	C₂²×C₅₃	—	2	1	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	4	1	no	—
245	C₂²×C₅₃	—	4	1	no	—
246	C₂²×C₅₃	—	4	2	no	—
247	C₂²×C₅₃	—	4	2	no	—
248	C₂²×C₅₃	—	4	2	no	—
249	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂²×C₅₃	#547	1	1	yes	characteristic
251	C₂²×C₅₃	#553	1	1	yes	characteristic
252	C₄×C₅₃	—	2	2	no	—
253	C₄×C₅₃	—	2	1	no	—
254	C₄×C₅₃	—	2	1	no	—
255	C₄×C₅₃	—	2	1	no	—
256	C₄×C₅₃	—	4	2	no	—
257	C₄×C₅₃	—	4	1	no	—
258	C₄×C₅₃	—	4	1	no	—
259	C₄×C₅₃	#444	1	2	yes	complemented
260	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
261	C₄×C₅₃	#520	1	1	yes	characteristic
262	C₄×C₅₃	D₄²	1	1	yes	characteristic
263	C₄×C₅₃	#548	1	1	yes	characteristic
264	Dic₅₃	—	4	2	no	—
265	Dic₅₃	—	4	2	no	—
266	#3035	D₅₃	1	1	yes	characteristic
267	C₂×#444	—	53	2	no	—
268	#2935	—	53	1	no	—
269	#2906	—	53	1	no	—
270	#1962	—	53	1	no	—
271	#1950	—	53	1	no	—
272	#1938	—	53	2	no	—
273	#1876	—	53	2	no	—
274	#1866	—	53	2	no	—
275	#1767	—	53	1	no	—
276	C₄×#104	—	53	1	no	—
277	C₂×C₅₃	—	2	1	no	—
278	C₂×C₅₃	—	2	1	no	—
279	C₂×C₅₃	—	4	2	no	—
280	C₂×C₅₃	—	8	2	no	—
281	C₂×C₅₃	#2361	1	1	yes	characteristic
282	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
283	C₂×C₅₃	#2956	1	1	yes	characteristic
284	C₂×#135	C₂×D₅₃	1	2	yes	—
285	C₂×#131	—	53	2	no	—
286	C₂×#131	—	53	2	no	—
287	#532	C₂×D₅₃	1	1	yes	characteristic
288	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
289	#464	Dic₅₃	1	2	yes	complemented
290	#463	Dic₅₃	1	2	yes	complemented
291	#462	Dic₅₃	1	1	yes	characteristic, complemented
292	#461	Dic₅₃	1	1	yes	characteristic, complemented
293	#460	Dic₅₃	1	1	yes	characteristic, complemented
294	#459	Dic₅₃	1	1	yes	characteristic, complemented
295	#444	—	106	8	no	—
296	#436	—	106	1	no	—
297	#435	—	106	2	no	—
298	C₈×D₄	—	106	1	no	—
299	C₂×#108	—	53	2	no	—
300	#424	—	53	2	no	—
301	#424	—	106	2	no	—
302	#423	—	53	1	no	—
303	#423	—	106	1	no	—
304	C₂×C₄⋊C₈	—	53	1	no	—
305	C₂×C₄⋊C₈	—	106	1	no	—
306	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
307	C₂×#104	—	53	1	no	—
308	C₂×#104	—	53	1	no	—
309	C₂×#104	—	53	2	no	—
310	C₂×#104	—	53	2	no	—
311	#416	C₂×D₅₃	1	2	yes	—
312	#408	—	53	2	no	—
313	C₄×M₁₆	—	53	1	no	—
314	C₂×#97	—	53	2	no	—
315	C₄³	—	53	1	no	—
316	#341	—	53	2	no	—
317	#339	—	53	1	no	—
318	#339	—	53	1	no	—
319	#336	—	53	2	no	—
320	C₅₃	#23016	1	1	yes	characteristic, complemented, Sylow
321	C₂×#41	C₂²×D₅₃	1	2	yes	—
322	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
323	#135	—	2	4	no	—
324	#135	—	2	4	no	—
325	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
326	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
327	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
328	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
329	#131	—	106	4	no	—
330	#131	—	106	4	no	—
331	#131	—	212	2	no	—
332	#131	—	212	2	no	—
333	C₂×M₁₆	—	53	1	no	—
334	C₂×M₁₆	—	53	2	no	—
335	C₂×M₁₆	—	53	1	no	—
336	C₂×M₁₆	—	106	2	no	—
337	C₂×M₁₆	—	106	2	no	—
338	C₂²×C₈	—	53	2	no	—
339	C₂²×C₈	—	106	2	no	—
340	C₂²×C₈	—	106	2	no	—
341	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
342	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
343	#127	C₂×Dic₅₃	1	1	yes	characteristic
344	#124	C₂×Dic₅₃	1	2	yes	—
345	#121	—	2	1	no	—
346	#121	—	2	1	no	—
347	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
348	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
354	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
355	#117	—	2	1	no	—
356	#117	C₂²×D₅₃	1	1	yes	characteristic
357	C₂×C₄²	—	2	1	no	—
358	C₂×C₄²	—	53	1	no	—
359	C₂×C₄²	—	53	1	no	—
360	C₂×C₄²	—	106	1	no	—
361	C₂×C₄²	—	106	1	no	—
362	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
363	#108	—	106	4	no	—
364	C₄⋊C₈	—	53	1	no	—
365	C₄⋊C₈	—	53	1	no	—
366	C₄⋊C₈	—	53	2	no	—
367	C₄⋊C₈	—	106	2	no	—
368	C₄⋊C₈	—	106	2	no	—
369	C₄⋊C₈	—	106	1	no	—
370	C₄⋊C₈	—	106	1	no	—
371	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
372	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
373	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
374	#104	—	53	4	no	—
375	#104	—	106	4	no	—
376	#104	—	106	2	no	—
377	#104	—	212	2	no	—
378	#103	C₂×Dic₅₃	1	2	yes	—
379	#103	C₂×Dic₅₃	1	2	yes	—
380	#102	C₂×Dic₅₃	1	2	yes	—
381	#102	C₂×Dic₅₃	1	2	yes	—
382	C₂²⋊C₈	—	106	2	no	—
383	C₂²⋊C₈	—	106	2	no	—
384	#97	—	106	4	no	—
385	#97	—	106	1	no	—
386	C₄×C₈	—	106	1	no	—
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	2	2	no	—
390	#41	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	D₄×D₅₃	1	2	yes	—
394	#41	#11719947	1	2	yes	—
395	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
396	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
402	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	106	1	no	—
408	C₂²×C₄	—	106	1	no	—
409	C₂²×C₄	#11719947	1	1	yes	characteristic
410	C₂²×C₄	#11719947	1	1	yes	characteristic
411	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
412	Q₁₆	—	2	4	no	—
413	SD₁₆	—	2	4	no	—
414	SD₁₆	—	2	4	no	—
415	D₈	—	2	4	no	—
416	M₁₆	—	53	2	no	—
417	M₁₆	—	106	1	no	—
418	M₁₆	—	106	2	no	—
419	M₁₆	—	106	1	no	—
420	M₁₆	—	106	2	no	—
421	M₁₆	—	106	1	no	—
422	M₁₆	—	212	4	no	—
423	M₁₆	—	212	4	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	53	2	no	—
427	C₂×C₈	—	53	2	no	—
428	C₂×C₈	—	106	2	no	—
429	C₂×C₈	—	106	1	no	—
430	C₂×C₈	—	106	2	no	—
431	C₂×C₈	—	106	1	no	—
432	C₂×C₈	—	106	2	no	—
433	C₂×C₈	—	212	4	no	—
434	C₂×C₈	—	212	4	no	—
435	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
436	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
442	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
443	C₂²⋊C₄	—	2	1	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	2	1	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₄²	—	2	1	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	53	2	no	—
452	C₄²	—	106	1	no	—
453	C₄²	—	106	1	no	—
454	C₄²	—	106	1	no	—
455	C₄²	—	212	2	no	—
456	C₄²	—	212	2	no	—
457	C₄²	D₄×D₅₃	1	1	yes	characteristic
458	C₄²	D₄×D₅₃	1	1	yes	characteristic
459	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
460	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	C₂×#11719947	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	D₄×Dic₅₃	1	2	yes	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	1	no	—
472	D₄	—	2	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	1	no	—
477	D₄	D₄×Dic₅₃	1	2	yes	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	8	1	no	—
490	C₂×C₄	—	106	2	no	—
491	C₂×C₄	—	106	2	no	—
492	C₂×C₄	—	212	1	no	—
493	C₂×C₄	—	212	1	no	—
494	C₂×C₄	—	212	1	no	—
495	C₂×C₄	—	212	1	no	—
496	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
497	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
498	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
499	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
500	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
501	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
502	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
503	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
504	C₂×C₄	#13133263	1	1	yes	characteristic
505	C₂×C₄	#13133264	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	2	2	no	—
508	C₈	—	106	2	no	—
509	C₈	—	106	2	no	—
510	C₈	—	106	2	no	—
511	C₈	—	212	1	no	—
512	C₈	—	212	2	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
522	C₂²	#16811273	1	1	yes	characteristic
523	C₂²	#16811325	1	1	yes	characteristic
524	C₄	—	2	1	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	1	no	—
531	C₄	—	212	2	no	—
532	C₄	—	212	2	no	—
533	C₄	#16810745	1	2	yes	complemented
534	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
535	C₄	#16811106	1	1	yes	characteristic
536	C₄	#16811257	1	1	yes	characteristic
537	C₄	#16811279	1	1	yes	characteristic
538	C₂	—	2	1	no	—
539	C₂	—	2	1	no	—
540	C₂	—	4	2	no	—
541	C₂	—	8	2	no	—
542	C₂	#20277735	1	1	yes	characteristic
543	C₂	C₂×#16810745	1	1	yes	characteristic
544	C₂	#20283625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3035	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3035	D₅₃
hypocenter	C₅₃	#23016
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3035

Complements

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

N	H ≅ G/N	type
1. C₂×#16810745	C₂	semidirect
2. #20283459	C₂	semidirect
3. #20283283	C₂	semidirect
4. #20275544	C₂	semidirect
5. #20275491	C₂	semidirect
6. #20275428	C₂	semidirect
7. #20275044	C₂	semidirect
8. #20274992	C₂	semidirect
9. #20274570	C₂	semidirect
10. C₄×#13133144	C₂	semidirect
40. C₅₃×#464	C₄	semidirect
41. C₅₃×#463	C₄	semidirect
42. C₅₃×#462	C₄	semidirect
43. C₅₃×#461	C₄	semidirect
44. C₅₃×#460	C₄	semidirect
45. C₅₃×#459	C₄	semidirect
75. #13133144	D₄	semidirect
259. C₄×C₅₃	#444	semidirect
289. #464	Dic₅₃	semidirect
290. #463	Dic₅₃	semidirect
291. #462	Dic₅₃	semidirect
292. #461	Dic₅₃	semidirect
293. #460	Dic₅₃	semidirect
294. #459	Dic₅₃	semidirect
320. C₅₃	#23016	semidirect
533. C₄	#16810745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810745	2
#20283459	2
#20283283	2
#20275544	2
#20275491	2
#20275428	2
#20275044	2
#20274992	2
#20274570	2
C₄×#13133144	2
C₅₃×#3035	2
#23016	53