Subgroups — #27353668

Order 13568.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283681	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
4	#20273595	C₂	1	2	yes	maximal, complemented
5	#20273588	C₂	1	2	yes	maximal, complemented
6	#20273581	C₂	1	2	yes	maximal, complemented
7	#20273574	C₂	1	2	yes	maximal, complemented
8	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273098	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273098	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₄	1	1	yes	characteristic
13	C₄×#11719945	C₂²	1	1	yes	characteristic
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	#16810753	C₂²	1	2	yes	—
16	C₈×Dic₁₀₆	C₂²	1	2	yes	—
17	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
18	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
19	C₄×#11719913	C₄	1	1	yes	characteristic
20	C₄×#11719913	C₄	1	1	yes	characteristic
21	C₄×#11719913	C₂²	1	1	yes	characteristic
22	C₄×#11719913	C₂²	1	1	yes	characteristic
23	C₄×D₄₂₄	C₄	1	1	yes	characteristic
24	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
25	#16810702	C₂²	1	2	yes	—
26	#16810694	—	2	2	no	—
27	C₈×D₂₁₂	C₂²	1	2	yes	—
28	C₈×C₂²⋊D₅₃	—	2	2	no	—
29	#16810676	C₂²	1	1	yes	characteristic
30	#16810676	C₂²	1	1	yes	characteristic
31	#16810674	—	2	1	no	—
32	#16810674	—	2	1	no	—
33	#16810661	C₄	1	1	yes	characteristic
34	#16810620	C₄	1	1	yes	characteristic
35	#16810620	C₄	1	1	yes	characteristic
36	C₄×#11719912	—	2	1	no	—
37	C₄×C₈×D₅₃	—	2	1	no	—
38	#16810376	C₂²	1	2	yes	—
39	#16810374	C₂²	1	2	yes	—
40	#16810355	C₂²	1	2	yes	—
41	#16810355	C₂²	1	2	yes	—
42	#16810352	C₂²	1	2	yes	—
43	#16810352	C₂²	1	2	yes	—
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#322	C₂²	1	2	yes	—
48	C₅₃×#322	C₂²	1	2	yes	—
49	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
51	#13133239	—	2	4	no	—
52	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
55	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
56	C₂×#11719912	—	2	2	no	—
57	C₂×C₈×D₅₃	—	2	2	no	—
58	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
62	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
65	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
66	C₄×C₂²⋊D₅₃	—	2	1	no	—
67	C₄×C₂²⋊D₅₃	—	2	1	no	—
68	C₄×C₂²⋊D₅₃	—	2	1	no	—
69	C₄×C₂²⋊D₅₃	—	2	1	no	—
70	#13133176	C₂×C₄	1	1	yes	characteristic
71	#13133176	C₂×C₄	1	1	yes	characteristic
72	#13133174	—	2	1	no	—
73	#13133174	—	2	1	no	—
74	C₄²×D₅₃	—	2	1	no	—
75	C₄²×D₅₃	—	2	1	no	—
76	#13133151	C₂×C₄	1	1	yes	characteristic
77	#13133151	C₂×C₄	1	1	yes	characteristic
78	#13133149	C₂×C₄	1	1	yes	characteristic
79	#13133149	C₂×C₄	1	1	yes	characteristic
80	C₄⋊#88322	C₂³	1	2	yes	—
81	C₄⋊#88322	C₂³	1	2	yes	—
82	#13133146	—	2	2	no	—
83	#13133146	—	2	2	no	—
84	#13133142	C₂×C₄	1	1	yes	characteristic
85	#13133142	C₂×C₄	1	1	yes	characteristic
86	#13133142	C₂×C₄	1	1	yes	characteristic
87	#13133142	C₂×C₄	1	1	yes	characteristic
88	#13133139	C₂×C₄	1	1	yes	characteristic
89	#13133139	C₂×C₄	1	1	yes	characteristic
90	#13133139	C₂×C₄	1	1	yes	characteristic
91	#13133139	C₂×C₄	1	1	yes	characteristic
92	C₂²⋊#88322	—	2	1	no	—
93	C₂²⋊#88322	—	2	1	no	—
94	#13133129	—	2	2	no	—
95	#13133129	—	2	2	no	—
96	#13133127	—	2	2	no	—
97	C₄×#88322	—	2	1	no	—
98	C₄×#88322	—	2	1	no	—
99	C₈×Dic₅₃	—	2	2	no	—
100	C₂²×C₈×C₅₃	—	2	1	no	—
101	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂²×C₈×C₅₃	D₄	1	2	yes	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₄×C₈×C₅₃	—	2	1	no	—
106	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
109	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
110	C₄×C₈×C₅₃	C₂³	1	2	yes	—
111	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₅₃	C₂³	1	2	yes	—
114	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
117	C₂×C₂²⋊D₅₃	—	2	1	no	—
118	C₂×C₂²⋊D₅₃	—	2	1	no	—
119	#11719945	—	2	2	no	—
120	#11719945	—	2	2	no	—
121	#11719945	—	2	2	no	—
122	#11719945	M₁₆	1	4	yes	complemented
123	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
124	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	—	2	1	no	—
126	C₂×C₄×D₅₃	—	2	1	no	—
127	C₂×C₄×D₅₃	—	2	1	no	—
128	C₂×C₄×D₅₃	—	2	1	no	—
129	C₂×C₄×D₅₃	—	2	1	no	—
130	C₂×C₄×D₅₃	—	2	1	no	—
131	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₅₃	—	2	4	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	4	2	no	—
138	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
144	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₅₃	#41	1	1	yes	characteristic
147	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₂²⋊Dic₅₃	—	2	1	no	—
149	C₂²⋊Dic₅₃	—	2	1	no	—
150	#11719921	—	2	1	no	—
151	#11719921	—	2	1	no	—
152	#11719921	—	2	1	no	—
153	#11719921	—	2	1	no	—
154	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
158	#11719919	—	2	1	no	—
159	#11719919	—	2	1	no	—
160	#11719919	—	2	1	no	—
161	#11719919	—	2	1	no	—
162	C₄×Dic₅₃	—	2	1	no	—
163	C₄×Dic₅₃	—	2	1	no	—
164	C₄×Dic₅₃	—	2	1	no	—
165	C₄×Dic₅₃	—	2	1	no	—
166	C₄×Dic₅₃	—	2	1	no	—
167	C₄×Dic₅₃	—	2	1	no	—
168	C₂×#88322	—	2	2	no	—
169	C₂×#88322	—	2	2	no	—
170	Dic₂₁₂	—	2	2	no	—
171	D₄₂₄	—	2	2	no	—
172	#11719913	—	2	2	no	—
173	#11719913	—	2	2	no	—
174	#11719912	—	4	4	no	—
175	C₈×D₅₃	—	4	4	no	—
176	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₅₃	—	2	1	no	—
178	C₂²×D₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
185	C₂×C₄×C₅₃	#131	1	2	yes	—
186	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
189	C₂²⋊D₅₃	—	2	4	no	—
190	C₂²⋊D₅₃	—	4	2	no	—
191	C₂×Dic₅₃	—	2	1	no	—
192	C₂×Dic₅₃	—	2	1	no	—
193	C₂×Dic₅₃	—	2	1	no	—
194	C₂×Dic₅₃	—	2	1	no	—
195	C₂×Dic₅₃	—	2	1	no	—
196	C₂×Dic₅₃	—	2	1	no	—
197	D₂₁₂	—	2	1	no	—
198	D₂₁₂	—	2	1	no	—
199	D₂₁₂	—	2	1	no	—
200	D₂₁₂	C₂×M₁₆	1	2	yes	—
201	C₄×D₅₃	—	2	4	no	—
202	C₄×D₅₃	—	4	2	no	—
203	C₄×D₅₃	—	4	2	no	—
204	C₄×D₅₃	—	4	2	no	—
205	C₄×D₅₃	—	4	2	no	—
206	C₄×D₅₃	—	4	2	no	—
207	Dic₁₀₆	—	2	1	no	—
208	Dic₁₀₆	—	2	1	no	—
209	Dic₁₀₆	—	2	1	no	—
210	Dic₁₀₆	C₂×M₁₆	1	2	yes	—
211	C₈×C₅₃	—	2	4	no	—
212	C₈×C₅₃	—	2	1	no	—
213	C₈×C₅₃	—	2	1	no	—
214	C₈×C₅₃	—	2	1	no	—
215	C₈×C₅₃	—	2	1	no	—
216	C₈×C₅₃	—	2	2	no	—
217	C₈×C₅₃	—	2	2	no	—
218	#88322	—	4	2	no	—
219	#88322	—	4	2	no	—
220	#17868	—	53	1	no	maximal, Sylow
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	—	2	1	no	—
223	C₂²×C₅₃	#435	1	2	yes	—
224	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂×D₅₃	—	2	2	no	—
226	C₂×D₅₃	—	4	1	no	—
227	C₂×D₅₃	—	4	1	no	—
228	C₂×D₅₃	—	4	1	no	—
229	C₂×D₅₃	—	4	1	no	—
230	C₂×D₅₃	—	4	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	#435	1	2	yes	—
234	C₄×C₅₃	#443	1	2	yes	—
235	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
236	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
237	C₄×C₅₃	#575	1	2	yes	—
238	Dic₅₃	—	2	2	no	—
239	Dic₅₃	—	4	1	no	—
240	Dic₅₃	—	4	1	no	—
241	Dic₅₃	—	4	1	no	—
242	Dic₅₃	—	4	1	no	—
243	Dic₅₃	—	4	1	no	—
244	#2965	—	53	1	no	—
245	C₈×#41	—	53	1	no	—
246	C₄×#135	—	53	1	no	—
247	#1588	—	53	2	no	—
248	#1587	—	53	2	no	—
249	#1586	—	53	2	no	—
250	#1585	—	53	2	no	—
251	#1564	—	53	1	no	—
252	#1476	—	53	1	no	—
253	#1476	—	53	1	no	—
254	C₂×#322	D₅₃	1	1	yes	characteristic, complemented
255	C₂×C₅₃	—	2	2	no	—
256	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
257	C₂×C₅₃	#2943	1	1	yes	characteristic
258	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
259	D₅₃	—	4	2	no	—
260	D₅₃	—	8	1	no	—
261	C₂×#135	—	53	1	no	—
262	C₄×#41	—	53	1	no	—
263	C₄×#41	—	53	1	no	—
264	#446	—	53	2	no	—
265	C₈×Q₈	—	53	2	no	—
266	C₄×Q₁₆	—	53	1	no	—
267	C₄×Q₁₆	—	53	1	no	—
268	C₄×SD₁₆	—	53	1	no	—
269	C₄×SD₁₆	—	53	1	no	—
270	C₄×SD₁₆	—	53	1	no	—
271	C₄×SD₁₆	—	53	1	no	—
272	C₄×D₈	—	53	1	no	—
273	C₄×D₈	—	53	1	no	—
274	#436	—	53	2	no	—
275	#435	—	106	2	no	—
276	C₈×D₄	—	53	2	no	—
277	C₈×D₄	—	106	2	no	—
278	#431	—	53	1	no	—
279	#431	—	53	1	no	—
280	#431	—	106	1	no	—
281	#431	—	106	1	no	—
282	#427	—	53	1	no	—
283	#416	—	53	1	no	—
284	#416	—	53	1	no	—
285	C₄×M₁₆	—	106	1	no	—
286	C₂×C₄×C₈	—	106	1	no	—
287	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
288	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
289	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
290	#335	—	53	2	no	—
291	#334	—	53	2	no	—
292	Q₈⋊C₈	—	53	2	no	—
293	Q₈⋊C₈	—	53	2	no	—
294	D₄⋊C₈	—	53	2	no	—
295	D₄⋊C₈	—	53	2	no	—
296	#322	C₂×D₅₃	1	2	yes	—
297	#322	C₂×D₅₃	1	2	yes	—
298	C₅₃	#17868	1	1	yes	characteristic, complemented, Sylow
299	C₂×#41	—	53	1	no	—
300	C₂×#41	—	53	1	no	—
301	#135	—	106	4	no	—
302	C₂×Q₁₆	—	53	1	no	—
303	C₂×SD₁₆	—	53	1	no	—
304	C₂×SD₁₆	—	53	1	no	—
305	C₂×D₈	—	53	1	no	—
306	C₂×M₁₆	—	106	2	no	—
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	106	2	no	—
309	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
310	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
311	C₂²×C₈	D₂₁₂	1	2	yes	—
312	C₄×Q₈	—	53	1	no	—
313	C₄×Q₈	—	53	1	no	—
314	C₄×Q₈	—	53	1	no	—
315	C₄×Q₈	—	53	1	no	—
316	C₄×D₄	—	53	1	no	—
317	C₄×D₄	—	53	1	no	—
318	C₄×D₄	—	53	1	no	—
319	C₄×D₄	—	53	1	no	—
320	C₄×D₄	—	106	1	no	—
321	C₄×D₄	—	106	1	no	—
322	C₄×D₄	—	106	1	no	—
323	C₄×D₄	—	106	1	no	—
324	#117	—	53	1	no	—
325	#117	—	53	1	no	—
326	#117	—	106	1	no	—
327	#117	—	106	1	no	—
328	C₂×C₄²	—	106	1	no	—
329	C₂×C₄²	—	106	1	no	—
330	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
331	#107	—	53	1	no	—
332	#107	—	53	1	no	—
333	#106	—	53	1	no	—
334	#106	—	53	1	no	—
335	C₄⋊C₈	—	53	2	no	—
336	C₄⋊C₈	—	53	2	no	—
337	C₄⋊C₈	—	106	2	no	—
338	C₄⋊C₈	—	106	2	no	—
339	#103	—	53	1	no	—
340	#103	—	53	1	no	—
341	#103	—	53	1	no	—
342	#103	—	53	1	no	—
343	#102	—	53	1	no	—
344	#102	—	53	1	no	—
345	#102	—	53	1	no	—
346	#102	—	53	1	no	—
347	C₂²⋊C₈	—	106	1	no	—
348	C₂²⋊C₈	—	106	1	no	—
349	C₂²⋊C₈	—	106	2	no	—
350	C₂²⋊C₈	—	106	2	no	—
351	#97	—	106	2	no	—
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	—	106	2	no	—
354	C₄×C₈	—	106	1	no	—
355	C₄×C₈	—	106	1	no	—
356	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
357	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
358	C₄×C₈	D₂₁₂	1	1	yes	characteristic
359	C₄×C₈	D₂₁₂	1	1	yes	characteristic
360	C₄×C₈	C₂²×D₅₃	1	2	yes	—
361	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
362	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
363	C₄×C₈	C₂²×D₅₃	1	2	yes	—
364	#41	—	53	4	no	—
365	#41	—	106	2	no	—
366	#41	—	106	2	no	—
367	#41	—	106	2	no	—
368	C₂×Q₈	—	53	1	no	—
369	C₂×Q₈	—	53	1	no	—
370	C₂×D₄	—	53	1	no	—
371	C₂×D₄	—	53	1	no	—
372	C₂×D₄	—	106	1	no	—
373	C₂×D₄	—	106	1	no	—
374	C₂²×C₄	—	106	1	no	—
375	C₂²×C₄	—	106	1	no	—
376	C₂²×C₄	—	106	1	no	—
377	C₂²×C₄	—	106	1	no	—
378	C₂²×C₄	—	106	1	no	—
379	C₂²×C₄	—	106	1	no	—
380	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
382	C₂²×C₄	#11719945	1	1	yes	characteristic
383	Q₁₆	—	106	2	no	—
384	SD₁₆	—	106	2	no	—
385	SD₁₆	—	106	2	no	—
386	D₈	—	106	2	no	—
387	M₁₆	—	212	4	no	—
388	C₂×C₈	—	2	4	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	4	2	no	—
393	C₂×C₈	—	106	2	no	—
394	C₂×C₈	—	106	2	no	—
395	C₂×C₈	—	212	4	no	—
396	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
397	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
398	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
399	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
400	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
401	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
402	C₄⋊C₄	—	53	1	no	—
403	C₄⋊C₄	—	53	1	no	—
404	C₄⋊C₄	—	53	1	no	—
405	C₄⋊C₄	—	53	1	no	—
406	C₄⋊C₄	—	106	1	no	—
407	C₄⋊C₄	—	106	1	no	—
408	C₄⋊C₄	—	106	1	no	—
409	C₄⋊C₄	—	106	1	no	—
410	C₂²⋊C₄	—	106	1	no	—
411	C₂²⋊C₄	—	106	1	no	—
412	C₂²⋊C₄	—	106	1	no	—
413	C₂²⋊C₄	—	106	1	no	—
414	C₂²⋊C₄	—	106	1	no	—
415	C₂²⋊C₄	—	106	1	no	—
416	C₄²	—	106	1	no	—
417	C₄²	—	106	1	no	—
418	C₄²	—	106	1	no	—
419	C₄²	—	106	1	no	—
420	C₄²	—	106	1	no	—
421	C₄²	—	106	1	no	—
422	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
423	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
424	C₄²	#11719945	1	1	yes	characteristic
425	C₄²	C₂³×D₅₃	1	1	yes	characteristic
426	C₂³	—	106	1	no	—
427	C₂³	—	106	1	no	—
428	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
429	Q₈	—	53	2	no	—
430	Q₈	—	106	1	no	—
431	Q₈	—	106	1	no	—
432	Q₈	—	106	1	no	—
433	D₄	—	53	2	no	—
434	D₄	—	106	4	no	—
435	D₄	—	106	1	no	—
436	D₄	—	106	1	no	—
437	D₄	—	106	1	no	—
438	D₄	—	212	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	106	1	no	—
442	C₂×C₄	—	106	4	no	—
443	C₂×C₄	—	106	1	no	—
444	C₂×C₄	—	106	1	no	—
445	C₂×C₄	—	106	1	no	—
446	C₂×C₄	—	106	1	no	—
447	C₂×C₄	—	106	1	no	—
448	C₂×C₄	—	212	2	no	—
449	C₂×C₄	—	212	2	no	—
450	C₂×C₄	—	212	2	no	—
451	C₂×C₄	—	212	2	no	—
452	C₂×C₄	—	212	2	no	—
453	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
454	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
455	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
456	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
457	C₂×C₄	#13133225	1	2	yes	—
458	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
460	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
461	C₈	—	2	4	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	1	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	1	no	—
466	C₈	—	2	2	no	—
467	C₈	—	2	2	no	—
468	C₈	—	212	2	no	—
469	C₈	—	212	2	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	106	2	no	—
473	C₂²	—	212	1	no	—
474	C₂²	—	212	1	no	—
475	C₂²	—	212	1	no	—
476	C₂²	—	212	1	no	—
477	C₂²	—	212	1	no	—
478	C₂²	#16810696	1	2	yes	—
479	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	106	2	no	—
483	C₄	—	212	1	no	—
484	C₄	—	212	1	no	—
485	C₄	—	212	1	no	—
486	C₄	—	212	1	no	—
487	C₄	—	212	1	no	—
488	C₄	#16810696	1	2	yes	—
489	C₄	#16810741	1	2	yes	—
490	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
491	C₄	C₂×#13133225	1	1	yes	characteristic
492	C₄	#16811423	1	2	yes	—
493	C₂	—	2	2	no	—
494	C₂	—	424	1	no	—
495	C₂	—	212	2	no	—
496	C₂	C₂×#16810696	1	1	yes	characteristic
497	C₂	#20283517	1	1	yes	characteristic
498	C₂	C₂×#16810741	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#17868
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
1. #20283681	C₂	semidirect
2. C₈×#11719945	C₂	semidirect
4. #20273595	C₂	semidirect
5. #20273588	C₂	semidirect
6. #20273581	C₂	semidirect
7. #20273574	C₂	semidirect
8. #20273463	C₂	semidirect
9. #20273098	C₂	semidirect
10. #20273098	C₂	semidirect
11. C₂×C₅₃×#322	C₂	semidirect
122. #11719945	M₁₆	semidirect
254. C₂×#322	D₅₃	semidirect
298. C₅₃	#17868	semidirect

Maximal subgroups

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

subgroup	index
#20283681	2
C₈×#11719945	2
C₄×#13133239	2
#20273595	2
#20273588	2
#20273581	2
#20273574	2
#20273463	2
#20273098	2
C₂×C₅₃×#322	2
#17868	53