Subgroups — #27353803

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283828	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283823	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	#20273574	C₂	1	2	yes	maximal, complemented
6	#20273567	C₂	1	2	yes	maximal, complemented
7	#20273546	C₂	1	2	yes	maximal, complemented
8	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273103	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1452	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	#16810691	—	2	2	no	—
27	C₈×D₂₁₂	C₂²	1	2	yes	—
28	#16810686	—	2	2	no	—
29	#16810679	—	2	1	no	—
30	#16810675	—	2	1	no	—
31	#16810661	C₄	1	1	yes	characteristic
32	#16810645	—	2	1	no	—
33	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×D₅₃	—	2	1	no	—
36	#16810620	C₄	1	1	yes	characteristic
37	#16810620	C₄	1	1	yes	characteristic
38	#16810376	C₂²	1	2	yes	—
39	#16810376	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	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₅₃×#431	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	C₂²×#88322	—	2	1	no	—
58	C₂²×#88322	—	2	1	no	—
59	C₂×C₈×D₅₃	—	2	2	no	—
60	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
63	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
64	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
66	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
67	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
68	C₄×C₂²⋊D₅₃	—	2	1	no	—
69	C₄×C₂²⋊D₅₃	—	2	1	no	—
70	C₄×C₂²⋊D₅₃	—	2	1	no	—
71	C₄×C₂²⋊D₅₃	—	2	1	no	—
72	#13133176	C₂×C₄	1	1	yes	characteristic
73	#13133176	C₂×C₄	1	1	yes	characteristic
74	#13133174	—	2	1	no	—
75	#13133174	—	2	1	no	—
76	C₄²×D₅₃	—	2	1	no	—
77	C₄²×D₅₃	—	2	1	no	—
78	#13133151	C₂×C₄	1	1	yes	characteristic
79	#13133151	C₂×C₄	1	1	yes	characteristic
80	#13133149	C₂×C₄	1	1	yes	characteristic
81	#13133149	C₂×C₄	1	1	yes	characteristic
82	C₄⋊#88322	—	2	1	no	—
83	C₄⋊#88322	—	2	1	no	—
84	C₄⋊#88322	C₂³	1	2	yes	—
85	C₄⋊#88322	C₂³	1	2	yes	—
86	#13133146	—	2	2	no	—
87	#13133146	—	2	2	no	—
88	#13133142	C₂×C₄	1	1	yes	characteristic
89	#13133142	C₂×C₄	1	1	yes	characteristic
90	#13133142	C₂×C₄	1	1	yes	characteristic
91	#13133142	C₂×C₄	1	1	yes	characteristic
92	#13133139	C₂×C₄	1	1	yes	characteristic
93	#13133139	C₂×C₄	1	1	yes	characteristic
94	#13133139	C₂×C₄	1	1	yes	characteristic
95	#13133139	C₂×C₄	1	1	yes	characteristic
96	#13133129	—	2	2	no	—
97	#13133129	—	2	2	no	—
98	#13133127	—	2	2	no	—
99	C₈×Dic₅₃	—	2	2	no	—
100	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₅₃	—	2	1	no	—
104	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
105	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₂²⋊C₈×C₅₃	—	2	1	no	—
107	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
108	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₄×C₈×C₅₃	C₂×C₄	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	2	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
142	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₅₃	#41	1	1	yes	characteristic
145	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
146	C₂²⋊Dic₅₃	—	2	1	no	—
147	C₂²⋊Dic₅₃	—	2	1	no	—
148	#11719921	—	2	1	no	—
149	#11719921	—	2	1	no	—
150	#11719921	—	2	1	no	—
151	#11719921	—	2	1	no	—
152	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
156	#11719919	—	2	1	no	—
157	#11719919	—	2	1	no	—
158	#11719919	—	2	1	no	—
159	#11719919	—	2	1	no	—
160	C₄×Dic₅₃	—	2	1	no	—
161	C₄×Dic₅₃	—	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	C₂×#88322	—	2	2	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	—	8	1	no	—
169	C₂×#88322	—	8	1	no	—
170	Dic₂₁₂	—	2	2	no	—
171	D₄₂₄	—	2	2	no	—
172	#11719913	—	2	2	no	—
173	#11719913	—	2	2	no	—
174	#11719912	—	8	2	no	—
175	C₈×D₅₃	—	8	2	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	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
186	C₂×C₄×C₅₃	#131	1	2	yes	—
187	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
190	C₂²⋊D₅₃	—	2	4	no	—
191	C₂²⋊D₅₃	—	4	2	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	C₂×Dic₅₃	—	2	1	no	—
198	D₂₁₂	—	2	1	no	—
199	D₂₁₂	—	2	1	no	—
200	D₂₁₂	—	2	1	no	—
201	D₂₁₂	C₂×M₁₆	1	2	yes	—
202	C₄×D₅₃	—	2	4	no	—
203	C₄×D₅₃	—	4	1	no	—
204	C₄×D₅₃	—	4	1	no	—
205	C₄×D₅₃	—	4	1	no	—
206	C₄×D₅₃	—	4	1	no	—
207	C₄×D₅₃	—	4	2	no	—
208	C₄×D₅₃	—	8	1	no	—
209	C₄×D₅₃	—	8	1	no	—
210	Dic₁₀₆	—	2	1	no	—
211	Dic₁₀₆	—	2	1	no	—
212	Dic₁₀₆	—	2	1	no	—
213	Dic₁₀₆	C₂×M₁₆	1	2	yes	—
214	C₈×C₅₃	—	2	1	no	—
215	C₈×C₅₃	—	2	1	no	—
216	C₈×C₅₃	—	2	1	no	—
217	C₈×C₅₃	—	2	1	no	—
218	C₈×C₅₃	—	2	2	no	—
219	C₈×C₅₃	—	2	2	no	—
220	C₈×C₅₃	—	4	2	no	—
221	#88322	—	4	2	no	—
222	#88322	—	4	2	no	—
223	#17881	—	53	1	no	maximal, Sylow
224	C₂²×C₅₃	—	2	1	no	—
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	#435	1	2	yes	—
227	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂×D₅₃	—	2	2	no	—
229	C₂×D₅₃	—	4	1	no	—
230	C₂×D₅₃	—	4	1	no	—
231	C₂×D₅₃	—	4	1	no	—
232	C₂×D₅₃	—	4	1	no	—
233	C₂×D₅₃	—	4	1	no	—
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	2	1	no	—
237	C₄×C₅₃	#435	1	2	yes	—
238	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
239	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
240	C₄×C₅₃	#576	1	1	yes	characteristic
241	C₄×C₅₃	#578	1	1	yes	characteristic
242	Dic₅₃	—	2	2	no	—
243	Dic₅₃	—	4	1	no	—
244	Dic₅₃	—	4	1	no	—
245	Dic₅₃	—	4	1	no	—
246	Dic₅₃	—	4	1	no	—
247	Dic₅₃	—	4	1	no	—
248	#2989	—	53	1	no	—
249	#2988	—	53	1	no	—
250	C₄×#135	—	53	1	no	—
251	#1588	—	53	2	no	—
252	#1585	—	53	2	no	—
253	#1584	—	53	2	no	—
254	#1581	—	53	2	no	—
255	C₂×#335	—	53	1	no	—
256	#1477	—	53	1	no	—
257	#1477	—	53	1	no	—
258	#1452	D₅₃	1	1	yes	characteristic, complemented
259	C₂×C₅₃	—	2	2	no	—
260	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
261	C₂×C₅₃	#2951	1	1	yes	characteristic
262	C₂×C₅₃	#2957	1	1	yes	characteristic
263	D₅₃	—	4	2	no	—
264	D₅₃	—	8	1	no	—
265	C₂×#135	—	53	1	no	—
266	C₄×#41	—	53	1	no	—
267	C₄×#41	—	53	1	no	—
268	#446	—	53	2	no	—
269	C₈×Q₈	—	53	2	no	—
270	C₄×Q₁₆	—	53	1	no	—
271	C₄×Q₁₆	—	53	1	no	—
272	C₄×SD₁₆	—	53	1	no	—
273	C₄×SD₁₆	—	53	1	no	—
274	C₄×SD₁₆	—	53	1	no	—
275	C₄×SD₁₆	—	53	1	no	—
276	C₄×D₈	—	53	1	no	—
277	C₄×D₈	—	53	1	no	—
278	#436	—	53	2	no	—
279	#435	—	106	2	no	—
280	C₈×D₄	—	53	2	no	—
281	C₈×D₄	—	106	2	no	—
282	#432	—	106	1	no	—
283	#431	—	106	1	no	—
284	#431	C₂×D₅₃	1	1	yes	characteristic
285	#431	C₂×D₅₃	1	1	yes	characteristic
286	#427	—	53	1	no	—
287	#423	—	106	1	no	—
288	C₂×C₄⋊C₈	—	53	1	no	—
289	C₂×C₄⋊C₈	—	53	1	no	—
290	C₂×C₄⋊C₈	—	106	1	no	—
291	#416	—	53	1	no	—
292	#416	—	53	1	no	—
293	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
294	#335	—	53	2	no	—
295	#335	—	53	2	no	—
296	Q₈⋊C₈	—	53	2	no	—
297	Q₈⋊C₈	—	53	2	no	—
298	D₄⋊C₈	—	53	2	no	—
299	D₄⋊C₈	—	53	2	no	—
300	#322	C₂×D₅₃	1	2	yes	—
301	#322	C₂×D₅₃	1	2	yes	—
302	C₅₃	#17881	1	1	yes	characteristic, complemented, Sylow
303	C₂×#41	—	53	1	no	—
304	C₂×#41	—	53	1	no	—
305	#135	—	106	4	no	—
306	C₂×Q₁₆	—	53	1	no	—
307	C₂×SD₁₆	—	53	1	no	—
308	C₂×SD₁₆	—	53	1	no	—
309	C₂×D₈	—	53	1	no	—
310	C₂×M₁₆	—	106	2	no	—
311	C₂²×C₈	—	106	1	no	—
312	C₂²×C₈	—	106	1	no	—
313	C₂²×C₈	—	106	2	no	—
314	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
315	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
316	C₄×Q₈	—	53	1	no	—
317	C₄×Q₈	—	53	1	no	—
318	C₄×Q₈	—	53	1	no	—
319	C₄×Q₈	—	53	1	no	—
320	C₄×D₄	—	53	1	no	—
321	C₄×D₄	—	53	1	no	—
322	C₄×D₄	—	53	1	no	—
323	C₄×D₄	—	53	1	no	—
324	C₄×D₄	—	106	1	no	—
325	C₄×D₄	—	106	1	no	—
326	C₄×D₄	—	106	1	no	—
327	C₄×D₄	—	106	1	no	—
328	#117	—	53	1	no	—
329	#117	—	53	1	no	—
330	#117	—	106	1	no	—
331	#117	—	106	1	no	—
332	C₂×C₄²	—	106	1	no	—
333	C₂×C₄²	—	106	1	no	—
334	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
335	#107	—	53	1	no	—
336	#107	—	53	1	no	—
337	#106	—	53	1	no	—
338	#106	—	53	1	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	—	53	2	no	—
341	C₄⋊C₈	—	53	2	no	—
342	C₄⋊C₈	—	106	2	no	—
343	C₄⋊C₈	—	106	2	no	—
344	C₄⋊C₈	—	106	1	no	—
345	C₄⋊C₈	—	106	1	no	—
346	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
347	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
348	#103	—	53	1	no	—
349	#103	—	53	1	no	—
350	#103	—	53	1	no	—
351	#103	—	53	1	no	—
352	#102	—	53	1	no	—
353	#102	—	53	1	no	—
354	#102	—	53	1	no	—
355	#102	—	53	1	no	—
356	C₂²⋊C₈	—	2	1	no	—
357	C₂²⋊C₈	—	106	2	no	—
358	C₂²⋊C₈	—	106	2	no	—
359	C₂²⋊C₈	D₂₁₂	1	2	yes	—
360	#97	—	106	2	no	—
361	C₄×C₈	—	106	2	no	—
362	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
363	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
364	C₄×C₈	C₂²×D₅₃	1	2	yes	—
365	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
366	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
367	C₄×C₈	C₂²×D₅₃	1	2	yes	—
368	#41	—	53	4	no	—
369	#41	—	106	2	no	—
370	#41	—	106	2	no	—
371	#41	—	106	2	no	—
372	C₂×Q₈	—	53	1	no	—
373	C₂×Q₈	—	53	1	no	—
374	C₂×D₄	—	53	1	no	—
375	C₂×D₄	—	53	1	no	—
376	C₂×D₄	—	106	1	no	—
377	C₂×D₄	—	106	1	no	—
378	C₂²×C₄	—	106	1	no	—
379	C₂²×C₄	—	106	1	no	—
380	C₂²×C₄	—	106	1	no	—
381	C₂²×C₄	—	106	1	no	—
382	C₂²×C₄	—	106	1	no	—
383	C₂²×C₄	—	106	1	no	—
384	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
386	C₂²×C₄	#11719945	1	1	yes	characteristic
387	Q₁₆	—	106	2	no	—
388	SD₁₆	—	106	2	no	—
389	SD₁₆	—	106	2	no	—
390	D₈	—	106	2	no	—
391	M₁₆	—	424	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	106	2	no	—
396	C₂×C₈	—	106	2	no	—
397	C₂×C₈	—	424	2	no	—
398	C₂×C₈	—	424	1	no	—
399	C₂×C₈	—	424	1	no	—
400	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
401	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
402	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
403	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
404	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
405	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
406	C₄⋊C₄	—	53	1	no	—
407	C₄⋊C₄	—	53	1	no	—
408	C₄⋊C₄	—	53	1	no	—
409	C₄⋊C₄	—	53	1	no	—
410	C₄⋊C₄	—	106	1	no	—
411	C₄⋊C₄	—	106	1	no	—
412	C₄⋊C₄	—	106	1	no	—
413	C₄⋊C₄	—	106	1	no	—
414	C₂²⋊C₄	—	106	1	no	—
415	C₂²⋊C₄	—	106	1	no	—
416	C₂²⋊C₄	—	106	1	no	—
417	C₂²⋊C₄	—	106	1	no	—
418	C₂²⋊C₄	—	106	1	no	—
419	C₂²⋊C₄	—	106	1	no	—
420	C₄²	—	106	1	no	—
421	C₄²	—	106	1	no	—
422	C₄²	—	106	1	no	—
423	C₄²	—	106	1	no	—
424	C₄²	—	106	1	no	—
425	C₄²	—	106	1	no	—
426	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
427	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
428	C₄²	#11719945	1	1	yes	characteristic
429	C₄²	C₂³×D₅₃	1	1	yes	characteristic
430	C₂³	—	106	1	no	—
431	C₂³	—	106	1	no	—
432	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
433	Q₈	—	53	2	no	—
434	Q₈	—	106	1	no	—
435	Q₈	—	106	1	no	—
436	Q₈	—	106	1	no	—
437	D₄	—	53	2	no	—
438	D₄	—	106	4	no	—
439	D₄	—	106	1	no	—
440	D₄	—	106	1	no	—
441	D₄	—	106	1	no	—
442	D₄	—	212	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	106	1	no	—
447	C₂×C₄	—	106	4	no	—
448	C₂×C₄	—	106	1	no	—
449	C₂×C₄	—	106	1	no	—
450	C₂×C₄	—	106	1	no	—
451	C₂×C₄	—	106	1	no	—
452	C₂×C₄	—	106	1	no	—
453	C₂×C₄	—	424	1	no	—
454	C₂×C₄	—	424	1	no	—
455	C₂×C₄	—	212	1	no	—
456	C₂×C₄	—	212	1	no	—
457	C₂×C₄	—	212	1	no	—
458	C₂×C₄	—	212	1	no	—
459	C₂×C₄	—	212	2	no	—
460	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
461	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
462	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
463	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
464	C₂×C₄	#13133225	1	2	yes	—
465	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
467	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
468	C₈	—	2	1	no	—
469	C₈	—	2	1	no	—
470	C₈	—	2	1	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	2	no	—
474	C₈	—	4	2	no	—
475	C₈	—	212	2	no	—
476	C₈	—	212	2	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	106	2	no	—
480	C₂²	—	212	1	no	—
481	C₂²	—	212	1	no	—
482	C₂²	—	212	1	no	—
483	C₂²	—	212	1	no	—
484	C₂²	—	212	1	no	—
485	C₂²	#16810696	1	2	yes	—
486	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	106	2	no	—
491	C₄	—	212	1	no	—
492	C₄	—	212	1	no	—
493	C₄	—	212	1	no	—
494	C₄	—	212	1	no	—
495	C₄	—	212	1	no	—
496	C₄	#16810696	1	2	yes	—
497	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
498	C₄	C₂×#13133225	1	1	yes	characteristic
499	C₄	#16811428	1	1	yes	characteristic
500	C₄	#16811440	1	1	yes	characteristic
501	C₂	—	2	2	no	—
502	C₂	—	424	1	no	—
503	C₂	—	212	2	no	—
504	C₂	C₂×#16810696	1	1	yes	characteristic
505	C₂	#20283592	1	1	yes	characteristic
506	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#17881
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
1. #20283828	C₂	semidirect
2. #20283823	C₂	semidirect
4. #20273595	C₂	semidirect
5. #20273574	C₂	semidirect
6. #20273567	C₂	semidirect
7. #20273546	C₂	semidirect
8. C₂×#16810376	C₂	semidirect
9. #20273103	C₂	semidirect
10. #20273103	C₂	semidirect
11. C₅₃×#1452	C₂	semidirect
122. #11719945	M₁₆	semidirect
258. #1452	D₅₃	semidirect
302. C₅₃	#17881	semidirect

Maximal subgroups

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

subgroup	index
#20283828	2
#20283823	2
C₄×#13133239	2
#20273595	2
#20273574	2
#20273567	2
#20273546	2
C₂×#16810376	2
#20273103	2
C₅₃×#1452	2
#17881	53