Subgroups — #27354086

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284775	C₂	1	1	yes	maximal, characteristic
2	#20283428	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273593	C₂	1	2	yes	maximal, complemented
4	#20273579	C₂	1	2	yes	maximal, complemented
5	C₈×#11719924	C₂	1	2	yes	maximal, complemented
6	C₈×#11719922	C₂	1	2	yes	maximal, complemented
7	#20273503	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273154	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	—	2	1	no	—
13	#16811173	C₂²	1	1	yes	characteristic
14	#16810947	C₄	1	2	yes	—
15	#16810918	C₄	1	1	yes	characteristic
16	#16810913	C₄	1	1	yes	characteristic
17	#16810823	C₄	1	1	yes	characteristic
18	#16810818	C₄	1	1	yes	characteristic
19	#16810768	C₄	1	2	yes	—
20	C₄×#11719924	C₂²	1	2	yes	—
21	C₄×#11719922	C₂²	1	2	yes	—
22	#16810702	—	2	1	no	—
23	#16810702	C₂²	1	2	yes	—
24	C₈×D₂₁₂	—	2	1	no	—
25	C₈×D₂₁₂	C₂²	1	2	yes	—
26	#16810676	C₂²	1	1	yes	characteristic
27	#16810647	C₂²	1	1	yes	characteristic
28	#16810589	—	2	1	no	—
29	C₄×#11719917	C₂²	1	1	yes	characteristic
30	#16810375	C₂²	1	2	yes	—
31	#16810373	C₂²	1	2	yes	—
32	Q₈⋊#88322	C₂²	1	2	yes	—
33	D₄⋊#88322	C₂²	1	2	yes	—
34	#16810350	C₂²	1	2	yes	—
35	#16810350	C₂²	1	2	yes	—
36	#16810345	C₂²	1	2	yes	—
37	C₈×#88322	C₂²	1	2	yes	—
38	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₅₃	—	2	1	no	—
42	C₈×D₄×C₅₃	—	2	1	no	—
43	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#431	—	2	1	no	—
46	C₅₃×#431	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×#11719945	—	2	1	no	—
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×#11719922	C₂×C₄	1	2	yes	—
52	#13133223	—	4	4	no	—
53	C₂×#11719917	—	2	1	no	—
54	C₂×#11719917	—	2	1	no	—
55	C₂×#11719912	—	2	2	no	—
56	C₂×C₈×D₅₃	—	2	2	no	—
57	#13133217	C₂×C₄	1	1	yes	characteristic
58	#13133214	C₂×C₄	1	1	yes	characteristic
59	#13133205	—	2	1	no	—
60	#13133191	—	2	1	no	—
61	#13133191	C₂×C₄	1	2	yes	—
62	C₄×D₂₁₂	—	2	1	no	—
63	C₄×D₂₁₂	C₂³	1	2	yes	—
64	#13133150	C₂×C₄	1	2	yes	—
65	#13133148	C₂×C₄	1	2	yes	—
66	C₄⋊#88322	—	2	1	no	—
67	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
68	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
69	C₄⋊#88322	C₂³	1	2	yes	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133138	C₂×C₄	1	2	yes	—
72	#13133137	C₂×C₄	1	2	yes	—
73	#13133137	C₂×C₄	1	2	yes	—
74	C₂²⋊#88322	C₂×C₄	1	2	yes	—
75	#13133129	—	2	2	no	—
76	#13133129	—	2	2	no	—
77	#13133128	—	2	1	no	—
78	C₄×#88322	C₂³	1	2	yes	—
79	C₄×#88322	C₂³	1	2	yes	—
80	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂²×C₈×C₅₃	—	2	1	no	—
82	C₂²×C₈×C₅₃	—	2	1	no	—
83	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
84	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×D₄×C₅₃	—	2	1	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
91	#117×C₅₃	—	2	1	no	—
92	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	—	2	1	no	—
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₅₃	—	2	1	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	C₂²⋊C₈×C₅₃	—	2	1	no	—
101	C₂²⋊C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	—	2	1	no	—
103	C₄×C₈×C₅₃	—	2	1	no	—
104	C₄×C₈×C₅₃	—	2	1	no	—
105	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	#41×C₅₃	—	2	2	no	—
110	#41×C₅₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
120	C₂×C₂²⋊D₅₃	—	2	2	no	—
121	#11719945	—	4	2	no	—
122	#11719945	—	4	2	no	—
123	C₂×D₂₁₂	—	2	1	no	—
124	C₂×D₂₁₂	—	2	1	no	—
125	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
126	C₂×C₄×D₅₃	—	2	2	no	—
127	C₂×Dic₁₀₆	—	2	1	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	4	1	no	—
132	C₂×C₈×C₅₃	—	4	2	no	—
133	C₂×C₈×C₅₃	—	4	2	no	—
134	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²⋊C₄×C₅₃	—	2	1	no	—
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₂²⋊C₄×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	—	2	1	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₅₃	#41	1	1	yes	characteristic
151	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
152	#11719924	—	2	4	no	—
153	#11719922	—	2	4	no	—
154	#11719921	—	2	2	no	—
155	#11719921	—	2	2	no	—
156	C₄⋊Dic₅₃	—	2	1	no	—
157	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
158	#11719917	—	2	4	no	—
159	#11719917	—	4	2	no	—
160	C₂×#88322	—	2	2	no	—
161	C₂×#88322	—	2	2	no	—
162	C₂×#88322	C₂²×C₄	1	2	yes	—
163	C₂×#88322	C₂²×C₄	1	2	yes	—
164	#11719912	—	4	4	no	—
165	C₈×D₅₃	—	4	4	no	—
166	C₂³×C₅₃	—	2	1	no	—
167	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
168	C₂²×D₅₃	—	2	2	no	—
169	Q₈×C₅₃	—	2	1	no	—
170	Q₈×C₅₃	#131	1	2	yes	complemented
171	D₄×C₅₃	—	2	4	no	—
172	D₄×C₅₃	—	2	1	no	—
173	D₄×C₅₃	#131	1	2	yes	complemented
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	2	no	—
186	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
192	C₂²⋊D₅₃	—	4	4	no	—
193	C₂×Dic₅₃	—	2	2	no	—
194	D₂₁₂	—	2	2	no	—
195	D₂₁₂	—	2	2	no	—
196	D₂₁₂	—	4	2	no	—
197	D₂₁₂	—	4	2	no	—
198	C₄×D₅₃	—	4	2	no	—
199	C₄×D₅₃	—	4	2	no	—
200	Dic₁₀₆	—	4	2	no	—
201	C₈×C₅₃	—	2	1	no	—
202	C₈×C₅₃	—	2	1	no	—
203	C₈×C₅₃	—	2	1	no	—
204	C₈×C₅₃	—	4	1	no	—
205	C₈×C₅₃	—	4	1	no	—
206	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
207	#88322	—	2	4	no	—
208	#88322	—	2	2	no	—
209	#88322	—	2	2	no	—
210	#88322	—	4	2	no	—
211	#17912	—	53	1	no	maximal, Sylow
212	C₂²×C₅₃	—	2	1	no	—
213	C₂²×C₅₃	—	2	2	no	—
214	C₂²×C₅₃	—	2	1	no	—
215	C₂²×C₅₃	—	2	1	no	—
216	C₂²×C₅₃	—	4	1	no	—
217	C₂²×C₅₃	—	4	1	no	—
218	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂×D₅₃	—	4	2	no	—
220	C₂×D₅₃	—	4	2	no	—
221	C₂×D₅₃	—	4	2	no	—
222	C₄×C₅₃	—	2	1	no	—
223	C₄×C₅₃	—	2	2	no	—
224	C₄×C₅₃	—	2	1	no	—
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	4	1	no	—
227	C₄×C₅₃	—	4	1	no	—
228	C₄×C₅₃	#443	1	2	yes	—
229	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
230	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
231	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₄×C₅₃	#575	1	1	yes	characteristic
233	Dic₅₃	—	4	2	no	—
234	#3106	—	53	1	no	—
235	C₈×#41	D₅₃	1	1	yes	characteristic, complemented
236	#2931	—	53	1	no	—
237	#1588	—	53	2	no	—
238	#1586	—	53	2	no	—
239	C₈×SD₁₆	—	53	2	no	—
240	C₈×D₈	—	53	2	no	—
241	#1574	—	53	1	no	—
242	#1489	—	53	1	no	—
243	#1484	—	53	1	no	—
244	C₈×M₁₆	—	53	1	no	—
245	C₂×C₅₃	—	4	1	no	—
246	C₂×C₅₃	—	4	2	no	—
247	C₂×C₅₃	#2929	1	1	yes	characteristic
248	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
250	D₅₃	—	8	2	no	—
251	C₂×#131	—	106	1	no	—
252	#532	—	53	1	no	—
253	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
254	#482	—	53	2	no	—
255	#477	—	53	1	no	—
256	#476	—	53	1	no	—
257	#460	—	53	1	no	—
258	#459	—	53	1	no	—
259	#449	—	53	2	no	—
260	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
261	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
262	C₄×SD₁₆	—	53	2	no	—
263	C₄×D₈	—	53	2	no	—
264	#436	—	53	2	no	—
265	#436	—	106	1	no	—
266	C₈×D₄	—	2	1	no	—
267	C₈×D₄	—	2	1	no	—
268	C₈×D₄	—	53	2	no	—
269	C₈×D₄	—	106	1	no	—
270	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
271	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
272	#431	—	2	1	no	—
273	#431	—	53	1	no	—
274	#431	C₂×D₅₃	1	1	yes	characteristic
275	#423	—	53	1	no	—
276	#408	—	106	1	no	—
277	C₄×M₁₆	—	53	1	no	—
278	C₂×C₄×C₈	—	2	1	no	—
279	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
280	#335	—	53	2	no	—
281	#334	—	53	2	no	—
282	Q₈⋊C₈	—	53	2	no	—
283	D₄⋊C₈	—	53	2	no	—
284	D₄⋊C₈	—	53	2	no	—
285	D₄⋊C₈	—	53	2	no	—
286	#322	—	53	2	no	—
287	C₈²	—	53	2	no	—
288	C₅₃	#17912	1	1	yes	characteristic, complemented, Sylow
289	C₂×#41	—	106	1	no	—
290	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
291	C₂×SD₁₆	—	53	2	no	—
292	C₂×D₈	—	53	2	no	—
293	#131	—	212	4	no	—
294	C₂×M₁₆	—	106	1	no	—
295	C₂×M₁₆	—	106	1	no	—
296	C₂×M₁₆	—	106	2	no	—
297	C₂²×C₈	—	2	1	no	—
298	C₂²×C₈	—	2	1	no	—
299	C₂²×C₈	—	106	2	no	—
300	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
301	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
302	C₄⋊Q₈	—	53	1	no	—
303	#127	—	53	1	no	—
304	#124	—	106	1	no	—
305	#121	—	53	2	no	—
306	#121	—	106	1	no	—
307	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
308	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	53	2	no	—
312	C₄×D₄	—	106	1	no	—
313	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
314	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	C₄×D₅₃	1	1	yes	characteristic
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
319	#107	—	53	2	no	—
320	#106	—	53	2	no	—
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	—	53	1	no	—
324	C₄⋊C₈	—	53	1	no	—
325	C₄⋊C₈	—	53	2	no	—
326	C₄⋊C₈	—	106	1	no	—
327	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
328	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
329	#103	—	53	2	no	—
330	#102	—	53	2	no	—
331	#102	—	53	2	no	—
332	#102	—	53	2	no	—
333	C₂²⋊C₈	—	2	1	no	—
334	C₂²⋊C₈	—	2	1	no	—
335	C₂²⋊C₈	—	2	1	no	—
336	C₂²⋊C₈	—	53	2	no	—
337	C₂²⋊C₈	—	106	2	no	—
338	C₂²⋊C₈	—	106	2	no	—
339	#97	—	106	1	no	—
340	C₄×C₈	—	2	1	no	—
341	C₄×C₈	—	2	1	no	—
342	C₄×C₈	—	2	1	no	—
343	C₄×C₈	—	53	2	no	—
344	C₄×C₈	—	53	2	no	—
345	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
346	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
347	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
348	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
349	#41	—	2	2	no	—
350	#41	—	2	2	no	—
351	#41	—	212	2	no	—
352	#41	—	212	2	no	—
353	C₂×Q₈	—	106	1	no	—
354	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	53	2	no	—
357	C₂×D₄	—	106	2	no	—
358	C₂×D₄	—	106	1	no	—
359	C₂×D₄	—	106	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	2	no	—
365	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
366	C₂²×C₄	#11719945	1	1	yes	characteristic
367	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
368	SD₁₆	—	106	4	no	—
369	D₈	—	106	4	no	—
370	M₁₆	—	106	4	no	—
371	M₁₆	—	212	4	no	—
372	M₁₆	—	212	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	4	1	no	—
377	C₂×C₈	—	4	2	no	—
378	C₂×C₈	—	4	2	no	—
379	C₂×C₈	—	53	2	no	—
380	C₂×C₈	—	53	2	no	—
381	C₂×C₈	—	106	2	no	—
382	C₂×C₈	—	106	2	no	—
383	C₂×C₈	—	212	4	no	—
384	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
385	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
386	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
387	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	53	2	no	—
391	C₄⋊C₄	—	106	1	no	—
392	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
393	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
394	C₂²⋊C₄	—	2	1	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₂²⋊C₄	—	2	1	no	—
397	C₂²⋊C₄	—	106	2	no	—
398	C₂²⋊C₄	—	106	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
403	C₄²	#11719945	1	1	yes	characteristic
404	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
405	C₄²	C₂³×D₅₃	1	1	yes	characteristic
406	C₂³	—	2	1	no	—
407	C₂³	—	106	2	no	—
408	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
409	Q₈	—	2	1	no	—
410	Q₈	—	212	2	no	—
411	Q₈	#13133223	1	2	yes	complemented
412	D₄	—	2	4	no	—
413	D₄	—	2	1	no	—
414	D₄	—	106	2	no	—
415	D₄	—	106	2	no	—
416	D₄	—	212	2	no	—
417	D₄	—	212	4	no	—
418	D₄	—	212	2	no	—
419	D₄	#13133223	1	2	yes	complemented
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	106	2	no	—
433	C₂×C₄	—	212	2	no	—
434	C₂×C₄	—	212	2	no	—
435	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
436	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
437	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
438	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
439	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
440	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
441	C₈	—	2	1	no	—
442	C₈	—	2	1	no	—
443	C₈	—	2	1	no	—
444	C₈	—	4	1	no	—
445	C₈	—	4	1	no	—
446	C₈	—	106	4	no	—
447	C₈	—	106	2	no	—
448	C₈	—	106	2	no	—
449	C₈	—	212	2	no	—
450	C₈	#13133242	1	2	yes	—
451	C₂²	—	2	1	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	212	2	no	—
458	C₂²	—	212	2	no	—
459	C₂²	—	212	2	no	—
460	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
461	C₄	—	2	1	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	1	no	—
464	C₄	—	2	1	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	212	2	no	—
468	C₄	#16810738	1	2	yes	—
469	C₄	C₂×#13133223	1	1	yes	characteristic
470	C₄	C₂×#13133223	1	1	yes	characteristic
471	C₄	C₂×#13133242	1	1	yes	characteristic
472	C₄	#16811421	1	1	yes	characteristic
473	C₂	—	4	1	no	—
474	C₂	—	4	2	no	—
475	C₂	—	424	2	no	—
476	C₂	#20283409	1	1	yes	characteristic
477	C₂	C₄×#13133242	1	1	yes	characteristic
478	C₂	C₂×#16810738	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20284775	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#17912
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #20283428	C₂	semidirect
3. #20273593	C₂	semidirect
4. #20273579	C₂	semidirect
5. C₈×#11719924	C₂	semidirect
6. C₈×#11719922	C₂	semidirect
7. #20273503	C₂	semidirect
8. #20273154	C₂	semidirect
9. #20273131	C₂	semidirect
10. C₈×#11719917	C₂	semidirect
11. C₈×#41×C₅₃	C₂	semidirect
170. Q₈×C₅₃	#131	semidirect
173. D₄×C₅₃	#131	semidirect
235. C₈×#41	D₅₃	semidirect
288. C₅₃	#17912	semidirect
411. Q₈	#13133223	semidirect
419. D₄	#13133223	semidirect

Maximal subgroups

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

subgroup	index
#20284775	2
#20283428	2
#20273593	2
#20273579	2
C₈×#11719924	2
C₈×#11719922	2
#20273503	2
#20273154	2
#20273131	2
C₈×#11719917	2
C₈×#41×C₅₃	2
#17912	53