Subgroups — #27353248

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283345	C₂	1	1	yes	maximal, characteristic
2	#432×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
4	#20273494	C₂	1	2	yes	maximal, complemented
5	#20273492	C₂	1	2	yes	maximal, complemented
6	#20273488	C₂	1	2	yes	maximal, complemented
7	#20273484	C₂	1	2	yes	maximal, complemented
8	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273476	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	#16810680	C₂²	1	1	yes	characteristic
14	#16810680	C₂²	1	1	yes	characteristic
15	#16810679	C₂²	1	2	yes	—
16	#16810678	—	2	1	no	—
17	#16810678	—	2	1	no	—
18	#16810677	—	2	1	no	—
19	#16810675	C₂²	1	2	yes	—
20	#16810674	C₄	1	2	yes	—
21	#16810674	C₄	1	2	yes	—
22	#16810674	C₂²	1	1	yes	characteristic
23	#16810674	C₂²	1	1	yes	characteristic
24	#16810674	C₂²	1	1	yes	characteristic
25	#16810674	C₂²	1	1	yes	characteristic
26	#16810673	—	2	1	no	—
27	#16810645	C₂²	1	2	yes	—
28	C₂×#13133146	C₄	1	2	yes	—
29	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
30	#16810585	—	2	1	no	—
31	C₂×#13133129	C₄	1	2	yes	—
32	C₂²⋊C₈×D₅₃	—	2	1	no	—
33	#16810575	—	2	1	no	—
34	#97×D₅₃	—	2	1	no	—
35	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
36	#16810376	C₂²	1	2	yes	complemented
37	#16810375	C₂²	1	2	yes	complemented
38	#16810375	C₂²	1	2	yes	complemented
39	#16810374	C₂²	1	2	yes	complemented
40	#16810373	C₂²	1	2	yes	complemented
41	#16810373	C₂²	1	2	yes	complemented
42	C₅₃×#432	C₂²	1	1	yes	characteristic
43	C₅₃×#432	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	2	yes	complemented
46	C₅₃×#334	C₂²	1	2	yes	complemented
47	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
50	C₂×#11719912	—	2	2	no	—
51	C₂²×#88322	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₅₃	—	2	2	no	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
56	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₅₃	D₄	1	1	yes	characteristic
59	C₄²×D₅₃	D₄	1	1	yes	characteristic
60	C₄²×D₅₃	Q₈	1	1	yes	characteristic
61	C₄²×D₅₃	Q₈	1	1	yes	characteristic
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	C₄²×D₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊#88322	C₂³	1	2	yes	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	#13133146	—	2	2	no	—
67	#13133146	—	2	2	no	—
68	#13133146	C₂×C₄	1	2	yes	—
69	#13133146	C₂×C₄	1	2	yes	—
70	#13133146	C₂×C₄	1	2	yes	—
71	#13133146	C₂×C₄	1	2	yes	—
72	C₂²⋊#88322	—	2	1	no	—
73	C₂²⋊#88322	—	2	1	no	—
74	#13133129	—	2	2	no	—
75	#13133129	—	2	2	no	—
76	#13133129	C₂×C₄	1	2	yes	—
77	#13133129	C₂×C₄	1	2	yes	—
78	#13133129	C₂×C₄	1	2	yes	—
79	#13133129	C₂×C₄	1	2	yes	—
80	#13133128	—	2	1	no	—
81	#13133128	—	2	1	no	—
82	#13133127	—	2	2	no	—
83	C₄×#88322	C₂×C₄	1	2	yes	—
84	C₄×#88322	C₂³	1	1	yes	characteristic
85	C₄×#88322	C₂³	1	1	yes	characteristic
86	C₈×Dic₅₃	—	2	2	no	—
87	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
90	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
91	C₂²⋊C₈×C₅₃	—	2	1	no	—
92	C₂²⋊C₈×C₅₃	—	2	1	no	—
93	#97×C₅₃	—	2	1	no	—
94	#97×C₅₃	—	2	1	no	—
95	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
96	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
102	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	—	2	4	no	—
106	C₂×C₄×D₅₃	—	4	2	no	—
107	C₂×C₄×D₅₃	—	4	2	no	—
108	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	4	2	no	—
123	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
125	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
128	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
133	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
135	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
140	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
141	C₂×#88322	—	2	2	no	—
142	C₂×#88322	—	2	2	no	—
143	C₂×#88322	—	4	2	no	—
144	C₂×#88322	C₂²×C₄	1	2	yes	—
145	C₂×#88322	C₂²×C₄	1	2	yes	—
146	#11719912	—	4	4	no	—
147	C₈×D₅₃	—	4	4	no	—
148	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×D₅₃	—	2	2	no	—
150	C₂²×D₅₃	—	2	2	no	—
151	C₂²×D₅₃	—	4	1	no	—
152	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
163	C₂×Dic₅₃	—	2	2	no	—
164	C₂×Dic₅₃	—	2	2	no	—
165	C₂×Dic₅₃	—	4	1	no	—
166	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₄×D₅₃	—	2	2	no	—
173	C₄×D₅₃	—	2	2	no	—
174	C₄×D₅₃	—	2	2	no	—
175	C₄×D₅₃	—	2	2	no	—
176	C₄×D₅₃	—	2	2	no	—
177	C₄×D₅₃	—	4	1	no	—
178	C₄×D₅₃	—	4	1	no	—
179	C₄×D₅₃	—	4	1	no	—
180	C₄×D₅₃	—	4	1	no	—
181	C₄×D₅₃	—	4	1	no	—
182	C₄×D₅₃	—	4	1	no	—
183	C₄×D₅₃	#135	1	4	yes	—
184	C₈×C₅₃	—	2	2	no	—
185	C₈×C₅₃	—	2	2	no	—
186	C₈×C₅₃	—	4	2	no	—
187	C₈×C₅₃	—	4	2	no	—
188	#88322	—	2	2	no	—
189	#88322	—	2	2	no	—
190	#88322	—	4	2	no	—
191	#88322	—	4	2	no	—
192	#17816	—	53	1	no	maximal, Sylow
193	C₂²×C₅₃	—	2	1	no	—
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	#430	1	2	yes	—
196	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂×D₅₃	—	2	1	no	—
198	C₂×D₅₃	—	2	1	no	—
199	C₂×D₅₃	—	2	1	no	—
200	C₂×D₅₃	—	2	1	no	—
201	C₂×D₅₃	—	2	1	no	—
202	C₂×D₅₃	—	4	4	no	—
203	C₂×D₅₃	#427	1	2	yes	—
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	#568	1	2	yes	—
209	C₄×C₅₃	C₂×#135	1	2	yes	—
210	Dic₅₃	—	2	1	no	—
211	Dic₅₃	—	2	1	no	—
212	Dic₅₃	—	2	1	no	—
213	Dic₅₃	—	2	1	no	—
214	Dic₅₃	—	2	1	no	—
215	Dic₅₃	#427	1	2	yes	—
216	#2920	—	53	1	no	—
217	C₂×#432	—	53	1	no	—
218	C₂×#431	—	53	1	no	—
219	#1572	—	53	2	no	—
220	#1572	—	53	2	no	—
221	#1571	—	53	2	no	—
222	#1570	—	53	2	no	—
223	#1568	—	53	1	no	—
224	#1568	—	53	1	no	—
225	#1568	—	53	1	no	—
226	#1568	D₅₃	1	1	yes	characteristic, complemented
227	C₂×C₅₃	—	2	2	no	—
228	C₂×C₅₃	#2905	1	1	yes	characteristic
229	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
230	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
231	D₅₃	—	2	2	no	—
232	D₅₃	—	4	1	no	—
233	C₂²×C₄²	—	53	1	no	—
234	#432	—	53	1	no	—
235	#432	—	53	1	no	—
236	#432	—	53	2	no	—
237	#432	—	106	1	no	—
238	#432	—	106	1	no	—
239	#432	—	106	1	no	—
240	#432	C₂×D₅₃	1	1	yes	characteristic
241	#432	C₂×D₅₃	1	1	yes	characteristic
242	#431	—	53	2	no	—
243	#431	—	53	2	no	—
244	#431	—	53	1	no	—
245	#431	—	53	1	no	—
246	#431	—	53	1	no	—
247	#431	—	53	1	no	—
248	#431	—	53	2	no	—
249	#431	—	106	1	no	—
250	#423	—	53	2	no	—
251	C₂×C₄⋊C₈	—	53	2	no	—
252	C₂×C₄⋊C₈	—	53	2	no	—
253	#407	—	106	1	no	—
254	C₂×C₂²⋊C₈	—	53	2	no	—
255	C₂×C₂²⋊C₈	—	106	1	no	—
256	C₄×M₁₆	—	106	1	no	—
257	C₂×#97	—	106	1	no	—
258	C₂×C₄×C₈	—	53	1	no	—
259	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
260	#335	—	53	2	no	—
261	#335	—	53	2	no	—
262	#335	—	53	2	no	—
263	#335	C₂×D₅₃	1	2	yes	complemented
264	#334	—	53	2	no	—
265	#334	—	53	2	no	—
266	#334	—	53	2	no	—
267	#334	C₂×D₅₃	1	2	yes	complemented
268	C₅₃	#17816	1	1	yes	characteristic, complemented, Sylow
269	C₂³×C₄	—	53	1	no	—
270	C₂³×C₄	—	53	1	no	—
271	C₂³×C₄	—	53	1	no	—
272	C₂×M₁₆	—	106	2	no	—
273	C₂²×C₈	—	53	2	no	—
274	C₂²×C₈	—	106	2	no	—
275	C₂²×C₈	C₄×D₅₃	1	2	yes	—
276	C₂×C₄²	—	53	1	no	—
277	C₂×C₄²	—	53	1	no	—
278	C₂×C₄²	—	53	1	no	—
279	C₂×C₄²	—	53	1	no	—
280	C₂×C₄²	—	53	1	no	—
281	C₂×C₄²	—	53	1	no	—
282	C₂×C₄²	—	53	1	no	—
283	C₂×C₄²	—	53	1	no	—
284	C₂×C₄²	—	53	1	no	—
285	C₂×C₄²	—	53	1	no	—
286	C₂×C₄²	—	53	1	no	—
287	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
288	C₄⋊C₈	—	53	2	no	—
289	C₄⋊C₈	—	53	2	no	—
290	C₄⋊C₈	—	53	2	no	—
291	C₄⋊C₈	—	53	2	no	—
292	C₄⋊C₈	—	53	2	no	—
293	C₄⋊C₈	—	53	2	no	—
294	C₄⋊C₈	—	106	2	no	—
295	C₄⋊C₈	—	106	2	no	—
296	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
297	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	53	2	no	—
301	C₂²⋊C₈	—	53	2	no	—
302	C₂²⋊C₈	—	53	2	no	—
303	C₂²⋊C₈	—	53	2	no	—
304	C₂²⋊C₈	—	106	2	no	—
305	C₂²⋊C₈	—	106	2	no	—
306	C₂²⋊C₈	—	106	1	no	—
307	C₂²⋊C₈	—	106	1	no	—
308	#97	—	2	1	no	—
309	#97	—	2	1	no	—
310	#97	—	106	2	no	—
311	#97	—	106	1	no	—
312	#97	—	106	1	no	—
313	C₄×C₈	—	53	2	no	—
314	C₄×C₈	—	53	1	no	—
315	C₄×C₈	—	53	1	no	—
316	C₄×C₈	—	106	2	no	—
317	C₄×C₈	C₄×D₅₃	1	2	yes	—
318	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
319	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
320	C₂⁴	—	53	1	no	—
321	C₂²×C₄	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	53	1	no	—
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	53	1	no	—
328	C₂²×C₄	—	53	1	no	—
329	C₂²×C₄	—	53	1	no	—
330	C₂²×C₄	—	53	1	no	—
331	C₂²×C₄	—	53	1	no	—
332	C₂²×C₄	—	53	1	no	—
333	C₂²×C₄	—	53	1	no	—
334	C₂²×C₄	—	53	1	no	—
335	C₂²×C₄	—	53	1	no	—
336	C₂²×C₄	—	106	4	no	—
337	C₂²×C₄	—	212	2	no	—
338	C₂²×C₄	—	212	2	no	—
339	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
340	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
341	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
342	M₁₆	—	212	4	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	4	2	no	—
346	C₂×C₈	—	53	2	no	—
347	C₂×C₈	—	53	2	no	—
348	C₂×C₈	—	106	2	no	—
349	C₂×C₈	—	106	2	no	—
350	C₂×C₈	—	212	4	no	—
351	C₂×C₈	—	212	2	no	—
352	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
353	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
354	C₄²	—	53	1	no	—
355	C₄²	—	53	1	no	—
356	C₄²	—	53	1	no	—
357	C₄²	—	53	1	no	—
358	C₄²	—	53	1	no	—
359	C₄²	—	53	1	no	—
360	C₄²	—	53	1	no	—
361	C₄²	—	53	1	no	—
362	C₄²	—	53	1	no	—
363	C₄²	—	53	1	no	—
364	C₄²	—	53	1	no	—
365	C₄²	—	53	1	no	—
366	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
367	C₄²	D₄×D₅₃	1	1	yes	characteristic
368	C₄²	Q₈×D₅₃	1	1	yes	characteristic
369	C₄²	C₂³×D₅₃	1	1	yes	characteristic
370	C₂³	—	53	1	no	—
371	C₂³	—	53	1	no	—
372	C₂³	—	106	2	no	—
373	C₂³	—	106	2	no	—
374	C₂³	—	212	1	no	—
375	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	53	1	no	—
380	C₂×C₄	—	53	4	no	—
381	C₂×C₄	—	53	1	no	—
382	C₂×C₄	—	53	1	no	—
383	C₂×C₄	—	53	1	no	—
384	C₂×C₄	—	53	1	no	—
385	C₂×C₄	—	53	1	no	—
386	C₂×C₄	—	106	2	no	—
387	C₂×C₄	—	106	2	no	—
388	C₂×C₄	—	106	2	no	—
389	C₂×C₄	—	106	2	no	—
390	C₂×C₄	—	106	2	no	—
391	C₂×C₄	—	106	2	no	—
392	C₂×C₄	—	106	2	no	—
393	C₂×C₄	—	212	1	no	—
394	C₂×C₄	—	212	1	no	—
395	C₂×C₄	—	212	1	no	—
396	C₂×C₄	—	212	1	no	—
397	C₂×C₄	—	212	1	no	—
398	C₂×C₄	—	212	1	no	—
399	C₂×C₄	—	212	1	no	—
400	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
401	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
402	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
403	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
405	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	2	no	—
410	C₈	—	106	2	no	—
411	C₈	—	106	2	no	—
412	C₈	—	212	2	no	—
413	C₈	—	212	2	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	1	no	—
416	C₂²	—	53	2	no	—
417	C₂²	—	106	1	no	—
418	C₂²	—	106	1	no	—
419	C₂²	—	106	1	no	—
420	C₂²	—	106	1	no	—
421	C₂²	—	106	1	no	—
422	C₂²	—	212	4	no	—
423	C₂²	#16810670	1	2	yes	—
424	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	53	2	no	—
430	C₄	—	106	1	no	—
431	C₄	—	106	1	no	—
432	C₄	—	106	1	no	—
433	C₄	—	106	1	no	—
434	C₄	—	106	1	no	—
435	C₄	#16811389	1	2	yes	—
436	C₄	#135×D₅₃	1	2	yes	—
437	C₂	—	2	2	no	—
438	C₂	—	106	2	no	—
439	C₂	—	212	1	no	—
440	C₂	#20283277	1	1	yes	characteristic
441	C₂	#427×D₅₃	1	1	yes	characteristic
442	C₂	C₂×#16810670	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1568	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#17816
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
4. #20273494	C₂	semidirect
5. #20273492	C₂	semidirect
6. #20273488	C₂	semidirect
7. #20273484	C₂	semidirect
8. #20273478	C₂	semidirect
9. #20273476	C₂	semidirect
10. #20273476	C₂	semidirect
11. C₅₃×#1568	C₂	semidirect
36. #16810376	C₂²	semidirect
37. #16810375	C₂²	semidirect
38. #16810375	C₂²	semidirect
39. #16810374	C₂²	semidirect
40. #16810373	C₂²	semidirect
41. #16810373	C₂²	semidirect
45. C₅₃×#335	C₂²	semidirect
46. C₅₃×#334	C₂²	semidirect
226. #1568	D₅₃	semidirect
263. #335	C₂×D₅₃	semidirect
267. #334	C₂×D₅₃	semidirect
268. C₅₃	#17816	semidirect

Maximal subgroups

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

subgroup	index
#20283345	2
#432×D₅₃	2
C₂×#16810674	2
#20273494	2
#20273492	2
#20273488	2
#20273484	2
#20273478	2
#20273476	2
C₅₃×#1568	2
#17816	53