Subgroups — #27274342

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283347	C₂	1	1	yes	maximal, characteristic
2	#20283343	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272612	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272611	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272610	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	—	2	1	no	—
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	#16810679	—	2	1	no	—
11	#16810679	—	2	1	no	—
12	#16810679	C₄	1	2	yes	—
13	#16810677	C₄	1	2	yes	—
14	#16810676	C₂²	1	1	yes	characteristic
15	#16810675	—	2	1	no	—
16	#16810675	C₄	1	2	yes	—
17	#16810674	—	2	1	no	—
18	#16810647	C₂²	1	1	yes	characteristic
19	#16810645	—	2	1	no	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×#13133146	—	2	1	no	—
22	C₄⋊C₈×D₅₃	—	2	1	no	—
23	C₄⋊C₈×D₅₃	—	2	1	no	—
24	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
25	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
26	#16810586	—	2	1	no	—
27	#16810585	C₄	1	2	yes	—
28	M₁₆×Dic₅₃	—	2	1	no	—
29	C₄×#11719912	—	2	1	no	—
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
36	C₂×#11719917	—	2	1	no	—
37	C₂×#11719912	—	2	2	no	—
38	C₂²×#88322	—	2	1	no	—
39	C₂×C₈×D₅₃	—	2	2	no	—
40	C₂×C₈×D₅₃	—	2	2	no	—
41	C₂×C₈×D₅₃	—	4	2	no	—
42	C₂×C₈×D₅₃	—	4	2	no	—
43	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
46	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₅₃	D₄	1	1	yes	characteristic
51	C₄²×D₅₃	D₄	1	1	yes	characteristic
52	C₄²×D₅₃	D₄	1	1	yes	characteristic
53	C₄²×D₅₃	D₄	1	1	yes	characteristic
54	C₄⋊#88322	—	2	1	no	—
55	C₄⋊#88322	—	2	1	no	—
56	C₄⋊#88322	—	2	1	no	—
57	C₄⋊#88322	C₂×C₄	1	2	yes	—
58	C₄⋊#88322	C₂×C₄	1	2	yes	—
59	#13133146	—	2	2	no	—
60	#13133146	—	2	2	no	—
61	#13133146	—	2	2	no	—
62	C₂²⋊#88322	C₂×C₄	1	2	yes	—
63	#13133129	—	2	2	no	—
64	#13133129	—	2	2	no	—
65	#13133129	—	2	2	no	—
66	#13133127	—	2	2	no	—
67	#13133127	—	2	2	no	—
68	C₄×#88322	—	2	1	no	—
69	C₈×Dic₅₃	—	2	2	no	—
70	C₂×M₁₆×C₅₃	—	2	1	no	—
71	C₂²×C₈×C₅₃	—	2	1	no	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₅₃	—	2	1	no	—
74	C₄⋊C₈×C₅₃	—	2	1	no	—
75	C₄⋊C₈×C₅₃	—	2	1	no	—
76	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
78	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₅₃	—	2	1	no	—
80	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₅₃	—	2	1	no	—
88	C₂×C₄×D₅₃	—	2	1	no	—
89	C₂×C₄×D₅₃	—	2	1	no	—
90	C₂×C₄×D₅₃	—	2	1	no	—
91	C₂×C₄×D₅₃	—	2	4	no	—
92	C₂×C₄×D₅₃	—	4	2	no	—
93	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
105	M₁₆×C₅₃	—	4	1	no	—
106	M₁₆×C₅₃	—	4	1	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	4	1	no	—
111	C₂×C₈×C₅₃	—	4	1	no	—
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
117	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
118	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
128	#11719917	—	4	1	no	—
129	#11719917	—	4	1	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	2	2	no	—
133	C₂×#88322	—	4	1	no	—
134	C₂×#88322	—	4	1	no	—
135	#11719912	—	4	4	no	—
136	C₈×D₅₃	—	4	4	no	—
137	C₈×D₅₃	—	4	4	no	—
138	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×D₅₃	—	2	1	no	—
140	C₂²×D₅₃	—	2	1	no	—
141	C₂²×D₅₃	—	2	2	no	—
142	C₂²×D₅₃	—	2	2	no	—
143	C₂²×D₅₃	#95	1	1	yes	characteristic
144	C₂²×D₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	4	1	no	—
147	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
157	C₂×Dic₅₃	—	2	1	no	—
158	C₂×Dic₅₃	—	2	1	no	—
159	C₂×Dic₅₃	—	2	2	no	—
160	C₂×Dic₅₃	—	4	1	no	—
161	C₂×Dic₅₃	#95	1	1	yes	characteristic
162	C₂×Dic₅₃	#95	1	1	yes	characteristic
163	C₂×Dic₅₃	#95	1	1	yes	characteristic
164	C₂×Dic₅₃	#95	1	1	yes	characteristic
165	C₂×Dic₅₃	#95	1	1	yes	characteristic
166	C₂×Dic₅₃	#95	1	1	yes	characteristic
167	C₄×D₅₃	—	2	2	no	—
168	C₄×D₅₃	—	2	2	no	—
169	C₄×D₅₃	—	2	2	no	—
170	C₄×D₅₃	—	2	1	no	—
171	C₄×D₅₃	—	2	1	no	—
172	C₄×D₅₃	—	2	1	no	—
173	C₄×D₅₃	—	2	1	no	—
174	C₄×D₅₃	—	2	1	no	—
175	C₄×D₅₃	—	2	1	no	—
176	C₄×D₅₃	—	2	1	no	—
177	C₄×D₅₃	—	2	1	no	—
178	C₄×D₅₃	—	2	2	no	—
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	4	1	no	—
181	C₄×D₅₃	—	4	1	no	—
182	C₄×D₅₃	#108	1	4	yes	—
183	C₈×C₅₃	—	4	2	no	—
184	C₈×C₅₃	—	4	2	no	—
185	C₈×C₅₃	—	4	2	no	—
186	#88322	—	4	2	no	—
187	#88322	—	4	2	no	—
188	#88322	—	4	2	no	—
189	#8082	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
193	C₂²×C₅₃	#428	1	1	yes	characteristic
194	C₂²×C₅₃	#428	1	1	yes	characteristic
195	C₂×D₅₃	—	2	4	no	—
196	C₂×D₅₃	—	2	4	no	—
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	#341	1	2	yes	—
202	C₂×D₅₃	#344	1	2	yes	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	#417	1	1	yes	characteristic
207	C₄×C₅₃	#417	1	1	yes	characteristic
208	C₄×C₅₃	#418	1	1	yes	characteristic
209	C₄×C₅₃	#419	1	1	yes	characteristic
210	C₄×C₅₃	C₂×#108	1	2	yes	—
211	Dic₅₃	—	2	1	no	—
212	Dic₅₃	—	2	1	no	—
213	Dic₅₃	—	2	1	no	—
214	Dic₅₃	—	2	1	no	—
215	Dic₅₃	—	2	1	no	—
216	Dic₅₃	#341	1	2	yes	—
217	#2920	—	53	1	no	—
218	#2920	—	53	1	no	—
219	C₂²×C₄⋊C₈	—	53	1	no	—
220	#1301	—	53	1	no	—
221	#1301	—	53	1	no	—
222	#1301	—	53	1	no	—
223	#1301	D₅₃	1	1	yes	characteristic, complemented
224	C₂×C₅₃	—	2	1	no	—
225	C₂×C₅₃	—	2	1	no	—
226	C₂×C₅₃	#1736	1	1	yes	characteristic
227	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
228	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
229	D₅₃	—	2	2	no	—
230	D₅₃	—	2	2	no	—
231	C₂³×C₈	—	106	1	no	—
232	C₂²×C₄²	—	53	1	no	—
233	#432	—	53	2	no	—
234	#432	—	53	2	no	—
235	#432	—	106	1	no	—
236	#432	—	106	1	no	—
237	#431	—	53	1	no	—
238	#431	—	53	2	no	—
239	#431	—	106	1	no	—
240	#431	—	106	1	no	—
241	#431	C₂×D₅₃	1	1	yes	characteristic
242	#423	—	53	1	no	—
243	#423	—	106	1	no	—
244	#423	C₂×D₅₃	1	1	yes	characteristic
245	C₂×C₄⋊C₈	—	53	1	no	—
246	C₂×C₄⋊C₈	—	53	2	no	—
247	C₂×C₄⋊C₈	—	53	2	no	—
248	C₂×C₄⋊C₈	—	106	1	no	—
249	C₂×C₄⋊C₈	—	106	1	no	—
250	C₂×C₄⋊C₈	—	106	1	no	—
251	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
252	#407	—	53	2	no	—
253	#407	—	106	1	no	—
254	C₄×M₁₆	—	106	1	no	—
255	C₄×M₁₆	—	106	1	no	—
256	C₅₃	#8082	1	1	yes	characteristic, complemented, Sylow
257	C₂³×C₄	—	53	1	no	—
258	C₂³×C₄	—	53	1	no	—
259	C₂³×C₄	—	53	1	no	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	106	2	no	—
262	C₂×M₁₆	—	106	1	no	—
263	C₂²×C₈	—	2	1	no	—
264	C₂²×C₈	—	106	2	no	—
265	C₂²×C₈	—	106	2	no	—
266	C₂²×C₈	—	106	1	no	—
267	C₂²×C₈	—	212	2	no	—
268	C₂²×C₈	—	212	2	no	—
269	C₂×C₄²	—	53	1	no	—
270	C₂×C₄²	—	53	1	no	—
271	C₂×C₄²	—	53	1	no	—
272	C₂×C₄²	—	53	1	no	—
273	C₂×C₄²	—	53	1	no	—
274	C₂×C₄²	—	53	1	no	—
275	C₂×C₄²	—	53	1	no	—
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	C₂²×D₅₃	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	2	1	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	53	2	no	—
285	C₄⋊C₈	—	53	2	no	—
286	C₄⋊C₈	—	106	2	no	—
287	C₄⋊C₈	—	106	2	no	—
288	C₄⋊C₈	—	106	2	no	—
289	C₄⋊C₈	—	106	1	no	—
290	C₄⋊C₈	—	106	1	no	—
291	C₄⋊C₈	—	106	1	no	—
292	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
293	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
294	C₂²⋊C₈	—	53	2	no	—
295	C₂²⋊C₈	—	106	2	no	—
296	C₂²⋊C₈	—	106	2	no	—
297	C₂²⋊C₈	—	106	2	no	—
298	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
299	#97	—	106	2	no	—
300	#97	—	106	2	no	—
301	C₄×C₈	—	2	1	no	—
302	C₄×C₈	—	106	2	no	—
303	C₄×C₈	—	106	1	no	—
304	C₂⁴	—	53	1	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	1	no	—
308	C₂²×C₄	—	53	1	no	—
309	C₂²×C₄	—	53	1	no	—
310	C₂²×C₄	—	53	1	no	—
311	C₂²×C₄	—	53	1	no	—
312	C₂²×C₄	—	53	1	no	—
313	C₂²×C₄	—	53	1	no	—
314	C₂²×C₄	—	53	1	no	—
315	C₂²×C₄	—	53	1	no	—
316	C₂²×C₄	—	53	1	no	—
317	C₂²×C₄	—	53	1	no	—
318	C₂²×C₄	—	53	1	no	—
319	C₂²×C₄	—	53	1	no	—
320	C₂²×C₄	—	106	4	no	—
321	C₂²×C₄	—	106	1	no	—
322	C₂²×C₄	—	106	1	no	—
323	C₂²×C₄	—	106	1	no	—
324	C₂²×C₄	—	106	1	no	—
325	C₂²×C₄	—	212	2	no	—
326	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
327	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
328	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
329	M₁₆	—	4	1	no	—
330	M₁₆	—	4	1	no	—
331	M₁₆	—	212	4	no	—
332	M₁₆	—	212	1	no	—
333	M₁₆	—	212	1	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	4	1	no	—
338	C₂×C₈	—	4	1	no	—
339	C₂×C₈	—	106	2	no	—
340	C₂×C₈	—	106	2	no	—
341	C₂×C₈	—	106	2	no	—
342	C₂×C₈	—	212	4	no	—
343	C₂×C₈	—	212	4	no	—
344	C₂×C₈	—	212	1	no	—
345	C₂×C₈	—	212	1	no	—
346	C₄²	—	53	1	no	—
347	C₄²	—	53	1	no	—
348	C₄²	—	53	1	no	—
349	C₄²	—	53	1	no	—
350	C₄²	—	53	1	no	—
351	C₄²	—	53	1	no	—
352	C₄²	—	53	1	no	—
353	C₄²	—	53	1	no	—
354	C₄²	—	53	1	no	—
355	C₄²	—	53	1	no	—
356	C₄²	—	53	1	no	—
357	C₄²	—	53	1	no	—
358	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
359	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
360	C₄²	D₄×D₅₃	1	1	yes	characteristic
361	C₄²	D₄×D₅₃	1	1	yes	characteristic
362	C₂³	—	53	1	no	—
363	C₂³	—	53	1	no	—
364	C₂³	—	106	2	no	—
365	C₂³	—	106	2	no	—
366	C₂³	—	106	1	no	—
367	C₂³	—	106	1	no	—
368	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	53	1	no	—
372	C₂×C₄	—	53	4	no	—
373	C₂×C₄	—	53	1	no	—
374	C₂×C₄	—	53	1	no	—
375	C₂×C₄	—	53	1	no	—
376	C₂×C₄	—	53	1	no	—
377	C₂×C₄	—	53	1	no	—
378	C₂×C₄	—	106	2	no	—
379	C₂×C₄	—	106	2	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	2	no	—
382	C₂×C₄	—	106	2	no	—
383	C₂×C₄	—	106	1	no	—
384	C₂×C₄	—	106	1	no	—
385	C₂×C₄	—	106	1	no	—
386	C₂×C₄	—	106	1	no	—
387	C₂×C₄	—	106	1	no	—
388	C₂×C₄	—	106	1	no	—
389	C₂×C₄	—	106	1	no	—
390	C₂×C₄	—	106	1	no	—
391	C₂×C₄	—	106	1	no	—
392	C₂×C₄	—	106	1	no	—
393	C₂×C₄	—	106	2	no	—
394	C₂×C₄	—	212	1	no	—
395	C₂×C₄	—	212	1	no	—
396	C₂×C₄	—	212	1	no	—
397	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
398	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂²⋊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	#13133241	1	1	yes	characteristic
404	C₂×C₄	#13133244	1	1	yes	characteristic
405	C₂×C₄	#13133246	1	1	yes	characteristic
406	C₂×C₄	#13133249	1	1	yes	characteristic
407	C₈	—	4	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	2	no	—
410	C₈	—	212	2	no	—
411	C₈	—	212	2	no	—
412	C₈	—	212	2	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	53	2	no	—
416	C₂²	—	53	2	no	—
417	C₂²	—	106	4	no	—
418	C₂²	—	106	4	no	—
419	C₂²	—	106	1	no	—
420	C₂²	—	106	1	no	—
421	C₂²	—	106	1	no	—
422	C₂²	—	106	1	no	—
423	C₂²	#95×D₅₃	1	1	yes	characteristic
424	C₂²	#16810662	1	1	yes	characteristic
425	C₂²	#16810664	1	1	yes	characteristic
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	#16810621	1	1	yes	characteristic
436	C₄	#16810623	1	1	yes	characteristic
437	C₄	#16810626	1	1	yes	characteristic
438	C₄	#16810630	1	1	yes	characteristic
439	C₄	#108×D₅₃	1	2	yes	—
440	C₂	—	2	1	no	—
441	C₂	—	2	1	no	—
442	C₂	—	106	2	no	—
443	C₂	—	106	2	no	—
444	C₂	#20274433	1	1	yes	characteristic
445	C₂	#341×D₅₃	1	1	yes	characteristic
446	C₂	#344×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1301	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1301	D₅₃
hypocenter	C₅₃	#8082
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
4. #20272612	C₂	semidirect
5. #20272611	C₂	semidirect
6. #20272610	C₂	semidirect
7. C₅₃×#1301	C₂	semidirect
223. #1301	D₅₃	semidirect
256. C₅₃	#8082	semidirect

Maximal subgroups

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

subgroup	index
#20283347	2
#20283343	2
C₂×C₄⋊C₈×D₅₃	2
#20272612	2
#20272611	2
#20272610	2
C₅₃×#1301	2
#8082	53