Subgroups — #27349719

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283346	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	#20273006	C₂	1	2	yes	maximal, complemented
5	#20273005	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273003	C₂	1	2	yes	maximal, complemented
7	#20273002	C₂	1	1	yes	maximal, characteristic, complemented
8	#20272997	C₂	1	2	yes	maximal, complemented
9	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11719912	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₅₃	C₄	1	2	yes	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	#16810679	C₄	1	2	yes	—
15	#16810678	C₄	1	2	yes	—
16	#16810678	C₄	1	2	yes	—
17	#16810677	C₄	1	2	yes	—
18	#16810676	C₂²	1	1	yes	characteristic
19	#16810674	C₂²	1	2	yes	—
20	#16810674	C₂²	1	2	yes	—
21	#16810674	C₂²	1	2	yes	—
22	#16810646	C₄	1	2	yes	—
23	#16810645	C₄	1	2	yes	—
24	#16810586	C₄	1	2	yes	—
25	#16810585	C₄	1	2	yes	—
26	C₄×#11719917	C₂²	1	1	yes	characteristic
27	C₄×#11719912	C₂²	1	2	yes	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
30	C₄×C₈×D₅₃	C₄	1	2	yes	—
31	C₄×C₈×D₅₃	C₄	1	2	yes	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	#16810345	C₂²	1	2	yes	complemented
37	#16810345	C₂²	1	2	yes	complemented
38	#16810345	C₂²	1	2	yes	complemented
39	#16810345	C₂²	1	2	yes	complemented
40	#16810344	C₂²	1	2	yes	complemented
41	C₈×#88322	C₂²	1	2	yes	complemented
42	C₅₃×#431	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₅₃	C₂²	1	2	yes	complemented
47	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719917	C₂×C₄	1	2	yes	—
51	C₂×#11719912	C₂×C₄	1	4	yes	—
52	C₂²×#88322	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₅₃	—	2	8	no	—
54	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₅₃	C₂³	1	1	yes	characteristic
68	C₄²×D₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊#88322	C₂×C₄	1	2	yes	—
70	#13133146	C₂×C₄	1	4	yes	—
71	#13133146	C₂×C₄	1	4	yes	—
72	#13133146	C₂×C₄	1	4	yes	—
73	C₂²⋊#88322	C₂×C₄	1	2	yes	—
74	#13133129	C₂×C₄	1	4	yes	—
75	#13133129	C₂×C₄	1	4	yes	—
76	#13133129	C₂×C₄	1	4	yes	—
77	#13133128	C₂×C₄	1	2	yes	—
78	#13133127	C₂×C₄	1	4	yes	—
79	C₄×#88322	C₂×C₄	1	2	yes	—
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₄×#88322	C₂³	1	1	yes	characteristic
82	C₄×#88322	C₂³	1	2	yes	—
83	C₄×#88322	C₂³	1	2	yes	—
84	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
85	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
86	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
87	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
88	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
89	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
92	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
93	#97×C₅₃	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
95	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₄×C₈×C₅₃	C₂³	1	2	yes	—
97	C₄×C₈×C₅₃	C₂³	1	2	yes	—
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₂³×D₅₃	C₄²	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
104	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
105	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
106	C₂×C₄×D₅₃	—	2	4	no	—
107	C₂×C₄×D₅₃	—	2	4	no	—
108	C₂×C₄×D₅₃	—	2	4	no	—
109	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
111	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
121	M₁₆×C₅₃	—	2	4	no	—
122	C₂×C₈×C₅₃	—	2	4	no	—
123	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
132	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
133	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
134	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
135	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
136	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
137	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
143	#11719917	—	2	4	no	—
144	C₂×#88322	—	2	4	no	—
145	C₂×#88322	C₂²×C₄	1	2	yes	—
146	C₂×#88322	C₂²×C₄	1	4	yes	—
147	C₂×#88322	C₂²×C₄	1	4	yes	—
148	C₂×#88322	C₂²×C₄	1	2	yes	—
149	#11719912	—	2	8	no	—
150	C₈×D₅₃	—	2	4	no	—
151	C₈×D₅₃	—	2	4	no	—
152	C₈×D₅₃	—	2	4	no	—
153	C₈×D₅₃	M₁₆	1	8	yes	complemented
154	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂²×D₅₃	—	2	2	no	—
156	C₂²×D₅₃	—	2	2	no	—
157	C₂²×D₅₃	—	2	2	no	—
158	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
159	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
165	C₂×C₄×C₅₃	#131	1	4	yes	—
166	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
169	C₂×Dic₅₃	—	2	2	no	—
170	C₂×Dic₅₃	—	2	2	no	—
171	C₂×Dic₅₃	—	2	2	no	—
172	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
173	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
174	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
175	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
176	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
177	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
178	C₄×D₅₃	—	2	2	no	—
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	2	2	no	—
181	C₄×D₅₃	—	2	2	no	—
182	C₄×D₅₃	—	2	2	no	—
183	C₄×D₅₃	—	2	2	no	—
184	C₄×D₅₃	—	2	2	no	—
185	C₄×D₅₃	—	2	2	no	—
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	2	2	no	—
188	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
189	C₄×D₅₃	#131	1	4	yes	—
190	C₈×C₅₃	—	2	2	no	—
191	C₈×C₅₃	—	2	4	no	—
192	C₈×C₅₃	—	2	4	no	—
193	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
194	#88322	—	2	2	no	—
195	#88322	—	2	4	no	—
196	#88322	—	2	4	no	—
197	#88322	C₂×M₁₆	1	4	yes	—
198	#17323	—	53	1	no	maximal, Sylow
199	C₂²×C₅₃	—	2	1	no	—
200	C₂²×C₅₃	—	2	1	no	—
201	C₂²×C₅₃	#405	1	2	yes	—
202	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
203	C₂×D₅₃	—	2	8	no	—
204	C₂×D₅₃	—	2	1	no	—
205	C₂×D₅₃	—	2	1	no	—
206	C₂×D₅₃	—	2	1	no	—
207	C₂×D₅₃	—	2	1	no	—
208	C₂×D₅₃	C₄×M₁₆	1	2	yes	—
209	C₂×D₅₃	#405	1	2	yes	—
210	C₄×C₅₃	—	2	1	no	—
211	C₄×C₅₃	—	2	1	no	—
212	C₄×C₅₃	#405	1	2	yes	—
213	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
214	C₄×C₅₃	C₂×#131	1	2	yes	—
215	C₄×C₅₃	C₂×#131	1	2	yes	—
216	Dic₅₃	—	2	1	no	—
217	Dic₅₃	—	2	1	no	—
218	Dic₅₃	—	2	1	no	—
219	Dic₅₃	—	2	1	no	—
220	Dic₅₃	C₄×M₁₆	1	2	yes	—
221	Dic₅₃	#405	1	2	yes	—
222	#2920	—	53	1	no	—
223	#2920	—	53	1	no	—
224	C₂²×C₄×C₈	—	53	1	no	—
225	#1452	—	53	2	no	—
226	#1451	—	53	1	no	—
227	#1451	—	53	1	no	—
228	#1451	—	53	2	no	—
229	#1450	—	53	2	no	—
230	C₈×M₁₆	—	53	1	no	—
231	C₈×M₁₆	—	53	2	no	—
232	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
233	C₂×C₅₃	—	2	2	no	—
234	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
235	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
236	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
237	D₅₃	—	2	2	no	—
238	D₅₃	—	2	2	no	—
239	C₂³×C₈	—	53	2	no	—
240	C₂²×C₄²	—	53	1	no	—
241	#432	—	53	2	no	—
242	#432	—	53	2	no	—
243	#432	—	53	2	no	—
244	#432	—	53	2	no	—
245	#431	—	53	1	no	—
246	#431	—	53	2	no	—
247	#431	—	53	2	no	—
248	#431	—	53	2	no	—
249	#431	C₂×D₅₃	1	1	yes	characteristic
250	#423	—	53	2	no	—
251	#423	—	53	2	no	—
252	#407	—	53	2	no	—
253	#407	—	53	2	no	—
254	C₄×M₁₆	—	53	1	no	—
255	C₄×M₁₆	—	53	2	no	—
256	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
257	C₂×C₄×C₈	—	53	2	no	—
258	C₂×C₄×C₈	—	53	1	no	—
259	C₂×C₄×C₈	—	53	2	no	—
260	C₂×C₄×C₈	—	53	2	no	—
261	C₂×C₄×C₈	—	53	1	no	—
262	C₂×C₄×C₈	—	53	1	no	—
263	C₂×C₄×C₈	—	53	1	no	—
264	C₂×C₄×C₈	—	53	1	no	—
265	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
266	#322	—	53	2	no	—
267	#322	—	53	2	no	—
268	#322	—	53	2	no	—
269	#322	—	53	2	no	—
270	#322	—	53	2	no	—
271	#322	C₂×D₅₃	1	2	yes	complemented
272	C₈²	—	53	2	no	—
273	C₈²	C₂×D₅₃	1	2	yes	complemented
274	C₅₃	#17323	1	1	yes	characteristic, complemented, Sylow
275	C₂³×C₄	—	53	1	no	—
276	C₂³×C₄	—	53	1	no	—
277	C₂³×C₄	—	53	1	no	—
278	C₂×M₁₆	—	53	2	no	—
279	C₂×M₁₆	—	53	4	no	—
280	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
281	C₂²×C₈	—	53	2	no	—
282	C₂²×C₈	—	53	2	no	—
283	C₂²×C₈	—	53	2	no	—
284	C₂²×C₈	—	53	2	no	—
285	C₂²×C₈	—	53	2	no	—
286	C₂²×C₈	—	106	8	no	—
287	C₂²×C₈	C₄×D₅₃	1	2	yes	—
288	C₂×C₄²	—	53	1	no	—
289	C₂×C₄²	—	53	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₂×C₄²	—	53	1	no	—
293	C₂×C₄²	—	53	1	no	—
294	C₂×C₄²	—	53	1	no	—
295	C₂×C₄²	—	53	1	no	—
296	C₂×C₄²	—	53	1	no	—
297	C₂×C₄²	—	53	1	no	—
298	C₂×C₄²	—	53	1	no	—
299	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
300	C₄⋊C₈	—	53	4	no	—
301	C₄⋊C₈	—	53	4	no	—
302	C₄⋊C₈	—	53	4	no	—
303	C₄⋊C₈	—	53	2	no	—
304	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
305	C₂²⋊C₈	—	53	2	no	—
306	C₂²⋊C₈	—	53	4	no	—
307	C₂²⋊C₈	—	53	4	no	—
308	C₂²⋊C₈	—	53	4	no	—
309	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
310	#97	—	53	4	no	—
311	#97	—	53	2	no	—
312	#97	C₄×D₅₃	1	2	yes	—
313	C₄×C₈	—	53	2	no	—
314	C₄×C₈	—	53	2	no	—
315	C₄×C₈	—	53	2	no	—
316	C₄×C₈	—	53	2	no	—
317	C₄×C₈	—	53	2	no	—
318	C₄×C₈	—	53	2	no	—
319	C₄×C₈	—	53	2	no	—
320	C₄×C₈	—	53	1	no	—
321	C₄×C₈	—	53	1	no	—
322	C₄×C₈	C₄×D₅₃	1	2	yes	—
323	C₄×C₈	C₂²×D₅₃	1	2	yes	—
324	C₄×C₈	C₂²×D₅₃	1	2	yes	—
325	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
326	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
327	C₂⁴	—	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	—	53	1	no	—
337	C₂²×C₄	—	53	1	no	—
338	C₂²×C₄	—	53	1	no	—
339	C₂²×C₄	—	53	1	no	—
340	C₂²×C₄	—	53	1	no	—
341	C₂²×C₄	—	53	1	no	—
342	C₂²×C₄	—	53	1	no	—
343	C₂²×C₄	—	106	4	no	—
344	C₂²×C₄	—	106	4	no	—
345	C₂²×C₄	—	106	4	no	—
346	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
347	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
348	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
349	M₁₆	—	2	4	no	—
350	M₁₆	—	106	8	no	—
351	M₁₆	—	106	4	no	—
352	C₂×C₈	—	2	4	no	—
353	C₂×C₈	—	53	2	no	—
354	C₂×C₈	—	53	8	no	—
355	C₂×C₈	—	53	2	no	—
356	C₂×C₈	—	53	4	no	—
357	C₂×C₈	—	53	4	no	—
358	C₂×C₈	—	106	4	no	—
359	C₂×C₈	—	106	4	no	—
360	C₂×C₈	—	106	4	no	—
361	C₂×C₈	—	106	4	no	—
362	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
363	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
364	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
365	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
366	C₄²	—	53	1	no	—
367	C₄²	—	53	1	no	—
368	C₄²	—	53	1	no	—
369	C₄²	—	53	1	no	—
370	C₄²	—	53	1	no	—
371	C₄²	—	53	1	no	—
372	C₄²	—	53	1	no	—
373	C₄²	—	53	1	no	—
374	C₄²	—	53	1	no	—
375	C₄²	—	53	1	no	—
376	C₄²	—	53	1	no	—
377	C₄²	—	53	1	no	—
378	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
379	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
380	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
381	C₄²	C₂³×D₅₃	1	1	yes	characteristic
382	C₂³	—	53	1	no	—
383	C₂³	—	53	1	no	—
384	C₂³	—	106	2	no	—
385	C₂³	—	106	2	no	—
386	C₂³	—	106	2	no	—
387	C₂³	C₄²×D₅₃	1	1	yes	characteristic
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	53	1	no	—
391	C₂×C₄	—	53	1	no	—
392	C₂×C₄	—	53	4	no	—
393	C₂×C₄	—	53	4	no	—
394	C₂×C₄	—	53	1	no	—
395	C₂×C₄	—	53	1	no	—
396	C₂×C₄	—	53	1	no	—
397	C₂×C₄	—	53	1	no	—
398	C₂×C₄	—	106	2	no	—
399	C₂×C₄	—	106	2	no	—
400	C₂×C₄	—	106	2	no	—
401	C₂×C₄	—	106	2	no	—
402	C₂×C₄	—	106	2	no	—
403	C₂×C₄	—	106	2	no	—
404	C₂×C₄	—	106	2	no	—
405	C₂×C₄	—	106	2	no	—
406	C₂×C₄	—	106	2	no	—
407	C₂×C₄	—	106	2	no	—
408	C₂×C₄	—	106	2	no	—
409	C₂×C₄	—	106	2	no	—
410	C₂×C₄	—	106	2	no	—
411	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
412	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
413	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
414	C₂×C₄	#13133223	1	4	yes	—
415	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	4	no	—
420	C₈	—	2	4	no	—
421	C₈	—	53	4	no	—
422	C₈	—	106	2	no	—
423	C₈	—	106	4	no	—
424	C₈	—	106	4	no	—
425	C₈	M₁₆×D₅₃	1	4	yes	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	53	2	no	—
429	C₂²	—	53	2	no	—
430	C₂²	—	106	8	no	—
431	C₂²	—	106	1	no	—
432	C₂²	—	106	1	no	—
433	C₂²	—	106	1	no	—
434	C₂²	—	106	1	no	—
435	C₂²	#16810577	1	2	yes	—
436	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	53	2	no	—
440	C₄	—	53	2	no	—
441	C₄	—	106	1	no	—
442	C₄	—	106	1	no	—
443	C₄	—	106	1	no	—
444	C₄	—	106	1	no	—
445	C₄	#16810577	1	2	yes	—
446	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
447	C₄	#131×D₅₃	1	2	yes	—
448	C₄	C₂×#13133223	1	2	yes	—
449	C₂	—	2	2	no	—
450	C₂	—	106	2	no	—
451	C₂	—	106	2	no	—
452	C₂	C₄×M₁₆×D₅₃	1	1	yes	characteristic
453	C₂	#405×D₅₃	1	1	yes	characteristic
454	C₂	C₂×#16810577	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#17323
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20273006	C₂	semidirect
5. #20273005	C₂	semidirect
6. #20273003	C₂	semidirect
7. #20273002	C₂	semidirect
8. #20272997	C₂	semidirect
9. C₈×#11719917	C₂	semidirect
10. C₈×#11719912	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
36. #16810345	C₂²	semidirect
37. #16810345	C₂²	semidirect
38. #16810345	C₂²	semidirect
39. #16810345	C₂²	semidirect
40. #16810344	C₂²	semidirect
41. C₈×#88322	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₈²×C₅₃	C₂²	semidirect
153. C₈×D₅₃	M₁₆	semidirect
232. C₈×M₁₆	D₅₃	semidirect
271. #322	C₂×D₅₃	semidirect
273. C₈²	C₂×D₅₃	semidirect
274. C₅₃	#17323	semidirect

Maximal subgroups

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

subgroup	index
#20283346	2
#20283343	2
C₂×C₄×C₈×D₅₃	2
#20273006	2
#20273005	2
#20273003	2
#20273002	2
#20272997	2
C₈×#11719917	2
C₈×#11719912	2
C₈×M₁₆×C₅₃	2
#17323	53