Subgroups — #27354713

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285321	C₂	1	1	yes	maximal, characteristic
2	#20283690	C₂	1	1	yes	maximal, characteristic
3	#20283649	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273613	C₂	1	2	yes	maximal, complemented
5	#20273607	C₂	1	2	yes	maximal, complemented
6	#20273535	C₂	1	2	yes	maximal, complemented
7	#20273529	C₂	1	2	yes	maximal, complemented
8	#20273482	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273189	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273181	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811180	C₂²	1	1	yes	characteristic
13	#16811157	C₂²	1	1	yes	characteristic
14	#16811032	C₄	1	1	yes	characteristic
15	#16811017	C₄	1	1	yes	characteristic
16	#16810992	C₄	1	1	yes	characteristic
17	#16810983	C₄	1	1	yes	characteristic
18	#16810881	C₄	1	2	yes	—
19	#16810869	C₄	1	2	yes	—
20	#16810753	—	2	1	no	—
21	#16810753	C₂²	1	2	yes	—
22	#16810753	C₂²	1	1	yes	characteristic
23	C₈×Dic₁₀₆	—	2	1	no	—
24	C₈×Dic₁₀₆	C₂²	1	2	yes	—
25	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₁₂	C₂²	1	2	yes	—
27	C₄×#11719913	C₂²	1	2	yes	—
28	#16810702	C₂²	1	1	yes	characteristic
29	#16810694	—	2	1	no	—
30	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
31	C₈×C₂²⋊D₅₃	—	2	1	no	—
32	#16810682	—	2	1	no	—
33	#16810680	C₂²	1	1	yes	characteristic
34	#16810649	—	2	1	no	—
35	#16810647	C₂²	1	1	yes	characteristic
36	#16810578	—	2	1	no	—
37	#16810577	—	2	1	no	—
38	#16810376	C₂²	1	2	yes	—
39	#16810374	C₂²	1	2	yes	—
40	#16810355	C₂²	1	2	yes	—
41	#16810355	C₂²	1	2	yes	—
42	#16810355	C₂²	1	2	yes	—
43	#16810352	C₂²	1	2	yes	—
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#322	C₂²	1	2	yes	—
48	C₈²×C₅₃	C₂²	1	2	yes	—
49	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
50	C₂×#11719913	C₂×C₄	1	2	yes	—
51	C₂×#11719917	—	2	1	no	—
52	C₂×#11719912	—	2	1	no	—
53	C₂×C₈×D₅₃	—	2	1	no	—
54	#13133217	C₂×C₄	1	1	yes	characteristic
55	#13133217	C₂×C₄	1	1	yes	characteristic
56	#13133211	—	2	1	no	—
57	#13133209	—	2	1	no	—
58	#13133209	C₂×C₄	1	1	yes	characteristic
59	#13133205	C₂×C₄	1	1	yes	characteristic
60	#13133200	—	2	1	no	—
61	#13133196	—	2	1	no	—
62	#13133196	C₂×C₄	1	2	yes	—
63	#13133196	C₂×C₄	1	1	yes	characteristic
64	#13133191	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₀₆	—	2	1	no	—
66	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₀₆	C₂³	1	2	yes	—
68	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
69	C₄×C₂²⋊D₅₃	—	2	1	no	—
70	#13133176	—	2	1	no	—
71	#13133174	—	2	1	no	—
72	#13133151	C₂×C₄	1	2	yes	—
73	#13133149	C₂×C₄	1	2	yes	—
74	C₄⋊#88322	—	2	1	no	—
75	C₄⋊#88322	C₂³	1	2	yes	—
76	C₄⋊#88322	C₂³	1	1	yes	characteristic
77	C₄⋊#88322	C₂³	1	1	yes	characteristic
78	#13133146	—	2	2	no	—
79	#13133146	—	2	2	no	—
80	#13133146	—	2	1	no	—
81	#13133146	—	2	1	no	—
82	#13133142	C₂×C₄	1	2	yes	—
83	#13133142	C₂×C₄	1	2	yes	—
84	#13133142	C₂×C₄	1	2	yes	—
85	#13133139	C₂×C₄	1	2	yes	—
86	C₂²⋊#88322	—	2	1	no	—
87	#13133129	—	2	1	no	—
88	#13133129	—	2	1	no	—
89	#13133128	—	2	1	no	—
90	#13133127	—	2	2	no	—
91	#13133127	—	2	1	no	—
92	C₈×Dic₅₃	—	2	1	no	—
93	C₈×Dic₅₃	—	2	2	no	—
94	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
95	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
96	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	—	2	1	no	—
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
101	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
103	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
104	C₄×C₈×C₅₃	C₂³	1	2	yes	—
105	C₄×C₈×C₅₃	C₂³	1	2	yes	—
106	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
111	C₂×C₂²⋊D₅₃	—	2	1	no	—
112	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	—	2	1	no	—
114	C₂×Dic₁₀₆	—	2	1	no	—
115	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
117	M₁₆×C₅₃	—	2	4	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	4	1	no	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₅₃	#41	1	1	yes	characteristic
130	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₂²⋊Dic₅₃	—	2	2	no	—
132	C₂²⋊Dic₅₃	—	2	1	no	—
133	#11719921	—	2	1	no	—
134	#11719921	—	2	1	no	—
135	#11719921	—	2	1	no	—
136	#11719921	—	2	1	no	—
137	C₄⋊Dic₅₃	—	2	1	no	—
138	C₄⋊Dic₅₃	—	2	1	no	—
139	C₄⋊Dic₅₃	—	2	1	no	—
140	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
141	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
143	#11719919	—	2	1	no	—
144	#11719919	—	2	1	no	—
145	#11719919	—	2	1	no	—
146	#11719919	—	2	2	no	—
147	#11719919	—	2	2	no	—
148	#11719919	—	2	1	no	—
149	C₄×Dic₅₃	—	2	1	no	—
150	C₄×Dic₅₃	—	2	2	no	—
151	#11719917	—	4	2	no	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	—	2	1	no	—
154	C₂×#88322	—	2	1	no	—
155	Dic₂₁₂	—	2	4	no	—
156	#11719913	—	2	4	no	—
157	#11719912	—	4	2	no	—
158	C₈×D₅₃	—	4	2	no	—
159	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₂²×D₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	4	1	no	—
165	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
171	C₂²⋊D₅₃	—	4	2	no	—
172	C₂×Dic₅₃	—	2	2	no	—
173	C₂×Dic₅₃	—	2	2	no	—
174	C₂×Dic₅₃	—	2	1	no	—
175	C₂×Dic₅₃	—	2	1	no	—
176	C₂×Dic₅₃	—	2	1	no	—
177	D₂₁₂	—	2	2	no	—
178	C₄×D₅₃	—	4	1	no	—
179	C₄×D₅₃	—	4	1	no	—
180	Dic₁₀₆	—	2	2	no	—
181	Dic₁₀₆	—	2	2	no	—
182	Dic₁₀₆	—	2	2	no	—
183	Dic₁₀₆	—	4	2	no	—
184	C₈×C₅₃	—	2	2	no	—
185	C₈×C₅₃	—	2	1	no	—
186	C₈×C₅₃	—	2	1	no	—
187	C₈×C₅₃	—	2	1	no	—
188	C₈×C₅₃	—	4	2	no	—
189	C₈×C₅₃	#131	1	4	yes	—
190	C₈×C₅₃	#137	1	2	yes	—
191	#88322	—	4	1	no	—
192	#88322	—	4	1	no	—
193	#88322	—	4	2	no	—
194	#17967	—	53	1	no	maximal, Sylow
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	2	1	no	—
197	C₂²×C₅₃	—	2	1	no	—
198	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂×D₅₃	—	4	1	no	—
200	C₂×D₅₃	—	4	1	no	—
201	C₂×D₅₃	—	4	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	C₂×#131	1	2	yes	—
207	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
208	C₄×C₅₃	#575	1	1	yes	characteristic
209	Dic₅₃	—	4	2	no	—
210	Dic₅₃	—	4	1	no	—
211	Dic₅₃	—	4	1	no	—
212	Dic₅₃	—	4	2	no	—
213	Dic₅₃	—	4	1	no	—
214	#3172	—	53	1	no	—
215	#2966	—	53	1	no	—
216	#2960	—	53	1	no	—
217	#1591	—	53	2	no	—
218	#1590	—	53	2	no	—
219	#1579	—	53	2	no	—
220	SD₁₆⋊C₈	—	53	2	no	—
221	#1569	—	53	1	no	—
222	#1497	—	53	1	no	—
223	#1495	—	53	1	no	—
224	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
225	C₂×C₅₃	—	4	1	no	—
226	C₂×C₅₃	#2931	1	1	yes	characteristic
227	C₂×C₅₃	C₄×#137	1	1	yes	characteristic
228	C₂×C₅₃	#2955	1	1	yes	characteristic
229	D₅₃	—	8	1	no	—
230	#533	—	53	1	no	—
231	#529	—	53	1	no	—
232	#501	—	53	1	no	—
233	#497	—	53	1	no	—
234	#491	—	53	1	no	—
235	#489	—	53	1	no	—
236	#470	—	53	2	no	—
237	#468	—	53	2	no	—
238	#446	—	53	2	no	—
239	#446	—	53	1	no	—
240	#446	—	106	1	no	—
241	C₈×Q₈	—	53	2	no	—
242	C₈×Q₈	—	53	1	no	—
243	C₈×Q₈	—	106	1	no	—
244	C₄×Q₁₆	—	53	2	no	—
245	C₄×SD₁₆	—	53	2	no	—
246	#436	—	53	1	no	—
247	#435	—	106	1	no	—
248	C₈×D₄	—	53	1	no	—
249	C₈×D₄	—	106	1	no	—
250	#433	—	106	1	no	—
251	#432	—	53	1	no	—
252	#431	C₂×D₅₃	1	1	yes	characteristic
253	#424	—	106	1	no	—
254	#423	—	53	1	no	—
255	#405	—	106	1	no	—
256	#405	—	106	1	no	—
257	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
259	#335	—	53	2	no	—
260	#334	—	53	2	no	—
261	Q₈⋊C₈	—	53	2	no	—
262	Q₈⋊C₈	—	53	2	no	—
263	Q₈⋊C₈	—	53	2	no	—
264	D₄⋊C₈	—	53	2	no	—
265	#322	C₂×D₅₃	1	2	yes	—
266	C₈²	C₂×D₅₃	1	2	yes	—
267	C₅₃	#17967	1	1	yes	characteristic, complemented, Sylow
268	C₂×Q₁₆	—	53	2	no	—
269	C₂×SD₁₆	—	53	2	no	—
270	C₂×M₁₆	—	106	1	no	—
271	C₂×M₁₆	—	106	1	no	—
272	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
273	C₂²×C₈	—	106	1	no	—
274	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
275	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
276	C₄⋊Q₈	—	53	1	no	—
277	C₄⋊Q₈	—	53	1	no	—
278	#126	—	106	1	no	—
279	#125	—	53	1	no	—
280	#125	—	106	1	no	—
281	#124	—	53	1	no	—
282	#123	—	106	1	no	—
283	C₂²⋊Q₈	—	53	1	no	—
284	C₂²⋊Q₈	—	53	2	no	—
285	C₂²⋊Q₈	—	106	1	no	—
286	#121	—	53	1	no	—
287	C₄×Q₈	—	53	1	no	—
288	C₄×Q₈	—	53	2	no	—
289	C₄×Q₈	—	106	1	no	—
290	C₄×D₄	—	53	1	no	—
291	C₄×D₄	—	106	1	no	—
292	#117	—	106	1	no	—
293	#117	—	106	1	no	—
294	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
295	#107	—	53	2	no	—
296	#106	—	53	2	no	—
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	—	53	2	no	—
299	C₄⋊C₈	—	53	1	no	—
300	C₄⋊C₈	—	53	1	no	—
301	C₄⋊C₈	—	106	2	no	—
302	C₄⋊C₈	—	106	2	no	—
303	C₄⋊C₈	—	106	1	no	—
304	C₄⋊C₈	—	106	1	no	—
305	C₄⋊C₈	—	106	1	no	—
306	#103	—	53	2	no	—
307	#103	—	53	2	no	—
308	#103	—	53	2	no	—
309	#102	—	53	2	no	—
310	C₂²⋊C₈	—	2	1	no	—
311	C₂²⋊C₈	—	106	1	no	—
312	C₂²⋊C₈	—	106	1	no	—
313	C₂²⋊C₈	—	106	1	no	—
314	#97	—	106	1	no	—
315	#97	—	106	2	no	—
316	#97	—	106	1	no	—
317	#97	C₄×D₅₃	1	1	yes	characteristic
318	#97	C₄×D₅₃	1	1	yes	characteristic
319	C₄×C₈	—	106	1	no	—
320	C₄×C₈	—	106	2	no	—
321	C₄×C₈	D₂₁₂	1	1	yes	characteristic
322	C₄×C₈	D₂₁₂	1	1	yes	characteristic
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	C₂×Q₈	—	53	1	no	—
328	C₂×Q₈	—	53	2	no	—
329	C₂×Q₈	—	106	1	no	—
330	C₂×D₄	—	53	1	no	—
331	C₂×D₄	—	106	1	no	—
332	C₂²×C₄	—	106	1	no	—
333	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
335	C₂²×C₄	#11719945	1	1	yes	characteristic
336	Q₁₆	—	106	4	no	—
337	SD₁₆	—	106	4	no	—
338	M₁₆	—	2	4	no	—
339	M₁₆	—	212	2	no	—
340	M₁₆	—	212	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	4	1	no	—
344	C₂×C₈	—	106	1	no	—
345	C₂×C₈	—	106	2	no	—
346	C₂×C₈	—	106	1	no	—
347	C₂×C₈	—	212	2	no	—
348	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
349	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
350	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
351	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
352	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
353	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
354	C₄⋊C₄	—	53	2	no	—
355	C₄⋊C₄	—	53	1	no	—
356	C₄⋊C₄	—	53	1	no	—
357	C₄⋊C₄	—	106	1	no	—
358	C₄⋊C₄	—	106	1	no	—
359	C₄⋊C₄	—	106	1	no	—
360	C₄⋊C₄	—	106	1	no	—
361	C₄⋊C₄	—	106	1	no	—
362	C₄⋊C₄	—	106	2	no	—
363	C₄⋊C₄	—	106	1	no	—
364	C₄⋊C₄	—	106	2	no	—
365	C₄⋊C₄	—	106	1	no	—
366	C₂²⋊C₄	—	106	2	no	—
367	C₂²⋊C₄	—	106	1	no	—
368	C₂²⋊C₄	—	106	1	no	—
369	C₂²⋊C₄	—	106	1	no	—
370	C₂²⋊C₄	—	106	1	no	—
371	C₂²⋊C₄	—	106	1	no	—
372	C₄²	—	106	1	no	—
373	C₄²	—	106	2	no	—
374	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
375	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
376	C₄²	#11719945	1	1	yes	characteristic
377	C₄²	C₂³×D₅₃	1	1	yes	characteristic
378	C₂³	—	106	1	no	—
379	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
380	Q₈	—	106	2	no	—
381	Q₈	—	106	2	no	—
382	Q₈	—	106	2	no	—
383	Q₈	—	212	2	no	—
384	D₄	—	106	2	no	—
385	D₄	—	212	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	106	1	no	—
391	C₂×C₄	—	106	1	no	—
392	C₂×C₄	—	106	2	no	—
393	C₂×C₄	—	106	2	no	—
394	C₂×C₄	—	106	1	no	—
395	C₂×C₄	—	212	1	no	—
396	C₂×C₄	—	212	1	no	—
397	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
398	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
399	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
400	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
402	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
403	C₈	—	2	2	no	—
404	C₈	—	2	1	no	—
405	C₈	—	2	1	no	—
406	C₈	—	2	1	no	—
407	C₈	—	4	2	no	—
408	C₈	—	212	1	no	—
409	C₈	—	212	1	no	—
410	C₈	—	212	2	no	—
411	C₈	#13133223	1	4	yes	—
412	C₈	#13133250	1	2	yes	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	1	no	—
416	C₂²	—	212	1	no	—
417	C₂²	—	212	1	no	—
418	C₂²	—	212	1	no	—
419	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	2	1	no	—
424	C₄	—	212	1	no	—
425	C₄	—	212	2	no	—
426	C₄	—	212	1	no	—
427	C₄	—	212	1	no	—
428	C₄	—	212	2	no	—
429	C₄	C₂×#13133223	1	2	yes	—
430	C₄	C₂×#13133250	1	1	yes	characteristic
431	C₄	#16811423	1	1	yes	characteristic
432	C₂	—	4	1	no	—
433	C₂	—	424	1	no	—
434	C₂	#20283428	1	1	yes	characteristic
435	C₂	C₄×#13133250	1	1	yes	characteristic
436	C₂	#20283619	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#13133191	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#17967
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20283649	C₂	semidirect
4. #20273613	C₂	semidirect
5. #20273607	C₂	semidirect
6. #20273535	C₂	semidirect
7. #20273529	C₂	semidirect
8. #20273482	C₂	semidirect
9. #20273189	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
224. C₈×M₁₆	D₅₃	semidirect
267. C₅₃	#17967	semidirect

Maximal subgroups

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

subgroup	index
#20285321	2
#20283690	2
#20283649	2
#20273613	2
#20273607	2
#20273535	2
#20273529	2
#20273482	2
#20273189	2
#20273181	2
C₈×M₁₆×C₅₃	2
#17967	53