Subgroups — #27537291

Order 13568.

692 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283407	C₂	1	1	yes	maximal, characteristic
3	#20276545	C₂	1	2	yes	maximal, complemented
4	#20276540	C₂	1	2	yes	maximal, complemented
5	#20276538	C₂	1	2	yes	maximal, complemented
6	C₁₆×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
7	#20276473	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276329	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16810402	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276314	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16811130	C₄	1	1	yes	characteristic
14	C₂×#13133157	C₂²	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	2	yes	—
16	M₃₂×D₅₃	—	2	4	no	—
17	C₂²×#11719908	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
19	#16810692	C₄	1	2	yes	—
20	#16810692	C₄	1	2	yes	—
21	C₈×C₂²⋊D₅₃	C₂²	1	2	yes	—
22	C₈×C₂²⋊D₅₃	C₂²	1	2	yes	—
23	#16810651	C₄	1	1	yes	characteristic
24	#16810590	C₄	1	1	yes	characteristic
25	#16810586	C₄	1	1	yes	characteristic
26	#16810578	C₂²	1	1	yes	characteristic
27	#16810441	C₂²	1	2	yes	complemented
28	#16810441	C₂²	1	2	yes	complemented
29	C₂²⋊#11719908	C₂²	1	2	yes	complemented
30	C₂²⋊#11719908	C₂²	1	2	yes	complemented
31	#16810402	C₂²	1	1	yes	characteristic
32	#16810402	C₂²	1	1	yes	characteristic
33	#16810402	C₂²	1	1	yes	characteristic
34	#16810402	C₂²	1	1	yes	characteristic
35	#16810398	C₂²	1	2	yes	complemented
36	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
37	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#504	—	2	4	no	—
39	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×M₃₂×C₅₃	C₂²	1	2	yes	complemented
41	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	complemented
43	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
45	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	—	2	2	no	—
47	C₂×C₈×D₅₃	—	2	2	no	—
48	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
52	#13133198	C₈	1	2	yes	—
53	#13133193	C₈	1	1	yes	characteristic
54	#13133193	C₈	1	1	yes	characteristic
55	#13133188	C₈	1	1	yes	characteristic
56	#13133188	C₈	1	1	yes	characteristic
57	#13133185	C₈	1	2	yes	—
58	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
60	#13133176	C₂×C₄	1	1	yes	characteristic
61	#13133157	—	2	2	no	—
62	#13133156	—	2	4	no	—
63	C₂×#11719908	—	2	2	no	—
64	C₂×#11719908	C₂³	1	2	yes	—
65	C₂×#11719908	C₂³	1	1	yes	characteristic
66	C₂×#11719908	C₂³	1	1	yes	characteristic
67	C₁₆×D₅₃	—	2	4	no	—
68	#13133146	C₂×C₄	1	2	yes	—
69	#13133146	C₂×C₄	1	2	yes	—
70	C₂²⋊#88322	C₂×C₄	1	2	yes	—
71	C₂²⋊#88322	C₂×C₄	1	2	yes	—
72	#13133129	C₂×C₄	1	2	yes	—
73	#13133129	C₂×C₄	1	2	yes	—
74	#13133127	C₂×C₄	1	2	yes	—
75	C₈×Dic₅₃	C₂³	1	2	yes	—
76	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
77	#131×C₅₃	—	2	2	no	—
78	#131×C₅₃	—	2	2	no	—
79	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
81	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
82	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	M₃₂×C₅₃	—	2	4	no	—
84	M₃₂×C₅₃	D₄	1	4	yes	complemented
85	C₂×C₁₆×C₅₃	—	2	2	no	—
86	C₂×C₁₆×C₅₃	—	2	4	no	—
87	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
90	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
91	#41×C₅₃	—	2	2	no	—
92	#41×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
94	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
95	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
99	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
100	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
101	C₂×C₄×D₅₃	—	2	2	no	—
102	C₂×C₄×D₅₃	—	2	2	no	—
103	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₅₃	—	2	2	no	—
108	M₁₆×C₅₃	—	2	4	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₅₃	#41	1	2	yes	—
115	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
117	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
118	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
119	#11719921	C₂×C₈	1	2	yes	—
120	#11719921	C₂×C₈	1	2	yes	—
121	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
122	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
123	#11719919	C₂×C₈	1	2	yes	—
124	#11719919	C₂×C₈	1	2	yes	—
125	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
126	#11719917	—	2	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	C₂²×C₄	1	2	yes	—
129	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
130	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
131	C₈×D₅₃	—	2	2	no	—
132	C₈×D₅₃	—	2	2	no	—
133	C₈×D₅₃	—	2	1	no	—
134	C₈×D₅₃	—	2	1	no	—
135	C₈×D₅₃	—	2	1	no	—
136	C₈×D₅₃	—	2	1	no	—
137	C₁₆×C₅₃	—	2	1	no	—
138	C₁₆×C₅₃	—	2	1	no	—
139	C₁₆×C₅₃	C₂×D₄	1	4	yes	—
140	#11719908	—	2	2	no	—
141	#11719908	—	2	1	no	—
142	#11719908	—	2	1	no	—
143	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
144	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
145	C₂²×D₅₃	—	2	2	no	—
146	C₂²×D₅₃	—	2	2	no	—
147	C₂²×D₅₃	—	4	1	no	—
148	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
149	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	2	no	—
151	D₄×C₅₃	—	2	4	no	—
152	D₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
156	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#131	1	2	yes	—
159	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
160	C₂²⋊D₅₃	—	2	4	no	—
161	C₂²⋊D₅₃	—	4	2	no	—
162	C₂×Dic₅₃	—	2	2	no	—
163	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
164	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
165	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
166	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
167	C₄×D₅₃	—	2	1	no	—
168	C₄×D₅₃	—	2	1	no	—
169	C₄×D₅₃	—	2	1	no	—
170	C₄×D₅₃	—	2	1	no	—
171	C₄×D₅₃	—	2	2	no	—
172	C₄×D₅₃	—	2	2	no	—
173	C₈×C₅₃	—	2	2	no	—
174	C₈×C₅₃	C₄×D₄	1	2	yes	—
175	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
176	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
177	#88322	—	2	1	no	—
178	#88322	—	2	1	no	—
179	#88322	—	2	2	no	—
180	#33436	—	53	1	no	maximal, Sylow
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	—	2	2	no	—
184	C₂²×C₅₃	—	2	2	no	—
185	C₂²×C₅₃	—	2	2	no	—
186	C₂²×C₅₃	C₈×D₄	1	2	yes	—
187	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
188	C₂×D₅₃	—	2	1	no	—
189	C₂×D₅₃	—	2	1	no	—
190	C₂×D₅₃	—	2	1	no	—
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	4	4	no	—
194	C₂×D₅₃	#504	1	2	yes	complemented
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	C₈×D₄	1	2	yes	—
197	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
199	Dic₅₃	—	2	2	no	—
200	Dic₅₃	—	2	2	no	—
201	Dic₅₃	—	2	1	no	—
202	Dic₅₃	—	2	1	no	—
203	C₂×#504	D₅₃	1	1	yes	characteristic, complemented
204	C₂²×M₃₂	—	53	1	no	—
205	#2929	—	53	1	no	—
206	#2169	—	53	2	no	—
207	#2168	—	53	2	no	—
208	#2168	—	53	2	no	—
209	D₄×C₁₆	—	53	2	no	—
210	#2151	—	53	1	no	—
211	#2113	—	53	1	no	—
212	C₂×C₂²⋊C₁₆	—	53	1	no	—
213	#2108	—	53	1	no	—
214	C₂×C₅₃	—	2	2	no	—
215	C₂×C₅₃	—	4	2	no	—
216	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
217	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
218	C₂×C₅₃	#3407	1	1	yes	characteristic
219	D₅₃	—	2	2	no	—
220	D₅₃	—	4	1	no	—
221	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
222	C₂³×C₈	—	53	1	no	—
223	#525	—	53	1	no	—
224	#504	—	2	4	no	—
225	C₂×M₃₂	—	53	1	no	—
226	C₂×M₃₂	—	53	2	no	—
227	C₂×M₃₂	—	106	4	no	—
228	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
229	C₂×M₃₂	C₂×D₅₃	1	2	yes	complemented
230	C₂²×C₁₆	—	53	1	no	—
231	C₂²×C₁₆	—	53	2	no	—
232	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
233	C₂²×C₁₆	C₂×D₅₃	1	2	yes	complemented
234	#435	—	53	2	no	—
235	#435	—	53	2	no	—
236	C₈×D₄	—	53	2	no	—
237	C₈×D₄	—	53	2	no	—
238	#424	—	53	1	no	—
239	#408	—	53	1	no	—
240	#407	—	53	1	no	—
241	#405	—	53	1	no	—
242	C₄⋊C₁₆	—	53	2	no	—
243	C₄⋊C₁₆	—	53	2	no	—
244	C₂²⋊C₁₆	—	53	2	no	—
245	C₂²⋊C₁₆	—	53	2	no	—
246	C₂²⋊C₁₆	—	53	1	no	—
247	C₂²⋊C₁₆	—	53	1	no	—
248	C₂²⋊C₁₆	—	53	1	no	—
249	C₂²⋊C₁₆	—	53	1	no	—
250	#346	—	53	2	no	—
251	C₄×C₁₆	—	53	2	no	—
252	C₅₃	#33436	1	1	yes	characteristic, complemented, Sylow
253	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
254	C₂³×C₄	—	53	1	no	—
255	#131	—	2	2	no	—
256	#131	—	2	2	no	—
257	C₂×M₁₆	—	53	1	no	—
258	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
259	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
260	C₂²×C₈	—	53	1	no	—
261	C₂²×C₈	—	53	1	no	—
262	C₂²×C₈	—	53	1	no	—
263	C₂²×C₈	—	53	1	no	—
264	C₂²×C₈	—	53	1	no	—
265	C₂²×C₈	—	106	2	no	—
266	C₂²×C₈	—	106	2	no	—
267	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
268	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
269	#123	—	53	2	no	—
270	C₂²⋊Q₈	—	53	1	no	—
271	C₂²⋊Q₈	—	53	1	no	—
272	#121	—	53	1	no	—
273	#121	—	53	1	no	—
274	#120	—	53	2	no	—
275	C₄×D₄	—	53	2	no	—
276	C₄×D₄	—	53	2	no	—
277	#117	—	53	1	no	—
278	M₃₂	—	2	4	no	—
279	M₃₂	—	106	2	no	—
280	M₃₂	—	106	4	no	—
281	M₃₂	C₂²⋊D₅₃	1	4	yes	complemented
282	C₂×C₁₆	—	2	2	no	—
283	C₂×C₁₆	—	2	4	no	—
284	C₂×C₁₆	—	53	1	no	—
285	C₂×C₁₆	—	53	1	no	—
286	C₂×C₁₆	—	53	2	no	—
287	C₂×C₁₆	—	106	2	no	—
288	C₂×C₁₆	—	106	4	no	—
289	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
290	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
291	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
292	C₄⋊C₈	—	53	2	no	—
293	C₄⋊C₈	—	53	2	no	—
294	C₂²⋊C₈	—	53	2	no	—
295	C₂²⋊C₈	—	53	2	no	—
296	C₂²⋊C₈	—	53	2	no	—
297	C₂²⋊C₈	—	53	2	no	—
298	#97	—	53	2	no	—
299	C₄×C₈	—	53	2	no	—
300	C₂⁴	—	53	1	no	—
301	#41	—	2	2	no	—
302	#41	—	2	2	no	—
303	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
304	C₂×D₄	—	53	2	no	—
305	C₂×D₄	—	53	2	no	—
306	C₂×D₄	C₈×D₅₃	1	2	yes	—
307	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
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	—	106	2	no	—
314	C₂²×C₄	—	106	2	no	—
315	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
316	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	4	no	—
319	M₁₆	—	106	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	53	2	no	—
323	C₂×C₈	—	53	1	no	—
324	C₂×C₈	—	53	1	no	—
325	C₂×C₈	—	106	2	no	—
326	C₂×C₈	—	106	2	no	—
327	C₂×C₈	—	106	2	no	—
328	C₂×C₈	—	106	1	no	—
329	C₂×C₈	—	106	1	no	—
330	C₂×C₈	—	106	1	no	—
331	C₂×C₈	—	106	1	no	—
332	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
333	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
334	C₂×C₈	#11719945	1	2	yes	—
335	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
336	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
337	C₄⋊C₄	—	53	1	no	—
338	C₄⋊C₄	—	53	1	no	—
339	C₄⋊C₄	—	53	2	no	—
340	C₄⋊C₄	—	53	2	no	—
341	C₂²⋊C₄	—	53	2	no	—
342	C₂²⋊C₄	—	53	2	no	—
343	C₂²⋊C₄	—	53	2	no	—
344	C₂²⋊C₄	—	53	2	no	—
345	C₂²⋊C₄	—	53	2	no	—
346	C₄²	—	53	2	no	—
347	C₁₆	—	2	1	no	—
348	C₁₆	—	2	1	no	—
349	C₁₆	—	106	2	no	—
350	C₁₆	—	106	1	no	—
351	C₁₆	—	106	1	no	—
352	C₁₆	C₂×C₂²⋊D₅₃	1	4	yes	—
353	C₂³	—	53	1	no	—
354	C₂³	—	53	1	no	—
355	C₂³	—	106	2	no	—
356	C₂³	—	106	2	no	—
357	C₂³	—	212	1	no	—
358	C₂³	C₂×C₈×D₅₃	1	2	yes	—
359	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	2	no	—
363	D₄	—	106	4	no	—
364	D₄	—	212	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	53	2	no	—
368	C₂×C₄	—	53	2	no	—
369	C₂×C₄	—	53	1	no	—
370	C₂×C₄	—	53	1	no	—
371	C₂×C₄	—	106	2	no	—
372	C₂×C₄	—	106	1	no	—
373	C₂×C₄	—	106	1	no	—
374	C₂×C₄	—	106	1	no	—
375	C₂×C₄	—	106	1	no	—
376	C₂×C₄	—	106	2	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
379	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
380	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
381	C₂×C₄	#13133223	1	2	yes	—
382	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	106	1	no	—
385	C₈	—	106	1	no	—
386	C₈	—	106	2	no	—
387	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
388	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
389	C₈	C₂×#11719945	1	1	yes	characteristic
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	2	no	—
394	C₂²	—	2	2	no	—
395	C₂²	—	53	2	no	—
396	C₂²	—	106	1	no	—
397	C₂²	—	106	1	no	—
398	C₂²	—	106	1	no	—
399	C₂²	—	106	1	no	—
400	C₂²	—	106	1	no	—
401	C₂²	—	212	4	no	—
402	C₂²	C₈×C₂²⋊D₅₃	1	2	yes	—
403	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	106	2	no	—
406	C₄	—	106	2	no	—
407	C₄	—	106	1	no	—
408	C₄	—	106	1	no	—
409	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
410	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
411	C₄	C₂×#13133223	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	4	2	no	—
414	C₂	—	106	2	no	—
415	C₂	—	212	1	no	—
416	C₂	C₂×C₈×C₂²⋊D₅₃	1	1	yes	characteristic
417	C₂	#504×D₅₃	1	1	yes	characteristic
418	C₂	#20287816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×#504	C₂
prime radical	#16811130	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#33436
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×M₃₂×D₅₃	C₂	semidirect
3. #20276545	C₂	semidirect
4. #20276540	C₂	semidirect
5. #20276538	C₂	semidirect
6. C₁₆×C₂²⋊D₅₃	C₂	semidirect
7. #20276473	C₂	semidirect
8. #20276329	C₂	semidirect
9. C₂×#16810402	C₂	semidirect
10. #20276314	C₂	semidirect
11. C₂×C₅₃×#504	C₂	semidirect
27. #16810441	C₂²	semidirect
28. #16810441	C₂²	semidirect
29. C₂²⋊#11719908	C₂²	semidirect
30. C₂²⋊#11719908	C₂²	semidirect
35. #16810398	C₂²	semidirect
36. C₁₆×Dic₅₃	C₂²	semidirect
40. C₂×M₃₂×C₅₃	C₂²	semidirect
42. C₂²×C₁₆×C₅₃	C₂²	semidirect
84. M₃₂×C₅₃	D₄	semidirect
194. C₂×D₅₃	#504	semidirect
203. C₂×#504	D₅₃	semidirect
229. C₂×M₃₂	C₂×D₅₃	semidirect
233. C₂²×C₁₆	C₂×D₅₃	semidirect
252. C₅₃	#33436	semidirect
281. M₃₂	C₂²⋊D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₅₃	2
#20283407	2
#20276545	2
#20276540	2
#20276538	2
C₁₆×C₂²⋊D₅₃	2
#20276473	2
#20276329	2
C₂×#16810402	2
#20276314	2
C₂×C₅₃×#504	2
#33436	53