Subgroups — #27382693

Order 13568.

998 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810740	C₂	1	1	yes	maximal, characteristic
2	#135×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20283141	C₂	1	2	yes	maximal, complemented
4	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274597	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274560	C₂	1	2	yes	maximal, complemented
8	#20274557	C₂	1	2	yes	maximal, complemented
9	#20274527	C₂	1	2	yes	maximal, complemented
10	#20274401	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	C₂²	1	2	yes	—
13	#41×Dic₅₃	C₂²	1	2	yes	—
14	#16810740	C₄	1	8	yes	complemented
15	Q₁₆×Dic₅₃	C₄	1	2	yes	complemented
16	SD₁₆×Dic₅₃	C₄	1	4	yes	complemented
17	D₈×Dic₅₃	C₄	1	2	yes	complemented
18	C₂×#13133153	C₂²	1	1	yes	characteristic
19	#16810661	C₂²	1	1	yes	characteristic
20	C₂×#13133144	C₂²	1	2	yes	—
21	#16810619	C₂²	1	2	yes	—
22	#16810581	—	2	2	no	—
23	#16810581	C₂²	1	2	yes	—
24	#16810578	—	2	2	no	—
25	#16810578	C₂²	1	2	yes	—
26	M₁₆×Dic₅₃	—	2	2	no	—
27	M₁₆×Dic₅₃	C₂²	1	2	yes	—
28	C₂×#13133128	C₂²	1	1	yes	characteristic
29	C₂×#13133127	—	2	2	no	—
30	C₂×#13133127	C₂²	1	1	yes	characteristic
31	C₂×#13133127	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
34	#16810469	C₂²	1	1	yes	characteristic
35	#16810387	C₂²	1	2	yes	—
36	#16810384	C₂²	1	2	yes	—
37	#16810383	C₂²	1	2	yes	—
38	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#131×C₅₃	C₂²	1	2	yes	—
40	C₅₃×#443	C₄	1	8	yes	complemented
41	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#104×C₅₃	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	#13133224	—	2	4	no	—
45	#13133224	C₂×C₄	1	8	yes	complemented
46	C₂×#11719917	—	2	2	no	—
47	C₂×#11719917	C₂³	1	2	yes	—
48	C₂²×#88322	—	2	2	no	—
49	C₂²×#88322	C₂³	1	1	yes	characteristic
50	C₂²×#88322	C₂³	1	1	yes	characteristic
51	Q₈×Dic₅₃	C₂×C₄	1	4	yes	—
52	D₄×Dic₅₃	—	2	4	no	—
53	D₄×Dic₅₃	C₂×C₄	1	4	yes	—
54	#13133176	—	2	2	no	—
55	#13133176	C₂³	1	2	yes	—
56	C₂×C₄×Dic₅₃	—	2	2	no	—
57	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
59	#13133153	C₂×C₄	1	4	yes	complemented
60	#13133151	C₂×C₄	1	2	yes	—
61	#13133149	C₂×C₄	1	2	yes	—
62	#13133144	C₂×C₄	1	8	yes	complemented
63	#13133141	C₂×C₄	1	4	yes	—
64	#13133138	C₂×C₄	1	4	yes	—
65	#13133128	—	2	2	no	—
66	#13133127	—	2	2	no	—
67	#13133127	—	2	2	no	—
68	#13133127	—	2	2	no	—
69	#13133127	—	2	2	no	—
70	#13133127	D₄	1	1	yes	characteristic
71	#13133127	D₄	1	1	yes	characteristic
72	#13133127	D₄	1	1	yes	characteristic
73	#13133127	D₄	1	1	yes	characteristic
74	C₈×Dic₅₃	—	2	2	no	—
75	C₈×Dic₅₃	—	2	2	no	—
76	C₈×Dic₅₃	—	2	2	no	—
77	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
78	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
79	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
80	#13133124	—	2	2	no	—
81	C₂×#41×C₅₃	C₂³	1	2	yes	—
82	#135×C₅₃	C₂×C₄	1	4	yes	—
83	#135×C₅₃	C₂×C₄	1	4	yes	—
84	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
86	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	#131×C₅₃	—	2	4	no	—
88	#131×C₅₃	C₂×C₄	1	8	yes	complemented
89	C₂×M₁₆×C₅₃	—	2	2	no	—
90	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
91	C₂²×C₈×C₅₃	—	2	2	no	—
92	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	#108×C₅₃	C₂×C₄	1	4	yes	complemented
96	#104×C₅₃	C₂×C₄	1	8	yes	complemented
97	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
98	#41×C₅₃	—	2	2	no	—
99	#41×C₅₃	—	2	2	no	—
100	#41×C₅₃	C₂²×C₄	1	4	yes	—
101	#41×C₅₃	C₂²×C₄	1	4	yes	—
102	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×D₄×C₅₃	—	2	2	no	—
104	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂²×Dic₅₃	—	2	1	no	—
109	C₂²×Dic₅₃	—	2	4	no	—
110	C₂²×Dic₅₃	#41	1	2	yes	—
111	Q₁₆×C₅₃	C₄²	1	4	yes	—
112	SD₁₆×C₅₃	C₄²	1	8	yes	—
113	D₈×C₅₃	C₄²	1	4	yes	—
114	M₁₆×C₅₃	—	2	2	no	—
115	M₁₆×C₅₃	—	2	8	no	—
116	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	8	no	—
119	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	#41	1	2	yes	—
127	C₄²×C₅₃	—	2	1	no	—
128	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂²⋊Dic₅₃	—	2	2	no	—
130	C₂²⋊Dic₅₃	—	2	4	no	—
131	C₄⋊Dic₅₃	—	2	4	no	—
132	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
133	C₄×Dic₅₃	—	2	2	no	—
134	C₄×Dic₅₃	—	2	1	no	—
135	C₄×Dic₅₃	—	2	4	no	—
136	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
137	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
138	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
140	#11719917	—	2	2	no	—
141	#11719917	—	2	8	no	—
142	#11719917	C₂²×C₄	1	4	yes	—
143	C₂×#88322	—	2	2	no	—
144	C₂×#88322	—	2	1	no	—
145	C₂×#88322	—	2	1	no	—
146	C₂×#88322	—	2	8	no	—
147	C₂×#88322	C₂²×C₄	1	2	yes	—
148	C₂×#88322	C₂²×C₄	1	2	yes	—
149	C₂×#88322	C₂×D₄	1	1	yes	characteristic
150	C₂×#88322	C₂×D₄	1	1	yes	characteristic
151	C₂×#88322	#41	1	2	yes	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
154	Q₈×C₅₃	—	2	2	no	—
155	Q₈×C₅₃	C₂×C₄²	1	4	yes	—
156	D₄×C₅₃	—	2	4	no	—
157	D₄×C₅₃	—	2	4	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	C₂×C₄²	1	4	yes	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	2	4	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
171	C₂×Dic₅₃	—	2	4	no	—
172	C₂×Dic₅₃	—	2	2	no	—
173	C₂×Dic₅₃	—	4	2	no	—
174	C₂×Dic₅₃	—	4	2	no	—
175	C₂×Dic₅₃	—	8	4	no	—
176	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
177	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
178	C₈×C₅₃	—	2	4	no	—
179	C₈×C₅₃	C₂×C₄²	1	2	yes	—
180	C₈×C₅₃	C₂×C₄²	1	2	yes	—
181	C₈×C₅₃	C₄×D₄	1	2	yes	—
182	C₈×C₅₃	C₄×D₄	1	2	yes	—
183	#88322	—	2	4	no	—
184	#88322	—	2	2	no	—
185	#88322	C₄×D₄	1	2	yes	—
186	#88322	C₄×D₄	1	2	yes	—
187	#20514	—	53	1	no	maximal, Sylow
188	C₂²×C₅₃	—	2	2	no	—
189	C₂²×C₅₃	—	2	4	no	—
190	C₂²×C₅₃	—	4	4	no	—
191	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
193	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
194	C₄×C₅₃	—	2	4	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
197	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
199	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
200	Dic₅₃	—	2	2	no	—
201	Dic₅₃	—	2	2	no	—
202	Dic₅₃	—	4	2	no	—
203	Dic₅₃	—	4	4	no	—
204	C₂×#443	—	53	1	no	—
205	C₂×#443	D₅₃	1	1	yes	characteristic
206	C₄×#135	—	53	1	no	—
207	#2875	—	53	2	no	—
208	C₄×#108	—	53	1	no	—
209	#1772	—	53	1	no	—
210	#1772	—	53	1	no	—
211	#1765	—	53	2	no	—
212	#1765	—	53	2	no	—
213	#1759	—	53	2	no	—
214	#1726	—	53	1	no	—
215	C₂×C₅₃	—	2	1	no	—
216	C₂×C₅₃	—	2	1	no	—
217	C₂×C₅₃	—	4	4	no	—
218	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
219	C₂×C₅₃	#2957	1	2	yes	—
220	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
221	C₂×#131	—	53	2	no	—
222	C₂×#131	C₂×D₅₃	1	2	yes	—
223	C₄×#41	—	53	2	no	—
224	#443	—	53	8	no	—
225	#443	Dic₅₃	1	8	yes	complemented
226	C₄×Q₁₆	—	53	2	no	—
227	C₄×SD₁₆	—	53	4	no	—
228	C₄×D₈	—	53	2	no	—
229	C₂×#108	—	53	1	no	—
230	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
231	#427	—	53	1	no	—
232	C₂×#104	—	53	2	no	—
233	C₂×#104	C₂×D₅₃	1	2	yes	—
234	#416	—	53	2	no	—
235	#405	—	53	2	no	—
236	#405	—	53	2	no	—
237	#405	—	106	2	no	—
238	#405	—	106	2	no	—
239	C₄×M₁₆	—	53	2	no	—
240	C₄×M₁₆	—	106	2	no	—
241	C₂×#97	—	53	1	no	—
242	C₂×#97	—	53	1	no	—
243	C₂×#97	—	53	1	no	—
244	C₂×#97	—	106	2	no	—
245	C₂×C₄×C₈	—	53	1	no	—
246	C₂×C₄×C₈	—	53	1	no	—
247	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
248	#376	—	53	1	no	—
249	#341	—	53	2	no	—
250	#339	—	53	2	no	—
251	#339	—	53	2	no	—
252	C₅₃	#20514	1	1	yes	characteristic, complemented, Sylow
253	C₂×#41	C₂²×D₅₃	1	2	yes	—
254	#135	C₄×D₅₃	1	4	yes	—
255	#135	C₂×Dic₅₃	1	4	yes	—
256	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
257	C₂×SD₁₆	C₄×D₅₃	1	2	yes	—
258	C₂×D₈	C₄×D₅₃	1	1	yes	characteristic
259	#131	—	2	4	no	—
260	#131	—	53	8	no	—
261	#131	—	106	4	no	—
262	#131	C₂×Dic₅₃	1	8	yes	complemented
263	C₂×M₁₆	—	2	2	no	—
264	C₂×M₁₆	—	53	2	no	—
265	C₂×M₁₆	—	106	2	no	—
266	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	—	53	1	no	—
269	C₂²×C₈	—	53	1	no	—
270	C₂²×C₈	—	106	2	no	—
271	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
272	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
273	C₄×Q₈	—	53	4	no	—
274	C₄×D₄	—	53	4	no	—
275	C₄×D₄	—	106	4	no	—
276	#117	—	53	2	no	—
277	#117	—	106	2	no	—
278	C₂×C₄²	—	53	1	no	—
279	C₂×C₄²	—	53	1	no	—
280	C₂×C₄²	—	106	2	no	—
281	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
282	#108	—	53	4	no	—
283	#108	C₂×Dic₅₃	1	4	yes	complemented
284	#107	—	53	2	no	—
285	#106	—	53	2	no	—
286	#104	—	53	8	no	—
287	#104	C₂×Dic₅₃	1	8	yes	complemented
288	#103	—	53	4	no	—
289	#102	—	53	4	no	—
290	#97	—	53	1	no	—
291	#97	—	53	1	no	—
292	#97	—	53	1	no	—
293	#97	—	53	1	no	—
294	#97	—	106	2	no	—
295	#97	—	106	2	no	—
296	#97	—	106	2	no	—
297	#97	—	106	2	no	—
298	#97	—	106	2	no	—
299	C₄×C₈	—	53	2	no	—
300	C₄×C₈	—	53	2	no	—
301	C₄×C₈	—	53	4	no	—
302	C₄×C₈	—	106	2	no	—
303	C₄×C₈	—	106	2	no	—
304	C₄×C₈	—	106	2	no	—
305	C₄×C₈	C₂×Dic₅₃	1	4	yes	complemented
306	#95	—	106	2	no	—
307	#41	—	2	2	no	—
308	#41	—	2	2	no	—
309	#41	C₂×C₄×D₅₃	1	4	yes	—
310	#41	C₂²×Dic₅₃	1	4	yes	—
311	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	C₂×C₄×D₅₃	1	2	yes	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	53	2	no	—
317	C₂²×C₄	—	106	4	no	—
318	C₂²×C₄	—	106	1	no	—
319	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
320	Q₁₆	C₄×Dic₅₃	1	4	yes	—
321	SD₁₆	C₄×Dic₅₃	1	8	yes	—
322	D₈	C₄×Dic₅₃	1	4	yes	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	8	no	—
325	M₁₆	—	53	4	no	—
326	M₁₆	—	106	8	no	—
327	M₁₆	—	106	2	no	—
328	M₁₆	C₂²×Dic₅₃	1	4	yes	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	8	no	—
331	C₂×C₈	—	53	2	no	—
332	C₂×C₈	—	53	1	no	—
333	C₂×C₈	—	53	1	no	—
334	C₂×C₈	—	53	2	no	—
335	C₂×C₈	—	53	2	no	—
336	C₂×C₈	—	106	8	no	—
337	C₂×C₈	—	106	2	no	—
338	C₂×C₈	—	106	1	no	—
339	C₂×C₈	—	106	1	no	—
340	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
341	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
342	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
343	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
344	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
345	C₂×C₈	#11719947	1	2	yes	—
346	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
347	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
348	C₄⋊C₄	—	53	4	no	—
349	C₄⋊C₄	—	106	4	no	—
350	C₂²⋊C₄	—	106	4	no	—
351	C₂²⋊C₄	—	106	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	53	2	no	—
354	C₄²	—	53	1	no	—
355	C₄²	—	53	1	no	—
356	C₄²	—	53	2	no	—
357	C₄²	—	106	4	no	—
358	C₄²	—	106	2	no	—
359	C₄²	—	106	1	no	—
360	C₄²	C₂²×Dic₅₃	1	2	yes	—
361	C₂³	—	2	2	no	—
362	C₂³	#41×D₅₃	1	1	yes	characteristic
363	Q₈	—	2	2	no	—
364	Q₈	C₂×C₄×Dic₅₃	1	4	yes	—
365	D₄	—	2	4	no	—
366	D₄	—	2	4	no	—
367	D₄	—	2	2	no	—
368	D₄	C₂×C₄×Dic₅₃	1	4	yes	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	53	2	no	—
375	C₂×C₄	—	53	2	no	—
376	C₂×C₄	—	106	4	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	—	424	4	no	—
379	C₂×C₄	—	212	2	no	—
380	C₂×C₄	—	212	2	no	—
381	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
383	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
384	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
385	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
386	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
387	C₈	—	2	4	no	—
388	C₈	—	53	2	no	—
389	C₈	—	53	2	no	—
390	C₈	—	106	4	no	—
391	C₈	—	106	2	no	—
392	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
393	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
394	C₈	D₄×Dic₅₃	1	2	yes	—
395	C₈	D₄×Dic₅₃	1	2	yes	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
400	C₂²	C₄×#11719947	1	1	yes	characteristic
401	C₂²	#41×Dic₅₃	1	1	yes	characteristic
402	C₄	—	2	4	no	—
403	C₄	—	4	2	no	—
404	C₄	—	106	2	no	—
405	C₄	—	106	2	no	—
406	C₄	—	212	2	no	—
407	C₄	—	212	4	no	—
408	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
409	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
410	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
411	C₄	C₄×#11719947	1	1	yes	characteristic
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	4	4	no	—
415	C₂	C₄×D₄×Dic₅₃	1	1	yes	characteristic
416	C₂	#20283627	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#20514
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
3. #20283141	C₂	semidirect
4. #108×Dic₅₃	C₂	semidirect
5. #20274597	C₂	semidirect
6. #20274594	C₂	semidirect
7. #20274560	C₂	semidirect
8. #20274557	C₂	semidirect
9. #20274527	C₂	semidirect
10. #20274401	C₂	semidirect
14. #16810740	C₄	semidirect
15. Q₁₆×Dic₅₃	C₄	semidirect
16. SD₁₆×Dic₅₃	C₄	semidirect
17. D₈×Dic₅₃	C₄	semidirect
40. C₅₃×#443	C₄	semidirect
45. #13133224	C₂×C₄	semidirect
59. #13133153	C₂×C₄	semidirect
62. #13133144	C₂×C₄	semidirect
79. C₈×Dic₅₃	C₂×C₄	semidirect
88. #131×C₅₃	C₂×C₄	semidirect
95. #108×C₅₃	C₂×C₄	semidirect
96. #104×C₅₃	C₂×C₄	semidirect
97. C₄×C₈×C₅₃	C₂×C₄	semidirect
225. #443	Dic₅₃	semidirect
252. C₅₃	#20514	semidirect
262. #131	C₂×Dic₅₃	semidirect
283. #108	C₂×Dic₅₃	semidirect
287. #104	C₂×Dic₅₃	semidirect
305. C₄×C₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810740	2
#135×Dic₅₃	2
#20283141	2
#108×Dic₅₃	2
#20274597	2
#20274594	2
#20274560	2
#20274557	2
#20274527	2
#20274401	2
C₂×C₅₃×#443	2
#20514	53