Subgroups — #27382115

Order 13568.

1042 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810670	C₂	1	1	yes	maximal, characteristic
2	#20283303	C₂	1	1	yes	maximal, characteristic
3	#20283131	C₂	1	2	yes	maximal
4	C₂×C₄×#11719912	C₂	1	1	yes	maximal, characteristic
5	#20274602	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274601	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274600	C₂	1	2	yes	maximal, complemented
8	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274593	C₂	1	2	yes	maximal, complemented
10	#339×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719912	C₂²	1	1	yes	characteristic
13	C₂²×#11719912	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
15	#117×D₅₃	C₂²	1	2	yes	—
16	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
17	#16810670	C₄	1	8	yes	complemented
18	C₂×#13133153	C₂²	1	1	yes	characteristic
19	C₂×#13133152	C₂²	1	2	yes	—
20	#16810664	C₄	1	4	yes	complemented
21	#16810662	C₄	1	4	yes	complemented
22	#16810661	C₂²	1	1	yes	characteristic
23	#16810659	C₂²	1	2	yes	—
24	#16810581	C₂²	1	2	yes	—
25	#16810580	—	2	2	no	—
26	#16810579	C₂²	1	2	yes	—
27	#16810578	C₂²	1	2	yes	—
28	#16810577	—	2	2	no	—
29	#16810575	—	2	2	no	—
30	C₄×#11719912	—	2	2	no	—
31	C₄×#11719912	C₄	1	8	yes	complemented
32	C₂×#13133127	C₂²	1	1	yes	characteristic
33	C₂×#13133127	C₂²	1	1	yes	characteristic
34	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
35	#16810384	C₂²	1	2	yes	—
36	#16810383	C₂²	1	2	yes	—
37	#16810382	C₂²	1	2	yes	—
38	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#427	C₂²	1	1	yes	characteristic
40	C₅₃×#405	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#339	C₂²	1	2	yes	—
43	C₂²×C₄×D₅₃	—	2	1	no	—
44	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
45	C₂×#11719917	C₂³	1	2	yes	—
46	C₂×#11719912	—	2	2	no	—
47	C₂×#11719912	C₂×C₄	1	4	yes	—
48	C₂×#11719912	C₂×C₄	1	2	yes	—
49	C₂×#11719912	C₂×C₄	1	2	yes	—
50	C₂×#11719912	C₂×C₄	1	4	yes	—
51	C₂×#11719912	C₂×C₄	1	4	yes	—
52	C₂×#11719912	D₄	1	2	yes	—
53	C₂×#11719912	D₄	1	2	yes	—
54	C₂×#11719912	Q₈	1	4	yes	—
55	M₁₆×D₅₃	—	2	4	no	—
56	M₁₆×D₅₃	C₂×C₄	1	8	yes	—
57	C₂²×#88322	C₂³	1	1	yes	characteristic
58	C₂²×#88322	C₂³	1	1	yes	characteristic
59	C₂×C₈×D₅₃	—	2	2	no	—
60	#13133176	C₂³	1	2	yes	—
61	#13133175	C₂×C₄	1	4	yes	—
62	#13133174	—	2	2	no	—
63	#13133173	—	2	2	no	—
64	C₄⋊C₄×D₅₃	C₂×C₄	1	4	yes	—
65	C₂²⋊C₄×D₅₃	—	2	2	no	—
66	C₂×C₄×Dic₅₃	—	2	1	no	—
67	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₄²×D₅₃	—	2	2	no	—
69	C₄²×D₅₃	—	2	2	no	—
70	C₄²×D₅₃	C₂×C₄	1	4	yes	—
71	#13133153	C₂×C₄	1	4	yes	complemented
72	#13133152	C₂×C₄	1	8	yes	complemented
73	#13133151	C₂×C₄	1	2	yes	complemented
74	#13133150	C₂×C₄	1	4	yes	complemented
75	#13133149	C₂×C₄	1	2	yes	complemented
76	#13133148	C₂×C₄	1	4	yes	complemented
77	#13133128	—	2	2	no	—
78	#13133127	—	2	2	no	—
79	#13133127	—	2	2	no	—
80	#13133127	C₂×C₄	1	4	yes	complemented
81	#13133127	C₂×C₄	1	4	yes	complemented
82	C₄×#88322	—	2	2	no	—
83	C₄×#88322	C₂×C₄	1	4	yes	complemented
84	C₈×Dic₅₃	—	2	2	no	—
85	C₈×Dic₅₃	—	2	2	no	—
86	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
87	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	#117×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	#108×C₅₃	C₂×C₄	1	4	yes	complemented
92	#107×C₅₃	C₂×C₄	1	2	yes	complemented
93	#106×C₅₃	C₂×C₄	1	2	yes	complemented
94	#97×C₅₃	—	2	2	no	—
95	C₄×C₈×C₅₃	—	2	2	no	—
96	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
97	C₂³×D₅₃	#41	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂²×Dic₅₃	—	2	1	no	—
101	C₂²×Dic₅₃	#41	1	1	yes	characteristic
102	C₂×C₄×D₅₃	—	2	4	no	—
103	C₂×C₄×D₅₃	—	2	4	no	—
104	C₂×C₄×D₅₃	—	4	4	no	—
105	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
110	C₂×C₄×D₅₃	#41	1	2	yes	—
111	M₁₆×C₅₃	—	2	2	no	—
112	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
122	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	1	no	—
126	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂²⋊Dic₅₃	—	2	2	no	—
128	#11719921	—	2	4	no	—
129	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
130	#11719919	—	2	4	no	—
131	C₄×Dic₅₃	—	2	2	no	—
132	C₄×Dic₅₃	—	2	1	no	—
133	C₄×Dic₅₃	—	2	4	no	—
134	C₄×Dic₅₃	—	2	4	no	—
135	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
136	#11719917	—	2	2	no	—
137	#11719917	C₂²×C₄	1	4	yes	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	C₂²×C₄	1	2	yes	—
140	C₂×#88322	C₂²×C₄	1	2	yes	—
141	C₂×#88322	C₂²×C₄	1	2	yes	—
142	C₂×#88322	C₂²×C₄	1	2	yes	—
143	C₂×#88322	C₂×D₄	1	2	yes	—
144	C₂×#88322	C₂×Q₈	1	2	yes	—
145	#11719912	—	2	8	no	—
146	#11719912	C₄²	1	8	yes	—
147	#11719912	C₄²	1	8	yes	—
148	#11719912	C₄⋊C₄	1	8	yes	—
149	#11719912	C₄⋊C₄	1	8	yes	—
150	C₈×D₅₃	—	2	8	no	—
151	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
152	C₂²×D₅₃	—	4	2	no	—
153	C₂²×D₅₃	#117	1	2	yes	—
154	C₂²×D₅₃	C₄×D₄	1	2	yes	—
155	C₂²×D₅₃	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂×Dic₅₃	—	2	2	no	—
166	C₂×Dic₅₃	—	2	4	no	—
167	C₂×Dic₅₃	—	4	2	no	—
168	C₂×Dic₅₃	#117	1	2	yes	—
169	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
170	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
171	C₄×D₅₃	—	4	8	no	—
172	C₄×D₅₃	—	8	4	no	—
173	C₄×D₅₃	C₂×C₄²	1	4	yes	—
174	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
175	C₄×D₅₃	C₄×D₄	1	4	yes	—
176	C₄×D₅₃	C₄×Q₈	1	4	yes	—
177	C₈×C₅₃	—	2	4	no	—
178	C₈×C₅₃	C₂×C₄²	1	2	yes	—
179	C₈×C₅₃	C₂×C₄²	1	2	yes	—
180	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
181	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
182	#88322	—	2	4	no	—
183	#88322	C₂×C₄²	1	4	yes	—
184	#88322	C₂×C₄⋊C₄	1	4	yes	—
185	#20445	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	—	2	2	no	—
187	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
190	C₂×D₅₃	—	2	4	no	—
191	C₂×D₅₃	—	4	2	no	—
192	C₂×D₅₃	—	4	2	no	—
193	C₂×D₅₃	C₄×C₄⋊C₄	1	4	yes	—
194	C₄×C₅₃	—	2	4	no	—
195	C₄×C₅₃	—	4	4	no	—
196	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
197	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
198	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
199	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
200	Dic₅₃	—	4	2	no	—
201	Dic₅₃	—	4	4	no	—
202	Dic₅₃	C₄×C₄⋊C₄	1	4	yes	—
203	C₂×#430	—	53	1	no	—
204	#2911	—	53	1	no	—
205	#2873	—	53	2	no	—
206	C₂×C₄×M₁₆	—	53	1	no	—
207	#1773	—	53	1	no	—
208	#1773	—	53	1	no	—
209	#1773	—	53	2	no	—
210	#1772	—	53	1	no	—
211	#1772	—	53	2	no	—
212	#1772	D₅₃	1	1	yes	characteristic, complemented
213	C₂×#339	—	53	2	no	—
214	C₂×C₅₃	—	2	1	no	—
215	C₂×C₅₃	—	2	1	no	—
216	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₅₃	#2956	1	2	yes	—
218	D₅₃	—	2	4	no	—
219	C₂²×M₁₆	—	53	1	no	—
220	C₂²×M₁₆	—	53	1	no	—
221	C₂²×M₁₆	—	53	2	no	—
222	C₂×#117	—	53	2	no	—
223	C₂²×C₄²	—	53	1	no	—
224	#430	—	53	8	no	—
225	C₂×#108	—	53	2	no	—
226	C₂×#108	—	53	1	no	—
227	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
228	#428	—	53	4	no	—
229	#428	—	53	4	no	—
230	#427	—	53	1	no	—
231	#427	—	53	2	no	—
232	#427	C₂×D₅₃	1	1	yes	characteristic
233	#405	—	53	2	no	—
234	#405	—	53	2	no	—
235	#405	—	53	2	no	—
236	#405	—	106	2	no	—
237	#405	—	106	2	no	—
238	#405	C₂×D₅₃	1	2	yes	—
239	C₄×M₁₆	—	53	8	no	—
240	C₄×M₁₆	—	106	2	no	—
241	C₄×M₁₆	—	106	2	no	—
242	C₂×#97	—	53	1	no	—
243	C₂×#97	—	53	1	no	—
244	C₂×C₄×C₈	—	53	1	no	—
245	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
246	#339	—	53	2	no	—
247	#339	—	53	2	no	—
248	#339	—	53	2	no	—
249	#339	C₂×D₅₃	1	2	yes	—
250	C₅₃	#20445	1	1	yes	characteristic, complemented, Sylow
251	C₂³×C₄	—	53	1	no	—
252	C₂³×C₄	—	106	1	no	—
253	C₂×M₁₆	—	53	4	no	—
254	C₂×M₁₆	—	53	4	no	—
255	C₂×M₁₆	—	53	2	no	—
256	C₂×M₁₆	—	53	2	no	—
257	C₂×M₁₆	—	53	4	no	—
258	C₂×M₁₆	—	53	2	no	—
259	C₂×M₁₆	—	53	2	no	—
260	C₂×M₁₆	—	53	4	no	—
261	C₂×M₁₆	—	53	2	no	—
262	C₂×M₁₆	—	53	8	no	—
263	C₂×M₁₆	—	106	2	no	—
264	C₂×M₁₆	—	106	4	no	—
265	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
266	C₂²×C₈	—	53	1	no	—
267	C₂²×C₈	—	53	1	no	—
268	C₂²×C₈	—	106	2	no	—
269	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
271	#117	—	53	2	no	—
272	#117	—	53	4	no	—
273	#117	—	106	2	no	—
274	#117	—	106	2	no	—
275	#117	C₂²×D₅₃	1	2	yes	—
276	C₂×C₄⋊C₄	—	53	4	no	—
277	C₂×C₂²⋊C₄	—	106	2	no	—
278	C₂×C₄²	—	53	1	no	—
279	C₂×C₄²	—	53	4	no	—
280	C₂×C₄²	—	106	2	no	—
281	C₂×C₄²	—	106	2	no	—
282	C₂×C₄²	—	106	1	no	—
283	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
284	#108	—	53	8	no	—
285	#108	—	53	4	no	—
286	#108	C₄×D₅₃	1	4	yes	complemented
287	#107	—	53	2	no	—
288	#107	—	53	4	no	—
289	#107	C₄×D₅₃	1	2	yes	complemented
290	#106	—	53	2	no	—
291	#106	—	53	4	no	—
292	#106	C₄×D₅₃	1	2	yes	complemented
293	#97	—	2	2	no	—
294	#97	—	53	4	no	—
295	#97	—	53	4	no	—
296	#97	—	106	2	no	—
297	#97	—	106	2	no	—
298	#97	—	106	2	no	—
299	C₄×C₈	—	2	2	no	—
300	C₄×C₈	—	53	4	no	—
301	C₄×C₈	—	106	2	no	—
302	C₄×C₈	—	106	2	no	—
303	C₄×C₈	—	106	2	no	—
304	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
305	C₂⁴	—	53	1	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	53	1	no	—
308	C₂²×C₄	—	53	2	no	—
309	C₂²×C₄	—	53	2	no	—
310	C₂²×C₄	—	53	2	no	—
311	C₂²×C₄	—	53	2	no	—
312	C₂²×C₄	—	53	2	no	—
313	C₂²×C₄	—	53	2	no	—
314	C₂²×C₄	—	106	4	no	—
315	C₂²×C₄	—	106	4	no	—
316	C₂²×C₄	—	106	1	no	—
317	C₂²×C₄	—	212	4	no	—
318	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
319	M₁₆	—	2	2	no	—
320	M₁₆	—	53	8	no	—
321	M₁₆	—	53	8	no	—
322	M₁₆	—	53	8	no	—
323	M₁₆	—	53	8	no	—
324	M₁₆	—	53	4	no	—
325	M₁₆	—	106	8	no	—
326	M₁₆	—	106	2	no	—
327	M₁₆	C₂×C₄×D₅₃	1	4	yes	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	53	2	no	—
330	C₂×C₈	—	53	2	no	—
331	C₂×C₈	—	53	2	no	—
332	C₂×C₈	—	53	2	no	—
333	C₂×C₈	—	53	2	no	—
334	C₂×C₈	—	53	2	no	—
335	C₂×C₈	—	106	8	no	—
336	C₂×C₈	—	106	2	no	—
337	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
338	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
339	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
340	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
341	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
342	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
343	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
344	C₂×C₈	Q₈×D₅₃	1	2	yes	—
345	C₄⋊C₄	—	53	4	no	—
346	C₄⋊C₄	—	106	4	no	—
347	C₄⋊C₄	C₂×C₄×D₅₃	1	4	yes	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	106	4	no	—
350	C₂²⋊C₄	—	106	2	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	53	2	no	—
354	C₄²	—	106	4	no	—
355	C₄²	—	106	4	no	—
356	C₄²	—	106	2	no	—
357	C₄²	—	106	1	no	—
358	C₄²	C₂×C₄×D₅₃	1	2	yes	—
359	C₂³	—	53	2	no	—
360	C₂³	—	53	2	no	—
361	C₂³	—	53	2	no	—
362	C₂³	—	212	2	no	—
363	C₂³	#41×D₅₃	1	1	yes	characteristic
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	53	2	no	—
368	C₂×C₄	—	53	2	no	—
369	C₂×C₄	—	53	2	no	—
370	C₂×C₄	—	53	4	no	—
371	C₂×C₄	—	53	4	no	—
372	C₂×C₄	—	53	4	no	—
373	C₂×C₄	—	53	4	no	—
374	C₂×C₄	—	106	2	no	—
375	C₂×C₄	—	106	4	no	—
376	C₂×C₄	—	424	4	no	—
377	C₂×C₄	—	212	8	no	—
378	C₂×C₄	—	212	2	no	—
379	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
383	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
384	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	53	4	no	—
387	C₈	—	53	4	no	—
388	C₈	—	106	4	no	—
389	C₈	C₄²×D₅₃	1	2	yes	—
390	C₈	C₄²×D₅₃	1	2	yes	—
391	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
392	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
393	C₂²	—	2	2	no	—
394	C₂²	—	53	4	no	—
395	C₂²	—	106	4	no	—
396	C₂²	—	212	2	no	—
397	C₂²	—	212	2	no	—
398	C₂²	#117×D₅₃	1	1	yes	characteristic
399	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
400	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
401	C₄	—	2	4	no	—
402	C₄	—	4	4	no	—
403	C₄	—	53	4	no	—
404	C₄	—	212	2	no	—
405	C₄	—	212	4	no	—
406	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
407	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
408	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
409	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	106	4	no	—
413	C₂	C₄×C₄⋊C₄×D₅₃	1	1	yes	characteristic
414	C₂	#20283620	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×D₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#20445
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
5. #20274602	C₂	semidirect
6. #20274601	C₂	semidirect
7. #20274600	C₂	semidirect
8. #20274594	C₂	semidirect
9. #20274593	C₂	semidirect
11. C₅₃×#1772	C₂	semidirect
17. #16810670	C₄	semidirect
20. #16810664	C₄	semidirect
21. #16810662	C₄	semidirect
31. C₄×#11719912	C₄	semidirect
71. #13133153	C₂×C₄	semidirect
72. #13133152	C₂×C₄	semidirect
73. #13133151	C₂×C₄	semidirect
74. #13133150	C₂×C₄	semidirect
75. #13133149	C₂×C₄	semidirect
76. #13133148	C₂×C₄	semidirect
80. #13133127	C₂×C₄	semidirect
81. #13133127	C₂×C₄	semidirect
83. C₄×#88322	C₂×C₄	semidirect
91. #108×C₅₃	C₂×C₄	semidirect
92. #107×C₅₃	C₂×C₄	semidirect
93. #106×C₅₃	C₂×C₄	semidirect
96. C₄×C₈×C₅₃	C₂×C₄	semidirect
212. #1772	D₅₃	semidirect
250. C₅₃	#20445	semidirect
286. #108	C₄×D₅₃	semidirect
289. #107	C₄×D₅₃	semidirect
292. #106	C₄×D₅₃	semidirect
304. C₄×C₈	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810670	2
#20283303	2
#20283131	2
C₂×C₄×#11719912	2
#20274602	2
#20274601	2
#20274600	2
#20274594	2
#20274593	2
#339×D₅₃	2
C₅₃×#1772	2
#20445	53