Subgroups — #27382122

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#427×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20283131	C₂	1	2	yes	maximal
4	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
5	#20274600	C₂	1	2	yes	maximal, complemented
6	#20274598	C₂	1	2	yes	maximal, complemented
7	#20274597	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274596	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274423	C₂	1	2	yes	maximal
11	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719912	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×#13133174	C₂²	1	1	yes	characteristic
16	#117×D₅₃	C₂²	1	2	yes	—
17	C₂×#13133153	C₂²	1	1	yes	characteristic
18	C₂×#13133152	C₂²	1	2	yes	—
19	#108×D₅₃	C₄	1	8	yes	complemented
20	#16810661	C₂²	1	1	yes	characteristic
21	#16810660	C₄	1	2	yes	complemented
22	#16810659	C₂²	1	2	yes	—
23	#16810658	C₄	1	2	yes	complemented
24	#107×D₅₃	C₄	1	2	yes	complemented
25	#106×D₅₃	C₄	1	2	yes	complemented
26	#16810581	C₂²	1	2	yes	—
27	#16810580	—	2	2	no	—
28	#16810579	C₂²	1	2	yes	—
29	#16810578	C₂²	1	2	yes	—
30	#16810577	—	2	2	no	—
31	#16810577	C₄	1	8	yes	complemented
32	#16810575	—	2	2	no	—
33	C₄×#11719912	—	2	2	no	—
34	C₂×#13133128	C₂²	1	1	yes	characteristic
35	C₂×#13133127	C₂²	1	1	yes	characteristic
36	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
37	#16810384	C₂²	1	2	yes	—
38	#16810383	C₂²	1	2	yes	—
39	#16810382	C₂²	1	2	yes	—
40	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#427	C₂²	1	1	yes	characteristic
42	C₅₃×#405	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#339	C₂²	1	2	yes	—
45	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
46	C₂×#11719917	C₂³	1	2	yes	—
47	C₂×#11719912	—	2	2	no	—
48	C₂×#11719912	C₂×C₄	1	4	yes	—
49	C₂×#11719912	D₄	1	1	yes	characteristic
50	C₂×#11719912	D₄	1	1	yes	characteristic
51	C₂×#11719912	D₄	1	1	yes	characteristic
52	C₂×#11719912	D₄	1	1	yes	characteristic
53	C₂×#11719912	Q₈	1	4	yes	—
54	M₁₆×D₅₃	—	2	4	no	—
55	M₁₆×D₅₃	C₂×C₄	1	8	yes	—
56	C₂²×#88322	C₂³	1	1	yes	characteristic
57	C₂²×#88322	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₅₃	—	2	2	no	—
59	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
60	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
65	#13133176	C₂³	1	2	yes	—
66	#13133175	C₂×C₄	1	4	yes	—
67	#13133174	—	2	2	no	—
68	#13133174	—	2	2	no	—
69	#13133174	C₂×C₄	1	4	yes	—
70	#13133173	—	2	2	no	—
71	C₂×#11719919	—	2	1	no	—
72	C₄⋊C₄×D₅₃	C₂×C₄	1	4	yes	—
73	C₂×#11719921	—	2	1	no	—
74	C₂²⋊C₄×D₅₃	—	2	2	no	—
75	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
76	C₄²×D₅₃	—	2	2	no	—
77	#13133153	C₂×C₄	1	4	yes	complemented
78	#13133152	C₂×C₄	1	8	yes	complemented
79	#13133151	C₂×C₄	1	2	yes	—
80	#13133150	C₂×C₄	1	4	yes	—
81	#13133149	C₂×C₄	1	2	yes	—
82	#13133148	C₂×C₄	1	4	yes	—
83	#13133128	—	2	2	no	—
84	#13133128	C₂×C₄	1	4	yes	complemented
85	#13133127	—	2	2	no	—
86	#13133127	—	2	2	no	—
87	#13133127	C₂×C₄	1	4	yes	complemented
88	C₄×#88322	—	2	2	no	—
89	C₈×Dic₅₃	—	2	2	no	—
90	C₈×Dic₅₃	—	2	2	no	—
91	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
92	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
93	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	#117×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	#108×C₅₃	C₂×C₄	1	4	yes	complemented
98	#107×C₅₃	C₂×C₄	1	2	yes	—
99	#106×C₅₃	C₂×C₄	1	2	yes	—
100	#97×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	—	2	2	no	—
102	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
103	C₂³×D₅₃	#41	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₂²×Dic₅₃	—	2	1	no	—
107	C₂²×Dic₅₃	#41	1	1	yes	characteristic
108	C₂×C₄×D₅₃	—	2	4	no	—
109	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
110	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
116	C₂×C₄×D₅₃	#41	1	2	yes	—
117	M₁₆×C₅₃	—	2	2	no	—
118	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
128	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
129	C₂²⋊C₄×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	1	no	—
132	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
133	C₂²⋊Dic₅₃	—	2	2	no	—
134	#11719921	—	2	4	no	—
135	#11719921	—	2	4	no	—
136	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
137	#11719919	—	2	4	no	—
138	#11719919	—	2	4	no	—
139	C₄×Dic₅₃	—	2	2	no	—
140	C₄×Dic₅₃	—	2	1	no	—
141	C₄×Dic₅₃	—	2	4	no	—
142	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
143	#11719917	—	2	2	no	—
144	#11719917	C₂²×C₄	1	4	yes	—
145	C₂×#88322	—	2	2	no	—
146	C₂×#88322	C₂²×C₄	1	2	yes	—
147	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
148	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
149	C₂×#88322	C₂²×C₄	1	2	yes	—
150	C₂×#88322	C₂²×C₄	1	2	yes	—
151	C₂×#88322	C₂×D₄	1	1	yes	characteristic
152	C₂×#88322	C₂×D₄	1	1	yes	characteristic
153	C₂×#88322	C₂×Q₈	1	2	yes	—
154	#11719912	—	2	8	no	—
155	#11719912	C₄⋊C₄	1	4	yes	—
156	#11719912	C₄⋊C₄	1	4	yes	—
157	#11719912	C₄⋊C₄	1	4	yes	—
158	#11719912	C₄⋊C₄	1	4	yes	—
159	C₈×D₅₃	—	2	8	no	—
160	C₈×D₅₃	C₄²	1	4	yes	—
161	C₈×D₅₃	C₄²	1	4	yes	—
162	C₈×D₅₃	C₄²	1	4	yes	—
163	C₈×D₅₃	C₄²	1	4	yes	—
164	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂²×D₅₃	—	4	2	no	—
166	C₂²×D₅₃	#117	1	2	yes	—
167	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
168	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
169	C₂²×D₅₃	C₄×Q₈	1	2	yes	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₂×Dic₅₃	—	2	2	no	—
180	C₂×Dic₅₃	—	2	4	no	—
181	C₂×Dic₅₃	—	4	2	no	—
182	C₂×Dic₅₃	#117	1	2	yes	—
183	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
185	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
186	C₄×D₅₃	—	8	4	no	—
187	C₄×D₅₃	C₂×C₄²	1	2	yes	—
188	C₄×D₅₃	C₂×C₄²	1	2	yes	—
189	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
190	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
191	C₄×D₅₃	C₄×D₄	1	2	yes	—
192	C₄×D₅₃	C₄×D₄	1	2	yes	—
193	C₄×D₅₃	C₄×Q₈	1	2	yes	—
194	C₄×D₅₃	C₄×Q₈	1	2	yes	—
195	C₈×C₅₃	—	2	4	no	—
196	C₈×C₅₃	C₂×C₄²	1	2	yes	—
197	C₈×C₅₃	C₂×C₄²	1	2	yes	—
198	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
199	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
200	#88322	—	2	4	no	—
201	#88322	C₂×C₄²	1	2	yes	—
202	#88322	C₂×C₄²	1	2	yes	—
203	#88322	C₂×C₄⋊C₄	1	2	yes	—
204	#88322	C₂×C₄⋊C₄	1	2	yes	—
205	#20446	—	53	1	no	maximal, Sylow
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
208	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
210	C₂×D₅₃	—	2	2	no	—
211	C₂×D₅₃	—	2	2	no	—
212	C₂×D₅₃	—	4	2	no	—
213	C₂×D₅₃	—	4	2	no	—
214	C₂×D₅₃	C₄×C₄⋊C₄	1	2	yes	—
215	C₂×D₅₃	C₄×C₄⋊C₄	1	2	yes	—
216	C₄×C₅₃	—	4	2	no	—
217	C₄×C₅₃	—	4	4	no	—
218	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
219	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
220	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
221	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
222	Dic₅₃	—	4	2	no	—
223	Dic₅₃	—	4	4	no	—
224	Dic₅₃	C₄×C₄⋊C₄	1	2	yes	—
225	Dic₅₃	C₄×C₄⋊C₄	1	2	yes	—
226	C₂²×#108	—	53	1	no	—
227	C₂×#427	—	53	1	no	—
228	#2873	—	53	2	no	—
229	C₂×#405	—	53	1	no	—
230	#1773	—	53	2	no	—
231	#1773	—	53	2	no	—
232	#1772	—	53	1	no	—
233	#1772	—	53	1	no	—
234	#1772	—	53	1	no	—
235	#1772	D₅₃	1	1	yes	characteristic, complemented
236	#1733	—	53	2	no	—
237	C₂×C₅₃	—	2	1	no	—
238	C₂×C₅₃	—	2	1	no	—
239	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₅₃	#2957	1	2	yes	—
241	D₅₃	—	2	2	no	—
242	D₅₃	—	2	2	no	—
243	C₂²×M₁₆	—	53	1	no	—
244	C₂²×M₁₆	—	53	2	no	—
245	C₂³×C₈	—	53	1	no	—
246	C₂×#117	—	53	1	no	—
247	C₂×#117	—	53	2	no	—
248	C₂×#108	—	53	2	no	—
249	C₂×#108	—	53	1	no	—
250	C₂×#108	—	53	8	no	—
251	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
252	#427	—	53	1	no	—
253	#427	—	53	2	no	—
254	#427	—	53	2	no	—
255	#427	—	53	2	no	—
256	#427	C₂×D₅₃	1	1	yes	characteristic
257	C₂×#107	—	53	2	no	—
258	C₂×#106	—	53	2	no	—
259	#405	—	53	8	no	—
260	#405	—	53	2	no	—
261	#405	—	53	2	no	—
262	#405	—	53	2	no	—
263	#405	—	106	2	no	—
264	#405	—	106	2	no	—
265	#405	C₂×D₅₃	1	2	yes	—
266	C₄×M₁₆	—	106	2	no	—
267	C₄×M₁₆	—	106	2	no	—
268	C₂×#97	—	53	1	no	—
269	C₂×#97	—	53	1	no	—
270	C₂×C₄×C₈	—	53	1	no	—
271	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
272	#339	—	53	2	no	—
273	#339	—	53	2	no	—
274	#339	—	53	2	no	—
275	#339	C₂×D₅₃	1	2	yes	—
276	C₅₃	#20446	1	1	yes	characteristic, complemented, Sylow
277	C₂³×C₄	—	53	1	no	—
278	C₂×M₁₆	—	53	4	no	—
279	C₂×M₁₆	—	53	1	no	—
280	C₂×M₁₆	—	53	1	no	—
281	C₂×M₁₆	—	53	1	no	—
282	C₂×M₁₆	—	53	1	no	—
283	C₂×M₁₆	—	53	4	no	—
284	C₂×M₁₆	—	53	2	no	—
285	C₂×M₁₆	—	53	8	no	—
286	C₂×M₁₆	—	106	2	no	—
287	C₂×M₁₆	—	106	4	no	—
288	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
289	C₂²×C₈	—	53	1	no	—
290	C₂²×C₈	—	53	1	no	—
291	C₂²×C₈	—	53	4	no	—
292	C₂²×C₈	—	53	1	no	—
293	C₂²×C₈	—	53	1	no	—
294	C₂²×C₈	—	53	1	no	—
295	C₂²×C₈	—	53	1	no	—
296	C₂²×C₈	—	53	4	no	—
297	C₂²×C₈	—	106	2	no	—
298	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
299	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
300	#117	—	53	4	no	—
301	#117	—	53	2	no	—
302	#117	—	53	4	no	—
303	#117	—	106	2	no	—
304	#117	—	106	2	no	—
305	#117	—	106	2	no	—
306	#117	C₂²×D₅₃	1	2	yes	—
307	C₂×C₄⋊C₄	—	53	4	no	—
308	C₂×C₄⋊C₄	—	106	1	no	—
309	C₂×C₂²⋊C₄	—	106	2	no	—
310	C₂×C₂²⋊C₄	—	106	1	no	—
311	C₂×C₄²	—	53	1	no	—
312	C₂×C₄²	—	106	2	no	—
313	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
314	#108	—	53	8	no	—
315	#108	—	53	4	no	—
316	#108	C₄×D₅₃	1	4	yes	complemented
317	#107	—	53	2	no	—
318	#107	—	53	4	no	—
319	#107	C₄×D₅₃	1	2	yes	—
320	#106	—	53	2	no	—
321	#106	—	53	4	no	—
322	#106	C₄×D₅₃	1	2	yes	—
323	#97	—	2	2	no	—
324	#97	—	53	4	no	—
325	#97	—	53	4	no	—
326	#97	—	106	2	no	—
327	#97	—	106	2	no	—
328	#97	—	106	2	no	—
329	C₄×C₈	—	2	2	no	—
330	C₄×C₈	—	53	4	no	—
331	C₄×C₈	—	106	2	no	—
332	C₄×C₈	—	106	2	no	—
333	C₄×C₈	—	106	2	no	—
334	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
335	C₂⁴	—	53	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	53	1	no	—
338	C₂²×C₄	—	53	1	no	—
339	C₂²×C₄	—	53	1	no	—
340	C₂²×C₄	—	53	2	no	—
341	C₂²×C₄	—	53	2	no	—
342	C₂²×C₄	—	53	1	no	—
343	C₂²×C₄	—	53	1	no	—
344	C₂²×C₄	—	53	2	no	—
345	C₂²×C₄	—	53	2	no	—
346	C₂²×C₄	—	106	4	no	—
347	C₂²×C₄	—	106	1	no	—
348	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
349	M₁₆	—	2	2	no	—
350	M₁₆	—	53	4	no	—
351	M₁₆	—	53	4	no	—
352	M₁₆	—	53	4	no	—
353	M₁₆	—	53	4	no	—
354	M₁₆	—	53	4	no	—
355	M₁₆	—	106	8	no	—
356	M₁₆	—	106	2	no	—
357	M₁₆	C₂×C₄×D₅₃	1	4	yes	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	53	2	no	—
360	C₂×C₈	—	53	2	no	—
361	C₂×C₈	—	53	1	no	—
362	C₂×C₈	—	53	1	no	—
363	C₂×C₈	—	53	2	no	—
364	C₂×C₈	—	53	1	no	—
365	C₂×C₈	—	53	1	no	—
366	C₂×C₈	—	53	2	no	—
367	C₂×C₈	—	53	4	no	—
368	C₂×C₈	—	53	4	no	—
369	C₂×C₈	—	53	4	no	—
370	C₂×C₈	—	53	4	no	—
371	C₂×C₈	—	106	8	no	—
372	C₂×C₈	—	106	2	no	—
373	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
374	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
375	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
376	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
377	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
378	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
379	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
380	C₂×C₈	Q₈×D₅₃	1	2	yes	—
381	C₄⋊C₄	—	53	4	no	—
382	C₄⋊C₄	—	106	4	no	—
383	C₄⋊C₄	—	106	4	no	—
384	C₄⋊C₄	C₂×C₄×D₅₃	1	4	yes	—
385	C₂²⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	106	4	no	—
387	C₂²⋊C₄	—	106	4	no	—
388	C₂²⋊C₄	—	106	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	53	2	no	—
392	C₄²	—	106	4	no	—
393	C₄²	—	106	2	no	—
394	C₄²	—	106	1	no	—
395	C₄²	C₂×C₄×D₅₃	1	2	yes	—
396	C₂³	—	53	1	no	—
397	C₂³	—	53	1	no	—
398	C₂³	—	53	2	no	—
399	C₂³	—	53	2	no	—
400	C₂³	—	212	2	no	—
401	C₂³	#41×D₅₃	1	1	yes	characteristic
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	53	2	no	—
406	C₂×C₄	—	53	1	no	—
407	C₂×C₄	—	53	1	no	—
408	C₂×C₄	—	53	2	no	—
409	C₂×C₄	—	53	2	no	—
410	C₂×C₄	—	53	2	no	—
411	C₂×C₄	—	53	2	no	—
412	C₂×C₄	—	53	2	no	—
413	C₂×C₄	—	53	2	no	—
414	C₂×C₄	—	53	2	no	—
415	C₂×C₄	—	53	2	no	—
416	C₂×C₄	—	53	2	no	—
417	C₂×C₄	—	106	2	no	—
418	C₂×C₄	—	106	4	no	—
419	C₂×C₄	—	424	4	no	—
420	C₂×C₄	—	212	2	no	—
421	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
426	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	—	53	2	no	—
429	C₈	—	53	2	no	—
430	C₈	—	53	2	no	—
431	C₈	—	53	2	no	—
432	C₈	—	106	4	no	—
433	C₈	C₄²×D₅₃	1	2	yes	—
434	C₈	C₄²×D₅₃	1	2	yes	—
435	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
436	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
437	C₂²	—	2	2	no	—
438	C₂²	—	53	2	no	—
439	C₂²	—	53	2	no	—
440	C₂²	—	106	2	no	—
441	C₂²	—	106	2	no	—
442	C₂²	—	212	2	no	—
443	C₂²	—	212	2	no	—
444	C₂²	#117×D₅₃	1	1	yes	characteristic
445	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
446	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
447	C₄	—	4	2	no	—
448	C₄	—	4	4	no	—
449	C₄	—	53	2	no	—
450	C₄	—	53	2	no	—
451	C₄	—	212	2	no	—
452	C₄	—	212	4	no	—
453	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
454	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
455	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
456	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
457	C₂	—	2	1	no	—
458	C₂	—	2	1	no	—
459	C₂	—	106	2	no	—
460	C₂	—	106	2	no	—
461	C₂	C₄×C₄⋊C₄×D₅₃	1	1	yes	characteristic
462	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	C₅₃×#1772	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×D₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#20446
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. #20274600	C₂	semidirect
6. #20274598	C₂	semidirect
7. #20274597	C₂	semidirect
8. #20274596	C₂	semidirect
9. #20274594	C₂	semidirect
11. C₅₃×#1772	C₂	semidirect
19. #108×D₅₃	C₄	semidirect
21. #16810660	C₄	semidirect
23. #16810658	C₄	semidirect
24. #107×D₅₃	C₄	semidirect
25. #106×D₅₃	C₄	semidirect
31. #16810577	C₄	semidirect
77. #13133153	C₂×C₄	semidirect
78. #13133152	C₂×C₄	semidirect
84. #13133128	C₂×C₄	semidirect
87. #13133127	C₂×C₄	semidirect
91. C₈×Dic₅₃	C₂×C₄	semidirect
97. #108×C₅₃	C₂×C₄	semidirect
102. C₄×C₈×C₅₃	C₂×C₄	semidirect
235. #1772	D₅₃	semidirect
276. C₅₃	#20446	semidirect
316. #108	C₄×D₅₃	semidirect
334. 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×#108×D₅₃	2
#427×D₅₃	2
#20283131	2
C₂×#16810577	2
#20274600	2
#20274598	2
#20274597	2
#20274596	2
#20274594	2
#20274423	2
C₅₃×#1772	2
#20446	53