Subgroups — #27394713

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283347	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
3	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20275487	C₂	1	4	yes	maximal, complemented
5	#20274918	C₂	1	4	yes	maximal
6	#20274886	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274853	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274852	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
11	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	#16810679	—	2	1	no	—
13	#16810679	—	2	1	no	—
14	#16810675	—	2	1	no	—
15	#16810675	—	2	1	no	—
16	#16810674	C₄	1	2	yes	—
17	#16810674	C₄	1	2	yes	—
18	#16810674	C₄	1	2	yes	—
19	#16810674	C₄	1	2	yes	—
20	C₂×#13133152	C₂²	1	4	yes	—
21	#16810647	C₂²	1	1	yes	characteristic
22	#16810647	C₂²	1	1	yes	characteristic
23	#16810646	C₂²	1	2	yes	—
24	#16810645	—	2	1	no	—
25	#16810645	—	2	1	no	—
26	C₂×#13133146	C₂²	1	2	yes	—
27	C₄⋊C₈×D₅₃	—	2	1	no	—
28	C₄⋊C₈×D₅₃	—	2	1	no	—
29	#16810586	C₂²	1	2	yes	—
30	C₂×#13133129	C₂²	1	2	yes	—
31	M₁₆×Dic₅₃	C₂²	1	2	yes	—
32	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
33	#16810388	C₂²	1	4	yes	—
34	#16810387	C₂²	1	4	yes	—
35	C₂×#108×C₅₃	C₂²	1	4	yes	—
36	C₅₃×#423	C₂²	1	1	yes	characteristic
37	C₅₃×#423	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
40	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
41	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
42	C₂×#11719917	C₂³	1	2	yes	—
43	C₂×#11719917	C₂³	1	2	yes	—
44	C₂×#11719912	—	2	2	no	—
45	M₁₆×D₅₃	—	4	4	no	—
46	C₂²×#88322	C₂³	1	2	yes	—
47	C₂×C₈×D₅₃	—	2	2	no	—
48	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
49	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
50	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
51	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×D₅₃	D₄	1	1	yes	characteristic
56	C₄²×D₅₃	D₄	1	1	yes	characteristic
57	C₄²×D₅₃	D₄	1	1	yes	characteristic
58	C₄²×D₅₃	D₄	1	1	yes	characteristic
59	#13133152	—	4	4	no	—
60	C₄⋊#88322	—	2	1	no	—
61	C₄⋊#88322	—	2	1	no	—
62	C₄⋊#88322	—	2	1	no	—
63	C₄⋊#88322	—	2	1	no	—
64	#13133146	—	2	2	no	—
65	#13133146	—	2	2	no	—
66	#13133146	C₂×C₄	1	4	yes	—
67	#13133146	C₂×C₄	1	4	yes	—
68	#13133129	—	2	2	no	—
69	#13133129	—	2	2	no	—
70	#13133129	C₂×C₄	1	4	yes	—
71	#13133129	C₂×C₄	1	4	yes	—
72	#13133127	—	2	2	no	—
73	C₄×#88322	C₂×C₄	1	2	yes	—
74	C₄×#88322	C₂×C₄	1	2	yes	—
75	C₈×Dic₅₃	—	2	2	no	—
76	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
77	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
78	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	#108×C₅₃	—	4	4	no	—
81	C₄⋊C₈×C₅₃	—	2	1	no	—
82	C₄⋊C₈×C₅₃	—	2	1	no	—
83	C₄⋊C₈×C₅₃	—	2	1	no	—
84	C₄⋊C₈×C₅₃	—	2	1	no	—
85	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
86	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
87	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₄×D₅₃	—	2	2	no	—
95	C₂×C₄×D₅₃	—	2	2	no	—
96	C₂×C₄×D₅₃	—	2	4	no	—
97	C₂×C₄×D₅₃	—	4	2	no	—
98	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
110	M₁₆×C₅₃	—	2	4	no	—
111	M₁₆×C₅₃	—	2	4	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
117	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
129	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
133	#11719917	—	2	4	no	—
134	#11719917	—	2	4	no	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	2	no	—
137	C₂×#88322	—	2	2	no	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	C₂²×C₄	1	4	yes	—
140	#11719912	—	4	4	no	—
141	C₈×D₅₃	—	4	4	no	—
142	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂²×D₅₃	—	2	2	no	—
144	C₂²×D₅₃	—	2	2	no	—
145	C₂²×D₅₃	—	4	1	no	—
146	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
157	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
158	C₂×Dic₅₃	—	2	2	no	—
159	C₂×Dic₅₃	—	2	2	no	—
160	C₂×Dic₅₃	—	4	1	no	—
161	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₄×D₅₃	—	2	2	no	—
168	C₄×D₅₃	—	2	2	no	—
169	C₄×D₅₃	—	2	2	no	—
170	C₄×D₅₃	—	2	1	no	—
171	C₄×D₅₃	—	2	1	no	—
172	C₄×D₅₃	—	2	1	no	—
173	C₄×D₅₃	—	2	1	no	—
174	C₄×D₅₃	—	2	2	no	—
175	C₄×D₅₃	—	2	2	no	—
176	C₄×D₅₃	—	2	2	no	—
177	C₄×D₅₃	—	2	2	no	—
178	C₄×D₅₃	—	4	1	no	—
179	C₄×D₅₃	—	4	1	no	—
180	C₄×D₅₃	#108	1	4	yes	complemented
181	C₈×C₅₃	—	2	4	no	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	—	4	2	no	—
184	#88322	—	2	4	no	—
185	#88322	—	4	2	no	—
186	#88322	—	4	2	no	—
187	#21993	—	53	1	no	maximal, Sylow
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	2	1	no	—
190	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₅₃	#544	1	2	yes	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	2	1	no	—
194	C₂×D₅₃	—	2	1	no	—
195	C₂×D₅₃	—	2	1	no	—
196	C₂×D₅₃	—	2	1	no	—
197	C₂×D₅₃	—	4	4	no	—
198	C₂×D₅₃	C₂×#108	1	2	yes	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	2	1	no	—
201	C₄×C₅₃	C₂×#108	1	2	yes	—
202	C₄×C₅₃	#430	1	2	yes	—
203	C₄×C₅₃	#520	1	2	yes	—
204	C₄×C₅₃	#535	1	1	yes	characteristic
205	C₄×C₅₃	#536	1	1	yes	characteristic
206	Dic₅₃	—	2	1	no	—
207	Dic₅₃	—	2	1	no	—
208	Dic₅₃	—	2	1	no	—
209	Dic₅₃	—	2	1	no	—
210	Dic₅₃	—	2	1	no	—
211	Dic₅₃	C₂×#108	1	2	yes	—
212	#2920	—	53	1	no	—
213	C₂×#431	—	53	1	no	—
214	C₂×#423	—	53	1	no	—
215	#1950	—	53	4	no	—
216	#1855	—	53	4	no	—
217	#1848	—	53	1	no	—
218	#1848	D₅₃	1	1	yes	characteristic, complemented
219	#1837	—	53	1	no	—
220	#1837	—	53	1	no	—
221	C₂×C₅₃	—	2	2	no	—
222	C₂×C₅₃	#2317	1	1	yes	characteristic
223	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
224	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
225	D₅₃	—	2	2	no	—
226	D₅₃	—	4	1	no	—
227	C₂²×M₁₆	—	53	2	no	—
228	C₂²×C₄²	—	53	1	no	—
229	#432	—	106	1	no	—
230	#432	—	106	1	no	—
231	#431	—	53	2	no	—
232	#431	—	53	2	no	—
233	#431	—	53	2	no	—
234	#431	—	53	2	no	—
235	#431	—	106	1	no	—
236	#431	—	106	1	no	—
237	C₂×#108	—	53	4	no	—
238	C₂×#108	C₂×D₅₃	1	4	yes	—
239	#423	—	53	2	no	—
240	#423	—	53	1	no	—
241	#423	—	53	1	no	—
242	#423	—	106	1	no	—
243	#423	—	106	1	no	—
244	#423	C₂×D₅₃	1	1	yes	characteristic
245	#423	C₂×D₅₃	1	1	yes	characteristic
246	C₂×C₄⋊C₈	—	53	2	no	—
247	C₂×C₄⋊C₈	—	106	1	no	—
248	C₂×C₄⋊C₈	—	106	1	no	—
249	#407	—	53	2	no	—
250	C₂×C₂²⋊C₈	—	53	2	no	—
251	C₄×M₁₆	—	53	2	no	—
252	C₂×C₄×C₈	—	53	1	no	—
253	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
254	#341	—	53	4	no	—
255	#341	—	53	4	no	—
256	C₅₃	#21993	1	1	yes	characteristic, complemented, Sylow
257	C₂³×C₄	—	53	1	no	—
258	C₂³×C₄	—	53	1	no	—
259	C₂³×C₄	—	53	1	no	—
260	C₂×M₁₆	—	53	2	no	—
261	C₂×M₁₆	—	53	2	no	—
262	C₂×M₁₆	—	106	2	no	—
263	C₂×M₁₆	—	212	4	no	—
264	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
265	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
266	C₂²×C₈	—	53	2	no	—
267	C₂²×C₈	—	106	2	no	—
268	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
269	C₂×C₄²	—	53	1	no	—
270	C₂×C₄²	—	53	1	no	—
271	C₂×C₄²	—	53	1	no	—
272	C₂×C₄²	—	53	1	no	—
273	C₂×C₄²	—	53	1	no	—
274	C₂×C₄²	—	53	1	no	—
275	C₂×C₄²	—	53	1	no	—
276	C₂×C₄²	—	53	1	no	—
277	C₂×C₄²	—	53	1	no	—
278	C₂×C₄²	—	53	1	no	—
279	C₂×C₄²	—	53	1	no	—
280	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
281	#108	—	4	4	no	—
282	#108	—	212	4	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	—	53	4	no	—
288	C₄⋊C₈	—	53	4	no	—
289	C₄⋊C₈	—	106	2	no	—
290	C₄⋊C₈	—	106	2	no	—
291	C₄⋊C₈	—	106	1	no	—
292	C₄⋊C₈	—	106	1	no	—
293	C₄⋊C₈	—	106	1	no	—
294	C₄⋊C₈	—	106	1	no	—
295	C₂²⋊C₈	—	53	4	no	—
296	C₂²⋊C₈	—	53	4	no	—
297	C₂²⋊C₈	—	106	2	no	—
298	C₂²⋊C₈	—	106	2	no	—
299	#97	—	106	2	no	—
300	C₄×C₈	—	53	2	no	—
301	C₄×C₈	—	53	2	no	—
302	C₄×C₈	—	106	2	no	—
303	C₄×C₈	C₄×D₅₃	1	2	yes	—
304	C₄×C₈	C₄×D₅₃	1	2	yes	—
305	C₂⁴	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	1	no	—
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	—	53	1	no	—
314	C₂²×C₄	—	53	1	no	—
315	C₂²×C₄	—	53	1	no	—
316	C₂²×C₄	—	53	1	no	—
317	C₂²×C₄	—	53	1	no	—
318	C₂²×C₄	—	53	1	no	—
319	C₂²×C₄	—	53	1	no	—
320	C₂²×C₄	—	53	1	no	—
321	C₂²×C₄	—	106	4	no	—
322	C₂²×C₄	—	106	2	no	—
323	C₂²×C₄	—	106	2	no	—
324	C₂²×C₄	—	212	2	no	—
325	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
326	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
327	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
328	M₁₆	—	2	4	no	—
329	M₁₆	—	2	4	no	—
330	M₁₆	—	106	4	no	—
331	M₁₆	—	106	4	no	—
332	M₁₆	—	212	4	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	53	4	no	—
338	C₂×C₈	—	106	2	no	—
339	C₂×C₈	—	106	2	no	—
340	C₂×C₈	—	106	2	no	—
341	C₂×C₈	—	106	2	no	—
342	C₂×C₈	—	212	4	no	—
343	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
344	C₄²	—	53	1	no	—
345	C₄²	—	53	1	no	—
346	C₄²	—	53	1	no	—
347	C₄²	—	53	1	no	—
348	C₄²	—	53	1	no	—
349	C₄²	—	53	1	no	—
350	C₄²	—	53	1	no	—
351	C₄²	—	53	1	no	—
352	C₄²	—	53	1	no	—
353	C₄²	—	53	1	no	—
354	C₄²	—	53	1	no	—
355	C₄²	—	53	1	no	—
356	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
357	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
358	C₄²	D₄×D₅₃	1	1	yes	characteristic
359	C₄²	D₄×D₅₃	1	1	yes	characteristic
360	C₂³	—	53	1	no	—
361	C₂³	—	53	1	no	—
362	C₂³	—	106	2	no	—
363	C₂³	—	106	2	no	—
364	C₂³	—	212	1	no	—
365	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	53	1	no	—
369	C₂×C₄	—	53	4	no	—
370	C₂×C₄	—	53	1	no	—
371	C₂×C₄	—	53	1	no	—
372	C₂×C₄	—	53	1	no	—
373	C₂×C₄	—	53	1	no	—
374	C₂×C₄	—	53	1	no	—
375	C₂×C₄	—	106	2	no	—
376	C₂×C₄	—	106	2	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	—	106	2	no	—
379	C₂×C₄	—	106	2	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	2	no	—
382	C₂×C₄	—	106	2	no	—
383	C₂×C₄	—	106	2	no	—
384	C₂×C₄	—	106	1	no	—
385	C₂×C₄	—	106	1	no	—
386	C₂×C₄	—	106	1	no	—
387	C₂×C₄	—	106	1	no	—
388	C₂×C₄	—	212	1	no	—
389	C₂×C₄	—	212	1	no	—
390	C₂×C₄	—	212	1	no	—
391	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
392	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
393	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
394	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
397	C₂×C₄	#13133262	1	2	yes	—
398	C₂×C₄	#13133265	1	2	yes	—
399	C₈	—	2	4	no	—
400	C₈	—	4	2	no	—
401	C₈	—	4	2	no	—
402	C₈	—	106	4	no	—
403	C₈	—	212	2	no	—
404	C₈	—	212	2	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	53	2	no	—
408	C₂²	—	106	1	no	—
409	C₂²	—	106	1	no	—
410	C₂²	—	106	1	no	—
411	C₂²	—	106	1	no	—
412	C₂²	—	106	1	no	—
413	C₂²	—	212	4	no	—
414	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
415	C₂²	#16811250	1	2	yes	—
416	C₄	—	2	1	no	—
417	C₄	—	2	1	no	—
418	C₄	—	53	2	no	—
419	C₄	—	106	1	no	—
420	C₄	—	106	1	no	—
421	C₄	—	106	1	no	—
422	C₄	—	106	1	no	—
423	C₄	—	106	1	no	—
424	C₄	#108×D₅₃	1	2	yes	—
425	C₄	#16810670	1	2	yes	—
426	C₄	#16811104	1	2	yes	—
427	C₄	#16811186	1	1	yes	characteristic
428	C₄	#16811198	1	1	yes	characteristic
429	C₂	—	2	2	no	—
430	C₂	—	106	2	no	—
431	C₂	—	212	1	no	—
432	C₂	#20277336	1	1	yes	characteristic
433	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
434	C₂	C₂×#16810670	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#21993
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
4. #20275487	C₂	semidirect
6. #20274886	C₂	semidirect
7. #20274853	C₂	semidirect
8. #20274852	C₂	semidirect
9. C₅₃×#1848	C₂	semidirect
180. C₄×D₅₃	#108	semidirect
218. #1848	D₅₃	semidirect
256. C₅₃	#21993	semidirect

Maximal subgroups

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

subgroup	index
#20283347	2
C₂×#16810674	2
#423×D₅₃	2
#20275487	2
#20274918	2
#20274886	2
#20274853	2
#20274852	2
C₅₃×#1848	2
#21993	53