Subgroups — #27389026

Order 13568.

780 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283346	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20275489	C₂	1	2	yes	maximal, complemented
5	#20275487	C₂	1	2	yes	maximal, complemented
6	#20274921	C₂	1	2	yes	maximal
7	#20274918	C₂	1	2	yes	maximal
8	C₄×#13133152	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133152	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	#16810678	—	2	1	no	—
15	#16810678	—	2	1	no	—
16	#16810674	—	2	1	no	—
17	#16810674	—	2	1	no	—
18	#16810674	C₄	1	2	yes	—
19	#16810674	C₄	1	2	yes	—
20	#16810674	C₄	1	2	yes	—
21	#16810674	C₄	1	2	yes	—
22	C₂×#13133153	C₂²	1	2	yes	—
23	C₂×#13133152	C₂²	1	2	yes	—
24	C₂×#13133152	C₂²	1	2	yes	—
25	#16810646	C₂²	1	2	yes	—
26	C₂×#13133146	C₂²	1	1	yes	characteristic
27	C₂×#13133146	C₂²	1	1	yes	characteristic
28	#16810586	C₂²	1	2	yes	—
29	C₂×#13133129	C₂²	1	1	yes	characteristic
30	C₂×#13133129	C₂²	1	1	yes	characteristic
31	C₄×#11719917	C₂²	1	1	yes	characteristic
32	C₄×#11719917	C₂²	1	1	yes	characteristic
33	#16810575	—	2	1	no	—
34	M₁₆×Dic₅₃	C₂²	1	2	yes	—
35	C₄×#11719912	—	2	1	no	—
36	#97×D₅₃	—	2	1	no	—
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	C₄×C₈×D₅₃	—	2	1	no	—
39	#16810388	C₂²	1	2	yes	—
40	#16810387	C₂²	1	2	yes	—
41	#16810387	C₂²	1	2	yes	—
42	C₂×#108×C₅₃	C₂²	1	2	yes	—
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#341	C₂²	1	2	yes	—
47	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
48	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
49	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
50	C₂×#11719917	C₂³	1	2	yes	—
51	C₂×#11719917	C₂³	1	2	yes	—
52	C₂×#11719912	—	2	2	no	—
53	M₁₆×D₅₃	—	2	4	no	—
54	M₁₆×D₅₃	—	2	4	no	—
55	C₂²×#88322	C₂³	1	1	yes	characteristic
56	C₂²×#88322	C₂³	1	1	yes	characteristic
57	C₂×C₈×D₅₃	—	2	2	no	—
58	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₅₃	D₄	1	1	yes	characteristic
66	C₄²×D₅₃	D₄	1	1	yes	characteristic
67	C₄²×D₅₃	D₄	1	1	yes	characteristic
68	C₄²×D₅₃	D₄	1	1	yes	characteristic
69	#13133153	—	2	4	no	—
70	#13133152	—	2	4	no	—
71	#13133152	—	2	4	no	—
72	#13133146	—	2	2	no	—
73	#13133146	—	2	2	no	—
74	#13133146	C₂×C₄	1	2	yes	—
75	#13133146	C₂×C₄	1	2	yes	—
76	#13133146	C₂×C₄	1	2	yes	—
77	#13133146	C₂×C₄	1	2	yes	—
78	#13133129	—	2	2	no	—
79	#13133129	—	2	2	no	—
80	#13133129	C₂×C₄	1	2	yes	—
81	#13133129	C₂×C₄	1	2	yes	—
82	#13133129	C₂×C₄	1	2	yes	—
83	#13133129	C₂×C₄	1	2	yes	—
84	#13133128	—	2	1	no	—
85	#13133128	—	2	1	no	—
86	#13133127	—	2	2	no	—
87	C₄×#88322	—	2	1	no	—
88	C₄×#88322	—	2	1	no	—
89	C₄×#88322	C₂×C₄	1	2	yes	—
90	C₄×#88322	C₂×C₄	1	2	yes	—
91	C₈×Dic₅₃	—	2	2	no	—
92	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
93	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
94	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	#108×C₅₃	—	2	4	no	—
98	#97×C₅₃	—	2	1	no	—
99	#97×C₅₃	—	2	1	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
103	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
104	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
109	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
110	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₄×D₅₃	—	2	2	no	—
112	C₂×C₄×D₅₃	—	2	2	no	—
113	C₂×C₄×D₅₃	—	2	4	no	—
114	C₂×C₄×D₅₃	—	4	2	no	—
115	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
127	M₁₆×C₅₃	—	2	2	no	—
128	M₁₆×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	2	2	no	—
130	M₁₆×C₅₃	#41	1	4	yes	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
136	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
137	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
142	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
143	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
144	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
145	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
146	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
148	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
153	#11719917	—	2	2	no	—
154	#11719917	—	2	2	no	—
155	#11719917	—	2	2	no	—
156	#11719917	#41	1	4	yes	—
157	C₂×#88322	—	2	2	no	—
158	C₂×#88322	—	2	2	no	—
159	C₂×#88322	—	2	2	no	—
160	C₂×#88322	—	2	2	no	—
161	C₂×#88322	C₂²×C₄	1	2	yes	—
162	C₂×#88322	C₂²×C₄	1	2	yes	—
163	#11719912	—	2	8	no	—
164	C₈×D₅₃	—	2	8	no	—
165	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
166	C₂²×D₅₃	—	2	2	no	—
167	C₂²×D₅₃	—	2	2	no	—
168	C₂²×D₅₃	—	4	1	no	—
169	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
180	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
181	C₂×Dic₅₃	—	2	2	no	—
182	C₂×Dic₅₃	—	2	2	no	—
183	C₂×Dic₅₃	—	4	1	no	—
184	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₄×D₅₃	—	2	2	no	—
191	C₄×D₅₃	—	2	1	no	—
192	C₄×D₅₃	—	2	1	no	—
193	C₄×D₅₃	—	2	1	no	—
194	C₄×D₅₃	—	2	1	no	—
195	C₄×D₅₃	—	2	2	no	—
196	C₄×D₅₃	—	2	2	no	—
197	C₄×D₅₃	—	2	2	no	—
198	C₄×D₅₃	—	2	2	no	—
199	C₄×D₅₃	—	2	2	no	—
200	C₄×D₅₃	—	2	2	no	—
201	C₄×D₅₃	—	4	1	no	—
202	C₄×D₅₃	—	4	1	no	—
203	C₄×D₅₃	#108	1	4	yes	complemented
204	C₈×C₅₃	—	2	2	no	—
205	C₈×C₅₃	—	2	2	no	—
206	C₈×C₅₃	—	2	4	no	—
207	C₈×C₅₃	—	4	2	no	—
208	#88322	—	2	2	no	—
209	#88322	—	2	2	no	—
210	#88322	—	2	4	no	—
211	#88322	—	4	2	no	—
212	#21370	—	53	1	no	maximal, Sylow
213	C₂²×C₅₃	—	2	1	no	—
214	C₂²×C₅₃	—	2	1	no	—
215	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
216	C₂²×C₅₃	#533	1	2	yes	—
217	C₂×D₅₃	—	2	1	no	—
218	C₂×D₅₃	—	2	1	no	—
219	C₂×D₅₃	—	2	1	no	—
220	C₂×D₅₃	—	2	1	no	—
221	C₂×D₅₃	—	2	1	no	—
222	C₂×D₅₃	—	4	4	no	—
223	C₂×D₅₃	C₂×#108	1	2	yes	—
224	C₄×C₅₃	—	2	1	no	—
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	C₂×#108	1	2	yes	—
227	C₄×C₅₃	#430	1	2	yes	—
228	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
229	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
230	C₄×C₅₃	#525	1	2	yes	—
231	Dic₅₃	—	2	1	no	—
232	Dic₅₃	—	2	1	no	—
233	Dic₅₃	—	2	1	no	—
234	Dic₅₃	—	2	1	no	—
235	Dic₅₃	—	2	1	no	—
236	Dic₅₃	C₂×#108	1	2	yes	—
237	#2920	—	53	1	no	—
238	C₂×#431	—	53	1	no	—
239	C₂×C₄×M₁₆	—	53	1	no	—
240	#1950	—	53	2	no	—
241	#1950	—	53	2	no	—
242	#1855	—	53	2	no	—
243	#1855	—	53	2	no	—
244	C₄×#108	—	53	1	no	—
245	C₄×#108	—	53	1	no	—
246	C₄×#108	—	53	1	no	—
247	C₄×#108	D₅₃	1	1	yes	characteristic, complemented
248	C₂×C₅₃	—	2	2	no	—
249	C₂×C₅₃	#2296	1	1	yes	characteristic
250	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
251	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
252	D₅₃	—	2	2	no	—
253	D₅₃	—	4	1	no	—
254	C₂²×M₁₆	—	53	2	no	—
255	C₂²×C₄²	—	53	1	no	—
256	#432	—	106	1	no	—
257	#432	—	106	1	no	—
258	#431	—	53	2	no	—
259	#431	—	53	2	no	—
260	#431	—	53	2	no	—
261	#431	—	53	2	no	—
262	#431	—	106	1	no	—
263	#431	—	106	1	no	—
264	C₂×#108	—	53	2	no	—
265	C₂×#108	—	53	2	no	—
266	C₂×#108	—	53	2	no	—
267	C₂×#108	C₂×D₅₃	1	2	yes	—
268	#423	—	53	2	no	—
269	C₂×C₄⋊C₈	—	53	1	no	—
270	C₂×C₄⋊C₈	—	53	1	no	—
271	#407	—	53	2	no	—
272	C₂×C₂²⋊C₈	—	53	1	no	—
273	C₂×C₂²⋊C₈	—	53	1	no	—
274	C₄×M₁₆	—	53	2	no	—
275	C₄×M₁₆	—	53	1	no	—
276	C₄×M₁₆	—	53	1	no	—
277	C₄×M₁₆	—	106	1	no	—
278	C₄×M₁₆	—	106	1	no	—
279	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
280	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
281	C₂×#97	—	106	1	no	—
282	C₂×C₄×C₈	—	53	1	no	—
283	C₂×C₄×C₈	—	106	1	no	—
284	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
285	#341	—	53	2	no	—
286	#341	—	53	2	no	—
287	#341	—	53	2	no	—
288	#341	C₂×D₅₃	1	2	yes	—
289	C₅₃	#21370	1	1	yes	characteristic, complemented, Sylow
290	C₂³×C₄	—	53	1	no	—
291	C₂³×C₄	—	53	1	no	—
292	C₂³×C₄	—	53	1	no	—
293	C₂×M₁₆	—	53	2	no	—
294	C₂×M₁₆	—	53	2	no	—
295	C₂×M₁₆	—	106	2	no	—
296	C₂×M₁₆	—	106	4	no	—
297	C₂×M₁₆	—	106	4	no	—
298	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
299	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
300	C₂²×C₈	—	53	1	no	—
301	C₂²×C₈	—	53	1	no	—
302	C₂²×C₈	—	106	2	no	—
303	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
304	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
305	C₂×C₄²	—	53	1	no	—
306	C₂×C₄²	—	53	1	no	—
307	C₂×C₄²	—	53	1	no	—
308	C₂×C₄²	—	53	1	no	—
309	C₂×C₄²	—	53	1	no	—
310	C₂×C₄²	—	53	1	no	—
311	C₂×C₄²	—	53	1	no	—
312	C₂×C₄²	—	53	1	no	—
313	C₂×C₄²	—	53	1	no	—
314	C₂×C₄²	—	53	1	no	—
315	C₂×C₄²	—	53	1	no	—
316	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
317	#108	—	2	4	no	—
318	#108	—	106	4	no	—
319	#108	—	106	4	no	—
320	#108	—	106	4	no	—
321	C₄⋊C₈	—	53	2	no	—
322	C₄⋊C₈	—	53	2	no	—
323	C₄⋊C₈	—	53	2	no	—
324	C₄⋊C₈	—	53	2	no	—
325	C₄⋊C₈	—	106	2	no	—
326	C₄⋊C₈	—	106	2	no	—
327	C₂²⋊C₈	—	53	2	no	—
328	C₂²⋊C₈	—	53	2	no	—
329	C₂²⋊C₈	—	53	2	no	—
330	C₂²⋊C₈	—	53	2	no	—
331	C₂²⋊C₈	—	106	2	no	—
332	C₂²⋊C₈	—	106	2	no	—
333	#97	—	2	1	no	—
334	#97	—	2	1	no	—
335	#97	—	106	2	no	—
336	#97	—	106	1	no	—
337	#97	—	106	1	no	—
338	C₄×C₈	—	2	1	no	—
339	C₄×C₈	—	2	1	no	—
340	C₄×C₈	—	53	2	no	—
341	C₄×C₈	—	53	2	no	—
342	C₄×C₈	—	106	2	no	—
343	C₄×C₈	—	106	1	no	—
344	C₄×C₈	—	106	1	no	—
345	C₄×C₈	C₄×D₅₃	1	2	yes	—
346	C₄×C₈	C₄×D₅₃	1	2	yes	—
347	C₂⁴	—	53	1	no	—
348	C₂²×C₄	—	53	1	no	—
349	C₂²×C₄	—	53	1	no	—
350	C₂²×C₄	—	53	1	no	—
351	C₂²×C₄	—	53	1	no	—
352	C₂²×C₄	—	53	1	no	—
353	C₂²×C₄	—	53	1	no	—
354	C₂²×C₄	—	53	1	no	—
355	C₂²×C₄	—	53	1	no	—
356	C₂²×C₄	—	53	1	no	—
357	C₂²×C₄	—	53	1	no	—
358	C₂²×C₄	—	53	1	no	—
359	C₂²×C₄	—	53	1	no	—
360	C₂²×C₄	—	53	1	no	—
361	C₂²×C₄	—	53	1	no	—
362	C₂²×C₄	—	53	1	no	—
363	C₂²×C₄	—	106	4	no	—
364	C₂²×C₄	—	106	2	no	—
365	C₂²×C₄	—	106	2	no	—
366	C₂²×C₄	—	212	2	no	—
367	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
368	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
370	M₁₆	—	2	2	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	2	2	no	—
373	M₁₆	—	53	4	no	—
374	M₁₆	—	106	8	no	—
375	M₁₆	—	106	2	no	—
376	M₁₆	—	106	2	no	—
377	M₁₆	—	106	2	no	—
378	M₁₆	#11719945	1	4	yes	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	53	2	no	—
384	C₂×C₈	—	53	2	no	—
385	C₂×C₈	—	106	8	no	—
386	C₂×C₈	—	106	2	no	—
387	C₂×C₈	—	106	2	no	—
388	C₂×C₈	—	106	2	no	—
389	C₂×C₈	—	106	2	no	—
390	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
391	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
392	C₄²	—	53	1	no	—
393	C₄²	—	53	1	no	—
394	C₄²	—	53	1	no	—
395	C₄²	—	53	1	no	—
396	C₄²	—	53	1	no	—
397	C₄²	—	53	1	no	—
398	C₄²	—	53	1	no	—
399	C₄²	—	53	1	no	—
400	C₄²	—	53	1	no	—
401	C₄²	—	53	1	no	—
402	C₄²	—	53	1	no	—
403	C₄²	—	53	1	no	—
404	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
405	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
406	C₄²	D₄×D₅₃	1	1	yes	characteristic
407	C₄²	D₄×D₅₃	1	1	yes	characteristic
408	C₂³	—	53	1	no	—
409	C₂³	—	53	1	no	—
410	C₂³	—	106	2	no	—
411	C₂³	—	106	2	no	—
412	C₂³	—	212	1	no	—
413	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	53	1	no	—
417	C₂×C₄	—	53	4	no	—
418	C₂×C₄	—	53	1	no	—
419	C₂×C₄	—	53	1	no	—
420	C₂×C₄	—	53	1	no	—
421	C₂×C₄	—	53	1	no	—
422	C₂×C₄	—	53	1	no	—
423	C₂×C₄	—	106	2	no	—
424	C₂×C₄	—	106	2	no	—
425	C₂×C₄	—	106	2	no	—
426	C₂×C₄	—	106	2	no	—
427	C₂×C₄	—	106	1	no	—
428	C₂×C₄	—	106	1	no	—
429	C₂×C₄	—	106	1	no	—
430	C₂×C₄	—	106	1	no	—
431	C₂×C₄	—	106	2	no	—
432	C₂×C₄	—	106	2	no	—
433	C₂×C₄	—	106	2	no	—
434	C₂×C₄	—	106	2	no	—
435	C₂×C₄	—	106	2	no	—
436	C₂×C₄	—	212	1	no	—
437	C₂×C₄	—	212	1	no	—
438	C₂×C₄	—	212	1	no	—
439	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
440	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
441	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
442	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
443	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
444	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
445	C₂×C₄	#41×D₅₃	1	2	yes	—
446	C₂×C₄	C₂×#11719945	1	2	yes	—
447	C₈	—	2	2	no	—
448	C₈	—	2	2	no	—
449	C₈	—	2	4	no	—
450	C₈	—	4	2	no	—
451	C₈	—	106	2	no	—
452	C₈	—	106	2	no	—
453	C₈	—	106	4	no	—
454	C₈	—	212	2	no	—
455	C₂²	—	2	1	no	—
456	C₂²	—	2	1	no	—
457	C₂²	—	53	2	no	—
458	C₂²	—	106	1	no	—
459	C₂²	—	106	1	no	—
460	C₂²	—	106	1	no	—
461	C₂²	—	106	1	no	—
462	C₂²	—	106	1	no	—
463	C₂²	—	212	4	no	—
464	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
465	C₂²	#16811177	1	2	yes	—
466	C₄	—	2	1	no	—
467	C₄	—	2	1	no	—
468	C₄	—	53	2	no	—
469	C₄	—	106	1	no	—
470	C₄	—	106	1	no	—
471	C₄	—	106	1	no	—
472	C₄	—	106	1	no	—
473	C₄	—	106	1	no	—
474	C₄	#108×D₅₃	1	2	yes	—
475	C₄	#16810670	1	2	yes	—
476	C₄	#16811091	1	1	yes	characteristic
477	C₄	C₄×#11719945	1	1	yes	characteristic
478	C₄	#16811132	1	2	yes	—
479	C₂	—	2	2	no	—
480	C₂	—	106	2	no	—
481	C₂	—	212	1	no	—
482	C₂	#20277195	1	1	yes	characteristic
483	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
484	C₂	C₂×#16810670	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#21370
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #20275489	C₂	semidirect
5. #20275487	C₂	semidirect
8. C₄×#13133152	C₂	semidirect
9. C₄×#13133152	C₂	semidirect
10. C₄×#13133153	C₂	semidirect
11. C₄×#108×C₅₃	C₂	semidirect
203. C₄×D₅₃	#108	semidirect
247. C₄×#108	D₅₃	semidirect
289. C₅₃	#21370	semidirect

Maximal subgroups

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

subgroup	index
#20283346	2
C₂×#16810674	2
C₄×M₁₆×D₅₃	2
#20275489	2
#20275487	2
#20274921	2
#20274918	2
C₄×#13133152	2
C₄×#13133153	2
C₄×#108×C₅₃	2
#21370	53