Subgroups — #27389027

Order 13568.

780 conjugacy classes of subgroups.

The subgroup lattice has 526 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×#16810675	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275490	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275488	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274921	C₂	1	2	yes	maximal
8	#20274918	C₂	1	2	yes	maximal
9	#108×Dic₅₃	C₂	1	2	yes	maximal, complemented
10	#20274616	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#16810679	—	2	1	no	—
16	#16810679	—	2	1	no	—
17	#16810675	—	2	1	no	—
18	#16810675	—	2	1	no	—
19	#16810675	C₄	1	2	yes	—
20	#16810675	C₄	1	2	yes	—
21	#16810675	C₄	1	2	yes	—
22	#16810675	C₄	1	2	yes	—
23	C₂×#13133153	C₂²	1	2	yes	—
24	C₂×#13133152	C₂²	1	2	yes	—
25	C₂×#13133152	C₂²	1	2	yes	—
26	#16810647	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	#16810586	C₂²	1	2	yes	—
29	C₂×#13133129	C₂²	1	2	yes	—
30	C₄×#11719917	C₂²	1	1	yes	characteristic
31	#16810575	—	2	1	no	—
32	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
33	M₁₆×Dic₅₃	C₂²	1	2	yes	—
34	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₄×#11719912	—	2	1	no	—
36	#97×D₅₃	—	2	1	no	—
37	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
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	C₅₃×#423	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×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	1	yes	characteristic
52	C₂×#11719917	C₂³	1	1	yes	characteristic
53	C₂×#11719912	—	2	1	no	—
54	C₂×#11719912	—	2	1	no	—
55	M₁₆×D₅₃	—	2	2	no	—
56	M₁₆×D₅₃	—	2	2	no	—
57	M₁₆×D₅₃	—	2	2	no	—
58	M₁₆×D₅₃	—	2	2	no	—
59	C₂²×#88322	C₂³	1	2	yes	—
60	C₂×C₈×D₅₃	—	2	1	no	—
61	C₂×C₈×D₅₃	—	2	1	no	—
62	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
63	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
64	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×D₅₃	D₄	1	1	yes	characteristic
70	C₄²×D₅₃	D₄	1	1	yes	characteristic
71	C₄²×D₅₃	D₄	1	1	yes	characteristic
72	C₄²×D₅₃	D₄	1	1	yes	characteristic
73	#13133153	—	2	4	no	—
74	#13133152	—	2	4	no	—
75	#13133152	—	2	4	no	—
76	C₄⋊#88322	—	2	1	no	—
77	C₄⋊#88322	—	2	1	no	—
78	C₄⋊#88322	C₂×C₄	1	2	yes	—
79	C₄⋊#88322	C₂×C₄	1	2	yes	—
80	#13133129	—	2	2	no	—
81	#13133129	—	2	2	no	—
82	#13133129	C₂×C₄	1	4	yes	—
83	#13133129	C₂×C₄	1	4	yes	—
84	#13133128	—	2	1	no	—
85	#13133127	—	2	1	no	—
86	#13133127	—	2	1	no	—
87	#13133127	—	2	2	no	—
88	C₄×#88322	—	2	1	no	—
89	C₈×Dic₅₃	—	2	1	no	—
90	C₈×Dic₅₃	—	2	1	no	—
91	C₈×Dic₅₃	—	2	2	no	—
92	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
93	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
94	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
95	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	#108×C₅₃	—	2	4	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	#97×C₅₃	—	2	1	no	—
105	C₄×C₈×C₅₃	—	2	1	no	—
106	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₄×D₅₃	—	2	1	no	—
114	C₂×C₄×D₅₃	—	2	1	no	—
115	C₂×C₄×D₅₃	—	2	1	no	—
116	C₂×C₄×D₅₃	—	2	1	no	—
117	C₂×C₄×D₅₃	—	2	4	no	—
118	C₂×C₄×D₅₃	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
131	M₁₆×C₅₃	—	2	1	no	—
132	M₁₆×C₅₃	—	2	1	no	—
133	M₁₆×C₅₃	—	2	2	no	—
134	M₁₆×C₅₃	—	2	2	no	—
135	M₁₆×C₅₃	#41	1	2	yes	—
136	M₁₆×C₅₃	#41	1	2	yes	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
143	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₅₃	C₂×D₄	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	C₄⋊C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
153	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
154	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
155	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
158	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
159	#11719917	—	2	1	no	—
160	#11719917	—	2	1	no	—
161	#11719917	—	2	2	no	—
162	#11719917	—	2	2	no	—
163	#11719917	#41	1	2	yes	—
164	#11719917	#41	1	2	yes	—
165	C₂×#88322	—	2	1	no	—
166	C₂×#88322	—	2	1	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	—	2	2	no	—
169	C₂×#88322	—	2	2	no	—
170	C₂×#88322	C₂²×C₄	1	4	yes	—
171	#11719912	—	2	4	no	—
172	#11719912	—	2	4	no	—
173	C₈×D₅₃	—	2	4	no	—
174	C₈×D₅₃	—	2	4	no	—
175	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
176	C₂²×D₅₃	—	2	1	no	—
177	C₂²×D₅₃	—	2	1	no	—
178	C₂²×D₅₃	—	2	2	no	—
179	C₂²×D₅₃	—	4	1	no	—
180	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
194	C₂×Dic₅₃	—	2	1	no	—
195	C₂×Dic₅₃	—	2	1	no	—
196	C₂×Dic₅₃	—	2	2	no	—
197	C₂×Dic₅₃	—	2	2	no	—
198	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
203	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₄×D₅₃	—	2	2	no	—
205	C₄×D₅₃	—	2	1	no	—
206	C₄×D₅₃	—	2	1	no	—
207	C₄×D₅₃	—	2	1	no	—
208	C₄×D₅₃	—	2	1	no	—
209	C₄×D₅₃	—	2	1	no	—
210	C₄×D₅₃	—	2	1	no	—
211	C₄×D₅₃	—	2	1	no	—
212	C₄×D₅₃	—	2	1	no	—
213	C₄×D₅₃	—	2	2	no	—
214	C₄×D₅₃	—	2	2	no	—
215	C₄×D₅₃	—	2	2	no	—
216	C₄×D₅₃	—	2	2	no	—
217	C₄×D₅₃	—	4	1	no	—
218	C₄×D₅₃	—	4	1	no	—
219	C₄×D₅₃	#108	1	4	yes	complemented
220	C₈×C₅₃	—	2	4	no	—
221	C₈×C₅₃	—	2	2	no	—
222	C₈×C₅₃	—	2	2	no	—
223	C₈×C₅₃	—	4	2	no	—
224	#88322	—	2	4	no	—
225	#88322	—	2	2	no	—
226	#88322	—	2	2	no	—
227	#88322	—	4	2	no	—
228	#21370	—	53	1	no	maximal, Sylow
229	C₂²×C₅₃	—	2	1	no	—
230	C₂²×C₅₃	—	2	1	no	—
231	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
232	C₂²×C₅₃	#533	1	1	yes	characteristic
233	C₂²×C₅₃	#533	1	1	yes	characteristic
234	C₂×D₅₃	—	2	1	no	—
235	C₂×D₅₃	—	2	1	no	—
236	C₂×D₅₃	—	2	1	no	—
237	C₂×D₅₃	—	2	1	no	—
238	C₂×D₅₃	—	2	1	no	—
239	C₂×D₅₃	—	4	2	no	—
240	C₂×D₅₃	—	4	2	no	—
241	C₂×D₅₃	C₂×#108	1	2	yes	—
242	C₄×C₅₃	—	2	1	no	—
243	C₄×C₅₃	—	2	1	no	—
244	C₄×C₅₃	—	2	1	no	—
245	C₄×C₅₃	C₂×#108	1	2	yes	—
246	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
247	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
248	C₄×C₅₃	#525	1	1	yes	characteristic
249	C₄×C₅₃	#525	1	1	yes	characteristic
250	Dic₅₃	—	2	1	no	—
251	Dic₅₃	—	2	1	no	—
252	Dic₅₃	—	2	1	no	—
253	Dic₅₃	—	2	1	no	—
254	Dic₅₃	C₂×#108	1	2	yes	—
255	Dic₅₃	#430	1	2	yes	—
256	#2920	—	53	1	no	—
257	C₂×#431	—	53	1	no	—
258	C₂×C₄×M₁₆	—	53	1	no	—
259	#1950	—	53	1	no	—
260	#1950	—	53	1	no	—
261	#1950	—	53	1	no	—
262	#1950	D₅₃	1	1	yes	characteristic, complemented
263	#1855	—	53	2	no	—
264	#1855	—	53	2	no	—
265	C₄×#108	—	53	2	no	—
266	C₄×#108	—	53	2	no	—
267	C₂×C₅₃	—	2	1	no	—
268	C₂×C₅₃	—	2	1	no	—
269	C₂×C₅₃	#2296	1	1	yes	characteristic
270	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
271	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
272	D₅₃	—	2	2	no	—
273	D₅₃	—	4	1	no	—
274	C₂²×M₁₆	—	53	1	no	—
275	C₂²×M₁₆	—	53	1	no	—
276	C₂²×C₄²	—	53	1	no	—
277	#432	—	106	1	no	—
278	#432	—	106	1	no	—
279	#431	—	53	2	no	—
280	#431	—	53	2	no	—
281	#431	—	53	2	no	—
282	#431	—	53	2	no	—
283	#431	—	106	1	no	—
284	#431	—	106	1	no	—
285	C₂×#108	—	53	2	no	—
286	C₂×#108	—	53	2	no	—
287	C₂×#108	—	53	2	no	—
288	C₂×#108	C₂×D₅₃	1	2	yes	—
289	#423	—	53	1	no	—
290	#423	C₂×D₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₈	—	53	1	no	—
292	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
293	#407	—	53	2	no	—
294	C₂×C₂²⋊C₈	—	53	2	no	—
295	C₄×M₁₆	—	53	1	no	—
296	C₄×M₁₆	—	53	1	no	—
297	C₄×M₁₆	—	53	2	no	—
298	C₄×M₁₆	—	53	1	no	—
299	C₄×M₁₆	—	106	1	no	—
300	C₄×M₁₆	—	106	1	no	—
301	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
302	C₂×#97	—	106	1	no	—
303	C₂×C₄×C₈	—	53	2	no	—
304	C₂×C₄×C₈	—	106	1	no	—
305	#341	—	53	2	no	—
306	#341	—	53	2	no	—
307	#341	—	53	2	no	—
308	#341	C₂×D₅₃	1	2	yes	—
309	C₅₃	#21370	1	1	yes	characteristic, complemented, Sylow
310	C₂³×C₄	—	53	1	no	—
311	C₂³×C₄	—	53	1	no	—
312	C₂³×C₄	—	53	1	no	—
313	C₂×M₁₆	—	53	1	no	—
314	C₂×M₁₆	—	53	1	no	—
315	C₂×M₁₆	—	53	2	no	—
316	C₂×M₁₆	—	106	1	no	—
317	C₂×M₁₆	—	106	1	no	—
318	C₂×M₁₆	—	106	2	no	—
319	C₂×M₁₆	—	106	2	no	—
320	C₂×M₁₆	—	106	2	no	—
321	C₂×M₁₆	—	106	2	no	—
322	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
323	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
324	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
325	C₂²×C₈	—	53	2	no	—
326	C₂²×C₈	—	106	1	no	—
327	C₂²×C₈	—	106	1	no	—
328	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
329	C₂×C₄²	—	53	1	no	—
330	C₂×C₄²	—	53	1	no	—
331	C₂×C₄²	—	53	1	no	—
332	C₂×C₄²	—	53	1	no	—
333	C₂×C₄²	—	53	1	no	—
334	C₂×C₄²	—	53	1	no	—
335	C₂×C₄²	—	53	1	no	—
336	C₂×C₄²	—	53	1	no	—
337	C₂×C₄²	—	53	1	no	—
338	C₂×C₄²	—	53	1	no	—
339	C₂×C₄²	—	53	1	no	—
340	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
341	#108	—	2	4	no	—
342	#108	—	106	4	no	—
343	#108	—	106	4	no	—
344	#108	—	106	4	no	—
345	C₄⋊C₈	—	2	1	no	—
346	C₄⋊C₈	—	2	1	no	—
347	C₄⋊C₈	—	53	2	no	—
348	C₄⋊C₈	—	53	2	no	—
349	C₄⋊C₈	—	106	1	no	—
350	C₄⋊C₈	—	106	1	no	—
351	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
352	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
353	C₂²⋊C₈	—	53	4	no	—
354	C₂²⋊C₈	—	53	4	no	—
355	C₂²⋊C₈	—	106	2	no	—
356	C₂²⋊C₈	—	106	2	no	—
357	#97	—	2	1	no	—
358	#97	—	106	1	no	—
359	#97	—	106	1	no	—
360	#97	—	106	2	no	—
361	#97	—	106	1	no	—
362	C₄×C₈	—	2	1	no	—
363	C₄×C₈	—	53	4	no	—
364	C₄×C₈	—	53	4	no	—
365	C₄×C₈	—	106	1	no	—
366	C₄×C₈	—	106	1	no	—
367	C₄×C₈	—	106	2	no	—
368	C₄×C₈	—	106	1	no	—
369	C₂⁴	—	53	1	no	—
370	C₂²×C₄	—	53	1	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	1	no	—
374	C₂²×C₄	—	53	1	no	—
375	C₂²×C₄	—	53	1	no	—
376	C₂²×C₄	—	53	1	no	—
377	C₂²×C₄	—	53	1	no	—
378	C₂²×C₄	—	53	1	no	—
379	C₂²×C₄	—	53	1	no	—
380	C₂²×C₄	—	53	1	no	—
381	C₂²×C₄	—	53	1	no	—
382	C₂²×C₄	—	53	1	no	—
383	C₂²×C₄	—	53	1	no	—
384	C₂²×C₄	—	53	1	no	—
385	C₂²×C₄	—	106	4	no	—
386	C₂²×C₄	—	106	1	no	—
387	C₂²×C₄	—	106	1	no	—
388	C₂²×C₄	—	106	1	no	—
389	C₂²×C₄	—	106	1	no	—
390	C₂²×C₄	—	212	2	no	—
391	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
392	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
393	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
394	M₁₆	—	2	1	no	—
395	M₁₆	—	2	1	no	—
396	M₁₆	—	2	2	no	—
397	M₁₆	—	2	2	no	—
398	M₁₆	—	53	2	no	—
399	M₁₆	—	53	2	no	—
400	M₁₆	—	106	4	no	—
401	M₁₆	—	106	4	no	—
402	M₁₆	—	106	1	no	—
403	M₁₆	—	106	1	no	—
404	M₁₆	—	106	2	no	—
405	M₁₆	—	106	2	no	—
406	M₁₆	#11719947	1	2	yes	—
407	M₁₆	#11719949	1	2	yes	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	53	4	no	—
414	C₂×C₈	—	106	4	no	—
415	C₂×C₈	—	106	4	no	—
416	C₂×C₈	—	106	1	no	—
417	C₂×C₈	—	106	1	no	—
418	C₂×C₈	—	106	2	no	—
419	C₂×C₈	—	106	2	no	—
420	C₂×C₈	—	106	2	no	—
421	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
422	C₄²	—	53	1	no	—
423	C₄²	—	53	1	no	—
424	C₄²	—	53	1	no	—
425	C₄²	—	53	1	no	—
426	C₄²	—	53	1	no	—
427	C₄²	—	53	1	no	—
428	C₄²	—	53	1	no	—
429	C₄²	—	53	1	no	—
430	C₄²	—	53	1	no	—
431	C₄²	—	53	1	no	—
432	C₄²	—	53	1	no	—
433	C₄²	—	53	1	no	—
434	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
435	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
436	C₄²	D₄×D₅₃	1	1	yes	characteristic
437	C₄²	D₄×D₅₃	1	1	yes	characteristic
438	C₂³	—	53	1	no	—
439	C₂³	—	53	1	no	—
440	C₂³	—	106	2	no	—
441	C₂³	—	106	1	no	—
442	C₂³	—	106	1	no	—
443	C₂³	—	212	1	no	—
444	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	53	1	no	—
448	C₂×C₄	—	53	1	no	—
449	C₂×C₄	—	53	4	no	—
450	C₂×C₄	—	53	1	no	—
451	C₂×C₄	—	53	1	no	—
452	C₂×C₄	—	53	1	no	—
453	C₂×C₄	—	53	1	no	—
454	C₂×C₄	—	106	2	no	—
455	C₂×C₄	—	106	2	no	—
456	C₂×C₄	—	106	2	no	—
457	C₂×C₄	—	106	2	no	—
458	C₂×C₄	—	106	2	no	—
459	C₂×C₄	—	106	2	no	—
460	C₂×C₄	—	106	1	no	—
461	C₂×C₄	—	106	1	no	—
462	C₂×C₄	—	106	1	no	—
463	C₂×C₄	—	106	1	no	—
464	C₂×C₄	—	106	1	no	—
465	C₂×C₄	—	106	1	no	—
466	C₂×C₄	—	106	1	no	—
467	C₂×C₄	—	106	1	no	—
468	C₂×C₄	—	106	1	no	—
469	C₂×C₄	—	106	1	no	—
470	C₂×C₄	—	106	2	no	—
471	C₂×C₄	—	212	1	no	—
472	C₂×C₄	—	212	1	no	—
473	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
474	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
475	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
476	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
477	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
478	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
479	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
480	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
481	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
482	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
483	C₈	—	2	4	no	—
484	C₈	—	2	2	no	—
485	C₈	—	2	2	no	—
486	C₈	—	4	2	no	—
487	C₈	—	106	4	no	—
488	C₈	—	106	2	no	—
489	C₈	—	106	2	no	—
490	C₈	—	212	2	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	53	2	no	—
494	C₂²	—	106	1	no	—
495	C₂²	—	106	1	no	—
496	C₂²	—	106	1	no	—
497	C₂²	—	106	1	no	—
498	C₂²	—	106	1	no	—
499	C₂²	—	212	2	no	—
500	C₂²	—	212	2	no	—
501	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
502	C₂²	#16811178	1	1	yes	characteristic
503	C₂²	#16811179	1	1	yes	characteristic
504	C₄	—	2	1	no	—
505	C₄	—	2	1	no	—
506	C₄	—	2	1	no	—
507	C₄	—	53	2	no	—
508	C₄	—	53	2	no	—
509	C₄	—	106	1	no	—
510	C₄	—	106	1	no	—
511	C₄	—	106	1	no	—
512	C₄	—	106	1	no	—
513	C₄	#108×D₅₃	1	2	yes	—
514	C₄	C₄×#11719947	1	1	yes	characteristic
515	C₄	C₄×#11719949	1	1	yes	characteristic
516	C₄	#16811133	1	1	yes	characteristic
517	C₄	#16811136	1	1	yes	characteristic
518	C₂	—	2	1	no	—
519	C₂	—	2	1	no	—
520	C₂	—	106	2	no	—
521	C₂	—	212	1	no	—
522	C₂	#20277196	1	1	yes	characteristic
523	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
524	C₂	#430×D₅₃	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×#1950	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1950	D₅₃
hypocenter	C₅₃	#21370
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
4. #20275491	C₂	semidirect
5. #20275490	C₂	semidirect
6. #20275488	C₂	semidirect
9. #108×Dic₅₃	C₂	semidirect
10. #20274616	C₂	semidirect
11. C₅₃×#1950	C₂	semidirect
219. C₄×D₅₃	#108	semidirect
262. #1950	D₅₃	semidirect
309. C₅₃	#21370	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×#16810675	2
C₄×M₁₆×D₅₃	2
#20275491	2
#20275490	2
#20275488	2
#20274921	2
#20274918	2
#108×Dic₅₃	2
#20274616	2
C₅₃×#1950	2
#21370	53