Subgroups — #28247510

Order 13568.

3178 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811388	C₂	1	1	yes	maximal, characteristic
2	C₂×#131×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20287914	C₂	1	1	yes	maximal, characteristic
4	#20283144	C₂	1	3	yes	maximal, complemented
5	#20283143	C₂	1	1	yes	maximal, characteristic, complemented
6	#20283142	C₂	1	3	yes	maximal, complemented
7	#20283141	C₂	1	1	yes	maximal, characteristic, complemented
8	#20283137	C₂	1	3	yes	maximal, complemented
9	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133223	C₂	1	3	yes	maximal, complemented
11	#20283132	C₂	1	3	yes	maximal, complemented
12	#20283131	C₂	1	3	yes	maximal, complemented
13	C₂×#16810577	C₂	1	3	yes	maximal, complemented
14	#405×D₅₃	C₂	1	3	yes	maximal, complemented
15	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×D₅₃	C₂²	1	1	yes	characteristic
17	#16811388	C₄	1	16	yes	complemented
18	C₂×#13133225	C₂²	1	3	yes	—
19	C₂×#13133225	C₂²	1	3	yes	—
20	C₂×#13133224	C₂²	1	1	yes	characteristic
21	C₂×#13133224	C₂²	1	1	yes	characteristic
22	C₂×#13133223	C₂²	1	3	yes	—
23	C₂×#13133223	C₂²	1	3	yes	—
24	#131×D₅₃	C₄	1	16	yes	complemented
25	C₂²×#11719912	C₂²	1	3	yes	—
26	C₂×M₁₆×D₅₃	C₂²	1	3	yes	—
27	C₂×M₁₆×D₅₃	C₂²	1	3	yes	—
28	C₂²×C₈×D₅₃	C₂²	1	3	yes	—
29	#16811105	C₄	1	2	yes	—
30	#16811102	C₄	1	6	yes	—
31	#16811099	C₄	1	2	yes	—
32	#16811095	C₄	1	6	yes	—
33	#41×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₄×#11719945	C₂²	1	3	yes	—
35	#16811091	C₂²	1	3	yes	—
36	C₂×#13133174	C₂²	1	3	yes	—
37	#117×D₅₃	C₂²	1	3	yes	—
38	#16810581	C₂²	1	3	yes	complemented
39	#16810581	C₂²	1	3	yes	complemented
40	#16810580	C₂²	1	6	yes	—
41	#16810580	C₂²	1	6	yes	complemented
42	#16810579	C₂²	1	3	yes	complemented
43	#16810579	C₂²	1	3	yes	complemented
44	#16810578	C₂²	1	3	yes	complemented
45	#16810578	C₂²	1	3	yes	complemented
46	#16810577	C₂²	1	3	yes	—
47	#16810577	C₂²	1	1	yes	characteristic
48	#16810577	C₂²	1	3	yes	—
49	#16810577	C₂²	1	3	yes	—
50	#16810577	C₂²	1	3	yes	complemented
51	#16810577	C₂²	1	3	yes	complemented
52	#16810577	C₂²	1	3	yes	complemented
53	#16810577	C₂²	1	6	yes	complemented
54	#16810577	C₂²	1	3	yes	complemented
55	C₄×#11719917	C₂²	1	3	yes	complemented
56	#16810575	C₂²	1	6	yes	complemented
57	M₁₆×Dic₅₃	C₂²	1	3	yes	complemented
58	M₁₆×Dic₅₃	C₂²	1	3	yes	complemented
59	C₄×#11719912	C₂²	1	6	yes	complemented
60	C₂×#13133128	C₂²	1	3	yes	complemented
61	C₂×#13133127	C₂²	1	3	yes	complemented
62	#97×D₅₃	C₂²	1	6	yes	complemented
63	C₂×C₈×Dic₅₃	C₂²	1	3	yes	complemented
64	C₄×C₈×D₅₃	C₂²	1	6	yes	complemented
65	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
66	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
67	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
68	C₅₃×#405	C₂²	1	3	yes	complemented
69	C₅₃×#405	C₂²	1	3	yes	complemented
70	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
71	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	complemented
72	C₂×#11719949	C₂×C₄	1	1	yes	characteristic
73	C₂×#11719945	C₂³	1	3	yes	—
74	C₂×#11719947	C₂×C₄	1	3	yes	—
75	#41×D₅₃	C₂×C₄	1	8	yes	—
76	#41×D₅₃	C₂×C₄	1	8	yes	—
77	C₂×Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×D₄×D₅₃	C₂×C₄	1	3	yes	—
79	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
80	#13133225	C₂×C₄	1	24	yes	complemented
81	#13133225	C₂×C₄	1	24	yes	complemented
82	#13133224	C₂×C₄	1	8	yes	complemented
83	#13133224	C₂×C₄	1	8	yes	complemented
84	#13133223	C₂×C₄	1	24	yes	complemented
85	#13133223	C₂×C₄	1	24	yes	complemented
86	C₂×#11719917	C₂³	1	3	yes	—
87	C₂×#11719917	C₂³	1	3	yes	—
88	C₂×#11719912	C₂×C₄	1	24	yes	complemented
89	C₂×#11719912	C₂³	1	3	yes	—
90	C₂×#11719912	C₂³	1	6	yes	—
91	C₂×#11719912	C₂³	1	3	yes	—
92	C₂×#11719912	C₂³	1	1	yes	characteristic
93	C₂×#11719912	C₂³	1	3	yes	—
94	M₁₆×D₅₃	C₂×C₄	1	24	yes	complemented
95	M₁₆×D₅₃	C₂×C₄	1	24	yes	complemented
96	C₂²×#88322	C₂³	1	3	yes	—
97	C₂²×#88322	C₂³	1	3	yes	—
98	C₂×C₈×D₅₃	C₂×C₄	1	24	yes	complemented
99	C₂×C₈×D₅₃	C₂³	1	3	yes	—
100	C₂×C₈×D₅₃	C₂³	1	6	yes	—
101	C₂×C₈×D₅₃	C₂³	1	3	yes	—
102	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×D₅₃	C₂³	1	3	yes	—
104	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
105	C₄×Dic₁₀₆	C₂×C₄	1	6	yes	—
106	#13133182	C₂×C₄	1	6	yes	—
107	C₄×D₂₁₂	C₂×C₄	1	6	yes	—
108	D₄×Dic₅₃	C₂×C₄	1	6	yes	—
109	#13133179	C₂×C₄	1	6	yes	—
110	C₄×C₂²⋊D₅₃	C₂×C₄	1	12	yes	—
111	#13133177	C₂×C₄	1	12	yes	—
112	#13133176	C₂³	1	3	yes	—
113	#13133175	C₂×C₄	1	6	yes	—
114	#13133174	C₂×C₄	1	6	yes	—
115	#13133174	C₂×C₄	1	6	yes	—
116	#13133174	C₂³	1	3	yes	—
117	#13133174	C₂³	1	1	yes	characteristic
118	#13133174	C₂³	1	3	yes	—
119	#13133174	C₂³	1	3	yes	—
120	#13133173	C₂×C₄	1	6	yes	—
121	C₂×#11719919	C₂×C₄	1	6	yes	—
122	C₄⋊C₄×D₅₃	C₂×C₄	1	6	yes	—
123	C₂×#11719921	C₂×C₄	1	6	yes	—
124	C₂²⋊C₄×D₅₃	C₂×C₄	1	6	yes	—
125	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
126	C₄²×D₅₃	C₂³	1	6	yes	—
127	#13133128	C₂³	1	3	yes	—
128	#13133128	C₂³	1	1	yes	characteristic
129	#13133128	C₂³	1	3	yes	—
130	#13133128	C₂³	1	3	yes	—
131	#13133127	C₂³	1	3	yes	—
132	#13133127	C₂³	1	6	yes	—
133	#13133127	C₂³	1	1	yes	characteristic
134	#13133127	C₂³	1	3	yes	—
135	#13133127	C₂³	1	3	yes	—
136	C₄×#88322	C₂³	1	6	yes	—
137	C₈×Dic₅₃	C₂³	1	3	yes	—
138	C₈×Dic₅₃	C₂³	1	6	yes	—
139	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
140	C₈×Dic₅₃	C₂³	1	3	yes	—
141	C₈×Dic₅₃	C₂³	1	3	yes	—
142	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
143	#131×C₅₃	C₂×C₄	1	8	yes	complemented
144	#131×C₅₃	C₂×C₄	1	8	yes	complemented
145	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
146	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
147	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
148	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
149	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
150	C₄×D₄×C₅₃	C₂×C₄	1	6	yes	—
151	#117×C₅₃	C₂³	1	3	yes	—
152	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
153	#97×C₅₃	C₂³	1	6	yes	—
154	C₄×C₈×C₅₃	C₂³	1	3	yes	—
155	C₄×C₈×C₅₃	C₂³	1	3	yes	—
156	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
157	C₄×C₈×C₅₃	C₂³	1	3	yes	—
158	C₂³×D₅₃	C₂²×C₄	1	3	yes	—
159	#41×C₅₃	C₂²×C₄	1	4	yes	—
160	#41×C₅₃	C₂²×C₄	1	4	yes	—
161	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₂×D₄×C₅₃	C₂²×C₄	1	3	yes	—
163	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
165	C₂×C₂²⋊D₅₃	C₂²×C₄	1	6	yes	—
166	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
167	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
168	#11719949	C₄²	1	8	yes	—
169	Q₈×D₅₃	C₄²	1	8	yes	—
170	#11719947	C₄²	1	24	yes	—
171	D₄×D₅₃	C₄²	1	24	yes	—
172	#11719945	C₂²×C₄	1	12	yes	—
173	#11719945	C₂²×C₄	1	12	yes	—
174	C₂×D₂₁₂	C₂²×C₄	1	3	yes	—
175	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
176	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
177	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
178	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
179	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
180	C₂×C₄×D₅₃	C₂⁴	1	3	yes	—
181	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
182	C₂×Dic₁₀₆	C₂²×C₄	1	3	yes	—
183	M₁₆×C₅₃	C₂²×C₄	1	12	yes	—
184	M₁₆×C₅₃	C₂²×C₄	1	12	yes	—
185	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
186	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
187	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
188	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
189	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
190	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
191	C₄⋊C₄×C₅₃	C₂²×C₄	1	6	yes	—
192	C₂²⋊C₄×C₅₃	C₂²×C₄	1	6	yes	—
193	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
194	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
195	C₄²×C₅₃	C₂⁴	1	3	yes	—
196	C₂²⋊Dic₅₃	C₂²×C₄	1	6	yes	—
197	#11719921	C₂²×C₄	1	6	yes	—
198	#11719921	C₂²×C₄	1	2	yes	—
199	#11719921	C₂²×C₄	1	6	yes	—
200	#11719921	C₂²×C₄	1	6	yes	—
201	C₄⋊Dic₅₃	C₂²×C₄	1	6	yes	—
202	#11719919	C₂²×C₄	1	6	yes	—
203	#11719919	C₂²×C₄	1	2	yes	—
204	#11719919	C₂²×C₄	1	6	yes	—
205	#11719919	C₂²×C₄	1	6	yes	—
206	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
207	C₄×Dic₅₃	C₂²×C₄	1	6	yes	—
208	C₄×Dic₅₃	C₂⁴	1	3	yes	—
209	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
210	#11719917	C₂²×C₄	1	12	yes	—
211	#11719917	C₂²×C₄	1	12	yes	—
212	C₂×#88322	C₂²×C₄	1	12	yes	—
213	C₂×#88322	C₂²×C₄	1	12	yes	—
214	C₂×#88322	C₂⁴	1	1	yes	characteristic
215	C₂×#88322	C₂⁴	1	3	yes	—
216	C₂×#88322	C₂⁴	1	3	yes	—
217	C₂×#88322	C₂⁴	1	1	yes	characteristic
218	#11719912	C₂²×C₄	1	12	yes	—
219	#11719912	C₂²×C₄	1	24	yes	—
220	#11719912	C₂²×C₄	1	4	yes	—
221	#11719912	C₂²×C₄	1	12	yes	—
222	#11719912	C₂²×C₄	1	12	yes	—
223	C₈×D₅₃	C₂²×C₄	1	4	yes	—
224	C₈×D₅₃	C₂²×C₄	1	12	yes	—
225	C₈×D₅₃	C₂²×C₄	1	12	yes	—
226	C₈×D₅₃	C₂²×C₄	1	12	yes	—
227	C₈×D₅₃	C₂²×C₄	1	24	yes	—
228	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
229	C₂²×D₅₃	—	4	6	no	—
230	C₂²×D₅₃	C₂×C₄²	1	12	yes	—
231	C₂²×D₅₃	C₂³×C₄	1	3	yes	—
232	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
233	Q₈×C₅₃	C₂×C₄²	1	4	yes	—
234	D₄×C₅₃	C₂×C₄²	1	12	yes	—
235	C₂×C₄×C₅₃	—	4	6	no	—
236	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
237	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
239	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
240	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
241	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
242	C₂²⋊D₅₃	C₂×C₄²	1	24	yes	—
243	C₂×Dic₅₃	—	4	6	no	—
244	C₂×Dic₅₃	C₂×C₄²	1	12	yes	—
245	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
246	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
247	C₂×Dic₅₃	C₂³×C₄	1	3	yes	—
248	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
249	D₂₁₂	C₂×C₄²	1	12	yes	—
250	C₄×D₅₃	—	4	12	no	—
251	C₄×D₅₃	C₂×C₄²	1	12	yes	—
252	C₄×D₅₃	C₂×C₄²	1	12	yes	—
253	C₄×D₅₃	C₂³×C₄	1	2	yes	—
254	C₄×D₅₃	C₂³×C₄	1	2	yes	—
255	C₄×D₅₃	C₂³×C₄	1	6	yes	—
256	C₄×D₅₃	C₂³×C₄	1	6	yes	—
257	Dic₁₀₆	C₂×C₄²	1	12	yes	—
258	C₈×C₅₃	C₂³×C₄	1	2	yes	—
259	C₈×C₅₃	C₂³×C₄	1	6	yes	—
260	C₈×C₅₃	C₂³×C₄	1	6	yes	—
261	C₈×C₅₃	C₂³×C₄	1	2	yes	—
262	#88322	C₂³×C₄	1	2	yes	—
263	#88322	C₂³×C₄	1	6	yes	—
264	#88322	C₂³×C₄	1	6	yes	—
265	#88322	C₂³×C₄	1	2	yes	—
266	#60058	—	53	1	no	maximal, Sylow
267	C₂²×C₅₃	—	2	6	no	—
268	C₂²×C₅₃	C₂²×C₄²	1	6	yes	—
269	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
270	C₂×D₅₃	—	2	6	no	—
271	C₂×D₅₃	—	2	2	no	—
272	C₂×D₅₃	—	4	12	no	—
273	C₂×D₅₃	C₂²×C₄²	1	2	yes	—
274	C₂×D₅₃	C₂²×C₄²	1	6	yes	—
275	C₄×C₅₃	—	2	6	no	—
276	C₄×C₅₃	—	2	2	no	—
277	C₄×C₅₃	C₂²×C₄²	1	6	yes	—
278	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
279	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
280	Dic₅₃	—	2	6	no	—
281	Dic₅₃	—	2	2	no	—
282	Dic₅₃	C₂²×C₄²	1	6	yes	—
283	Dic₅₃	C₂²×C₄²	1	2	yes	—
284	C₂×#568	—	53	1	no	—
285	C₂²×#131	—	53	1	no	—
286	#3428	—	53	1	no	—
287	#2875	—	53	3	no	—
288	#2875	—	53	3	no	—
289	#2875	—	53	1	no	—
290	#2875	—	53	1	no	—
291	C₄×#131	—	53	3	no	—
292	C₄×#131	—	53	3	no	—
293	C₄×#131	—	53	1	no	—
294	C₄×#131	D₅₃	1	1	yes	characteristic, complemented
295	#2873	—	53	3	no	—
296	#2873	—	53	3	no	—
297	C₂×#405	—	53	3	no	—
298	C₂×#405	—	53	3	no	—
299	C₂×C₅₃	—	2	6	no	—
300	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
301	C₂×C₅₃	#3573	1	2	yes	—
302	D₅₃	—	2	2	no	—
303	D₅₃	—	2	6	no	—
304	C₂²×#41	—	53	1	no	—
305	#568	—	53	16	no	—
306	C₂×#131	—	53	1	no	—
307	C₂×#131	—	53	1	no	—
308	C₂×#131	—	53	16	no	—
309	C₂×#131	—	53	3	no	—
310	C₂×#131	—	53	3	no	—
311	C₂×#131	—	53	3	no	—
312	C₂×#131	—	53	3	no	—
313	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
314	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
315	C₂²×M₁₆	—	53	3	no	—
316	C₂²×M₁₆	—	53	3	no	—
317	C₂²×M₁₆	—	53	3	no	—
318	C₂³×C₈	—	53	3	no	—
319	#520	—	53	2	no	—
320	#520	—	53	6	no	—
321	#519	—	53	2	no	—
322	#518	—	53	6	no	—
323	C₄×#41	—	53	1	no	—
324	C₄×#41	—	53	3	no	—
325	C₄×#41	—	53	3	no	—
326	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
327	C₂×#117	—	53	3	no	—
328	C₂×#117	—	53	3	no	—
329	#405	—	53	3	no	—
330	#405	—	53	3	no	—
331	#405	—	53	3	no	—
332	#405	—	53	3	no	—
333	#405	—	53	3	no	—
334	#405	—	53	3	no	—
335	#405	—	53	6	no	—
336	#405	—	53	6	no	—
337	#405	—	53	3	no	—
338	#405	—	53	1	no	—
339	#405	—	53	3	no	—
340	#405	—	53	3	no	—
341	#405	—	53	3	no	—
342	#405	—	53	3	no	—
343	#405	—	53	3	no	—
344	#405	—	53	3	no	—
345	#405	—	53	6	no	—
346	#405	C₂×D₅₃	1	3	yes	complemented
347	#405	C₂×D₅₃	1	3	yes	complemented
348	C₄×M₁₆	—	53	3	no	—
349	C₄×M₁₆	—	53	3	no	—
350	C₄×M₁₆	—	53	3	no	—
351	C₄×M₁₆	—	53	6	no	—
352	C₄×M₁₆	—	53	6	no	—
353	C₄×M₁₆	C₂×D₅₃	1	3	yes	complemented
354	C₂×#97	—	53	3	no	—
355	C₂×#97	—	53	3	no	—
356	C₂×#97	—	53	6	no	—
357	C₂×C₄×C₈	—	53	3	no	—
358	C₂×C₄×C₈	—	53	6	no	—
359	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	complemented
360	C₅₃	#60058	1	1	yes	characteristic, complemented, Sylow
361	C₂×#41	—	53	8	no	—
362	C₂×#41	—	53	8	no	—
363	C₂×#41	—	53	1	no	—
364	C₂×#41	—	53	3	no	—
365	C₂×#41	—	53	3	no	—
366	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
367	C₂²×Q₈	—	53	1	no	—
368	C₂²×D₄	—	53	3	no	—
369	C₂³×C₄	—	53	3	no	—
370	#131	—	53	8	no	—
371	#131	—	53	8	no	—
372	#131	—	53	24	no	—
373	#131	—	53	24	no	—
374	#131	—	53	24	no	—
375	#131	—	53	24	no	—
376	#131	C₄×D₅₃	1	8	yes	complemented
377	#131	C₄×D₅₃	1	8	yes	complemented
378	C₂×M₁₆	—	53	3	no	—
379	C₂×M₁₆	—	53	3	no	—
380	C₂×M₁₆	—	53	24	no	—
381	C₂×M₁₆	—	53	24	no	—
382	C₂×M₁₆	—	53	24	no	—
383	C₂×M₁₆	—	53	3	no	—
384	C₂×M₁₆	—	53	1	no	—
385	C₂×M₁₆	—	53	6	no	—
386	C₂×M₁₆	—	53	3	no	—
387	C₂×M₁₆	—	53	3	no	—
388	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
389	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
390	C₂²×C₈	—	53	3	no	—
391	C₂²×C₈	—	53	3	no	—
392	C₂²×C₈	—	53	24	no	—
393	C₂²×C₈	—	53	3	no	—
394	C₂²×C₈	—	53	3	no	—
395	C₂²×C₈	—	53	6	no	—
396	C₂²×C₈	—	53	1	no	—
397	C₂²×C₈	—	53	3	no	—
398	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
399	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
400	C₄×Q₈	—	53	2	no	—
401	C₄×Q₈	—	53	6	no	—
402	C₄×Q₈	—	53	6	no	—
403	C₄×Q₈	C₄×D₅₃	1	2	yes	—
404	C₄×D₄	—	53	6	no	—
405	C₄×D₄	—	53	12	no	—
406	C₄×D₄	—	53	6	no	—
407	C₄×D₄	—	53	12	no	—
408	C₄×D₄	—	53	6	no	—
409	C₄×D₄	C₄×D₅₃	1	6	yes	—
410	#117	—	53	3	no	—
411	#117	—	53	6	no	—
412	#117	—	53	6	no	—
413	#117	—	53	6	no	—
414	#117	—	53	6	no	—
415	#117	—	53	3	no	—
416	#117	—	53	1	no	—
417	#117	—	53	3	no	—
418	#117	—	53	3	no	—
419	#117	C₂²×D₅₃	1	3	yes	—
420	C₂×C₄⋊C₄	—	53	6	no	—
421	C₂×C₄⋊C₄	—	53	6	no	—
422	C₂×C₂²⋊C₄	—	53	6	no	—
423	C₂×C₂²⋊C₄	—	53	6	no	—
424	C₂×C₄²	—	53	3	no	—
425	C₂×C₄²	—	53	6	no	—
426	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
427	#97	—	53	3	no	—
428	#97	—	53	1	no	—
429	#97	—	53	6	no	—
430	#97	—	53	3	no	—
431	#97	—	53	3	no	—
432	#97	—	53	3	no	—
433	#97	—	53	1	no	—
434	#97	—	53	3	no	—
435	#97	—	53	3	no	—
436	#97	C₂²×D₅₃	1	6	yes	—
437	C₄×C₈	—	53	6	no	—
438	C₄×C₈	—	53	3	no	—
439	C₄×C₈	—	53	3	no	—
440	C₄×C₈	—	53	3	no	—
441	C₄×C₈	—	53	1	no	—
442	C₄×C₈	—	53	6	no	—
443	C₄×C₈	C₂²×D₅₃	1	3	yes	—
444	C₄×C₈	C₂²×D₅₃	1	3	yes	—
445	C₄×C₈	C₂²×D₅₃	1	3	yes	—
446	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
447	C₂⁴	—	53	3	no	—
448	#41	—	53	8	no	—
449	#41	—	53	24	no	—
450	#41	—	53	12	no	—
451	#41	—	53	12	no	—
452	#41	C₂×C₄×D₅₃	1	4	yes	—
453	#41	C₂×C₄×D₅₃	1	4	yes	—
454	C₂×Q₈	—	53	8	no	—
455	C₂×Q₈	—	53	3	no	—
456	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
457	C₂×D₄	—	53	6	no	—
458	C₂×D₄	—	53	24	no	—
459	C₂×D₄	—	53	3	no	—
460	C₂×D₄	C₂×C₄×D₅₃	1	3	yes	—
461	C₂²×C₄	—	53	3	no	—
462	C₂²×C₄	—	53	3	no	—
463	C₂²×C₄	—	53	3	no	—
464	C₂²×C₄	—	53	12	no	—
465	C₂²×C₄	—	53	12	no	—
466	C₂²×C₄	—	53	3	no	—
467	C₂²×C₄	—	53	1	no	—
468	C₂²×C₄	—	53	6	no	—
469	C₂²×C₄	—	53	12	no	—
470	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
471	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
472	M₁₆	—	53	12	no	—
473	M₁₆	—	53	12	no	—
474	M₁₆	—	53	12	no	—
475	M₁₆	—	53	4	no	—
476	M₁₆	—	53	24	no	—
477	M₁₆	—	53	12	no	—
478	M₁₆	—	53	12	no	—
479	M₁₆	C₂×C₄×D₅₃	1	12	yes	—
480	M₁₆	C₂×C₄×D₅₃	1	12	yes	—
481	C₂×C₈	—	53	12	no	—
482	C₂×C₈	—	53	12	no	—
483	C₂×C₈	—	53	1	no	—
484	C₂×C₈	—	53	3	no	—
485	C₂×C₈	—	53	3	no	—
486	C₂×C₈	—	53	1	no	—
487	C₂×C₈	—	53	4	no	—
488	C₂×C₈	—	53	12	no	—
489	C₂×C₈	—	53	12	no	—
490	C₂×C₈	—	53	12	no	—
491	C₂×C₈	—	53	24	no	—
492	C₂×C₈	C₂×C₄×D₅₃	1	12	yes	—
493	C₂×C₈	C₂×C₄×D₅₃	1	12	yes	—
494	C₂×C₈	C₂³×D₅₃	1	3	yes	—
495	C₂×C₈	C₂³×D₅₃	1	3	yes	—
496	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
497	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
498	C₄⋊C₄	—	53	6	no	—
499	C₄⋊C₄	—	53	2	no	—
500	C₄⋊C₄	—	53	6	no	—
501	C₄⋊C₄	—	53	6	no	—
502	C₄⋊C₄	—	53	6	no	—
503	C₄⋊C₄	C₂×C₄×D₅₃	1	6	yes	—
504	C₂²⋊C₄	—	53	6	no	—
505	C₂²⋊C₄	—	53	6	no	—
506	C₂²⋊C₄	—	53	2	no	—
507	C₂²⋊C₄	—	53	6	no	—
508	C₂²⋊C₄	—	53	6	no	—
509	C₂²⋊C₄	C₂×C₄×D₅₃	1	6	yes	—
510	C₄²	—	53	12	no	—
511	C₄²	—	53	6	no	—
512	C₄²	—	53	1	no	—
513	C₄²	—	53	3	no	—
514	C₄²	C₂×C₄×D₅₃	1	6	yes	—
515	C₄²	C₂³×D₅₃	1	3	yes	—
516	C₄²	C₂³×D₅₃	1	1	yes	characteristic
517	C₂³	—	53	1	no	—
518	C₂³	—	53	3	no	—
519	C₂³	—	53	12	no	—
520	C₂³	—	212	6	no	—
521	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
522	Q₈	—	53	12	no	—
523	Q₈	C₄²×D₅₃	1	4	yes	—
524	D₄	—	53	24	no	—
525	D₄	—	53	12	no	—
526	D₄	C₄²×D₅₃	1	12	yes	—
527	C₂×C₄	—	4	6	no	—
528	C₂×C₄	—	53	12	no	—
529	C₂×C₄	—	53	3	no	—
530	C₂×C₄	—	53	1	no	—
531	C₂×C₄	—	53	12	no	—
532	C₂×C₄	—	53	12	no	—
533	C₂×C₄	—	53	2	no	—
534	C₂×C₄	—	53	6	no	—
535	C₂×C₄	—	53	6	no	—
536	C₂×C₄	—	53	2	no	—
537	C₂×C₄	—	53	6	no	—
538	C₂×C₄	—	53	2	no	—
539	C₂×C₄	—	212	12	no	—
540	C₂×C₄	—	212	6	no	—
541	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
542	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
543	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
544	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
545	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
546	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
547	C₈	—	53	2	no	—
548	C₈	—	53	6	no	—
549	C₈	—	53	6	no	—
550	C₈	—	53	2	no	—
551	C₈	C₂²×C₄×D₅₃	1	2	yes	—
552	C₈	C₂²×C₄×D₅₃	1	6	yes	—
553	C₈	C₂²×C₄×D₅₃	1	6	yes	—
554	C₈	C₂²×C₄×D₅₃	1	2	yes	—
555	C₂²	—	2	6	no	—
556	C₂²	—	53	2	no	—
557	C₂²	—	53	6	no	—
558	C₂²	—	106	6	no	—
559	C₂²	—	106	2	no	—
560	C₂²	—	212	12	no	—
561	C₂²	C₂×C₄²×D₅₃	1	6	yes	—
562	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
563	C₄	—	2	2	no	—
564	C₄	—	2	6	no	—
565	C₄	—	53	2	no	—
566	C₄	—	53	6	no	—
567	C₄	—	106	6	no	—
568	C₄	—	106	2	no	—
569	C₄	C₂×C₄²×D₅₃	1	6	yes	—
570	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
571	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
572	C₂	—	2	6	no	—
573	C₂	—	106	2	no	—
574	C₂	—	106	6	no	—
575	C₂	C₂²×C₄²×D₅₃	1	1	yes	characteristic
576	C₂	#20289426	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂⁴×C₄
prime radical	C₂×#41×D₅₃	C₂²
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#60058
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
4. #20283144	C₂	semidirect
5. #20283143	C₂	semidirect
6. #20283142	C₂	semidirect
7. #20283141	C₂	semidirect
8. #20283137	C₂	semidirect
9. #131×Dic₅₃	C₂	semidirect
10. C₄×#13133223	C₂	semidirect
11. #20283132	C₂	semidirect
12. #20283131	C₂	semidirect
13. C₂×#16810577	C₂	semidirect
14. #405×D₅₃	C₂	semidirect
15. C₄×#131×C₅₃	C₂	semidirect
17. #16811388	C₄	semidirect
24. #131×D₅₃	C₄	semidirect
38. #16810581	C₂²	semidirect
39. #16810581	C₂²	semidirect
41. #16810580	C₂²	semidirect
42. #16810579	C₂²	semidirect
43. #16810579	C₂²	semidirect
44. #16810578	C₂²	semidirect
45. #16810578	C₂²	semidirect
50. #16810577	C₂²	semidirect
51. #16810577	C₂²	semidirect
52. #16810577	C₂²	semidirect
53. #16810577	C₂²	semidirect
54. #16810577	C₂²	semidirect
55. C₄×#11719917	C₂²	semidirect
56. #16810575	C₂²	semidirect
57. M₁₆×Dic₅₃	C₂²	semidirect
58. M₁₆×Dic₅₃	C₂²	semidirect
59. C₄×#11719912	C₂²	semidirect
60. C₂×#13133128	C₂²	semidirect
61. C₂×#13133127	C₂²	semidirect
62. #97×D₅₃	C₂²	semidirect
63. C₂×C₈×Dic₅₃	C₂²	semidirect
64. C₄×C₈×D₅₃	C₂²	semidirect
68. C₅₃×#405	C₂²	semidirect
69. C₅₃×#405	C₂²	semidirect
70. C₄×M₁₆×C₅₃	C₂²	semidirect
71. C₂×C₄×C₈×C₅₃	C₂²	semidirect
80. #13133225	C₂×C₄	semidirect
81. #13133225	C₂×C₄	semidirect
82. #13133224	C₂×C₄	semidirect
83. #13133224	C₂×C₄	semidirect
84. #13133223	C₂×C₄	semidirect
85. #13133223	C₂×C₄	semidirect
88. C₂×#11719912	C₂×C₄	semidirect
94. M₁₆×D₅₃	C₂×C₄	semidirect
95. M₁₆×D₅₃	C₂×C₄	semidirect
98. C₂×C₈×D₅₃	C₂×C₄	semidirect
143. #131×C₅₃	C₂×C₄	semidirect
144. #131×C₅₃	C₂×C₄	semidirect
294. C₄×#131	D₅₃	semidirect
346. #405	C₂×D₅₃	semidirect
347. #405	C₂×D₅₃	semidirect
353. C₄×M₁₆	C₂×D₅₃	semidirect
359. C₂×C₄×C₈	C₂×D₅₃	semidirect
360. C₅₃	#60058	semidirect
376. #131	C₄×D₅₃	semidirect
377. #131	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×#16811388	2
C₂×#131×D₅₃	2
#20287914	2
#20283144	2
#20283143	2
#20283142	2
#20283141	2
#20283137	2
#131×Dic₅₃	2
C₄×#13133223	2
#20283132	2
#20283131	2
C₂×#16810577	2
#405×D₅₃	2
C₄×#131×C₅₃	2
#60058	53