Subgroups — #27394597

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810684	C₂	1	1	yes	maximal, characteristic
2	#20277213	C₂	1	1	yes	maximal, characteristic
3	#20275436	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275436	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275431	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275431	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274898	C₂	1	1	yes	maximal, characteristic
8	#20274898	C₂	1	1	yes	maximal, characteristic
9	#20274893	C₂	1	1	yes	maximal, characteristic
10	#20274893	C₂	1	1	yes	maximal, characteristic
11	#20274868	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274866	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274834	C₂	1	1	yes	maximal, characteristic, complemented
14	#20274831	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₅₃×#424	C₂	1	1	yes	maximal, characteristic
16	#16810684	—	2	4	no	—
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133146	C₂²	1	1	yes	characteristic
20	C₂×#13133146	C₂²	1	1	yes	characteristic
21	C₂×#13133146	C₂²	1	1	yes	characteristic
22	C₂×#13133146	C₂²	1	1	yes	characteristic
23	C₂×#13133146	C₂²	1	1	yes	characteristic
24	C₂×#13133146	C₂²	1	1	yes	characteristic
25	C₂×#13133146	C₂²	1	1	yes	characteristic
26	C₂×#13133146	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
29	C₂×#13133128	C₂²	1	1	yes	characteristic
30	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
31	#16810516	C₄	1	1	yes	characteristic
32	#16810516	C₄	1	1	yes	characteristic
33	#16810516	C₄	1	1	yes	characteristic
34	#16810516	C₄	1	1	yes	characteristic
35	#16810509	C₄	1	1	yes	characteristic
36	#16810509	C₄	1	1	yes	characteristic
37	#16810509	C₄	1	1	yes	characteristic
38	#16810509	C₄	1	1	yes	characteristic
39	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
43	#16810469	C₂²	1	1	yes	characteristic
44	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
45	#16810377	C₂²	1	1	yes	characteristic
46	#16810377	C₂²	1	1	yes	characteristic
47	#16810377	C₂²	1	1	yes	characteristic
48	#16810377	C₂²	1	1	yes	characteristic
49	#16810377	C₂²	1	1	yes	characteristic
50	#16810377	C₂²	1	1	yes	characteristic
51	#16810377	C₂²	1	1	yes	characteristic
52	#16810377	C₂²	1	1	yes	characteristic
53	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
54	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
56	C₅₃×#424	—	2	4	no	—
57	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
60	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
61	C₂²×#88322	C₂³	1	1	yes	characteristic
62	C₂²×#88322	C₂³	1	1	yes	characteristic
63	C₂²×#88322	C₂³	1	1	yes	characteristic
64	C₂²×#88322	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
81	C₄⋊#88322	—	2	2	no	—
82	C₄⋊#88322	—	2	2	no	—
83	#13133146	—	4	1	no	—
84	#13133146	—	4	1	no	—
85	#13133146	—	4	1	no	—
86	#13133146	—	4	1	no	—
87	#13133146	—	4	1	no	—
88	#13133146	—	4	1	no	—
89	#13133146	—	4	1	no	—
90	#13133146	—	4	1	no	—
91	C₂²⋊#88322	—	2	2	no	—
92	C₂²⋊#88322	—	2	2	no	—
93	#13133128	—	2	2	no	—
94	C₄×#88322	—	2	2	no	—
95	#13133124	C₂×C₄	1	1	yes	characteristic
96	#13133124	C₂×C₄	1	1	yes	characteristic
97	#13133124	C₂×C₄	1	1	yes	characteristic
98	#13133124	C₂×C₄	1	1	yes	characteristic
99	#13133124	C₂×C₄	1	1	yes	characteristic
100	#13133124	C₂×C₄	1	1	yes	characteristic
101	#13133124	C₂×C₄	1	1	yes	characteristic
102	#13133124	C₂×C₄	1	1	yes	characteristic
103	#13133124	C₂×C₄	1	1	yes	characteristic
104	#13133124	C₂×C₄	1	1	yes	characteristic
105	#13133124	C₂×C₄	1	1	yes	characteristic
106	#13133124	C₂×C₄	1	1	yes	characteristic
107	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×M₁₆×C₅₃	—	2	2	no	—
109	C₂×M₁₆×C₅₃	—	4	1	no	—
110	C₂×M₁₆×C₅₃	—	4	1	no	—
111	C₂²×C₈×C₅₃	—	2	2	no	—
112	C₂²×C₈×C₅₃	—	4	1	no	—
113	C₂²×C₈×C₅₃	—	4	1	no	—
114	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
115	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
116	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
117	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
118	#117×C₅₃	D₄	1	4	yes	—
119	#117×C₅₃	Q₈	1	4	yes	—
120	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
122	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
123	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
126	C₄⋊C₈×C₅₃	—	2	2	no	—
127	C₄⋊C₈×C₅₃	—	2	2	no	—
128	C₄⋊C₈×C₅₃	—	2	2	no	—
129	C₄⋊C₈×C₅₃	—	2	2	no	—
130	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	—	2	1	no	—
132	C₂²×C₄×C₅₃	—	2	1	no	—
133	C₂²×C₄×C₅₃	—	2	1	no	—
134	C₂²×C₄×C₅₃	—	2	1	no	—
135	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	M₁₆×C₅₃	—	4	4	no	—
152	C₂×C₈×C₅₃	—	2	1	no	—
153	C₂×C₈×C₅₃	—	2	1	no	—
154	C₂×C₈×C₅₃	—	2	1	no	—
155	C₂×C₈×C₅₃	—	2	1	no	—
156	C₂×C₈×C₅₃	—	2	1	no	—
157	C₂×C₈×C₅₃	—	2	1	no	—
158	C₂×C₈×C₅₃	—	2	1	no	—
159	C₂×C₈×C₅₃	—	2	1	no	—
160	C₂×C₈×C₅₃	—	2	1	no	—
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	2	1	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	4	4	no	—
165	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
166	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
167	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
168	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
169	C₄²×C₅₃	C₂×D₄	1	2	yes	—
170	C₄²×C₅₃	C₂×D₄	1	2	yes	—
171	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
172	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
173	C₂²⋊Dic₅₃	—	2	2	no	—
174	C₂²⋊Dic₅₃	—	2	2	no	—
175	C₂²⋊Dic₅₃	—	4	1	no	—
176	C₂²⋊Dic₅₃	—	4	1	no	—
177	C₄⋊Dic₅₃	—	2	2	no	—
178	C₄⋊Dic₅₃	—	2	2	no	—
179	C₄⋊Dic₅₃	—	4	1	no	—
180	C₄⋊Dic₅₃	—	4	1	no	—
181	C₄×Dic₅₃	—	2	1	no	—
182	C₄×Dic₅₃	—	2	1	no	—
183	C₄×Dic₅₃	—	2	1	no	—
184	C₄×Dic₅₃	—	2	1	no	—
185	C₄×Dic₅₃	—	2	1	no	—
186	C₄×Dic₅₃	—	2	1	no	—
187	C₄×Dic₅₃	—	2	1	no	—
188	C₄×Dic₅₃	—	2	1	no	—
189	C₄×Dic₅₃	—	2	2	no	—
190	C₄×Dic₅₃	—	2	2	no	—
191	C₄×Dic₅₃	—	4	1	no	—
192	C₄×Dic₅₃	—	4	1	no	—
193	C₂×#88322	—	2	1	no	—
194	C₂×#88322	—	2	1	no	—
195	C₂×#88322	—	2	1	no	—
196	C₂×#88322	—	2	1	no	—
197	C₂×#88322	—	2	1	no	—
198	C₂×#88322	—	2	1	no	—
199	C₂×#88322	—	2	1	no	—
200	C₂×#88322	—	2	1	no	—
201	C₂×#88322	—	2	1	no	—
202	C₂×#88322	—	2	1	no	—
203	C₂×#88322	—	2	1	no	—
204	C₂×#88322	—	2	1	no	—
205	C₂³×C₅₃	—	2	1	no	—
206	C₂³×C₅₃	—	2	1	no	—
207	C₂³×C₅₃	—	2	1	no	—
208	C₂³×C₅₃	—	2	1	no	—
209	C₂³×C₅₃	—	2	1	no	—
210	C₂³×C₅₃	—	2	1	no	—
211	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
212	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	1	no	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	2	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	2	no	—
227	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
228	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
229	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
230	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
231	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
232	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
234	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
237	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
238	C₂×Dic₅₃	—	2	1	no	—
239	C₂×Dic₅₃	—	2	1	no	—
240	C₂×Dic₅₃	—	2	1	no	—
241	C₂×Dic₅₃	—	2	1	no	—
242	C₂×Dic₅₃	—	2	1	no	—
243	C₂×Dic₅₃	—	2	1	no	—
244	C₂×Dic₅₃	—	2	1	no	—
245	C₂×Dic₅₃	—	2	1	no	—
246	C₂×Dic₅₃	—	2	1	no	—
247	C₂×Dic₅₃	—	2	1	no	—
248	C₂×Dic₅₃	—	2	1	no	—
249	C₂×Dic₅₃	—	2	1	no	—
250	C₂×Dic₅₃	—	2	1	no	—
251	C₂×Dic₅₃	—	2	1	no	—
252	C₂×Dic₅₃	—	2	1	no	—
253	C₂×Dic₅₃	—	2	1	no	—
254	C₂×Dic₅₃	—	2	1	no	—
255	C₂×Dic₅₃	—	2	1	no	—
256	C₂×Dic₅₃	—	2	1	no	—
257	C₂×Dic₅₃	—	2	1	no	—
258	C₂×Dic₅₃	—	2	1	no	—
259	C₂×Dic₅₃	—	2	1	no	—
260	C₂×Dic₅₃	#131	1	2	yes	—
261	C₂×Dic₅₃	#131	1	2	yes	—
262	C₈×C₅₃	—	4	1	no	—
263	C₈×C₅₃	—	4	1	no	—
264	C₈×C₅₃	—	4	1	no	—
265	C₈×C₅₃	—	4	1	no	—
266	#88322	—	4	1	no	—
267	#88322	—	4	1	no	—
268	#88322	—	4	1	no	—
269	#88322	—	4	1	no	—
270	#21980	—	53	1	no	maximal, Sylow
271	C₂²×C₅₃	—	2	2	no	—
272	C₂²×C₅₃	—	2	2	no	—
273	C₂²×C₅₃	—	2	2	no	—
274	C₂²×C₅₃	—	4	1	no	—
275	C₂²×C₅₃	—	4	1	no	—
276	C₂²×C₅₃	—	4	1	no	—
277	C₂²×C₅₃	—	4	1	no	—
278	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
279	C₂²×C₅₃	#520	1	1	yes	characteristic
280	C₂²×C₅₃	#520	1	1	yes	characteristic
281	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
282	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
283	C₂²×C₅₃	#568	1	1	yes	characteristic
284	C₂²×C₅₃	#568	1	1	yes	characteristic
285	C₄×C₅₃	—	2	2	no	—
286	C₄×C₅₃	—	2	2	no	—
287	C₄×C₅₃	—	2	2	no	—
288	C₄×C₅₃	—	2	2	no	—
289	C₄×C₅₃	—	2	2	no	—
290	C₄×C₅₃	#535	1	1	yes	characteristic
291	C₄×C₅₃	#536	1	1	yes	characteristic
292	C₄×C₅₃	#544	1	1	yes	characteristic
293	C₄×C₅₃	#544	1	1	yes	characteristic
294	Dic₅₃	—	2	2	no	—
295	Dic₅₃	—	2	2	no	—
296	Dic₅₃	—	4	1	no	—
297	Dic₅₃	—	4	1	no	—
298	Dic₅₃	—	4	1	no	—
299	Dic₅₃	—	4	1	no	—
300	Dic₅₃	—	4	1	no	—
301	Dic₅₃	—	4	1	no	—
302	C₂×#433	—	53	1	no	—
303	C₂×#424	D₅₃	1	1	yes	characteristic
304	#2298	—	53	1	no	—
305	#1940	—	53	1	no	—
306	#1940	—	53	1	no	—
307	#1939	—	53	1	no	—
308	#1939	—	53	1	no	—
309	#1851	—	53	1	no	—
310	#1851	—	53	1	no	—
311	#1850	—	53	1	no	—
312	#1850	—	53	1	no	—
313	#1843	—	53	1	no	—
314	#1842	—	53	1	no	—
315	#1832	—	53	1	no	—
316	#1831	—	53	1	no	—
317	C₂×C₅₃	—	4	2	no	—
318	C₂×C₅₃	#2317	1	1	yes	characteristic
319	C₂×C₅₃	#2906	1	1	yes	characteristic
320	C₂×C₅₃	#2906	1	1	yes	characteristic
321	C₂×C₅₃	#2992	1	1	yes	characteristic
322	C₂×C₅₃	#2993	1	1	yes	characteristic
323	C₂×C₅₃	#2994	1	1	yes	characteristic
324	C₂×C₅₃	#2995	1	1	yes	characteristic
325	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
326	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
327	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
328	#433	—	106	4	no	—
329	#424	—	2	4	no	—
330	C₂×C₄⋊C₈	—	53	1	no	—
331	C₂×C₄⋊C₈	—	53	1	no	—
332	C₂×C₄⋊C₈	—	53	1	no	—
333	C₂×C₄⋊C₈	—	53	1	no	—
334	C₂×C₄⋊C₈	—	53	1	no	—
335	C₂×C₄⋊C₈	—	53	1	no	—
336	C₂×C₄⋊C₈	—	53	1	no	—
337	C₂×C₄⋊C₈	—	53	1	no	—
338	C₂×C₄⋊C₈	—	53	1	no	—
339	C₂×C₄⋊C₈	—	53	1	no	—
340	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
343	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
344	C₂×C₂²⋊C₈	—	53	1	no	—
345	C₂×C₂²⋊C₈	—	53	1	no	—
346	C₂×#97	—	53	1	no	—
347	C₂×C₄×C₈	—	53	1	no	—
348	#388	—	53	1	no	—
349	#388	—	53	1	no	—
350	#388	—	53	1	no	—
351	#388	—	53	1	no	—
352	#387	—	53	1	no	—
353	#387	—	53	1	no	—
354	#387	—	53	1	no	—
355	#387	—	53	1	no	—
356	C₄×C₄⋊C₄	—	53	1	no	—
357	C₄×C₄⋊C₄	—	53	1	no	—
358	C₄×C₂²⋊C₄	—	53	1	no	—
359	C₄×C₂²⋊C₄	—	53	1	no	—
360	#376	—	53	1	no	—
361	C₄³	—	53	1	no	—
362	#336	—	53	1	no	—
363	#336	—	53	1	no	—
364	#336	—	53	1	no	—
365	#336	—	53	1	no	—
366	#336	—	53	1	no	—
367	#336	—	53	1	no	—
368	#336	—	53	1	no	—
369	#336	—	53	1	no	—
370	C₅₃	#21980	1	1	yes	characteristic, complemented, Sylow
371	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
372	C₂×M₁₆	—	2	2	no	—
373	C₂×M₁₆	—	4	1	no	—
374	C₂×M₁₆	—	4	1	no	—
375	C₂²×C₈	—	2	2	no	—
376	C₂²×C₈	—	4	1	no	—
377	C₂²×C₈	—	4	1	no	—
378	C₂²×C₈	—	53	1	no	—
379	C₂²×C₈	—	53	1	no	—
380	C₂²×C₈	—	53	1	no	—
381	C₂²×C₈	—	53	1	no	—
382	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
383	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
384	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
385	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
386	#117	Dic₁₀₆	1	4	yes	—
387	#117	C₂²⋊D₅₃	1	4	yes	—
388	C₂×C₄⋊C₄	—	53	1	no	—
389	C₂×C₄⋊C₄	—	53	1	no	—
390	C₂×C₄⋊C₄	—	53	1	no	—
391	C₂×C₄⋊C₄	—	53	1	no	—
392	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	—	53	1	no	—
395	C₂×C₂²⋊C₄	—	53	1	no	—
396	C₂×C₂²⋊C₄	—	53	1	no	—
397	C₂×C₂²⋊C₄	—	53	1	no	—
398	C₂×C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
400	C₂×C₄²	—	53	1	no	—
401	C₂×C₄²	—	53	1	no	—
402	C₂×C₄²	—	53	1	no	—
403	C₂×C₄²	—	53	1	no	—
404	C₂×C₄²	—	53	1	no	—
405	C₂×C₄²	—	53	1	no	—
406	C₂×C₄²	—	53	1	no	—
407	C₂×C₄²	—	53	1	no	—
408	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
409	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
410	C₄⋊C₈	—	2	2	no	—
411	C₄⋊C₈	—	2	2	no	—
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	—	2	2	no	—
414	C₄⋊C₈	—	106	2	no	—
415	C₄⋊C₈	—	106	2	no	—
416	C₄⋊C₈	—	212	1	no	—
417	C₄⋊C₈	—	212	1	no	—
418	C₄⋊C₈	—	212	1	no	—
419	C₄⋊C₈	—	212	1	no	—
420	C₄⋊C₈	—	212	1	no	—
421	C₄⋊C₈	—	212	1	no	—
422	C₄⋊C₈	—	212	1	no	—
423	C₄⋊C₈	—	212	1	no	—
424	C₂²⋊C₈	—	106	2	no	—
425	C₂²⋊C₈	—	106	2	no	—
426	#97	—	106	2	no	—
427	C₄×C₈	—	106	2	no	—
428	#95	—	53	1	no	—
429	#95	—	53	1	no	—
430	#95	—	53	1	no	—
431	#95	—	53	1	no	—
432	#95	—	53	1	no	—
433	#95	—	53	1	no	—
434	#95	—	53	1	no	—
435	#95	—	53	1	no	—
436	#95	—	53	1	no	—
437	#95	—	53	1	no	—
438	#95	—	53	1	no	—
439	#95	—	53	1	no	—
440	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	53	1	no	—
446	C₂²×C₄	—	53	1	no	—
447	C₂²×C₄	—	53	1	no	—
448	C₂²×C₄	—	53	1	no	—
449	C₂²×C₄	—	53	1	no	—
450	C₂²×C₄	—	53	1	no	—
451	C₂²×C₄	—	53	1	no	—
452	C₂²×C₄	—	53	1	no	—
453	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
454	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
456	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
458	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
459	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
460	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
461	M₁₆	—	4	4	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	4	4	no	—
475	C₂×C₈	—	106	1	no	—
476	C₂×C₈	—	106	1	no	—
477	C₂×C₈	—	106	1	no	—
478	C₂×C₈	—	106	1	no	—
479	C₂×C₈	—	106	1	no	—
480	C₂×C₈	—	106	1	no	—
481	C₂×C₈	—	106	1	no	—
482	C₂×C₈	—	106	1	no	—
483	C₂×C₈	—	106	1	no	—
484	C₂×C₈	—	106	1	no	—
485	C₂×C₈	—	106	1	no	—
486	C₂×C₈	—	106	1	no	—
487	C₄⋊C₄	—	106	2	no	—
488	C₄⋊C₄	—	106	2	no	—
489	C₄⋊C₄	—	212	1	no	—
490	C₄⋊C₄	—	212	1	no	—
491	C₄⋊C₄	#11719919	1	4	yes	—
492	C₄⋊C₄	#11719919	1	4	yes	—
493	C₂²⋊C₄	—	106	2	no	—
494	C₂²⋊C₄	—	106	2	no	—
495	C₂²⋊C₄	—	212	1	no	—
496	C₂²⋊C₄	—	212	1	no	—
497	C₂²⋊C₄	#11719919	1	4	yes	—
498	C₂²⋊C₄	#11719919	1	4	yes	—
499	C₄²	—	106	2	no	—
500	C₄²	—	106	2	no	—
501	C₄²	—	106	1	no	—
502	C₄²	—	106	1	no	—
503	C₄²	—	106	1	no	—
504	C₄²	—	106	1	no	—
505	C₄²	—	106	1	no	—
506	C₄²	—	106	1	no	—
507	C₄²	—	106	1	no	—
508	C₄²	—	106	1	no	—
509	C₄²	—	212	1	no	—
510	C₄²	—	212	1	no	—
511	C₄²	C₂×Dic₁₀₆	1	2	yes	—
512	C₄²	C₂×Dic₁₀₆	1	2	yes	—
513	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
514	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	C₂×#11719919	1	2	yes	—
522	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	53	2	no	—
538	C₂×C₄	—	53	2	no	—
539	C₂×C₄	—	106	1	no	—
540	C₂×C₄	—	106	1	no	—
541	C₂×C₄	—	106	1	no	—
542	C₂×C₄	—	106	1	no	—
543	C₂×C₄	—	106	1	no	—
544	C₂×C₄	—	106	1	no	—
545	C₂×C₄	—	106	1	no	—
546	C₂×C₄	—	106	1	no	—
547	C₂×C₄	—	106	1	no	—
548	C₂×C₄	—	106	1	no	—
549	C₂×C₄	—	106	1	no	—
550	C₂×C₄	—	106	1	no	—
551	C₂×C₄	—	106	1	no	—
552	C₂×C₄	—	106	1	no	—
553	C₂×C₄	—	106	1	no	—
554	C₂×C₄	—	106	1	no	—
555	C₂×C₄	—	106	1	no	—
556	C₂×C₄	—	106	1	no	—
557	C₂×C₄	—	106	1	no	—
558	C₂×C₄	—	106	1	no	—
559	C₂×C₄	—	106	1	no	—
560	C₂×C₄	—	106	1	no	—
561	C₂×C₄	C₂×#11719919	1	2	yes	—
562	C₂×C₄	C₂×#11719919	1	2	yes	—
563	C₂×C₄	C₂×#11719919	1	2	yes	—
564	C₂×C₄	C₂×#11719919	1	2	yes	—
565	C₂×C₄	C₂×#11719919	1	2	yes	—
566	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
567	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
568	C₂×C₄	#13133262	1	1	yes	characteristic
569	C₂×C₄	#13133262	1	1	yes	characteristic
570	C₂×C₄	#13133265	1	1	yes	characteristic
571	C₂×C₄	#13133265	1	1	yes	characteristic
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	—	212	1	no	—
577	C₈	—	212	1	no	—
578	C₈	—	212	1	no	—
579	C₈	—	212	1	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	C₂²×#11719919	1	1	yes	characteristic
588	C₂²	#16811104	1	1	yes	characteristic
589	C₂²	#16811104	1	1	yes	characteristic
590	C₂²	#131×D₅₃	1	1	yes	characteristic
591	C₂²	#131×D₅₃	1	1	yes	characteristic
592	C₂²	#16811388	1	1	yes	characteristic
593	C₂²	#16811388	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	106	2	no	—
600	C₄	—	106	2	no	—
601	C₄	—	212	1	no	—
602	C₄	—	212	1	no	—
603	C₄	—	212	1	no	—
604	C₄	—	212	1	no	—
605	C₄	—	212	1	no	—
606	C₄	—	212	1	no	—
607	C₄	#16811190	1	1	yes	characteristic
608	C₄	#16811197	1	1	yes	characteristic
609	C₄	#16811254	1	1	yes	characteristic
610	C₄	#16811254	1	1	yes	characteristic
611	C₂	—	4	2	no	—
612	C₂	#20277339	1	1	yes	characteristic
613	C₂	#20283281	1	1	yes	characteristic
614	C₂	#20283281	1	1	yes	characteristic
615	C₂	#20283852	1	1	yes	characteristic
616	C₂	#20283862	1	1	yes	characteristic
617	C₂	#20283873	1	1	yes	characteristic
618	C₂	#20283883	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₅₃×#424	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#424	D₅₃
hypocenter	C₅₃	#21980
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
3. #20275436	C₂	semidirect
4. #20275436	C₂	semidirect
5. #20275431	C₂	semidirect
6. #20275431	C₂	semidirect
11. #20274868	C₂	semidirect
12. #20274866	C₂	semidirect
13. #20274834	C₂	semidirect
14. #20274831	C₂	semidirect
370. C₅₃	#21980	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810684	2
#20277213	2
#20275436	2
#20275431	2
#20274898	2
#20274893	2
#20274868	2
#20274866	2
#20274834	2
#20274831	2
C₂×C₅₃×#424	2
#21980	53