Subgroups — #27403624

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275323	C₂	1	1	yes	maximal, characteristic, complemented
2	#20275323	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275281	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275274	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275013	C₂	1	1	yes	maximal, characteristic
6	#20275008	C₂	1	1	yes	maximal, characteristic
7	#20274858	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274786	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274699	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274499	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274481	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274461	C₂	1	1	yes	maximal, characteristic, complemented
14	#20274412	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#2293	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂³×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×#13133128	C₂²	1	1	yes	characteristic
27	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
30	#16810378	C₂²	1	1	yes	characteristic
31	#16810378	C₂²	1	1	yes	characteristic
32	#16810378	C₂²	1	1	yes	characteristic
33	#16810378	C₂²	1	1	yes	characteristic
34	#16810378	C₂²	1	1	yes	characteristic
35	#16810378	C₂²	1	1	yes	characteristic
36	#16810378	C₂²	1	1	yes	characteristic
37	#16810378	C₂²	1	1	yes	characteristic
38	#16810378	C₂²	1	1	yes	characteristic
39	#16810378	C₂²	1	1	yes	characteristic
40	#16810378	C₂²	1	1	yes	characteristic
41	#16810378	C₂²	1	1	yes	characteristic
42	#16810378	C₂²	1	1	yes	characteristic
43	#16810378	C₂²	1	1	yes	characteristic
44	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#390	C₄	1	1	yes	characteristic
48	C₅₃×#390	C₄	1	1	yes	characteristic
49	C₅₃×#389	C₄	1	1	yes	characteristic
50	C₅₃×#389	C₄	1	1	yes	characteristic
51	C₅₃×#388	C₄	1	1	yes	characteristic
52	C₅₃×#388	C₄	1	1	yes	characteristic
53	C₅₃×#388	C₄	1	1	yes	characteristic
54	C₅₃×#388	C₄	1	1	yes	characteristic
55	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
58	C₄³×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×#11719917	—	2	2	no	—
60	C₂×#11719917	—	2	2	no	—
61	C₂×#11719917	—	4	1	no	—
62	C₂²×#88322	—	2	2	no	—
63	C₂²×#88322	—	2	2	no	—
64	C₂²×#88322	—	4	1	no	—
65	C₂²×#88322	C₂³	1	1	yes	characteristic
66	C₂²×#88322	C₂³	1	1	yes	characteristic
67	C₂²×#88322	C₂³	1	1	yes	characteristic
68	C₂²×#88322	C₂³	1	1	yes	characteristic
69	C₂²×#88322	C₂³	1	1	yes	characteristic
70	C₂²×#88322	C₂³	1	1	yes	characteristic
71	C₂²×#88322	C₂³	1	1	yes	characteristic
72	C₂²×#88322	C₂³	1	1	yes	characteristic
73	C₄⋊#88322	—	4	1	no	—
74	C₄⋊#88322	—	4	1	no	—
75	C₂²⋊#88322	—	2	2	no	—
76	C₂²⋊#88322	—	2	2	no	—
77	C₂²⋊#88322	—	2	2	no	—
78	C₂²⋊#88322	—	2	2	no	—
79	C₂²⋊#88322	—	4	1	no	—
80	C₂²⋊#88322	—	4	1	no	—
81	#13133128	—	4	1	no	—
82	C₄×#88322	—	2	2	no	—
83	C₄×#88322	—	2	2	no	—
84	C₄×#88322	—	4	1	no	—
85	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×D₄×C₅₃	—	2	2	no	—
89	C₄×D₄×C₅₃	—	2	2	no	—
90	#117×C₅₃	—	4	1	no	—
91	#117×C₅₃	—	4	1	no	—
92	#117×C₅₃	—	4	1	no	—
93	#117×C₅₃	—	4	1	no	—
94	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
121	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
122	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
123	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
124	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
125	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
126	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄×C₅₃	—	2	2	no	—
129	C₂×D₄×C₅₃	—	2	2	no	—
130	C₂×D₄×C₅₃	—	4	1	no	—
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	—	2	1	no	—
136	C₂²×C₄×C₅₃	—	2	1	no	—
137	C₂²×C₄×C₅₃	—	2	1	no	—
138	C₂²×C₄×C₅₃	—	2	1	no	—
139	C₂²×C₄×C₅₃	—	2	1	no	—
140	C₂²×C₄×C₅₃	—	2	1	no	—
141	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	#41	1	2	yes	—
154	C₂²×C₄×C₅₃	#41	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₄⋊C₄×C₅₃	—	2	2	no	—
157	C₄⋊C₄×C₅₃	—	2	2	no	—
158	C₄⋊C₄×C₅₃	—	4	1	no	—
159	C₄⋊C₄×C₅₃	—	4	1	no	—
160	C₄⋊C₄×C₅₃	—	4	1	no	—
161	C₄⋊C₄×C₅₃	—	4	1	no	—
162	C₄⋊C₄×C₅₃	—	4	1	no	—
163	C₂²⋊C₄×C₅₃	—	2	2	no	—
164	C₂²⋊C₄×C₅₃	—	2	2	no	—
165	C₂²⋊C₄×C₅₃	—	2	2	no	—
166	C₂²⋊C₄×C₅₃	—	2	2	no	—
167	C₂²⋊C₄×C₅₃	—	2	2	no	—
168	C₂²⋊C₄×C₅₃	—	2	2	no	—
169	C₂²⋊C₄×C₅₃	—	4	1	no	—
170	C₂²⋊C₄×C₅₃	—	4	1	no	—
171	C₂²⋊C₄×C₅₃	—	4	1	no	—
172	C₂²⋊C₄×C₅₃	—	4	1	no	—
173	C₄²×C₅₃	—	2	2	no	—
174	C₄²×C₅₃	—	2	2	no	—
175	C₄²×C₅₃	—	2	2	no	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	—	2	1	no	—
178	C₄²×C₅₃	—	2	1	no	—
179	C₄²×C₅₃	—	2	1	no	—
180	C₄²×C₅₃	—	2	1	no	—
181	C₄²×C₅₃	—	2	1	no	—
182	C₄²×C₅₃	—	2	1	no	—
183	C₄²×C₅₃	—	2	1	no	—
184	C₄²×C₅₃	—	2	1	no	—
185	C₄²×C₅₃	—	2	1	no	—
186	C₄²×C₅₃	—	4	1	no	—
187	#11719917	—	4	4	no	—
188	C₂×#88322	—	2	1	no	—
189	C₂×#88322	—	2	1	no	—
190	C₂×#88322	—	2	1	no	—
191	C₂×#88322	—	2	1	no	—
192	C₂×#88322	—	2	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	C₂×#88322	—	2	1	no	—
206	C₂×#88322	—	2	1	no	—
207	C₂×#88322	—	2	1	no	—
208	C₂×#88322	—	2	1	no	—
209	C₂×#88322	—	2	1	no	—
210	C₂×#88322	—	4	4	no	—
211	C₂×#88322	#41	1	2	yes	—
212	C₂×#88322	#41	1	2	yes	—
213	C₂³×C₅₃	—	2	1	no	—
214	C₂³×C₅₃	—	2	1	no	—
215	C₂³×C₅₃	—	2	1	no	—
216	C₂³×C₅₃	—	2	1	no	—
217	C₂³×C₅₃	—	2	1	no	—
218	C₂³×C₅₃	—	2	1	no	—
219	C₂³×C₅₃	—	2	1	no	—
220	C₂³×C₅₃	—	2	1	no	—
221	C₂³×C₅₃	—	2	1	no	—
222	C₂³×C₅₃	—	2	1	no	—
223	C₂³×C₅₃	—	2	1	no	—
224	C₂³×C₅₃	—	2	1	no	—
225	C₂³×C₅₃	—	2	1	no	—
226	C₂³×C₅₃	#117	1	2	yes	—
227	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
228	D₄×C₅₃	—	4	4	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	1	no	—
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	1	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	—	2	1	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	1	no	—
251	C₂×C₄×C₅₃	—	2	1	no	—
252	C₂×C₄×C₅₃	—	2	1	no	—
253	C₂×C₄×C₅₃	—	2	1	no	—
254	C₂×C₄×C₅₃	—	2	1	no	—
255	C₂×C₄×C₅₃	—	2	1	no	—
256	C₂×C₄×C₅₃	—	2	1	no	—
257	C₂×C₄×C₅₃	—	2	1	no	—
258	C₂×C₄×C₅₃	—	2	1	no	—
259	C₂×C₄×C₅₃	—	2	1	no	—
260	C₂×C₄×C₅₃	—	2	1	no	—
261	C₂×C₄×C₅₃	—	2	1	no	—
262	C₂×C₄×C₅₃	—	2	1	no	—
263	C₂×C₄×C₅₃	—	2	1	no	—
264	C₂×C₄×C₅₃	—	2	1	no	—
265	C₂×C₄×C₅₃	—	4	1	no	—
266	C₂×C₄×C₅₃	—	4	1	no	—
267	C₂×C₄×C₅₃	—	4	1	no	—
268	C₂×C₄×C₅₃	—	4	1	no	—
269	C₂×C₄×C₅₃	#117	1	2	yes	—
270	C₂×C₄×C₅₃	#131	1	2	yes	—
271	C₂×C₄×C₅₃	#131	1	2	yes	—
272	C₂×C₄×C₅₃	#131	1	2	yes	—
273	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
275	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
276	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
278	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
279	#88322	—	2	2	no	—
280	#88322	—	2	2	no	—
281	#88322	—	4	1	no	—
282	#88322	—	4	1	no	—
283	#88322	—	4	1	no	—
284	#88322	—	4	1	no	—
285	#88322	—	4	1	no	—
286	#88322	—	4	1	no	—
287	#22789	—	53	1	no	maximal, Sylow
288	C₂²×C₅₃	—	2	2	no	—
289	C₂²×C₅₃	—	2	2	no	—
290	C₂²×C₅₃	—	2	2	no	—
291	C₂²×C₅₃	—	4	1	no	—
292	C₂²×C₅₃	—	4	1	no	—
293	C₂²×C₅₃	—	4	1	no	—
294	C₂²×C₅₃	—	4	1	no	—
295	C₂²×C₅₃	—	4	1	no	—
296	C₂²×C₅₃	—	4	1	no	—
297	C₂²×C₅₃	—	4	1	no	—
298	C₂²×C₅₃	—	4	1	no	—
299	C₂²×C₅₃	—	4	1	no	—
300	C₂²×C₅₃	—	4	1	no	—
301	C₂²×C₅₃	—	4	1	no	—
302	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
303	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
304	C₂²×C₅₃	#518	1	1	yes	characteristic
305	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
306	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
307	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
308	C₂²×C₅₃	#568	1	1	yes	characteristic
309	C₄×C₅₃	—	2	2	no	—
310	C₄×C₅₃	—	2	2	no	—
311	C₄×C₅₃	—	2	2	no	—
312	C₄×C₅₃	—	2	2	no	—
313	C₄×C₅₃	—	4	1	no	—
314	C₄×C₅₃	—	4	1	no	—
315	C₄×C₅₃	—	4	1	no	—
316	C₄×C₅₃	—	4	1	no	—
317	C₄×C₅₃	—	4	1	no	—
318	C₄×C₅₃	—	4	1	no	—
319	C₄×C₅₃	—	4	1	no	—
320	C₄×C₅₃	—	4	1	no	—
321	C₄×C₅₃	#551	1	1	yes	characteristic
322	C₄×C₅₃	#551	1	1	yes	characteristic
323	C₄×C₅₃	#553	1	1	yes	characteristic
324	C₄×C₅₃	#553	1	1	yes	characteristic
325	#2293	D₅₃	1	1	yes	characteristic
326	#1923	—	53	1	no	—
327	#1923	—	53	1	no	—
328	#1917	—	53	1	no	—
329	#1916	—	53	1	no	—
330	#1870	—	53	1	no	—
331	#1869	—	53	1	no	—
332	#1839	—	53	1	no	—
333	#1818	—	53	1	no	—
334	#1798	—	53	1	no	—
335	#1753	—	53	1	no	—
336	C₈×C₂²⋊C₄	—	53	1	no	—
337	#1750	—	53	1	no	—
338	#1744	—	53	1	no	—
339	#1729	—	53	1	no	—
340	C₂×C₅₃	—	4	2	no	—
341	C₂×C₅₃	—	8	1	no	—
342	C₂×C₅₃	#2353	1	1	yes	characteristic
343	C₂×C₅₃	#2922	1	1	yes	characteristic
344	C₂×C₅₃	#2924	1	1	yes	characteristic
345	C₂×C₅₃	#2966	1	1	yes	characteristic
346	C₂×C₅₃	#2968	1	1	yes	characteristic
347	C₂×C₅₃	#2979	1	1	yes	characteristic
348	C₂×C₅₃	#2981	1	1	yes	characteristic
349	C₂²×M₁₆	—	53	1	no	—
350	C₂³×C₈	—	53	1	no	—
351	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
352	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
353	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
354	C₂×C₄⋊C₈	—	53	1	no	—
355	C₂×C₄⋊C₈	—	53	1	no	—
356	C₂×C₂²⋊C₈	—	53	1	no	—
357	C₂×C₂²⋊C₈	—	53	1	no	—
358	C₂×C₂²⋊C₈	—	53	1	no	—
359	C₂×C₂²⋊C₈	—	53	1	no	—
360	C₂×C₂²⋊C₈	—	53	1	no	—
361	C₂×C₂²⋊C₈	—	53	1	no	—
362	C₂×#97	—	53	1	no	—
363	C₂×C₄×C₈	—	53	1	no	—
364	C₂×C₄×C₈	—	53	1	no	—
365	C₂×C₄×C₈	—	53	1	no	—
366	#390	Dic₅₃	1	1	yes	characteristic
367	#390	Dic₅₃	1	1	yes	characteristic
368	#389	Dic₅₃	1	1	yes	characteristic
369	#389	Dic₅₃	1	1	yes	characteristic
370	#388	Dic₅₃	1	1	yes	characteristic
371	#388	Dic₅₃	1	1	yes	characteristic
372	#388	Dic₅₃	1	1	yes	characteristic
373	#388	Dic₅₃	1	1	yes	characteristic
374	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
375	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
376	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
377	C₄³	C₂×D₅₃	1	1	yes	characteristic
378	#336	—	53	1	no	—
379	#336	—	53	1	no	—
380	#336	—	53	1	no	—
381	#336	—	53	1	no	—
382	#336	—	53	1	no	—
383	#336	—	53	1	no	—
384	#336	—	53	1	no	—
385	#336	—	53	1	no	—
386	#336	—	53	1	no	—
387	#336	—	53	1	no	—
388	#336	—	53	1	no	—
389	#336	—	53	1	no	—
390	#336	—	53	1	no	—
391	#336	—	53	1	no	—
392	C₅₃	#22789	1	1	yes	characteristic, complemented, Sylow
393	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
394	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
395	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
396	C₂×M₁₆	—	106	2	no	—
397	C₂×M₁₆	—	106	2	no	—
398	C₂×M₁₆	—	212	1	no	—
399	C₂²×C₈	—	53	1	no	—
400	C₂²×C₈	—	53	1	no	—
401	C₂²×C₈	—	53	1	no	—
402	C₂²×C₈	—	53	1	no	—
403	C₂²×C₈	—	53	1	no	—
404	C₂²×C₈	—	53	1	no	—
405	C₂²×C₈	—	53	1	no	—
406	C₂²×C₈	—	53	1	no	—
407	C₂²×C₈	—	106	2	no	—
408	C₂²×C₈	—	106	2	no	—
409	C₂²×C₈	—	212	1	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	#117	—	4	1	no	—
413	#117	—	4	1	no	—
414	#117	—	4	1	no	—
415	#117	—	4	1	no	—
416	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
424	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
428	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
429	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
433	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
434	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
435	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
436	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
437	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
438	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
439	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
440	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
441	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
442	C₄⋊C₈	—	212	1	no	—
443	C₄⋊C₈	—	212	1	no	—
444	C₂²⋊C₈	—	106	2	no	—
445	C₂²⋊C₈	—	106	2	no	—
446	C₂²⋊C₈	—	106	2	no	—
447	C₂²⋊C₈	—	106	2	no	—
448	C₂²⋊C₈	—	212	1	no	—
449	C₂²⋊C₈	—	212	1	no	—
450	#97	—	212	1	no	—
451	C₄×C₈	—	106	2	no	—
452	C₄×C₈	—	106	2	no	—
453	C₄×C₈	—	212	1	no	—
454	#95	C₂×Dic₅₃	1	1	yes	characteristic
455	#95	C₂×Dic₅₃	1	1	yes	characteristic
456	#95	C₂×Dic₅₃	1	1	yes	characteristic
457	#95	C₂×Dic₅₃	1	1	yes	characteristic
458	#95	C₂×Dic₅₃	1	1	yes	characteristic
459	#95	C₂×Dic₅₃	1	1	yes	characteristic
460	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
461	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	4	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	#11719945	1	2	yes	—
476	C₂²×C₄	#11719945	1	2	yes	—
477	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
478	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
479	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
480	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
481	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
482	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
483	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
484	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
485	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
486	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
487	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
488	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
489	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
490	M₁₆	—	212	4	no	—
491	C₂×C₈	—	53	2	no	—
492	C₂×C₈	—	53	2	no	—
493	C₂×C₈	—	106	1	no	—
494	C₂×C₈	—	106	1	no	—
495	C₂×C₈	—	106	1	no	—
496	C₂×C₈	—	106	1	no	—
497	C₂×C₈	—	106	1	no	—
498	C₂×C₈	—	106	1	no	—
499	C₂×C₈	—	106	1	no	—
500	C₂×C₈	—	106	1	no	—
501	C₂×C₈	—	106	1	no	—
502	C₂×C₈	—	106	1	no	—
503	C₂×C₈	—	106	1	no	—
504	C₂×C₈	—	106	1	no	—
505	C₂×C₈	—	106	1	no	—
506	C₂×C₈	—	106	1	no	—
507	C₂×C₈	—	106	1	no	—
508	C₂×C₈	—	106	1	no	—
509	C₂×C₈	—	106	1	no	—
510	C₂×C₈	—	106	1	no	—
511	C₂×C₈	—	106	1	no	—
512	C₂×C₈	—	106	1	no	—
513	C₂×C₈	—	106	1	no	—
514	C₂×C₈	—	106	1	no	—
515	C₂×C₈	—	212	4	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	4	1	no	—
519	C₄⋊C₄	—	4	1	no	—
520	C₄⋊C₄	—	4	1	no	—
521	C₄⋊C₄	—	4	1	no	—
522	C₄⋊C₄	—	4	1	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	4	1	no	—
530	C₂²⋊C₄	—	4	1	no	—
531	C₂²⋊C₄	—	4	1	no	—
532	C₂²⋊C₄	—	4	1	no	—
533	C₄²	—	2	2	no	—
534	C₄²	—	2	2	no	—
535	C₄²	—	2	2	no	—
536	C₄²	—	2	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	4	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	1	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	2	1	no	—
560	C₂³	#13133176	1	2	yes	—
561	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
562	D₄	—	4	4	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	#13133176	1	2	yes	—
604	C₂×C₄	#13133224	1	2	yes	—
605	C₂×C₄	#13133224	1	2	yes	—
606	C₂×C₄	#13133224	1	2	yes	—
607	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
608	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
609	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
610	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
611	C₂×C₄	#13133262	1	1	yes	characteristic
612	C₂×C₄	#13133262	1	1	yes	characteristic
613	C₈	—	106	2	no	—
614	C₈	—	106	2	no	—
615	C₈	—	212	1	no	—
616	C₈	—	212	1	no	—
617	C₈	—	212	1	no	—
618	C₈	—	212	1	no	—
619	C₈	—	212	1	no	—
620	C₈	—	212	1	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	4	1	no	—
625	C₂²	—	4	1	no	—
626	C₂²	—	4	1	no	—
627	C₂²	—	4	1	no	—
628	C₂²	—	4	1	no	—
629	C₂²	—	4	1	no	—
630	C₂²	—	4	1	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	C₂×#13133176	1	1	yes	characteristic
636	C₂²	#41×Dic₅₃	1	1	yes	characteristic
637	C₂²	#16811098	1	1	yes	characteristic
638	C₂²	C₂×#13133224	1	1	yes	characteristic
639	C₂²	C₂×#13133224	1	1	yes	characteristic
640	C₂²	C₂×#13133224	1	1	yes	characteristic
641	C₂²	#16811390	1	1	yes	characteristic
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	2	2	no	—
645	C₄	—	2	2	no	—
646	C₄	—	4	1	no	—
647	C₄	—	4	1	no	—
648	C₄	—	4	1	no	—
649	C₄	—	4	1	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	1	no	—
652	C₄	—	4	1	no	—
653	C₄	—	4	1	no	—
654	C₄	#16811302	1	1	yes	characteristic
655	C₄	#16811302	1	1	yes	characteristic
656	C₄	#16811322	1	1	yes	characteristic
657	C₄	#16811322	1	1	yes	characteristic
658	C₂	—	4	2	no	—
659	C₂	—	8	1	no	—
660	C₂	#20277655	1	1	yes	characteristic
661	C₂	#20283359	1	1	yes	characteristic
662	C₂	#20283373	1	1	yes	characteristic
663	C₂	#20283691	1	1	yes	characteristic
664	C₂	#20283711	1	1	yes	characteristic
665	C₂	#20283782	1	1	yes	characteristic
666	C₂	#20283791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2293	C₂
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2293	D₅₃
hypocenter	C₅₃	#22789
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2293

Complements

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

N	H ≅ G/N	type
1. #20275323	C₂	semidirect
2. #20275323	C₂	semidirect
3. #20275281	C₂	semidirect
4. #20275274	C₂	semidirect
7. #20274858	C₂	semidirect
8. #20274786	C₂	semidirect
9. #20274699	C₂	semidirect
10. #20274499	C₂	semidirect
11. C₂²⋊C₄×#88322	C₂	semidirect
12. #20274481	C₂	semidirect
13. #20274461	C₂	semidirect
14. #20274412	C₂	semidirect
392. C₅₃	#22789	semidirect

Maximal subgroups

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

subgroup	index
#20275323	2
#20275281	2
#20275274	2
#20275013	2
#20275008	2
#20274858	2
#20274786	2
#20274699	2
#20274499	2
C₂²⋊C₄×#88322	2
#20274481	2
#20274461	2
#20274412	2
C₅₃×#2293	2
#22789	53