Subgroups — #27401631

Order 13568.

932 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810680	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
3	#20275439	C₂	1	2	yes	maximal, complemented
4	#20275281	C₂	1	2	yes	maximal, complemented
5	#20275274	C₂	1	2	yes	maximal, complemented
6	#20274783	C₂	1	1	yes	maximal, characteristic
7	#20274783	C₂	1	1	yes	maximal, characteristic
8	#20274695	C₂	1	1	yes	maximal, characteristic
9	#20274590	C₂	1	2	yes	maximal, complemented
10	#20274477	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	#16810680	—	2	4	no	—
13	#16810676	—	2	4	no	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×C₄⋊#88322	C₂²	1	2	yes	—
17	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×#13133128	C₂²	1	2	yes	—
22	C₂×#13133128	C₂²	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	2	yes	—
24	#16810378	C₂²	1	2	yes	—
25	#16810378	C₂²	1	2	yes	—
26	#16810378	C₂²	1	2	yes	—
27	#16810378	C₂²	1	2	yes	—
28	#16810378	C₂²	1	2	yes	—
29	#16810378	C₂²	1	2	yes	—
30	C₂²×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
32	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×#95×C₅₃	C₄	1	1	yes	characteristic
44	C₂×#95×C₅₃	C₄	1	1	yes	characteristic
45	C₂²×#88322	C₂³	1	2	yes	—
46	C₂²×#88322	C₂³	1	2	yes	—
47	C₂²×#88322	C₂³	1	2	yes	—
48	C₂²×#88322	C₂³	1	2	yes	—
49	C₄⋊#88322	—	2	4	no	—
50	C₄⋊#88322	—	2	4	no	—
51	C₄⋊#88322	—	2	4	no	—
52	C₂²⋊#88322	—	2	2	no	—
53	C₂²⋊#88322	—	2	2	no	—
54	C₂²⋊#88322	—	2	2	no	—
55	C₂²⋊#88322	D₄	1	4	yes	—
56	#13133128	—	2	4	no	—
57	#13133128	—	2	4	no	—
58	C₄×#88322	—	2	4	no	—
59	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	—	2	4	no	—
77	C₂×C₄²×C₅₃	—	2	2	no	—
78	C₂×C₄²×C₅₃	—	2	2	no	—
79	C₂×C₄²×C₅₃	—	2	2	no	—
80	C₂×C₄²×C₅₃	—	2	2	no	—
81	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
94	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
95	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
96	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
97	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
98	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
99	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
100	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	2	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	4	1	no	—
119	C₂²×C₄×C₅₃	—	4	1	no	—
120	C₂²×C₄×C₅₃	—	4	1	no	—
121	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₅₃	#41	1	2	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	2	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	2	2	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	—	2	2	no	—
151	C₄²×C₅₃	—	2	2	no	—
152	C₄²×C₅₃	—	2	1	no	—
153	C₄²×C₅₃	—	2	1	no	—
154	C₄²×C₅₃	—	2	1	no	—
155	C₄²×C₅₃	—	2	1	no	—
156	C₄²×C₅₃	—	2	2	no	—
157	C₄²×C₅₃	—	2	1	no	—
158	C₄²×C₅₃	—	2	1	no	—
159	C₄²×C₅₃	—	2	1	no	—
160	C₄²×C₅₃	—	2	1	no	—
161	C₄²×C₅₃	#41	1	4	yes	—
162	C₂×#88322	—	2	2	no	—
163	C₂×#88322	—	2	2	no	—
164	C₂×#88322	—	2	2	no	—
165	C₂×#88322	—	2	2	no	—
166	C₂×#88322	—	2	2	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	—	2	2	no	—
169	C₂×#88322	—	2	2	no	—
170	C₂×#88322	—	2	2	no	—
171	C₂×#88322	—	2	2	no	—
172	C₂×#88322	—	2	2	no	—
173	C₂×#88322	C₂×D₄	1	4	yes	—
174	C₂³×C₅₃	—	2	1	no	—
175	C₂³×C₅₃	—	2	1	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	—	2	1	no	—
178	C₂³×C₅₃	C₄×D₄	1	2	yes	—
179	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
180	C₂³×C₅₃	#131	1	2	yes	—
181	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	—	2	8	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	1	no	—
192	C₂×C₄×C₅₃	—	2	1	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	8	no	—
200	C₂×C₄×C₅₃	—	2	4	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	1	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
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	—	4	4	no	—
218	C₂×C₄×C₅₃	—	4	4	no	—
219	C₂×C₄×C₅₃	—	4	4	no	—
220	C₂×C₄×C₅₃	—	4	4	no	—
221	C₂×C₄×C₅₃	#117	1	2	yes	—
222	C₂×C₄×C₅₃	#117	1	2	yes	—
223	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
224	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
225	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
226	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
227	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
228	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
229	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
232	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
233	#88322	—	2	4	no	—
234	#88322	—	4	2	no	—
235	#88322	—	4	2	no	—
236	#88322	—	4	2	no	—
237	#22601	—	53	1	no	maximal, Sylow
238	C₂²×C₅₃	—	2	2	no	—
239	C₂²×C₅₃	—	2	2	no	—
240	C₂²×C₅₃	—	2	2	no	—
241	C₂²×C₅₃	—	2	2	no	—
242	C₂²×C₅₃	—	2	2	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	#435	1	4	yes	—
245	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
246	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
247	C₂²×C₅₃	#519	1	1	yes	characteristic
248	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
249	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
250	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
251	C₂²×C₅₃	#568	1	1	yes	characteristic
252	C₄×C₅₃	—	2	2	no	—
253	C₄×C₅₃	—	2	2	no	—
254	C₄×C₅₃	—	2	2	no	—
255	C₄×C₅₃	—	2	2	no	—
256	C₄×C₅₃	—	2	2	no	—
257	C₄×C₅₃	—	2	2	no	—
258	C₄×C₅₃	—	2	2	no	—
259	C₄×C₅₃	—	2	2	no	—
260	C₄×C₅₃	—	4	1	no	—
261	C₄×C₅₃	—	4	1	no	—
262	C₄×C₅₃	—	4	1	no	—
263	C₄×C₅₃	—	4	1	no	—
264	C₄×C₅₃	—	4	1	no	—
265	C₄×C₅₃	#547	1	2	yes	—
266	C₄×C₅₃	#552	1	2	yes	—
267	C₂×#432	—	53	1	no	—
268	C₂×#431	—	53	1	no	—
269	C₂×C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
270	#1940	—	53	2	no	—
271	#1917	—	53	2	no	—
272	#1916	—	53	2	no	—
273	#1817	—	53	1	no	—
274	#1817	—	53	1	no	—
275	#1797	—	53	1	no	—
276	#1771	—	53	2	no	—
277	#1749	—	53	1	no	—
278	C₂×C₅₃	—	2	4	no	—
279	C₂×C₅₃	#2344	1	1	yes	characteristic
280	C₂×C₅₃	#2920	1	1	yes	characteristic
281	C₂×C₅₃	#2924	1	1	yes	characteristic
282	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
283	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
284	C₂×C₅₃	#2983	1	1	yes	characteristic
285	C₂×C₅₃	#2984	1	1	yes	characteristic
286	C₂²×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
287	C₂²×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
288	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
289	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
290	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
291	#432	—	106	4	no	—
292	#431	—	106	4	no	—
293	C₂×C₄⋊C₈	—	53	2	no	—
294	C₂×C₄⋊C₈	—	53	2	no	—
295	C₂×C₄⋊C₈	—	53	2	no	—
296	C₂×C₂²⋊C₈	—	53	1	no	—
297	C₂×C₂²⋊C₈	—	53	1	no	—
298	C₂×C₂²⋊C₈	—	53	1	no	—
299	C₂×C₂²⋊C₈	—	53	1	no	—
300	C₂×#97	—	53	2	no	—
301	C₂×#97	—	53	2	no	—
302	C₂×C₄×C₈	—	53	2	no	—
303	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
304	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
305	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
306	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
307	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
308	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
309	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
310	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
311	C₂×#95	Dic₅₃	1	1	yes	characteristic
312	C₂×#95	Dic₅₃	1	1	yes	characteristic
313	#336	—	53	2	no	—
314	#336	—	53	2	no	—
315	#336	—	53	2	no	—
316	#336	—	53	2	no	—
317	#336	—	53	2	no	—
318	#336	—	53	2	no	—
319	C₅₃	#22601	1	1	yes	characteristic, complemented, Sylow
320	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
321	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
322	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
323	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
324	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
325	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
326	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
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	C₂×C₄⋊C₄	—	2	4	no	—
332	C₂×C₄⋊C₄	—	2	4	no	—
333	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
341	C₂×C₄²	—	2	2	no	—
342	C₂×C₄²	—	2	2	no	—
343	C₂×C₄²	—	2	2	no	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	—	2	4	no	—
346	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
347	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
348	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
349	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
350	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
351	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
352	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
353	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
354	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
355	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
356	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
357	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
358	C₄⋊C₈	—	106	4	no	—
359	C₄⋊C₈	—	106	4	no	—
360	C₄⋊C₈	—	106	4	no	—
361	C₂²⋊C₈	—	53	4	no	—
362	C₂²⋊C₈	—	106	2	no	—
363	C₂²⋊C₈	—	106	2	no	—
364	C₂²⋊C₈	—	106	2	no	—
365	#97	—	106	4	no	—
366	#97	—	106	4	no	—
367	C₄×C₈	—	106	4	no	—
368	#95	C₂×Dic₅₃	1	1	yes	characteristic
369	#95	C₂×Dic₅₃	1	1	yes	characteristic
370	#95	C₂×Dic₅₃	1	1	yes	characteristic
371	#95	C₂×Dic₅₃	1	1	yes	characteristic
372	#95	C₂×Dic₅₃	1	1	yes	characteristic
373	#95	C₂×Dic₅₃	1	1	yes	characteristic
374	#95	C₂×Dic₅₃	1	1	yes	characteristic
375	#95	C₂×Dic₅₃	1	1	yes	characteristic
376	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	4	1	no	—
394	C₂²×C₄	—	4	1	no	—
395	C₂²×C₄	—	4	1	no	—
396	C₂²×C₄	#11719917	1	4	yes	—
397	C₂²×C₄	D₄×D₅₃	1	2	yes	—
398	C₂²×C₄	#11719947	1	2	yes	—
399	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
400	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
401	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
402	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
403	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
404	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
405	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
406	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
407	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
408	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
409	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
410	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
411	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
412	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
413	C₂×C₈	—	53	4	no	—
414	C₂×C₈	—	106	2	no	—
415	C₂×C₈	—	106	2	no	—
416	C₂×C₈	—	106	2	no	—
417	C₂×C₈	—	106	2	no	—
418	C₂×C₈	—	106	2	no	—
419	C₂×C₈	—	106	2	no	—
420	C₂×C₈	—	106	2	no	—
421	C₂×C₈	—	106	2	no	—
422	C₂×C₈	—	106	2	no	—
423	C₂×C₈	—	106	2	no	—
424	C₂×C₈	—	106	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	#11719917	1	4	yes	—
433	C₄²	—	2	2	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	2	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	2	2	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	1	no	—
447	C₄²	—	2	2	no	—
448	C₄²	#11719945	1	4	yes	—
449	C₂³	—	2	1	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	1	no	—
453	C₂³	D₄×Dic₅₃	1	2	yes	—
454	C₂³	C₂×#11719917	1	2	yes	—
455	C₂³	#13133224	1	2	yes	—
456	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	4	no	—
494	C₂×C₄	—	4	4	no	—
495	C₂×C₄	—	4	4	no	—
496	C₂×C₄	#13133176	1	2	yes	—
497	C₂×C₄	#13133176	1	2	yes	—
498	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
499	C₂×C₄	C₂×#11719917	1	2	yes	—
500	C₂×C₄	C₂×#11719917	1	2	yes	—
501	C₂×C₄	C₂×#11719917	1	2	yes	—
502	C₂×C₄	C₂×#11719917	1	2	yes	—
503	C₂×C₄	C₂×#11719917	1	2	yes	—
504	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
505	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
506	C₂×C₄	C₂×#11719945	1	2	yes	—
507	C₂×C₄	#13133265	1	2	yes	—
508	C₈	—	106	4	no	—
509	C₈	—	212	2	no	—
510	C₈	—	212	2	no	—
511	C₈	—	212	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	2	2	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	2	2	no	—
517	C₂²	—	2	2	no	—
518	C₂²	#16810695	1	4	yes	—
519	C₂²	C₂×#13133176	1	1	yes	characteristic
520	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
521	C₂²	#16811101	1	1	yes	characteristic
522	C₂²	C₂²×#11719917	1	1	yes	characteristic
523	C₂²	C₂²×#11719917	1	1	yes	characteristic
524	C₂²	C₂×#13133224	1	1	yes	characteristic
525	C₂²	#16811390	1	1	yes	characteristic
526	C₄	—	2	2	no	—
527	C₄	—	2	2	no	—
528	C₄	—	2	2	no	—
529	C₄	—	2	2	no	—
530	C₄	—	2	2	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	1	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	1	no	—
537	C₄	—	4	1	no	—
538	C₄	—	4	1	no	—
539	C₄	#16811274	1	2	yes	—
540	C₄	#16811308	1	2	yes	—
541	C₂	—	2	4	no	—
542	C₂	#20277584	1	1	yes	characteristic
543	C₂	#20283348	1	1	yes	characteristic
544	C₂	#20283373	1	1	yes	characteristic
545	C₂	C₂×#16810695	1	1	yes	characteristic
546	C₂	D₄×#11719917	1	1	yes	characteristic
547	C₂	#20283800	1	1	yes	characteristic
548	C₂	#20283804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#22601
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20275439	C₂	semidirect
4. #20275281	C₂	semidirect
5. #20275274	C₂	semidirect
9. #20274590	C₂	semidirect
319. C₅₃	#22601	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810680	2
C₂×#16810676	2
#20275439	2
#20275281	2
#20275274	2
#20274783	2
#20274695	2
#20274590	2
#20274477	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#22601	53