Subgroups — #27382866

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 518 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	#20277199	C₂	1	1	yes	maximal, characteristic
3	#20274592	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274592	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274587	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274587	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274585	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274582	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274580	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274483	C₂	1	1	yes	maximal, characteristic
12	#20274483	C₂	1	1	yes	maximal, characteristic
13	#20274479	C₂	1	1	yes	maximal, characteristic
14	C₂²⋊C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
15	C₂×C₅₃×#433	C₂	1	1	yes	maximal, characteristic
16	C₂×#13133176	C₂²	1	1	yes	characteristic
17	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810684	C₄	1	8	yes	complemented
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#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×#13133128	C₂²	1	1	yes	characteristic
24	C₂×#13133127	C₂²	1	1	yes	characteristic
25	C₂×#13133127	C₂²	1	1	yes	characteristic
26	C₂×#13133127	C₂²	1	1	yes	characteristic
27	C₂×#13133127	C₂²	1	1	yes	characteristic
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
33	#16810522	C₄	1	2	yes	—
34	#16810522	C₄	1	2	yes	—
35	#16810498	C₄	1	2	yes	—
36	#16810498	C₄	1	2	yes	—
37	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
41	#16810377	C₂²	1	1	yes	characteristic
42	#16810377	C₂²	1	1	yes	characteristic
43	#16810377	C₂²	1	1	yes	characteristic
44	#16810377	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	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
50	C₅₃×#433	C₄	1	8	yes	complemented
51	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
52	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
57	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₂²×#88322	C₂³	1	1	yes	characteristic
59	C₂²×#88322	C₂³	1	1	yes	characteristic
60	C₂²×#88322	C₂³	1	1	yes	characteristic
61	C₂²×#88322	C₂³	1	1	yes	characteristic
62	#13133176	—	4	1	no	—
63	#13133176	—	4	1	no	—
64	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
70	C₂×C₄×Dic₅₃	—	4	1	no	—
71	C₂×C₄×Dic₅₃	—	4	1	no	—
72	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
73	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
78	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
79	C₂²⋊#88322	C₂×C₄	1	4	yes	—
80	C₂²⋊#88322	C₂×C₄	1	4	yes	—
81	#13133128	C₂×C₄	1	4	yes	complemented
82	#13133127	—	4	1	no	—
83	#13133127	—	4	1	no	—
84	#13133127	—	4	1	no	—
85	#13133127	—	4	1	no	—
86	C₄×#88322	C₂×C₄	1	4	yes	complemented
87	C₈×Dic₅₃	—	4	1	no	—
88	C₈×Dic₅₃	—	4	1	no	—
89	C₈×Dic₅₃	—	4	1	no	—
90	C₈×Dic₅₃	—	4	1	no	—
91	#13133124	C₂×C₄	1	2	yes	—
92	#13133124	C₂×C₄	1	2	yes	—
93	#13133124	C₂×C₄	1	2	yes	—
94	#13133124	C₂×C₄	1	2	yes	—
95	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	#117×C₅₃	C₂×C₄	1	4	yes	—
101	#117×C₅₃	C₂×C₄	1	4	yes	—
102	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄⋊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×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
109	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
110	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
111	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
112	#97×C₅₃	C₂×C₄	1	4	yes	complemented
113	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
114	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
125	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂²×Dic₅₃	—	2	4	no	—
128	C₂²×Dic₅₃	—	4	2	no	—
129	C₂²×Dic₅₃	—	4	2	no	—
130	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
131	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
132	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
133	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	—	2	1	no	—
142	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
143	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
146	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
147	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
148	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
149	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
150	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
151	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
152	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
153	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
154	C₂²⋊Dic₅₃	—	4	2	no	—
155	C₄⋊Dic₅₃	—	2	4	no	—
156	C₄⋊Dic₅₃	—	2	4	no	—
157	C₄⋊Dic₅₃	—	4	2	no	—
158	C₄⋊Dic₅₃	—	4	2	no	—
159	C₄⋊Dic₅₃	—	4	2	no	—
160	C₄×Dic₅₃	—	2	1	no	—
161	C₄×Dic₅₃	—	2	1	no	—
162	C₄×Dic₅₃	—	2	1	no	—
163	C₄×Dic₅₃	—	2	1	no	—
164	C₄×Dic₅₃	—	2	1	no	—
165	C₄×Dic₅₃	—	2	1	no	—
166	C₄×Dic₅₃	—	2	1	no	—
167	C₄×Dic₅₃	—	2	1	no	—
168	C₄×Dic₅₃	—	4	2	no	—
169	C₂×#88322	—	2	1	no	—
170	C₂×#88322	—	2	1	no	—
171	C₂×#88322	—	2	1	no	—
172	C₂×#88322	—	2	1	no	—
173	C₂×#88322	—	2	1	no	—
174	C₂×#88322	—	2	1	no	—
175	C₂×#88322	—	2	1	no	—
176	C₂×#88322	—	2	1	no	—
177	C₂×#88322	C₂²×C₄	1	2	yes	—
178	C₂×#88322	C₂²×C₄	1	2	yes	—
179	C₂×#88322	C₂²×C₄	1	2	yes	—
180	C₂×#88322	C₂²×C₄	1	2	yes	—
181	C₂³×C₅₃	—	2	1	no	—
182	C₂³×C₅₃	—	2	1	no	—
183	C₂³×C₅₃	—	2	2	no	—
184	C₂³×C₅₃	—	2	2	no	—
185	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
186	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
198	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
199	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
200	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
201	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
202	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
205	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
207	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
208	C₂×Dic₅₃	—	2	2	no	—
209	C₂×Dic₅₃	—	2	2	no	—
210	C₂×Dic₅₃	—	2	2	no	—
211	C₂×Dic₅₃	—	2	2	no	—
212	C₂×Dic₅₃	—	2	2	no	—
213	C₂×Dic₅₃	—	2	2	no	—
214	C₂×Dic₅₃	—	2	2	no	—
215	C₂×Dic₅₃	—	2	2	no	—
216	C₂×Dic₅₃	—	2	2	no	—
217	C₂×Dic₅₃	—	2	2	no	—
218	C₂×Dic₅₃	—	2	2	no	—
219	C₂×Dic₅₃	—	4	8	no	—
220	C₂×Dic₅₃	#131	1	4	yes	—
221	C₈×C₅₃	—	2	2	no	—
222	C₈×C₅₃	—	2	2	no	—
223	C₈×C₅₃	—	2	2	no	—
224	C₈×C₅₃	—	2	2	no	—
225	#88322	—	2	2	no	—
226	#88322	—	2	2	no	—
227	#88322	—	2	2	no	—
228	#88322	—	2	2	no	—
229	#20541	—	53	1	no	maximal, Sylow
230	C₂²×C₅₃	—	2	2	no	—
231	C₂²×C₅₃	—	2	2	no	—
232	C₂²×C₅₃	—	2	2	no	—
233	C₂²×C₅₃	—	4	2	no	—
234	C₂²×C₅₃	—	4	2	no	—
235	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
236	C₂²×C₅₃	#518	1	1	yes	characteristic
237	C₂²×C₅₃	#519	1	1	yes	characteristic
238	C₂²×C₅₃	C₂×#131	1	2	yes	—
239	C₂²×C₅₃	#568	1	2	yes	—
240	C₄×C₅₃	—	2	2	no	—
241	C₄×C₅₃	—	2	2	no	—
242	C₄×C₅₃	—	2	2	no	—
243	C₄×C₅₃	—	2	2	no	—
244	C₄×C₅₃	—	2	2	no	—
245	C₄×C₅₃	#520	1	1	yes	characteristic
246	C₄×C₅₃	#520	1	1	yes	characteristic
247	C₄×C₅₃	#520	1	1	yes	characteristic
248	C₄×C₅₃	#520	1	1	yes	characteristic
249	Dic₅₃	—	2	4	no	—
250	Dic₅₃	—	4	2	no	—
251	Dic₅₃	—	4	2	no	—
252	Dic₅₃	—	4	2	no	—
253	C₂×#433	—	53	1	no	—
254	C₂×#433	D₅₃	1	1	yes	characteristic
255	#2296	—	53	1	no	—
256	#1771	—	53	1	no	—
257	#1771	—	53	1	no	—
258	#1771	—	53	1	no	—
259	#1771	—	53	1	no	—
260	#1770	—	53	1	no	—
261	#1770	—	53	1	no	—
262	C₈×C₄⋊C₄	—	53	1	no	—
263	C₈×C₄⋊C₄	—	53	1	no	—
264	#1750	—	53	1	no	—
265	#1750	—	53	1	no	—
266	#1749	—	53	1	no	—
267	C₄×C₂²⋊C₈	—	53	1	no	—
268	C₂×C₅₃	—	4	2	no	—
269	C₂×C₅₃	#2274	1	1	yes	characteristic
270	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
271	C₂×C₅₃	#2875	1	1	yes	characteristic
272	C₂×C₅₃	#2976	1	2	yes	—
273	C₂×C₅₃	#2985	1	2	yes	—
274	C₂×#117	—	53	1	no	—
275	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
276	C₂²×C₄²	—	53	1	no	—
277	#433	—	53	8	no	—
278	#433	Dic₅₃	1	8	yes	complemented
279	C₂×C₄⋊C₈	—	53	1	no	—
280	C₂×C₄⋊C₈	—	53	1	no	—
281	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
282	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
283	C₂×C₂²⋊C₈	—	53	1	no	—
284	C₂×C₂²⋊C₈	—	53	1	no	—
285	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
286	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
287	C₂×#97	—	53	1	no	—
288	C₂×#97	—	53	1	no	—
289	C₂×#97	—	53	1	no	—
290	C₂×#97	—	53	1	no	—
291	C₂×#97	—	53	1	no	—
292	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
293	C₂×C₄×C₈	—	53	1	no	—
294	C₂×C₄×C₈	—	53	1	no	—
295	C₂×C₄×C₈	—	53	1	no	—
296	C₂×C₄×C₈	—	53	1	no	—
297	C₂×C₄×C₈	—	53	1	no	—
298	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
299	#389	—	53	2	no	—
300	#389	—	53	2	no	—
301	#385	—	53	2	no	—
302	#385	—	53	2	no	—
303	C₄×C₄⋊C₄	—	53	1	no	—
304	C₄×C₄⋊C₄	—	53	1	no	—
305	C₄×C₄⋊C₄	—	53	1	no	—
306	C₄×C₄⋊C₄	—	53	1	no	—
307	#336	—	53	1	no	—
308	#336	—	53	1	no	—
309	#336	—	53	1	no	—
310	#336	—	53	1	no	—
311	#336	—	53	1	no	—
312	#336	—	53	1	no	—
313	#336	—	53	1	no	—
314	#336	—	53	1	no	—
315	C₅₃	#20541	1	1	yes	characteristic, complemented, Sylow
316	C₂³×C₄	—	53	2	no	—
317	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
318	C₂²×C₈	—	53	1	no	—
319	C₂²×C₈	—	53	1	no	—
320	C₂²×C₈	—	53	1	no	—
321	C₂²×C₈	—	53	1	no	—
322	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
323	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
324	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
325	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
326	#117	—	212	1	no	—
327	#117	—	212	1	no	—
328	#117	C₄×D₅₃	1	4	yes	—
329	#117	C₂×Dic₅₃	1	4	yes	—
330	C₂×C₄⋊C₄	—	53	2	no	—
331	C₂×C₄⋊C₄	—	53	2	no	—
332	C₂×C₄⋊C₄	—	53	2	no	—
333	C₂×C₄⋊C₄	—	53	2	no	—
334	C₂×C₄⋊C₄	—	53	2	no	—
335	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
337	C₂×C₂²⋊C₄	—	53	2	no	—
338	C₂×C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
340	C₂×C₄²	—	53	1	no	—
341	C₂×C₄²	—	53	1	no	—
342	C₂×C₄²	—	53	1	no	—
343	C₂×C₄²	—	53	1	no	—
344	C₂×C₄²	—	53	2	no	—
345	C₂×C₄²	—	212	1	no	—
346	C₂×C₄²	—	212	1	no	—
347	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
348	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
349	C₄⋊C₈	—	53	4	no	—
350	C₄⋊C₈	—	53	4	no	—
351	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	complemented
352	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	complemented
353	C₂²⋊C₈	—	53	4	no	—
354	C₂²⋊C₈	—	53	4	no	—
355	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
356	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
357	#97	—	53	4	no	—
358	#97	—	212	1	no	—
359	#97	—	212	1	no	—
360	#97	—	212	1	no	—
361	#97	—	212	1	no	—
362	#97	C₂×Dic₅₃	1	4	yes	complemented
363	C₄×C₈	—	53	4	no	—
364	C₄×C₈	—	212	1	no	—
365	C₄×C₈	—	212	1	no	—
366	C₄×C₈	—	212	1	no	—
367	C₄×C₈	—	212	1	no	—
368	C₄×C₈	C₂×Dic₅₃	1	4	yes	complemented
369	#95	—	53	2	no	—
370	#95	—	53	2	no	—
371	#95	—	53	2	no	—
372	#95	—	53	2	no	—
373	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	53	2	no	—
379	C₂²×C₄	—	53	2	no	—
380	C₂²×C₄	—	53	2	no	—
381	C₂²×C₄	—	53	2	no	—
382	C₂²×C₄	—	106	4	no	—
383	C₂²×C₄	—	212	2	no	—
384	C₂²×C₄	—	212	2	no	—
385	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
386	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
387	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
388	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
390	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
391	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
392	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	53	2	no	—
402	C₂×C₈	—	53	2	no	—
403	C₂×C₈	—	53	2	no	—
404	C₂×C₈	—	53	2	no	—
405	C₂×C₈	—	106	1	no	—
406	C₂×C₈	—	106	1	no	—
407	C₂×C₈	—	106	1	no	—
408	C₂×C₈	—	106	1	no	—
409	C₂×C₈	—	106	1	no	—
410	C₂×C₈	—	106	1	no	—
411	C₂×C₈	—	106	1	no	—
412	C₂×C₈	—	106	1	no	—
413	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
414	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
415	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
416	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
417	C₄⋊C₄	—	106	4	no	—
418	C₄⋊C₄	—	106	4	no	—
419	C₄⋊C₄	—	212	2	no	—
420	C₄⋊C₄	—	212	2	no	—
421	C₄⋊C₄	—	212	2	no	—
422	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
423	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
424	C₂²⋊C₄	—	212	2	no	—
425	C₂²⋊C₄	C₄×Dic₅₃	1	4	yes	—
426	C₂²⋊C₄	C₄×Dic₅₃	1	4	yes	—
427	C₄²	—	106	1	no	—
428	C₄²	—	106	1	no	—
429	C₄²	—	106	1	no	—
430	C₄²	—	106	1	no	—
431	C₄²	—	106	1	no	—
432	C₄²	—	106	1	no	—
433	C₄²	—	106	1	no	—
434	C₄²	—	106	1	no	—
435	C₄²	—	212	2	no	—
436	C₄²	C₂×C₄×D₅₃	1	2	yes	—
437	C₄²	C₂×C₄×D₅₃	1	2	yes	—
438	C₄²	C₂²×Dic₅₃	1	2	yes	—
439	C₄²	C₂²×Dic₅₃	1	2	yes	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
445	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	53	4	no	—
457	C₂×C₄	—	106	2	no	—
458	C₂×C₄	—	106	2	no	—
459	C₂×C₄	—	106	2	no	—
460	C₂×C₄	—	106	2	no	—
461	C₂×C₄	—	106	2	no	—
462	C₂×C₄	—	106	2	no	—
463	C₂×C₄	—	106	2	no	—
464	C₂×C₄	—	106	2	no	—
465	C₂×C₄	—	106	2	no	—
466	C₂×C₄	—	106	2	no	—
467	C₂×C₄	—	106	2	no	—
468	C₂×C₄	—	212	8	no	—
469	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
470	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
471	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
472	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
473	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
474	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
475	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
476	C₂×C₄	#13133263	1	1	yes	characteristic
477	C₂×C₄	#13133263	1	1	yes	characteristic
478	C₂×C₄	#13133264	1	1	yes	characteristic
479	C₂×C₄	#13133264	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	2	2	no	—
482	C₈	—	2	2	no	—
483	C₈	—	2	2	no	—
484	C₈	—	106	2	no	—
485	C₈	—	106	2	no	—
486	C₈	—	106	2	no	—
487	C₈	—	106	2	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
494	C₂²	#16811097	1	1	yes	characteristic
495	C₂²	#16811100	1	1	yes	characteristic
496	C₂²	#131×D₅₃	1	2	yes	—
497	C₂²	#16811388	1	2	yes	—
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	106	4	no	—
504	C₄	—	212	2	no	—
505	C₄	—	212	2	no	—
506	C₄	—	212	2	no	—
507	C₄	#16811103	1	1	yes	characteristic
508	C₄	#16811103	1	1	yes	characteristic
509	C₄	#16811106	1	1	yes	characteristic
510	C₄	#16811106	1	1	yes	characteristic
511	C₂	—	4	2	no	—
512	C₂	#20277071	1	1	yes	characteristic
513	C₂	#131×Dic₅₃	1	1	yes	characteristic
514	C₂	#20283141	1	1	yes	characteristic
515	C₂	#20283761	1	2	yes	—
516	C₂	#20283807	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₂×C₅₃×#433	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#433	D₅₃
hypocenter	C₅₃	#20541
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #20274592	C₂	semidirect
4. #20274592	C₂	semidirect
5. #20274587	C₂	semidirect
6. #20274587	C₂	semidirect
7. #20274585	C₂	semidirect
8. #20274582	C₂	semidirect
9. #20274580	C₂	semidirect
10. C₈×C₄⋊Dic₅₃	C₂	semidirect
18. #16810684	C₄	semidirect
50. C₅₃×#433	C₄	semidirect
77. C₄⋊#88322	C₂×C₄	semidirect
78. C₄⋊#88322	C₂×C₄	semidirect
81. #13133128	C₂×C₄	semidirect
86. C₄×#88322	C₂×C₄	semidirect
108. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
109. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
112. #97×C₅₃	C₂×C₄	semidirect
113. C₄×C₈×C₅₃	C₂×C₄	semidirect
278. #433	Dic₅₃	semidirect
315. C₅₃	#20541	semidirect
351. C₄⋊C₈	C₂×Dic₅₃	semidirect
352. C₄⋊C₈	C₂×Dic₅₃	semidirect
362. #97	C₂×Dic₅₃	semidirect
368. C₄×C₈	C₂×Dic₅₃	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
#20277199	2
#20274592	2
#20274587	2
#20274585	2
#20274582	2
#20274580	2
C₈×C₄⋊Dic₅₃	2
#20274483	2
#20274479	2
C₂²⋊C₈×Dic₅₃	2
C₂×C₅₃×#433	2
#20541	53