Subgroups — #27387318

Order 13568.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283359	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13133224	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275365	C₂	1	2	yes	maximal, complemented
4	#20275316	C₂	1	2	yes	maximal, complemented
5	#20275044	C₂	1	2	yes	maximal
6	#20274992	C₂	1	2	yes	maximal, complemented
7	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13133144	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133144	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274460	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	C₂²	1	2	yes	—
13	#16810684	—	2	2	no	—
14	#16810676	—	2	1	no	—
15	#16810676	—	2	1	no	—
16	C₂×#13133153	C₂²	1	2	yes	—
17	#16810650	C₂²	1	2	yes	—
18	C₂×#13133144	C₂²	1	2	yes	—
19	C₂×#13133144	C₂²	1	2	yes	—
20	#16810590	C₂²	1	2	yes	—
21	#16810579	—	2	2	no	—
22	#16810579	C₂²	1	2	yes	—
23	C₄×#11719917	—	2	1	no	—
24	C₄×#11719917	C₂²	1	1	yes	characteristic
25	C₄×#11719917	C₂²	1	1	yes	characteristic
26	C₂×C₄×#88322	—	2	1	no	—
27	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
28	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
29	#16810388	C₂²	1	2	yes	—
30	#16810382	C₂²	1	2	yes	—
31	#16810382	C₂²	1	2	yes	—
32	#16810377	C₂²	1	2	yes	—
33	#16810377	C₂²	1	2	yes	—
34	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
38	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
39	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
40	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
41	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
42	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
43	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
44	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
45	C₅₃×#427	C₂²	1	1	yes	characteristic
46	C₅₃×#416	C₂²	1	1	yes	characteristic
47	C₅₃×#416	C₂²	1	1	yes	characteristic
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	#13133224	—	2	4	no	—
50	#13133224	—	2	4	no	—
51	C₂×#11719917	—	2	2	no	—
52	C₂×#11719917	C₂³	1	2	yes	—
53	C₂×#11719917	C₂³	1	2	yes	—
54	C₂²×#88322	—	2	2	no	—
55	C₂²×#88322	C₂³	1	2	yes	—
56	C₂×C₄×Dic₅₃	—	2	1	no	—
57	C₂×C₄×Dic₅₃	—	2	1	no	—
58	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
59	#13133153	—	2	4	no	—
60	C₄⋊#88322	—	2	2	no	—
61	C₄⋊#88322	—	2	2	no	—
62	#13133144	—	2	4	no	—
63	#13133144	—	2	4	no	—
64	C₂²⋊#88322	—	2	2	no	—
65	C₂²⋊#88322	—	2	2	no	—
66	#13133128	—	2	1	no	—
67	#13133128	—	2	1	no	—
68	#13133128	—	2	2	no	—
69	C₄×#88322	—	2	1	no	—
70	C₄×#88322	—	2	1	no	—
71	C₄×#88322	—	2	2	no	—
72	C₄×#88322	D₄	1	1	yes	characteristic
73	C₄×#88322	D₄	1	1	yes	characteristic
74	C₄×#88322	D₄	1	2	yes	—
75	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
77	#135×C₅₃	—	2	4	no	—
78	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	—	2	1	no	—
90	C₄×D₄×C₅₃	—	2	1	no	—
91	C₄×D₄×C₅₃	—	2	1	no	—
92	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
96	#117×C₅₃	—	2	1	no	—
97	#117×C₅₃	—	2	1	no	—
98	#117×C₅₃	C₂³	1	1	yes	characteristic
99	#117×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	—	2	1	no	—
101	C₂×C₄²×C₅₃	—	2	1	no	—
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
104	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
105	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
106	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
107	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
108	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
109	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
110	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
111	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
112	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
113	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
114	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
115	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
116	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
117	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
118	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	—	2	2	no	—
121	#41×C₅₃	—	2	2	no	—
122	#41×C₅₃	#41	1	4	yes	—
123	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₄×C₅₃	—	2	1	no	—
126	C₂×D₄×C₅₃	—	2	1	no	—
127	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	—	2	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	#41	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂²×Dic₅₃	—	2	2	no	—
139	Q₁₆×C₅₃	—	2	2	no	—
140	SD₁₆×C₅₃	—	2	2	no	—
141	SD₁₆×C₅₃	—	2	2	no	—
142	D₈×C₅₃	—	2	2	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	—	2	1	no	—
152	C₄⋊C₄×C₅₃	—	2	1	no	—
153	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²⋊C₄×C₅₃	—	2	1	no	—
158	C₂²⋊C₄×C₅₃	—	2	1	no	—
159	C₂²⋊C₄×C₅₃	—	2	1	no	—
160	C₂²⋊C₄×C₅₃	—	2	1	no	—
161	C₂²⋊C₄×C₅₃	—	2	1	no	—
162	C₂²⋊C₄×C₅₃	—	2	1	no	—
163	C₄²×C₅₃	—	2	1	no	—
164	C₄²×C₅₃	—	2	1	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	—	2	1	no	—
168	C₄²×C₅₃	—	2	1	no	—
169	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
173	C₄×Dic₅₃	—	2	4	no	—
174	C₄×Dic₅₃	—	2	2	no	—
175	C₄×Dic₅₃	—	2	2	no	—
176	C₄×Dic₅₃	—	4	2	no	—
177	#11719917	—	2	8	no	—
178	#11719917	—	2	2	no	—
179	#11719917	—	2	2	no	—
180	#11719917	—	2	2	no	—
181	#11719917	#41	1	4	yes	—
182	C₂×#88322	—	2	8	no	—
183	C₂×#88322	—	2	2	no	—
184	C₂×#88322	—	2	2	no	—
185	C₂×#88322	—	2	2	no	—
186	C₂×#88322	—	2	2	no	—
187	C₂×#88322	C₂×D₄	1	2	yes	—
188	C₂×#88322	C₂×D₄	1	2	yes	—
189	C₂³×C₅₃	—	2	1	no	—
190	C₂³×C₅₃	—	2	1	no	—
191	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
192	Q₈×C₅₃	—	2	1	no	—
193	Q₈×C₅₃	—	2	1	no	—
194	Q₈×C₅₃	—	2	1	no	—
195	Q₈×C₅₃	#117	1	2	yes	—
196	D₄×C₅₃	—	2	4	no	—
197	D₄×C₅₃	—	2	1	no	—
198	D₄×C₅₃	—	2	1	no	—
199	D₄×C₅₃	—	2	1	no	—
200	D₄×C₅₃	—	4	2	no	—
201	D₄×C₅₃	#117	1	2	yes	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	2	2	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	—	4	2	no	—
212	C₂×C₄×C₅₃	—	4	2	no	—
213	C₂×C₄×C₅₃	—	4	2	no	—
214	C₂×C₄×C₅₃	—	4	2	no	—
215	C₂×C₄×C₅₃	—	4	2	no	—
216	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
223	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
224	C₂×Dic₅₃	—	2	4	no	—
225	C₂×Dic₅₃	—	4	2	no	—
226	C₂×Dic₅₃	—	4	2	no	—
227	C₈×C₅₃	—	2	1	no	—
228	C₈×C₅₃	—	2	1	no	—
229	C₈×C₅₃	—	2	1	no	—
230	C₈×C₅₃	—	2	1	no	—
231	#88322	—	2	2	no	—
232	#88322	—	2	4	no	—
233	#88322	—	2	2	no	—
234	#88322	—	4	2	no	—
235	#21137	—	53	1	no	maximal, Sylow
236	C₂²×C₅₃	—	2	1	no	—
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	2	2	no	—
239	C₂²×C₅₃	—	4	1	no	—
240	C₂²×C₅₃	—	4	1	no	—
241	C₂²×C₅₃	—	4	1	no	—
242	C₂²×C₅₃	—	4	1	no	—
243	C₂²×C₅₃	—	4	1	no	—
244	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₅₃	#529	1	2	yes	—
246	C₄×C₅₃	—	2	1	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	4	1	no	—
250	C₄×C₅₃	—	4	1	no	—
251	C₄×C₅₃	—	4	1	no	—
252	C₄×C₅₃	—	4	1	no	—
253	C₄×C₅₃	—	4	1	no	—
254	C₄×C₅₃	#443	1	2	yes	—
255	C₄×C₅₃	#444	1	2	yes	—
256	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
257	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
258	C₄×C₅₃	#525	1	2	yes	—
259	Dic₅₃	—	4	4	no	—
260	C₄×#135	D₅₃	1	1	yes	characteristic
261	#2922	—	53	1	no	—
262	C₄×#131	—	53	1	no	—
263	#1929	—	53	2	no	—
264	#1922	—	53	2	no	—
265	#1876	—	53	2	no	—
266	#1866	—	53	2	no	—
267	C₄×#108	—	53	1	no	—
268	C₄×#104	—	53	1	no	—
269	C₄×#104	—	53	1	no	—
270	#1744	—	53	1	no	—
271	C₂×C₅₃	—	2	2	no	—
272	C₂×C₅₃	—	4	2	no	—
273	C₂×C₅₃	—	8	1	no	—
274	C₂×C₅₃	#2290	1	1	yes	characteristic
275	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
276	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
277	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
278	C₂×#131	—	53	2	no	—
279	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
280	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
281	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
282	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
283	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic, complemented
284	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic, complemented
285	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic, complemented
286	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic, complemented
287	C₄×D₈	Dic₅₃	1	1	yes	characteristic, complemented
288	C₄×D₈	Dic₅₃	1	1	yes	characteristic, complemented
289	#433	—	106	2	no	—
290	#431	—	106	1	no	—
291	#431	—	106	1	no	—
292	C₂×#108	—	53	2	no	—
293	#427	C₂×D₅₃	1	1	yes	characteristic
294	#424	—	53	2	no	—
295	C₂×#104	—	53	2	no	—
296	C₂×#104	—	53	2	no	—
297	#416	C₂×D₅₃	1	1	yes	characteristic
298	#416	C₂×D₅₃	1	1	yes	characteristic
299	#408	—	53	2	no	—
300	#405	—	53	2	no	—
301	#405	—	106	2	no	—
302	C₄×M₁₆	—	53	1	no	—
303	C₄×M₁₆	—	53	1	no	—
304	C₄×M₁₆	—	106	1	no	—
305	C₂×C₄×C₈	—	53	1	no	—
306	C₂×C₄×C₈	—	106	1	no	—
307	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
308	C₄³	—	53	1	no	—
309	#341	—	53	2	no	—
310	#339	—	53	2	no	—
311	#339	—	53	2	no	—
312	#336	—	53	2	no	—
313	#336	—	53	2	no	—
314	C₅₃	#21137	1	1	yes	characteristic, complemented, Sylow
315	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
316	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
317	#135	—	2	4	no	—
318	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
319	C₂×SD₁₆	C₂×Dic₅₃	1	1	yes	characteristic
320	C₂×SD₁₆	C₂×Dic₅₃	1	1	yes	characteristic
321	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
322	#131	—	106	4	no	—
323	#131	—	106	4	no	—
324	C₂×M₁₆	—	53	2	no	—
325	C₂×M₁₆	—	53	2	no	—
326	C₂×M₁₆	—	106	2	no	—
327	C₂²×C₈	—	53	2	no	—
328	C₂²×C₈	—	106	2	no	—
329	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
330	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
331	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
332	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
333	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
334	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
340	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
341	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
342	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
343	#117	—	2	1	no	—
344	#117	—	2	1	no	—
345	#117	C₂²×D₅₃	1	1	yes	characteristic
346	#117	C₂²×D₅₃	1	1	yes	characteristic
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	53	2	no	—
350	C₂×C₄²	—	106	1	no	—
351	C₂×C₄²	—	106	1	no	—
352	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
353	#108	—	106	4	no	—
354	#107	C₂×Dic₅₃	1	1	yes	characteristic
355	#107	C₂×Dic₅₃	1	1	yes	characteristic
356	#106	C₂×Dic₅₃	1	1	yes	characteristic
357	#106	C₂×Dic₅₃	1	1	yes	characteristic
358	C₄⋊C₈	—	106	2	no	—
359	C₄⋊C₈	—	106	2	no	—
360	#104	—	106	4	no	—
361	#104	—	106	4	no	—
362	#103	C₂×Dic₅₃	1	1	yes	characteristic
363	#103	C₂×Dic₅₃	1	1	yes	characteristic
364	#103	C₂×Dic₅₃	1	1	yes	characteristic
365	#103	C₂×Dic₅₃	1	1	yes	characteristic
366	#102	C₂×Dic₅₃	1	1	yes	characteristic
367	#102	C₂×Dic₅₃	1	1	yes	characteristic
368	#102	C₂×Dic₅₃	1	1	yes	characteristic
369	#102	C₂×Dic₅₃	1	1	yes	characteristic
370	C₂²⋊C₈	—	106	2	no	—
371	C₂²⋊C₈	—	106	2	no	—
372	#97	—	106	2	no	—
373	#97	—	106	1	no	—
374	#97	—	106	1	no	—
375	C₄×C₈	—	53	2	no	—
376	C₄×C₈	—	53	1	no	—
377	C₄×C₈	—	53	1	no	—
378	C₄×C₈	—	106	2	no	—
379	C₄×C₈	—	106	1	no	—
380	C₄×C₈	—	106	1	no	—
381	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
382	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
383	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
384	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
385	#41	—	2	2	no	—
386	#41	—	2	2	no	—
387	#41	—	2	2	no	—
388	#41	#11719945	1	4	yes	—
389	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
390	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
394	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	106	2	no	—
402	C₂²×C₄	#11719947	1	1	yes	characteristic
403	C₂²×C₄	#11719947	1	1	yes	characteristic
404	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
405	Q₁₆	—	2	2	no	—
406	SD₁₆	—	2	2	no	—
407	SD₁₆	—	2	2	no	—
408	D₈	—	2	2	no	—
409	M₁₆	—	53	4	no	—
410	M₁₆	—	106	8	no	—
411	M₁₆	—	106	2	no	—
412	M₁₆	—	106	2	no	—
413	M₁₆	—	106	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	53	2	no	—
417	C₂×C₈	—	53	2	no	—
418	C₂×C₈	—	106	8	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	C₂²×Dic₅₃	1	1	yes	characteristic
424	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
425	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
426	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
432	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
433	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
434	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	2	1	no	—
438	C₂²⋊C₄	—	2	1	no	—
439	C₂²⋊C₄	—	2	1	no	—
440	C₂²⋊C₄	—	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	—	106	4	no	—
448	C₄²	—	106	2	no	—
449	C₄²	—	106	2	no	—
450	C₄²	—	212	2	no	—
451	C₄²	D₄×D₅₃	1	1	yes	characteristic
452	C₄²	D₄×D₅₃	1	1	yes	characteristic
453	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
454	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
455	C₂³	—	2	1	no	—
456	C₂³	—	2	1	no	—
457	C₂³	C₂×#11719947	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	—	2	1	no	—
460	Q₈	—	2	1	no	—
461	Q₈	#13133176	1	2	yes	—
462	D₄	—	2	4	no	—
463	D₄	—	2	1	no	—
464	D₄	—	2	1	no	—
465	D₄	—	2	1	no	—
466	D₄	—	4	2	no	—
467	D₄	#13133176	1	2	yes	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	106	4	no	—
483	C₂×C₄	—	212	2	no	—
484	C₂×C₄	—	212	2	no	—
485	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
486	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
487	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
488	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
489	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
491	C₂×C₄	#41×D₅₃	1	2	yes	—
492	C₂×C₄	C₂×#11719945	1	2	yes	—
493	C₈	—	2	1	no	—
494	C₈	—	2	1	no	—
495	C₈	—	2	1	no	—
496	C₈	—	2	1	no	—
497	C₈	—	106	2	no	—
498	C₈	—	106	2	no	—
499	C₈	—	106	4	no	—
500	C₈	—	212	2	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	1	no	—
509	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
510	C₂²	#16811153	1	2	yes	—
511	C₄	—	2	2	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	4	1	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	4	1	no	—
519	C₄	—	212	4	no	—
520	C₄	#16810740	1	2	yes	—
521	C₄	#16810745	1	2	yes	—
522	C₄	C₂×#13133176	1	1	yes	characteristic
523	C₄	#41×Dic₅₃	1	1	yes	characteristic
524	C₄	#16811132	1	2	yes	—
525	C₂	—	2	2	no	—
526	C₂	—	4	2	no	—
527	C₂	—	8	1	no	—
528	C₂	#20277160	1	1	yes	characteristic
529	C₂	C₂×#16810740	1	1	yes	characteristic
530	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	#21137
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #20283359	C₂	semidirect
2. C₄×#13133224	C₂	semidirect
3. #20275365	C₂	semidirect
4. #20275316	C₂	semidirect
6. #20274992	C₂	semidirect
7. C₄×#13133153	C₂	semidirect
8. C₄×#13133144	C₂	semidirect
9. C₄×#13133144	C₂	semidirect
10. #20274460	C₂	semidirect
37. C₄×Q₁₆×C₅₃	C₄	semidirect
38. C₄×Q₁₆×C₅₃	C₄	semidirect
39. C₄×SD₁₆×C₅₃	C₄	semidirect
40. C₄×SD₁₆×C₅₃	C₄	semidirect
41. C₄×SD₁₆×C₅₃	C₄	semidirect
42. C₄×SD₁₆×C₅₃	C₄	semidirect
43. C₄×D₈×C₅₃	C₄	semidirect
44. C₄×D₈×C₅₃	C₄	semidirect
281. C₄×Q₁₆	Dic₅₃	semidirect
282. C₄×Q₁₆	Dic₅₃	semidirect
283. C₄×SD₁₆	Dic₅₃	semidirect
284. C₄×SD₁₆	Dic₅₃	semidirect
285. C₄×SD₁₆	Dic₅₃	semidirect
286. C₄×SD₁₆	Dic₅₃	semidirect
287. C₄×D₈	Dic₅₃	semidirect
288. C₄×D₈	Dic₅₃	semidirect
314. C₅₃	#21137	semidirect

Maximal subgroups

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

subgroup	index
#20283359	2
C₄×#13133224	2
#20275365	2
#20275316	2
#20275044	2
#20274992	2
C₄×#13133153	2
C₄×#13133144	2
#20274460	2
C₄×#135×C₅₃	2
#21137	53