Subgroups — #27355544

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283735	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283729	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273606	C₂	1	2	yes	maximal, complemented
4	#20273566	C₂	1	2	yes	maximal, complemented
5	#20273545	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#88322	C₂	1	2	yes	maximal, complemented
7	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273149	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273149	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
12	#16810754	C₂²	1	2	yes	—
13	Q₈×#88322	C₂²	1	2	yes	—
14	#16810701	—	2	2	no	—
15	#16810701	C₂²	1	2	yes	—
16	#16810695	—	2	2	no	—
17	D₄×#88322	C₂²	1	2	yes	—
18	#16810680	—	2	1	no	—
19	#16810680	—	2	1	no	—
20	#16810680	C₂²	1	1	yes	characteristic
21	#16810680	C₂²	1	1	yes	characteristic
22	#16810676	C₂²	1	1	yes	characteristic
23	C₄×#11719917	—	2	1	no	—
24	C₄×#11719917	C₂²	1	1	yes	characteristic
25	C₂×#13133128	—	2	1	no	—
26	#16810376	C₂²	1	2	yes	—
27	#16810374	C₂²	1	2	yes	—
28	Q₈⋊#88322	C₂²	1	2	yes	—
29	Q₈⋊#88322	C₂²	1	2	yes	—
30	D₄⋊#88322	C₂²	1	2	yes	—
31	D₄⋊#88322	C₂²	1	2	yes	—
32	#16810345	C₂²	1	2	yes	—
33	C₈×#88322	C₂²	1	2	yes	—
34	C₂×#135×C₅₃	C₄	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
38	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
40	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
41	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
44	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#427	C₄	1	1	yes	characteristic
46	C₅₃×#416	C₄	1	1	yes	characteristic
47	C₅₃×#416	C₄	1	1	yes	characteristic
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#11719917	—	2	2	no	—
50	C₂×#11719917	D₄	1	2	yes	—
51	C₂²×#88322	—	2	2	no	—
52	C₄⋊#88322	—	2	1	no	—
53	C₄⋊#88322	—	2	2	no	—
54	C₄⋊#88322	—	2	2	no	—
55	C₄⋊#88322	C₂³	1	2	yes	—
56	C₄⋊#88322	C₂³	1	2	yes	—
57	C₂²⋊#88322	—	2	2	no	—
58	C₂²⋊#88322	—	2	2	no	—
59	C₂²⋊#88322	—	2	1	no	—
60	C₂²⋊#88322	—	2	1	no	—
61	C₂²⋊#88322	—	2	1	no	—
62	#13133128	—	2	1	no	—
63	#13133128	—	2	1	no	—
64	#13133128	—	2	2	no	—
65	#13133128	D₄	1	1	yes	characteristic
66	#13133128	D₄	1	1	yes	characteristic
67	C₄×#88322	—	2	2	no	—
68	C₄×#88322	C₂³	1	2	yes	—
69	C₄×#88322	C₂³	1	2	yes	—
70	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
72	#135×C₅₃	—	2	4	no	—
73	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₅₃	—	2	1	no	—
84	C₄×D₄×C₅₃	—	2	1	no	—
85	C₄×D₄×C₅₃	—	2	1	no	—
86	C₄×D₄×C₅₃	—	2	1	no	—
87	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
91	#117×C₅₃	—	2	1	no	—
92	#117×C₅₃	—	2	1	no	—
93	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
94	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	—	2	1	no	—
96	C₂×C₄²×C₅₃	—	2	1	no	—
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
99	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
100	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
101	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
102	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
103	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
104	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
105	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
106	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
107	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
108	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
109	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	#41×C₅₃	—	2	2	no	—
115	#41×C₅₃	—	2	2	no	—
116	#41×C₅₃	—	4	1	no	—
117	#41×C₅₃	—	4	1	no	—
118	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄×C₅₃	—	2	1	no	—
121	C₂×D₄×C₅₃	—	2	1	no	—
122	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	—	2	1	no	—
127	C₂²×C₄×C₅₃	—	2	1	no	—
128	C₂²×C₄×C₅₃	—	2	1	no	—
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
133	Q₁₆×C₅₃	—	2	2	no	—
134	SD₁₆×C₅₃	—	2	2	no	—
135	SD₁₆×C₅₃	M₁₆	1	4	yes	complemented
136	D₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	4	1	no	—
139	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²⋊C₄×C₅₃	—	2	1	no	—
152	C₂²⋊C₄×C₅₃	—	2	1	no	—
153	C₂²⋊C₄×C₅₃	—	2	1	no	—
154	C₂²⋊C₄×C₅₃	—	2	1	no	—
155	C₂²⋊C₄×C₅₃	—	2	1	no	—
156	C₂²⋊C₄×C₅₃	—	2	1	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	—	2	1	no	—
162	C₄²×C₅₃	—	2	1	no	—
163	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₅₃	#41	1	1	yes	characteristic
166	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
167	#11719917	—	2	4	no	—
168	#11719917	—	4	4	no	—
169	C₂×#88322	—	2	2	no	—
170	C₂×#88322	—	2	2	no	—
171	C₂×#88322	—	2	2	no	—
172	C₂×#88322	—	4	4	no	—
173	C₂×#88322	C₂×D₄	1	2	yes	—
174	C₂×#88322	C₂×D₄	1	2	yes	—
175	C₂³×C₅₃	—	2	1	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
178	Q₈×C₅₃	—	2	1	no	—
179	Q₈×C₅₃	—	2	1	no	—
180	Q₈×C₅₃	—	2	1	no	—
181	Q₈×C₅₃	C₂×M₁₆	1	2	yes	—
182	D₄×C₅₃	—	2	1	no	—
183	D₄×C₅₃	—	2	1	no	—
184	D₄×C₅₃	—	2	1	no	—
185	D₄×C₅₃	—	4	2	no	—
186	D₄×C₅₃	—	4	2	no	—
187	D₄×C₅₃	C₂×M₁₆	1	2	yes	—
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	2	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	—	4	1	no	—
198	C₂×C₄×C₅₃	—	4	2	no	—
199	C₂×C₄×C₅₃	—	4	2	no	—
200	C₂×C₄×C₅₃	—	4	2	no	—
201	C₂×C₄×C₅₃	—	4	2	no	—
202	C₂×C₄×C₅₃	—	8	1	no	—
203	C₂×C₄×C₅₃	—	8	1	no	—
204	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
210	C₈×C₅₃	—	2	1	no	—
211	C₈×C₅₃	—	2	1	no	—
212	C₈×C₅₃	—	2	1	no	—
213	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
214	#88322	—	2	2	no	—
215	#88322	—	4	2	no	—
216	#88322	—	4	2	no	—
217	#88322	—	4	2	no	—
218	#88322	#135	1	4	yes	complemented
219	#18037	—	53	1	no	maximal, Sylow
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	—	2	1	no	—
223	C₂²×C₅₃	—	2	1	no	—
224	C₂²×C₅₃	—	4	1	no	—
225	C₂²×C₅₃	—	4	1	no	—
226	C₂²×C₅₃	—	4	1	no	—
227	C₂²×C₅₃	—	4	1	no	—
228	C₂²×C₅₃	—	4	1	no	—
229	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	2	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	4	1	no	—
236	C₄×C₅₃	—	4	1	no	—
237	C₄×C₅₃	—	4	1	no	—
238	C₄×C₅₃	—	4	1	no	—
239	C₄×C₅₃	—	4	1	no	—
240	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
241	C₄×C₅₃	#568	1	1	yes	characteristic
242	C₄×C₅₃	C₂×#135	1	2	yes	—
243	#2971	—	53	1	no	—
244	#2970	—	53	1	no	—
245	C₄×#135	D₅₃	1	1	yes	characteristic
246	#1590	—	53	2	no	—
247	#1584	—	53	2	no	—
248	#1581	—	53	2	no	—
249	C₈×SD₁₆	—	53	2	no	—
250	#1568	—	53	1	no	—
251	#1488	—	53	1	no	—
252	#1488	—	53	1	no	—
253	C₈×M₁₆	—	53	1	no	—
254	C₂×C₅₃	—	4	1	no	—
255	C₂×C₅₃	—	4	2	no	—
256	C₂×C₅₃	—	8	1	no	—
257	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
258	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
259	C₂×C₅₃	#2957	1	1	yes	characteristic
260	C₂×#135	Dic₅₃	1	1	yes	characteristic
261	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
262	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
263	#446	—	53	2	no	—
264	C₈×Q₈	—	53	2	no	—
265	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
266	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
267	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic
268	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic
269	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
270	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
271	C₄×D₈	Dic₅₃	1	1	yes	characteristic
272	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
273	#436	—	53	2	no	—
274	#436	—	106	2	no	—
275	#435	—	106	2	no	—
276	C₈×D₄	—	53	2	no	—
277	#432	—	53	1	no	—
278	#432	—	53	1	no	—
279	#432	—	106	1	no	—
280	#432	—	106	1	no	—
281	#431	—	53	1	no	—
282	#427	Dic₅₃	1	1	yes	characteristic
283	#416	Dic₅₃	1	1	yes	characteristic
284	#416	Dic₅₃	1	1	yes	characteristic
285	C₄×M₁₆	—	53	1	no	—
286	C₄×M₁₆	—	106	1	no	—
287	C₂×#97	—	106	1	no	—
288	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
289	#335	—	53	2	no	—
290	#334	—	53	2	no	—
291	Q₈⋊C₈	—	53	2	no	—
292	Q₈⋊C₈	—	53	2	no	—
293	D₄⋊C₈	—	53	2	no	—
294	D₄⋊C₈	—	53	2	no	—
295	#322	—	53	2	no	—
296	C₈²	—	53	2	no	—
297	C₅₃	#18037	1	1	yes	characteristic, complemented, Sylow
298	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
299	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
300	#135	—	2	4	no	—
301	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
302	C₂×SD₁₆	C₂×Dic₅₃	1	1	yes	characteristic
303	C₂×SD₁₆	C₂×Dic₅₃	1	1	yes	characteristic
304	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
305	C₂×M₁₆	—	53	2	no	—
306	C₂×M₁₆	—	106	2	no	—
307	C₂²×C₈	—	106	2	no	—
308	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
309	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
310	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
311	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
312	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
313	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
319	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
320	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
321	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	—	2	1	no	—
324	#117	C₂×Dic₅₃	1	1	yes	characteristic
325	#117	C₂×Dic₅₃	1	1	yes	characteristic
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
329	#107	C₂×Dic₅₃	1	1	yes	characteristic
330	#107	C₂×Dic₅₃	1	1	yes	characteristic
331	#106	C₂×Dic₅₃	1	1	yes	characteristic
332	#106	C₂×Dic₅₃	1	1	yes	characteristic
333	C₄⋊C₈	—	53	2	no	—
334	C₄⋊C₈	—	53	2	no	—
335	C₄⋊C₈	—	106	2	no	—
336	C₄⋊C₈	—	106	2	no	—
337	C₄⋊C₈	—	106	1	no	—
338	#103	C₂×Dic₅₃	1	1	yes	characteristic
339	#103	C₂×Dic₅₃	1	1	yes	characteristic
340	#103	C₂×Dic₅₃	1	1	yes	characteristic
341	#103	C₂×Dic₅₃	1	1	yes	characteristic
342	#102	C₂×Dic₅₃	1	1	yes	characteristic
343	#102	C₂×Dic₅₃	1	1	yes	characteristic
344	#102	C₂×Dic₅₃	1	1	yes	characteristic
345	#102	C₂×Dic₅₃	1	1	yes	characteristic
346	C₂²⋊C₈	—	106	2	no	—
347	C₂²⋊C₈	—	106	2	no	—
348	C₂²⋊C₈	—	106	1	no	—
349	C₂²⋊C₈	—	106	1	no	—
350	C₂²⋊C₈	—	106	1	no	—
351	#97	—	53	1	no	—
352	#97	—	53	1	no	—
353	#97	—	106	2	no	—
354	#97	—	106	1	no	—
355	#97	—	106	1	no	—
356	C₄×C₈	—	53	2	no	—
357	C₄×C₈	—	53	2	no	—
358	C₄×C₈	—	106	2	no	—
359	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
360	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
361	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
362	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
363	#41	—	2	2	no	—
364	#41	—	2	2	no	—
365	#41	—	4	1	no	—
366	#41	—	4	1	no	—
367	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
368	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
372	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
373	C₂²×C₄	—	2	1	no	—
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	—	2	1	no	—
379	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
380	C₂²×C₄	#11719947	1	1	yes	characteristic
381	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
382	Q₁₆	—	2	2	no	—
383	SD₁₆	—	2	2	no	—
384	SD₁₆	#11719917	1	4	yes	complemented
385	D₈	—	2	2	no	—
386	M₁₆	—	106	4	no	—
387	M₁₆	—	212	4	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	4	1	no	—
390	C₂×C₈	—	53	2	no	—
391	C₂×C₈	—	53	2	no	—
392	C₂×C₈	—	106	2	no	—
393	C₂×C₈	—	106	2	no	—
394	C₂×C₈	—	106	2	no	—
395	C₂×C₈	—	212	4	no	—
396	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
397	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
398	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
399	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
405	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
406	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
407	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
408	C₂²⋊C₄	—	2	1	no	—
409	C₂²⋊C₄	—	2	1	no	—
410	C₂²⋊C₄	—	2	1	no	—
411	C₂²⋊C₄	—	2	1	no	—
412	C₂²⋊C₄	—	2	1	no	—
413	C₂²⋊C₄	—	2	1	no	—
414	C₄²	—	2	1	no	—
415	C₄²	—	2	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	D₄×D₅₃	1	1	yes	characteristic
421	C₄²	#11719947	1	1	yes	characteristic
422	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
423	C₄²	C₂³×D₅₃	1	1	yes	characteristic
424	C₂³	—	2	1	no	—
425	C₂³	—	2	1	no	—
426	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
427	Q₈	—	2	1	no	—
428	Q₈	—	2	1	no	—
429	Q₈	—	2	1	no	—
430	Q₈	C₂×#11719917	1	2	yes	—
431	D₄	—	2	1	no	—
432	D₄	—	2	1	no	—
433	D₄	—	2	1	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	C₂×#11719917	1	2	yes	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	8	1	no	—
452	C₂×C₄	—	8	1	no	—
453	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
454	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
455	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
456	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
457	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
458	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
459	C₈	—	2	1	no	—
460	C₈	—	2	1	no	—
461	C₈	—	2	1	no	—
462	C₈	—	53	4	no	—
463	C₈	—	106	2	no	—
464	C₈	—	212	2	no	—
465	C₈	—	212	2	no	—
466	C₈	—	212	2	no	—
467	C₈	C₂×#11719917	1	2	yes	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
478	C₄	—	2	1	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	1	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	—	4	1	no	—
488	C₄	C₂²×#11719917	1	1	yes	characteristic
489	C₄	#16811390	1	1	yes	characteristic
490	C₄	#135×D₅₃	1	2	yes	—
491	C₂	—	4	1	no	—
492	C₂	—	4	2	no	—
493	C₂	—	8	1	no	—
494	C₂	D₄×#11719917	1	1	yes	characteristic
495	C₂	#135×Dic₅₃	1	1	yes	characteristic
496	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	#18037
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #20283735	C₂	semidirect
2. #20283729	C₂	semidirect
3. #20273606	C₂	semidirect
4. #20273566	C₂	semidirect
5. #20273545	C₂	semidirect
6. SD₁₆×#88322	C₂	semidirect
7. #20273478	C₂	semidirect
8. #20273149	C₂	semidirect
9. #20273149	C₂	semidirect
10. C₈×#11719917	C₂	semidirect
135. SD₁₆×C₅₃	M₁₆	semidirect
218. #88322	#135	semidirect
297. C₅₃	#18037	semidirect
384. SD₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283735	2
#20283729	2
#20273606	2
#20273566	2
#20273545	2
SD₁₆×#88322	2
#20273478	2
#20273149	2
C₈×#11719917	2
C₄×#135×C₅₃	2
#18037	53