Subgroups — #27355810

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283847	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283840	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273618	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	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273154	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273154	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273004	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	#16810676	—	2	1	no	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810647	—	2	1	no	—
22	#16810647	C₂²	1	1	yes	characteristic
23	#16810647	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#88322	—	2	1	no	—
25	C₄×#11719917	C₂²	1	1	yes	characteristic
26	#16810376	C₂²	1	2	yes	—
27	#16810376	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	#16810344	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	—	2	1	no	—
51	C₂×#11719917	—	2	1	no	—
52	C₂×#11719917	—	2	1	no	—
53	C₂²×#88322	—	2	2	no	—
54	C₄⋊#88322	—	2	1	no	—
55	C₄⋊#88322	—	2	1	no	—
56	C₄⋊#88322	—	2	2	no	—
57	C₄⋊#88322	—	2	2	no	—
58	C₄⋊#88322	D₄	1	1	yes	characteristic
59	C₄⋊#88322	D₄	1	1	yes	characteristic
60	C₄⋊#88322	C₂³	1	2	yes	—
61	C₄⋊#88322	C₂³	1	2	yes	—
62	C₂²⋊#88322	—	2	2	no	—
63	C₂²⋊#88322	—	2	2	no	—
64	C₂²⋊#88322	D₄	1	2	yes	—
65	#13133128	—	2	1	no	—
66	#13133128	—	2	2	no	—
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	—	4	2	no	—
168	#11719917	—	8	2	no	—
169	#11719917	—	8	1	no	—
170	#11719917	—	8	1	no	—
171	C₂×#88322	—	2	2	no	—
172	C₂×#88322	—	2	2	no	—
173	C₂×#88322	—	2	2	no	—
174	C₂×#88322	—	8	2	no	—
175	C₂×#88322	C₂×D₄	1	2	yes	—
176	C₂×#88322	C₂×D₄	1	2	yes	—
177	C₂³×C₅₃	—	2	1	no	—
178	C₂³×C₅₃	—	2	1	no	—
179	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
180	Q₈×C₅₃	—	2	1	no	—
181	Q₈×C₅₃	—	2	1	no	—
182	Q₈×C₅₃	—	2	1	no	—
183	Q₈×C₅₃	C₂×M₁₆	1	2	yes	—
184	D₄×C₅₃	—	2	1	no	—
185	D₄×C₅₃	—	2	1	no	—
186	D₄×C₅₃	—	2	1	no	—
187	D₄×C₅₃	—	4	2	no	—
188	D₄×C₅₃	—	4	2	no	—
189	D₄×C₅₃	C₂×M₁₆	1	2	yes	—
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	2	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	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	4	2	no	—
202	C₂×C₄×C₅₃	—	4	2	no	—
203	C₂×C₄×C₅₃	—	4	1	no	—
204	C₂×C₄×C₅₃	—	4	1	no	—
205	C₂×C₄×C₅₃	—	4	1	no	—
206	C₂×C₄×C₅₃	—	4	1	no	—
207	C₂×C₄×C₅₃	—	4	2	no	—
208	C₂×C₄×C₅₃	—	4	2	no	—
209	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
215	C₈×C₅₃	—	2	1	no	—
216	C₈×C₅₃	—	2	1	no	—
217	C₈×C₅₃	—	2	1	no	—
218	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
219	#88322	—	2	2	no	—
220	#88322	—	2	2	no	—
221	#88322	—	4	2	no	—
222	#88322	—	4	2	no	—
223	#88322	—	4	2	no	—
224	#18061	—	53	1	no	maximal, Sylow
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	2	2	no	—
227	C₂²×C₅₃	—	2	1	no	—
228	C₂²×C₅₃	—	2	1	no	—
229	C₂²×C₅₃	—	4	1	no	—
230	C₂²×C₅₃	—	4	1	no	—
231	C₂²×C₅₃	—	4	1	no	—
232	C₂²×C₅₃	—	4	1	no	—
233	C₂²×C₅₃	—	4	1	no	—
234	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	2	2	no	—
237	C₄×C₅₃	—	2	1	no	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	4	1	no	—
240	C₄×C₅₃	—	4	1	no	—
241	C₄×C₅₃	—	4	1	no	—
242	C₄×C₅₃	—	4	1	no	—
243	C₄×C₅₃	—	4	1	no	—
244	C₄×C₅₃	#444	1	2	yes	—
245	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
246	C₄×C₅₃	#568	1	1	yes	characteristic
247	C₄×C₅₃	#576	1	1	yes	characteristic
248	C₄×C₅₃	#578	1	1	yes	characteristic
249	#2991	—	53	1	no	—
250	#2990	—	53	1	no	—
251	C₄×#135	D₅₃	1	1	yes	characteristic
252	#1592	—	53	2	no	—
253	#1584	—	53	2	no	—
254	#1581	—	53	2	no	—
255	SD₁₆⋊C₈	—	53	2	no	—
256	#1567	—	53	1	no	—
257	#1489	—	53	1	no	—
258	#1489	—	53	1	no	—
259	#1451	—	53	1	no	—
260	C₂×C₅₃	—	4	1	no	—
261	C₂×C₅₃	—	4	2	no	—
262	C₂×C₅₃	—	8	1	no	—
263	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
264	C₂×C₅₃	#2951	1	1	yes	characteristic
265	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
266	C₂×#135	Dic₅₃	1	1	yes	characteristic
267	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
268	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
269	#446	—	53	2	no	—
270	C₈×Q₈	—	53	2	no	—
271	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
272	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
273	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic
274	C₄×SD₁₆	Dic₅₃	1	1	yes	characteristic
275	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
276	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
277	C₄×D₈	Dic₅₃	1	1	yes	characteristic
278	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
279	#436	—	53	2	no	—
280	#436	—	106	2	no	—
281	#435	—	106	2	no	—
282	C₈×D₄	—	53	2	no	—
283	#432	—	106	1	no	—
284	#431	—	53	1	no	—
285	#431	—	106	1	no	—
286	#427	Dic₅₃	1	1	yes	characteristic
287	#423	—	53	1	no	—
288	#423	—	53	1	no	—
289	#423	—	106	1	no	—
290	C₂×C₄⋊C₈	—	106	1	no	—
291	#416	Dic₅₃	1	1	yes	characteristic
292	#416	Dic₅₃	1	1	yes	characteristic
293	C₄×M₁₆	—	53	1	no	—
294	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
295	#335	—	53	2	no	—
296	#335	—	53	2	no	—
297	Q₈⋊C₈	—	53	2	no	—
298	Q₈⋊C₈	—	53	2	no	—
299	D₄⋊C₈	—	53	2	no	—
300	D₄⋊C₈	—	53	2	no	—
301	#322	—	53	2	no	—
302	#322	—	53	2	no	—
303	C₅₃	#18061	1	1	yes	characteristic, complemented, Sylow
304	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
305	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
306	#135	—	2	4	no	—
307	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
308	C₂×SD₁₆	C₂×Dic₅₃	1	1	yes	characteristic
309	C₂×SD₁₆	C₂×Dic₅₃	1	1	yes	characteristic
310	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
311	C₂×M₁₆	—	106	2	no	—
312	C₂×M₁₆	—	106	1	no	—
313	C₂×M₁₆	—	106	1	no	—
314	C₂×M₁₆	—	106	1	no	—
315	C₂²×C₈	—	106	2	no	—
316	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
317	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
318	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
319	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
320	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
321	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
322	C₄×D₄	—	2	1	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
327	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
328	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
329	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
330	#117	—	2	1	no	—
331	#117	—	2	1	no	—
332	#117	C₂×Dic₅₃	1	1	yes	characteristic
333	#117	C₂×Dic₅₃	1	1	yes	characteristic
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	2	1	no	—
336	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
337	#107	C₂×Dic₅₃	1	1	yes	characteristic
338	#107	C₂×Dic₅₃	1	1	yes	characteristic
339	#106	C₂×Dic₅₃	1	1	yes	characteristic
340	#106	C₂×Dic₅₃	1	1	yes	characteristic
341	C₄⋊C₈	—	53	1	no	—
342	C₄⋊C₈	—	53	1	no	—
343	C₄⋊C₈	—	53	2	no	—
344	C₄⋊C₈	—	53	2	no	—
345	C₄⋊C₈	—	106	2	no	—
346	C₄⋊C₈	—	106	2	no	—
347	C₄⋊C₈	—	106	1	no	—
348	C₄⋊C₈	—	106	1	no	—
349	#103	C₂×Dic₅₃	1	1	yes	characteristic
350	#103	C₂×Dic₅₃	1	1	yes	characteristic
351	#103	C₂×Dic₅₃	1	1	yes	characteristic
352	#103	C₂×Dic₅₃	1	1	yes	characteristic
353	#102	C₂×Dic₅₃	1	1	yes	characteristic
354	#102	C₂×Dic₅₃	1	1	yes	characteristic
355	#102	C₂×Dic₅₃	1	1	yes	characteristic
356	#102	C₂×Dic₅₃	1	1	yes	characteristic
357	C₂²⋊C₈	—	53	2	no	—
358	C₂²⋊C₈	—	106	2	no	—
359	C₂²⋊C₈	—	106	2	no	—
360	#97	—	106	2	no	—
361	#97	—	106	1	no	—
362	C₄×C₈	—	53	2	no	—
363	C₄×C₈	—	53	2	no	—
364	C₄×C₈	—	106	2	no	—
365	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
366	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
367	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
368	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	#41	—	4	1	no	—
372	#41	—	4	1	no	—
373	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
374	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
378	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
386	C₂²×C₄	#11719947	1	1	yes	characteristic
387	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
388	Q₁₆	—	2	2	no	—
389	SD₁₆	—	2	2	no	—
390	SD₁₆	#11719917	1	4	yes	complemented
391	D₈	—	2	2	no	—
392	M₁₆	—	424	2	no	—
393	M₁₆	—	424	1	no	—
394	M₁₆	—	424	1	no	—
395	M₁₆	—	212	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	1	no	—
398	C₂×C₈	—	53	2	no	—
399	C₂×C₈	—	53	2	no	—
400	C₂×C₈	—	106	2	no	—
401	C₂×C₈	—	106	2	no	—
402	C₂×C₈	—	106	2	no	—
403	C₂×C₈	—	424	2	no	—
404	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
405	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
406	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
407	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
413	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
414	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
415	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	2	1	no	—
419	C₂²⋊C₄	—	2	1	no	—
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	1	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	D₄×D₅₃	1	1	yes	characteristic
429	C₄²	#11719947	1	1	yes	characteristic
430	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
431	C₄²	C₂³×D₅₃	1	1	yes	characteristic
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
435	Q₈	—	2	1	no	—
436	Q₈	—	2	1	no	—
437	Q₈	—	2	1	no	—
438	Q₈	C₂×#11719917	1	2	yes	—
439	D₄	—	2	1	no	—
440	D₄	—	2	1	no	—
441	D₄	—	2	1	no	—
442	D₄	—	4	2	no	—
443	D₄	—	4	2	no	—
444	D₄	C₂×#11719917	1	2	yes	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
465	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
466	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
467	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
468	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
469	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
470	C₈	—	2	1	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	106	2	no	—
474	C₈	—	106	2	no	—
475	C₈	—	212	2	no	—
476	C₈	—	212	2	no	—
477	C₈	—	212	2	no	—
478	C₈	C₂×#11719917	1	2	yes	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	4	1	no	—
487	C₂²	—	4	1	no	—
488	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
489	C₄	—	2	1	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	1	no	—
498	C₄	#16810745	1	2	yes	—
499	C₄	C₂²×#11719917	1	1	yes	characteristic
500	C₄	#16811390	1	1	yes	characteristic
501	C₄	#16811425	1	1	yes	characteristic
502	C₄	#16811437	1	1	yes	characteristic
503	C₂	—	4	1	no	—
504	C₂	—	4	2	no	—
505	C₂	—	8	1	no	—
506	C₂	D₄×#11719917	1	1	yes	characteristic
507	C₂	#20283591	1	1	yes	characteristic
508	C₂	C₂×#16810745	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	#18061
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. #20283847	C₂	semidirect
2. #20283840	C₂	semidirect
3. #20273618	C₂	semidirect
4. #20273566	C₂	semidirect
5. #20273545	C₂	semidirect
6. SD₁₆⋊#88322	C₂	semidirect
7. #20273474	C₂	semidirect
8. #20273154	C₂	semidirect
9. #20273154	C₂	semidirect
10. #20273004	C₂	semidirect
135. SD₁₆×C₅₃	M₁₆	semidirect
303. C₅₃	#18061	semidirect
390. SD₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283847	2
#20283840	2
#20273618	2
#20273566	2
#20273545	2
SD₁₆⋊#88322	2
#20273474	2
#20273154	2
#20273004	2
C₄×#135×C₅₃	2
#18061	53