Subgroups — #27354064

Order 13568.

720 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283691	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273580	C₂	1	2	yes	maximal, complemented
4	#20273573	C₂	1	2	yes	maximal, complemented
5	SD₁₆×#88322	C₂	1	2	yes	maximal, complemented
6	D₈×#88322	C₂	1	2	yes	maximal, complemented
7	#20273500	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273158	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273128	C₂	1	1	yes	maximal, characteristic, complemented
10	#20272996	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3105	C₂	1	1	yes	maximal, characteristic
12	C₂³×#88322	—	2	1	no	—
13	#16810754	C₂²	1	1	yes	characteristic, complemented
14	Q₈×#88322	C₂²	1	1	yes	characteristic, complemented
15	#16810701	C₂²	1	1	yes	characteristic, complemented
16	#16810695	—	2	1	no	—
17	D₄×#88322	—	2	1	no	—
18	D₄×#88322	—	2	1	no	—
19	D₄×#88322	—	2	1	no	—
20	D₄×#88322	C₂²	1	2	yes	—
21	D₄×#88322	C₂²	1	2	yes	—
22	D₄×#88322	C₂²	1	1	yes	characteristic, complemented
23	#16810684	—	2	1	no	—
24	#16810680	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊#88322	—	2	1	no	—
27	#16810579	—	2	1	no	—
28	C₄×#11719917	C₂²	1	1	yes	characteristic
29	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	2	yes	complemented
31	#16810374	C₂²	1	2	yes	complemented
32	Q₈⋊#88322	C₂²	1	2	yes	complemented
33	D₄⋊#88322	C₂²	1	2	yes	—
34	D₄⋊#88322	C₂²	1	2	yes	—
35	D₄⋊#88322	C₂²	1	2	yes	complemented
36	#16810345	C₂²	1	2	yes	complemented
37	C₈×#88322	C₂²	1	2	yes	complemented
38	C₅₃×#529	C₂²	1	1	yes	characteristic
39	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#484	C₄	1	1	yes	characteristic
41	C₅₃×#481	C₄	1	1	yes	characteristic
42	C₅₃×#478	C₄	1	2	yes	—
43	C₅₃×#463	C₄	1	2	yes	—
44	C₅₃×#450	C₄	1	1	yes	characteristic
45	C₅₃×#447	C₄	1	1	yes	characteristic
46	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
47	C₄×D₈×C₅₃	C₂²	1	2	yes	—
48	C₅₃×#431	C₂²	1	1	yes	characteristic
49	C₂×#11719917	—	2	1	no	—
50	C₂×#11719917	—	2	1	no	—
51	C₂²×#88322	—	2	1	no	—
52	C₂²×#88322	—	2	2	no	—
53	C₂²×#88322	—	2	2	no	—
54	C₂²×#88322	—	2	1	no	—
55	C₂²×#88322	—	4	4	no	—
56	C₂²×#88322	D₄	1	1	yes	characteristic
57	C₂²×#88322	D₄	1	1	yes	characteristic
58	C₄⋊#88322	—	2	1	no	—
59	C₄⋊#88322	—	2	1	no	—
60	C₄⋊#88322	—	2	1	no	—
61	C₄⋊#88322	C₂³	1	1	yes	characteristic
62	C₄⋊#88322	C₂³	1	1	yes	characteristic
63	C₄⋊#88322	C₂³	1	2	yes	—
64	C₂²⋊#88322	—	2	2	no	—
65	C₂²⋊#88322	—	2	2	no	—
66	C₂²⋊#88322	—	2	1	no	—
67	C₂²⋊#88322	—	2	1	no	—
68	C₂²⋊#88322	—	2	1	no	—
69	#13133128	—	2	1	no	—
70	#13133128	—	2	1	no	—
71	#13133128	—	2	1	no	—
72	C₄×#88322	—	2	1	no	—
73	C₄×#88322	D₄	1	1	yes	characteristic
74	C₄×#88322	D₄	1	1	yes	characteristic
75	C₄×#88322	C₂³	1	2	yes	—
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	—	2	1	no	—
80	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
81	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
82	#126×C₅₃	—	2	1	no	—
83	#125×C₅₃	C₂×C₄	1	1	yes	characteristic
84	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
85	#123×C₅₃	—	2	1	no	—
86	C₂²⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	#121×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
93	C₄×D₄×C₅₃	C₂³	1	2	yes	—
94	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
95	#117×C₅₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	#107×C₅₃	C₂×C₄	1	2	yes	—
100	#106×C₅₃	C₂×C₄	1	2	yes	—
101	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
102	#103×C₅₃	C₂×C₄	1	2	yes	—
103	#102×C₅₃	C₂×C₄	1	2	yes	—
104	#102×C₅₃	C₂×C₄	1	2	yes	—
105	#102×C₅₃	C₂×C₄	1	2	yes	—
106	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₄×C₈×C₅₃	C₂³	1	2	yes	—
109	C₂⁴×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×D₄×C₅₃	—	2	4	no	—
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	2	no	—
120	C₂²×C₄×C₅₃	—	4	2	no	—
121	C₂²×C₄×C₅₃	—	4	2	no	—
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
125	SD₁₆×C₅₃	—	2	4	no	—
126	D₈×C₅₃	—	2	4	no	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂²⋊C₄×C₅₃	—	2	1	no	—
138	C₂²⋊C₄×C₅₃	—	2	2	no	—
139	C₂²⋊C₄×C₅₃	—	2	1	no	—
140	C₂²⋊C₄×C₅₃	—	2	1	no	—
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₂²⋊C₄×C₅₃	—	2	2	no	—
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₅₃	#41	1	1	yes	characteristic
148	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
149	#11719917	—	2	4	no	—
150	#11719917	—	4	2	no	—
151	C₂×#88322	—	2	2	no	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	—	2	1	no	—
154	C₂×#88322	—	2	1	no	—
155	C₂×#88322	—	2	2	no	—
156	C₂×#88322	—	4	1	no	—
157	C₂×#88322	—	4	4	no	—
158	C₂×#88322	—	4	2	no	—
159	C₂×#88322	—	4	2	no	—
160	C₂×#88322	—	4	4	no	—
161	C₂×#88322	C₂×D₄	1	1	yes	characteristic
162	C₂×#88322	C₂×D₄	1	1	yes	characteristic
163	C₂×#88322	C₂×D₄	1	1	yes	characteristic
164	C₂×#88322	C₂×D₄	1	1	yes	characteristic
165	C₂³×C₅₃	—	2	4	no	—
166	C₂³×C₅₃	—	2	2	no	—
167	C₂³×C₅₃	—	2	1	no	—
168	C₂³×C₅₃	—	4	2	no	—
169	C₂³×C₅₃	—	4	2	no	—
170	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
171	Q₈×C₅₃	—	2	2	no	—
172	D₄×C₅₃	—	2	4	no	—
173	D₄×C₅₃	—	2	2	no	—
174	D₄×C₅₃	—	2	2	no	—
175	D₄×C₅₃	—	4	2	no	—
176	D₄×C₅₃	#131	1	4	yes	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	4	4	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₈×C₅₃	—	2	2	no	—
198	C₈×C₅₃	—	2	2	no	—
199	#88322	—	2	4	no	—
200	#88322	—	2	1	no	—
201	#88322	—	2	1	no	—
202	#88322	—	2	1	no	—
203	#88322	—	4	1	no	—
204	#88322	—	4	2	no	—
205	#88322	—	4	1	no	—
206	#88322	AGL(1,ℤ/8)	1	2	yes	—
207	#17910	—	53	1	no	maximal, Sylow
208	C₂²×C₅₃	—	2	1	no	—
209	C₂²×C₅₃	—	2	1	no	—
210	C₂²×C₅₃	—	2	4	no	—
211	C₂²×C₅₃	—	4	8	no	—
212	C₂²×C₅₃	—	4	2	no	—
213	C₂²×C₅₃	—	4	2	no	—
214	C₂²×C₅₃	—	4	1	no	—
215	C₂²×C₅₃	—	4	1	no	—
216	C₂²×C₅₃	—	4	1	no	—
217	C₂²×C₅₃	#443	1	2	yes	—
218	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	2	1	no	—
222	C₄×C₅₃	—	2	1	no	—
223	C₄×C₅₃	—	4	1	no	—
224	C₄×C₅₃	—	4	1	no	—
225	C₄×C₅₃	—	4	2	no	—
226	C₄×C₅₃	—	4	1	no	—
227	C₄×C₅₃	C₂×#131	1	2	yes	—
228	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₄×C₅₃	#575	1	1	yes	characteristic
230	#3105	D₅₃	1	1	yes	characteristic
231	#2966	—	53	1	no	—
232	C₂×C₈×D₄	—	53	1	no	—
233	#1586	—	53	2	no	—
234	#1585	—	53	2	no	—
235	C₈×SD₁₆	—	53	2	no	—
236	C₈×D₈	—	53	2	no	—
237	#1573	—	53	1	no	—
238	#1490	—	53	1	no	—
239	#1483	—	53	1	no	—
240	C₈×M₁₆	—	53	1	no	—
241	C₂×C₅₃	—	2	2	no	—
242	C₂×C₅₃	—	4	4	no	—
243	C₂×C₅₃	—	8	1	no	—
244	C₂×C₅₃	#2929	1	1	yes	characteristic
245	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
247	C₂³×C₈	—	106	1	no	—
248	#529	C₂×D₅₃	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
250	#484	Dic₅₃	1	1	yes	characteristic
251	#481	Dic₅₃	1	1	yes	characteristic
252	#478	Dic₅₃	1	2	yes	—
253	#463	Dic₅₃	1	2	yes	—
254	#450	Dic₅₃	1	1	yes	characteristic
255	#447	Dic₅₃	1	1	yes	characteristic
256	#446	—	53	1	no	—
257	C₈×Q₈	—	53	1	no	—
258	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
259	C₄×D₈	C₂×D₅₃	1	2	yes	—
260	#436	—	53	1	no	—
261	#435	—	106	1	no	—
262	C₈×D₄	—	53	1	no	—
263	C₈×D₄	—	53	2	no	—
264	C₈×D₄	—	53	2	no	—
265	C₈×D₄	—	106	1	no	—
266	C₈×D₄	—	106	1	no	—
267	C₈×D₄	—	106	1	no	—
268	#433	—	106	1	no	—
269	#432	—	53	1	no	—
270	#431	C₂×D₅₃	1	1	yes	characteristic
271	C₂×C₄⋊C₈	—	53	1	no	—
272	C₂×C₂²⋊C₈	—	106	1	no	—
273	#405	—	106	1	no	—
274	C₄×M₁₆	—	53	1	no	—
275	C₂×C₄×C₈	—	53	1	no	—
276	#335	—	53	2	no	—
277	#334	—	53	2	no	—
278	Q₈⋊C₈	—	53	2	no	—
279	D₄⋊C₈	—	53	2	no	—
280	D₄⋊C₈	—	53	2	no	—
281	D₄⋊C₈	—	53	2	no	—
282	#322	—	53	2	no	—
283	C₈²	—	53	2	no	—
284	C₅₃	#17910	1	1	yes	characteristic, complemented, Sylow
285	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
286	C₂³×C₄	—	2	1	no	—
287	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
288	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
289	C₂×M₁₆	—	106	1	no	—
290	C₂×M₁₆	—	106	1	no	—
291	C₂²×C₈	—	53	1	no	—
292	C₂²×C₈	—	53	1	no	—
293	C₂²×C₈	—	106	2	no	—
294	C₂²×C₈	—	106	2	no	—
295	C₂²×C₈	—	106	1	no	—
296	C₂²×C₈	—	106	1	no	—
297	C₂²×C₈	—	212	4	no	—
298	#126	—	2	1	no	—
299	#125	C₂×Dic₅₃	1	1	yes	characteristic
300	#124	C₂×Dic₅₃	1	1	yes	characteristic
301	#123	—	2	1	no	—
302	C₂²⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
303	#121	C₂×Dic₅₃	1	1	yes	characteristic
304	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
309	C₄×D₄	C₂²×D₅₃	1	2	yes	—
310	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
311	#117	—	2	1	no	—
312	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
315	#107	C₂×Dic₅₃	1	2	yes	—
316	#106	C₂×Dic₅₃	1	2	yes	—
317	C₄⋊C₈	—	53	1	no	—
318	C₄⋊C₈	—	53	1	no	—
319	C₄⋊C₈	—	53	2	no	—
320	C₄⋊C₈	—	106	1	no	—
321	C₄⋊C₈	—	106	1	no	—
322	C₄⋊C₈	—	106	1	no	—
323	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
324	#103	C₂×Dic₅₃	1	2	yes	—
325	#102	C₂×Dic₅₃	1	2	yes	—
326	#102	C₂×Dic₅₃	1	2	yes	—
327	#102	C₂×Dic₅₃	1	2	yes	—
328	C₂²⋊C₈	—	106	2	no	—
329	C₂²⋊C₈	—	106	2	no	—
330	C₂²⋊C₈	—	106	1	no	—
331	C₂²⋊C₈	—	106	1	no	—
332	C₂²⋊C₈	—	106	1	no	—
333	C₂²⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
334	C₂²⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
335	#97	—	106	1	no	—
336	#97	—	106	1	no	—
337	#97	—	106	1	no	—
338	C₄×C₈	—	53	1	no	—
339	C₄×C₈	—	53	1	no	—
340	C₄×C₈	—	53	1	no	—
341	C₄×C₈	—	53	1	no	—
342	C₄×C₈	—	53	2	no	—
343	C₄×C₈	—	106	1	no	—
344	C₄×C₈	C₂²×D₅₃	1	2	yes	—
345	C₂⁴	—	2	1	no	—
346	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
351	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
359	C₂²×C₄	#11719947	1	1	yes	characteristic
360	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
361	SD₁₆	—	2	4	no	—
362	D₈	—	2	4	no	—
363	M₁₆	—	106	4	no	—
364	M₁₆	—	212	2	no	—
365	C₂×C₈	—	53	1	no	—
366	C₂×C₈	—	53	1	no	—
367	C₂×C₈	—	53	1	no	—
368	C₂×C₈	—	53	1	no	—
369	C₂×C₈	—	106	2	no	—
370	C₂×C₈	—	106	2	no	—
371	C₂×C₈	—	106	1	no	—
372	C₂×C₈	—	106	1	no	—
373	C₂×C₈	—	106	2	no	—
374	C₂×C₈	—	212	4	no	—
375	C₂×C₈	—	212	2	no	—
376	C₂×C₈	—	212	1	no	—
377	C₂×C₈	—	212	4	no	—
378	C₂×C₈	—	212	2	no	—
379	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
380	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
387	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
388	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
389	C₂²⋊C₄	—	2	1	no	—
390	C₂²⋊C₄	—	2	2	no	—
391	C₂²⋊C₄	—	2	1	no	—
392	C₂²⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	D₄×D₅₃	1	1	yes	characteristic
398	C₄²	#11719947	1	1	yes	characteristic
399	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
400	C₄²	C₂³×D₅₃	1	1	yes	characteristic
401	C₂³	—	2	4	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
407	Q₈	—	2	2	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	2	no	—
411	D₄	—	4	2	no	—
412	D₄	#13133224	1	4	yes	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
428	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
429	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
430	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
431	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
432	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
433	C₈	—	2	2	no	—
434	C₈	—	2	2	no	—
435	C₈	—	53	2	no	—
436	C₈	—	106	4	no	—
437	C₈	—	106	1	no	—
438	C₈	—	106	1	no	—
439	C₈	—	106	1	no	—
440	C₈	—	212	1	no	—
441	C₈	—	212	2	no	—
442	C₈	—	212	1	no	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	4	8	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	#16810740	1	2	yes	—
453	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	4	1	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	1	no	—
461	C₄	—	4	1	no	—
462	C₄	C₂×#13133224	1	2	yes	—
463	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
464	C₄	#16811419	1	1	yes	characteristic
465	C₂	—	2	2	no	—
466	C₂	—	4	4	no	—
467	C₂	—	8	1	no	—
468	C₂	#20283410	1	1	yes	characteristic
469	C₂	AGL(1,ℤ/8)×Dic₅₃	1	1	yes	characteristic
470	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3105	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#447	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3105	D₅₃
hypocenter	C₅₃	#17910
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3105

Complements

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

N	H ≅ G/N	type
1. #20283691	C₂	semidirect
2. C₂×D₄×#88322	C₂	semidirect
3. #20273580	C₂	semidirect
4. #20273573	C₂	semidirect
5. SD₁₆×#88322	C₂	semidirect
6. D₈×#88322	C₂	semidirect
7. #20273500	C₂	semidirect
8. #20273158	C₂	semidirect
9. #20273128	C₂	semidirect
10. #20272996	C₂	semidirect
13. #16810754	C₂²	semidirect
14. Q₈×#88322	C₂²	semidirect
15. #16810701	C₂²	semidirect
22. D₄×#88322	C₂²	semidirect
30. #16810376	C₂²	semidirect
31. #16810374	C₂²	semidirect
32. Q₈⋊#88322	C₂²	semidirect
35. D₄⋊#88322	C₂²	semidirect
36. #16810345	C₂²	semidirect
37. C₈×#88322	C₂²	semidirect
284. C₅₃	#17910	semidirect

Maximal subgroups

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

subgroup	index
#20283691	2
C₂×D₄×#88322	2
#20273580	2
#20273573	2
SD₁₆×#88322	2
D₈×#88322	2
#20273500	2
#20273158	2
#20273128	2
#20272996	2
C₅₃×#3105	2
#17910	53