Subgroups — #27396401

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283650	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283359	C₂	1	1	yes	maximal, characteristic
3	#20275660	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275464	C₂	1	2	yes	maximal, complemented
6	#20275421	C₂	1	2	yes	maximal
7	#20275365	C₂	1	2	yes	maximal
8	#20275316	C₂	1	2	yes	maximal, complemented
9	#20274735	C₂	1	1	yes	maximal, characteristic
10	#20274580	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3076	C₂	1	1	yes	maximal, characteristic
12	#16810754	—	2	1	no	—
13	#16810754	—	2	1	no	—
14	Q₈×#88322	—	2	1	no	—
15	Q₈×#88322	—	2	1	no	—
16	#16810684	—	2	1	no	—
17	#16810684	—	2	1	no	—
18	#16810676	—	2	1	no	—
19	#16810676	—	2	1	no	—
20	C₂×#13133153	C₂²	1	2	yes	—
21	#16810650	C₂²	1	2	yes	—
22	#16810650	C₂²	1	1	yes	characteristic
23	#16810650	C₂²	1	1	yes	characteristic
24	#16810647	C₂²	1	1	yes	characteristic
25	C₂×#13133144	C₂²	1	2	yes	—
26	#16810590	C₂²	1	1	yes	characteristic
27	#16810590	C₂²	1	1	yes	characteristic
28	#16810579	C₂²	1	2	yes	—
29	C₄×#11719917	C₂²	1	1	yes	characteristic
30	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
33	#16810388	C₂²	1	2	yes	—
34	#16810382	C₂²	1	2	yes	—
35	#16810382	C₂²	1	2	yes	—
36	#16810382	C₂²	1	2	yes	—
37	#16810377	C₂²	1	2	yes	—
38	C₅₃×#533	C₂²	1	1	yes	characteristic
39	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#479	C₄	1	2	yes	complemented
41	C₅₃×#478	C₄	1	2	yes	complemented
42	C₅₃×#477	C₄	1	1	yes	characteristic, complemented
43	C₅₃×#476	C₄	1	1	yes	characteristic, complemented
44	C₅₃×#475	C₄	1	1	yes	characteristic, complemented
45	C₅₃×#474	C₄	1	1	yes	characteristic, complemented
46	C₅₃×#427	C₂²	1	2	yes	—
47	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
48	C₅₃×#416	C₂²	1	2	yes	—
49	C₂×#11719917	C₂³	1	1	yes	characteristic
50	C₂×#11719917	C₂³	1	1	yes	characteristic
51	C₂×#11719917	C₂³	1	2	yes	—
52	C₂²×#88322	C₂³	1	1	yes	characteristic
53	C₂²×#88322	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊Dic₅₃	—	2	1	no	—
55	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
56	#13133153	—	4	2	no	—
57	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	—	2	1	no	—
59	C₄⋊#88322	—	2	2	no	—
60	C₄⋊#88322	—	2	2	no	—
61	C₄⋊#88322	—	2	1	no	—
62	C₄⋊#88322	—	2	1	no	—
63	C₄⋊#88322	—	2	1	no	—
64	C₄⋊#88322	—	2	1	no	—
65	#13133144	—	4	2	no	—
66	C₂²⋊#88322	—	2	1	no	—
67	C₂²⋊#88322	—	2	1	no	—
68	C₂²⋊#88322	—	2	1	no	—
69	C₂²⋊#88322	—	2	1	no	—
70	#13133128	—	2	1	no	—
71	#13133128	—	2	2	no	—
72	C₄×#88322	—	2	1	no	—
73	C₄×#88322	—	2	2	no	—
74	C₄×#88322	D₄	1	1	yes	characteristic
75	C₄×#88322	D₄	1	1	yes	characteristic
76	C₄×#88322	D₄	1	1	yes	characteristic
77	C₄×#88322	D₄	1	1	yes	characteristic
78	C₈×Dic₅₃	—	4	2	no	—
79	#13133124	—	2	1	no	—
80	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	#125×C₅₃	C₂×C₄	1	2	yes	—
85	C₂²⋊Q₈×C₅₃	—	2	1	no	—
86	C₂²⋊Q₈×C₅₃	—	2	1	no	—
87	C₄×Q₈×C₅₃	—	2	1	no	—
88	C₄×Q₈×C₅₃	—	2	1	no	—
89	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₅₃	—	2	1	no	—
92	C₄×D₄×C₅₃	—	2	1	no	—
93	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	#117×C₅₃	—	2	1	no	—
96	#117×C₅₃	C₂³	1	1	yes	characteristic
97	#117×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₅₃	—	2	1	no	—
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	#107×C₅₃	C₂×C₄	1	2	yes	—
101	#107×C₅₃	C₂×C₄	1	2	yes	—
102	#106×C₅₃	C₂×C₄	1	2	yes	—
103	#106×C₅₃	C₂×C₄	1	2	yes	—
104	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
107	#103×C₅₃	C₂×C₄	1	2	yes	—
108	#103×C₅₃	C₂×C₄	1	2	yes	—
109	#102×C₅₃	C₂×C₄	1	2	yes	—
110	#102×C₅₃	C₂×C₄	1	2	yes	—
111	#41×C₅₃	—	2	1	no	—
112	#41×C₅₃	—	2	1	no	—
113	#41×C₅₃	—	2	1	no	—
114	#41×C₅₃	—	2	1	no	—
115	C₂×Q₈×C₅₃	—	2	1	no	—
116	C₂×Q₈×C₅₃	—	2	1	no	—
117	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₅₃	—	2	1	no	—
119	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂²×Dic₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₂²⋊C₄×C₅₃	—	2	2	no	—
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₂²⋊C₄×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	—	2	1	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄⋊Dic₅₃	—	2	4	no	—
154	C₄×Dic₅₃	—	2	2	no	—
155	C₄×Dic₅₃	—	2	2	no	—
156	#11719917	—	2	2	no	—
157	#11719917	—	2	1	no	—
158	#11719917	—	2	1	no	—
159	#11719917	—	2	1	no	—
160	#11719917	—	2	1	no	—
161	#11719917	—	2	2	no	—
162	C₂×#88322	—	2	1	no	—
163	C₂×#88322	—	2	1	no	—
164	C₂×#88322	—	2	1	no	—
165	C₂×#88322	—	2	1	no	—
166	C₂×#88322	—	2	1	no	—
167	C₂×#88322	—	2	1	no	—
168	C₂×#88322	—	2	2	no	—
169	C₂×#88322	C₂×D₄	1	1	yes	characteristic
170	C₂×#88322	C₂×D₄	1	1	yes	characteristic
171	C₂×#88322	C₂×D₄	1	1	yes	characteristic
172	C₂×#88322	C₂×D₄	1	1	yes	characteristic
173	C₂³×C₅₃	—	2	1	no	—
174	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
175	Q₈×C₅₃	—	2	1	no	—
176	Q₈×C₅₃	—	2	1	no	—
177	Q₈×C₅₃	—	4	2	no	—
178	Q₈×C₅₃	—	4	2	no	—
179	D₄×C₅₃	—	2	1	no	—
180	D₄×C₅₃	—	2	1	no	—
181	D₄×C₅₃	—	4	1	no	—
182	D₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
202	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
203	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
204	C₂×Dic₅₃	—	2	4	no	—
205	C₂×Dic₅₃	—	4	2	no	—
206	C₂×Dic₅₃	—	4	2	no	—
207	C₈×C₅₃	—	2	2	no	—
208	C₈×C₅₃	—	2	2	no	—
209	#88322	—	2	1	no	—
210	#88322	—	2	1	no	—
211	#88322	—	2	1	no	—
212	#88322	—	2	1	no	—
213	#88322	—	4	1	no	—
214	#88322	—	4	1	no	—
215	#88322	—	4	2	no	—
216	#22130	—	53	1	no	maximal, Sylow
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	4	1	no	—
220	C₂²×C₅₃	—	4	1	no	—
221	C₂²×C₅₃	—	4	1	no	—
222	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂²×C₅₃	#541	1	1	yes	characteristic
224	C₂²×C₅₃	#542	1	1	yes	characteristic
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	2	1	no	—
228	C₄×C₅₃	—	4	1	no	—
229	C₄×C₅₃	—	4	1	no	—
230	C₄×C₅₃	—	4	2	no	—
231	C₄×C₅₃	—	4	1	no	—
232	C₄×C₅₃	—	4	2	no	—
233	C₄×C₅₃	#443	1	2	yes	—
234	C₄×C₅₃	#519	1	1	yes	characteristic
235	C₄×C₅₃	#520	1	1	yes	characteristic
236	C₄×C₅₃	#536	1	1	yes	characteristic
237	C₄×C₅₃	#537	1	1	yes	characteristic
238	Dic₅₃	—	4	4	no	—
239	#3076	D₅₃	1	1	yes	characteristic
240	#2960	—	53	1	no	—
241	#2922	—	53	1	no	—
242	#1986	—	53	1	no	—
243	#1950	—	53	1	no	—
244	#1945	—	53	2	no	—
245	#1937	—	53	2	no	—
246	#1929	—	53	2	no	—
247	#1922	—	53	2	no	—
248	#1806	—	53	1	no	—
249	C₈×C₄⋊C₄	—	53	1	no	—
250	C₂×C₅₃	—	2	1	no	—
251	C₂×C₅₃	—	2	1	no	—
252	C₂×C₅₃	—	8	1	no	—
253	C₂×C₅₃	#2323	1	1	yes	characteristic
254	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
255	C₂×C₅₃	#2955	1	1	yes	characteristic
256	#533	C₂×D₅₃	1	1	yes	characteristic
257	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
258	#479	Dic₅₃	1	2	yes	complemented
259	#478	Dic₅₃	1	2	yes	complemented
260	#477	Dic₅₃	1	1	yes	characteristic, complemented
261	#476	Dic₅₃	1	1	yes	characteristic, complemented
262	#475	Dic₅₃	1	1	yes	characteristic, complemented
263	#474	Dic₅₃	1	1	yes	characteristic, complemented
264	#446	—	106	1	no	—
265	#446	—	106	1	no	—
266	C₈×Q₈	—	106	1	no	—
267	C₈×Q₈	—	106	1	no	—
268	#433	—	106	1	no	—
269	#433	—	106	1	no	—
270	#431	—	106	1	no	—
271	#431	—	106	1	no	—
272	C₂×#108	—	53	2	no	—
273	#427	C₂×D₅₃	1	2	yes	—
274	#424	—	53	1	no	—
275	#424	—	53	1	no	—
276	#424	—	53	2	no	—
277	#423	—	53	1	no	—
278	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
279	C₂×#104	—	53	2	no	—
280	#416	C₂×D₅₃	1	2	yes	—
281	#408	—	53	1	no	—
282	#408	—	53	1	no	—
283	#405	—	53	2	no	—
284	C₄×M₁₆	—	53	1	no	—
285	C₂×C₄×C₈	—	53	1	no	—
286	C₂×C₄×C₈	—	53	2	no	—
287	C₄×C₄⋊C₄	—	53	1	no	—
288	#341	—	53	2	no	—
289	#339	—	53	2	no	—
290	#339	—	53	2	no	—
291	#339	—	53	2	no	—
292	#336	—	53	2	no	—
293	C₅₃	#22130	1	1	yes	characteristic, complemented, Sylow
294	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
295	C₂×M₁₆	—	53	1	no	—
296	C₂×M₁₆	—	53	1	no	—
297	C₂×M₁₆	—	53	2	no	—
298	C₂²×C₈	—	53	1	no	—
299	C₂²×C₈	—	53	1	no	—
300	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
301	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
302	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
303	#125	C₂×Dic₅₃	1	2	yes	—
304	C₂²⋊Q₈	—	2	1	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
309	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
313	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
314	#117	—	2	1	no	—
315	#117	C₂²×D₅₃	1	1	yes	characteristic
316	#117	C₂²×D₅₃	1	2	yes	—
317	C₂×C₄⋊C₄	—	106	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	53	2	no	—
320	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
321	#108	—	212	2	no	—
322	#107	C₂×Dic₅₃	1	2	yes	—
323	#107	C₂×Dic₅₃	1	2	yes	—
324	#106	C₂×Dic₅₃	1	2	yes	—
325	#106	C₂×Dic₅₃	1	2	yes	—
326	C₄⋊C₈	—	106	1	no	—
327	C₄⋊C₈	—	106	1	no	—
328	C₄⋊C₈	—	106	1	no	—
329	C₄⋊C₈	—	106	1	no	—
330	C₄⋊C₈	—	106	2	no	—
331	C₄⋊C₈	—	106	2	no	—
332	C₄⋊C₈	—	106	1	no	—
333	C₄⋊C₈	—	106	1	no	—
334	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
335	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
336	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
337	#104	—	212	2	no	—
338	#103	C₂×Dic₅₃	1	2	yes	—
339	#103	C₂×Dic₅₃	1	2	yes	—
340	#102	C₂×Dic₅₃	1	2	yes	—
341	#102	C₂×Dic₅₃	1	2	yes	—
342	C₂²⋊C₈	—	106	1	no	—
343	C₂²⋊C₈	—	106	1	no	—
344	C₂²⋊C₈	—	106	1	no	—
345	C₂²⋊C₈	—	106	1	no	—
346	#97	—	106	2	no	—
347	#97	—	106	1	no	—
348	C₄×C₈	—	53	1	no	—
349	C₄×C₈	—	53	1	no	—
350	C₄×C₈	—	53	1	no	—
351	C₄×C₈	—	53	1	no	—
352	C₄×C₈	—	106	2	no	—
353	C₄×C₈	—	106	1	no	—
354	C₄×C₈	—	212	2	no	—
355	#95	—	106	1	no	—
356	#41	—	2	1	no	—
357	#41	—	2	1	no	—
358	#41	—	2	1	no	—
359	#41	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	106	2	no	—
369	C₂²×C₄	#11719947	1	1	yes	characteristic
370	C₂²×C₄	#11719947	1	1	yes	characteristic
371	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
372	M₁₆	—	106	2	no	—
373	M₁₆	—	106	1	no	—
374	M₁₆	—	106	1	no	—
375	M₁₆	—	106	1	no	—
376	M₁₆	—	106	1	no	—
377	M₁₆	—	106	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	53	1	no	—
381	C₂×C₈	—	53	1	no	—
382	C₂×C₈	—	53	1	no	—
383	C₂×C₈	—	53	1	no	—
384	C₂×C₈	—	106	1	no	—
385	C₂×C₈	—	106	1	no	—
386	C₂×C₈	—	106	1	no	—
387	C₂×C₈	—	106	1	no	—
388	C₂×C₈	—	106	1	no	—
389	C₂×C₈	—	106	1	no	—
390	C₂×C₈	—	106	2	no	—
391	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
392	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	106	4	no	—
400	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
401	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
402	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
404	C₂²⋊C₄	—	2	1	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	2	1	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	106	2	no	—
413	C₄²	—	106	2	no	—
414	C₄²	D₄×D₅₃	1	1	yes	characteristic
415	C₄²	D₄×D₅₃	1	1	yes	characteristic
416	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
417	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
418	C₂³	—	2	1	no	—
419	C₂³	C₂×#11719947	1	1	yes	characteristic
420	Q₈	—	2	1	no	—
421	Q₈	—	2	1	no	—
422	Q₈	—	4	2	no	—
423	Q₈	—	4	2	no	—
424	D₄	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	4	1	no	—
427	D₄	—	4	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	106	4	no	—
440	C₂×C₄	—	212	2	no	—
441	C₂×C₄	—	212	2	no	—
442	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
443	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
444	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
445	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
446	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
447	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
448	C₂×C₄	#13133263	1	1	yes	characteristic
449	C₂×C₄	#13133264	1	1	yes	characteristic
450	C₂×C₄	#13133265	1	1	yes	characteristic
451	C₂×C₄	#13133265	1	1	yes	characteristic
452	C₈	—	2	2	no	—
453	C₈	—	2	2	no	—
454	C₈	—	106	1	no	—
455	C₈	—	106	1	no	—
456	C₈	—	106	1	no	—
457	C₈	—	106	1	no	—
458	C₈	—	212	1	no	—
459	C₈	—	212	1	no	—
460	C₈	—	212	2	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
467	C₂²	#16811233	1	1	yes	characteristic
468	C₂²	#16811246	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	1	no	—
473	C₄	—	4	1	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	2	no	—
477	C₄	—	212	4	no	—
478	C₄	#16810740	1	2	yes	—
479	C₄	#16811101	1	1	yes	characteristic
480	C₄	#16811106	1	1	yes	characteristic
481	C₄	#16811196	1	1	yes	characteristic
482	C₄	#16811203	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	8	1	no	—
486	C₂	#20277399	1	1	yes	characteristic
487	C₂	C₂×#16810740	1	1	yes	characteristic
488	C₂	#20283618	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3076	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#41×C₅₃	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3076	D₅₃
hypocenter	C₅₃	#22130
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3076

Complements

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

N	H ≅ G/N	type
1. #20283650	C₂	semidirect
3. #20275660	C₂	semidirect
4. #20275491	C₂	semidirect
5. #20275464	C₂	semidirect
8. #20275316	C₂	semidirect
10. #20274580	C₂	semidirect
40. C₅₃×#479	C₄	semidirect
41. C₅₃×#478	C₄	semidirect
42. C₅₃×#477	C₄	semidirect
43. C₅₃×#476	C₄	semidirect
44. C₅₃×#475	C₄	semidirect
45. C₅₃×#474	C₄	semidirect
258. #479	Dic₅₃	semidirect
259. #478	Dic₅₃	semidirect
260. #477	Dic₅₃	semidirect
261. #476	Dic₅₃	semidirect
262. #475	Dic₅₃	semidirect
263. #474	Dic₅₃	semidirect
293. C₅₃	#22130	semidirect

Maximal subgroups

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

subgroup	index
#20283650	2
#20283359	2
#20275660	2
#20275491	2
#20275464	2
#20275421	2
#20275365	2
#20275316	2
#20274735	2
#20274580	2
C₅₃×#3076	2
#22130	53