Subgroups — #27402554

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810740	C₂	1	1	yes	maximal, characteristic
2	#135×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20283281	C₂	1	2	yes	maximal, complemented
4	#20275463	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275461	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274990	C₂	1	2	yes	maximal, complemented
7	#20274988	C₂	1	2	yes	maximal, complemented
8	#20274885	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274733	C₂	1	2	yes	maximal, complemented
10	C₈×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	C₂²	1	2	yes	—
13	#41×Dic₅₃	C₂²	1	2	yes	—
14	#16810740	—	2	4	no	—
15	Q₁₆×Dic₅₃	C₄	1	1	yes	characteristic, complemented
16	Q₁₆×Dic₅₃	C₄	1	1	yes	characteristic, complemented
17	SD₁₆×Dic₅₃	C₄	1	2	yes	complemented
18	SD₁₆×Dic₅₃	C₄	1	2	yes	complemented
19	D₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
20	D₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
21	C₂×#13133153	C₂²	1	1	yes	characteristic
22	C₂×#13133152	C₂²	1	2	yes	—
23	#16810661	C₂²	1	1	yes	characteristic
24	#16810651	—	2	2	no	—
25	#16810651	C₂²	1	2	yes	—
26	#16810649	—	2	2	no	—
27	#16810649	C₂²	1	2	yes	—
28	#16810646	—	2	2	no	—
29	#16810646	C₂²	1	2	yes	—
30	C₂×#13133146	—	2	2	no	—
31	C₂×#13133146	C₂²	1	1	yes	characteristic
32	C₂×#13133144	C₂²	1	2	yes	—
33	#16810619	C₂²	1	2	yes	—
34	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₄×#11719919	C₂²	1	1	yes	characteristic
37	#16810384	C₂²	1	2	yes	—
38	#16810383	C₂²	1	2	yes	—
39	#16810378	C₂²	1	1	yes	characteristic
40	#16810377	C₂²	1	1	yes	characteristic
41	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#131×C₅₃	C₂²	1	2	yes	—
43	C₅₃×#443	—	2	4	no	—
44	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#104×C₅₃	C₂²	1	2	yes	—
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	#13133224	—	2	4	no	—
48	#13133224	—	4	2	no	—
49	C₂×#11719917	—	2	2	no	—
50	C₂×#11719917	C₂³	1	2	yes	—
51	C₂²×#88322	—	2	2	no	—
52	C₂²×#88322	C₂³	1	1	yes	characteristic
53	C₂²×#88322	C₂³	1	1	yes	characteristic
54	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
55	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
56	D₄×Dic₅₃	—	2	2	no	—
57	D₄×Dic₅₃	—	2	2	no	—
58	D₄×Dic₅₃	C₂×C₄	1	2	yes	—
59	D₄×Dic₅₃	C₂×C₄	1	2	yes	—
60	#13133176	—	2	2	no	—
61	#13133176	C₂³	1	2	yes	—
62	C₂×#11719919	—	2	1	no	—
63	C₂×C₄×Dic₅₃	—	2	2	no	—
64	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
66	#13133153	—	2	2	no	—
67	#13133152	—	4	2	no	—
68	#13133151	C₂×C₄	1	1	yes	characteristic
69	#13133151	C₂×C₄	1	1	yes	characteristic
70	#13133149	C₂×C₄	1	2	yes	—
71	#13133146	—	2	2	no	—
72	#13133146	—	2	2	no	—
73	#13133146	—	2	2	no	—
74	#13133146	—	2	2	no	—
75	#13133146	—	2	2	no	—
76	#13133146	—	2	2	no	—
77	#13133146	D₄	1	1	yes	characteristic
78	#13133146	D₄	1	1	yes	characteristic
79	#13133146	D₄	1	1	yes	characteristic
80	#13133146	D₄	1	1	yes	characteristic
81	#13133144	—	2	4	no	—
82	#13133141	C₂×C₄	1	2	yes	—
83	#13133141	C₂×C₄	1	2	yes	—
84	#13133138	C₂×C₄	1	2	yes	—
85	#13133138	C₂×C₄	1	2	yes	—
86	C₈×Dic₅₃	—	2	2	no	—
87	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
90	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
91	#13133124	—	2	1	no	—
92	C₂×#41×C₅₃	C₂³	1	2	yes	—
93	#135×C₅₃	D₄	1	4	yes	—
94	#135×C₅₃	Q₈	1	4	yes	—
95	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
97	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	#131×C₅₃	—	2	4	no	—
99	#131×C₅₃	—	4	2	no	—
100	C₂×M₁₆×C₅₃	—	2	2	no	—
101	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
102	C₂²×C₈×C₅₃	—	2	2	no	—
103	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	#108×C₅₃	—	2	2	no	—
107	#104×C₅₃	—	2	4	no	—
108	C₄×C₈×C₅₃	—	2	2	no	—
109	#41×C₅₃	—	2	2	no	—
110	#41×C₅₃	—	2	2	no	—
111	#41×C₅₃	C₂×D₄	1	4	yes	—
112	#41×C₅₃	C₂×Q₈	1	4	yes	—
113	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂²×Dic₅₃	—	2	1	no	—
120	C₂²×Dic₅₃	—	2	2	no	—
121	C₂²×Dic₅₃	—	2	2	no	—
122	C₂²×Dic₅₃	#41	1	1	yes	characteristic
123	C₂²×Dic₅₃	#41	1	1	yes	characteristic
124	Q₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
125	SD₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
126	D₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
127	M₁₆×C₅₃	—	2	2	no	—
128	M₁₆×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	4	4	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	4	4	no	—
134	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₂²⋊Dic₅₃	—	2	2	no	—
143	C₂²⋊Dic₅₃	—	2	2	no	—
144	C₂²⋊Dic₅₃	—	2	2	no	—
145	C₄⋊Dic₅₃	—	2	2	no	—
146	C₄⋊Dic₅₃	—	2	2	no	—
147	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
148	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
149	#11719919	—	2	4	no	—
150	C₄×Dic₅₃	—	2	2	no	—
151	C₄×Dic₅₃	—	2	2	no	—
152	C₄×Dic₅₃	—	2	2	no	—
153	C₄×Dic₅₃	—	2	1	no	—
154	C₄×Dic₅₃	—	2	1	no	—
155	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
158	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
159	#11719917	—	2	2	no	—
160	#11719917	—	2	2	no	—
161	#11719917	—	4	4	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	2	no	—
167	C₂×#88322	—	4	4	no	—
168	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
169	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
170	C₂×#88322	C₂×D₄	1	1	yes	characteristic
171	C₂×#88322	C₂×D₄	1	1	yes	characteristic
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
174	Q₈×C₅₃	—	2	2	no	—
175	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
176	D₄×C₅₃	—	2	4	no	—
177	D₄×C₅₃	—	2	4	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	4	no	—
182	C₂×C₄×C₅₃	—	2	4	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
190	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
192	C₂×Dic₅₃	—	2	2	no	—
193	C₂×Dic₅₃	—	2	2	no	—
194	C₂×Dic₅₃	—	2	2	no	—
195	C₂×Dic₅₃	—	2	2	no	—
196	C₂×Dic₅₃	—	4	1	no	—
197	C₂×Dic₅₃	—	4	1	no	—
198	C₂×Dic₅₃	—	4	1	no	—
199	C₂×Dic₅₃	—	4	4	no	—
200	C₂×Dic₅₃	—	8	2	no	—
201	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
202	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
203	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
204	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
205	C₈×C₅₃	—	2	1	no	—
206	C₈×C₅₃	—	2	1	no	—
207	C₈×C₅₃	—	4	2	no	—
208	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
209	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
210	#88322	—	2	1	no	—
211	#88322	—	2	1	no	—
212	#88322	—	2	1	no	—
213	#88322	—	2	1	no	—
214	#88322	—	4	2	no	—
215	#22688	—	53	1	no	maximal, Sylow
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	4	no	—
219	C₂²×C₅₃	—	4	2	no	—
220	C₂²×C₅₃	—	4	2	no	—
221	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₅₃	#547	1	1	yes	characteristic
223	C₂²×C₅₃	#554	1	1	yes	characteristic
224	C₄×C₅₃	—	2	4	no	—
225	C₄×C₅₃	—	4	2	no	—
226	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
227	C₄×C₅₃	#520	1	1	yes	characteristic
228	C₄×C₅₃	D₄²	1	1	yes	characteristic
229	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
230	Dic₅₃	—	2	1	no	—
231	Dic₅₃	—	2	1	no	—
232	Dic₅₃	—	2	1	no	—
233	Dic₅₃	—	4	2	no	—
234	Dic₅₃	—	4	2	no	—
235	Dic₅₃	—	4	2	no	—
236	Dic₅₃	#443	1	2	yes	complemented
237	C₂×#443	—	53	1	no	—
238	C₂×#443	D₅₃	1	1	yes	characteristic
239	C₄×#135	—	53	1	no	—
240	#2906	—	53	2	no	—
241	#1945	—	53	1	no	—
242	#1945	—	53	1	no	—
243	#1866	—	53	2	no	—
244	#1866	—	53	2	no	—
245	#1848	—	53	1	no	—
246	#1806	—	53	2	no	—
247	C₈×C₄⋊C₄	—	53	1	no	—
248	C₂×C₅₃	—	2	1	no	—
249	C₂×C₅₃	—	2	1	no	—
250	C₂×C₅₃	—	4	4	no	—
251	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
252	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
253	C₂×C₅₃	#2957	1	1	yes	characteristic
254	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
255	C₂×#131	—	53	2	no	—
256	C₂×#131	C₂×D₅₃	1	2	yes	—
257	C₄×#41	—	53	2	no	—
258	#443	—	2	4	no	—
259	#443	—	106	4	no	—
260	C₄×Q₁₆	—	53	1	no	—
261	C₄×Q₁₆	—	53	1	no	—
262	C₄×SD₁₆	—	53	2	no	—
263	C₄×SD₁₆	—	53	2	no	—
264	C₄×D₈	—	53	1	no	—
265	C₄×D₈	—	53	1	no	—
266	C₂×#108	—	53	1	no	—
267	C₂×#108	—	53	2	no	—
268	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
269	#427	—	53	1	no	—
270	#424	—	53	2	no	—
271	#424	—	53	2	no	—
272	#424	—	106	2	no	—
273	#424	—	106	2	no	—
274	#423	—	53	2	no	—
275	#423	—	106	2	no	—
276	C₂×C₄⋊C₈	—	53	1	no	—
277	C₂×C₄⋊C₈	—	106	2	no	—
278	C₂×#104	—	53	2	no	—
279	C₂×#104	C₂×D₅₃	1	2	yes	—
280	#416	—	53	2	no	—
281	C₂×C₄×C₈	—	53	1	no	—
282	C₂×C₄×C₈	—	53	1	no	—
283	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
284	C₄×C₄⋊C₄	—	53	1	no	—
285	#339	—	53	2	no	—
286	#339	—	53	2	no	—
287	#336	—	53	1	no	—
288	#336	—	53	1	no	—
289	C₅₃	#22688	1	1	yes	characteristic, complemented, Sylow
290	C₂×#41	C₂²×D₅₃	1	2	yes	—
291	#135	Dic₁₀₆	1	4	yes	—
292	#135	C₂²⋊D₅₃	1	4	yes	—
293	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
294	C₂×SD₁₆	C₄×D₅₃	1	2	yes	—
295	C₂×D₈	C₄×D₅₃	1	1	yes	characteristic
296	#131	—	2	4	no	—
297	#131	—	4	2	no	—
298	#131	—	106	4	no	—
299	#131	—	212	2	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂×M₁₆	—	53	2	no	—
302	C₂×M₁₆	—	106	2	no	—
303	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
304	C₂²×C₈	—	2	2	no	—
305	C₂²×C₈	—	53	1	no	—
306	C₂²×C₈	—	53	1	no	—
307	C₂²×C₈	—	106	2	no	—
308	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
309	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
310	C₄×Q₈	—	53	2	no	—
311	C₄×Q₈	—	53	2	no	—
312	C₄×D₄	—	53	2	no	—
313	C₄×D₄	—	53	2	no	—
314	C₄×D₄	—	106	2	no	—
315	C₄×D₄	—	106	2	no	—
316	#117	—	53	2	no	—
317	#117	—	106	2	no	—
318	C₂×C₄⋊C₄	—	106	1	no	—
319	C₂×C₄²	—	53	1	no	—
320	C₂×C₄²	—	53	1	no	—
321	C₂×C₄²	—	106	2	no	—
322	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
323	#108	—	2	2	no	—
324	#108	—	106	2	no	—
325	#108	—	212	2	no	—
326	#107	—	53	1	no	—
327	#107	—	53	1	no	—
328	#106	—	53	2	no	—
329	C₄⋊C₈	—	53	1	no	—
330	C₄⋊C₈	—	53	1	no	—
331	C₄⋊C₈	—	53	1	no	—
332	C₄⋊C₈	—	53	1	no	—
333	C₄⋊C₈	—	106	2	no	—
334	C₄⋊C₈	—	106	2	no	—
335	C₄⋊C₈	—	106	2	no	—
336	C₄⋊C₈	—	106	2	no	—
337	C₄⋊C₈	—	106	2	no	—
338	C₄⋊C₈	—	106	2	no	—
339	#104	—	2	4	no	—
340	#104	—	106	4	no	—
341	#103	—	53	2	no	—
342	#103	—	53	2	no	—
343	#102	—	53	2	no	—
344	#102	—	53	2	no	—
345	C₄×C₈	—	2	2	no	—
346	C₄×C₈	—	53	1	no	—
347	C₄×C₈	—	53	1	no	—
348	C₄×C₈	—	53	1	no	—
349	C₄×C₈	—	53	1	no	—
350	C₄×C₈	—	106	2	no	—
351	#95	—	106	1	no	—
352	#41	—	2	2	no	—
353	#41	—	2	2	no	—
354	#41	C₂×Dic₁₀₆	1	4	yes	—
355	#41	C₂×C₂²⋊D₅₃	1	4	yes	—
356	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	C₂×C₄×D₅₃	1	2	yes	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	53	1	no	—
362	C₂²×C₄	—	53	1	no	—
363	C₂²×C₄	—	106	2	no	—
364	C₂²×C₄	—	106	2	no	—
365	C₂²×C₄	—	106	1	no	—
366	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
367	Q₁₆	#11719919	1	4	yes	complemented
368	SD₁₆	#11719919	1	8	yes	complemented
369	D₈	#11719919	1	4	yes	complemented
370	M₁₆	—	2	2	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	4	4	no	—
373	M₁₆	—	106	2	no	—
374	M₁₆	—	106	2	no	—
375	M₁₆	—	212	4	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	4	4	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	2	no	—
389	C₂×C₈	—	212	4	no	—
390	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
391	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
392	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
393	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
394	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
395	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
396	C₄⋊C₄	—	53	2	no	—
397	C₄⋊C₄	—	53	2	no	—
398	C₄⋊C₄	—	106	2	no	—
399	C₄⋊C₄	—	106	2	no	—
400	C₄⋊C₄	—	106	4	no	—
401	C₂²⋊C₄	—	106	2	no	—
402	C₂²⋊C₄	—	106	2	no	—
403	C₂²⋊C₄	—	106	2	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	53	1	no	—
407	C₄²	—	53	1	no	—
408	C₄²	—	53	1	no	—
409	C₄²	—	53	1	no	—
410	C₄²	—	106	2	no	—
411	C₄²	—	106	2	no	—
412	C₄²	—	106	2	no	—
413	C₄²	—	106	1	no	—
414	C₄²	—	106	1	no	—
415	C₂³	—	2	2	no	—
416	C₂³	#41×D₅₃	1	1	yes	characteristic
417	Q₈	—	2	2	no	—
418	Q₈	C₂×#11719919	1	4	yes	—
419	D₄	—	2	4	no	—
420	D₄	—	2	4	no	—
421	D₄	—	2	2	no	—
422	D₄	C₂×#11719919	1	4	yes	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	53	1	no	—
430	C₂×C₄	—	53	1	no	—
431	C₂×C₄	—	53	1	no	—
432	C₂×C₄	—	53	1	no	—
433	C₂×C₄	—	106	2	no	—
434	C₂×C₄	—	106	2	no	—
435	C₂×C₄	—	106	2	no	—
436	C₂×C₄	—	106	2	no	—
437	C₂×C₄	—	424	2	no	—
438	C₂×C₄	—	212	4	no	—
439	C₂×C₄	—	212	1	no	—
440	C₂×C₄	—	212	1	no	—
441	C₂×C₄	—	212	1	no	—
442	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
443	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
444	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
445	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
446	C₂×C₄	#13133262	1	1	yes	characteristic
447	C₂×C₄	#13133264	1	1	yes	characteristic
448	C₈	—	2	1	no	—
449	C₈	—	2	1	no	—
450	C₈	—	4	2	no	—
451	C₈	—	106	1	no	—
452	C₈	—	106	1	no	—
453	C₈	—	106	1	no	—
454	C₈	—	106	1	no	—
455	C₈	—	212	2	no	—
456	C₈	C₂×#11719919	1	2	yes	—
457	C₈	C₂×#11719919	1	2	yes	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	1	no	—
460	C₂²	—	2	4	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
464	C₂²	#16811272	1	1	yes	characteristic
465	C₂²	#16811332	1	1	yes	characteristic
466	C₄	—	2	4	no	—
467	C₄	—	4	2	no	—
468	C₄	—	53	2	no	—
469	C₄	—	106	1	no	—
470	C₄	—	106	1	no	—
471	C₄	—	106	1	no	—
472	C₄	—	212	2	no	—
473	C₄	—	212	2	no	—
474	C₄	—	212	2	no	—
475	C₄	C₂²×#11719919	1	1	yes	characteristic
476	C₄	#16811102	1	1	yes	characteristic
477	C₄	D₄×C₂²⋊D₅₃	1	1	yes	characteristic
478	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
479	C₂	—	2	1	no	—
480	C₂	—	2	1	no	—
481	C₂	—	4	4	no	—
482	C₂	D₄×#11719919	1	1	yes	characteristic
483	C₂	#443×D₅₃	1	1	yes	characteristic
484	C₂	#20283627	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	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#22688
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
3. #20283281	C₂	semidirect
4. #20275463	C₂	semidirect
5. #20275461	C₂	semidirect
6. #20274990	C₂	semidirect
7. #20274988	C₂	semidirect
8. #20274885	C₂	semidirect
9. #20274733	C₂	semidirect
10. C₈×#11719919	C₂	semidirect
15. Q₁₆×Dic₅₃	C₄	semidirect
16. Q₁₆×Dic₅₃	C₄	semidirect
17. SD₁₆×Dic₅₃	C₄	semidirect
18. SD₁₆×Dic₅₃	C₄	semidirect
19. D₈×Dic₅₃	C₄	semidirect
20. D₈×Dic₅₃	C₄	semidirect
124. Q₁₆×C₅₃	C₄⋊C₄	semidirect
125. SD₁₆×C₅₃	C₄⋊C₄	semidirect
126. D₈×C₅₃	C₄⋊C₄	semidirect
236. Dic₅₃	#443	semidirect
289. C₅₃	#22688	semidirect
367. Q₁₆	#11719919	semidirect
368. SD₁₆	#11719919	semidirect
369. D₈	#11719919	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810740	2
#135×Dic₅₃	2
#20283281	2
#20275463	2
#20275461	2
#20274990	2
#20274988	2
#20274885	2
#20274733	2
C₈×#11719919	2
C₂×C₅₃×#443	2
#22688	53