Subgroups — #28318607

Order 13568.

1014 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#443	C₂	1	1	yes	maximal, characteristic
2	#20291469	C₂	1	1	yes	maximal, characteristic
3	#20291285	C₂	1	2	yes	maximal, complemented
4	#20290536	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290521	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290520	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290495	C₂	1	2	yes	maximal, complemented
8	#20290489	C₂	1	2	yes	maximal, complemented
9	#104×#6157	C₂	1	2	yes	maximal, complemented
10	#97×#6157	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×D₅₃	C₂	1	1	yes	maximal, characteristic
12	D₅₃⋊#131	C₂²	1	2	yes	—
13	#41×#6157	C₂²	1	2	yes	—
14	#16811667	C₄	1	2	yes	complemented
15	#16811666	C₄	1	4	yes	complemented
16	#16811665	C₄	1	2	yes	complemented
17	#16811664	C₄	1	2	yes	complemented
18	#16811663	C₄	1	2	yes	complemented
19	#16811662	C₄	1	4	yes	complemented
20	D₅₃⋊#108	C₂²	1	1	yes	characteristic
21	#16811633	C₂²	1	1	yes	characteristic
22	D₅₃⋊#104	C₂²	1	2	yes	—
23	#16811613	C₂²	1	2	yes	—
24	#16811589	—	2	2	no	—
25	#16811589	C₂²	1	2	yes	—
26	#16811586	—	2	2	no	—
27	#16811586	C₂²	1	2	yes	—
28	M₁₆×#6157	—	2	2	no	—
29	M₁₆×#6157	C₂²	1	2	yes	—
30	#16811582	C₂²	1	1	yes	characteristic
31	C₂×#13133271	—	2	2	no	—
32	C₂×#13133271	C₂²	1	1	yes	characteristic
33	C₂×#13133271	C₂²	1	1	yes	characteristic
34	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
35	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
36	C₄²×#6157	C₂²	1	1	yes	characteristic
37	#16811498	C₂²	1	2	yes	—
38	#16811495	C₂²	1	2	yes	—
39	#16811494	C₂²	1	2	yes	—
40	#135×D₅₃	C₂²	1	1	yes	characteristic
41	#131×D₅₃	C₂²	1	2	yes	—
42	#16810746	C₄	1	1	yes	characteristic, complemented
43	#16810745	C₄	1	1	yes	characteristic, complemented
44	#16810744	C₄	1	1	yes	characteristic, complemented
45	#16810743	C₄	1	2	yes	complemented
46	#16810742	C₄	1	2	yes	complemented
47	#108×D₅₃	C₂²	1	1	yes	characteristic
48	#104×D₅₃	C₂²	1	2	yes	—
49	#97×D₅₃	C₂²	1	1	yes	characteristic
50	C₅₃×#444	C₄	1	1	yes	characteristic, complemented
51	#13133305	C₂×C₄	1	4	yes	complemented
52	#13133304	—	2	4	no	—
53	#13133304	C₂×C₄	1	4	yes	complemented
54	D₅₃⋊M₁₆	—	2	2	no	—
55	D₅₃⋊M₁₆	C₂³	1	2	yes	—
56	C₂×D₅₃⋊C₈	—	2	2	no	—
57	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
59	Q₈×#6157	C₂×C₄	1	4	yes	—
60	D₄×#6157	—	2	4	no	—
61	D₄×#6157	C₂×C₄	1	4	yes	—
62	#13133296	—	2	2	no	—
63	#13133296	C₂³	1	2	yes	—
64	C₂×C₄×#6157	—	2	2	no	—
65	C₂×C₄×#6157	—	2	2	no	—
66	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
67	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
68	#13133288	C₂×C₄	1	2	yes	complemented
69	#13133287	C₂×C₄	1	2	yes	complemented
70	#13133286	C₂×C₄	1	2	yes	—
71	#13133285	C₂×C₄	1	2	yes	—
72	#13133282	C₂×C₄	1	4	yes	complemented
73	#13133281	C₂×C₄	1	4	yes	complemented
74	#13133280	C₂×C₄	1	4	yes	—
75	#13133279	C₂×C₄	1	4	yes	—
76	#13133272	—	2	2	no	—
77	#13133271	—	2	2	no	—
78	#13133271	—	2	2	no	—
79	#13133271	—	2	2	no	—
80	#13133271	C₂×C₄	1	2	yes	—
81	#13133271	C₂×C₄	1	2	yes	—
82	#13133271	D₄	1	2	yes	—
83	#13133271	D₄	1	2	yes	—
84	C₈×#6157	—	2	2	no	—
85	C₈×#6157	—	2	2	no	—
86	C₈×#6157	—	2	2	no	—
87	C₈×#6157	—	2	2	no	—
88	C₈×#6157	C₂×C₄	1	2	yes	complemented
89	C₈×#6157	C₂×C₄	1	2	yes	complemented
90	#41×D₅₃	C₂³	1	2	yes	—
91	#13133240	C₂×C₄	1	1	yes	characteristic
92	#13133239	C₂×C₄	1	1	yes	characteristic
93	#13133238	C₂×C₄	1	2	yes	—
94	#13133237	C₂×C₄	1	2	yes	—
95	#13133236	C₂×C₄	1	1	yes	characteristic
96	Q₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
97	SD₁₆×D₅₃	C₂×C₄	1	2	yes	—
98	D₈×D₅₃	C₂×C₄	1	1	yes	characteristic
99	#13133225	—	2	2	no	—
100	#13133225	C₂×C₄	1	2	yes	complemented
101	#13133224	C₂×C₄	1	2	yes	complemented
102	#13133223	—	2	2	no	—
103	#13133223	C₂×C₄	1	2	yes	complemented
104	M₁₆×D₅₃	—	2	2	no	—
105	M₁₆×D₅₃	C₂³	1	2	yes	—
106	C₂×C₈×D₅₃	—	2	2	no	—
107	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
108	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
109	C₄²×D₅₃	C₂³	1	1	yes	characteristic
110	#13133153	C₂×C₄	1	1	yes	characteristic, complemented
111	#13133152	C₂×C₄	1	2	yes	complemented
112	#13133145	C₂×C₄	1	2	yes	complemented
113	#13133144	C₂×C₄	1	2	yes	complemented
114	#13133143	C₂×C₄	1	2	yes	complemented
115	#13133128	C₂×C₄	1	1	yes	characteristic, complemented
116	#13133127	C₂×C₄	1	1	yes	characteristic, complemented
117	#13133127	C₂×C₄	1	1	yes	characteristic, complemented
118	#135×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#131×C₅₃	C₂×C₄	1	2	yes	complemented
120	#108×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
121	#104×C₅₃	C₂×C₄	1	2	yes	complemented
122	#97×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
123	C₂²×#6157	—	2	1	no	—
124	C₂²×#6157	—	2	4	no	—
125	C₂²×#6157	#41	1	2	yes	—
126	#41×C₅₃	C₂²×C₄	1	2	yes	—
127	#11719949	C₂²×C₄	1	2	yes	—
128	Q₈×D₅₃	C₂²×C₄	1	2	yes	—
129	#11719947	—	2	2	no	—
130	#11719947	C₂²×C₄	1	2	yes	—
131	D₄×D₅₃	—	2	2	no	—
132	D₄×D₅₃	C₂²×C₄	1	2	yes	—
133	#11719945	—	2	2	no	—
134	#11719945	C₂²×C₄	1	2	yes	—
135	C₂×C₄×D₅₃	—	2	2	no	—
136	C₂×C₄×D₅₃	—	2	1	no	—
137	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
138	C₂²⋊#6157	—	2	2	no	—
139	C₂²⋊#6157	—	2	4	no	—
140	#11719940	—	2	2	no	—
141	#11719940	—	2	4	no	—
142	C₂×#88324	—	2	1	no	—
143	C₂×#88324	—	2	4	no	—
144	C₂×#88324	#41	1	2	yes	—
145	C₄⋊#6157	—	2	4	no	—
146	C₄⋊#6157	C₂²×C₄	1	4	yes	—
147	C₄×#6157	—	2	4	no	—
148	C₄×#6157	—	2	2	no	—
149	C₄×#6157	—	2	1	no	—
150	C₄×#6157	—	4	4	no	—
151	C₄×#6157	C₂²×C₄	1	2	yes	—
152	C₄×#6157	C₂²×C₄	1	2	yes	—
153	C₄×#6157	C₂×D₄	1	2	yes	—
154	#11719936	—	2	4	no	—
155	#11719936	C₂²×C₄	1	4	yes	—
156	D₅₃⋊C₈	—	2	2	no	—
157	D₅₃⋊C₈	—	2	1	no	—
158	D₅₃⋊C₈	—	2	4	no	—
159	D₅₃⋊C₈	C₂²×C₄	1	2	yes	—
160	D₅₃⋊C₈	C₂²×C₄	1	2	yes	—
161	D₅₃⋊C₈	C₂×D₄	1	1	yes	characteristic
162	D₅₃⋊C₈	C₂×D₄	1	1	yes	characteristic
163	Q₁₆×C₅₃	C₄²	1	1	yes	characteristic, complemented
164	SD₁₆×C₅₃	C₄²	1	2	yes	complemented
165	D₈×C₅₃	C₄²	1	1	yes	characteristic, complemented
166	M₁₆×C₅₃	—	2	2	no	—
167	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
168	C₂×C₈×C₅₃	—	2	2	no	—
169	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
172	#11719925	C₄²	1	2	yes	complemented
173	#11719924	C₄²	1	2	yes	complemented
174	#11719923	C₄²	1	2	yes	complemented
175	#11719922	C₄²	1	2	yes	complemented
176	C₄×Dic₅₃	—	2	1	no	—
177	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
178	#11719917	—	2	2	no	—
179	#11719917	C₂²×C₄	1	2	yes	—
180	C₂×#88322	—	2	2	no	—
181	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
182	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
183	Dic₂₁₂	C₄²	1	1	yes	characteristic, complemented
184	D₄₂₄	C₄²	1	1	yes	characteristic, complemented
185	#11719913	C₄²	1	2	yes	complemented
186	#11719912	—	2	2	no	—
187	#11719912	—	2	2	no	—
188	#11719912	—	2	2	no	—
189	C₈×D₅₃	—	2	2	no	—
190	C₈×D₅₃	—	2	2	no	—
191	C₈×D₅₃	—	2	2	no	—
192	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
195	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
196	C₈×D₅₃	C₂×D₄	1	2	yes	—
197	C₈×D₅₃	#41	1	2	yes	—
198	C₂²×D₅₃	—	2	2	no	—
199	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
200	C₂×#6157	—	2	4	no	—
201	C₂×#6157	—	2	2	no	—
202	C₂×#6157	—	2	4	no	—
203	C₂×#6157	—	4	2	no	—
204	C₂×#6157	—	8	4	no	—
205	C₂×#6157	C₄×D₄	1	4	yes	—
206	Q₈×C₅₃	C₂×C₄²	1	2	yes	—
207	D₄×C₅₃	—	2	2	no	—
208	D₄×C₅₃	C₂×C₄²	1	2	yes	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
212	C₂²⋊D₅₃	—	2	2	no	—
213	C₂²⋊D₅₃	—	2	2	no	—
214	C₂²⋊D₅₃	—	2	2	no	—
215	C₂×Dic₅₃	—	2	2	no	—
216	C₂×Dic₅₃	—	2	1	no	—
217	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
218	D₂₁₂	—	2	2	no	—
219	D₂₁₂	C₂×C₄²	1	2	yes	—
220	C₄×D₅₃	—	2	2	no	—
221	C₄×D₅₃	—	2	2	no	—
222	C₄×D₅₃	—	2	2	no	—
223	C₄×D₅₃	—	4	2	no	—
224	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
225	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
226	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
227	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
228	Dic₁₀₆	—	2	2	no	—
229	Dic₁₀₆	C₂×C₄²	1	2	yes	—
230	#88324	—	2	4	no	—
231	#88324	—	2	2	no	—
232	#88324	C₄×D₄	1	2	yes	—
233	#88324	C₄×D₄	1	2	yes	—
234	C₈×C₅₃	—	2	2	no	—
235	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
236	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
237	C₈×C₅₃	C₄×D₄	1	2	yes	—
238	#88322	—	2	2	no	—
239	#88322	C₂×C₄²	1	1	yes	characteristic
240	#88322	C₂×C₄²	1	1	yes	characteristic
241	#88322	C₄×D₄	1	2	yes	—
242	#20512	—	53	1	no	maximal, Sylow
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
245	C₂×D₅₃	—	2	2	no	—
246	C₂×D₅₃	—	2	2	no	—
247	C₂×D₅₃	—	4	4	no	—
248	C₂×D₅₃	C₂×C₄×D₄	1	1	yes	characteristic
249	C₂×D₅₃	C₄×#41	1	1	yes	characteristic
250	#6157	—	2	4	no	—
251	#6157	—	2	4	no	—
252	#6157	—	4	4	no	—
253	C₄×C₅₃	—	2	2	no	—
254	C₄×C₅₃	—	4	1	no	—
255	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
256	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
257	Dic₅₃	—	2	2	no	—
258	Dic₅₃	—	4	1	no	—
259	Dic₅₃	C₂×C₄×D₄	1	1	yes	characteristic
260	Dic₅₃	C₄×#41	1	1	yes	characteristic
261	C₂×#444	—	53	1	no	—
262	C₂×#443	—	53	1	no	—
263	#2943	—	53	1	no	—
264	#2875	—	53	2	no	—
265	#1778	—	53	1	no	—
266	#1773	—	53	1	no	—
267	#1772	—	53	1	no	—
268	#1765	—	53	2	no	—
269	#1764	—	53	2	no	—
270	C₄×#104	—	53	2	no	—
271	C₄×#97	—	53	1	no	—
272	C₂×C₅₃	—	2	1	no	—
273	C₂×C₅₃	—	4	2	no	—
274	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
275	D₅₃	—	2	1	no	—
276	D₅₃	—	4	2	no	—
277	D₅₃	#2956	1	2	yes	—
278	C₂×#135	—	53	1	no	—
279	C₂×#131	—	53	2	no	—
280	C₂×#131	—	53	2	no	—
281	C₄×#41	—	53	2	no	—
282	#444	—	53	1	no	—
283	#444	—	53	2	no	—
284	#444	—	53	1	no	—
285	#444	—	53	2	no	—
286	#444	—	53	1	no	—
287	#444	#6157	1	1	yes	characteristic, complemented
288	#443	—	53	2	no	—
289	#443	—	53	4	no	—
290	#443	—	53	2	no	—
291	#442	—	53	2	no	—
292	#441	—	53	2	no	—
293	#440	—	53	4	no	—
294	C₂×#108	—	53	1	no	—
295	C₂×#108	—	53	1	no	—
296	#427	—	53	1	no	—
297	C₂×#104	—	53	2	no	—
298	C₂×#104	—	53	2	no	—
299	#416	—	53	2	no	—
300	#405	—	53	2	no	—
301	#405	—	53	2	no	—
302	#405	—	106	2	no	—
303	#405	—	106	2	no	—
304	C₄×M₁₆	—	53	2	no	—
305	C₄×M₁₆	—	106	2	no	—
306	C₂×#97	—	53	1	no	—
307	C₂×#97	—	53	1	no	—
308	C₂×#97	—	53	1	no	—
309	C₂×#97	—	53	1	no	—
310	C₂×#97	—	106	2	no	—
311	C₂×C₄×C₈	—	53	1	no	—
312	C₂×C₄×C₈	—	53	1	no	—
313	C₄³	—	53	1	no	—
314	#341	—	53	2	no	—
315	#339	—	53	2	no	—
316	#339	—	53	2	no	—
317	C₅₃	#20512	1	1	yes	characteristic, complemented, Sylow
318	C₂×#41	—	53	2	no	—
319	#135	—	53	2	no	—
320	#135	—	53	1	no	—
321	#135	—	53	2	no	—
322	#135	—	53	1	no	—
323	#135	—	53	1	no	—
324	#135	C₂×#6157	1	1	yes	characteristic
325	C₂×Q₁₆	—	53	1	no	—
326	C₂×SD₁₆	—	53	2	no	—
327	C₂×D₈	—	53	1	no	—
328	#131	—	53	4	no	—
329	#131	—	53	4	no	—
330	#131	—	53	2	no	—
331	#131	—	53	2	no	—
332	#131	—	53	2	no	—
333	#131	—	106	2	no	—
334	#131	—	106	2	no	—
335	#131	—	106	4	no	—
336	#131	C₂×#6157	1	2	yes	complemented
337	C₂×M₁₆	—	53	2	no	—
338	C₂×M₁₆	—	53	2	no	—
339	C₂×M₁₆	—	106	2	no	—
340	C₂×M₁₆	—	106	2	no	—
341	C₂²×C₈	—	53	1	no	—
342	C₂²×C₈	—	53	1	no	—
343	C₂²×C₈	—	53	1	no	—
344	C₂²×C₈	—	53	1	no	—
345	C₂²×C₈	—	106	2	no	—
346	C₂²×C₈	—	106	2	no	—
347	C₄×Q₈	—	53	4	no	—
348	C₄×D₄	—	53	4	no	—
349	C₄×D₄	—	106	4	no	—
350	#117	—	53	2	no	—
351	#117	—	106	2	no	—
352	C₂×C₄²	—	53	1	no	—
353	C₂×C₄²	—	53	1	no	—
354	C₂×C₄²	—	53	1	no	—
355	C₂×C₄²	—	106	2	no	—
356	C₂×C₄²	—	106	2	no	—
357	#108	—	53	2	no	—
358	#108	—	53	2	no	—
359	#108	—	53	2	no	—
360	#108	—	53	1	no	—
361	#108	C₂×#6157	1	1	yes	characteristic, complemented
362	#107	—	53	2	no	—
363	#106	—	53	2	no	—
364	#104	—	53	4	no	—
365	#104	—	53	4	no	—
366	#104	—	53	2	no	—
367	#104	—	53	2	no	—
368	#104	—	53	2	no	—
369	#104	C₂×#6157	1	2	yes	complemented
370	#103	—	53	4	no	—
371	#102	—	53	4	no	—
372	#97	—	53	2	no	—
373	#97	—	53	2	no	—
374	#97	—	53	2	no	—
375	#97	—	53	2	no	—
376	#97	—	53	1	no	—
377	#97	—	53	1	no	—
378	#97	—	53	1	no	—
379	#97	—	106	2	no	—
380	#97	—	106	2	no	—
381	#97	—	106	2	no	—
382	#97	—	106	2	no	—
383	#97	C₂×#6157	1	1	yes	characteristic, complemented
384	C₄×C₈	—	53	2	no	—
385	C₄×C₈	—	53	2	no	—
386	C₄×C₈	—	106	2	no	—
387	C₄×C₈	—	106	2	no	—
388	C₄×C₈	—	106	2	no	—
389	C₄×C₈	—	106	2	no	—
390	#41	—	53	2	no	—
391	#41	—	53	2	no	—
392	#41	—	53	2	no	—
393	#41	—	106	2	no	—
394	#41	—	106	2	no	—
395	#41	C₂²×#6157	1	2	yes	—
396	C₂×Q₈	—	53	2	no	—
397	C₂×D₄	—	53	2	no	—
398	C₂×D₄	—	106	2	no	—
399	C₂²×C₄	—	53	1	no	—
400	C₂²×C₄	—	53	2	no	—
401	C₂²×C₄	—	106	2	no	—
402	C₂²×C₄	—	106	1	no	—
403	C₂²×C₄	—	106	4	no	—
404	C₂²×C₄	—	106	1	no	—
405	Q₁₆	—	53	2	no	—
406	Q₁₆	—	53	1	no	—
407	Q₁₆	C₄×#6157	1	1	yes	characteristic, complemented
408	SD₁₆	—	53	2	no	—
409	SD₁₆	—	53	2	no	—
410	SD₁₆	—	53	2	no	—
411	SD₁₆	C₄×#6157	1	2	yes	complemented
412	D₈	—	53	2	no	—
413	D₈	—	53	1	no	—
414	D₈	C₄×#6157	1	1	yes	characteristic, complemented
415	M₁₆	—	2	2	no	—
416	M₁₆	—	53	4	no	—
417	M₁₆	—	53	2	no	—
418	M₁₆	—	106	2	no	—
419	M₁₆	—	106	2	no	—
420	M₁₆	—	106	2	no	—
421	M₁₆	—	106	4	no	—
422	M₁₆	—	106	4	no	—
423	M₁₆	—	106	2	no	—
424	M₁₆	—	106	2	no	—
425	M₁₆	C₂²×#6157	1	2	yes	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	53	2	no	—
428	C₂×C₈	—	53	2	no	—
429	C₂×C₈	—	53	1	no	—
430	C₂×C₈	—	53	1	no	—
431	C₂×C₈	—	53	1	no	—
432	C₂×C₈	—	53	1	no	—
433	C₂×C₈	—	53	1	no	—
434	C₂×C₈	—	53	1	no	—
435	C₂×C₈	—	53	1	no	—
436	C₂×C₈	—	53	1	no	—
437	C₂×C₈	—	53	2	no	—
438	C₂×C₈	—	53	2	no	—
439	C₂×C₈	—	53	2	no	—
440	C₂×C₈	—	106	2	no	—
441	C₂×C₈	—	106	2	no	—
442	C₂×C₈	—	106	2	no	—
443	C₂×C₈	—	106	4	no	—
444	C₂×C₈	—	106	4	no	—
445	C₂×C₈	—	106	1	no	—
446	C₂×C₈	—	106	2	no	—
447	C₂×C₈	—	106	1	no	—
448	C₂×C₈	—	106	2	no	—
449	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
450	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
451	C₄⋊C₄	—	53	4	no	—
452	C₄⋊C₄	—	106	4	no	—
453	C₂²⋊C₄	—	106	4	no	—
454	C₂²⋊C₄	—	106	2	no	—
455	C₄²	—	53	2	no	—
456	C₄²	—	53	2	no	—
457	C₄²	—	53	2	no	—
458	C₄²	—	53	1	no	—
459	C₄²	—	106	1	no	—
460	C₄²	—	106	2	no	—
461	C₄²	—	106	1	no	—
462	C₄²	—	106	4	no	—
463	C₄²	—	212	4	no	—
464	C₄²	C₂²×#6157	1	1	yes	characteristic
465	C₂³	—	53	1	no	—
466	C₂³	—	106	2	no	—
467	Q₈	—	53	2	no	—
468	Q₈	—	106	2	no	—
469	Q₈	C₂×C₄×#6157	1	2	yes	—
470	D₄	—	2	2	no	—
471	D₄	—	53	2	no	—
472	D₄	—	106	2	no	—
473	D₄	—	106	2	no	—
474	D₄	—	106	2	no	—
475	D₄	—	106	2	no	—
476	D₄	C₂×C₄×#6157	1	2	yes	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	53	1	no	—
480	C₂×C₄	—	53	1	no	—
481	C₂×C₄	—	53	1	no	—
482	C₂×C₄	—	53	1	no	—
483	C₂×C₄	—	53	1	no	—
484	C₂×C₄	—	53	4	no	—
485	C₂×C₄	—	106	2	no	—
486	C₂×C₄	—	106	2	no	—
487	C₂×C₄	—	106	2	no	—
488	C₂×C₄	—	106	2	no	—
489	C₂×C₄	—	106	4	no	—
490	C₂×C₄	—	106	2	no	—
491	C₂×C₄	—	106	4	no	—
492	C₂×C₄	—	106	1	no	—
493	C₂×C₄	—	424	4	no	—
494	C₂×C₄	—	212	2	no	—
495	C₂×C₄	—	212	2	no	—
496	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	53	2	no	—
499	C₈	—	53	2	no	—
500	C₈	—	53	1	no	—
501	C₈	—	53	1	no	—
502	C₈	—	53	2	no	—
503	C₈	—	106	2	no	—
504	C₈	—	106	4	no	—
505	C₈	—	106	2	no	—
506	C₈	C₂×C₄×#6157	1	1	yes	characteristic
507	C₈	C₂×C₄×#6157	1	1	yes	characteristic
508	C₈	D₄×#6157	1	2	yes	—
509	C₂²	—	2	2	no	—
510	C₂²	—	53	1	no	—
511	C₂²	—	53	1	no	—
512	C₂²	—	106	2	no	—
513	C₂²	—	106	2	no	—
514	C₂²	—	212	4	no	—
515	C₂²	#41×#6157	1	1	yes	characteristic
516	C₄	—	2	2	no	—
517	C₄	—	4	1	no	—
518	C₄	—	53	1	no	—
519	C₄	—	53	1	no	—
520	C₄	—	106	4	no	—
521	C₄	—	106	4	no	—
522	C₄	—	106	2	no	—
523	C₄	—	212	1	no	—
524	C₄	—	212	4	no	—
525	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
526	C₄	C₂×D₄×#6157	1	1	yes	characteristic
527	C₂	—	2	1	no	—
528	C₂	—	4	2	no	—
529	C₂	—	53	2	no	—
530	C₂	—	106	1	no	—
531	C₂	—	212	2	no	—
532	C₂	C₄×D₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#444	C₄
socle	C₂×C₅₃	C₄²×D₄
prime radical	#135×D₅₃	C₂²
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#444	#6157
hypocenter	C₅₃	#20512
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
3. #20291285	C₂	semidirect
4. #20290536	C₂	semidirect
5. #20290521	C₂	semidirect
6. #20290520	C₂	semidirect
7. #20290495	C₂	semidirect
8. #20290489	C₂	semidirect
9. #104×#6157	C₂	semidirect
10. #97×#6157	C₂	semidirect
14. #16811667	C₄	semidirect
15. #16811666	C₄	semidirect
16. #16811665	C₄	semidirect
17. #16811664	C₄	semidirect
18. #16811663	C₄	semidirect
19. #16811662	C₄	semidirect
42. #16810746	C₄	semidirect
43. #16810745	C₄	semidirect
44. #16810744	C₄	semidirect
45. #16810743	C₄	semidirect
46. #16810742	C₄	semidirect
50. C₅₃×#444	C₄	semidirect
51. #13133305	C₂×C₄	semidirect
53. #13133304	C₂×C₄	semidirect
68. #13133288	C₂×C₄	semidirect
69. #13133287	C₂×C₄	semidirect
72. #13133282	C₂×C₄	semidirect
73. #13133281	C₂×C₄	semidirect
88. C₈×#6157	C₂×C₄	semidirect
89. C₈×#6157	C₂×C₄	semidirect
100. #13133225	C₂×C₄	semidirect
101. #13133224	C₂×C₄	semidirect
103. #13133223	C₂×C₄	semidirect
110. #13133153	C₂×C₄	semidirect
111. #13133152	C₂×C₄	semidirect
112. #13133145	C₂×C₄	semidirect
113. #13133144	C₂×C₄	semidirect
114. #13133143	C₂×C₄	semidirect
115. #13133128	C₂×C₄	semidirect
116. #13133127	C₂×C₄	semidirect
117. #13133127	C₂×C₄	semidirect
119. #131×C₅₃	C₂×C₄	semidirect
120. #108×C₅₃	C₂×C₄	semidirect
121. #104×C₅₃	C₂×C₄	semidirect
122. #97×C₅₃	C₂×C₄	semidirect
163. Q₁₆×C₅₃	C₄²	semidirect
164. SD₁₆×C₅₃	C₄²	semidirect
165. D₈×C₅₃	C₄²	semidirect
172. #11719925	C₄²	semidirect
173. #11719924	C₄²	semidirect
174. #11719923	C₄²	semidirect
175. #11719922	C₄²	semidirect
183. Dic₂₁₂	C₄²	semidirect
184. D₄₂₄	C₄²	semidirect
185. #11719913	C₄²	semidirect
287. #444	#6157	semidirect
317. C₅₃	#20512	semidirect
336. #131	C₂×#6157	semidirect
361. #108	C₂×#6157	semidirect
369. #104	C₂×#6157	semidirect
383. #97	C₂×#6157	semidirect
407. Q₁₆	C₄×#6157	semidirect
411. SD₁₆	C₄×#6157	semidirect
414. D₈	C₄×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#443	2
#20291469	2
#20291285	2
#20290536	2
#20290521	2
#20290520	2
#20290495	2
#20290489	2
#104×#6157	2
#97×#6157	2
#444×D₅₃	2
#20512	53