Subgroups — #28306920

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811586	C₂	1	1	yes	maximal, characteristic
3	#20290423	C₂	1	2	yes	maximal
4	#20290396	C₂	1	2	yes	maximal
5	#427×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊M₁₆	C₂²	1	2	yes	—
7	C₂×D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
8	C₂×#13133296	C₂²	1	1	yes	characteristic
9	C₂×#13133288	C₄	1	2	yes	complemented
10	D₅₃⋊#108	—	2	1	no	—
11	D₅₃⋊#108	—	2	1	no	—
12	D₅₃⋊#108	—	2	1	no	—
13	D₅₃⋊#108	—	2	1	no	—
14	C₂×#13133287	C₄	1	2	yes	complemented
15	#16811586	—	2	1	no	—
16	#16811586	—	2	1	no	—
17	#16811586	C₄	1	2	yes	complemented
18	#16811586	C₄	1	2	yes	complemented
19	C₂×#13133271	—	2	1	no	—
20	C₂×C₈×#6157	—	2	1	no	—
21	#16811504	—	2	4	no	—
22	#16811494	—	2	4	no	—
23	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
24	#117×D₅₃	C₂²	1	2	yes	—
25	#16810661	C₄	1	1	yes	characteristic, complemented
26	#16810660	—	2	1	no	—
27	#16810659	C₄	1	2	yes	complemented
28	#16810658	—	2	1	no	—
29	#107×D₅₃	—	2	1	no	—
30	#106×D₅₃	—	2	1	no	—
31	C₅₃×#427	C₄	1	1	yes	characteristic, complemented
32	C₂×#11719936	—	2	1	no	—
33	C₂×#11719936	C₂×C₄	1	4	yes	—
34	D₅₃⋊M₁₆	—	2	1	no	—
35	D₅₃⋊M₁₆	—	2	1	no	—
36	D₅₃⋊M₁₆	—	2	2	no	—
37	D₅₃⋊M₁₆	—	2	2	no	—
38	D₅₃⋊M₁₆	—	2	2	no	—
39	D₅₃⋊M₁₆	—	2	2	no	—
40	D₅₃⋊M₁₆	C₂×C₄	1	2	yes	—
41	D₅₃⋊M₁₆	Q₈	1	2	yes	—
42	C₂×#11719940	—	2	2	no	—
43	C₂×#11719940	D₄	1	2	yes	—
44	C₂×D₅₃⋊C₈	—	2	1	no	—
45	C₂×D₅₃⋊C₈	—	2	2	no	—
46	#13133296	—	2	1	no	—
47	#13133296	—	2	1	no	—
48	#13133296	C₂×C₄	1	2	yes	—
49	#13133296	D₄	1	2	yes	—
50	C₂×C₄⋊#6157	Q₈	1	2	yes	—
51	C₂×C₂²⋊#6157	—	2	1	no	—
52	C₂×C₄×#6157	—	2	1	no	—
53	#13133288	—	2	2	no	—
54	#13133288	—	2	2	no	—
55	#13133287	—	2	2	no	—
56	#13133287	—	2	2	no	—
57	#13133271	—	2	2	no	—
58	#13133271	—	2	2	no	—
59	C₈×#6157	—	2	2	no	—
60	C₈×#6157	—	2	2	no	—
61	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
62	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
68	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
69	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
70	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
71	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
72	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
73	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
74	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
75	#13133176	C₂×C₄	1	2	yes	—
76	#13133175	—	2	2	no	—
77	#13133174	—	2	2	no	—
78	#13133173	—	2	2	no	—
79	C₄⋊C₄×D₅₃	—	2	2	no	—
80	C₂²⋊C₄×D₅₃	—	2	2	no	—
81	C₄²×D₅₃	—	2	2	no	—
82	#13133151	—	2	1	no	—
83	#13133150	—	2	2	no	—
84	#13133149	—	2	1	no	—
85	#13133148	—	2	2	no	—
86	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	#117×C₅₃	C₂×C₄	1	2	yes	—
88	#107×C₅₃	—	2	1	no	—
89	#106×C₅₃	—	2	1	no	—
90	C₂³×D₅₃	#41	1	1	yes	characteristic
91	C₂²×#6157	—	2	2	no	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₅₃	—	4	2	no	—
95	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
105	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
106	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
107	C₂²⋊#6157	—	2	2	no	—
108	C₂²⋊#6157	—	2	2	no	—
109	#11719940	—	2	2	no	—
110	#11719940	—	4	2	no	—
111	#11719940	—	4	2	no	—
112	#11719940	C₄⋊C₄	1	4	yes	—
113	C₂×#88324	—	2	2	no	—
114	C₂×#88324	—	2	4	no	—
115	C₄⋊#6157	—	2	2	no	—
116	C₄⋊#6157	C₄⋊C₄	1	4	yes	—
117	C₄×#6157	—	2	2	no	—
118	C₄×#6157	—	2	2	no	—
119	#11719936	—	2	2	no	—
120	#11719936	—	2	2	no	—
121	#11719936	—	2	4	no	—
122	#11719936	—	2	4	no	—
123	D₅₃⋊C₈	—	2	2	no	—
124	D₅₃⋊C₈	—	2	2	no	—
125	D₅₃⋊C₈	—	4	2	no	—
126	D₅₃⋊C₈	—	4	2	no	—
127	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₂²⋊C₄×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₂²⋊Dic₅₃	—	2	2	no	—
137	#11719921	—	4	2	no	—
138	C₄⋊Dic₅₃	—	2	2	no	—
139	#11719919	—	4	2	no	—
140	C₄×Dic₅₃	—	2	2	no	—
141	C₄×Dic₅₃	—	4	2	no	—
142	C₂×#88322	C₄²	1	1	yes	characteristic
143	C₂×#88322	C₄²	1	1	yes	characteristic
144	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
145	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
147	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
148	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
150	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
151	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
152	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
153	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
154	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
157	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
158	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
159	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
160	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
161	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
162	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
163	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
164	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
165	C₂²×D₅₃	—	4	2	no	—
166	C₂²×D₅₃	#117	1	1	yes	characteristic
167	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
168	C₂²×D₅₃	#121	1	1	yes	characteristic
169	C₂²×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
170	C₂²×D₅₃	#124	1	1	yes	characteristic
171	C₂²×D₅₃	#125	1	1	yes	characteristic
172	C₂×#6157	—	2	4	no	—
173	C₂×#6157	—	4	4	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
181	C₂×Dic₅₃	—	4	2	no	—
182	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂×Dic₅₃	#121	1	1	yes	characteristic
185	C₂×Dic₅₃	#121	1	1	yes	characteristic
186	C₂×Dic₅₃	#127	1	1	yes	characteristic
187	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
188	C₄×D₅₃	—	8	4	no	—
189	C₄×D₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₄×D₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₄×D₅₃	#117	1	1	yes	characteristic
195	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
196	C₄×D₅₃	C₄×Q₈	1	1	yes	characteristic
197	C₄×D₅₃	#121	1	1	yes	characteristic
198	C₄×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
199	C₄×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
200	C₄×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
201	C₄×D₅₃	#124	1	1	yes	characteristic
202	C₄×D₅₃	#125	1	1	yes	characteristic
203	C₄×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
204	C₄×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
205	#88324	—	2	4	no	—
206	#88324	—	4	4	no	—
207	C₈×C₅₃	#95	1	1	yes	characteristic
208	C₈×C₅₃	#95	1	1	yes	characteristic
209	C₈×C₅₃	#95	1	1	yes	characteristic
210	C₈×C₅₃	#95	1	1	yes	characteristic
211	#88322	#95	1	1	yes	characteristic
212	#88322	#95	1	1	yes	characteristic
213	#88322	#95	1	1	yes	characteristic
214	#88322	#95	1	1	yes	characteristic
215	#10872	—	53	1	no	maximal, Sylow
216	C₂²×C₅₃	—	4	1	no	—
217	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
218	C₂²×C₅₃	#389	1	1	yes	characteristic
219	C₂²×C₅₃	#390	1	1	yes	characteristic
220	C₂×D₅₃	—	2	2	no	—
221	C₂×D₅₃	—	2	2	no	—
222	C₂×D₅₃	—	2	2	no	—
223	C₂×D₅₃	—	4	1	no	—
224	C₂×D₅₃	—	4	1	no	—
225	C₂×D₅₃	—	4	1	no	—
226	C₂×D₅₃	#380	1	1	yes	characteristic
227	C₂×D₅₃	#382	1	1	yes	characteristic
228	C₂×D₅₃	#389	1	1	yes	characteristic
229	C₂×D₅₃	#391	1	1	yes	characteristic
230	#6157	—	4	4	no	—
231	C₄×C₅₃	—	8	2	no	—
232	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
233	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
234	C₄×C₅₃	#389	1	1	yes	characteristic
235	C₄×C₅₃	#391	1	1	yes	characteristic
236	Dic₅₃	—	8	2	no	—
237	Dic₅₃	#380	1	1	yes	characteristic
238	Dic₅₃	#384	1	1	yes	characteristic
239	Dic₅₃	#389	1	1	yes	characteristic
240	Dic₅₃	#390	1	1	yes	characteristic
241	C₂²×#108	—	53	1	no	—
242	C₂×#427	—	53	1	no	—
243	C₂×#405	—	53	1	no	—
244	C₂×#344	—	53	2	no	—
245	#1733	—	53	2	no	—
246	C₂×C₅₃	—	2	1	no	—
247	C₂×C₅₃	—	2	1	no	—
248	C₂×C₅₃	—	2	1	no	—
249	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
250	D₅₃	—	2	1	no	—
251	D₅₃	—	2	1	no	—
252	D₅₃	—	2	1	no	—
253	D₅₃	#1814	1	2	yes	—
254	C₂²×M₁₆	—	53	1	no	—
255	C₂²×M₁₆	—	53	2	no	—
256	C₂³×C₈	—	53	1	no	—
257	C₂×#117	—	53	1	no	—
258	C₂×#117	—	53	2	no	—
259	C₂×#108	—	53	2	no	—
260	C₂×#108	—	53	2	no	—
261	C₂×#108	—	106	1	no	—
262	C₂×#108	—	106	1	no	—
263	C₂×#108	—	106	1	no	—
264	C₂×#108	—	106	1	no	—
265	#427	—	53	1	no	—
266	#427	—	53	2	no	—
267	#427	—	106	1	no	—
268	#427	—	106	1	no	—
269	#427	#6157	1	1	yes	characteristic, complemented
270	C₂×#107	—	106	1	no	—
271	C₂×#106	—	106	1	no	—
272	#405	—	53	2	no	—
273	#405	—	53	2	no	—
274	#405	—	106	1	no	—
275	#405	—	106	1	no	—
276	C₂×#97	—	106	1	no	—
277	C₂×C₄×C₈	—	106	1	no	—
278	#344	—	106	4	no	—
279	#339	—	106	4	no	—
280	C₅₃	#10872	1	1	yes	characteristic, complemented, Sylow
281	C₂³×C₄	—	53	1	no	—
282	C₂×M₁₆	—	53	2	no	—
283	C₂×M₁₆	—	53	2	no	—
284	C₂×M₁₆	—	53	2	no	—
285	C₂×M₁₆	—	53	4	no	—
286	C₂×M₁₆	—	106	1	no	—
287	C₂×M₁₆	—	106	1	no	—
288	C₂×M₁₆	—	106	1	no	—
289	C₂×M₁₆	—	106	2	no	—
290	C₂×M₁₆	—	106	2	no	—
291	C₂×M₁₆	—	106	2	no	—
292	C₂×M₁₆	—	106	2	no	—
293	C₂×M₁₆	—	106	2	no	—
294	C₂²×C₈	—	53	1	no	—
295	C₂²×C₈	—	53	1	no	—
296	C₂²×C₈	—	53	1	no	—
297	C₂²×C₈	—	53	1	no	—
298	C₂²×C₈	—	53	1	no	—
299	C₂²×C₈	—	53	1	no	—
300	C₂²×C₈	—	53	1	no	—
301	C₂²×C₈	—	53	1	no	—
302	C₂²×C₈	—	53	1	no	—
303	C₂²×C₈	—	53	1	no	—
304	C₂²×C₈	—	53	1	no	—
305	C₂²×C₈	—	53	1	no	—
306	C₂²×C₈	—	53	1	no	—
307	C₂²×C₈	—	106	1	no	—
308	C₂²×C₈	—	106	2	no	—
309	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
310	#117	—	53	2	no	—
311	#117	—	53	2	no	—
312	#117	—	53	2	no	—
313	#117	—	106	2	no	—
314	#117	—	106	1	no	—
315	#117	—	106	1	no	—
316	#117	—	106	2	no	—
317	#117	—	106	2	no	—
318	#117	C₂×#6157	1	2	yes	—
319	C₂×C₄⋊C₄	—	53	2	no	—
320	C₂×C₄⋊C₄	—	106	2	no	—
321	C₂×C₂²⋊C₄	—	106	2	no	—
322	C₂×C₂²⋊C₄	—	106	1	no	—
323	C₂×C₄²	—	106	2	no	—
324	C₂×C₄²	—	106	1	no	—
325	#108	—	106	2	no	—
326	#108	—	106	2	no	—
327	#108	—	106	2	no	—
328	#108	—	106	2	no	—
329	#107	—	2	1	no	—
330	#107	—	106	2	no	—
331	#107	—	106	1	no	—
332	#106	—	2	1	no	—
333	#106	—	106	2	no	—
334	#106	—	106	1	no	—
335	#97	—	106	2	no	—
336	#97	—	106	2	no	—
337	C₄×C₈	—	106	2	no	—
338	C₄×C₈	—	106	2	no	—
339	C₂⁴	—	53	1	no	—
340	C₂²×C₄	—	53	1	no	—
341	C₂²×C₄	—	53	1	no	—
342	C₂²×C₄	—	53	1	no	—
343	C₂²×C₄	—	53	1	no	—
344	C₂²×C₄	—	53	1	no	—
345	C₂²×C₄	—	53	1	no	—
346	C₂²×C₄	—	53	1	no	—
347	C₂²×C₄	—	53	1	no	—
348	C₂²×C₄	—	53	1	no	—
349	C₂²×C₄	—	53	1	no	—
350	C₂²×C₄	—	53	1	no	—
351	C₂²×C₄	—	53	1	no	—
352	C₂²×C₄	—	53	1	no	—
353	C₂²×C₄	—	106	2	no	—
354	C₂²×C₄	—	212	2	no	—
355	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
356	M₁₆	—	53	4	no	—
357	M₁₆	—	106	2	no	—
358	M₁₆	—	106	2	no	—
359	M₁₆	—	106	2	no	—
360	M₁₆	—	106	4	no	—
361	M₁₆	—	106	4	no	—
362	M₁₆	—	212	2	no	—
363	M₁₆	—	212	2	no	—
364	C₂×C₈	—	53	1	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	—	53	1	no	—
370	C₂×C₈	—	53	1	no	—
371	C₂×C₈	—	53	1	no	—
372	C₂×C₈	—	53	1	no	—
373	C₂×C₈	—	53	1	no	—
374	C₂×C₈	—	53	1	no	—
375	C₂×C₈	—	53	1	no	—
376	C₂×C₈	—	53	1	no	—
377	C₂×C₈	—	53	1	no	—
378	C₂×C₈	—	53	1	no	—
379	C₂×C₈	—	53	1	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	—	53	1	no	—
385	C₂×C₈	—	53	1	no	—
386	C₂×C₈	—	106	2	no	—
387	C₂×C₈	—	106	2	no	—
388	C₂×C₈	—	106	2	no	—
389	C₂×C₈	—	106	4	no	—
390	C₂×C₈	—	212	2	no	—
391	C₂×C₈	—	212	2	no	—
392	C₂×C₈	C₄×#6157	1	1	yes	characteristic
393	C₂×C₈	C₄×#6157	1	1	yes	characteristic
394	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
395	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
396	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
397	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	53	4	no	—
400	C₄⋊C₄	—	106	2	no	—
401	C₄⋊C₄	—	106	2	no	—
402	C₄⋊C₄	—	212	2	no	—
403	C₂²⋊C₄	—	2	2	no	—
404	C₂²⋊C₄	—	106	2	no	—
405	C₂²⋊C₄	—	106	2	no	—
406	C₂²⋊C₄	—	106	2	no	—
407	C₂²⋊C₄	—	212	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	106	2	no	—
410	C₄²	—	106	2	no	—
411	C₄²	—	106	2	no	—
412	C₄²	—	212	2	no	—
413	C₂³	—	53	1	no	—
414	C₂³	—	53	1	no	—
415	C₂³	—	53	1	no	—
416	C₂³	—	53	1	no	—
417	C₂³	—	53	1	no	—
418	C₂³	—	53	1	no	—
419	C₂³	—	212	2	no	—
420	C₂³	D₄×#6157	1	1	yes	characteristic
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	53	1	no	—
423	C₂×C₄	—	53	1	no	—
424	C₂×C₄	—	53	1	no	—
425	C₂×C₄	—	53	1	no	—
426	C₂×C₄	—	53	1	no	—
427	C₂×C₄	—	53	1	no	—
428	C₂×C₄	—	53	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	—	53	1	no	—
434	C₂×C₄	—	53	1	no	—
435	C₂×C₄	—	53	1	no	—
436	C₂×C₄	—	53	1	no	—
437	C₂×C₄	—	53	1	no	—
438	C₂×C₄	—	53	1	no	—
439	C₂×C₄	—	53	1	no	—
440	C₂×C₄	—	53	1	no	—
441	C₂×C₄	—	53	1	no	—
442	C₂×C₄	—	53	1	no	—
443	C₂×C₄	—	53	1	no	—
444	C₂×C₄	—	106	4	no	—
445	C₂×C₄	—	424	4	no	—
446	C₂×C₄	—	212	2	no	—
447	C₂×C₄	—	212	4	no	—
448	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
449	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
450	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
451	C₂×C₄	D₄×#6157	1	1	yes	characteristic
452	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
453	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
454	C₈	—	53	1	no	—
455	C₈	—	53	1	no	—
456	C₈	—	53	1	no	—
457	C₈	—	53	1	no	—
458	C₈	—	106	4	no	—
459	C₈	—	212	4	no	—
460	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
461	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
462	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
463	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
464	C₂²	—	4	1	no	—
465	C₂²	—	53	1	no	—
466	C₂²	—	53	1	no	—
467	C₂²	—	53	1	no	—
468	C₂²	—	53	1	no	—
469	C₂²	—	106	2	no	—
470	C₂²	—	106	2	no	—
471	C₂²	—	106	2	no	—
472	C₂²	—	212	1	no	—
473	C₂²	—	212	1	no	—
474	C₂²	—	212	1	no	—
475	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
476	C₂²	#16811574	1	1	yes	characteristic
477	C₂²	#16811575	1	1	yes	characteristic
478	C₄	—	8	2	no	—
479	C₄	—	53	1	no	—
480	C₄	—	53	1	no	—
481	C₄	—	53	1	no	—
482	C₄	—	53	1	no	—
483	C₄	—	424	2	no	—
484	C₄	—	212	4	no	—
485	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
486	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
487	C₄	#16811574	1	1	yes	characteristic
488	C₄	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
489	C₂	—	2	1	no	—
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	53	2	no	—
493	C₂	—	106	1	no	—
494	C₂	—	106	1	no	—
495	C₂	—	106	1	no	—
496	C₂	#20289892	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811574
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#427	C₄
socle	C₂×C₅₃	C₄⋊#95
prime radical	C₂³×D₅₃	#41
cosocle	C₂²×C₄×D₅₃	C₂³
hypercenter	#427	#6157
hypocenter	C₅₃	#10872
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
9. C₂×#13133288	C₄	semidirect
14. C₂×#13133287	C₄	semidirect
17. #16811586	C₄	semidirect
18. #16811586	C₄	semidirect
25. #16810661	C₄	semidirect
27. #16810659	C₄	semidirect
31. C₅₃×#427	C₄	semidirect
269. #427	#6157	semidirect
280. C₅₃	#10872	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊#108	2
C₂×#16811586	2
#20290423	2
#20290396	2
#427×D₅₃	2
#10872	53