Subgroups — #27353331

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719945	C₂	1	1	yes	maximal, characteristic
2	#20283376	C₂	1	2	yes	maximal
3	#20273493	C₂	1	2	yes	maximal, complemented
4	#20273492	C₂	1	2	yes	maximal, complemented
5	#20273484	C₂	1	2	yes	maximal, complemented
6	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273477	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273472	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273462	C₂	1	1	yes	maximal, characteristic, complemented
10	#335×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11719945	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
15	C₈×D₂₁₂	C₄	1	1	yes	characteristic
16	C₈×D₂₁₂	C₄	1	1	yes	characteristic
17	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
18	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
19	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
20	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
21	#16810682	—	2	2	no	—
22	#16810681	—	2	2	no	—
23	#16810680	C₂²	1	2	yes	—
24	#16810679	C₂²	1	2	yes	—
25	#16810676	C₂²	1	1	yes	characteristic
26	#16810675	C₂²	1	2	yes	—
27	#16810674	C₂²	1	1	yes	characteristic
28	#16810674	C₂²	1	1	yes	characteristic
29	#16810645	C₂²	1	2	yes	—
30	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
31	#16810588	—	2	2	no	—
32	#16810577	—	2	2	no	—
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	#16810376	C₂²	1	1	yes	characteristic
36	#16810376	C₂²	1	1	yes	characteristic
37	#16810375	C₂²	1	2	yes	—
38	#16810375	C₂²	1	2	yes	—
39	#16810374	C₂²	1	2	yes	—
40	#16810373	C₂²	1	2	yes	—
41	#16810373	C₂²	1	2	yes	—
42	C₅₃×#431	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#335	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	1	yes	characteristic
46	C₅₃×#334	C₂²	1	2	yes	—
47	C₂×#11719945	D₄	1	1	yes	characteristic
48	C₂×#11719912	—	2	2	no	—
49	C₂×C₈×D₅₃	—	2	2	no	—
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₀₆	Q₈	1	1	yes	characteristic
56	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
57	C₄×D₂₁₂	Q₈	1	1	yes	characteristic
58	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
61	C₄×C₂²⋊D₅₃	Q₈	1	1	yes	characteristic
62	#13133176	D₄	1	1	yes	characteristic
63	#13133174	D₄	1	1	yes	characteristic
64	#13133174	D₄	1	1	yes	characteristic
65	C₄²×D₅₃	C₂³	1	1	yes	characteristic
66	C₄²×D₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
68	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
69	C₄⋊#88322	C₂³	1	2	yes	—
70	C₄⋊#88322	C₂³	1	2	yes	—
71	#13133146	—	2	2	no	—
72	#13133146	—	2	2	no	—
73	#13133146	C₂×C₄	1	1	yes	characteristic
74	#13133146	C₂×C₄	1	1	yes	characteristic
75	#13133146	C₂×C₄	1	1	yes	characteristic
76	#13133146	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊#88322	—	2	2	no	—
78	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
80	#13133129	—	2	2	no	—
81	#13133129	—	2	2	no	—
82	#13133129	C₂×C₄	1	1	yes	characteristic
83	#13133129	C₂×C₄	1	1	yes	characteristic
84	#13133129	C₂×C₄	1	1	yes	characteristic
85	#13133129	C₂×C₄	1	1	yes	characteristic
86	#13133128	—	2	2	no	—
87	#13133127	—	2	2	no	—
88	C₄×#88322	C₂³	1	1	yes	characteristic
89	C₄×#88322	C₂³	1	1	yes	characteristic
90	C₈×Dic₅₃	—	2	2	no	—
91	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
92	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
93	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
94	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
100	C₂²⋊C₈×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	—	2	2	no	—
102	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	1	yes	characteristic
111	#11719945	—	2	2	no	—
112	#11719945	—	2	2	no	—
113	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
120	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	4	1	no	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
132	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
134	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
135	#11719921	C₄⋊C₄	1	1	yes	characteristic
136	#11719921	C₄⋊C₄	1	1	yes	characteristic
137	#11719921	C₄⋊C₄	1	1	yes	characteristic
138	#11719921	C₄⋊C₄	1	1	yes	characteristic
139	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
140	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
141	#11719919	C₄⋊C₄	1	1	yes	characteristic
142	#11719919	C₄⋊C₄	1	1	yes	characteristic
143	#11719919	C₄⋊C₄	1	1	yes	characteristic
144	#11719919	C₄⋊C₄	1	1	yes	characteristic
145	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
151	C₂×#88322	—	2	2	no	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
154	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
155	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
156	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
157	#11719912	—	4	4	no	—
158	C₈×D₅₃	—	2	2	no	—
159	C₈×D₅₃	—	2	2	no	—
160	C₈×D₅₃	—	4	1	no	—
161	C₈×D₅₃	—	4	1	no	—
162	C₈×D₅₃	—	4	4	no	—
163	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
177	C₂²⋊D₅₃	—	2	2	no	—
178	C₂²⋊D₅₃	—	2	2	no	—
179	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
185	D₂₁₂	—	2	2	no	—
186	C₄×D₅₃	—	2	1	no	—
187	C₄×D₅₃	—	2	1	no	—
188	C₄×D₅₃	—	2	1	no	—
189	C₄×D₅₃	—	2	1	no	—
190	C₄×D₅₃	—	2	1	no	—
191	C₄×D₅₃	—	2	1	no	—
192	C₄×D₅₃	—	4	1	no	—
193	C₄×D₅₃	—	4	1	no	—
194	C₄×D₅₃	#135	1	2	yes	—
195	C₄×D₅₃	#135	1	2	yes	—
196	Dic₁₀₆	—	2	2	no	—
197	C₈×C₅₃	—	2	1	no	—
198	C₈×C₅₃	—	2	1	no	—
199	C₈×C₅₃	—	2	1	no	—
200	C₈×C₅₃	—	4	2	no	—
201	C₈×C₅₃	—	4	2	no	—
202	C₈×C₅₃	#131	1	2	yes	—
203	#88322	—	2	1	no	—
204	#88322	—	2	1	no	—
205	#88322	—	2	1	no	—
206	#88322	—	2	1	no	—
207	#88322	—	4	2	no	—
208	#88322	—	4	2	no	—
209	#17829	—	53	1	no	maximal, Sylow
210	C₂²×C₅₃	—	2	1	no	—
211	C₂²×C₅₃	—	2	1	no	—
212	C₂²×C₅₃	—	2	1	no	—
213	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
214	C₂×D₅₃	—	2	1	no	—
215	C₂×D₅₃	—	2	1	no	—
216	C₂×D₅₃	—	2	1	no	—
217	C₂×D₅₃	—	2	1	no	—
218	C₂×D₅₃	—	2	1	no	—
219	C₂×D₅₃	#427	1	2	yes	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	2	1	no	—
222	C₄×C₅₃	—	2	1	no	—
223	C₄×C₅₃	—	2	1	no	—
224	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
225	C₄×C₅₃	#568	1	1	yes	characteristic
226	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
227	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
228	Dic₅₃	—	2	1	no	—
229	Dic₅₃	—	2	1	no	—
230	Dic₅₃	—	2	1	no	—
231	Dic₅₃	—	2	1	no	—
232	Dic₅₃	—	2	1	no	—
233	Dic₅₃	#427	1	2	yes	—
234	C₈×#41	—	53	1	no	—
235	#2925	—	53	2	no	—
236	#1572	—	53	2	no	—
237	#1572	—	53	2	no	—
238	#1570	—	53	2	no	—
239	#1568	—	53	1	no	—
240	#1568	—	53	1	no	—
241	#1567	—	53	1	no	—
242	#1564	—	53	1	no	—
243	#1564	D₅₃	1	1	yes	characteristic, complemented
244	C₂×#335	—	53	1	no	—
245	C₂×C₅₃	—	4	1	no	—
246	C₂×C₅₃	#2906	1	1	yes	characteristic
247	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
248	C₂×C₅₃	#2916	1	1	yes	characteristic
249	D₅₃	—	2	2	no	—
250	D₅₃	—	4	1	no	—
251	C₄×#41	—	53	1	no	—
252	C₈×Q₈	—	53	1	no	—
253	C₈×Q₈	—	53	1	no	—
254	C₈×D₄	—	53	1	no	—
255	C₈×D₄	—	53	1	no	—
256	C₈×D₄	—	53	1	no	—
257	C₈×D₄	—	53	1	no	—
258	C₈×D₄	—	53	1	no	—
259	C₈×D₄	—	53	1	no	—
260	#433	—	106	2	no	—
261	#433	—	106	2	no	—
262	#432	—	53	2	no	—
263	#432	—	53	2	no	—
264	#431	—	53	1	no	—
265	#431	—	53	1	no	—
266	#431	—	53	1	no	—
267	#431	—	53	2	no	—
268	#431	C₂×D₅₃	1	2	yes	—
269	#423	—	53	2	no	—
270	C₂×C₄⋊C₈	—	53	2	no	—
271	#408	—	106	2	no	—
272	#405	—	106	2	no	—
273	C₂×C₄×C₈	—	53	1	no	—
274	C₂×C₄×C₈	—	53	1	no	—
275	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
276	#335	—	53	2	no	—
277	#335	—	53	2	no	—
278	#335	—	53	1	no	—
279	#335	—	53	1	no	—
280	#335	C₂×D₅₃	1	1	yes	characteristic
281	#335	C₂×D₅₃	1	1	yes	characteristic
282	#334	—	53	2	no	—
283	#334	—	53	2	no	—
284	#334	—	53	2	no	—
285	#334	C₂×D₅₃	1	2	yes	—
286	C₅₃	#17829	1	1	yes	characteristic, complemented, Sylow
287	C₂×#41	—	53	1	no	—
288	C₂×M₁₆	—	106	2	no	—
289	C₂²×C₈	—	53	1	no	—
290	C₂²×C₈	—	53	1	no	—
291	C₂²×C₈	—	53	1	no	—
292	C₂²×C₈	—	53	1	no	—
293	C₂²×C₈	—	106	2	no	—
294	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
295	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
296	C₄×Q₈	—	53	1	no	—
297	C₄×Q₈	—	53	1	no	—
298	C₄×D₄	—	53	1	no	—
299	C₄×D₄	—	53	1	no	—
300	C₄×D₄	—	53	1	no	—
301	C₄×D₄	—	53	1	no	—
302	C₄×D₄	—	53	1	no	—
303	C₄×D₄	—	53	1	no	—
304	#117	—	53	1	no	—
305	#117	—	53	1	no	—
306	#117	—	53	1	no	—
307	C₂×C₄²	—	53	1	no	—
308	C₂×C₄²	—	53	1	no	—
309	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
310	C₄⋊C₈	—	53	1	no	—
311	C₄⋊C₈	—	53	1	no	—
312	C₄⋊C₈	—	53	1	no	—
313	C₄⋊C₈	—	53	1	no	—
314	C₄⋊C₈	—	53	1	no	—
315	C₄⋊C₈	—	53	1	no	—
316	C₄⋊C₈	—	53	2	no	—
317	C₄⋊C₈	—	53	2	no	—
318	C₄⋊C₈	—	106	2	no	—
319	C₄⋊C₈	—	106	2	no	—
320	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
321	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
322	C₂²⋊C₈	—	2	2	no	—
323	C₂²⋊C₈	—	53	1	no	—
324	C₂²⋊C₈	—	53	1	no	—
325	C₂²⋊C₈	—	53	1	no	—
326	C₂²⋊C₈	—	53	1	no	—
327	C₂²⋊C₈	—	53	1	no	—
328	C₂²⋊C₈	—	53	1	no	—
329	C₂²⋊C₈	—	106	2	no	—
330	C₂²⋊C₈	—	106	2	no	—
331	C₂²⋊C₈	—	106	2	no	—
332	#97	—	106	2	no	—
333	#97	—	106	2	no	—
334	C₄×C₈	—	2	2	no	—
335	C₄×C₈	—	53	1	no	—
336	C₄×C₈	—	53	1	no	—
337	C₄×C₈	—	53	1	no	—
338	C₄×C₈	—	53	1	no	—
339	C₄×C₈	—	53	1	no	—
340	C₄×C₈	—	53	1	no	—
341	C₄×C₈	—	106	2	no	—
342	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
343	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
344	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
345	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
346	#41	—	106	2	no	—
347	#41	—	106	2	no	—
348	C₂×Q₈	—	53	1	no	—
349	C₂×D₄	—	53	1	no	—
350	C₂×D₄	—	53	1	no	—
351	C₂×D₄	—	53	1	no	—
352	C₂²×C₄	—	53	1	no	—
353	C₂²×C₄	—	53	1	no	—
354	C₂²×C₄	—	53	1	no	—
355	C₂²×C₄	—	53	1	no	—
356	C₂²×C₄	—	53	1	no	—
357	C₂²×C₄	—	53	1	no	—
358	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
359	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
360	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
361	M₁₆	—	212	4	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	4	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	—	106	2	no	—
371	C₂×C₈	—	106	2	no	—
372	C₂×C₈	—	106	2	no	—
373	C₂×C₈	—	106	2	no	—
374	C₂×C₈	—	212	1	no	—
375	C₂×C₈	—	212	1	no	—
376	C₂×C₈	—	212	4	no	—
377	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
378	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
379	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
380	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
381	C₄⋊C₄	—	53	1	no	—
382	C₄⋊C₄	—	53	1	no	—
383	C₄⋊C₄	—	53	1	no	—
384	C₄⋊C₄	—	53	1	no	—
385	C₄⋊C₄	—	53	1	no	—
386	C₄⋊C₄	—	53	1	no	—
387	C₂²⋊C₄	—	53	1	no	—
388	C₂²⋊C₄	—	53	1	no	—
389	C₂²⋊C₄	—	53	1	no	—
390	C₂²⋊C₄	—	53	1	no	—
391	C₂²⋊C₄	—	53	1	no	—
392	C₂²⋊C₄	—	53	1	no	—
393	C₄²	—	53	1	no	—
394	C₄²	—	53	1	no	—
395	C₄²	—	53	1	no	—
396	C₄²	—	53	1	no	—
397	C₄²	—	53	1	no	—
398	C₄²	—	53	1	no	—
399	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
400	C₄²	D₄×D₅₃	1	1	yes	characteristic
401	C₄²	Q₈×D₅₃	1	1	yes	characteristic
402	C₄²	C₂³×D₅₃	1	1	yes	characteristic
403	C₂³	—	53	1	no	—
404	C₂³	—	53	1	no	—
405	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
406	Q₈	—	106	2	no	—
407	D₄	—	106	2	no	—
408	D₄	—	106	2	no	—
409	D₄	—	106	2	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	53	1	no	—
416	C₂×C₄	—	53	2	no	—
417	C₂×C₄	—	53	2	no	—
418	C₂×C₄	—	53	1	no	—
419	C₂×C₄	—	53	1	no	—
420	C₂×C₄	—	53	1	no	—
421	C₂×C₄	—	53	1	no	—
422	C₂×C₄	—	53	1	no	—
423	C₂×C₄	—	106	1	no	—
424	C₂×C₄	—	106	1	no	—
425	C₂×C₄	—	106	1	no	—
426	C₂×C₄	—	106	1	no	—
427	C₂×C₄	—	106	1	no	—
428	C₂×C₄	—	106	1	no	—
429	C₂×C₄	—	212	1	no	—
430	C₂×C₄	—	212	1	no	—
431	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
432	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
433	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
434	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
435	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
436	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
437	C₈	—	2	1	no	—
438	C₈	—	2	1	no	—
439	C₈	—	2	1	no	—
440	C₈	—	4	2	no	—
441	C₈	—	4	2	no	—
442	C₈	—	106	1	no	—
443	C₈	—	106	1	no	—
444	C₈	—	106	1	no	—
445	C₈	—	106	1	no	—
446	C₈	—	212	2	no	—
447	C₈	—	212	2	no	—
448	C₈	#13133223	1	2	yes	—
449	C₂²	—	2	1	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	2	1	no	—
452	C₂²	—	53	2	no	—
453	C₂²	—	106	1	no	—
454	C₂²	—	106	1	no	—
455	C₂²	—	106	1	no	—
456	C₂²	—	106	1	no	—
457	C₂²	—	106	1	no	—
458	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
459	C₄	—	2	1	no	—
460	C₄	—	2	1	no	—
461	C₄	—	2	1	no	—
462	C₄	—	2	1	no	—
463	C₄	—	53	2	no	—
464	C₄	—	106	1	no	—
465	C₄	—	106	1	no	—
466	C₄	—	106	1	no	—
467	C₄	—	106	1	no	—
468	C₄	—	106	1	no	—
469	C₄	C₂×#13133223	1	1	yes	characteristic
470	C₄	#16811389	1	1	yes	characteristic
471	C₄	#135×D₅₃	1	1	yes	characteristic
472	C₄	#135×D₅₃	1	1	yes	characteristic
473	C₂	—	4	1	no	—
474	C₂	—	106	2	no	—
475	C₂	—	212	1	no	—
476	C₂	#20283280	1	1	yes	characteristic
477	C₂	#427×D₅₃	1	1	yes	characteristic
478	C₂	#20283323	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×#1564	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#17829
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #20273493	C₂	semidirect
4. #20273492	C₂	semidirect
5. #20273484	C₂	semidirect
6. #20273478	C₂	semidirect
7. #20273477	C₂	semidirect
8. #20273472	C₂	semidirect
9. #20273462	C₂	semidirect
10. #335×D₅₃	C₂	semidirect
11. C₅₃×#1564	C₂	semidirect
243. #1564	D₅₃	semidirect
286. C₅₃	#17829	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719945	2
#20283376	2
#20273493	2
#20273492	2
#20273484	2
#20273478	2
#20273477	2
#20273472	2
#20273462	2
#335×D₅₃	2
C₅₃×#1564	2
#17829	53