Subgroups — #27402616

Order 13568.

808 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283628	C₂	1	2	yes	maximal
2	C₂×#16810743	C₂	1	1	yes	maximal, characteristic
3	#20283326	C₂	1	1	yes	maximal, characteristic, complemented
4	#20283323	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275504	C₂	1	2	yes	maximal, complemented
6	#20274958	C₂	1	2	yes	maximal, complemented
7	#20274953	C₂	1	2	yes	maximal, complemented
8	#20274888	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274766	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274601	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133237	C₂²	1	1	yes	characteristic
13	#16811390	C₂²	1	2	yes	—
14	#16811389	C₂²	1	2	yes	—
15	C₂×#13133223	C₂²	1	1	yes	characteristic
16	#16810743	C₄	1	4	yes	complemented
17	#16810743	C₄	1	4	yes	complemented
18	#16810742	—	2	4	no	—
19	#16810737	—	2	4	no	—
20	#16810672	—	2	2	no	—
21	#16810672	C₂²	1	2	yes	—
22	#16810671	C₂²	1	2	yes	—
23	#16810670	—	2	2	no	—
24	C₂×#13133153	—	2	2	no	—
25	C₂×#13133153	C₂²	1	1	yes	characteristic
26	C₂×#13133153	C₂²	1	1	yes	characteristic
27	C₂×#13133152	C₂²	1	2	yes	—
28	C₂×#13133152	C₂²	1	1	yes	characteristic
29	#108×D₅₃	—	2	2	no	—
30	#16810646	C₂²	1	2	yes	—
31	#16810641	C₂²	1	2	yes	—
32	C₂×#13133144	C₂²	1	1	yes	characteristic
33	C₂×#13133143	C₂²	1	1	yes	characteristic
34	#16810579	C₂²	1	2	yes	—
35	C₂×#13133127	C₂²	1	1	yes	characteristic
36	#16810393	C₂²	1	2	yes	—
37	#16810382	C₂²	1	2	yes	—
38	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#427	C₂²	1	1	yes	characteristic
41	C₅₃×#421	C₂²	1	2	yes	—
42	C₅₃×#344	C₂²	1	2	yes	—
43	C₂×#11719945	C₂³	1	1	yes	characteristic
44	#13133237	C₂×C₄	1	4	yes	—
45	#13133237	D₄	1	4	yes	—
46	C₂×#11719925	Q₈	1	1	yes	characteristic
47	C₂×#11719923	Q₈	1	1	yes	characteristic
48	C₂×#11719924	Q₈	1	1	yes	characteristic
49	C₂×#11719922	Q₈	1	1	yes	characteristic
50	#13133225	—	2	4	no	—
51	#13133224	—	2	4	no	—
52	#13133224	—	4	2	no	—
53	#13133223	—	2	1	no	—
54	#13133223	—	2	1	no	—
55	#13133223	—	4	2	no	—
56	#13133223	C₂×C₄	1	2	yes	—
57	#13133223	C₂×C₄	1	2	yes	—
58	C₂×#11719917	—	2	4	no	—
59	C₂×#11719917	C₂³	1	2	yes	—
60	C₂×#11719917	C₂³	1	1	yes	characteristic
61	C₂×#11719917	C₂³	1	2	yes	—
62	C₂×#11719912	—	2	1	no	—
63	M₁₆×D₅₃	—	2	4	no	—
64	C₂²×#88322	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₅₃	—	2	1	no	—
66	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
67	#13133153	—	2	1	no	—
68	#13133153	—	2	1	no	—
69	#13133153	—	2	4	no	—
70	#13133153	D₄	1	2	yes	—
71	#13133153	D₄	1	2	yes	—
72	#13133152	—	2	4	no	—
73	#13133152	—	2	2	no	—
74	#13133152	—	2	2	no	—
75	#13133152	C₂×C₄	1	4	yes	—
76	#13133146	—	4	2	no	—
77	#13133145	—	2	4	no	—
78	#13133144	C₂×C₄	1	4	yes	—
79	#13133143	C₂×C₄	1	4	yes	—
80	#13133128	—	2	2	no	—
81	#13133127	C₂×C₄	1	2	yes	—
82	#13133127	C₂×C₄	1	2	yes	—
83	C₄×#88322	—	2	2	no	—
84	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
85	#131×C₅₃	—	2	1	no	—
86	#131×C₅₃	—	2	1	no	—
87	#131×C₅₃	C₂×C₄	1	2	yes	—
88	#131×C₅₃	C₂×C₄	1	2	yes	—
89	C₂×M₁₆×C₅₃	—	2	1	no	—
90	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
91	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₅₃	—	2	1	no	—
93	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	#117×C₅₃	C₂³	1	2	yes	—
95	#108×C₅₃	—	2	1	no	—
96	#108×C₅₃	—	2	1	no	—
97	#107×C₅₃	—	2	1	no	—
98	#106×C₅₃	—	2	1	no	—
99	#104×C₅₃	—	2	4	no	—
100	#41×C₅₃	—	2	1	no	—
101	#41×C₅₃	—	2	1	no	—
102	#41×C₅₃	C₂²×C₄	1	2	yes	—
103	#41×C₅₃	C₂×D₄	1	2	yes	—
104	C₂×Q₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
105	C₂×D₄×C₅₃	—	2	1	no	—
106	C₂×D₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₂²⋊D₅₃	—	2	1	no	—
110	C₂²×Dic₅₃	—	2	1	no	—
111	#11719945	—	2	1	no	—
112	#11719945	—	2	1	no	—
113	#11719945	C₂²×C₄	1	2	yes	—
114	#11719945	C₂×D₄	1	2	yes	—
115	C₂×D₂₁₂	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₄×D₅₃	—	2	1	no	—
117	C₂×Dic₁₀₆	C₂×Q₈	1	1	yes	characteristic
118	M₁₆×C₅₃	—	2	1	no	—
119	M₁₆×C₅₃	—	2	2	no	—
120	M₁₆×C₅₃	—	2	2	no	—
121	M₁₆×C₅₃	—	2	2	no	—
122	M₁₆×C₅₃	—	2	2	no	—
123	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
133	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₂²⋊C₄×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	#11719925	C₄⋊C₄	1	4	yes	—
138	#11719924	C₄⋊C₄	1	4	yes	—
139	#11719923	C₄⋊C₄	1	4	yes	—
140	#11719922	C₄⋊C₄	1	4	yes	—
141	C₄×Dic₅₃	—	2	1	no	—
142	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
143	#11719917	—	2	1	no	—
144	#11719917	—	2	2	no	—
145	#11719917	—	2	2	no	—
146	#11719917	—	2	2	no	—
147	#11719917	—	4	4	no	—
148	#11719917	C₂²×C₄	1	2	yes	—
149	#11719917	C₂×D₄	1	4	yes	—
150	C₂×#88322	—	2	2	no	—
151	C₂×#88322	—	2	1	no	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	—	4	4	no	—
154	C₂×#88322	C₂²×C₄	1	2	yes	—
155	C₂×#88322	C₂×D₄	1	2	yes	—
156	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
157	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
158	#11719912	—	2	2	no	—
159	#11719912	—	2	2	no	—
160	#11719912	—	4	4	no	—
161	C₈×D₅₃	—	2	2	no	—
162	C₈×D₅₃	—	2	2	no	—
163	C₈×D₅₃	—	4	4	no	—
164	C₂³×C₅₃	—	2	1	no	—
165	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂²×D₅₃	—	2	1	no	—
167	Q₈×C₅₃	—	2	1	no	—
168	Q₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
169	D₄×C₅₃	—	2	2	no	—
170	D₄×C₅₃	—	2	2	no	—
171	D₄×C₅₃	—	2	1	no	—
172	D₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂²⋊D₅₃	—	2	2	no	—
184	C₂²⋊D₅₃	—	2	2	no	—
185	C₂×Dic₅₃	—	2	1	no	—
186	C₂×Dic₅₃	—	2	2	no	—
187	C₂×Dic₅₃	—	8	1	no	—
188	D₂₁₂	—	2	1	no	—
189	D₂₁₂	C₂×C₄⋊C₄	1	2	yes	—
190	C₄×D₅₃	—	2	2	no	—
191	C₄×D₅₃	—	2	2	no	—
192	Dic₁₀₆	—	2	1	no	—
193	Dic₁₀₆	C₂×C₄⋊C₄	1	2	yes	—
194	C₈×C₅₃	—	2	2	no	—
195	C₈×C₅₃	—	4	2	no	—
196	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
197	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
198	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
199	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
200	#88322	—	2	2	no	—
201	#88322	—	2	4	no	—
202	#88322	—	4	2	no	—
203	#88322	C₂×C₄⋊C₄	1	2	yes	—
204	#88322	C₂×C₄⋊C₄	1	2	yes	—
205	#22695	—	53	1	no	maximal, Sylow
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	—	4	1	no	—
208	C₂²×C₅₃	—	8	1	no	—
209	C₂²×C₅₃	#520	1	1	yes	characteristic
210	C₂²×C₅₃	#547	1	1	yes	characteristic
211	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
212	C₂×D₅₃	—	2	2	no	—
213	C₂×D₅₃	—	8	1	no	—
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	8	2	no	—
216	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
217	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₄×C₅₃	D₄²	1	1	yes	characteristic
219	C₄×C₅₃	#554	1	1	yes	characteristic
220	Dic₅₃	—	2	2	no	—
221	Dic₅₃	—	4	2	no	—
222	#2957	—	53	2	no	—
223	C₂×#444	—	53	1	no	—
224	#2916	—	53	1	no	—
225	#2916	—	53	1	no	—
226	#1953	—	53	2	no	—
227	#1861	—	53	2	no	—
228	#1861	—	53	2	no	—
229	#1849	—	53	1	no	—
230	#1814	—	53	1	no	—
231	#1814	D₅₃	1	1	yes	characteristic, complemented
232	#1773	—	53	1	no	—
233	C₂×C₅₃	—	2	1	no	—
234	C₂×C₅₃	—	2	1	no	—
235	C₂×C₅₃	—	2	1	no	—
236	C₂×C₅₃	—	4	2	no	—
237	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
238	D₅₃	—	4	2	no	—
239	C₂×#135	—	53	1	no	—
240	#568	—	53	2	no	—
241	#568	—	53	2	no	—
242	C₂×#131	—	53	1	no	—
243	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
244	#444	—	53	4	no	—
245	#444	—	53	4	no	—
246	#444	—	106	4	no	—
247	#443	—	106	4	no	—
248	#430	—	53	2	no	—
249	#430	—	53	2	no	—
250	#430	—	106	2	no	—
251	#430	—	106	2	no	—
252	C₂×#108	—	53	1	no	—
253	C₂×#108	—	53	1	no	—
254	C₂×#108	—	53	2	no	—
255	C₂×#108	—	53	1	no	—
256	C₂×#108	—	106	2	no	—
257	C₂×#108	—	106	2	no	—
258	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
259	#427	C₂×D₅₃	1	1	yes	characteristic
260	#423	—	53	2	no	—
261	#421	—	53	2	no	—
262	#421	C₂×D₅₃	1	2	yes	—
263	C₂×#104	—	53	1	no	—
264	C₂×#104	—	53	1	no	—
265	#405	—	53	2	no	—
266	C₂×#97	—	53	1	no	—
267	#344	—	53	2	no	—
268	#344	C₂×D₅₃	1	2	yes	—
269	#339	—	53	2	no	—
270	C₅₃	#22695	1	1	yes	characteristic, complemented, Sylow
271	C₂×#41	—	53	1	no	—
272	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
273	#135	—	53	4	no	—
274	#135	—	53	4	no	—
275	C₂×Q₁₆	—	53	1	no	—
276	C₂×SD₁₆	—	53	1	no	—
277	C₂×SD₁₆	—	53	1	no	—
278	C₂×D₈	—	53	1	no	—
279	#131	—	2	1	no	—
280	#131	—	2	1	no	—
281	#131	—	53	2	no	—
282	#131	—	53	2	no	—
283	#131	—	106	1	no	—
284	#131	—	106	1	no	—
285	#131	—	106	4	no	—
286	#131	—	106	4	no	—
287	#131	—	212	2	no	—
288	#131	—	212	2	no	—
289	#131	C₄×D₅₃	1	2	yes	—
290	#131	C₄×D₅₃	1	2	yes	—
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	—	53	1	no	—
293	C₂×M₁₆	—	53	2	no	—
294	C₂×M₁₆	—	53	2	no	—
295	C₂×M₁₆	—	106	1	no	—
296	C₂×M₁₆	—	106	4	no	—
297	C₂×M₁₆	—	106	4	no	—
298	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
299	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
300	C₂²×C₈	—	2	1	no	—
301	C₂²×C₈	—	53	1	no	—
302	C₂²×C₈	—	106	1	no	—
303	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
304	#117	C₂²×D₅₃	1	2	yes	—
305	C₂×C₄²	—	53	1	no	—
306	#108	—	2	1	no	—
307	#108	—	2	1	no	—
308	#108	—	53	4	no	—
309	#108	—	53	2	no	—
310	#108	—	53	2	no	—
311	#108	—	106	4	no	—
312	#108	—	106	2	no	—
313	#108	—	106	2	no	—
314	#108	—	106	4	no	—
315	#108	—	106	1	no	—
316	#108	—	106	1	no	—
317	#107	—	2	1	no	—
318	#106	—	2	1	no	—
319	C₄⋊C₈	—	212	2	no	—
320	#104	—	2	4	no	—
321	#104	—	53	4	no	—
322	#104	—	53	4	no	—
323	#104	—	106	4	no	—
324	#97	—	53	2	no	—
325	#97	—	53	2	no	—
326	#97	—	106	2	no	—
327	C₄×C₈	—	106	2	no	—
328	#41	—	2	1	no	—
329	#41	—	2	1	no	—
330	#41	—	53	2	no	—
331	#41	—	53	2	no	—
332	#41	—	106	1	no	—
333	#41	—	106	1	no	—
334	#41	C₂×C₄×D₅₃	1	2	yes	—
335	#41	D₄×D₅₃	1	2	yes	—
336	C₂×Q₈	—	53	1	no	—
337	C₂×Q₈	Q₈×D₅₃	1	1	yes	characteristic
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	53	1	no	—
340	C₂×D₄	—	106	1	no	—
341	C₂×D₄	Q₈×D₅₃	1	1	yes	characteristic
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	106	1	no	—
344	C₂²×C₄	—	106	1	no	—
345	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
346	Q₁₆	—	53	4	no	—
347	SD₁₆	—	53	4	no	—
348	SD₁₆	—	53	4	no	—
349	D₈	—	53	4	no	—
350	M₁₆	—	2	1	no	—
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	2	no	—
353	M₁₆	—	2	2	no	—
354	M₁₆	—	2	2	no	—
355	M₁₆	—	53	2	no	—
356	M₁₆	—	53	4	no	—
357	M₁₆	—	106	1	no	—
358	M₁₆	—	106	2	no	—
359	M₁₆	—	106	2	no	—
360	M₁₆	—	106	2	no	—
361	M₁₆	—	106	2	no	—
362	M₁₆	—	106	2	no	—
363	M₁₆	—	212	4	no	—
364	M₁₆	—	212	4	no	—
365	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	53	2	no	—
371	C₂×C₈	—	53	2	no	—
372	C₂×C₈	—	53	1	no	—
373	C₂×C₈	—	53	1	no	—
374	C₂×C₈	—	106	2	no	—
375	C₂×C₈	—	106	1	no	—
376	C₂×C₈	—	106	2	no	—
377	C₂×C₈	—	106	2	no	—
378	C₂×C₈	—	106	2	no	—
379	C₂×C₈	—	212	4	no	—
380	C₂×C₈	—	212	4	no	—
381	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
382	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
383	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
384	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
385	C₂×C₈	#11719947	1	1	yes	characteristic
386	C₂×C₈	#11719947	1	1	yes	characteristic
387	C₄⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	53	2	no	—
391	C₄²	—	106	1	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	106	1	no	—
394	C₂³	#13133264	1	1	yes	characteristic
395	Q₈	—	2	1	no	—
396	Q₈	—	53	2	no	—
397	Q₈	—	106	1	no	—
398	Q₈	C₄⋊C₄×D₅₃	1	2	yes	—
399	D₄	—	2	2	no	—
400	D₄	—	2	2	no	—
401	D₄	—	2	1	no	—
402	D₄	—	53	2	no	—
403	D₄	—	106	2	no	—
404	D₄	—	106	2	no	—
405	D₄	—	106	1	no	—
406	D₄	C₄⋊C₄×D₅₃	1	2	yes	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	106	1	no	—
412	C₂×C₄	—	106	2	no	—
413	C₂×C₄	—	106	2	no	—
414	C₂×C₄	—	106	2	no	—
415	C₂×C₄	—	424	1	no	—
416	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
418	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
419	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
420	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
421	C₂×C₄	#13133263	1	1	yes	characteristic
422	C₈	—	2	2	no	—
423	C₈	—	4	2	no	—
424	C₈	—	53	2	no	—
425	C₈	—	53	2	no	—
426	C₈	—	106	2	no	—
427	C₈	—	106	4	no	—
428	C₈	—	212	2	no	—
429	C₈	#13133177	1	1	yes	characteristic
430	C₈	#13133177	1	1	yes	characteristic
431	C₈	#13133177	1	1	yes	characteristic
432	C₈	#13133177	1	1	yes	characteristic
433	C₂²	—	2	2	no	—
434	C₂²	—	4	1	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	106	2	no	—
437	C₂²	—	424	1	no	—
438	C₂²	#16811103	1	1	yes	characteristic
439	C₂²	#16811273	1	1	yes	characteristic
440	C₂²	#16811286	1	1	yes	characteristic
441	C₄	—	2	2	no	—
442	C₄	—	8	2	no	—
443	C₄	—	106	2	no	—
444	C₄	—	212	2	no	—
445	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
446	C₄	C₂×#13133177	1	1	yes	characteristic
447	C₄	#16811257	1	1	yes	characteristic
448	C₄	#16811331	1	1	yes	characteristic
449	C₂	—	2	1	no	—
450	C₂	—	2	1	no	—
451	C₂	—	2	1	no	—
452	C₂	—	4	2	no	—
453	C₂	—	212	2	no	—
454	C₂	#20277611	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811257
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1814	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1814	D₅₃
hypocenter	C₅₃	#22695
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1814

Complements

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

N	H ≅ G/N	type
3. #20283326	C₂	semidirect
4. #20283323	C₂	semidirect
5. #20275504	C₂	semidirect
6. #20274958	C₂	semidirect
7. #20274953	C₂	semidirect
8. #20274888	C₂	semidirect
9. #20274766	C₂	semidirect
10. #20274601	C₂	semidirect
11. C₅₃×#1814	C₂	semidirect
16. #16810743	C₄	semidirect
17. #16810743	C₄	semidirect
231. #1814	D₅₃	semidirect
270. C₅₃	#22695	semidirect

Maximal subgroups

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

subgroup	index
#20283628	2
C₂×#16810743	2
#20283326	2
#20283323	2
#20275504	2
#20274958	2
#20274953	2
#20274888	2
#20274766	2
#20274601	2
C₅₃×#1814	2
#22695	53