Subgroups — #27430065

Order 13568.

720 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284702	C₂	1	2	yes	maximal, complemented
2	#20284385	C₂	1	1	yes	maximal, characteristic
3	#20283592	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276162	C₂	1	2	yes	maximal, complemented
5	#20275943	C₂	1	2	yes	maximal, complemented
6	#20275603	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275500	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274949	C₂	1	2	yes	maximal, complemented
9	#20274879	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274779	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	#16811201	C₂²	1	2	yes	—
14	#16811103	C₂²	1	2	yes	—
15	#16810962	—	2	2	no	—
16	#16810952	—	2	2	no	—
17	#16810946	—	2	2	no	—
18	#16810940	—	2	2	no	—
19	#16810869	—	2	2	no	—
20	#16810864	—	2	1	no	—
21	#16810859	—	2	1	no	—
22	#16810725	—	2	2	no	—
23	C₄×Dic₂₁₂	—	2	1	no	—
24	C₄×D₄₂₄	—	2	1	no	—
25	C₂×#13133153	C₂²	1	1	yes	characteristic
26	#16810666	C₂²	1	2	yes	—
27	C₂×#13133151	C₂²	1	1	yes	characteristic
28	C₂×#13133151	C₂²	1	1	yes	characteristic
29	C₂×#13133149	C₂²	1	2	yes	—
30	C₂×#13133145	C₂²	1	2	yes	—
31	#16810625	C₂²	1	2	yes	—
32	#16810625	C₂²	1	2	yes	—
33	#16810624	C₂²	1	2	yes	—
34	#16810589	C₂²	1	2	yes	—
35	#16810489	C₂²	1	2	yes	—
36	#16810391	C₂²	1	2	yes	—
37	#16810391	C₂²	1	2	yes	—
38	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#416	C₂²	1	1	yes	characteristic
41	C₅₃×#405	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#341	C₂²	1	1	yes	characteristic
44	C₅₃×#339	C₂²	1	1	yes	characteristic
45	C₂×#11719945	C₂³	1	2	yes	—
46	#13133239	—	2	2	no	—
47	#13133239	—	2	2	no	—
48	C₂×Dic₂₁₂	D₄	1	1	yes	characteristic
49	C₂×#11719913	D₄	1	2	yes	—
50	C₂×D₄₂₄	D₄	1	1	yes	characteristic
51	C₂×#11719917	C₂³	1	2	yes	—
52	#13133196	—	2	2	no	—
53	#13133193	—	2	2	no	—
54	#13133191	—	2	2	no	—
55	#13133189	—	4	2	no	—
56	#13133188	—	2	2	no	—
57	C₄×Dic₁₀₆	—	2	2	no	—
58	C₄×D₂₁₂	—	2	2	no	—
59	#13133177	—	4	2	no	—
60	#13133175	—	2	2	no	—
61	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
62	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
63	C₄⋊C₄×D₅₃	—	2	2	no	—
64	#13133153	—	2	2	no	—
65	#13133151	—	2	1	no	—
66	#13133151	—	2	1	no	—
67	#13133151	—	2	1	no	—
68	#13133151	—	2	1	no	—
69	#13133150	—	2	2	no	—
70	#13133149	—	2	2	no	—
71	#13133149	—	2	2	no	—
72	#13133148	—	2	2	no	—
73	#13133145	—	2	4	no	—
74	#13133142	—	2	2	no	—
75	#13133142	—	2	2	no	—
76	#13133141	—	2	2	no	—
77	#13133139	—	2	2	no	—
78	#13133139	—	2	2	no	—
79	#13133138	—	2	2	no	—
80	#13133129	—	2	2	no	—
81	#13133129	—	2	2	no	—
82	#13133124	—	4	2	no	—
83	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
84	#131×C₅₃	—	2	2	no	—
85	#131×C₅₃	—	2	2	no	—
86	C₂×M₁₆×C₅₃	—	2	1	no	—
87	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	—	2	1	no	—
90	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	#117×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	#104×C₅₃	—	2	2	no	—
95	#103×C₅₃	D₄	1	1	yes	characteristic
96	#103×C₅₃	D₄	1	1	yes	characteristic
97	#102×C₅₃	D₄	1	1	yes	characteristic
98	#102×C₅₃	D₄	1	1	yes	characteristic
99	#97×C₅₃	—	2	1	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	—	2	2	no	—
102	#41×C₅₃	—	2	1	no	—
103	#41×C₅₃	—	2	1	no	—
104	#41×C₅₃	—	2	1	no	—
105	#41×C₅₃	—	2	1	no	—
106	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₂²⋊D₅₃	—	2	2	no	—
113	C₂×C₂²⋊D₅₃	—	4	2	no	—
114	C₂²×Dic₅₃	—	2	2	no	—
115	C₂²×Dic₅₃	—	2	2	no	—
116	#11719945	—	2	2	no	—
117	#11719945	—	2	2	no	—
118	#11719945	—	2	2	no	—
119	#11719945	—	2	2	no	—
120	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
121	C₂×C₄×D₅₃	—	2	2	no	—
122	C₂×C₄×D₅₃	—	4	2	no	—
123	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
124	M₁₆×C₅₃	—	2	1	no	—
125	M₁₆×C₅₃	—	2	1	no	—
126	M₁₆×C₅₃	—	2	1	no	—
127	M₁₆×C₅₃	—	2	1	no	—
128	M₁₆×C₅₃	—	4	2	no	—
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	4	2	no	—
134	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂×D₄	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	#41	1	2	yes	—
139	C₂×C₈×C₅₃	#41	1	2	yes	—
140	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₂²⋊C₄×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	#41	1	2	yes	—
146	C₂²⋊Dic₅₃	—	4	2	no	—
147	#11719921	—	4	2	no	—
148	#11719921	—	4	2	no	—
149	C₄⋊Dic₅₃	—	2	2	no	—
150	C₄⋊Dic₅₃	—	2	2	no	—
151	C₄⋊Dic₅₃	—	2	2	no	—
152	C₄⋊Dic₅₃	—	2	2	no	—
153	#11719919	—	4	2	no	—
154	#11719919	—	4	2	no	—
155	C₄×Dic₅₃	—	4	2	no	—
156	#11719917	—	2	2	no	—
157	#11719917	—	2	2	no	—
158	C₂×#88322	—	2	2	no	—
159	Dic₂₁₂	—	2	2	no	—
160	D₄₂₄	—	2	2	no	—
161	#11719913	—	2	4	no	—
162	C₂³×C₅₃	—	2	1	no	—
163	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×D₅₃	—	2	2	no	—
165	Q₈×C₅₃	—	2	1	no	—
166	Q₈×C₅₃	—	2	1	no	—
167	D₄×C₅₃	—	2	1	no	—
168	D₄×C₅₃	—	2	1	no	—
169	D₄×C₅₃	—	4	1	no	—
170	D₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂²⋊D₅₃	—	4	2	no	—
186	C₂²⋊D₅₃	—	4	2	no	—
187	C₂²⋊D₅₃	—	16	2	no	—
188	C₂×Dic₅₃	—	2	2	no	—
189	C₂×Dic₅₃	—	4	2	no	—
190	C₂×Dic₅₃	—	4	2	no	—
191	C₂×Dic₅₃	—	8	2	no	—
192	C₂×Dic₅₃	—	8	2	no	—
193	D₂₁₂	—	2	2	no	—
194	D₂₁₂	—	2	2	no	—
195	C₄×D₅₃	—	4	2	no	—
196	C₄×D₅₃	—	4	2	no	—
197	C₄×D₅₃	—	16	2	no	—
198	Dic₁₀₆	—	2	2	no	—
199	Dic₁₀₆	—	2	2	no	—
200	C₈×C₅₃	—	2	1	no	—
201	C₈×C₅₃	—	2	1	no	—
202	C₈×C₅₃	—	4	1	no	—
203	C₈×C₅₃	—	4	1	no	—
204	C₈×C₅₃	#123	1	2	yes	—
205	C₈×C₅₃	#123	1	2	yes	—
206	#88322	—	4	2	no	—
207	#25052	—	53	1	no	maximal, Sylow
208	C₂²×C₅₃	—	2	1	no	—
209	C₂²×C₅₃	—	2	1	no	—
210	C₂²×C₅₃	—	4	1	no	—
211	C₂²×C₅₃	—	4	1	no	—
212	C₂²×C₅₃	—	4	1	no	—
213	C₂²×C₅₃	#542	1	1	yes	characteristic
214	C₂²×C₅₃	D₄²	1	1	yes	characteristic
215	C₂²×C₅₃	#553	1	1	yes	characteristic
216	C₂×D₅₃	—	4	2	no	—
217	C₂×D₅₃	—	4	2	no	—
218	C₂×D₅₃	—	4	2	no	—
219	C₄×C₅₃	—	4	2	no	—
220	C₄×C₅₃	—	4	1	no	—
221	C₄×C₅₃	—	4	1	no	—
222	C₄×C₅₃	—	4	1	no	—
223	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
224	C₄×C₅₃	#532	1	1	yes	characteristic
225	C₄×C₅₃	#547	1	1	yes	characteristic
226	C₄×C₅₃	#547	1	1	yes	characteristic
227	Dic₅₃	—	4	2	no	—
228	Dic₅₃	—	8	2	no	—
229	Dic₅₃	—	8	2	no	—
230	#3099	—	53	2	no	—
231	#3061	—	53	1	no	—
232	#2951	—	53	1	no	—
233	#2079	—	53	2	no	—
234	#2040	—	53	2	no	—
235	#1974	—	53	1	no	—
236	#1952	—	53	1	no	—
237	#1860	—	53	2	no	—
238	#1846	—	53	1	no	—
239	#1816	—	53	1	no	—
240	#1764	D₅₃	1	1	yes	characteristic, complemented
241	C₂×C₅₃	—	2	1	no	—
242	C₂×C₅₃	—	2	1	no	—
243	C₂×C₅₃	—	8	1	no	—
244	C₂×C₅₃	#2422	1	1	yes	characteristic
245	C₂×C₅₃	#3290	1	1	yes	characteristic
246	C₂×C₅₃	#3290	1	1	yes	characteristic
247	D₅₃	—	8	2	no	—
248	C₂×#135	—	53	1	no	—
249	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
250	#537	—	53	2	no	—
251	#520	—	53	2	no	—
252	#485	—	106	2	no	—
253	#483	—	106	2	no	—
254	#482	—	106	2	no	—
255	#481	—	106	2	no	—
256	#468	—	106	2	no	—
257	#467	—	106	1	no	—
258	#466	—	106	1	no	—
259	#440	—	106	2	no	—
260	C₄×Q₁₆	—	106	1	no	—
261	C₄×D₈	—	106	1	no	—
262	C₂×#108	—	53	1	no	—
263	#428	—	53	2	no	—
264	C₂×#107	—	53	1	no	—
265	C₂×#107	—	53	1	no	—
266	C₂×#106	—	53	2	no	—
267	C₂×#104	—	53	2	no	—
268	C₂×#104	C₂×D₅₃	1	1	yes	characteristic
269	#417	—	53	2	no	—
270	#417	—	53	2	no	—
271	#417	—	53	2	no	—
272	#416	C₂×D₅₃	1	1	yes	characteristic
273	#408	—	53	2	no	—
274	#405	C₂×D₅₃	1	1	yes	characteristic
275	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
276	#382	—	53	2	no	—
277	#343	—	53	2	no	—
278	#343	—	53	2	no	—
279	#341	C₂×D₅₃	1	1	yes	characteristic
280	#339	C₂×D₅₃	1	1	yes	characteristic
281	C₅₃	#25052	1	1	yes	characteristic, complemented, Sylow
282	C₂×#41	—	53	2	no	—
283	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
284	#135	—	106	2	no	—
285	#135	—	106	2	no	—
286	C₂×Q₁₆	—	53	1	no	—
287	C₂×SD₁₆	—	53	2	no	—
288	C₂×D₈	—	53	1	no	—
289	#131	—	2	2	no	—
290	#131	—	2	2	no	—
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	—	53	2	no	—
293	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
294	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
295	C₂²×C₈	—	2	1	no	—
296	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
298	C₂²⋊Q₈	—	106	2	no	—
299	C₂²⋊Q₈	—	106	2	no	—
300	#121	—	106	2	no	—
301	#121	—	106	2	no	—
302	#121	—	212	2	no	—
303	C₄×Q₈	—	106	2	no	—
304	C₄×D₄	—	106	2	no	—
305	C₄×D₄	—	212	2	no	—
306	#117	—	106	2	no	—
307	#117	C₂²×D₅₃	1	1	yes	characteristic
308	C₂×C₄⋊C₄	—	53	2	no	—
309	C₂×C₄⋊C₄	—	53	2	no	—
310	C₂×C₄⋊C₄	—	106	2	no	—
311	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
312	#108	—	106	2	no	—
313	#107	—	106	1	no	—
314	#107	—	106	1	no	—
315	#107	—	106	1	no	—
316	#107	—	106	1	no	—
317	#107	—	106	2	no	—
318	#106	—	106	2	no	—
319	#106	—	106	2	no	—
320	#106	—	106	2	no	—
321	#104	—	2	2	no	—
322	#104	—	106	4	no	—
323	#103	—	106	2	no	—
324	#103	—	106	2	no	—
325	#103	—	106	2	no	—
326	#103	D₂₁₂	1	1	yes	characteristic
327	#103	D₂₁₂	1	1	yes	characteristic
328	#102	—	106	2	no	—
329	#102	—	106	2	no	—
330	#102	—	106	2	no	—
331	#102	D₂₁₂	1	1	yes	characteristic
332	#102	D₂₁₂	1	1	yes	characteristic
333	C₂²⋊C₈	—	106	2	no	—
334	C₂²⋊C₈	—	106	2	no	—
335	#97	—	2	1	no	—
336	C₄×C₈	—	2	1	no	—
337	C₄×C₈	—	2	2	no	—
338	#95	—	212	2	no	—
339	#41	—	2	1	no	—
340	#41	—	2	1	no	—
341	#41	—	2	1	no	—
342	#41	—	2	1	no	—
343	#41	—	106	2	no	—
344	#41	—	106	2	no	—
345	#41	—	106	2	no	—
346	#41	—	106	2	no	—
347	C₂×Q₈	—	53	2	no	—
348	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	53	2	no	—
351	C₂×D₄	—	106	2	no	—
352	C₂×D₄	—	212	2	no	—
353	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	106	2	no	—
357	C₂²×C₄	—	106	2	no	—
358	C₂²×C₄	—	106	2	no	—
359	C₂²×C₄	—	212	2	no	—
360	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
361	Q₁₆	—	106	2	no	—
362	SD₁₆	—	106	4	no	—
363	D₈	—	106	2	no	—
364	M₁₆	—	2	1	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	—	2	1	no	—
368	M₁₆	—	4	2	no	—
369	M₁₆	—	106	2	no	—
370	M₁₆	—	106	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	4	2	no	—
376	C₂×C₈	—	106	2	no	—
377	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
378	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
379	C₂×C₈	#11719945	1	2	yes	—
380	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
381	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
382	C₂×C₈	#11719947	1	2	yes	—
383	C₄⋊C₄	—	106	2	no	—
384	C₄⋊C₄	—	106	2	no	—
385	C₄⋊C₄	—	106	2	no	—
386	C₄⋊C₄	—	106	2	no	—
387	C₄⋊C₄	—	212	2	no	—
388	C₄⋊C₄	—	212	2	no	—
389	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
390	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
391	C₂²⋊C₄	—	2	1	no	—
392	C₂²⋊C₄	—	212	2	no	—
393	C₂²⋊C₄	—	212	2	no	—
394	C₂²⋊C₄	—	212	2	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	212	2	no	—
398	C₄²	#11719945	1	2	yes	—
399	C₂³	—	2	1	no	—
400	C₂³	—	106	2	no	—
401	C₂³	#13133263	1	1	yes	characteristic
402	Q₈	—	2	1	no	—
403	Q₈	—	2	1	no	—
404	Q₈	—	106	2	no	—
405	Q₈	—	106	2	no	—
406	D₄	—	2	1	no	—
407	D₄	—	2	1	no	—
408	D₄	—	4	1	no	—
409	D₄	—	4	1	no	—
410	D₄	—	848	2	no	—
411	D₄	—	106	2	no	—
412	D₄	—	106	2	no	—
413	D₄	—	212	2	no	—
414	D₄	—	212	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	848	2	no	—
424	C₂×C₄	—	106	2	no	—
425	C₂×C₄	—	424	2	no	—
426	C₂×C₄	—	424	2	no	—
427	C₂×C₄	—	212	2	no	—
428	C₂×C₄	—	212	2	no	—
429	C₂×C₄	—	212	2	no	—
430	C₂×C₄	—	212	2	no	—
431	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
433	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
434	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
435	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
436	C₂×C₄	#13133264	1	1	yes	characteristic
437	C₈	—	2	1	no	—
438	C₈	—	2	1	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	1	no	—
441	C₈	—	212	2	no	—
442	C₈	#13133199	1	2	yes	—
443	C₈	#13133199	1	2	yes	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	1	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	212	2	no	—
450	C₂²	—	212	2	no	—
451	C₂²	—	212	2	no	—
452	C₂²	#16811244	1	1	yes	characteristic
453	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
454	C₂²	#16811319	1	1	yes	characteristic
455	C₄	—	4	2	no	—
456	C₄	—	4	1	no	—
457	C₄	—	4	1	no	—
458	C₄	—	4	1	no	—
459	C₄	—	424	2	no	—
460	C₄	—	424	2	no	—
461	C₄	—	212	2	no	—
462	C₄	C₂×#13133199	1	1	yes	characteristic
463	C₄	#16811173	1	1	yes	characteristic
464	C₄	#16811277	1	1	yes	characteristic
465	C₄	#16811278	1	1	yes	characteristic
466	C₂	—	2	1	no	—
467	C₂	—	2	1	no	—
468	C₂	—	8	1	no	—
469	C₂	—	424	2	no	—
470	C₂	#20278229	1	1	yes	characteristic
471	C₂	#20286502	1	1	yes	characteristic
472	C₂	#20286502	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1764	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20284385	C₂
hypercenter	#1764	D₅₃
hypocenter	C₅₃	#25052
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764

Complements

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

N	H ≅ G/N	type
1. #20284702	C₂	semidirect
3. #20283592	C₂	semidirect
4. #20276162	C₂	semidirect
5. #20275943	C₂	semidirect
6. #20275603	C₂	semidirect
7. #20275500	C₂	semidirect
8. #20274949	C₂	semidirect
9. #20274879	C₂	semidirect
10. #20274779	C₂	semidirect
11. C₅₃×#1764	C₂	semidirect
240. #1764	D₅₃	semidirect
281. C₅₃	#25052	semidirect

Maximal subgroups

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

subgroup	index
#20284702	2
#20284385	2
#20283592	2
#20276162	2
#20275943	2
#20275603	2
#20275500	2
#20274949	2
#20274879	2
#20274779	2
C₅₃×#1764	2
#25052	53