Subgroups — #27354113

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284795	C₂	1	1	yes	maximal, characteristic
2	#20283684	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273562	C₂	1	2	yes	maximal, complemented
5	#20273555	C₂	1	2	yes	maximal, complemented
6	#20273531	C₂	1	2	yes	maximal, complemented
7	#20273530	C₂	1	2	yes	maximal, complemented
8	#20273501	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273129	C₂	1	1	yes	maximal, characteristic
10	#20273002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1491	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811157	C₂²	1	1	yes	characteristic
13	#16810953	C₄	1	1	yes	characteristic
14	#16810936	C₄	1	1	yes	characteristic
15	#16810930	C₄	1	2	yes	—
16	#16810845	C₄	1	2	yes	—
17	#16810779	C₄	1	1	yes	characteristic
18	#16810762	C₄	1	1	yes	characteristic
19	#16810750	C₂²	1	1	yes	characteristic
20	Q₈×#88322	—	2	1	no	—
21	Q₈×#88322	—	2	1	no	—
22	Q₈×#88322	C₂²	1	2	yes	—
23	Q₈×#88322	C₂²	1	2	yes	—
24	#16810747	C₂²	1	1	yes	characteristic
25	C₄×#11719925	C₂²	1	2	yes	—
26	C₄×#11719924	C₂²	1	2	yes	—
27	#16810700	C₂²	1	1	yes	characteristic
28	#16810691	—	2	1	no	—
29	#16810688	C₂²	1	1	yes	characteristic
30	#16810686	—	2	1	no	—
31	#16810685	—	2	1	no	—
32	#16810676	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊#88322	—	2	1	no	—
34	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
35	#16810580	—	2	1	no	—
36	C₂×C₄×#88322	—	2	1	no	—
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	#16810375	C₂²	1	2	yes	—
39	#16810373	C₂²	1	2	yes	—
40	#16810355	C₂²	1	2	yes	—
41	#16810352	C₂²	1	2	yes	—
42	#16810345	C₂²	1	2	yes	—
43	#16810345	C₂²	1	2	yes	—
44	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#432	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₂×#11719925	C₂×C₄	1	2	yes	—
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×#11719912	—	2	1	no	—
52	C₂²×#88322	—	2	1	no	—
53	C₂²×#88322	D₄	1	1	yes	characteristic
54	C₂²×#88322	D₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	—	2	1	no	—
56	#13133211	—	2	1	no	—
57	#13133209	C₂×C₄	1	1	yes	characteristic
58	#13133205	C₂×C₄	1	1	yes	characteristic
59	#13133200	—	2	1	no	—
60	#13133196	C₂×C₄	1	1	yes	characteristic
61	#13133191	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
63	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
64	C₄×C₂²⋊D₅₃	—	2	1	no	—
65	#13133174	—	2	1	no	—
66	#13133150	C₂×C₄	1	2	yes	—
67	#13133148	C₂×C₄	1	2	yes	—
68	C₄⋊#88322	—	2	1	no	—
69	C₄⋊#88322	—	2	2	no	—
70	C₄⋊#88322	—	2	2	no	—
71	C₄⋊#88322	C₂×C₄	1	2	yes	—
72	C₄⋊#88322	C₂³	1	2	yes	—
73	#13133146	—	2	1	no	—
74	#13133146	—	2	1	no	—
75	#13133141	C₂×C₄	1	2	yes	—
76	#13133141	C₂×C₄	1	2	yes	—
77	#13133140	C₂×C₄	1	2	yes	—
78	#13133137	C₂×C₄	1	2	yes	—
79	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
81	#13133129	—	2	1	no	—
82	#13133129	—	2	1	no	—
83	#13133127	—	2	1	no	—
84	C₄×#88322	—	2	2	no	—
85	C₄×#88322	—	2	2	no	—
86	C₄×#88322	D₄	1	1	yes	characteristic
87	C₄×#88322	D₄	1	1	yes	characteristic
88	C₄×#88322	C₂³	1	2	yes	—
89	C₄×#88322	C₂³	1	1	yes	characteristic
90	C₄×#88322	C₂³	1	1	yes	characteristic
91	C₈×Dic₅₃	—	2	1	no	—
92	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×M₁₆×C₅₃	—	2	1	no	—
94	C₄×Q₈×C₅₃	—	2	1	no	—
95	C₄×Q₈×C₅₃	—	2	1	no	—
96	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
97	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	—	2	1	no	—
101	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
104	C₂²⋊C₈×C₅₃	—	2	1	no	—
105	#97×C₅₃	—	2	1	no	—
106	#97×C₅₃	—	2	1	no	—
107	C₄×C₈×C₅₃	C₂³	1	2	yes	—
108	C₂×Q₈×C₅₃	—	2	4	no	—
109	C₂×Q₈×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
116	C₂×C₂²⋊D₅₃	—	2	1	no	—
117	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	—	2	1	no	—
119	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
120	M₁₆×C₅₃	—	4	2	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	#41	1	1	yes	characteristic
133	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂²⋊Dic₅₃	—	2	1	no	—
135	#11719925	—	2	4	no	—
136	#11719924	—	2	4	no	—
137	#11719921	—	2	1	no	—
138	#11719921	—	2	1	no	—
139	#11719921	—	2	1	no	—
140	#11719921	—	2	1	no	—
141	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
143	#11719919	—	2	1	no	—
144	#11719919	—	2	1	no	—
145	#11719919	—	2	1	no	—
146	#11719919	—	2	1	no	—
147	C₄×Dic₅₃	—	2	1	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	—	2	2	no	—
150	C₂×#88322	—	4	1	no	—
151	C₂×#88322	—	4	2	no	—
152	C₂×#88322	C₂²×C₄	1	2	yes	—
153	C₂×#88322	C₂²×C₄	1	2	yes	—
154	C₂×#88322	C₂×D₄	1	1	yes	characteristic
155	C₂×#88322	C₂×D₄	1	1	yes	characteristic
156	C₂×#88322	C₂×D₄	1	1	yes	characteristic
157	C₂×#88322	C₂×D₄	1	1	yes	characteristic
158	#11719912	—	4	2	no	—
159	C₈×D₅₃	—	4	2	no	—
160	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₂²×D₅₃	—	2	1	no	—
162	Q₈×C₅₃	—	2	4	no	—
163	Q₈×C₅₃	—	2	2	no	—
164	Q₈×C₅₃	#131	1	4	yes	—
165	C₂×C₄×C₅₃	—	2	4	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₂²⋊D₅₃	—	4	2	no	—
180	C₂×Dic₅₃	—	2	1	no	—
181	C₂×Dic₅₃	—	2	1	no	—
182	C₂×Dic₅₃	—	2	1	no	—
183	D₂₁₂	—	2	2	no	—
184	C₄×D₅₃	—	4	1	no	—
185	C₄×D₅₃	—	4	1	no	—
186	Dic₁₀₆	—	2	2	no	—
187	C₈×C₅₃	—	4	2	no	—
188	C₈×C₅₃	—	4	1	no	—
189	C₈×C₅₃	—	4	1	no	—
190	#88322	—	2	2	no	—
191	#88322	—	2	2	no	—
192	#88322	—	2	1	no	—
193	#88322	—	2	1	no	—
194	#88322	—	2	1	no	—
195	#88322	—	4	2	no	—
196	#88322	#137	1	2	yes	—
197	#17915	—	53	1	no	maximal, Sylow
198	C₂²×C₅₃	—	2	1	no	—
199	C₂²×C₅₃	—	2	1	no	—
200	C₂²×C₅₃	#444	1	2	yes	—
201	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂×D₅₃	—	4	1	no	—
203	C₂×D₅₃	—	4	1	no	—
204	C₂×D₅₃	—	4	1	no	—
205	C₄×C₅₃	—	2	4	no	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	2	1	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	4	2	no	—
211	C₄×C₅₃	C₂×#131	1	2	yes	—
212	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
213	C₄×C₅₃	#575	1	1	yes	characteristic
214	Dic₅₃	—	4	1	no	—
215	Dic₅₃	—	4	1	no	—
216	Dic₅₃	—	4	1	no	—
217	#3108	—	53	1	no	—
218	#2966	—	53	1	no	—
219	C₂×C₈×Q₈	—	53	1	no	—
220	#1584	—	53	2	no	—
221	#1583	—	53	2	no	—
222	#1579	—	53	2	no	—
223	SD₁₆⋊C₈	—	53	2	no	—
224	#1573	—	53	1	no	—
225	#1491	D₅₃	1	1	yes	characteristic, complemented
226	#1483	—	53	1	no	—
227	#1451	—	53	1	no	—
228	C₂×C₅₃	—	2	2	no	—
229	C₂×C₅₃	#2929	1	1	yes	characteristic
230	C₂×C₅₃	C₄×#137	1	1	yes	characteristic
231	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
232	D₅₃	—	8	1	no	—
233	#529	—	53	1	no	—
234	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
235	#483	—	53	1	no	—
236	#480	—	53	1	no	—
237	#479	—	53	2	no	—
238	#464	—	53	2	no	—
239	#451	—	53	1	no	—
240	#448	—	53	1	no	—
241	#446	—	53	1	no	—
242	C₈×Q₈	—	53	2	no	—
243	C₈×Q₈	—	53	2	no	—
244	C₈×Q₈	—	53	1	no	—
245	C₈×Q₈	—	106	1	no	—
246	C₈×Q₈	—	106	1	no	—
247	C₄×Q₁₆	—	53	2	no	—
248	C₄×SD₁₆	—	53	2	no	—
249	#436	—	53	1	no	—
250	#435	—	106	1	no	—
251	C₈×D₄	—	53	1	no	—
252	C₈×D₄	—	106	1	no	—
253	#433	—	106	1	no	—
254	#432	C₂×D₅₃	1	1	yes	characteristic
255	#431	—	53	1	no	—
256	C₂×C₄⋊C₈	—	53	1	no	—
257	C₂×C₄⋊C₈	—	106	1	no	—
258	#405	—	106	1	no	—
259	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
260	C₂×C₄×C₈	—	53	1	no	—
261	C₂×C₄×C₈	—	106	1	no	—
262	#335	—	53	2	no	—
263	#334	—	53	2	no	—
264	Q₈⋊C₈	—	53	2	no	—
265	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
266	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
267	D₄⋊C₈	—	53	2	no	—
268	#322	—	53	2	no	—
269	#322	—	53	2	no	—
270	C₅₃	#17915	1	1	yes	characteristic, complemented, Sylow
271	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
272	C₂×Q₁₆	—	53	2	no	—
273	C₂×SD₁₆	—	53	2	no	—
274	C₂×M₁₆	—	2	1	no	—
275	C₂×M₁₆	—	106	1	no	—
276	C₂²×C₈	—	53	1	no	—
277	C₂²×C₈	—	53	1	no	—
278	C₂²×C₈	—	106	1	no	—
279	C₂²×C₈	—	106	1	no	—
280	#126	—	106	1	no	—
281	#125	—	53	1	no	—
282	#124	—	53	1	no	—
283	#123	—	106	1	no	—
284	C₂²⋊Q₈	—	53	1	no	—
285	#121	—	53	1	no	—
286	C₄×Q₈	—	2	1	no	—
287	C₄×Q₈	—	2	1	no	—
288	C₄×Q₈	—	53	1	no	—
289	C₄×Q₈	C₄×D₅₃	1	2	yes	—
290	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
291	C₄×D₄	—	53	1	no	—
292	C₄×D₄	—	106	1	no	—
293	#117	—	106	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
298	#107	—	53	2	no	—
299	#106	—	53	2	no	—
300	C₄⋊C₈	—	53	2	no	—
301	C₄⋊C₈	—	53	2	no	—
302	C₄⋊C₈	—	106	2	no	—
303	C₄⋊C₈	—	106	2	no	—
304	C₄⋊C₈	—	106	1	no	—
305	C₄⋊C₈	—	106	1	no	—
306	C₄⋊C₈	—	106	1	no	—
307	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
308	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
309	#103	—	53	2	no	—
310	#103	—	53	2	no	—
311	#103	—	53	2	no	—
312	#102	—	53	2	no	—
313	C₂²⋊C₈	—	2	1	no	—
314	C₂²⋊C₈	—	53	1	no	—
315	C₂²⋊C₈	—	53	1	no	—
316	C₂²⋊C₈	—	106	1	no	—
317	C₂²⋊C₈	—	106	1	no	—
318	#97	—	2	1	no	—
319	#97	—	2	1	no	—
320	#97	—	106	1	no	—
321	C₄×C₈	—	53	1	no	—
322	C₄×C₈	—	53	1	no	—
323	C₄×C₈	—	53	2	no	—
324	C₄×C₈	—	53	1	no	—
325	C₄×C₈	—	53	1	no	—
326	C₄×C₈	—	106	2	no	—
327	C₄×C₈	—	106	2	no	—
328	C₄×C₈	—	106	1	no	—
329	C₄×C₈	C₂²×D₅₃	1	2	yes	—
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	53	1	no	—
333	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
334	C₂×D₄	—	53	1	no	—
335	C₂×D₄	—	106	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	106	1	no	—
339	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
340	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
341	C₂²×C₄	#11719947	1	1	yes	characteristic
342	Q₁₆	—	106	4	no	—
343	SD₁₆	—	106	4	no	—
344	M₁₆	—	4	2	no	—
345	M₁₆	—	212	2	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	53	1	no	—
350	C₂×C₈	—	53	2	no	—
351	C₂×C₈	—	53	2	no	—
352	C₂×C₈	—	53	1	no	—
353	C₂×C₈	—	53	1	no	—
354	C₂×C₈	—	53	1	no	—
355	C₂×C₈	—	106	2	no	—
356	C₂×C₈	—	106	2	no	—
357	C₂×C₈	—	212	2	no	—
358	C₂×C₈	—	212	1	no	—
359	C₂×C₈	—	212	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	53	1	no	—
364	C₄⋊C₄	—	53	1	no	—
365	C₄⋊C₄	—	106	1	no	—
366	C₄⋊C₄	—	106	1	no	—
367	C₄⋊C₄	—	106	1	no	—
368	C₄⋊C₄	—	106	1	no	—
369	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
370	C₂²⋊C₄	—	106	1	no	—
371	C₂²⋊C₄	—	106	1	no	—
372	C₂²⋊C₄	—	106	1	no	—
373	C₂²⋊C₄	—	106	1	no	—
374	C₂²⋊C₄	—	106	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	106	1	no	—
378	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
379	C₄²	D₄×D₅₃	1	1	yes	characteristic
380	C₄²	#11719947	1	1	yes	characteristic
381	C₄²	C₂³×D₅₃	1	1	yes	characteristic
382	C₂³	—	106	1	no	—
383	C₂³	#13133177	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	2	no	—
386	Q₈	—	106	2	no	—
387	Q₈	#13133225	1	4	yes	—
388	D₄	—	106	2	no	—
389	D₄	—	212	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	106	1	no	—
399	C₂×C₄	—	106	1	no	—
400	C₂×C₄	—	106	1	no	—
401	C₂×C₄	—	212	1	no	—
402	C₂×C₄	—	212	1	no	—
403	C₂×C₄	#13133177	1	1	yes	characteristic
404	C₂×C₄	#13133177	1	1	yes	characteristic
405	C₂×C₄	#13133177	1	1	yes	characteristic
406	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
407	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
408	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
409	C₈	—	4	2	no	—
410	C₈	—	4	1	no	—
411	C₈	—	4	1	no	—
412	C₈	—	53	2	no	—
413	C₈	—	106	2	no	—
414	C₈	—	106	2	no	—
415	C₈	—	106	1	no	—
416	C₈	—	106	1	no	—
417	C₈	—	106	1	no	—
418	C₈	—	212	2	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	212	1	no	—
422	C₂²	—	212	1	no	—
423	C₂²	—	212	1	no	—
424	C₂²	#16810742	1	2	yes	—
425	C₂²	C₂×#13133177	1	1	yes	characteristic
426	C₄	—	2	4	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	4	2	no	—
432	C₄	—	212	1	no	—
433	C₄	—	212	1	no	—
434	C₄	—	212	1	no	—
435	C₄	C₂×#13133225	1	2	yes	—
436	C₄	#137×D₅₃	1	1	yes	characteristic
437	C₄	#16811422	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	424	1	no	—
440	C₂	#20283405	1	1	yes	characteristic
441	C₂	#20283530	1	1	yes	characteristic
442	C₂	C₂×#16810742	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×#1491	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#13133191	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1491	D₅₃
hypocenter	C₅₃	#17915
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1491

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈×#88322	C₂	semidirect
4. #20273562	C₂	semidirect
5. #20273555	C₂	semidirect
6. #20273531	C₂	semidirect
7. #20273530	C₂	semidirect
8. #20273501	C₂	semidirect
10. #20273002	C₂	semidirect
11. C₅₃×#1491	C₂	semidirect
225. #1491	D₅₃	semidirect
270. C₅₃	#17915	semidirect

Maximal subgroups

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

subgroup	index
#20284795	2
#20283684	2
C₂×Q₈×#88322	2
#20273562	2
#20273555	2
#20273531	2
#20273530	2
#20273501	2
#20273129	2
#20273002	2
C₅₃×#1491	2
#17915	53