Subgroups — #27278675

Order 13568.

584 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	C₂×#16810747	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#11719913	C₂	1	1	yes	maximal, characteristic
3	#20275508	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810353	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810350	C₂	1	1	yes	maximal, characteristic
6	#20272542	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719913	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#16810747	—	2	2	no	—
12	#16810747	—	2	2	no	—
13	C₄×#11719913	—	2	2	no	—
14	C₄×#11719913	—	2	2	no	—
15	C₂×#13133149	C₄	1	1	yes	characteristic
16	C₂×#13133146	—	2	1	no	—
17	C₂×#13133142	C₄	1	1	yes	characteristic
18	C₂×#13133139	C₄	1	1	yes	characteristic
19	C₂×#13133129	—	2	1	no	—
20	C₂×#13133129	—	2	1	no	—
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	—	2	1	no	—
24	#16810353	—	2	2	no	—
25	#16810350	—	2	2	no	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₅₃	—	2	1	no	—
32	C₂×#11719913	—	2	2	no	—
33	C₂×#11719913	—	2	2	no	—
34	C₂×#11719913	C₈	1	4	yes	complemented
35	C₂²×#88322	—	2	1	no	—
36	C₂²×#88322	D₄	1	1	yes	characteristic
37	C₂²×#88322	D₄	1	1	yes	characteristic
38	C₄×Dic₁₀₆	—	2	1	no	—
39	C₄×Dic₁₀₆	—	2	1	no	—
40	C₄×Dic₁₀₆	D₄	1	2	yes	—
41	C₄×Dic₁₀₆	D₄	1	2	yes	—
42	C₄×D₂₁₂	—	2	1	no	—
43	C₄×D₂₁₂	—	2	1	no	—
44	C₄×D₂₁₂	—	2	1	no	—
45	C₄×D₂₁₂	—	2	1	no	—
46	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719919	—	2	1	no	—
49	C₂×#11719921	—	2	1	no	—
50	C₂×C₄×Dic₅₃	—	2	1	no	—
51	#13133149	—	2	2	no	—
52	#13133146	—	2	4	no	—
53	#13133142	—	2	2	no	—
54	#13133139	—	2	2	no	—
55	#13133129	—	4	2	no	—
56	#13133129	—	4	2	no	—
57	C₄×#88322	—	2	1	no	—
58	C₄×#88322	—	2	1	no	—
59	C₄×#88322	—	2	1	no	—
60	C₄×#88322	—	2	1	no	—
61	C₈×Dic₅₃	—	2	4	no	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	—	2	1	no	—
68	C₄⋊C₈×C₅₃	—	2	1	no	—
69	C₄⋊C₈×C₅₃	—	2	1	no	—
70	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
71	C₄×C₈×C₅₃	—	2	1	no	—
72	C₄×C₈×C₅₃	—	2	1	no	—
73	C₂³×D₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
77	C₂²×Dic₅₃	—	2	1	no	—
78	C₂²×Dic₅₃	—	2	1	no	—
79	C₂²×Dic₅₃	—	2	1	no	—
80	C₂×D₂₁₂	—	2	1	no	—
81	C₂×D₂₁₂	—	2	1	no	—
82	C₂×D₂₁₂	—	2	1	no	—
83	C₂×D₂₁₂	—	2	1	no	—
84	C₂×D₂₁₂	—	2	1	no	—
85	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₅₃	—	4	1	no	—
87	C₂×C₄×D₅₃	—	4	1	no	—
88	C₂×C₄×D₅₃	—	4	1	no	—
89	C₂×C₄×D₅₃	—	4	1	no	—
90	C₂×C₄×D₅₃	—	4	1	no	—
91	C₂×C₄×D₅₃	—	4	1	no	—
92	C₂×Dic₁₀₆	—	2	1	no	—
93	C₂×Dic₁₀₆	—	2	1	no	—
94	C₂×Dic₁₀₆	—	2	1	no	—
95	C₂×Dic₁₀₆	C₂²⋊C₄	1	2	yes	—
96	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
97	C₂×Dic₁₀₆	M₁₆	1	2	yes	—
98	C₂×C₈×C₅₃	—	2	1	no	—
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	4	1	no	—
105	C₂×C₈×C₅₃	—	4	1	no	—
106	C₂×C₈×C₅₃	—	4	1	no	—
107	C₂×C₈×C₅₃	—	4	1	no	—
108	C₂×C₈×C₅₃	—	4	1	no	—
109	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	#41	1	1	yes	characteristic
114	#11719921	—	4	1	no	—
115	#11719921	—	4	1	no	—
116	C₄⋊Dic₅₃	—	2	1	no	—
117	C₄⋊Dic₅₃	—	2	1	no	—
118	C₄⋊Dic₅₃	—	2	1	no	—
119	C₄⋊Dic₅₃	C₂²⋊C₄	1	2	yes	—
120	#11719919	—	2	2	no	—
121	#11719919	—	2	2	no	—
122	C₄×Dic₅₃	—	2	2	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₂×#88322	—	2	1	no	—
125	C₂×#88322	—	2	1	no	—
126	C₂×#88322	—	2	2	no	—
127	C₂×#88322	—	2	1	no	—
128	C₂×#88322	—	2	1	no	—
129	C₂×#88322	—	2	1	no	—
130	C₂×#88322	—	4	1	no	—
131	C₂×#88322	—	4	1	no	—
132	C₂×#88322	SD₁₆	1	2	yes	complemented
133	#11719913	—	2	8	no	—
134	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
135	C₂²×D₅₃	—	2	2	no	—
136	C₂²×D₅₃	—	4	1	no	—
137	C₂²×D₅₃	—	4	1	no	—
138	C₂²×D₅₃	—	4	1	no	—
139	C₂²×D₅₃	—	4	1	no	—
140	C₂²×D₅₃	—	4	1	no	—
141	C₂²×D₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
156	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
158	C₂×Dic₅₃	—	2	2	no	—
159	C₂×Dic₅₃	—	2	2	no	—
160	C₂×Dic₅₃	—	2	2	no	—
161	C₂×Dic₅₃	—	2	2	no	—
162	C₂×Dic₅₃	—	4	1	no	—
163	C₂×Dic₅₃	—	4	1	no	—
164	C₂×Dic₅₃	—	4	1	no	—
165	C₂×Dic₅₃	—	4	1	no	—
166	C₂×Dic₅₃	—	4	1	no	—
167	D₂₁₂	—	2	2	no	—
168	D₂₁₂	—	2	2	no	—
169	D₂₁₂	—	4	1	no	—
170	D₂₁₂	—	4	1	no	—
171	C₄×D₅₃	—	8	1	no	—
172	C₄×D₅₃	—	8	1	no	—
173	C₄×D₅₃	—	8	1	no	—
174	C₄×D₅₃	—	8	1	no	—
175	Dic₁₀₆	—	2	2	no	—
176	Dic₁₀₆	—	2	2	no	—
177	Dic₁₀₆	—	2	2	no	—
178	Dic₁₀₆	C₂²⋊C₈	1	4	yes	complemented
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	4	1	no	—
181	C₈×C₅₃	—	4	2	no	—
182	C₈×C₅₃	—	8	1	no	—
183	#88322	—	2	2	no	—
184	#88322	—	4	2	no	—
185	#88322	—	4	1	no	—
186	#8948	—	53	1	no	maximal, Sylow
187	C₂²×C₅₃	#386	1	1	yes	characteristic
188	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
189	C₂²×C₅₃	#435	1	1	yes	characteristic
190	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
191	C₂²×C₅₃	#440	1	1	yes	characteristic
192	C₂²×C₅₃	#443	1	1	yes	characteristic
193	C₂²×C₅₃	#444	1	1	yes	characteristic
194	C₂×D₅₃	—	4	2	no	—
195	C₂×D₅₃	—	4	2	no	—
196	C₂×D₅₃	—	4	2	no	—
197	C₂×D₅₃	—	8	1	no	—
198	C₂×D₅₃	—	8	1	no	—
199	C₂×D₅₃	—	8	1	no	—
200	C₂×D₅₃	—	8	1	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
204	C₄×C₅₃	#408	1	1	yes	characteristic
205	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
206	C₄×C₅₃	#435	1	1	yes	characteristic
207	C₄×C₅₃	#448	1	1	yes	characteristic
208	C₄×C₅₃	#449	1	1	yes	characteristic
209	C₄×C₅₃	#460	1	1	yes	characteristic
210	C₄×C₅₃	#464	1	1	yes	characteristic
211	Dic₅₃	—	2	4	no	—
212	Dic₅₃	—	4	2	no	—
213	Dic₅₃	—	8	1	no	—
214	C₂×C₈×Q₈	—	53	1	no	—
215	C₂×C₄×SD₁₆	—	53	1	no	—
216	#1954	—	53	1	no	—
217	C₂×Q₈⋊C₈	—	53	1	no	—
218	C₂×D₄⋊C₈	—	53	1	no	—
219	#1277	D₅₃	1	1	yes	characteristic, complemented
220	#1273	—	53	1	no	—
221	C₂×C₅₃	C₈×SD₁₆	1	1	yes	characteristic
222	C₂×C₅₃	SD₁₆⋊C₈	1	1	yes	characteristic
223	C₂×C₅₃	#1583	1	1	yes	characteristic
224	C₂×C₅₃	#1588	1	1	yes	characteristic
225	C₂×C₅₃	#1869	1	1	yes	characteristic
226	C₂×C₅₃	#1876	1	1	yes	characteristic
227	C₂×C₅₃	#1880	1	1	yes	characteristic
228	D₅₃	—	8	2	no	—
229	C₂²×SD₁₆	—	53	1	no	—
230	C₂×C₄×Q₈	—	53	1	no	—
231	C₂×C₄×D₄	—	53	1	no	—
232	C₈×Q₈	—	106	2	no	—
233	C₈×Q₈	—	106	2	no	—
234	C₄×SD₁₆	—	106	2	no	—
235	C₄×SD₁₆	—	106	2	no	—
236	C₂×#106	—	53	1	no	—
237	C₂×C₄⋊C₈	—	106	1	no	—
238	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
239	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
240	C₂×#103	—	53	1	no	—
241	C₂×#102	—	53	1	no	—
242	C₂×C₂²⋊C₈	—	106	1	no	—
243	C₂×C₂²⋊C₈	—	106	1	no	—
244	C₂×C₄×C₈	—	53	1	no	—
245	C₂×C₄×C₈	—	53	1	no	—
246	C₂×C₄×C₈	—	106	1	no	—
247	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
248	Q₈⋊C₈	—	106	2	no	—
249	D₄⋊C₈	—	106	2	no	—
250	C₅₃	#8948	1	1	yes	characteristic, complemented, Sylow
251	C₂²×Q₈	—	53	1	no	—
252	C₂²×D₄	—	53	1	no	—
253	C₂³×C₄	—	106	1	no	—
254	C₂×SD₁₆	—	53	4	no	—
255	C₂×SD₁₆	—	106	2	no	—
256	C₂×SD₁₆	—	106	2	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₂²×C₈	—	53	1	no	—
260	C₂²×C₈	—	53	1	no	—
261	C₂²×C₈	—	106	1	no	—
262	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
263	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
264	C₄×Q₈	—	53	2	no	—
265	C₄×Q₈	—	53	2	no	—
266	C₄×Q₈	—	106	1	no	—
267	C₄×Q₈	—	106	1	no	—
268	C₄×D₄	—	106	1	no	—
269	C₄×D₄	—	106	1	no	—
270	C₄×D₄	—	106	1	no	—
271	C₄×D₄	—	106	1	no	—
272	C₂×C₄⋊C₄	—	53	1	no	—
273	C₂×C₄⋊C₄	—	53	1	no	—
274	C₂×C₄⋊C₄	—	106	1	no	—
275	C₂×C₂²⋊C₄	—	106	1	no	—
276	C₂×C₄²	—	106	1	no	—
277	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
278	#106	—	106	2	no	—
279	C₄⋊C₈	—	2	1	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	106	4	no	—
283	C₄⋊C₈	D₂₁₂	1	2	yes	—
284	#103	—	106	2	no	—
285	#102	—	106	2	no	—
286	C₂²⋊C₈	—	212	2	no	—
287	C₂²⋊C₈	—	212	2	no	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	106	4	no	—
291	C₄×C₈	—	106	1	no	—
292	C₄×C₈	—	106	1	no	—
293	C₄×C₈	—	106	1	no	—
294	C₄×C₈	—	106	1	no	—
295	C₂⁴	—	106	1	no	—
296	C₂×Q₈	—	53	2	no	—
297	C₂×Q₈	—	53	2	no	—
298	C₂×Q₈	—	53	2	no	—
299	C₂×Q₈	—	106	1	no	—
300	C₂×Q₈	—	106	1	no	—
301	C₂×Q₈	—	106	1	no	—
302	C₂×D₄	—	53	2	no	—
303	C₂×D₄	—	106	1	no	—
304	C₂×D₄	—	106	1	no	—
305	C₂×D₄	—	106	1	no	—
306	C₂×D₄	—	106	1	no	—
307	C₂×D₄	—	106	1	no	—
308	C₂²×C₄	—	106	1	no	—
309	C₂²×C₄	—	106	1	no	—
310	C₂²×C₄	—	106	1	no	—
311	C₂²×C₄	—	212	1	no	—
312	C₂²×C₄	—	212	1	no	—
313	C₂²×C₄	—	212	1	no	—
314	C₂²×C₄	—	212	1	no	—
315	C₂²×C₄	—	212	1	no	—
316	C₂²×C₄	—	212	1	no	—
317	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
318	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
319	C₂²×C₄	#11719949	1	1	yes	characteristic
320	SD₁₆	—	106	8	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	4	1	no	—
328	C₂×C₈	—	4	1	no	—
329	C₂×C₈	—	4	1	no	—
330	C₂×C₈	—	4	1	no	—
331	C₂×C₈	—	4	1	no	—
332	C₂×C₈	—	53	2	no	—
333	C₂×C₈	—	106	1	no	—
334	C₂×C₈	—	106	1	no	—
335	C₂×C₈	—	106	2	no	—
336	C₂×C₈	—	106	1	no	—
337	C₂×C₈	—	106	1	no	—
338	C₂×C₈	—	106	1	no	—
339	C₂×C₈	—	212	1	no	—
340	C₂×C₈	—	212	1	no	—
341	C₂×C₈	C₈×D₅₃	1	2	yes	—
342	C₄⋊C₄	—	53	2	no	—
343	C₄⋊C₄	—	106	1	no	—
344	C₄⋊C₄	—	106	2	no	—
345	C₄⋊C₄	—	106	2	no	—
346	C₄⋊C₄	—	106	1	no	—
347	C₄⋊C₄	—	106	1	no	—
348	C₂²⋊C₄	—	212	1	no	—
349	C₂²⋊C₄	—	212	1	no	—
350	C₄²	—	106	2	no	—
351	C₄²	—	106	2	no	—
352	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
353	C₄²	#11719945	1	1	yes	characteristic
354	C₄²	D₄×D₅₃	1	1	yes	characteristic
355	C₄²	D₄×D₅₃	1	1	yes	characteristic
356	C₂³	—	106	2	no	—
357	C₂³	—	212	1	no	—
358	C₂³	—	212	1	no	—
359	C₂³	—	212	1	no	—
360	C₂³	—	212	1	no	—
361	C₂³	—	212	1	no	—
362	C₂³	—	212	1	no	—
363	C₂³	#13133179	1	1	yes	characteristic
364	Q₈	—	53	4	no	—
365	Q₈	—	106	2	no	—
366	Q₈	—	106	2	no	—
367	Q₈	—	106	2	no	—
368	D₄	—	106	2	no	—
369	D₄	—	106	2	no	—
370	D₄	—	212	1	no	—
371	D₄	—	212	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	106	2	no	—
375	C₂×C₄	—	106	2	no	—
376	C₂×C₄	—	106	2	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	—	424	1	no	—
379	C₂×C₄	—	424	1	no	—
380	C₂×C₄	—	424	1	no	—
381	C₂×C₄	—	424	1	no	—
382	C₂×C₄	—	212	1	no	—
383	C₂×C₄	—	212	1	no	—
384	C₂×C₄	—	212	1	no	—
385	C₂×C₄	—	212	1	no	—
386	C₂×C₄	—	212	1	no	—
387	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
388	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
389	C₂×C₄	#13133187	1	1	yes	characteristic
390	C₂×C₄	#13133190	1	1	yes	characteristic
391	C₂×C₄	#13133192	1	1	yes	characteristic
392	C₂×C₄	#13133202	1	1	yes	characteristic
393	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
394	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
395	C₂×C₄	#13133223	1	1	yes	characteristic
396	C₂×C₄	#13133225	1	1	yes	characteristic
397	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
398	C₂×C₄	#13133238	1	1	yes	characteristic
399	C₂×C₄	#13133241	1	1	yes	characteristic
400	C₂×C₄	#13133249	1	1	yes	characteristic
401	C₈	—	2	2	no	—
402	C₈	—	4	2	no	—
403	C₈	—	4	1	no	—
404	C₈	—	8	1	no	—
405	C₈	—	106	2	no	—
406	C₈	—	212	2	no	—
407	C₈	—	212	1	no	—
408	C₂²	—	424	1	no	—
409	C₂²	—	424	1	no	—
410	C₂²	—	424	1	no	—
411	C₂²	—	424	1	no	—
412	C₂²	—	212	2	no	—
413	C₂²	—	212	2	no	—
414	C₂²	—	212	2	no	—
415	C₂²	#16810501	1	1	yes	characteristic
416	C₂²	#16810688	1	1	yes	characteristic
417	C₂²	#16810693	1	1	yes	characteristic
418	C₂²	#16810710	1	1	yes	characteristic
419	C₂²	#16810722	1	1	yes	characteristic
420	C₂²	#16810739	1	1	yes	characteristic
421	C₂²	#16810744	1	1	yes	characteristic
422	C₄	—	2	1	no	—
423	C₄	—	2	1	no	—
424	C₄	—	106	4	no	—
425	C₄	—	424	1	no	—
426	C₄	—	212	2	no	—
427	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
428	C₄	#16810588	1	1	yes	characteristic
429	C₄	C₈×D₂₁₂	1	1	yes	characteristic
430	C₄	#16810696	1	1	yes	characteristic
431	C₄	#16810763	1	1	yes	characteristic
432	C₄	#16810769	1	1	yes	characteristic
433	C₄	#16810822	1	1	yes	characteristic
434	C₄	#16810848	1	1	yes	characteristic
435	C₂	—	424	2	no	—
436	C₂	#20273514	1	1	yes	characteristic
437	C₂	#20273526	1	1	yes	characteristic
438	C₂	#20273557	1	1	yes	characteristic
439	C₂	#20273592	1	1	yes	characteristic
440	C₂	#20275006	1	1	yes	characteristic
441	C₂	#20275043	1	1	yes	characteristic
442	C₂	#20275071	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1277	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1277	D₅₃
hypocenter	C₅₃	#8948
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
1. C₂×#16810747	C₂	semidirect
4. C₂×#16810353	C₂	semidirect
6. #20272542	C₂	semidirect
7. C₅₃×#1277	C₂	semidirect
34. C₂×#11719913	C₈	semidirect
132. C₂×#88322	SD₁₆	semidirect
178. Dic₁₀₆	C₂²⋊C₈	semidirect
219. #1277	D₅₃	semidirect
250. C₅₃	#8948	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810747	2
C₂×C₄×#11719913	2
#20275508	2
C₂×#16810353	2
C₂×#16810350	2
#20272542	2
C₅₃×#1277	2
#8948	53