Subgroups — D₅₃⋊#1922 #28319936

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#424	C₂	1	1	yes	maximal, characteristic
2	#20291275	C₂	1	1	yes	maximal, characteristic
3	#20290783	C₂	1	2	yes	maximal, complemented
4	#20290782	C₂	1	2	yes	maximal, complemented
5	#20290781	C₂	1	2	yes	maximal, complemented
6	#20290780	C₂	1	2	yes	maximal, complemented
7	#20290407	C₂	1	1	yes	maximal, characteristic
8	#20290391	C₂	1	1	yes	maximal, characteristic
9	#20290391	C₂	1	1	yes	maximal, characteristic
10	C₂×(C₂×#6157)⋊C₈	C₂	1	1	yes	maximal, characteristic
11	#427×D₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
13	C₂×D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₄×#6157	C₂²	1	1	yes	characteristic
16	#16811630	C₂²	1	2	yes	—
17	#16811630	C₂²	1	2	yes	—
18	#16811629	—	2	2	no	—
19	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
20	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
21	#16811588	C₂²	1	2	yes	—
22	#16811588	C₂²	1	2	yes	—
23	#16811587	—	2	2	no	—
24	#16811582	—	2	1	no	—
25	C₄×D₅₃⋊C₈	—	2	1	no	—
26	#16811499	C₂²	1	2	yes	—
27	#16811499	C₂²	1	2	yes	—
28	#16811493	C₂²	1	2	yes	—
29	#16811493	C₂²	1	2	yes	—
30	#16811493	C₂²	1	2	yes	—
31	#16811493	C₂²	1	2	yes	—
32	(C₂×#6157)⋊C₈	C₂²	1	2	yes	—
33	(C₂×#6157)⋊C₈	C₂²	1	2	yes	—
34	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	#117×D₅₃	C₂²	1	1	yes	characteristic
36	#117×D₅₃	C₂²	1	1	yes	characteristic
37	#16810661	C₄	1	1	yes	characteristic, complemented
38	#16810660	C₂²	1	1	yes	characteristic
39	#16810660	C₂²	1	1	yes	characteristic
40	#16810659	C₄	1	1	yes	characteristic, complemented
41	#16810659	C₄	1	1	yes	characteristic, complemented
42	#16810658	C₂²	1	1	yes	characteristic
43	#16810658	C₂²	1	1	yes	characteristic
44	#107×D₅₃	C₄	1	1	yes	characteristic, complemented
45	#107×D₅₃	C₄	1	1	yes	characteristic, complemented
46	#106×D₅₃	C₄	1	1	yes	characteristic, complemented
47	#106×D₅₃	C₄	1	1	yes	characteristic, complemented
48	C₅₃×#427	C₄	1	1	yes	characteristic, complemented
49	C₂³×#6157	—	2	1	no	—
50	C₂×#11719936	C₂³	1	2	yes	—
51	C₂×#11719936	C₂³	1	2	yes	—
52	D₅₃⋊M₁₆	—	2	2	no	—
53	D₅₃⋊M₁₆	—	2	2	no	—
54	D₅₃⋊M₁₆	—	2	2	no	—
55	D₅₃⋊M₁₆	Q₈	1	4	yes	—
56	C₂×#11719940	—	2	1	no	—
57	C₂×#11719940	—	2	1	no	—
58	C₂×D₅₃⋊C₈	—	2	1	no	—
59	C₂×D₅₃⋊C₈	—	2	1	no	—
60	C₂×D₅₃⋊C₈	—	2	2	no	—
61	C₂×D₅₃⋊C₈	—	2	2	no	—
62	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
63	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
64	C₂×D₅₃⋊C₈	C₂³	1	2	yes	—
65	C₂²×#88324	D₄	1	2	yes	—
66	C₂×C₄×#6157	—	2	2	no	—
67	C₂×C₄×#6157	—	2	2	no	—
68	C₂×C₄×#6157	—	2	1	no	—
69	C₂×C₄×#6157	C₂³	1	2	yes	—
70	#13133284	—	2	2	no	—
71	#13133284	—	2	2	no	—
72	C₄⋊#88324	—	2	2	no	—
73	C₄⋊#88324	—	2	2	no	—
74	#13133272	—	2	2	no	—
75	#13133272	—	2	2	no	—
76	C₄×#88324	—	2	2	no	—
77	C₄×#88324	—	2	2	no	—
78	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
79	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
80	C₂×C₈×D₅₃	—	2	2	no	—
81	C₂×C₈×D₅₃	—	2	2	no	—
82	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
85	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
86	#13133176	C₂×C₄	1	1	yes	characteristic
87	#13133176	C₂×C₄	1	1	yes	characteristic
88	#13133175	C₂³	1	1	yes	characteristic
89	#13133175	C₂³	1	1	yes	characteristic
90	#13133175	C₂³	1	1	yes	characteristic
91	#13133175	C₂³	1	1	yes	characteristic
92	#13133174	—	2	1	no	—
93	#13133174	—	2	1	no	—
94	#13133173	—	2	1	no	—
95	#13133173	—	2	1	no	—
96	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
98	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
99	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂²⋊C₄×D₅₃	—	2	1	no	—
101	C₂²⋊C₄×D₅₃	—	2	1	no	—
102	C₄²×D₅₃	—	2	1	no	—
103	C₄²×D₅₃	—	2	1	no	—
104	#13133151	C₂×C₄	1	1	yes	characteristic, complemented
105	#13133151	C₂×C₄	1	1	yes	characteristic, complemented
106	#13133150	C₂×C₄	1	1	yes	characteristic, complemented
107	#13133150	C₂×C₄	1	1	yes	characteristic, complemented
108	#13133150	C₂×C₄	1	1	yes	characteristic, complemented
109	#13133150	C₂×C₄	1	1	yes	characteristic, complemented
110	#13133149	C₂×C₄	1	1	yes	characteristic, complemented
111	#13133149	C₂×C₄	1	1	yes	characteristic, complemented
112	#13133148	C₂×C₄	1	1	yes	characteristic, complemented
113	#13133148	C₂×C₄	1	1	yes	characteristic, complemented
114	#13133148	C₂×C₄	1	1	yes	characteristic, complemented
115	#13133148	C₂×C₄	1	1	yes	characteristic, complemented
116	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
117	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
118	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#107×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
120	#107×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
121	#106×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
122	#106×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
123	C₂³×D₅₃	#41	1	1	yes	characteristic
124	C₂²×#6157	—	2	4	no	—
125	C₂²×#6157	—	2	2	no	—
126	C₂²×#6157	—	4	2	no	—
127	C₂²×#6157	—	4	2	no	—
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×D₅₃	—	2	1	no	—
131	C₂×C₄×D₅₃	—	2	1	no	—
132	C₂×C₄×D₅₃	—	2	1	no	—
133	C₂×C₄×D₅₃	—	2	1	no	—
134	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
137	C₂×C₄×D₅₃	#41	1	2	yes	—
138	C₂×C₄×D₅₃	#41	1	2	yes	—
139	C₂×C₄×D₅₃	#41	1	2	yes	—
140	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
141	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
142	#11719940	—	2	4	no	—
143	#11719940	—	4	2	no	—
144	C₂×#88324	—	2	4	no	—
145	C₂×#88324	—	2	2	no	—
146	C₂×#88324	—	2	2	no	—
147	C₂×#88324	C₂×D₄	1	2	yes	—
148	C₂×#88324	C₂×D₄	1	2	yes	—
149	C₄×#6157	—	2	2	no	—
150	C₄×#6157	—	2	2	no	—
151	C₄×#6157	—	4	2	no	—
152	#11719936	—	2	2	no	—
153	#11719936	—	2	2	no	—
154	#11719936	—	2	2	no	—
155	#11719936	C₂×Q₈	1	4	yes	—
156	D₅₃⋊C₈	—	2	4	no	—
157	D₅₃⋊C₈	—	2	2	no	—
158	D₅₃⋊C₈	—	2	2	no	—
159	D₅₃⋊C₈	—	2	2	no	—
160	D₅₃⋊C₈	—	2	2	no	—
161	D₅₃⋊C₈	—	4	2	no	—
162	C₂×C₈×C₅₃	—	2	2	no	—
163	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
167	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₂²⋊C₄×C₅₃	—	2	1	no	—
170	C₂²⋊C₄×C₅₃	—	2	1	no	—
171	C₄²×C₅₃	—	2	1	no	—
172	C₄²×C₅₃	—	2	1	no	—
173	C₂²⋊Dic₅₃	—	2	1	no	—
174	C₂²⋊Dic₅₃	—	2	1	no	—
175	#11719921	—	2	1	no	—
176	#11719921	—	2	1	no	—
177	#11719921	—	2	1	no	—
178	#11719921	—	2	1	no	—
179	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
180	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
181	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
183	#11719919	—	2	1	no	—
184	#11719919	—	2	1	no	—
185	#11719919	—	2	1	no	—
186	#11719919	—	2	1	no	—
187	C₄×Dic₅₃	—	2	1	no	—
188	C₄×Dic₅₃	—	2	1	no	—
189	C₄×Dic₅₃	—	2	1	no	—
190	C₄×Dic₅₃	—	2	1	no	—
191	C₄×Dic₅₃	—	2	1	no	—
192	C₄×Dic₅₃	—	2	1	no	—
193	C₂×#88322	—	2	2	no	—
194	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
195	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
196	C₈×D₅₃	—	2	2	no	—
197	C₈×D₅₃	—	2	2	no	—
198	C₈×D₅₃	—	2	1	no	—
199	C₈×D₅₃	—	2	1	no	—
200	C₈×D₅₃	—	2	1	no	—
201	C₈×D₅₃	—	2	1	no	—
202	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
203	C₂²×D₅₃	—	2	2	no	—
204	C₂²×D₅₃	—	2	2	no	—
205	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
206	C₂²×D₅₃	#125	1	2	yes	—
207	C₂²×D₅₃	#126	1	2	yes	—
208	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
209	C₂×#6157	—	2	4	no	—
210	C₂×#6157	—	4	2	no	—
211	C₂×#6157	—	4	8	no	—
212	C₂×#6157	—	4	2	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	#117	1	2	yes	—
218	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
220	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
221	C₂×Dic₅₃	—	2	1	no	—
222	C₂×Dic₅₃	—	2	1	no	—
223	C₂×Dic₅₃	—	2	1	no	—
224	C₂×Dic₅₃	—	2	1	no	—
225	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
226	C₂×Dic₅₃	C₂²⋊Q₈	1	2	yes	—
227	C₂×Dic₅₃	#123	1	2	yes	—
228	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
229	C₄×D₅₃	—	4	2	no	—
230	C₄×D₅₃	—	4	2	no	—
231	C₄×D₅₃	—	4	2	no	—
232	C₄×D₅₃	—	4	2	no	—
233	C₄×D₅₃	#117	1	2	yes	—
234	C₄×D₅₃	C₄×Q₈	1	2	yes	—
235	C₄×D₅₃	C₂²⋊Q₈	1	2	yes	—
236	C₄×D₅₃	#123	1	2	yes	—
237	C₄×D₅₃	#125	1	2	yes	—
238	C₄×D₅₃	#126	1	2	yes	—
239	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
240	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
241	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
242	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
243	#88324	—	2	4	no	—
244	#88324	—	2	2	no	—
245	#88324	—	2	2	no	—
246	#88324	—	4	2	no	—
247	C₈×C₅₃	—	2	1	no	—
248	C₈×C₅₃	—	2	1	no	—
249	#88322	—	2	1	no	—
250	#88322	—	2	1	no	—
251	C₂×#1922	—	53	1	no	maximal, Sylow
252	C₂²×C₅₃	—	2	1	no	—
253	C₂²×C₅₃	—	2	1	no	—
254	C₂²×C₅₃	#387	1	2	yes	—
255	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂×D₅₃	—	2	1	no	—
257	C₂×D₅₃	—	2	1	no	—
258	C₂×D₅₃	—	2	4	no	—
259	C₂×D₅₃	—	2	4	no	—
260	C₂×D₅₃	#387	1	2	yes	—
261	C₂×D₅₃	#443	1	2	yes	—
262	C₂×D₅₃	#444	1	2	yes	—
263	C₂×D₅₃	C₂×#125	1	1	yes	characteristic
264	C₂×D₅₃	C₂×#126	1	1	yes	characteristic
265	#6157	—	4	4	no	—
266	C₄×C₅₃	—	4	1	no	—
267	C₄×C₅₃	—	4	1	no	—
268	C₄×C₅₃	—	4	1	no	—
269	C₄×C₅₃	—	4	1	no	—
270	C₄×C₅₃	#387	1	2	yes	—
271	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
272	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
273	Dic₅₃	—	4	1	no	—
274	Dic₅₃	—	4	1	no	—
275	Dic₅₃	—	4	1	no	—
276	Dic₅₃	—	4	1	no	—
277	Dic₅₃	#387	1	2	yes	—
278	Dic₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
279	Dic₅₃	C₂×#123	1	1	yes	characteristic
280	C₂×#427	—	53	1	no	—
281	C₂×#424	—	53	1	no	—
282	C₂×#405	—	53	1	no	—
283	#1922	—	53	2	no	—
284	#1922	—	53	2	no	—
285	#1922	—	53	2	no	—
286	#1922	—	53	2	no	—
287	C₂×#341	—	53	1	no	—
288	C₂×#339	—	53	1	no	—
289	C₂×#339	—	53	1	no	—
290	C₂×#336	—	53	1	no	—
291	C₂×C₅₃	—	2	2	no	—
292	C₂×C₅₃	C₂×#387	1	1	yes	characteristic
293	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
294	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
295	D₅₃	—	2	1	no	—
296	D₅₃	—	2	1	no	—
297	D₅₃	#1922	1	4	yes	complemented
298	C₂²×M₁₆	—	53	1	no	—
299	C₂²×M₁₆	—	53	1	no	—
300	C₂³×C₈	—	53	1	no	—
301	C₂³×C₈	—	53	1	no	—
302	C₂×#117	—	53	1	no	—
303	C₂×#117	—	53	1	no	—
304	C₂²×C₄²	—	53	1	no	—
305	#427	—	53	1	no	—
306	#427	—	53	1	no	—
307	#427	—	53	1	no	—
308	#427	—	53	1	no	—
309	#427	—	53	1	no	—
310	#427	—	53	1	no	—
311	#427	—	53	1	no	—
312	#427	#6157	1	1	yes	characteristic, complemented
313	C₂×#107	—	53	1	no	—
314	C₂×#107	—	53	1	no	—
315	C₂×#106	—	53	1	no	—
316	C₂×#106	—	53	1	no	—
317	#424	—	53	2	no	—
318	#424	—	53	2	no	—
319	#424	—	106	2	no	—
320	C₂×C₄⋊C₈	—	106	1	no	—
321	C₂×C₄⋊C₈	—	106	1	no	—
322	#405	—	53	2	no	—
323	#405	—	53	2	no	—
324	#405	—	106	2	no	—
325	C₂×#97	—	106	1	no	—
326	C₂×C₄×C₈	—	106	1	no	—
327	#341	—	53	2	no	—
328	#341	—	53	2	no	—
329	#339	—	53	2	no	—
330	#339	—	53	2	no	—
331	#339	—	53	2	no	—
332	#339	—	53	2	no	—
333	#336	—	53	2	no	—
334	#336	—	53	2	no	—
335	C₅₃	C₂×#1922	1	1	yes	characteristic, complemented, Sylow
336	C₂³×C₄	—	53	1	no	—
337	C₂³×C₄	—	106	1	no	—
338	C₂×M₁₆	—	53	2	no	—
339	C₂×M₁₆	—	53	4	no	—
340	C₂×M₁₆	—	53	2	no	—
341	C₂×M₁₆	—	106	1	no	—
342	C₂×M₁₆	—	106	2	no	—
343	C₂×M₁₆	—	106	1	no	—
344	C₂×M₁₆	—	106	2	no	—
345	C₂×M₁₆	—	106	2	no	—
346	C₂²×C₈	—	53	2	no	—
347	C₂²×C₈	—	53	1	no	—
348	C₂²×C₈	—	53	1	no	—
349	C₂²×C₈	—	53	2	no	—
350	C₂²×C₈	—	53	1	no	—
351	C₂²×C₈	—	53	1	no	—
352	C₂²×C₈	—	53	1	no	—
353	C₂²×C₈	—	53	1	no	—
354	C₂²×C₈	—	53	1	no	—
355	C₂²×C₈	—	106	2	no	—
356	C₂²×C₈	—	106	2	no	—
357	C₂²×C₈	—	106	2	no	—
358	C₂²×C₈	—	106	2	no	—
359	C₂²×C₈	—	106	1	no	—
360	C₂²×C₈	—	106	1	no	—
361	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
362	#117	—	53	1	no	—
363	#117	—	53	1	no	—
364	#117	—	53	1	no	—
365	#117	—	53	1	no	—
366	#117	—	53	1	no	—
367	#117	—	53	1	no	—
368	#117	—	106	1	no	—
369	#117	—	106	1	no	—
370	#117	—	106	1	no	—
371	#117	—	106	1	no	—
372	#117	C₂×#6157	1	1	yes	characteristic
373	#117	C₂×#6157	1	1	yes	characteristic
374	C₂×C₄⋊C₄	—	53	1	no	—
375	C₂×C₄⋊C₄	—	53	1	no	—
376	C₂×C₄⋊C₄	—	53	1	no	—
377	C₂×C₄⋊C₄	—	53	1	no	—
378	C₂×C₂²⋊C₄	—	106	1	no	—
379	C₂×C₂²⋊C₄	—	106	1	no	—
380	C₂×C₄²	—	53	2	no	—
381	C₂×C₄²	—	106	1	no	—
382	C₂×C₄²	—	106	1	no	—
383	C₂×C₄²	—	106	2	no	—
384	C₂×C₄²	—	106	2	no	—
385	C₂×C₄²	—	106	1	no	—
386	#107	—	53	1	no	—
387	#107	—	53	1	no	—
388	#107	—	53	1	no	—
389	#107	—	53	1	no	—
390	#107	—	53	1	no	—
391	#107	—	53	1	no	—
392	#107	C₂×#6157	1	1	yes	characteristic, complemented
393	#107	C₂×#6157	1	1	yes	characteristic, complemented
394	#106	—	53	1	no	—
395	#106	—	53	1	no	—
396	#106	—	53	1	no	—
397	#106	—	53	1	no	—
398	#106	—	53	1	no	—
399	#106	—	53	1	no	—
400	#106	C₂×#6157	1	1	yes	characteristic, complemented
401	#106	C₂×#6157	1	1	yes	characteristic, complemented
402	C₄⋊C₈	—	106	2	no	—
403	C₄⋊C₈	—	106	2	no	—
404	C₄⋊C₈	—	106	2	no	—
405	C₄⋊C₈	—	106	2	no	—
406	#97	—	106	2	no	—
407	#97	—	106	2	no	—
408	C₄×C₈	—	106	2	no	—
409	C₄×C₈	—	106	2	no	—
410	C₂⁴	—	53	1	no	—
411	C₂²×C₄	—	53	1	no	—
412	C₂²×C₄	—	53	2	no	—
413	C₂²×C₄	—	53	2	no	—
414	C₂²×C₄	—	53	1	no	—
415	C₂²×C₄	—	53	1	no	—
416	C₂²×C₄	—	53	2	no	—
417	C₂²×C₄	—	53	2	no	—
418	C₂²×C₄	—	53	1	no	—
419	C₂²×C₄	—	53	1	no	—
420	C₂²×C₄	—	106	1	no	—
421	C₂²×C₄	—	106	1	no	—
422	C₂²×C₄	—	106	1	no	—
423	C₂²×C₄	—	106	1	no	—
424	C₂²×C₄	—	106	2	no	—
425	C₂²×C₄	—	106	4	no	—
426	C₂²×C₄	—	212	2	no	—
427	C₂²×C₄	—	212	2	no	—
428	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
429	M₁₆	—	53	4	no	—
430	M₁₆	—	106	2	no	—
431	M₁₆	—	106	4	no	—
432	M₁₆	—	106	2	no	—
433	M₁₆	—	106	2	no	—
434	M₁₆	—	212	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	53	2	no	—
437	C₂×C₈	—	53	2	no	—
438	C₂×C₈	—	53	1	no	—
439	C₂×C₈	—	53	1	no	—
440	C₂×C₈	—	106	2	no	—
441	C₂×C₈	—	106	2	no	—
442	C₂×C₈	—	106	1	no	—
443	C₂×C₈	—	106	1	no	—
444	C₂×C₈	—	106	1	no	—
445	C₂×C₈	—	106	1	no	—
446	C₂×C₈	—	106	2	no	—
447	C₂×C₈	—	106	4	no	—
448	C₂×C₈	—	106	2	no	—
449	C₂×C₈	—	106	2	no	—
450	C₂×C₈	—	106	2	no	—
451	C₂×C₈	—	106	2	no	—
452	C₂×C₈	—	106	4	no	—
453	C₂×C₈	—	106	2	no	—
454	C₂×C₈	—	106	2	no	—
455	C₂×C₈	—	212	2	no	—
456	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
457	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
458	C₄⋊C₄	—	53	1	no	—
459	C₄⋊C₄	—	53	1	no	—
460	C₄⋊C₄	—	53	1	no	—
461	C₄⋊C₄	—	53	1	no	—
462	C₄⋊C₄	—	106	1	no	—
463	C₄⋊C₄	—	106	1	no	—
464	C₄⋊C₄	—	106	1	no	—
465	C₄⋊C₄	—	106	1	no	—
466	C₄⋊C₄	C₂²×#6157	1	1	yes	characteristic
467	C₄⋊C₄	C₂²×#6157	1	1	yes	characteristic
468	C₄⋊C₄	C₂²×#6157	1	1	yes	characteristic
469	C₄⋊C₄	C₂²×#6157	1	1	yes	characteristic
470	C₂²⋊C₄	—	2	1	no	—
471	C₂²⋊C₄	—	2	1	no	—
472	C₂²⋊C₄	—	106	1	no	—
473	C₂²⋊C₄	—	106	1	no	—
474	C₂²⋊C₄	—	106	1	no	—
475	C₂²⋊C₄	—	106	1	no	—
476	C₂²⋊C₄	—	106	1	no	—
477	C₂²⋊C₄	—	106	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	106	1	no	—
481	C₄²	—	106	1	no	—
482	C₄²	—	106	1	no	—
483	C₄²	—	106	1	no	—
484	C₄²	—	106	2	no	—
485	C₄²	—	106	2	no	—
486	C₄²	—	106	1	no	—
487	C₄²	—	106	1	no	—
488	C₄²	—	212	2	no	—
489	C₂³	—	53	1	no	—
490	C₂³	—	53	2	no	—
491	C₂³	—	53	2	no	—
492	C₂³	—	53	1	no	—
493	C₂³	—	106	2	no	—
494	C₂³	—	106	2	no	—
495	C₂³	D₄×#6157	1	1	yes	characteristic
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	53	2	no	—
501	C₂×C₄	—	53	2	no	—
502	C₂×C₄	—	53	1	no	—
503	C₂×C₄	—	53	1	no	—
504	C₂×C₄	—	53	2	no	—
505	C₂×C₄	—	53	2	no	—
506	C₂×C₄	—	53	2	no	—
507	C₂×C₄	—	53	2	no	—
508	C₂×C₄	—	53	2	no	—
509	C₂×C₄	—	53	1	no	—
510	C₂×C₄	—	53	1	no	—
511	C₂×C₄	—	53	2	no	—
512	C₂×C₄	—	53	1	no	—
513	C₂×C₄	—	53	1	no	—
514	C₂×C₄	—	106	4	no	—
515	C₂×C₄	—	106	1	no	—
516	C₂×C₄	—	106	1	no	—
517	C₂×C₄	—	106	1	no	—
518	C₂×C₄	—	106	1	no	—
519	C₂×C₄	—	212	2	no	—
520	C₂×C₄	—	212	2	no	—
521	C₂×C₄	—	212	2	no	—
522	C₂×C₄	—	212	2	no	—
523	C₂×C₄	—	212	2	no	—
524	C₂×C₄	—	212	8	no	—
525	C₂×C₄	—	212	2	no	—
526	C₂×C₄	#13133296	1	2	yes	—
527	C₂×C₄	D₄×#6157	1	1	yes	characteristic
528	C₂×C₄	Q₈×#6157	1	2	yes	—
529	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	106	2	no	—
533	C₈	—	106	2	no	—
534	C₈	—	106	4	no	—
535	C₈	—	106	1	no	—
536	C₈	—	106	1	no	—
537	C₈	—	212	2	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	53	2	no	—
541	C₂²	—	53	2	no	—
542	C₂²	—	53	2	no	—
543	C₂²	—	53	1	no	—
544	C₂²	—	53	1	no	—
545	C₂²	—	106	4	no	—
546	C₂²	—	106	4	no	—
547	C₂²	—	106	1	no	—
548	C₂²	—	106	1	no	—
549	C₂²	#16811572	1	2	yes	—
550	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	53	1	no	—
556	C₄	—	53	1	no	—
557	C₄	—	53	2	no	—
558	C₄	—	212	4	no	—
559	C₄	—	212	1	no	—
560	C₄	—	212	1	no	—
561	C₄	—	212	1	no	—
562	C₄	—	212	1	no	—
563	C₄	#16811572	1	2	yes	—
564	C₄	C₂×#13133296	1	1	yes	characteristic
565	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
566	C₂	—	2	2	no	—
567	C₂	—	53	4	no	—
568	C₂	—	106	1	no	—
569	C₂	—	106	1	no	—
570	C₂	C₂×#16811572	1	1	yes	characteristic
571	C₂	D₅₃⋊#443	1	1	yes	characteristic
572	C₂	D₅₃⋊#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#427	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×D₅₃	#41
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#427	#6157
hypocenter	C₅₃	C₂×#1922
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
3. #20290783	C₂	semidirect
4. #20290782	C₂	semidirect
5. #20290781	C₂	semidirect
6. #20290780	C₂	semidirect
37. #16810661	C₄	semidirect
40. #16810659	C₄	semidirect
41. #16810659	C₄	semidirect
44. #107×D₅₃	C₄	semidirect
45. #107×D₅₃	C₄	semidirect
46. #106×D₅₃	C₄	semidirect
47. #106×D₅₃	C₄	semidirect
48. C₅₃×#427	C₄	semidirect
104. #13133151	C₂×C₄	semidirect
105. #13133151	C₂×C₄	semidirect
106. #13133150	C₂×C₄	semidirect
107. #13133150	C₂×C₄	semidirect
108. #13133150	C₂×C₄	semidirect
109. #13133150	C₂×C₄	semidirect
110. #13133149	C₂×C₄	semidirect
111. #13133149	C₂×C₄	semidirect
112. #13133148	C₂×C₄	semidirect
113. #13133148	C₂×C₄	semidirect
114. #13133148	C₂×C₄	semidirect
115. #13133148	C₂×C₄	semidirect
119. #107×C₅₃	C₂×C₄	semidirect
120. #107×C₅₃	C₂×C₄	semidirect
121. #106×C₅₃	C₂×C₄	semidirect
122. #106×C₅₃	C₂×C₄	semidirect
297. D₅₃	#1922	semidirect
312. #427	#6157	semidirect
335. C₅₃	C₂×#1922	semidirect
392. #107	C₂×#6157	semidirect
393. #107	C₂×#6157	semidirect
400. #106	C₂×#6157	semidirect
401. #106	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#424	2
#20291275	2
#20290783	2
#20290782	2
#20290781	2
#20290780	2
#20290407	2
#20290391	2
C₂×(C₂×#6157)⋊C₈	2
#427×D₅₃	2
C₂×#1922	53

Subgroups — D53⋊#1922 #28319936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃⋊#1922 #28319936