Subgroups — D₅₃×#3292 #28260313

Order 13568.

2766 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#576×D₅₃	C₂	1	4	yes	maximal, complemented
2	#20286515	C₂	1	2	yes	maximal, complemented
3	#20286514	C₂	1	1	yes	maximal, characteristic, complemented
4	#20286513	C₂	1	4	yes	maximal, complemented
5	#20286512	C₂	1	4	yes	maximal, complemented
6	#20286511	C₂	1	4	yes	maximal, complemented
7	#493×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#472×D₅₃	C₂	1	4	yes	maximal, complemented
9	#453×D₅₃	C₂	1	4	yes	maximal, complemented
10	#443×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#3292	C₂	1	1	yes	maximal, characteristic, complemented
12	H(5,𝔽₂)×D₅₃	C₂²	1	4	yes	complemented
13	#16811428	C₂²	1	4	yes	complemented
14	#16811427	—	2	8	no	—
15	#16811427	C₂²	1	8	yes	complemented
16	#16811426	C₂²	1	8	yes	complemented
17	#16811425	—	2	4	no	—
18	#16811425	C₂²	1	4	yes	complemented
19	#16811424	—	2	4	no	—
20	#16811424	C₂²	1	4	yes	complemented
21	AGL(1,ℤ/8)×D₅₃	—	2	8	no	—
22	AGL(1,ℤ/8)×D₅₃	C₂²	1	8	yes	complemented
23	#135×D₅₃	—	2	4	no	—
24	#135×D₅₃	C₂²	1	2	yes	—
25	C₂×D₈×D₅₃	—	2	4	no	—
26	C₂×D₈×D₅₃	C₂²	1	4	yes	complemented
27	#131×D₅₃	C₂²	1	4	yes	complemented
28	#127×D₅₃	C₂²	1	2	yes	complemented
29	#16811000	C₂²	1	1	yes	characteristic, complemented
30	#16810999	C₂²	1	2	yes	complemented
31	#16810998	C₂²	1	4	yes	complemented
32	#16810894	C₂²	1	4	yes	complemented
33	#16810893	C₂²	1	4	yes	complemented
34	#16810892	C₂²	1	8	yes	complemented
35	#16810891	C₂²	1	8	yes	complemented
36	#16810890	C₂²	1	4	yes	complemented
37	#16810793	C₂²	1	4	yes	complemented
38	#16810792	C₂²	1	8	yes	complemented
39	#16810791	C₂²	1	4	yes	complemented
40	#16810790	C₂²	1	4	yes	complemented
41	#16810789	C₂²	1	8	yes	complemented
42	#16810741	C₂²	1	2	yes	complemented
43	#16810740	C₂²	1	2	yes	complemented
44	#16810739	C₂²	1	2	yes	complemented
45	#16810738	C₂²	1	4	yes	complemented
46	#16810737	C₂²	1	4	yes	complemented
47	#108×D₅₃	C₂²	1	2	yes	complemented
48	#104×D₅₃	C₂²	1	4	yes	complemented
49	#100×D₅₃	C₂²	1	4	yes	complemented
50	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
51	C₅₃×#576	C₂²	1	4	yes	complemented
52	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
53	C₅₃×#472	C₂²	1	4	yes	complemented
54	C₅₃×#453	C₂²	1	4	yes	complemented
55	C₅₃×#443	C₂²	1	2	yes	complemented
56	#13133263	—	2	8	no	—
57	#13133263	C₂³	1	8	yes	complemented
58	#13133262	—	2	8	no	—
59	#13133262	—	4	4	no	—
60	#13133262	C₂³	1	4	yes	complemented
61	#41×D₅₃	—	2	8	no	—
62	#41×D₅₃	C₂³	1	4	yes	—
63	C₂×D₄×D₅₃	—	2	8	no	—
64	C₂×D₄×D₅₃	—	4	2	no	—
65	C₂×D₄×D₅₃	—	4	4	no	—
66	C₂×D₄×D₅₃	C₂³	1	4	yes	—
67	#13133245	—	2	8	no	—
68	#13133245	C₂³	1	8	yes	complemented
69	#13133244	—	2	8	no	—
70	#13133244	—	2	8	no	—
71	#13133244	—	2	8	no	—
72	#13133243	—	2	8	no	—
73	#13133243	C₂³	1	8	yes	complemented
74	#13133242	—	2	8	no	—
75	#13133242	C₂³	1	8	yes	complemented
76	#13133241	—	2	8	no	—
77	#13133241	—	2	8	no	—
78	#13133241	—	2	8	no	—
79	#13133240	—	2	4	no	—
80	#13133240	D₄	1	2	yes	complemented
81	#13133239	—	2	4	no	—
82	#13133239	C₂³	1	2	yes	—
83	#13133238	—	2	8	no	—
84	#13133238	D₄	1	4	yes	complemented
85	#13133237	—	2	8	no	—
86	#13133237	C₂³	1	4	yes	—
87	#13133236	—	2	4	no	—
88	#13133236	D₄	1	2	yes	complemented
89	Q₁₆×D₅₃	D₄	1	2	yes	complemented
90	SD₁₆×D₅₃	—	2	8	no	—
91	SD₁₆×D₅₃	D₄	1	4	yes	complemented
92	C₂×D₄₂₄	—	2	4	no	—
93	C₂×D₄₂₄	C₂³	1	4	yes	complemented
94	C₂×#11719922	—	2	8	no	—
95	C₂×#11719922	C₂³	1	8	yes	complemented
96	D₈×D₅₃	—	2	4	no	—
97	D₈×D₅₃	—	2	8	no	—
98	D₈×D₅₃	—	2	4	no	—
99	D₈×D₅₃	—	2	4	no	—
100	D₈×D₅₃	—	2	4	no	—
101	D₈×D₅₃	—	2	4	no	—
102	D₈×D₅₃	—	2	4	no	—
103	D₈×D₅₃	—	2	8	no	—
104	D₈×D₅₃	D₄	1	2	yes	complemented
105	#13133225	—	2	4	no	—
106	#13133225	C₂³	1	4	yes	complemented
107	#13133224	C₂³	1	4	yes	complemented
108	#13133223	—	2	4	no	—
109	#13133223	C₂³	1	4	yes	complemented
110	M₁₆×D₅₃	—	2	4	no	—
111	M₁₆×D₅₃	C₂³	1	4	yes	—
112	C₂×C₈×D₅₃	—	2	4	no	—
113	C₂×C₈×D₅₃	C₂³	1	2	yes	—
114	#13133214	C₂³	1	2	yes	complemented
115	#13133213	—	4	2	no	—
116	#13133213	C₂³	1	4	yes	complemented
117	C₄²×D₅₃	C₂³	1	1	yes	characteristic
118	#13133153	C₂³	1	2	yes	complemented
119	#13133152	C₂³	1	4	yes	complemented
120	#13133145	C₂³	1	4	yes	complemented
121	#13133144	C₂³	1	4	yes	complemented
122	#13133143	C₂³	1	4	yes	complemented
123	#13133134	C₂³	1	4	yes	complemented
124	#13133133	C₂³	1	8	yes	complemented
125	C₄×#88322	C₂³	1	1	yes	characteristic, complemented
126	C₈×Dic₅₃	C₂³	1	2	yes	complemented
127	H(5,𝔽₂)×C₅₃	C₂³	1	4	yes	complemented
128	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
129	AGL(1,ℤ/8)×C₅₃	C₂³	1	8	yes	complemented
130	#135×C₅₃	—	2	4	no	—
131	#135×C₅₃	C₂³	1	2	yes	—
132	C₂×D₈×C₅₃	—	2	4	no	—
133	C₂×D₈×C₅₃	C₂³	1	4	yes	complemented
134	#131×C₅₃	C₂³	1	4	yes	complemented
135	#127×C₅₃	C₂³	1	2	yes	complemented
136	#108×C₅₃	C₂³	1	2	yes	complemented
137	#104×C₅₃	C₂³	1	4	yes	complemented
138	#100×C₅₃	C₂³	1	4	yes	complemented
139	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic, complemented
140	C₂³×D₅₃	—	2	4	no	—
141	C₂³×D₅₃	—	4	4	no	—
142	#41×C₅₃	—	2	8	no	—
143	#41×C₅₃	C₂⁴	1	4	yes	—
144	C₂×D₄×C₅₃	—	2	8	no	—
145	C₂×D₄×C₅₃	—	4	2	no	—
146	C₂×D₄×C₅₃	—	4	4	no	—
147	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
148	C₂×C₂²⋊D₅₃	—	2	4	no	—
149	C₂×C₂²⋊D₅₃	—	4	8	no	—
150	C₂×C₂²⋊D₅₃	—	4	4	no	—
151	C₂×C₂²⋊D₅₃	—	4	4	no	—
152	C₂×C₂²⋊D₅₃	—	4	4	no	—
153	#11719949	—	2	8	no	—
154	#11719949	C₂×D₄	1	4	yes	—
155	Q₈×D₅₃	C₂×D₄	1	4	yes	—
156	#11719947	—	2	4	no	—
157	#11719947	—	2	8	no	—
158	#11719947	—	4	8	no	—
159	#11719947	C₂×D₄	1	4	yes	—
160	D₄×D₅₃	—	2	4	no	—
161	D₄×D₅₃	—	2	8	no	—
162	D₄×D₅₃	—	2	8	no	—
163	D₄×D₅₃	—	2	8	no	—
164	D₄×D₅₃	—	2	8	no	—
165	D₄×D₅₃	—	4	8	no	—
166	D₄×D₅₃	—	4	8	no	—
167	D₄×D₅₃	—	4	8	no	—
168	D₄×D₅₃	—	4	8	no	—
169	D₄×D₅₃	—	8	8	no	—
170	D₄×D₅₃	C₂×D₄	1	4	yes	—
171	#11719945	—	2	4	no	—
172	#11719945	—	2	8	no	—
173	#11719945	—	4	8	no	—
174	#11719945	C₂⁴	1	4	yes	—
175	C₂×D₂₁₂	—	2	8	no	—
176	C₂×D₂₁₂	—	4	4	no	—
177	C₂×D₂₁₂	—	4	2	no	—
178	C₂×D₂₁₂	C₂⁴	1	4	yes	—
179	C₂×C₄×D₅₃	—	2	1	no	—
180	C₂×C₄×D₅₃	—	2	4	no	—
181	C₂×C₄×D₅₃	—	4	4	no	—
182	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
183	Q₁₆×C₅₃	C₂×D₄	1	2	yes	complemented
184	SD₁₆×C₅₃	—	2	8	no	—
185	SD₁₆×C₅₃	C₂×D₄	1	4	yes	complemented
186	D₈×C₅₃	—	2	8	no	—
187	D₈×C₅₃	—	2	4	no	—
188	D₈×C₅₃	—	2	4	no	—
189	D₈×C₅₃	C₂×D₄	1	2	yes	complemented
190	M₁₆×C₅₃	—	2	4	no	—
191	M₁₆×C₅₃	C₂⁴	1	4	yes	—
192	C₂×C₈×C₅₃	—	2	4	no	—
193	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
194	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
195	#11719925	C₂×D₄	1	4	yes	complemented
196	#11719924	—	2	8	no	—
197	#11719924	C₂×D₄	1	4	yes	complemented
198	#11719923	—	2	8	no	—
199	#11719923	C₂×D₄	1	4	yes	complemented
200	#11719922	—	2	8	no	—
201	#11719922	—	2	8	no	—
202	#11719922	—	2	8	no	—
203	#11719922	—	2	8	no	—
204	#11719922	C₂×D₄	1	4	yes	complemented
205	C₄×Dic₅₃	—	2	1	no	—
206	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
207	#11719917	—	2	4	no	—
208	#11719917	C₂⁴	1	4	yes	—
209	C₂×#88322	—	2	4	no	—
210	C₂×#88322	C₂⁴	1	2	yes	—
211	Dic₂₁₂	C₂×D₄	1	2	yes	complemented
212	D₄₂₄	—	2	8	no	—
213	D₄₂₄	—	2	4	no	—
214	D₄₂₄	—	2	4	no	—
215	D₄₂₄	C₂×D₄	1	2	yes	complemented
216	#11719913	—	2	8	no	—
217	#11719913	C₂×D₄	1	4	yes	complemented
218	#11719912	—	2	4	no	—
219	#11719912	—	2	4	no	—
220	#11719912	—	2	4	no	—
221	C₈×D₅₃	—	2	4	no	—
222	C₈×D₅₃	—	2	4	no	—
223	C₈×D₅₃	—	2	4	no	—
224	C₈×D₅₃	C₂×D₄	1	2	yes	—
225	C₈×D₅₃	C₂×D₄	1	2	yes	—
226	C₈×D₅₃	C₂×D₄	1	2	yes	—
227	C₈×D₅₃	C₂×D₄	1	2	yes	—
228	C₂³×C₅₃	—	2	4	no	—
229	C₂³×C₅₃	—	4	4	no	—
230	C₂²×D₅₃	—	2	4	no	—
231	C₂²×D₅₃	—	2	4	no	—
232	C₂²×D₅₃	—	4	4	no	—
233	C₂²×D₅₃	—	4	8	no	—
234	C₂²×D₅₃	—	4	4	no	—
235	C₂²×D₅₃	—	4	4	no	—
236	C₂²×D₅₃	—	8	8	no	—
237	C₂²×D₅₃	—	16	8	no	—
238	C₂²×D₅₃	H(5,𝔽₂)	1	1	yes	characteristic
239	Q₈×C₅₃	C₂²×D₄	1	4	yes	—
240	D₄×C₅₃	—	2	4	no	—
241	D₄×C₅₃	—	2	8	no	—
242	D₄×C₅₃	—	4	8	no	—
243	D₄×C₅₃	—	8	4	no	—
244	D₄×C₅₃	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	2	4	no	—
247	C₂×C₄×C₅₃	—	4	4	no	—
248	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
249	C₂²⋊D₅₃	—	2	4	no	—
250	C₂²⋊D₅₃	—	2	4	no	—
251	C₂²⋊D₅₃	—	2	4	no	—
252	C₂²⋊D₅₃	—	4	8	no	—
253	C₂²⋊D₅₃	—	4	8	no	—
254	C₂²⋊D₅₃	—	4	8	no	—
255	C₂²⋊D₅₃	—	8	8	no	—
256	C₂×Dic₅₃	—	2	1	no	—
257	C₂×Dic₅₃	—	2	4	no	—
258	C₂×Dic₅₃	—	4	4	no	—
259	C₂×Dic₅₃	H(5,𝔽₂)	1	1	yes	characteristic
260	D₂₁₂	—	2	4	no	—
261	D₂₁₂	—	2	8	no	—
262	D₂₁₂	—	4	8	no	—
263	D₂₁₂	—	8	4	no	—
264	D₂₁₂	C₂²×D₄	1	4	yes	—
265	C₄×D₅₃	—	2	4	no	—
266	C₄×D₅₃	—	2	4	no	—
267	C₄×D₅₃	—	2	4	no	—
268	C₄×D₅₃	—	2	4	no	—
269	C₄×D₅₃	—	4	8	no	—
270	C₄×D₅₃	C₂²×D₄	1	2	yes	—
271	C₄×D₅₃	C₂²×D₄	1	2	yes	—
272	Dic₁₀₆	—	2	4	no	—
273	Dic₁₀₆	C₂²×D₄	1	4	yes	—
274	C₈×C₅₃	—	2	4	no	—
275	C₈×C₅₃	C₂²×D₄	1	2	yes	—
276	C₈×C₅₃	C₂²×D₄	1	2	yes	—
277	#88322	—	2	4	no	—
278	#88322	C₂²×D₄	1	2	yes	—
279	#88322	C₂²×D₄	1	2	yes	—
280	C₂×#3292	—	53	1	no	maximal, Sylow
281	C₂²×C₅₃	—	2	4	no	—
282	C₂²×C₅₃	—	4	4	no	—
283	C₂²×C₅₃	—	8	4	no	—
284	C₂²×C₅₃	—	16	4	no	—
285	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×D₅₃	—	2	2	no	—
287	C₂×D₅₃	—	2	4	no	—
288	C₂×D₅₃	—	4	8	no	—
289	C₂×D₅₃	—	4	4	no	—
290	C₂×D₅₃	—	8	8	no	—
291	C₂×D₅₃	—	8	4	no	—
292	C₂×D₅₃	—	16	4	no	—
293	C₂×D₅₃	—	16	4	no	—
294	C₂×D₅₃	—	16	8	no	—
295	C₂×D₅₃	D₄²	1	1	yes	characteristic
296	C₂×D₅₃	D₄²	1	1	yes	characteristic
297	C₄×C₅₃	—	2	2	no	—
298	C₄×C₅₃	—	2	4	no	—
299	C₄×C₅₃	C₂³×D₄	1	2	yes	—
300	Dic₅₃	—	2	2	no	—
301	Dic₅₃	—	2	4	no	—
302	Dic₅₃	D₄²	1	2	yes	—
303	C₂×#576	—	53	4	no	—
304	#3292	—	53	1	no	—
305	#3292	—	53	4	no	—
306	#3292	—	53	4	no	—
307	#3292	—	53	2	no	—
308	#3292	—	53	4	no	—
309	#3292	D₅₃	1	1	yes	characteristic, direct product factor
310	C₂×#493	—	53	1	no	—
311	C₂×#472	—	53	4	no	—
312	C₂×#453	—	53	4	no	—
313	C₂×#443	—	53	2	no	—
314	C₂×C₅₃	—	2	1	no	—
315	C₂×C₅₃	—	4	4	no	—
316	C₂×C₅₃	—	8	4	no	—
317	C₂×C₅₃	C₂×D₄²	1	1	yes	characteristic
318	D₅₃	—	2	1	no	—
319	D₅₃	—	4	4	no	—
320	D₅₃	—	8	4	no	—
321	D₅₃	#3292	1	2	yes	direct product factor
322	C₂×H(5,𝔽₂)	—	53	4	no	—
323	#576	—	53	8	no	—
324	#576	—	53	8	no	—
325	#576	—	53	4	no	—
326	#576	—	53	4	no	—
327	#576	—	53	4	no	—
328	#576	—	106	4	no	—
329	#576	—	106	4	no	—
330	#576	—	106	8	no	—
331	#576	C₂×D₅₃	1	4	yes	complemented
332	C₂×AGL(1,ℤ/8)	—	53	8	no	—
333	C₂×AGL(1,ℤ/8)	—	106	8	no	—
334	C₂×#135	—	53	2	no	—
335	C₂×#135	—	106	4	no	—
336	C₂²×D₈	—	53	4	no	—
337	C₂²×D₈	—	106	4	no	—
338	C₂×#131	—	53	4	no	—
339	C₂×#127	—	53	2	no	—
340	#493	—	53	4	no	—
341	#493	—	53	2	no	—
342	#493	—	53	1	no	—
343	#493	C₂×D₅₃	1	1	yes	characteristic, complemented
344	#472	—	53	8	no	—
345	#472	—	53	4	no	—
346	#472	—	53	4	no	—
347	#472	—	53	4	no	—
348	#472	—	53	8	no	—
349	#472	C₂×D₅₃	1	4	yes	complemented
350	#453	—	53	4	no	—
351	#453	—	53	8	no	—
352	#453	—	53	4	no	—
353	#453	—	53	8	no	—
354	#453	—	53	4	no	—
355	#453	C₂×D₅₃	1	4	yes	complemented
356	#443	—	53	2	no	—
357	#443	—	53	4	no	—
358	#443	—	53	2	no	—
359	#443	—	53	4	no	—
360	#443	—	53	2	no	—
361	#443	C₂×D₅₃	1	2	yes	complemented
362	C₂×#108	—	53	2	no	—
363	C₂×#104	—	53	4	no	—
364	C₂×#100	—	53	4	no	—
365	C₂×C₄×C₈	—	53	1	no	—
366	C₅₃	C₂×#3292	1	1	yes	characteristic, complemented, Sylow
367	H(5,𝔽₂)	—	53	8	no	—
368	H(5,𝔽₂)	—	53	4	no	—
369	H(5,𝔽₂)	—	106	8	no	—
370	H(5,𝔽₂)	—	106	8	no	—
371	H(5,𝔽₂)	—	212	4	no	—
372	H(5,𝔽₂)	C₂²×D₅₃	1	4	yes	complemented
373	C₂×#41	—	53	4	no	—
374	C₂×#41	—	106	8	no	—
375	C₂²×D₄	—	53	4	no	—
376	C₂²×D₄	—	106	8	no	—
377	C₂²×D₄	—	212	2	no	—
378	C₂²×D₄	—	212	4	no	—
379	AGL(1,ℤ/8)	—	2	8	no	—
380	AGL(1,ℤ/8)	—	53	8	no	—
381	AGL(1,ℤ/8)	—	53	8	no	—
382	AGL(1,ℤ/8)	—	53	8	no	—
383	AGL(1,ℤ/8)	—	106	8	no	—
384	AGL(1,ℤ/8)	—	106	8	no	—
385	AGL(1,ℤ/8)	—	106	8	no	—
386	AGL(1,ℤ/8)	—	106	8	no	—
387	AGL(1,ℤ/8)	—	106	8	no	—
388	AGL(1,ℤ/8)	—	106	8	no	—
389	AGL(1,ℤ/8)	—	106	8	no	—
390	AGL(1,ℤ/8)	—	106	8	no	—
391	AGL(1,ℤ/8)	—	106	8	no	—
392	AGL(1,ℤ/8)	C₂²×D₅₃	1	8	yes	complemented
393	#135	—	2	4	no	—
394	#135	—	53	4	no	—
395	#135	—	53	2	no	—
396	#135	—	53	4	no	—
397	#135	—	53	2	no	—
398	#135	—	53	2	no	—
399	#135	—	106	4	no	—
400	#135	—	106	4	no	—
401	#135	—	106	8	no	—
402	#135	—	106	8	no	—
403	#135	—	106	4	no	—
404	#135	C₂²×D₅₃	1	2	yes	—
405	C₂×Q₁₆	—	53	2	no	—
406	C₂×SD₁₆	—	53	4	no	—
407	C₂×SD₁₆	—	106	8	no	—
408	C₂×D₈	—	2	4	no	—
409	C₂×D₈	—	53	2	no	—
410	C₂×D₈	—	53	8	no	—
411	C₂×D₈	—	53	4	no	—
412	C₂×D₈	—	106	4	no	—
413	C₂×D₈	—	106	4	no	—
414	C₂×D₈	—	106	4	no	—
415	C₂×D₈	—	106	4	no	—
416	C₂×D₈	—	106	4	no	—
417	C₂×D₈	—	106	8	no	—
418	C₂×D₈	—	106	4	no	—
419	C₂×D₈	—	106	8	no	—
420	C₂×D₈	—	106	8	no	—
421	C₂×D₈	—	106	4	no	—
422	C₂×D₈	C₂²×D₅₃	1	4	yes	complemented
423	#131	—	53	4	no	—
424	#131	—	53	4	no	—
425	#131	—	53	4	no	—
426	#131	—	106	4	no	—
427	#131	—	106	4	no	—
428	#131	C₂²×D₅₃	1	4	yes	complemented
429	C₂×M₁₆	—	53	4	no	—
430	C₂×M₁₆	—	106	4	no	—
431	C₂²×C₈	—	53	2	no	—
432	C₂²×C₈	—	106	4	no	—
433	#127	—	53	4	no	—
434	#127	—	53	2	no	—
435	#127	—	212	2	no	—
436	#127	C₂²×D₅₃	1	2	yes	complemented
437	C₂×C₄²	—	53	1	no	—
438	#108	—	53	4	no	—
439	#108	—	53	2	no	—
440	#108	C₂²×D₅₃	1	2	yes	complemented
441	#104	—	53	4	no	—
442	#104	—	53	4	no	—
443	#104	—	53	4	no	—
444	#104	C₂²×D₅₃	1	4	yes	complemented
445	#100	—	53	4	no	—
446	#100	—	53	8	no	—
447	#100	C₂²×D₅₃	1	4	yes	complemented
448	C₄×C₈	—	53	2	no	—
449	C₄×C₈	—	53	1	no	—
450	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic, complemented
451	C₂⁴	—	106	4	no	—
452	C₂⁴	—	212	4	no	—
453	#41	—	2	8	no	—
454	#41	—	53	4	no	—
455	#41	—	53	4	no	—
456	#41	—	53	4	no	—
457	#41	—	106	4	no	—
458	#41	—	106	8	no	—
459	#41	—	106	4	no	—
460	#41	—	106	8	no	—
461	#41	—	106	8	no	—
462	#41	—	212	8	no	—
463	#41	—	212	8	no	—
464	#41	C₂³×D₅₃	1	4	yes	—
465	C₂×Q₈	—	53	4	no	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	4	2	no	—
468	C₂×D₄	—	4	4	no	—
469	C₂×D₄	—	53	4	no	—
470	C₂×D₄	—	53	4	no	—
471	C₂×D₄	—	106	8	no	—
472	C₂×D₄	—	106	8	no	—
473	C₂×D₄	—	106	8	no	—
474	C₂×D₄	—	106	4	no	—
475	C₂×D₄	—	106	8	no	—
476	C₂×D₄	—	106	4	no	—
477	C₂×D₄	—	106	8	no	—
478	C₂×D₄	—	424	8	no	—
479	C₂×D₄	—	212	2	no	—
480	C₂×D₄	—	212	4	no	—
481	C₂×D₄	—	212	8	no	—
482	C₂×D₄	—	212	8	no	—
483	C₂×D₄	—	212	8	no	—
484	C₂×D₄	—	212	4	no	—
485	C₂×D₄	—	212	8	no	—
486	C₂×D₄	—	212	8	no	—
487	C₂×D₄	—	212	4	no	—
488	C₂×D₄	—	212	4	no	—
489	C₂×D₄	C₂³×D₅₃	1	4	yes	—
490	C₂²×C₄	—	53	1	no	—
491	C₂²×C₄	—	106	4	no	—
492	C₂²×C₄	—	106	1	no	—
493	C₂²×C₄	—	212	4	no	—
494	Q₁₆	—	53	4	no	—
495	Q₁₆	—	53	2	no	—
496	Q₁₆	D₄×D₅₃	1	2	yes	complemented
497	SD₁₆	—	2	8	no	—
498	SD₁₆	—	53	4	no	—
499	SD₁₆	—	53	4	no	—
500	SD₁₆	—	53	4	no	—
501	SD₁₆	—	106	8	no	—
502	SD₁₆	—	106	8	no	—
503	SD₁₆	—	106	8	no	—
504	SD₁₆	D₄×D₅₃	1	4	yes	complemented
505	D₈	—	2	8	no	—
506	D₈	—	2	4	no	—
507	D₈	—	2	4	no	—
508	D₈	—	53	4	no	—
509	D₈	—	53	2	no	—
510	D₈	—	106	8	no	—
511	D₈	—	106	8	no	—
512	D₈	—	106	8	no	—
513	D₈	—	106	8	no	—
514	D₈	—	106	8	no	—
515	D₈	—	106	4	no	—
516	D₈	—	106	4	no	—
517	D₈	D₄×D₅₃	1	2	yes	complemented
518	M₁₆	—	2	4	no	—
519	M₁₆	—	53	4	no	—
520	M₁₆	—	106	4	no	—
521	M₁₆	—	106	4	no	—
522	M₁₆	—	106	4	no	—
523	M₁₆	—	106	4	no	—
524	M₁₆	C₂³×D₅₃	1	4	yes	—
525	C₂×C₈	—	2	4	no	—
526	C₂×C₈	—	53	2	no	—
527	C₂×C₈	—	53	2	no	—
528	C₂×C₈	—	53	2	no	—
529	C₂×C₈	—	53	2	no	—
530	C₂×C₈	—	53	2	no	—
531	C₂×C₈	—	106	4	no	—
532	C₂×C₈	—	106	4	no	—
533	C₂×C₈	—	106	4	no	—
534	C₂×C₈	—	106	4	no	—
535	C₂×C₈	C₂³×D₅₃	1	2	yes	—
536	C₄²	—	53	1	no	—
537	C₄²	—	106	1	no	—
538	C₄²	C₂³×D₅₃	1	1	yes	characteristic
539	C₂³	—	2	4	no	—
540	C₂³	—	4	4	no	—
541	C₂³	—	53	1	no	—
542	C₂³	—	848	8	no	—
543	C₂³	—	106	4	no	—
544	C₂³	—	106	4	no	—
545	C₂³	—	424	8	no	—
546	C₂³	—	212	8	no	—
547	C₂³	—	212	4	no	—
548	C₂³	—	212	4	no	—
549	C₂³	—	212	4	no	—
550	Q₈	—	53	4	no	—
551	Q₈	—	106	4	no	—
552	Q₈	C₂×D₄×D₅₃	1	4	yes	—
553	D₄	—	2	4	no	—
554	D₄	—	2	8	no	—
555	D₄	—	4	8	no	—
556	D₄	—	8	4	no	—
557	D₄	—	53	4	no	—
558	D₄	—	106	4	no	—
559	D₄	—	106	4	no	—
560	D₄	—	106	4	no	—
561	D₄	—	106	4	no	—
562	D₄	—	106	8	no	—
563	D₄	—	424	4	no	—
564	D₄	—	424	8	no	—
565	D₄	—	212	8	no	—
566	D₄	—	212	8	no	—
567	D₄	—	212	8	no	—
568	D₄	—	212	8	no	—
569	D₄	C₂×D₄×D₅₃	1	4	yes	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	53	1	no	—
574	C₂×C₄	—	53	2	no	—
575	C₂×C₄	—	53	2	no	—
576	C₂×C₄	—	106	4	no	—
577	C₂×C₄	—	106	4	no	—
578	C₂×C₄	—	106	4	no	—
579	C₂×C₄	—	106	4	no	—
580	C₂×C₄	—	106	1	no	—
581	C₂×C₄	—	106	4	no	—
582	C₂×C₄	—	212	8	no	—
583	C₂×C₄	—	212	4	no	—
584	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
585	C₈	—	2	4	no	—
586	C₈	—	53	2	no	—
587	C₈	—	53	2	no	—
588	C₈	—	106	4	no	—
589	C₈	C₂×D₄×D₅₃	1	2	yes	—
590	C₈	C₂×D₄×D₅₃	1	2	yes	—
591	C₂²	—	2	4	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	8	4	no	—
594	C₂²	—	16	4	no	—
595	C₂²	—	53	1	no	—
596	C₂²	—	53	1	no	—
597	C₂²	—	848	8	no	—
598	C₂²	—	848	4	no	—
599	C₂²	—	848	4	no	—
600	C₂²	—	106	2	no	—
601	C₂²	—	106	4	no	—
602	C₂²	—	424	4	no	—
603	C₂²	—	424	8	no	—
604	C₂²	—	212	8	no	—
605	C₂²	—	212	4	no	—
606	C₂²	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
607	C₄	—	2	2	no	—
608	C₄	—	2	4	no	—
609	C₄	—	53	2	no	—
610	C₄	—	106	2	no	—
611	C₄	—	106	4	no	—
612	C₄	C₂²×D₄×D₅₃	1	2	yes	—
613	C₂	—	2	1	no	—
614	C₂	—	4	4	no	—
615	C₂	—	8	4	no	—
616	C₂	—	53	2	no	—
617	C₂	—	106	1	no	—
618	C₂	—	424	4	no	—
619	C₂	—	212	4	no	—
620	C₂	D₄²×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄²×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₅₃×#3292	C₂
socle	C₂×C₅₃	C₂×D₄²
hypercenter	#3292	D₅₃
hypocenter	C₅₃	C₂×#3292
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3292

Complements

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

N	H ≅ G/N	type
1. #576×D₅₃	C₂	semidirect
2. #20286515	C₂	semidirect
3. #20286514	C₂	semidirect
4. #20286513	C₂	semidirect
5. #20286512	C₂	semidirect
6. #20286511	C₂	semidirect
7. #493×D₅₃	C₂	semidirect
8. #472×D₅₃	C₂	semidirect
9. #453×D₅₃	C₂	semidirect
10. #443×D₅₃	C₂	semidirect
11. C₅₃×#3292	C₂	semidirect
12. H(5,𝔽₂)×D₅₃	C₂²	semidirect
13. #16811428	C₂²	semidirect
15. #16811427	C₂²	semidirect
16. #16811426	C₂²	semidirect
18. #16811425	C₂²	semidirect
20. #16811424	C₂²	semidirect
22. AGL(1,ℤ/8)×D₅₃	C₂²	semidirect
26. C₂×D₈×D₅₃	C₂²	semidirect
27. #131×D₅₃	C₂²	semidirect
28. #127×D₅₃	C₂²	semidirect
29. #16811000	C₂²	semidirect
30. #16810999	C₂²	semidirect
31. #16810998	C₂²	semidirect
32. #16810894	C₂²	semidirect
33. #16810893	C₂²	semidirect
34. #16810892	C₂²	semidirect
35. #16810891	C₂²	semidirect
36. #16810890	C₂²	semidirect
37. #16810793	C₂²	semidirect
38. #16810792	C₂²	semidirect
39. #16810791	C₂²	semidirect
40. #16810790	C₂²	semidirect
41. #16810789	C₂²	semidirect
42. #16810741	C₂²	semidirect
43. #16810740	C₂²	semidirect
44. #16810739	C₂²	semidirect
45. #16810738	C₂²	semidirect
46. #16810737	C₂²	semidirect
47. #108×D₅₃	C₂²	semidirect
48. #104×D₅₃	C₂²	semidirect
49. #100×D₅₃	C₂²	semidirect
50. C₄×C₈×D₅₃	C₂²	semidirect
51. C₅₃×#576	C₂²	semidirect
52. C₅₃×#493	C₂²	semidirect
53. C₅₃×#472	C₂²	semidirect
54. C₅₃×#453	C₂²	semidirect
55. C₅₃×#443	C₂²	semidirect
57. #13133263	C₂³	semidirect
60. #13133262	C₂³	semidirect
68. #13133245	C₂³	semidirect
73. #13133243	C₂³	semidirect
75. #13133242	C₂³	semidirect
80. #13133240	D₄	semidirect
84. #13133238	D₄	semidirect
88. #13133236	D₄	semidirect
89. Q₁₆×D₅₃	D₄	semidirect
91. SD₁₆×D₅₃	D₄	semidirect
93. C₂×D₄₂₄	C₂³	semidirect
95. C₂×#11719922	C₂³	semidirect
104. D₈×D₅₃	D₄	semidirect
106. #13133225	C₂³	semidirect
107. #13133224	C₂³	semidirect
109. #13133223	C₂³	semidirect
114. #13133214	C₂³	semidirect
116. #13133213	C₂³	semidirect
118. #13133153	C₂³	semidirect
119. #13133152	C₂³	semidirect
120. #13133145	C₂³	semidirect
121. #13133144	C₂³	semidirect
122. #13133143	C₂³	semidirect
123. #13133134	C₂³	semidirect
124. #13133133	C₂³	semidirect
125. C₄×#88322	C₂³	semidirect
126. C₈×Dic₅₃	C₂³	semidirect
127. H(5,𝔽₂)×C₅₃	C₂³	semidirect
129. AGL(1,ℤ/8)×C₅₃	C₂³	semidirect
133. C₂×D₈×C₅₃	C₂³	semidirect
134. #131×C₅₃	C₂³	semidirect
135. #127×C₅₃	C₂³	semidirect
136. #108×C₅₃	C₂³	semidirect
137. #104×C₅₃	C₂³	semidirect
138. #100×C₅₃	C₂³	semidirect
139. C₄×C₈×C₅₃	C₂³	semidirect
183. Q₁₆×C₅₃	C₂×D₄	semidirect
185. SD₁₆×C₅₃	C₂×D₄	semidirect
189. D₈×C₅₃	C₂×D₄	semidirect
195. #11719925	C₂×D₄	semidirect
197. #11719924	C₂×D₄	semidirect
199. #11719923	C₂×D₄	semidirect
204. #11719922	C₂×D₄	semidirect
211. Dic₂₁₂	C₂×D₄	semidirect
215. D₄₂₄	C₂×D₄	semidirect
217. #11719913	C₂×D₄	semidirect
309. #3292	D₅₃	direct (N × H)
321. D₅₃	#3292	direct (N × H)
331. #576	C₂×D₅₃	semidirect
343. #493	C₂×D₅₃	semidirect
349. #472	C₂×D₅₃	semidirect
355. #453	C₂×D₅₃	semidirect
361. #443	C₂×D₅₃	semidirect
366. C₅₃	C₂×#3292	semidirect
372. H(5,𝔽₂)	C₂²×D₅₃	semidirect
392. AGL(1,ℤ/8)	C₂²×D₅₃	semidirect
422. C₂×D₈	C₂²×D₅₃	semidirect
428. #131	C₂²×D₅₃	semidirect
436. #127	C₂²×D₅₃	semidirect
440. #108	C₂²×D₅₃	semidirect
444. #104	C₂²×D₅₃	semidirect
447. #100	C₂²×D₅₃	semidirect
450. C₄×C₈	C₂²×D₅₃	semidirect
496. Q₁₆	D₄×D₅₃	semidirect
504. SD₁₆	D₄×D₅₃	semidirect
517. D₈	D₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#576×D₅₃	2
#20286515	2
#20286514	2
#20286513	2
#20286512	2
#20286511	2
#493×D₅₃	2
#472×D₅₃	2
#453×D₅₃	2
#443×D₅₃	2
C₅₃×#3292	2
C₂×#3292	53

Subgroups — D53×#3292 #28260313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#3292 #28260313