Subgroups — #27466287

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283434	C₂	1	1	yes	maximal, characteristic
2	#20283323	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276293	C₂	1	2	yes	maximal, complemented
4	#20276271	C₂	1	2	yes	maximal, complemented
5	#20275468	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275097	C₂	1	2	yes	maximal
7	#20274999	C₂	1	2	yes	maximal, complemented
8	#20274770	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274766	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274758	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1946	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811389	C₂²	1	2	yes	—
13	C₂×#13133223	C₂²	1	1	yes	characteristic
14	C₂×#13133223	C₂²	1	1	yes	characteristic
15	#16811173	C₂²	1	1	yes	characteristic
16	#16810696	—	2	2	no	—
17	#16810696	—	2	2	no	—
18	#16810671	C₂²	1	2	yes	—
19	#16810670	—	2	2	no	—
20	C₂×#13133153	C₂²	1	1	yes	characteristic
21	#108×D₅₃	—	2	2	no	—
22	#16810661	C₂²	1	1	yes	characteristic
23	#16810649	C₂²	1	2	yes	—
24	#16810647	C₂²	1	1	yes	characteristic
25	#16810641	C₂²	1	2	yes	—
26	#16810639	C₂²	1	2	yes	—
27	#16810596	C₂²	1	2	yes	—
28	#16810589	C₂²	1	1	yes	characteristic
29	#16810589	C₂²	1	1	yes	characteristic
30	#16810394	C₂²	1	2	yes	—
31	#16810393	C₂²	1	2	yes	—
32	#16810383	C₂²	1	2	yes	—
33	#16810379	C₂²	1	2	yes	—
34	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#427	C₂²	1	1	yes	characteristic
36	C₅₃×#424	C₂²	1	2	yes	—
37	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#339	C₂²	1	2	yes	—
39	C₂×#11719945	C₂³	1	1	yes	characteristic
40	C₂×#11719945	C₂³	1	1	yes	characteristic
41	#13133225	—	4	2	no	—
42	#13133223	—	2	1	no	—
43	#13133223	—	2	1	no	—
44	#13133223	—	2	1	no	—
45	#13133223	—	2	1	no	—
46	#13133223	—	2	1	no	—
47	#13133223	—	2	1	no	—
48	#13133223	—	2	1	no	—
49	#13133223	—	2	1	no	—
50	#13133223	—	4	2	no	—
51	C₂×#11719917	C₂³	1	1	yes	characteristic
52	C₂×#11719917	C₂³	1	2	yes	—
53	C₂×#11719917	C₂³	1	1	yes	characteristic
54	C₂×#11719912	—	2	1	no	—
55	C₂×#11719912	D₄	1	2	yes	—
56	M₁₆×D₅₃	—	2	4	no	—
57	C₂×C₈×D₅₃	—	2	1	no	—
58	C₂×C₈×D₅₃	D₄	1	2	yes	—
59	#13133217	Q₈	1	1	yes	characteristic
60	#13133214	Q₈	1	1	yes	characteristic
61	#13133205	Q₈	1	1	yes	characteristic
62	#13133205	Q₈	1	1	yes	characteristic
63	#13133191	—	2	2	no	—
64	C₄×D₂₁₂	—	2	2	no	—
65	#13133176	C₂³	1	2	yes	—
66	#13133153	—	2	1	no	—
67	#13133153	—	2	1	no	—
68	#13133152	—	2	4	no	—
69	#13133151	—	2	1	no	—
70	#13133149	—	2	1	no	—
71	C₄⋊#88322	—	2	2	no	—
72	#13133146	—	4	2	no	—
73	#13133145	—	4	2	no	—
74	#13133143	—	4	2	no	—
75	#13133131	—	2	4	no	—
76	#13133129	—	2	2	no	—
77	#13133129	—	2	2	no	—
78	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	#117×C₅₃	C₂³	1	2	yes	—
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	#108×C₅₃	—	2	1	no	—
84	#108×C₅₃	—	2	1	no	—
85	#107×C₅₃	—	2	1	no	—
86	#106×C₅₃	—	2	1	no	—
87	C₄⋊C₈×C₅₃	—	4	2	no	—
88	#97×C₅₃	—	2	1	no	—
89	#97×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₂²⋊D₅₃	—	2	1	no	—
93	C₂×C₂²⋊D₅₃	#41	1	2	yes	—
94	#11719945	—	2	1	no	—
95	#11719945	—	2	1	no	—
96	#11719945	—	2	1	no	—
97	#11719945	—	2	1	no	—
98	#11719945	—	2	1	no	—
99	#11719945	—	2	1	no	—
100	#11719945	—	2	1	no	—
101	#11719945	—	2	1	no	—
102	C₂×D₂₁₂	—	2	2	no	—
103	C₂×D₂₁₂	C₂×Q₈	1	1	yes	characteristic
104	C₂×D₂₁₂	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₄×D₅₃	—	2	1	no	—
106	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
107	C₂×Dic₁₀₆	C₂×Q₈	1	1	yes	characteristic
108	C₂×Dic₁₀₆	C₂×Q₈	1	1	yes	characteristic
109	M₁₆×C₅₃	—	2	2	no	—
110	M₁₆×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
118	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
120	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	—	2	1	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
126	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
127	C₂²⋊Dic₅₃	—	2	2	no	—
128	#11719921	—	2	2	no	—
129	#11719921	—	2	2	no	—
130	C₄⋊Dic₅₃	—	2	2	no	—
131	C₄⋊Dic₅₃	—	2	2	no	—
132	C₄×Dic₅₃	—	2	2	no	—
133	#11719917	—	2	1	no	—
134	#11719917	—	2	1	no	—
135	#11719917	—	2	2	no	—
136	#11719917	—	2	1	no	—
137	#11719917	—	2	1	no	—
138	#11719917	—	2	2	no	—
139	C₂×#88322	—	2	1	no	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	C₂×D₄	1	2	yes	—
142	#11719912	—	2	2	no	—
143	#11719912	—	2	2	no	—
144	#11719912	—	4	4	no	—
145	#11719912	—	4	1	no	—
146	#11719912	—	4	1	no	—
147	C₈×D₅₃	—	2	2	no	—
148	C₈×D₅₃	—	2	2	no	—
149	C₈×D₅₃	—	4	4	no	—
150	C₈×D₅₃	—	4	1	no	—
151	C₈×D₅₃	—	4	1	no	—
152	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
153	C₂²×D₅₃	—	2	1	no	—
154	C₂²×D₅₃	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	8	1	no	—
159	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂²⋊D₅₃	—	2	2	no	—
166	C₂²⋊D₅₃	—	2	2	no	—
167	C₂²⋊D₅₃	—	4	1	no	—
168	C₂²⋊D₅₃	—	4	1	no	—
169	C₂×Dic₅₃	—	2	1	no	—
170	C₂×Dic₅₃	—	4	2	no	—
171	C₂×Dic₅₃	C₂²⋊Q₈	1	2	yes	—
172	D₂₁₂	—	2	1	no	—
173	D₂₁₂	—	2	1	no	—
174	D₂₁₂	—	2	1	no	—
175	D₂₁₂	—	2	1	no	—
176	D₂₁₂	—	8	2	no	—
177	C₄×D₅₃	—	2	2	no	—
178	C₄×D₅₃	—	2	2	no	—
179	C₄×D₅₃	—	4	1	no	—
180	C₄×D₅₃	—	4	1	no	—
181	Dic₁₀₆	—	2	1	no	—
182	Dic₁₀₆	—	2	1	no	—
183	Dic₁₀₆	—	2	1	no	—
184	Dic₁₀₆	—	2	1	no	—
185	C₈×C₅₃	—	4	2	no	—
186	C₈×C₅₃	—	4	1	no	—
187	C₈×C₅₃	#123	1	1	yes	characteristic
188	C₈×C₅₃	#123	1	1	yes	characteristic
189	C₈×C₅₃	#123	1	1	yes	characteristic
190	C₈×C₅₃	#123	1	1	yes	characteristic
191	#88322	—	2	2	no	—
192	#88322	—	4	1	no	—
193	#88322	—	4	2	no	—
194	#27694	—	53	1	no	maximal, Sylow
195	C₂²×C₅₃	—	4	1	no	—
196	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
197	C₂²×C₅₃	#553	1	1	yes	characteristic
198	C₂²×C₅₃	#553	1	1	yes	characteristic
199	C₂×D₅₃	—	2	2	no	—
200	C₂×D₅₃	—	4	2	no	—
201	C₂×D₅₃	—	4	2	no	—
202	C₂×D₅₃	—	4	1	no	—
203	C₄×C₅₃	—	4	1	no	—
204	C₄×C₅₃	—	4	1	no	—
205	C₄×C₅₃	—	8	2	no	—
206	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
207	C₄×C₅₃	#535	1	1	yes	characteristic
208	C₄×C₅₃	#554	1	1	yes	characteristic
209	C₄×C₅₃	#554	1	1	yes	characteristic
210	Dic₅₃	—	2	2	no	—
211	Dic₅₃	—	4	1	no	—
212	Dic₅₃	—	8	2	no	—
213	#2932	—	53	1	no	—
214	#2916	—	53	1	no	—
215	#2102	—	53	2	no	—
216	#2098	—	53	2	no	—
217	#1946	—	53	1	no	—
218	#1946	D₅₃	1	1	yes	characteristic, complemented
219	#1884	—	53	2	no	—
220	#1868	—	53	2	no	—
221	#1814	—	53	1	no	—
222	#1814	—	53	1	no	—
223	#1812	—	53	1	no	—
224	C₂×C₅₃	—	2	1	no	—
225	C₂×C₅₃	—	2	1	no	—
226	C₂×C₅₃	—	2	1	no	—
227	C₂×C₅₃	#2487	1	1	yes	characteristic
228	D₅₃	—	4	2	no	—
229	D₅₃	—	8	1	no	—
230	#568	—	53	2	no	—
231	C₂×#131	—	53	1	no	—
232	C₂×#131	—	53	1	no	—
233	#532	—	53	1	no	—
234	#435	—	106	2	no	—
235	#435	—	106	2	no	—
236	#430	—	53	2	no	—
237	#430	—	106	2	no	—
238	C₂×#108	—	53	1	no	—
239	C₂×#108	—	106	2	no	—
240	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
241	#427	—	53	1	no	—
242	#427	C₂×D₅₃	1	1	yes	characteristic
243	#424	—	53	2	no	—
244	#424	C₂×D₅₃	1	2	yes	—
245	#423	—	53	1	no	—
246	#421	—	53	2	no	—
247	#421	—	53	2	no	—
248	#410	—	53	2	no	—
249	#408	—	53	1	no	—
250	#408	—	53	1	no	—
251	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
252	#344	—	53	2	no	—
253	#344	—	53	2	no	—
254	#339	—	53	2	no	—
255	#339	C₂×D₅₃	1	2	yes	—
256	H(3,ℤ/4)	—	53	2	no	—
257	C₅₃	#27694	1	1	yes	characteristic, complemented, Sylow
258	C₂×#41	—	53	1	no	—
259	C₂×#41	—	53	1	no	—
260	#131	—	106	1	no	—
261	#131	—	106	1	no	—
262	#131	—	106	1	no	—
263	#131	—	106	1	no	—
264	#131	—	106	1	no	—
265	#131	—	106	1	no	—
266	#131	—	106	1	no	—
267	#131	—	106	1	no	—
268	#131	—	212	2	no	—
269	#131	—	212	2	no	—
270	C₂×M₁₆	—	53	1	no	—
271	C₂×M₁₆	—	53	2	no	—
272	C₂×M₁₆	—	53	1	no	—
273	C₂×M₁₆	—	53	2	no	—
274	C₂×M₁₆	—	106	1	no	—
275	C₂×M₁₆	—	106	4	no	—
276	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
277	C₂²×C₈	—	53	2	no	—
278	C₂²×C₈	—	106	1	no	—
279	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
280	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
281	C₄⋊Q₈	—	53	1	no	—
282	#127	—	53	1	no	—
283	#124	—	53	1	no	—
284	#124	—	53	1	no	—
285	#121	—	106	2	no	—
286	C₄×D₄	—	106	2	no	—
287	#117	—	53	2	no	—
288	#117	C₂²×D₅₃	1	2	yes	—
289	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
290	#108	—	2	1	no	—
291	#108	—	2	1	no	—
292	#108	—	106	4	no	—
293	#108	—	106	1	no	—
294	#108	—	106	1	no	—
295	#107	—	2	1	no	—
296	#107	—	106	1	no	—
297	#106	—	2	1	no	—
298	#106	—	106	1	no	—
299	C₄⋊C₈	—	4	2	no	—
300	C₄⋊C₈	—	106	2	no	—
301	C₄⋊C₈	—	212	2	no	—
302	#104	—	212	2	no	—
303	#104	—	212	2	no	—
304	#99	—	106	4	no	—
305	C₂²⋊C₈	—	106	2	no	—
306	C₂²⋊C₈	—	106	2	no	—
307	#97	—	2	1	no	—
308	#97	—	2	1	no	—
309	#41	—	106	1	no	—
310	#41	—	106	1	no	—
311	#41	—	106	1	no	—
312	#41	—	106	1	no	—
313	#41	—	106	1	no	—
314	#41	—	106	1	no	—
315	#41	—	106	1	no	—
316	#41	—	106	1	no	—
317	C₂×Q₈	—	53	1	no	—
318	C₂×Q₈	—	53	1	no	—
319	C₂×D₄	—	53	2	no	—
320	C₂×D₄	—	53	1	no	—
321	C₂×D₄	—	53	1	no	—
322	C₂×D₄	—	106	2	no	—
323	C₂×D₄	—	106	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	53	2	no	—
326	C₂²×C₄	—	106	1	no	—
327	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
328	M₁₆	—	2	2	no	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	106	1	no	—
331	M₁₆	—	106	1	no	—
332	M₁₆	—	106	2	no	—
333	M₁₆	—	106	2	no	—
334	M₁₆	—	106	2	no	—
335	M₁₆	—	106	1	no	—
336	M₁₆	—	106	1	no	—
337	M₁₆	—	106	2	no	—
338	M₁₆	—	212	4	no	—
339	M₁₆	—	212	1	no	—
340	M₁₆	—	212	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	53	2	no	—
346	C₂×C₈	—	106	2	no	—
347	C₂×C₈	—	106	2	no	—
348	C₂×C₈	—	106	1	no	—
349	C₂×C₈	—	106	2	no	—
350	C₂×C₈	—	212	4	no	—
351	C₂×C₈	—	212	1	no	—
352	C₂×C₈	—	212	1	no	—
353	C₂×C₈	#11719945	1	1	yes	characteristic
354	C₂×C₈	#11719945	1	1	yes	characteristic
355	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
356	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
357	C₂×C₈	#11719949	1	1	yes	characteristic
358	C₂×C₈	#11719949	1	1	yes	characteristic
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	106	2	no	—
361	C₄⋊C₄	—	106	2	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	106	2	no	—
364	C₂²⋊C₄	—	106	2	no	—
365	C₂²⋊C₄	—	106	2	no	—
366	C₄²	—	2	1	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	106	2	no	—
369	C₄²	Q₈×D₅₃	1	1	yes	characteristic
370	C₄²	Q₈×D₅₃	1	1	yes	characteristic
371	C₂³	—	53	2	no	—
372	C₂³	—	106	1	no	—
373	C₂³	#41×D₅₃	1	1	yes	characteristic
374	Q₈	—	106	1	no	—
375	Q₈	—	106	1	no	—
376	Q₈	—	106	1	no	—
377	Q₈	—	106	1	no	—
378	D₄	—	106	2	no	—
379	D₄	—	106	2	no	—
380	D₄	—	106	1	no	—
381	D₄	—	106	1	no	—
382	D₄	—	106	1	no	—
383	D₄	—	106	1	no	—
384	D₄	—	424	2	no	—
385	D₄	—	212	1	no	—
386	D₄	—	212	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	8	1	no	—
391	C₂×C₄	—	53	2	no	—
392	C₂×C₄	—	106	2	no	—
393	C₂×C₄	—	106	2	no	—
394	C₂×C₄	—	106	1	no	—
395	C₂×C₄	—	212	2	no	—
396	C₂×C₄	—	212	1	no	—
397	C₂×C₄	—	212	1	no	—
398	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
399	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
401	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
402	C₂×C₄	#13133262	1	1	yes	characteristic
403	C₂×C₄	#13133263	1	1	yes	characteristic
404	C₈	—	4	1	no	—
405	C₈	—	4	2	no	—
406	C₈	—	106	2	no	—
407	C₈	—	212	1	no	—
408	C₈	—	212	2	no	—
409	C₈	#13133202	1	1	yes	characteristic
410	C₈	#13133202	1	1	yes	characteristic
411	C₈	#13133202	1	1	yes	characteristic
412	C₈	#13133202	1	1	yes	characteristic
413	C₂²	—	4	1	no	—
414	C₂²	—	106	2	no	—
415	C₂²	—	212	2	no	—
416	C₂²	—	212	2	no	—
417	C₂²	—	212	1	no	—
418	C₂²	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
419	C₂²	#16811323	1	1	yes	characteristic
420	C₂²	#16811326	1	1	yes	characteristic
421	C₄	—	4	1	no	—
422	C₄	—	4	1	no	—
423	C₄	—	8	2	no	—
424	C₄	—	106	2	no	—
425	C₄	—	424	2	no	—
426	C₄	—	212	1	no	—
427	C₄	C₂×#13133202	1	1	yes	characteristic
428	C₄	#16811189	1	1	yes	characteristic
429	C₄	#16811333	1	1	yes	characteristic
430	C₄	#16811334	1	1	yes	characteristic
431	C₂	—	2	1	no	—
432	C₂	—	2	1	no	—
433	C₂	—	2	1	no	—
434	C₂	—	424	1	no	—
435	C₂	—	212	2	no	—
436	C₂	#20279030	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1946	C₂
socle	C₂×C₅₃	#2487
prime radical	#16811173	C₂²
hypercenter	#1946	D₅₃
hypocenter	C₅₃	#27694
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1946

Complements

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

N	H ≅ G/N	type
2. #20283323	C₂	semidirect
3. #20276293	C₂	semidirect
4. #20276271	C₂	semidirect
5. #20275468	C₂	semidirect
7. #20274999	C₂	semidirect
8. #20274770	C₂	semidirect
9. #20274766	C₂	semidirect
10. #20274758	C₂	semidirect
11. C₅₃×#1946	C₂	semidirect
218. #1946	D₅₃	semidirect
257. C₅₃	#27694	semidirect

Maximal subgroups

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

subgroup	index
#20283434	2
#20283323	2
#20276293	2
#20276271	2
#20275468	2
#20275097	2
#20274999	2
#20274770	2
#20274766	2
#20274758	2
C₅₃×#1946	2
#27694	53