Subgroups — #27469916

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284371	C₂	1	1	yes	maximal, characteristic
2	#20283323	C₂	1	1	yes	maximal, characteristic
3	#20276185	C₂	1	2	yes	maximal, complemented
4	#20276184	C₂	1	2	yes	maximal, complemented
5	#20275630	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275628	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275266	C₂	1	2	yes	maximal
8	#20275236	C₂	1	2	yes	maximal
9	#20274838	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274516	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
13	#16811389	C₂²	1	2	yes	—
14	C₂×#13133223	C₂²	1	1	yes	characteristic
15	#16811199	C₂²	1	2	yes	—
16	#16810882	—	2	2	no	—
17	#16810865	—	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	#16810633	C₂²	1	2	yes	—
23	#16810631	C₂²	1	2	yes	—
24	#16810631	C₂²	1	2	yes	—
25	#16810605	C₂²	1	1	yes	characteristic
26	C₂×#13133135	C₂²	1	2	yes	—
27	C₂×#13133135	C₂²	1	2	yes	—
28	#16810596	C₂²	1	1	yes	characteristic
29	#16810578	C₂²	1	1	yes	characteristic
30	#16810395	C₂²	1	1	yes	characteristic
31	#16810393	C₂²	1	2	yes	—
32	#16810393	C₂²	1	1	yes	characteristic
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×#405	C₂²	1	1	yes	characteristic
37	C₅₃×#344	C₂²	1	2	yes	—
38	C₅₃×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#11719945	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
41	C₂×Dic₂₁₂	—	2	1	no	—
42	C₂×Dic₂₁₂	—	2	1	no	—
43	C₂×Dic₂₁₂	—	2	1	no	—
44	C₂×Dic₂₁₂	—	2	1	no	—
45	C₂×Dic₂₁₂	—	2	1	no	—
46	C₂×Dic₂₁₂	—	2	1	no	—
47	#13133225	—	4	2	no	—
48	#13133223	—	2	1	no	—
49	#13133223	—	2	1	no	—
50	#13133223	—	2	1	no	—
51	#13133223	—	2	1	no	—
52	#13133223	—	4	2	no	—
53	C₂×#11719917	C₂³	1	1	yes	characteristic
54	C₂×#11719917	C₂³	1	2	yes	—
55	C₂×#11719917	C₂³	1	1	yes	characteristic
56	C₂×#11719912	D₄	1	2	yes	—
57	M₁₆×D₅₃	—	2	4	no	—
58	C₂×C₈×D₅₃	D₄	1	2	yes	—
59	#13133217	—	2	2	no	—
60	#13133205	—	2	2	no	—
61	#13133201	—	2	4	no	—
62	#13133197	—	2	4	no	—
63	#13133176	C₂³	1	1	yes	characteristic
64	#13133153	—	2	1	no	—
65	#13133153	—	2	1	no	—
66	#13133152	—	2	4	no	—
67	#13133142	—	4	2	no	—
68	#13133140	—	2	4	no	—
69	#13133140	—	4	2	no	—
70	#13133135	—	2	1	no	—
71	#13133135	—	4	2	no	—
72	#13133135	—	4	2	no	—
73	#13133133	—	2	1	no	—
74	#13133131	—	2	2	no	—
75	#13133127	—	2	1	no	—
76	C₈×Dic₅₃	—	2	1	no	—
77	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	#117×C₅₃	C₂³	1	1	yes	characteristic
81	#117×C₅₃	C₂³	1	2	yes	—
82	#108×C₅₃	—	2	1	no	—
83	#108×C₅₃	—	2	1	no	—
84	#107×C₅₃	—	2	1	no	—
85	#106×C₅₃	—	2	1	no	—
86	#97×C₅₃	—	2	1	no	—
87	C₄×C₈×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₂²⋊D₅₃	#41	1	2	yes	—
90	C₂²×Dic₅₃	—	2	2	no	—
91	#11719945	—	2	1	no	—
92	#11719945	—	2	1	no	—
93	#11719945	—	2	1	no	—
94	#11719945	—	2	1	no	—
95	C₂×D₂₁₂	#41	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
97	C₂×Dic₁₀₆	—	2	2	no	—
98	C₂×Dic₁₀₆	—	2	2	no	—
99	C₂×Dic₁₀₆	—	2	2	no	—
100	C₂×Dic₁₀₆	—	2	2	no	—
101	C₂×Dic₁₀₆	—	2	2	no	—
102	C₂×Dic₁₀₆	—	2	2	no	—
103	C₂×Dic₁₀₆	#41	1	1	yes	characteristic
104	M₁₆×C₅₃	—	2	2	no	—
105	M₁₆×C₅₃	—	2	1	no	—
106	M₁₆×C₅₃	—	2	1	no	—
107	M₁₆×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
115	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	1	no	—
118	C₂²⋊C₄×C₅₃	—	2	1	no	—
119	C₂²⋊C₄×C₅₃	—	2	2	no	—
120	C₄²×C₅₃	—	2	1	no	—
121	C₄²×C₅₃	—	2	2	no	—
122	C₂²⋊Dic₅₃	—	2	1	no	—
123	#11719921	—	4	4	no	—
124	C₄⋊Dic₅₃	—	2	1	no	—
125	C₄⋊Dic₅₃	—	4	4	no	—
126	C₄×Dic₅₃	—	2	1	no	—
127	#11719917	—	2	1	no	—
128	#11719917	—	2	1	no	—
129	#11719917	—	2	2	no	—
130	#11719917	—	2	1	no	—
131	#11719917	—	2	1	no	—
132	#11719917	—	2	2	no	—
133	C₂×#88322	—	2	1	no	—
134	C₂×#88322	—	2	2	no	—
135	C₂×#88322	C₂×D₄	1	2	yes	—
136	Dic₂₁₂	—	4	1	no	—
137	Dic₂₁₂	—	4	1	no	—
138	Dic₂₁₂	—	4	1	no	—
139	Dic₂₁₂	—	4	1	no	—
140	#11719912	—	2	2	no	—
141	#11719912	—	2	2	no	—
142	#11719912	—	4	4	no	—
143	C₈×D₅₃	—	2	2	no	—
144	C₈×D₅₃	—	2	2	no	—
145	C₈×D₅₃	—	4	4	no	—
146	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₂²×D₅₃	#124	1	2	yes	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₅₃	—	2	2	no	—
157	C₂²⋊D₅₃	—	2	2	no	—
158	C₂×Dic₅₃	—	4	2	no	—
159	C₂×Dic₅₃	—	4	2	no	—
160	C₂×Dic₅₃	—	4	2	no	—
161	C₂×Dic₅₃	—	4	1	no	—
162	C₂×Dic₅₃	#124	1	2	yes	—
163	D₂₁₂	—	2	1	no	—
164	D₂₁₂	—	2	1	no	—
165	C₄×D₅₃	—	2	2	no	—
166	C₄×D₅₃	—	2	2	no	—
167	Dic₁₀₆	—	2	1	no	—
168	Dic₁₀₆	—	2	1	no	—
169	Dic₁₀₆	—	4	2	no	—
170	Dic₁₀₆	—	4	2	no	—
171	Dic₁₀₆	—	4	2	no	—
172	Dic₁₀₆	—	4	2	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	—	4	1	no	—
175	C₈×C₅₃	#123	1	1	yes	characteristic
176	C₈×C₅₃	#123	1	1	yes	characteristic
177	C₈×C₅₃	#123	1	1	yes	characteristic
178	C₈×C₅₃	#123	1	1	yes	characteristic
179	#88322	—	2	2	no	—
180	#88322	—	4	2	no	—
181	#88322	—	4	1	no	—
182	#27988	—	53	1	no	maximal, Sylow
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
185	C₂²×C₅₃	#553	1	1	yes	characteristic
186	C₂²×C₅₃	#553	1	1	yes	characteristic
187	C₂×D₅₃	—	2	2	no	—
188	C₂×D₅₃	—	4	2	no	—
189	C₄×C₅₃	—	8	2	no	—
190	C₄×C₅₃	—	8	1	no	—
191	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
192	C₄×C₅₃	#535	1	1	yes	characteristic
193	C₄×C₅₃	#553	1	1	yes	characteristic
194	C₄×C₅₃	#553	1	1	yes	characteristic
195	Dic₅₃	—	2	2	no	—
196	Dic₅₃	—	8	1	no	—
197	Dic₅₃	—	8	2	no	—
198	#3059	—	53	1	no	—
199	#2916	—	53	1	no	—
200	#2082	—	53	2	no	—
201	#2082	—	53	2	no	—
202	#1979	—	53	1	no	—
203	#1979	—	53	1	no	—
204	#1915	—	53	2	no	—
205	#1909	—	53	2	no	—
206	#1833	—	53	1	no	—
207	#1833	D₅₃	1	1	yes	characteristic, complemented
208	#1757	—	53	1	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	—	2	1	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	#2504	1	1	yes	characteristic
213	D₅₃	—	4	2	no	—
214	C₂²×Q₁₆	—	53	1	no	—
215	#568	—	53	2	no	—
216	C₂×#131	—	53	1	no	—
217	#536	—	53	2	no	—
218	#470	—	106	2	no	—
219	#467	—	106	2	no	—
220	#430	—	53	2	no	—
221	#430	—	106	2	no	—
222	C₂×#108	—	53	1	no	—
223	C₂×#108	—	106	2	no	—
224	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
225	#427	C₂×D₅₃	1	1	yes	characteristic
226	#419	—	53	2	no	—
227	#419	—	53	2	no	—
228	#419	—	53	2	no	—
229	#413	—	53	1	no	—
230	C₂×#101	—	53	2	no	—
231	C₂×#101	—	53	2	no	—
232	#410	—	53	1	no	—
233	#405	—	53	1	no	—
234	#405	C₂×D₅₃	1	1	yes	characteristic
235	#344	—	53	1	no	—
236	#344	—	53	1	no	—
237	#344	—	53	2	no	—
238	#344	C₂×D₅₃	1	2	yes	—
239	H(3,ℤ/4)	—	53	2	no	—
240	H(3,ℤ/4)	C₂×D₅₃	1	2	yes	—
241	C₅₃	#27988	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	53	1	no	—
243	C₂²×Q₈	—	53	2	no	—
244	C₂×Q₁₆	—	106	1	no	—
245	C₂×Q₁₆	—	106	1	no	—
246	C₂×Q₁₆	—	106	1	no	—
247	C₂×Q₁₆	—	106	1	no	—
248	C₂×Q₁₆	—	106	1	no	—
249	C₂×Q₁₆	—	106	1	no	—
250	#131	—	106	1	no	—
251	#131	—	106	1	no	—
252	#131	—	106	1	no	—
253	#131	—	106	1	no	—
254	#131	—	212	2	no	—
255	#131	—	212	2	no	—
256	C₂×M₁₆	—	53	1	no	—
257	C₂×M₁₆	—	53	2	no	—
258	C₂×M₁₆	—	53	2	no	—
259	C₂×M₁₆	—	53	1	no	—
260	C₂×M₁₆	—	106	4	no	—
261	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
262	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
263	C₂²×C₈	—	53	2	no	—
264	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
265	C₄⋊Q₈	—	106	2	no	—
266	#124	—	106	2	no	—
267	#123	—	106	4	no	—
268	C₂²⋊Q₈	—	106	4	no	—
269	#117	—	53	1	no	—
270	#117	C₂²×D₅₃	1	2	yes	—
271	#117	C₂²×D₅₃	1	1	yes	characteristic
272	#108	—	2	1	no	—
273	#108	—	2	1	no	—
274	#108	—	106	4	no	—
275	#108	—	106	1	no	—
276	#108	—	106	1	no	—
277	#107	—	2	1	no	—
278	#106	—	2	1	no	—
279	#103	—	106	4	no	—
280	#103	—	212	2	no	—
281	#103	—	212	2	no	—
282	#101	—	106	1	no	—
283	#101	—	212	2	no	—
284	#101	—	212	2	no	—
285	#100	—	106	1	no	—
286	#99	—	106	2	no	—
287	#97	—	2	1	no	—
288	#97	—	106	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	106	1	no	—
291	#41	—	106	1	no	—
292	#41	—	106	1	no	—
293	#41	—	106	1	no	—
294	#41	—	106	1	no	—
295	C₂×Q₈	—	53	1	no	—
296	C₂×Q₈	—	106	2	no	—
297	C₂×Q₈	—	106	2	no	—
298	C₂×Q₈	—	106	2	no	—
299	C₂×Q₈	—	106	2	no	—
300	C₂×Q₈	—	106	2	no	—
301	C₂×Q₈	—	106	2	no	—
302	C₂×D₄	—	53	2	no	—
303	C₂×D₄	—	53	1	no	—
304	C₂²×C₄	—	53	2	no	—
305	C₂²×C₄	—	106	2	no	—
306	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
307	Q₁₆	—	212	1	no	—
308	Q₁₆	—	212	1	no	—
309	Q₁₆	—	212	1	no	—
310	Q₁₆	—	212	1	no	—
311	M₁₆	—	2	2	no	—
312	M₁₆	—	2	1	no	—
313	M₁₆	—	2	1	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	106	1	no	—
316	M₁₆	—	106	1	no	—
317	M₁₆	—	106	2	no	—
318	M₁₆	—	106	2	no	—
319	M₁₆	—	106	2	no	—
320	M₁₆	—	106	1	no	—
321	M₁₆	—	106	1	no	—
322	M₁₆	—	106	2	no	—
323	M₁₆	—	212	4	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	53	2	no	—
327	C₂×C₈	—	106	2	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	106	1	no	—
330	C₂×C₈	—	106	2	no	—
331	C₂×C₈	—	212	4	no	—
332	C₂×C₈	#11719945	1	1	yes	characteristic
333	C₂×C₈	#11719945	1	1	yes	characteristic
334	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
335	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
336	C₂×C₈	#11719949	1	1	yes	characteristic
337	C₂×C₈	#11719949	1	1	yes	characteristic
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	106	1	no	—
341	C₄⋊C₄	—	212	4	no	—
342	C₂²⋊C₄	—	2	1	no	—
343	C₂²⋊C₄	—	2	2	no	—
344	C₂²⋊C₄	—	106	1	no	—
345	C₂²⋊C₄	—	212	4	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	1	no	—
348	C₄²	—	106	1	no	—
349	C₂³	—	53	2	no	—
350	C₂³	#41×D₅₃	1	1	yes	characteristic
351	Q₈	—	106	1	no	—
352	Q₈	—	106	1	no	—
353	Q₈	—	212	2	no	—
354	Q₈	—	212	2	no	—
355	Q₈	—	212	2	no	—
356	Q₈	—	212	2	no	—
357	D₄	—	106	2	no	—
358	D₄	—	106	2	no	—
359	D₄	—	106	1	no	—
360	D₄	—	106	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	53	2	no	—
364	C₂×C₄	—	106	2	no	—
365	C₂×C₄	—	106	2	no	—
366	C₂×C₄	—	212	2	no	—
367	C₂×C₄	—	212	1	no	—
368	C₂×C₄	—	212	2	no	—
369	C₂×C₄	—	212	2	no	—
370	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
371	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
372	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
373	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
374	C₂×C₄	#13133263	1	1	yes	characteristic
375	C₂×C₄	#13133263	1	1	yes	characteristic
376	C₈	—	4	1	no	—
377	C₈	—	4	2	no	—
378	C₈	—	106	2	no	—
379	C₈	—	212	1	no	—
380	C₈	—	212	2	no	—
381	C₈	#13133202	1	1	yes	characteristic
382	C₈	#13133202	1	1	yes	characteristic
383	C₈	#13133202	1	1	yes	characteristic
384	C₈	#13133202	1	1	yes	characteristic
385	C₂²	—	4	1	no	—
386	C₂²	—	106	2	no	—
387	C₂²	—	212	2	no	—
388	C₂²	#124×D₅₃	1	1	yes	characteristic
389	C₂²	#16811317	1	1	yes	characteristic
390	C₂²	#16811326	1	1	yes	characteristic
391	C₄	—	8	2	no	—
392	C₄	—	8	1	no	—
393	C₄	—	106	2	no	—
394	C₄	—	424	1	no	—
395	C₄	—	424	2	no	—
396	C₄	C₂×#13133202	1	1	yes	characteristic
397	C₄	#16811191	1	1	yes	characteristic
398	C₄	#16811317	1	1	yes	characteristic
399	C₄	#16811326	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	212	2	no	—
404	C₂	#20279204	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20279204
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1833	C₂
socle	C₂×C₅₃	#2504
prime radical	C₂×#11719945	C₂³
hypercenter	#1833	D₅₃
hypocenter	C₅₃	#27988
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1833

Complements

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

N	H ≅ G/N	type
3. #20276185	C₂	semidirect
4. #20276184	C₂	semidirect
5. #20275630	C₂	semidirect
6. #20275628	C₂	semidirect
9. #20274838	C₂	semidirect
11. C₅₃×#1833	C₂	semidirect
207. #1833	D₅₃	semidirect
241. C₅₃	#27988	semidirect

Maximal subgroups

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

subgroup	index
#20284371	2
#20283323	2
#20276185	2
#20276184	2
#20275630	2
#20275628	2
#20275266	2
#20275236	2
#20274838	2
#20274516	2
C₅₃×#1833	2
#27988	53