Subgroups — #27443004

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285251	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284582	C₂	1	2	yes	maximal
3	#20283311	C₂	1	1	yes	maximal, characteristic
4	#20276226	C₂	1	2	yes	maximal, complemented
5	#20276065	C₂	1	2	yes	maximal, complemented
6	#20275498	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275463	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274947	C₂	1	2	yes	maximal
9	#20274780	C₂	1	1	yes	maximal, characteristic
10	#20274552	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	C₂²	1	1	yes	characteristic
13	#16811254	C₂²	1	2	yes	—
14	#16811104	C₂²	1	2	yes	—
15	#16811031	—	2	2	no	—
16	#16811027	—	2	1	no	—
17	#16811024	—	2	1	no	—
18	#16810933	—	2	2	no	—
19	#16810922	—	2	2	no	—
20	#16810912	—	2	2	no	—
21	#16810907	—	2	2	no	—
22	C₂×#13133153	C₂²	1	1	yes	characteristic
23	#16810664	—	2	2	no	—
24	#16810663	C₂²	1	2	yes	—
25	#16810663	C₂²	1	2	yes	—
26	#16810663	C₂²	1	2	yes	—
27	#16810661	C₂²	1	1	yes	characteristic
28	#16810660	—	2	1	no	—
29	#107×D₅₃	—	2	1	no	—
30	#16810649	C₂²	1	2	yes	—
31	C₂×#13133145	C₂²	1	1	yes	characteristic
32	#16810622	C₂²	1	2	yes	—
33	#16810620	C₂²	1	1	yes	characteristic
34	#16810578	C₂²	1	1	yes	characteristic
35	#16810392	C₂²	1	2	yes	—
36	#16810392	C₂²	1	2	yes	—
37	#16810388	C₂²	1	1	yes	characteristic
38	#16810383	C₂²	1	2	yes	—
39	#16810383	C₂²	1	1	yes	characteristic
40	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#106×C₅₃	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#382	C₂²	1	2	yes	—
45	C₂×#11719945	C₂³	1	1	yes	characteristic
46	#13133223	—	2	2	no	—
47	#13133223	Q₈	1	4	yes	complemented
48	C₂×#11719917	C₂³	1	2	yes	—
49	C₂×#11719917	C₂³	1	1	yes	characteristic
50	C₂×#11719917	C₂³	1	1	yes	characteristic
51	C₂×#11719912	—	2	1	no	—
52	C₂×C₈×D₅₃	—	2	1	no	—
53	#13133216	—	2	2	no	—
54	#13133216	—	2	2	no	—
55	#13133208	—	2	2	no	—
56	#13133208	—	2	2	no	—
57	#13133196	—	4	2	no	—
58	#13133182	—	2	2	no	—
59	#13133179	—	2	2	no	—
60	C₄×C₂²⋊D₅₃	—	4	2	no	—
61	#13133176	C₂³	1	1	yes	characteristic
62	#13133176	C₂³	1	2	yes	—
63	#13133175	—	2	2	no	—
64	C₄⋊C₄×D₅₃	—	2	2	no	—
65	#13133153	—	2	2	no	—
66	#13133151	D₄	1	1	yes	characteristic
67	#13133151	D₄	1	1	yes	characteristic
68	#13133150	—	2	2	no	—
69	#13133150	—	2	2	no	—
70	#13133150	—	2	2	no	—
71	#13133149	D₄	1	2	yes	—
72	#13133148	—	2	2	no	—
73	#13133148	—	2	2	no	—
74	#13133148	—	2	2	no	—
75	#13133146	—	2	2	no	—
76	#13133146	—	2	2	no	—
77	#13133145	Q₈	1	4	yes	complemented
78	#13133142	D₄	1	1	yes	characteristic
79	#13133142	D₄	1	1	yes	characteristic
80	#13133140	—	2	2	no	—
81	#13133139	D₄	1	1	yes	characteristic
82	#13133139	D₄	1	1	yes	characteristic
83	#13133137	—	2	2	no	—
84	#13133127	—	2	1	no	—
85	C₈×Dic₅₃	—	2	1	no	—
86	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	#107×C₅₃	—	2	1	no	—
92	#107×C₅₃	—	2	1	no	—
93	#107×C₅₃	—	2	1	no	—
94	#107×C₅₃	—	2	1	no	—
95	#106×C₅₃	—	2	2	no	—
96	#106×C₅₃	—	2	2	no	—
97	C₄×C₈×C₅₃	—	2	2	no	—
98	#95×C₅₃	—	4	2	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
103	C₂×C₂²⋊D₅₃	—	2	1	no	—
104	#11719945	—	2	1	no	—
105	#11719945	—	2	1	no	—
106	#11719945	C₂×Q₈	1	2	yes	—
107	#11719945	C₂×Q₈	1	2	yes	—
108	C₂×D₂₁₂	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	—	2	1	no	—
110	C₂×C₄×D₅₃	—	4	2	no	—
111	C₂×Dic₁₀₆	—	4	2	no	—
112	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
120	C₂×C₈×C₅₃	#41	1	2	yes	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	1	no	—
126	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
127	C₂²⋊Dic₅₃	—	2	2	no	—
128	C₂²⋊Dic₅₃	—	2	1	no	—
129	#11719921	—	4	2	no	—
130	C₄⋊Dic₅₃	—	4	2	no	—
131	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
134	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
135	#11719919	—	4	2	no	—
136	#11719919	—	4	2	no	—
137	#11719919	—	4	2	no	—
138	C₄×Dic₅₃	—	2	1	no	—
139	C₄×Dic₅₃	—	2	2	no	—
140	C₄×Dic₅₃	—	4	2	no	—
141	#11719917	—	2	1	no	—
142	#11719917	—	2	1	no	—
143	#11719917	—	2	2	no	—
144	#11719917	—	2	2	no	—
145	#11719917	C₂×Q₈	1	2	yes	—
146	#11719917	C₂×Q₈	1	2	yes	—
147	C₂×#88322	—	2	1	no	—
148	C₂×#88322	—	2	1	no	—
149	C₂×#88322	—	2	2	no	—
150	#11719912	—	2	4	no	—
151	C₈×D₅₃	—	2	4	no	—
152	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂²×D₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	1	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	8	2	no	—
160	C₂×C₄×C₅₃	—	8	2	no	—
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂²⋊D₅₃	—	2	2	no	—
168	C₂²⋊D₅₃	—	2	2	no	—
169	C₂×Dic₅₃	—	2	1	no	—
170	C₂×Dic₅₃	—	2	1	no	—
171	C₂×Dic₅₃	—	2	2	no	—
172	C₂×Dic₅₃	—	2	2	no	—
173	C₂×Dic₅₃	—	2	1	no	—
174	D₂₁₂	—	2	1	no	—
175	D₂₁₂	C₄⋊Q₈	1	2	yes	—
176	C₄×D₅₃	—	2	2	no	—
177	C₄×D₅₃	—	2	2	no	—
178	C₄×D₅₃	—	16	2	no	—
179	Dic₁₀₆	—	2	1	no	—
180	Dic₁₀₆	—	16	2	no	—
181	Dic₁₀₆	C₄⋊Q₈	1	2	yes	—
182	C₈×C₅₃	—	2	1	no	—
183	C₈×C₅₃	—	2	1	no	—
184	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
185	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
186	#88322	—	2	2	no	—
187	#88322	—	2	2	no	—
188	#88322	—	4	2	no	—
189	#26082	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	#544	1	1	yes	characteristic
193	C₂²×C₅₃	D₄²	1	1	yes	characteristic
194	C₂²×C₅₃	#554	1	1	yes	characteristic
195	C₂×D₅₃	—	2	2	no	—
196	C₂×D₅₃	—	4	1	no	—
197	C₂×D₅₃	—	4	1	no	—
198	C₄×C₅₃	—	4	2	no	—
199	C₄×C₅₃	—	8	2	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
202	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
203	C₄×C₅₃	#547	1	1	yes	characteristic
204	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
205	Dic₅₃	—	2	2	no	—
206	Dic₅₃	—	4	1	no	—
207	Dic₅₃	—	4	2	no	—
208	Dic₅₃	—	4	2	no	—
209	Dic₅₃	—	4	1	no	—
210	#3164	—	53	1	no	—
211	#3084	—	53	2	no	—
212	#2913	—	53	1	no	—
213	#2090	—	53	2	no	—
214	#2061	—	53	2	no	—
215	#1952	—	53	1	no	—
216	#1945	—	53	1	no	—
217	#1860	—	53	2	no	—
218	#1846	D₅₃	1	1	yes	characteristic, complemented
219	#1816	—	53	1	no	—
220	#1764	—	53	1	no	—
221	C₂×C₅₃	—	2	1	no	—
222	C₂×C₅₃	—	2	1	no	—
223	C₂×C₅₃	#2440	1	1	yes	characteristic
224	C₂×C₅₃	#3290	1	1	yes	characteristic
225	C₂×C₅₃	#3290	1	1	yes	characteristic
226	D₅₃	—	4	2	no	—
227	C₂×#131	—	53	1	no	—
228	#544	—	53	2	no	—
229	#520	—	53	2	no	—
230	#501	—	106	2	no	—
231	#500	—	106	1	no	—
232	#499	—	106	1	no	—
233	#479	—	106	2	no	—
234	#478	—	106	2	no	—
235	#476	—	106	2	no	—
236	#475	—	106	2	no	—
237	C₂×#108	—	53	1	no	—
238	#428	—	53	2	no	—
239	#428	—	53	2	no	—
240	#428	—	53	2	no	—
241	#428	—	106	2	no	—
242	#427	—	53	1	no	—
243	#427	—	106	1	no	—
244	C₂×#107	—	106	1	no	—
245	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
246	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
247	C₂×#106	C₂×D₅₃	1	2	yes	—
248	#424	—	53	2	no	—
249	C₂×#104	—	53	1	no	—
250	#417	—	53	2	no	—
251	#416	—	53	1	no	—
252	#405	—	53	1	no	—
253	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
254	#382	C₂×D₅₃	1	2	yes	—
255	#343	—	53	2	no	—
256	#343	—	53	2	no	—
257	#341	—	53	1	no	—
258	#339	—	53	1	no	—
259	#339	—	53	2	no	—
260	C₅₃	#26082	1	1	yes	characteristic, complemented, Sylow
261	C₂×#41	—	53	1	no	—
262	#131	—	53	4	no	—
263	#131	—	106	2	no	—
264	C₂×M₁₆	—	53	1	no	—
265	C₂×M₁₆	—	53	1	no	—
266	C₂×M₁₆	—	53	2	no	—
267	C₂×M₁₆	—	106	1	no	—
268	C₂²×C₈	—	106	1	no	—
269	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
271	C₄⋊Q₈	—	106	2	no	—
272	C₄⋊Q₈	—	106	2	no	—
273	#125	—	106	2	no	—
274	#125	—	106	2	no	—
275	C₂²⋊Q₈	—	212	2	no	—
276	C₄×Q₈	—	106	2	no	—
277	C₄×D₄	—	106	2	no	—
278	C₄×D₄	—	212	2	no	—
279	#117	—	53	2	no	—
280	#117	—	53	1	no	—
281	#117	—	106	2	no	—
282	C₂×C₄⋊C₄	—	106	2	no	—
283	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
284	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
285	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
286	#108	—	106	2	no	—
287	#107	—	2	1	no	—
288	#107	—	2	1	no	—
289	#107	—	2	1	no	—
290	#107	—	2	1	no	—
291	#107	—	53	1	no	—
292	#107	—	53	1	no	—
293	#107	—	106	2	no	—
294	#107	—	106	2	no	—
295	#107	—	106	2	no	—
296	#106	—	2	2	no	—
297	#106	—	2	2	no	—
298	#106	—	53	2	no	—
299	#106	—	106	2	no	—
300	#106	—	106	2	no	—
301	#106	—	106	2	no	—
302	C₄⋊C₈	—	106	2	no	—
303	C₄⋊C₈	—	106	2	no	—
304	#104	—	53	4	no	—
305	#103	—	53	1	no	—
306	#103	—	53	1	no	—
307	#103	—	106	2	no	—
308	#102	—	53	1	no	—
309	#102	—	53	1	no	—
310	#102	—	106	2	no	—
311	#97	—	106	1	no	—
312	C₄×C₈	—	2	2	no	—
313	C₄×C₈	—	106	1	no	—
314	#95	—	4	2	no	—
315	#41	—	53	2	no	—
316	#41	—	53	2	no	—
317	#41	—	106	1	no	—
318	#41	—	106	1	no	—
319	C₂×Q₈	—	53	1	no	—
320	C₂×Q₈	—	212	2	no	—
321	C₂×D₄	—	53	1	no	—
322	C₂×D₄	—	106	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	106	1	no	—
327	C₂²×C₄	—	212	2	no	—
328	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
329	M₁₆	—	53	2	no	—
330	M₁₆	—	53	2	no	—
331	M₁₆	—	106	1	no	—
332	M₁₆	—	106	1	no	—
333	M₁₆	—	106	4	no	—
334	M₁₆	—	106	2	no	—
335	M₁₆	—	106	2	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	106	1	no	—
339	C₂×C₈	—	106	1	no	—
340	C₂×C₈	—	106	4	no	—
341	C₂×C₈	—	106	2	no	—
342	C₂×C₈	#11719945	1	2	yes	—
343	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
344	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
345	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
346	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
347	C₂×C₈	Q₈×D₅₃	1	2	yes	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	53	1	no	—
353	C₄⋊C₄	—	53	1	no	—
354	C₄⋊C₄	—	53	2	no	—
355	C₄⋊C₄	—	53	2	no	—
356	C₄⋊C₄	—	212	2	no	—
357	C₄⋊C₄	—	212	2	no	—
358	C₄⋊C₄	—	212	2	no	—
359	C₄⋊C₄	—	212	2	no	—
360	C₂²⋊C₄	—	106	2	no	—
361	C₂²⋊C₄	—	106	1	no	—
362	C₂²⋊C₄	—	212	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	106	1	no	—
365	C₄²	—	106	2	no	—
366	C₄²	—	212	2	no	—
367	C₄²	Q₈×D₅₃	1	2	yes	—
368	C₂³	—	106	1	no	—
369	C₂³	#13133262	1	1	yes	characteristic
370	Q₈	—	53	2	no	—
371	Q₈	—	848	2	no	—
372	Q₈	—	106	1	no	—
373	D₄	—	53	2	no	—
374	D₄	—	106	2	no	—
375	D₄	—	106	2	no	—
376	D₄	—	106	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	848	2	no	—
385	C₂×C₄	—	106	1	no	—
386	C₂×C₄	—	106	2	no	—
387	C₂×C₄	—	106	2	no	—
388	C₂×C₄	—	106	2	no	—
389	C₂×C₄	—	106	1	no	—
390	C₂×C₄	—	106	2	no	—
391	C₂×C₄	—	106	1	no	—
392	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
393	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
397	C₂×C₄	#13133265	1	1	yes	characteristic
398	C₈	—	2	1	no	—
399	C₈	—	2	1	no	—
400	C₈	—	106	2	no	—
401	C₈	—	106	2	no	—
402	C₈	—	212	2	no	—
403	C₈	#13133195	1	2	yes	—
404	C₈	#13133195	1	2	yes	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	106	2	no	—
408	C₂²	—	212	1	no	—
409	C₂²	—	212	1	no	—
410	C₂²	#16811250	1	1	yes	characteristic
411	C₂²	#16811258	1	1	yes	characteristic
412	C₂²	#16811328	1	1	yes	characteristic
413	C₄	—	4	2	no	—
414	C₄	—	8	2	no	—
415	C₄	—	8	2	no	—
416	C₄	—	106	2	no	—
417	C₄	—	212	2	no	—
418	C₄	—	212	1	no	—
419	C₄	—	212	2	no	—
420	C₄	—	212	1	no	—
421	C₄	C₂×#13133195	1	1	yes	characteristic
422	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
423	C₄	#16811274	1	1	yes	characteristic
424	C₄	#16811288	1	1	yes	characteristic
425	C₂	—	2	1	no	—
426	C₂	—	2	1	no	—
427	C₂	—	212	2	no	—
428	C₂	#20278420	1	1	yes	characteristic
429	C₂	#20286500	1	1	yes	characteristic
430	C₂	#20286500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16811258
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1846	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#11719945	C₂³
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#26082
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1846

Complements

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

N	H ≅ G/N	type
1. #20285251	C₂	semidirect
4. #20276226	C₂	semidirect
5. #20276065	C₂	semidirect
6. #20275498	C₂	semidirect
7. #20275463	C₂	semidirect
11. C₅₃×#1846	C₂	semidirect
47. #13133223	Q₈	semidirect
77. #13133145	Q₈	semidirect
218. #1846	D₅₃	semidirect
260. C₅₃	#26082	semidirect

Maximal subgroups

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

subgroup	index
#20285251	2
#20284582	2
#20283311	2
#20276226	2
#20276065	2
#20275498	2
#20275463	2
#20274947	2
#20274780	2
#20274552	2
C₅₃×#1846	2
#26082	53