Subgroups — #27484046

Order 13568.

612 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	#20285227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810739	C₂	1	1	yes	maximal, characteristic
3	#20275890	C₂	1	2	yes	maximal, complemented
4	#20275886	C₂	1	2	yes	maximal, complemented
5	#20275613	C₂	1	1	yes	maximal, characteristic
6	#20275611	C₂	1	1	yes	maximal, characteristic
7	#20274840	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274768	C₂	1	2	yes	maximal, complemented
9	#20274760	C₂	1	2	yes	maximal, complemented
10	#20274595	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	C₂×#13133225	C₂²	1	2	yes	—
14	#16811176	C₂²	1	2	yes	—
15	#16811027	—	2	1	no	—
16	#16811024	—	2	1	no	—
17	#16811023	—	2	1	no	—
18	#16811020	—	2	1	no	—
19	#16810739	—	2	1	no	—
20	#16810739	—	2	1	no	—
21	#16810739	—	2	1	no	—
22	#16810739	—	2	1	no	—
23	C₂×#13133152	C₂²	1	1	yes	characteristic
24	#16810661	C₂²	1	1	yes	characteristic
25	#16810659	C₂²	1	2	yes	—
26	#16810659	C₂²	1	1	yes	characteristic
27	#16810641	C₂²	1	2	yes	—
28	#16810639	C₂²	1	2	yes	—
29	C₂×#13133143	C₂²	1	2	yes	—
30	#16810596	C₂²	1	2	yes	—
31	#16810596	C₂²	1	2	yes	—
32	#16810578	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
34	#16810395	C₂²	1	2	yes	—
35	#16810384	C₂²	1	1	yes	characteristic
36	#16810382	C₂²	1	1	yes	characteristic
37	#16810380	C₂²	1	2	yes	—
38	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#427	C₂²	1	1	yes	characteristic
40	C₅₃×#405	C₂²	1	1	yes	characteristic
41	C₅₃×#344	C₂²	1	2	yes	—
42	C₅₃×H(3,ℤ/4)	C₂²	1	2	yes	—
43	C₂×#11719945	C₂³	1	2	yes	—
44	#13133239	—	2	1	no	—
45	#13133239	—	2	1	no	—
46	#13133239	—	2	1	no	—
47	#13133239	—	2	1	no	—
48	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic, complemented
49	C₂×#11719913	Q₈	1	2	yes	complemented
50	C₂×D₄₂₄	Q₈	1	1	yes	characteristic, complemented
51	#13133225	—	2	2	no	—
52	#13133225	—	2	2	no	—
53	#13133225	—	2	2	no	—
54	#13133225	—	2	2	no	—
55	C₂×#11719917	C₂³	1	1	yes	characteristic
56	C₂×#11719912	—	2	2	no	—
57	C₂²×#88322	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₅₃	—	2	2	no	—
59	#13133216	—	2	2	no	—
60	#13133208	—	2	2	no	—
61	#13133194	—	4	2	no	—
62	C₄×Dic₁₀₆	—	4	2	no	—
63	#13133176	C₂³	1	1	yes	characteristic
64	#13133176	C₂³	1	2	yes	—
65	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
66	#13133152	—	2	1	no	—
67	#13133152	—	2	1	no	—
68	#13133151	—	2	1	no	—
69	#13133150	—	2	2	no	—
70	#13133150	—	2	1	no	—
71	#13133149	—	2	1	no	—
72	#13133148	—	2	2	no	—
73	#13133148	—	2	1	no	—
74	#13133145	—	4	2	no	—
75	#13133143	—	2	2	no	—
76	#13133143	—	2	2	no	—
77	#13133143	—	4	2	no	—
78	#13133131	—	4	2	no	—
79	#13133131	—	4	2	no	—
80	#13133127	—	2	1	no	—
81	C₈×Dic₅₃	—	2	1	no	—
82	C₈×Dic₅₃	D₄	1	1	yes	characteristic
83	C₈×Dic₅₃	D₄	1	1	yes	characteristic
84	C₈×Dic₅₃	D₄	1	1	yes	characteristic
85	C₈×Dic₅₃	D₄	1	1	yes	characteristic
86	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
87	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	#117×C₅₃	C₂³	1	1	yes	characteristic
90	#117×C₅₃	C₂³	1	2	yes	—
91	#108×C₅₃	—	2	1	no	—
92	#108×C₅₃	—	2	1	no	—
93	#107×C₅₃	—	2	1	no	—
94	#106×C₅₃	—	2	1	no	—
95	#97×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₂²⋊D₅₃	—	2	2	no	—
99	C₂²×Dic₅₃	—	2	1	no	—
100	#11719945	—	2	2	no	—
101	#11719945	—	2	2	no	—
102	#11719945	—	2	2	no	—
103	#11719945	—	2	2	no	—
104	C₂×D₂₁₂	C₂×Q₈	1	2	yes	—
105	C₂×C₄×D₅₃	—	2	2	no	—
106	C₂×Dic₁₀₆	—	4	2	no	—
107	C₂×Dic₁₀₆	C₂×Q₈	1	2	yes	—
108	M₁₆×C₅₃	—	2	2	no	—
109	M₁₆×C₅₃	—	2	1	no	—
110	M₁₆×C₅₃	—	2	2	no	—
111	M₁₆×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₂²⋊C₄×C₅₃	—	2	1	no	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	1	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₂²⋊Dic₅₃	—	2	1	no	—
127	C₂²⋊Dic₅₃	—	2	2	no	—
128	C₄⋊Dic₅₃	—	2	2	no	—
129	C₄⋊Dic₅₃	—	2	1	no	—
130	C₄⋊Dic₅₃	—	4	2	no	—
131	#11719919	—	4	2	no	—
132	#11719919	—	4	2	no	—
133	C₄×Dic₅₃	—	2	1	no	—
134	C₄×Dic₅₃	—	2	1	no	—
135	C₄×Dic₅₃	—	2	2	no	—
136	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
138	#11719917	—	2	1	no	—
139	#11719917	—	2	1	no	—
140	C₂×#88322	—	2	1	no	—
141	C₂×#88322	C₂×D₄	1	1	yes	characteristic
142	C₂×#88322	C₂×D₄	1	1	yes	characteristic
143	C₂×#88322	#41	1	1	yes	characteristic
144	C₂×#88322	#41	1	1	yes	characteristic
145	C₂×#88322	#41	1	1	yes	characteristic
146	C₂×#88322	#41	1	1	yes	characteristic
147	Dic₂₁₂	—	2	1	no	—
148	Dic₂₁₂	—	2	1	no	—
149	D₄₂₄	—	2	1	no	—
150	D₄₂₄	—	2	1	no	—
151	#11719913	—	2	2	no	—
152	#11719913	—	2	2	no	—
153	#11719912	—	4	2	no	—
154	#11719912	—	4	2	no	—
155	C₈×D₅₃	—	4	2	no	—
156	C₈×D₅₃	—	4	2	no	—
157	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×D₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂²⋊D₅₃	—	4	2	no	—
168	C₂²⋊D₅₃	—	4	2	no	—
169	C₂×Dic₅₃	—	2	1	no	—
170	C₂×Dic₅₃	—	2	1	no	—
171	C₂×Dic₅₃	—	2	2	no	—
172	C₂×Dic₅₃	—	4	2	no	—
173	C₂×Dic₅₃	—	4	2	no	—
174	C₂×Dic₅₃	—	4	1	no	—
175	D₂₁₂	—	2	2	no	—
176	D₂₁₂	—	2	2	no	—
177	C₄×D₅₃	—	4	2	no	—
178	C₄×D₅₃	—	4	2	no	—
179	Dic₁₀₆	—	2	2	no	—
180	Dic₁₀₆	—	2	2	no	—
181	Dic₁₀₆	—	8	4	no	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	—	4	1	no	—
184	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
185	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
186	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
187	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
188	#88322	—	4	1	no	—
189	#88322	#124	1	1	yes	characteristic
190	#88322	#124	1	1	yes	characteristic
191	#88322	#124	1	1	yes	characteristic
192	#88322	#124	1	1	yes	characteristic
193	#29154	—	53	1	no	maximal, Sylow
194	C₂²×C₅₃	—	4	1	no	—
195	C₂²×C₅₃	#535	1	1	yes	characteristic
196	C₂²×C₅₃	#554	1	1	yes	characteristic
197	C₂²×C₅₃	#555	1	1	yes	characteristic
198	C₂×D₅₃	—	4	2	no	—
199	C₂×D₅₃	—	8	2	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	—	8	1	no	—
202	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
203	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
204	C₄×C₅₃	#554	1	1	yes	characteristic
205	C₄×C₅₃	#555	1	1	yes	characteristic
206	Dic₅₃	—	2	2	no	—
207	Dic₅₃	—	4	1	no	—
208	Dic₅₃	—	4	2	no	—
209	Dic₅₃	—	8	2	no	—
210	Dic₅₃	—	8	1	no	—
211	#3160	—	53	1	no	—
212	C₂×#443	—	53	1	no	—
213	#2031	—	53	2	no	—
214	#2031	—	53	2	no	—
215	#1976	—	53	1	no	—
216	#1976	—	53	1	no	—
217	#1833	—	53	1	no	—
218	#1833	D₅₃	1	1	yes	characteristic, complemented
219	#1814	—	53	2	no	—
220	#1812	—	53	2	no	—
221	#1772	—	53	1	no	—
222	C₂×C₅₃	—	2	1	no	—
223	C₂×C₅₃	—	2	1	no	—
224	C₂×C₅₃	—	2	1	no	—
225	C₂×C₅₃	#2563	1	1	yes	characteristic
226	D₅₃	—	8	2	no	—
227	C₂×#135	—	53	1	no	—
228	C₂×#131	—	53	2	no	—
229	#533	—	53	2	no	—
230	#500	—	106	1	no	—
231	#499	—	106	1	no	—
232	#499	—	106	1	no	—
233	#498	—	106	1	no	—
234	#443	—	106	1	no	—
235	#443	—	106	1	no	—
236	#443	—	106	1	no	—
237	#443	—	106	1	no	—
238	C₂×#108	—	53	1	no	—
239	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
240	#427	—	53	1	no	—
241	#427	—	53	1	no	—
242	#427	—	53	2	no	—
243	#427	C₂×D₅₃	1	1	yes	characteristic
244	#421	—	53	2	no	—
245	#421	—	53	2	no	—
246	C₂×#104	—	53	2	no	—
247	#410	—	53	2	no	—
248	#410	—	53	2	no	—
249	#405	—	53	1	no	—
250	#405	C₂×D₅₃	1	1	yes	characteristic
251	C₂×C₄×C₈	—	53	1	no	—
252	#344	—	53	2	no	—
253	#344	C₂×D₅₃	1	2	yes	—
254	#339	—	53	1	no	—
255	#339	—	53	1	no	—
256	H(3,ℤ/4)	—	53	2	no	—
257	H(3,ℤ/4)	C₂×D₅₃	1	2	yes	—
258	C₅₃	#29154	1	1	yes	characteristic, complemented, Sylow
259	C₂×#41	—	53	2	no	—
260	#135	—	106	1	no	—
261	#135	—	106	1	no	—
262	#135	—	106	1	no	—
263	#135	—	106	1	no	—
264	C₂×Q₁₆	—	53	1	no	—
265	C₂×SD₁₆	—	53	2	no	—
266	C₂×D₈	—	53	1	no	—
267	#131	—	106	2	no	—
268	#131	—	106	2	no	—
269	#131	—	106	2	no	—
270	#131	—	106	2	no	—
271	C₂×M₁₆	—	53	1	no	—
272	C₂×M₁₆	—	106	2	no	—
273	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
274	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
275	C₂²×C₈	—	53	1	no	—
276	C₂²×C₈	—	106	2	no	—
277	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₄⋊Q₈	—	106	2	no	—
279	#125	—	106	2	no	—
280	C₂²⋊Q₈	—	212	2	no	—
281	C₄×Q₈	—	212	2	no	—
282	#117	—	53	2	no	—
283	#117	—	53	1	no	—
284	#117	C₂²×D₅₃	1	2	yes	—
285	#117	C₂²×D₅₃	1	1	yes	characteristic
286	C₂×C₄²	—	53	1	no	—
287	#108	—	2	1	no	—
288	#108	—	2	1	no	—
289	#108	—	106	1	no	—
290	#108	—	106	1	no	—
291	#107	—	2	1	no	—
292	#107	—	106	1	no	—
293	#107	—	106	2	no	—
294	#107	—	106	1	no	—
295	#106	—	2	1	no	—
296	#106	—	106	1	no	—
297	#106	—	106	2	no	—
298	#106	—	106	1	no	—
299	#104	—	106	2	no	—
300	#104	—	106	2	no	—
301	#104	—	212	2	no	—
302	#104	—	212	2	no	—
303	#99	—	212	2	no	—
304	#99	—	212	2	no	—
305	#97	—	2	1	no	—
306	#97	—	106	1	no	—
307	C₄×C₈	—	2	1	no	—
308	C₄×C₈	—	53	1	no	—
309	C₄×C₈	—	53	1	no	—
310	C₄×C₈	—	53	1	no	—
311	C₄×C₈	—	53	1	no	—
312	C₄×C₈	—	106	1	no	—
313	#41	—	106	2	no	—
314	#41	—	106	2	no	—
315	#41	—	106	2	no	—
316	#41	—	106	2	no	—
317	C₂×Q₈	—	53	2	no	—
318	C₂×Q₈	—	212	2	no	—
319	C₂×D₄	—	53	2	no	—
320	C₂×D₄	—	106	2	no	—
321	C₂²×C₄	—	106	1	no	—
322	C₂²×C₄	—	106	2	no	—
323	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
324	Q₁₆	—	106	1	no	—
325	Q₁₆	—	106	1	no	—
326	SD₁₆	—	106	2	no	—
327	SD₁₆	—	106	2	no	—
328	D₈	—	106	1	no	—
329	D₈	—	106	1	no	—
330	M₁₆	—	2	2	no	—
331	M₁₆	—	2	1	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	2	1	no	—
334	M₁₆	—	106	1	no	—
335	M₁₆	—	106	1	no	—
336	M₁₆	—	212	2	no	—
337	M₁₆	—	212	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	53	1	no	—
341	C₂×C₈	—	53	1	no	—
342	C₂×C₈	—	53	1	no	—
343	C₂×C₈	—	53	1	no	—
344	C₂×C₈	—	53	1	no	—
345	C₂×C₈	—	53	1	no	—
346	C₂×C₈	—	106	1	no	—
347	C₂×C₈	—	212	2	no	—
348	C₂×C₈	—	212	2	no	—
349	C₂×C₈	#11719945	1	1	yes	characteristic
350	C₂×C₈	#11719945	1	1	yes	characteristic
351	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
352	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
353	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
354	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	106	2	no	—
358	C₄⋊C₄	—	106	1	no	—
359	C₄⋊C₄	—	212	2	no	—
360	C₄⋊C₄	—	212	2	no	—
361	C₄⋊C₄	—	212	2	no	—
362	C₂²⋊C₄	—	2	1	no	—
363	C₂²⋊C₄	—	2	2	no	—
364	C₂²⋊C₄	—	106	1	no	—
365	C₂²⋊C₄	—	106	2	no	—
366	C₄²	—	2	1	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	53	1	no	—
369	C₄²	—	53	1	no	—
370	C₄²	—	106	1	no	—
371	C₄²	—	106	1	no	—
372	C₄²	—	106	2	no	—
373	C₂³	—	106	2	no	—
374	C₂³	#13133263	1	1	yes	characteristic
375	Q₈	—	106	2	no	—
376	Q₈	—	106	2	no	—
377	Q₈	—	424	4	no	—
378	D₄	—	106	2	no	—
379	D₄	—	106	2	no	—
380	D₄	—	212	2	no	—
381	D₄	—	212	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	106	1	no	—
385	C₂×C₄	—	106	2	no	—
386	C₂×C₄	—	106	1	no	—
387	C₂×C₄	—	212	2	no	—
388	C₂×C₄	—	212	1	no	—
389	C₂×C₄	—	212	2	no	—
390	C₂×C₄	—	212	2	no	—
391	C₂×C₄	—	212	2	no	—
392	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
393	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
394	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
395	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
397	C₂×C₄	#13133263	1	1	yes	characteristic
398	C₈	—	4	1	no	—
399	C₈	—	4	2	no	—
400	C₈	—	53	1	no	—
401	C₈	—	53	1	no	—
402	C₈	—	53	1	no	—
403	C₈	—	53	1	no	—
404	C₈	—	212	1	no	—
405	C₈	#13133195	1	1	yes	characteristic
406	C₈	#13133195	1	1	yes	characteristic
407	C₈	#13133195	1	1	yes	characteristic
408	C₈	#13133195	1	1	yes	characteristic
409	C₂²	—	4	1	no	—
410	C₂²	—	424	2	no	—
411	C₂²	—	212	2	no	—
412	C₂²	#16811191	1	1	yes	characteristic
413	C₂²	#16811329	1	1	yes	characteristic
414	C₂²	#16811336	1	1	yes	characteristic
415	C₄	—	8	1	no	—
416	C₄	—	8	2	no	—
417	C₄	—	106	2	no	—
418	C₄	—	424	1	no	—
419	C₄	—	424	2	no	—
420	C₄	—	212	1	no	—
421	C₄	—	212	2	no	—
422	C₄	C₂×#13133195	1	1	yes	characteristic
423	C₄	#124×D₅₃	1	1	yes	characteristic
424	C₄	#16811329	1	1	yes	characteristic
425	C₄	#16811336	1	1	yes	characteristic
426	C₂	—	2	1	no	—
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	424	2	no	—
430	C₂	#20279887	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811336
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1833	C₂
socle	C₂×C₅₃	#2563
prime radical	C₂×#13133239	C₂²
hypercenter	#1833	D₅₃
hypocenter	C₅₃	#29154
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1833

Complements

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

N	H ≅ G/N	type
1. #20285227	C₂	semidirect
3. #20275890	C₂	semidirect
4. #20275886	C₂	semidirect
7. #20274840	C₂	semidirect
8. #20274768	C₂	semidirect
9. #20274760	C₂	semidirect
10. #20274595	C₂	semidirect
11. C₅₃×#1833	C₂	semidirect
48. C₂×Dic₂₁₂	Q₈	semidirect
49. C₂×#11719913	Q₈	semidirect
50. C₂×D₄₂₄	Q₈	semidirect
218. #1833	D₅₃	semidirect
258. C₅₃	#29154	semidirect

Maximal subgroups

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

subgroup	index
#20285227	2
C₂×#16810739	2
#20275890	2
#20275886	2
#20275613	2
#20275611	2
#20274840	2
#20274768	2
#20274760	2
#20274595	2
C₅₃×#1833	2
#29154	53