Subgroups — #27355598

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283731	C₂	1	2	yes	maximal, complemented
2	C₄×#13133236	C₂	1	1	yes	maximal, characteristic
3	#20273629	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273596	C₂	1	2	yes	maximal, complemented
5	#20273591	C₂	1	2	yes	maximal, complemented
6	#20273535	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273533	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273475	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273146	C₂	1	2	yes	maximal, complemented
10	#20273001	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1594	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133236	C₄	1	1	yes	characteristic
13	C₄×#11719947	C₂²	1	2	yes	—
14	C₈×Dic₁₀₆	C₂²	1	2	yes	—
15	#16810747	C₂²	1	2	yes	—
16	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
17	#16810717	C₄	1	1	yes	characteristic
18	#16810714	C₄	1	2	yes	—
19	C₄×#11719923	C₂²	1	2	yes	—
20	D₈×Dic₅₃	C₄	1	1	yes	characteristic
21	#16810701	C₂²	1	2	yes	—
22	#16810699	—	2	2	no	—
23	#16810698	—	2	2	no	—
24	#16810679	C₂²	1	2	yes	—
25	#16810677	—	2	2	no	—
26	#16810674	C₂²	1	1	yes	characteristic
27	#16810660	C₄	1	1	yes	characteristic
28	#16810616	C₄	1	2	yes	—
29	M₁₆×Dic₅₃	—	2	2	no	—
30	C₄×#11719912	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810375	C₂²	1	2	yes	—
33	#16810374	C₂²	1	1	yes	characteristic
34	#16810355	C₂²	1	2	yes	—
35	#16810353	C₂²	1	2	yes	—
36	D₄⋊#88322	C₂²	1	2	yes	—
37	#16810345	C₂²	1	1	yes	characteristic
38	#16810344	C₂²	1	1	yes	characteristic
39	#16810344	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#436	C₂²	1	2	yes	—
42	C₅₃×#334	C₂²	1	1	yes	characteristic
43	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
44	C₅₃×#322	C₂²	1	1	yes	characteristic
45	C₂×#11719947	C₂×C₄	1	2	yes	—
46	#13133236	—	2	4	no	—
47	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719923	C₂×C₄	1	2	yes	—
49	C₂×#11719917	—	2	2	no	—
50	C₂×#11719912	D₄	1	2	yes	—
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₀₆	C₂³	1	2	yes	—
54	#13133182	C₂×C₄	1	2	yes	—
55	D₄×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₄×C₂²⋊D₅₃	—	2	2	no	—
57	#13133177	—	2	2	no	—
58	#13133175	C₂×C₄	1	2	yes	—
59	#13133173	—	2	2	no	—
60	C₂×C₄×Dic₅₃	—	2	2	no	—
61	C₄²×D₅₃	C₂³	1	1	yes	characteristic
62	#13133151	C₂×C₄	1	1	yes	characteristic
63	#13133148	C₂×C₄	1	2	yes	—
64	C₄⋊#88322	C₂³	1	2	yes	—
65	#13133146	—	2	2	no	—
66	#13133146	—	2	1	no	—
67	#13133146	—	2	2	no	—
68	#13133142	C₂×C₄	1	2	yes	—
69	#13133140	C₂×C₄	1	2	yes	—
70	#13133138	C₂×C₄	1	2	yes	—
71	C₂²⋊#88322	—	2	2	no	—
72	#13133129	—	2	2	no	—
73	#13133129	—	2	1	no	—
74	#13133127	—	2	2	no	—
75	#13133127	D₄	1	2	yes	—
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₈×Dic₅₃	—	2	2	no	—
80	C₈×Dic₅₃	—	2	2	no	—
81	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×M₁₆×C₅₃	—	2	2	no	—
85	C₄×D₄×C₅₃	C₂³	1	2	yes	—
86	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
88	#102×C₅₃	C₂×C₄	1	2	yes	—
89	C₂²⋊C₈×C₅₃	—	2	2	no	—
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂×C₂²⋊D₅₃	—	2	2	no	—
96	C₂²×Dic₅₃	—	2	2	no	—
97	C₂²×Dic₅₃	—	2	2	no	—
98	#11719947	—	2	4	no	—
99	#11719947	—	4	2	no	—
100	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
103	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
104	D₈×C₅₃	—	2	2	no	—
105	M₁₆×C₅₃	—	4	4	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²⋊C₄×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊Dic₅₃	—	2	2	no	—
115	#11719923	—	2	4	no	—
116	#11719921	—	2	2	no	—
117	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
118	#11719919	—	2	2	no	—
119	#11719919	—	2	2	no	—
120	C₄×Dic₅₃	—	2	2	no	—
121	C₄×Dic₅₃	—	2	2	no	—
122	C₄×Dic₅₃	—	2	2	no	—
123	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₅₃	#41	1	1	yes	characteristic
126	#11719917	—	4	4	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	—	2	1	no	—
129	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
130	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
131	C₂×#88322	C₂×D₄	1	2	yes	—
132	Dic₂₁₂	M₁₆	1	4	yes	complemented
133	#11719912	—	4	2	no	—
134	C₈×D₅₃	—	2	2	no	—
135	C₈×D₅₃	—	4	1	no	—
136	C₂³×C₅₃	—	2	2	no	—
137	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
138	D₄×C₅₃	—	2	2	no	—
139	D₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₅₃	—	4	4	no	—
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	—	4	4	no	—
150	C₂×Dic₅₃	—	8	2	no	—
151	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
154	C₄×D₅₃	—	2	1	no	—
155	C₄×D₅₃	—	2	1	no	—
156	C₄×D₅₃	—	2	2	no	—
157	C₄×D₅₃	—	4	1	no	—
158	Dic₁₀₆	—	2	2	no	—
159	Dic₁₀₆	C₂×M₁₆	1	4	yes	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	2	2	no	—
162	C₈×C₅₃	—	4	1	no	—
163	C₈×C₅₃	—	4	2	no	—
164	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
165	#88322	—	2	1	no	—
166	#88322	—	2	1	no	—
167	#88322	—	2	2	no	—
168	#88322	—	4	1	no	—
169	#88322	—	4	2	no	—
170	#18043	—	53	1	no	maximal, Sylow
171	C₂²×C₅₃	—	4	2	no	—
172	C₂²×C₅₃	—	4	2	no	—
173	C₂²×C₅₃	—	4	2	no	—
174	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₅₃	—	2	1	no	—
176	C₂×D₅₃	—	2	1	no	—
177	C₂×D₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₅₃	#568	1	1	yes	characteristic
182	C₄×C₅₃	#575	1	2	yes	—
183	Dic₅₃	—	2	4	no	—
184	Dic₅₃	—	2	1	no	—
185	Dic₅₃	—	2	1	no	—
186	Dic₅₃	—	2	1	no	—
187	Dic₅₃	—	4	2	no	—
188	#2971	—	53	2	no	—
189	C₄×#135	—	53	1	no	—
190	#1594	—	53	1	no	—
191	#1594	D₅₃	1	1	yes	characteristic, complemented
192	#1588	—	53	2	no	—
193	#1588	—	53	2	no	—
194	#1579	—	53	1	no	—
195	#1579	—	53	1	no	—
196	#1568	—	53	1	no	—
197	#1488	—	53	2	no	—
198	#1451	—	53	1	no	—
199	C₂×C₅₃	—	8	2	no	—
200	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
201	C₂×C₅₃	#2943	1	1	yes	characteristic
202	C₂×C₅₃	#2957	1	1	yes	characteristic
203	D₅₃	—	4	1	no	—
204	C₂×#135	—	53	1	no	—
205	C₄×#41	—	53	2	no	—
206	C₈×Q₈	—	53	2	no	—
207	C₈×Q₈	—	53	2	no	—
208	C₄×Q₁₆	—	53	1	no	—
209	C₄×Q₁₆	—	53	1	no	—
210	C₄×SD₁₆	—	53	2	no	—
211	C₄×SD₁₆	—	53	2	no	—
212	C₄×D₈	—	53	1	no	—
213	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
214	#436	—	53	2	no	—
215	#436	—	106	2	no	—
216	#436	—	106	2	no	—
217	#436	C₂×D₅₃	1	2	yes	—
218	#432	—	53	2	no	—
219	#432	—	106	2	no	—
220	#431	—	53	1	no	—
221	#427	—	53	1	no	—
222	#416	—	53	2	no	—
223	C₄×M₁₆	—	53	1	no	—
224	C₄×M₁₆	—	106	2	no	—
225	C₂×C₄×C₈	—	53	1	no	—
226	#335	—	53	2	no	—
227	#334	—	53	1	no	—
228	#334	C₂×D₅₃	1	1	yes	characteristic
229	Q₈⋊C₈	—	53	2	no	—
230	Q₈⋊C₈	—	53	2	no	—
231	D₄⋊C₈	—	53	2	no	—
232	D₄⋊C₈	C₂×D₅₃	1	2	yes	—
233	#322	—	53	1	no	—
234	#322	—	53	1	no	—
235	#322	—	53	1	no	—
236	#322	C₂×D₅₃	1	1	yes	characteristic
237	C₅₃	#18043	1	1	yes	characteristic, complemented, Sylow
238	C₂×#41	—	53	2	no	—
239	#135	—	106	4	no	—
240	C₂×Q₁₆	—	53	1	no	—
241	C₂×SD₁₆	—	53	2	no	—
242	C₂×D₈	C₄×D₅₃	1	1	yes	characteristic
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	53	2	no	—
245	C₂×M₁₆	—	106	2	no	—
246	C₂²×C₈	—	53	1	no	—
247	C₂²×C₈	—	53	1	no	—
248	C₄×Q₈	—	53	2	no	—
249	C₄×Q₈	—	53	2	no	—
250	C₄×D₄	—	53	2	no	—
251	C₄×D₄	—	106	2	no	—
252	C₄×D₄	—	106	2	no	—
253	C₄×D₄	C₂²×D₅₃	1	2	yes	—
254	#117	—	53	2	no	—
255	#117	—	106	2	no	—
256	C₂×C₄²	—	53	1	no	—
257	C₂×C₄²	—	106	2	no	—
258	#107	—	53	1	no	—
259	#107	C₄×D₅₃	1	1	yes	characteristic
260	#106	—	53	2	no	—
261	C₄⋊C₈	—	53	2	no	—
262	C₄⋊C₈	—	106	1	no	—
263	C₄⋊C₈	—	106	2	no	—
264	C₄⋊C₈	—	106	2	no	—
265	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
266	#103	—	53	2	no	—
267	#103	—	53	2	no	—
268	#102	—	53	2	no	—
269	#102	C₄×D₅₃	1	2	yes	—
270	C₂²⋊C₈	—	2	2	no	—
271	C₂²⋊C₈	—	106	1	no	—
272	C₂²⋊C₈	—	106	2	no	—
273	C₂²⋊C₈	—	106	2	no	—
274	#97	—	53	2	no	—
275	#97	—	106	2	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	—	53	1	no	—
278	C₄×C₈	—	53	1	no	—
279	C₄×C₈	—	53	1	no	—
280	C₄×C₈	—	53	1	no	—
281	C₄×C₈	—	106	2	no	—
282	C₄×C₈	—	106	2	no	—
283	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
286	#41	—	106	4	no	—
287	#41	—	212	2	no	—
288	C₂×Q₈	—	53	2	no	—
289	C₂×D₄	—	106	2	no	—
290	C₂×D₄	C₂×C₄×D₅₃	1	2	yes	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	53	1	no	—
293	C₂²×C₄	—	53	1	no	—
294	C₂²×C₄	—	53	1	no	—
295	C₂²×C₄	—	106	2	no	—
296	C₂²×C₄	—	106	2	no	—
297	Q₁₆	—	53	4	no	—
298	SD₁₆	—	106	4	no	—
299	D₈	—	2	2	no	—
300	M₁₆	—	4	4	no	—
301	M₁₆	—	212	2	no	—
302	M₁₆	—	212	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	53	1	no	—
306	C₂×C₈	—	53	1	no	—
307	C₂×C₈	—	53	2	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	1	no	—
311	C₂×C₈	—	212	1	no	—
312	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₈	D₄×D₅₃	1	2	yes	—
315	C₄⋊C₄	—	53	2	no	—
316	C₄⋊C₄	—	106	2	no	—
317	C₄⋊C₄	—	106	2	no	—
318	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	106	2	no	—
321	C₂²⋊C₄	—	106	2	no	—
322	C₄²	—	53	1	no	—
323	C₄²	—	53	1	no	—
324	C₄²	—	53	1	no	—
325	C₄²	—	106	2	no	—
326	C₄²	—	106	2	no	—
327	C₄²	—	106	2	no	—
328	C₄²	C₂³×D₅₃	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	—	53	1	no	—
331	Q₈	—	53	4	no	—
332	Q₈	—	106	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	2	no	—
335	D₄	—	212	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	53	1	no	—
340	C₂×C₄	—	53	1	no	—
341	C₂×C₄	—	53	1	no	—
342	C₂×C₄	—	106	2	no	—
343	C₂×C₄	—	106	1	no	—
344	C₂×C₄	—	106	1	no	—
345	C₂×C₄	—	106	2	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	—	424	2	no	—
348	C₂×C₄	—	212	1	no	—
349	C₂×C₄	—	212	4	no	—
350	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
352	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	2	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	106	1	no	—
358	C₈	—	106	1	no	—
359	C₈	—	106	2	no	—
360	C₈	—	212	1	no	—
361	C₈	—	212	2	no	—
362	C₈	M₁₆×D₅₃	1	2	yes	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	106	1	no	—
367	C₂²	—	106	1	no	—
368	C₂²	—	106	1	no	—
369	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	106	1	no	—
373	C₄	—	106	4	no	—
374	C₄	—	106	1	no	—
375	C₄	—	106	1	no	—
376	C₄	—	212	2	no	—
377	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
378	C₄	#16811388	1	1	yes	characteristic
379	C₄	#16811419	1	2	yes	—
380	C₂	—	8	2	no	—
381	C₂	—	212	1	no	—
382	C₂	#20283451	1	1	yes	characteristic
383	C₂	#20283512	1	1	yes	characteristic
384	C₂	#20283629	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1594	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133236	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1594	D₅₃
hypocenter	C₅₃	#18043
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1594

Complements

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

N	H ≅ G/N	type
1. #20283731	C₂	semidirect
3. #20273629	C₂	semidirect
4. #20273596	C₂	semidirect
5. #20273591	C₂	semidirect
6. #20273535	C₂	semidirect
7. #20273533	C₂	semidirect
8. #20273475	C₂	semidirect
9. #20273146	C₂	semidirect
10. #20273001	C₂	semidirect
11. C₅₃×#1594	C₂	semidirect
132. Dic₂₁₂	M₁₆	semidirect
191. #1594	D₅₃	semidirect
237. C₅₃	#18043	semidirect

Maximal subgroups

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

subgroup	index
#20283731	2
C₄×#13133236	2
#20273629	2
#20273596	2
#20273591	2
#20273535	2
#20273533	2
#20273475	2
#20273146	2
#20273001	2
C₅₃×#1594	2
#18043	53