Subgroups — #27354946

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285274	C₂	1	1	yes	maximal, characteristic
2	#20283809	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283790	C₂	1	1	yes	maximal, characteristic
4	#20273595	C₂	1	2	yes	maximal, complemented
5	#20273574	C₂	1	2	yes	maximal, complemented
6	#20273560	C₂	1	2	yes	maximal, complemented
7	#20273553	C₂	1	2	yes	maximal, complemented
8	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273189	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273185	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811180	C₂²	1	1	yes	characteristic
13	#16811173	C₂²	1	1	yes	characteristic
14	#16811021	C₄	1	1	yes	characteristic
15	#16810996	C₄	1	1	yes	characteristic
16	#16810974	C₄	1	1	yes	characteristic
17	#16810970	C₄	1	1	yes	characteristic
18	#16810852	C₄	1	2	yes	—
19	#16810852	C₄	1	2	yes	—
20	#16810753	C₂²	1	2	yes	—
21	#16810752	—	2	2	no	—
22	C₈×Dic₁₀₆	C₂²	1	2	yes	—
23	C₄×#11719913	C₂²	1	2	yes	—
24	C₄×#11719913	C₂²	1	2	yes	—
25	#16810702	C₂²	1	2	yes	—
26	#16810693	—	2	2	no	—
27	C₈×D₂₁₂	C₂²	1	2	yes	—
28	#16810681	—	2	2	no	—
29	#16810680	C₂²	1	1	yes	characteristic
30	#16810680	C₂²	1	1	yes	characteristic
31	#16810588	—	2	2	no	—
32	#16810376	C₂²	1	2	yes	—
33	#16810374	C₂²	1	2	yes	—
34	#16810355	C₂²	1	2	yes	—
35	#16810355	C₂²	1	2	yes	—
36	#16810352	C₂²	1	2	yes	—
37	#16810352	C₂²	1	2	yes	—
38	C₅₃×#431	C₂²	1	1	yes	characteristic
39	C₅₃×#431	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#322	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	1	yes	characteristic
43	C₅₃×#322	C₂²	1	1	yes	characteristic
44	C₅₃×#322	C₂²	1	1	yes	characteristic
45	C₂×#11719945	—	2	1	no	—
46	C₂×#11719913	C₂×C₄	1	2	yes	—
47	C₂×#11719913	C₂×C₄	1	2	yes	—
48	C₂×#11719912	—	2	2	no	—
49	C₂×C₈×D₅₃	—	2	2	no	—
50	#13133217	C₂×C₄	1	1	yes	characteristic
51	#13133217	C₂×C₄	1	1	yes	characteristic
52	#13133217	C₂×C₄	1	1	yes	characteristic
53	#13133214	C₂×C₄	1	1	yes	characteristic
54	#13133209	—	2	1	no	—
55	#13133205	—	2	1	no	—
56	#13133196	—	2	1	no	—
57	#13133196	C₂×C₄	1	2	yes	—
58	#13133191	—	2	1	no	—
59	#13133191	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀₆	—	2	1	no	—
61	C₄×Dic₁₀₆	C₂³	1	2	yes	—
62	C₄×D₂₁₂	—	2	1	no	—
63	C₄×D₂₁₂	C₂³	1	2	yes	—
64	#13133176	—	2	1	no	—
65	#13133149	C₂×C₄	1	2	yes	—
66	#13133149	C₂×C₄	1	2	yes	—
67	C₄⋊#88322	C₂³	1	2	yes	—
68	C₄⋊#88322	C₂³	1	2	yes	—
69	#13133146	—	2	2	no	—
70	#13133146	—	2	2	no	—
71	#13133142	C₂×C₄	1	2	yes	—
72	#13133142	C₂×C₄	1	2	yes	—
73	#13133139	C₂×C₄	1	2	yes	—
74	#13133139	C₂×C₄	1	2	yes	—
75	C₂²⋊#88322	—	2	1	no	—
76	C₂²⋊#88322	—	2	1	no	—
77	#13133129	—	2	2	no	—
78	#13133129	—	2	2	no	—
79	#13133128	—	2	1	no	—
80	#13133128	—	2	1	no	—
81	#13133127	—	2	2	no	—
82	C₈×Dic₅₃	—	2	2	no	—
83	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	—	2	1	no	—
87	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
88	C₂²⋊C₈×C₅₃	—	2	1	no	—
89	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
90	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
101	C₂×C₂²⋊D₅₃	—	2	2	no	—
102	#11719945	—	8	2	no	—
103	C₂×D₂₁₂	—	2	1	no	—
104	C₂×D₂₁₂	—	2	1	no	—
105	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₅₃	—	2	2	no	—
107	C₂×Dic₁₀₆	—	2	1	no	—
108	C₂×Dic₁₀₆	—	2	1	no	—
109	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	4	1	no	—
113	C₂×C₈×C₅₃	—	4	1	no	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅₃	#41	1	1	yes	characteristic
123	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₅₃	—	2	2	no	—
125	#11719921	—	2	2	no	—
126	#11719921	—	2	2	no	—
127	C₄⋊Dic₅₃	—	2	1	no	—
128	C₄⋊Dic₅₃	—	2	1	no	—
129	C₄⋊Dic₅₃	—	2	1	no	—
130	C₄⋊Dic₅₃	—	2	1	no	—
131	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
132	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
133	#11719919	—	2	2	no	—
134	#11719919	—	2	2	no	—
135	C₄×Dic₅₃	—	2	2	no	—
136	C₂×#88322	—	2	2	no	—
137	C₂×#88322	—	2	2	no	—
138	#11719913	—	4	2	no	—
139	#11719913	—	4	2	no	—
140	#11719912	—	8	2	no	—
141	C₈×D₅₃	—	8	2	no	—
142	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂²×D₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	4	1	no	—
148	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₅₃	—	8	2	no	—
155	C₂×Dic₅₃	—	2	2	no	—
156	C₂×Dic₅₃	—	2	2	no	—
157	C₂×Dic₅₃	—	2	2	no	—
158	D₂₁₂	—	2	2	no	—
159	D₂₁₂	—	2	2	no	—
160	D₂₁₂	—	8	1	no	—
161	D₂₁₂	—	8	1	no	—
162	C₄×D₅₃	—	4	2	no	—
163	C₄×D₅₃	—	4	2	no	—
164	Dic₁₀₆	—	2	2	no	—
165	Dic₁₀₆	—	2	2	no	—
166	Dic₁₀₆	—	8	1	no	—
167	Dic₁₀₆	—	8	1	no	—
168	C₈×C₅₃	—	2	1	no	—
169	C₈×C₅₃	—	2	1	no	—
170	C₈×C₅₃	—	2	1	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	2	2	no	—
173	C₈×C₅₃	—	2	2	no	—
174	C₈×C₅₃	—	4	1	no	—
175	C₈×C₅₃	—	4	1	no	—
176	#88322	—	4	2	no	—
177	#88322	—	4	2	no	—
178	#17990	—	53	1	no	maximal, Sylow
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂×D₅₃	—	4	2	no	—
184	C₂×D₅₃	—	4	2	no	—
185	C₂×D₅₃	—	4	2	no	—
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	#568	1	1	yes	characteristic
191	C₄×C₅₃	#568	1	1	yes	characteristic
192	C₄×C₅₃	#577	1	2	yes	—
193	Dic₅₃	—	4	2	no	—
194	Dic₅₃	—	4	2	no	—
195	Dic₅₃	—	4	2	no	—
196	#3167	—	53	1	no	—
197	#2985	—	53	1	no	—
198	#2981	—	53	1	no	—
199	#1588	—	53	2	no	—
200	#1585	—	53	2	no	—
201	#1583	—	53	2	no	—
202	#1582	—	53	2	no	—
203	#1568	—	53	1	no	—
204	#1497	—	53	1	no	—
205	#1496	—	53	1	no	—
206	#1452	D₅₃	1	1	yes	characteristic, complemented
207	C₂×C₅₃	—	4	1	no	—
208	C₂×C₅₃	#2932	1	1	yes	characteristic
209	C₂×C₅₃	#2949	1	1	yes	characteristic
210	C₂×C₅₃	#2957	1	1	yes	characteristic
211	D₅₃	—	8	2	no	—
212	#533	—	53	1	no	—
213	#532	—	53	1	no	—
214	#498	—	53	1	no	—
215	#492	—	53	1	no	—
216	#487	—	53	1	no	—
217	#486	—	53	1	no	—
218	#465	—	53	2	no	—
219	#465	—	53	2	no	—
220	#446	—	53	2	no	—
221	#446	—	106	2	no	—
222	C₈×Q₈	—	53	2	no	—
223	C₄×SD₁₆	—	53	2	no	—
224	C₄×SD₁₆	—	53	2	no	—
225	#436	—	53	2	no	—
226	#435	—	106	2	no	—
227	C₈×D₄	—	53	2	no	—
228	#433	—	106	2	no	—
229	#432	—	53	1	no	—
230	#432	—	53	1	no	—
231	#431	C₂×D₅₃	1	1	yes	characteristic
232	#431	C₂×D₅₃	1	1	yes	characteristic
233	#408	—	106	2	no	—
234	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
235	#335	—	53	2	no	—
236	#334	—	53	2	no	—
237	Q₈⋊C₈	—	53	2	no	—
238	Q₈⋊C₈	—	53	2	no	—
239	D₄⋊C₈	—	53	2	no	—
240	D₄⋊C₈	—	53	2	no	—
241	#322	C₂×D₅₃	1	1	yes	characteristic
242	#322	C₂×D₅₃	1	1	yes	characteristic
243	#322	C₂×D₅₃	1	1	yes	characteristic
244	#322	C₂×D₅₃	1	1	yes	characteristic
245	C₅₃	#17990	1	1	yes	characteristic, complemented, Sylow
246	C₂×#41	—	106	1	no	—
247	C₂×SD₁₆	—	53	2	no	—
248	C₂×SD₁₆	—	53	2	no	—
249	C₂×M₁₆	—	106	2	no	—
250	C₂²×C₈	—	106	2	no	—
251	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
253	C₄⋊Q₈	—	53	1	no	—
254	C₄⋊Q₈	—	53	1	no	—
255	C₄⋊Q₈	—	53	1	no	—
256	#127	—	53	1	no	—
257	#125	—	106	1	no	—
258	#124	—	106	1	no	—
259	C₂²⋊Q₈	—	53	2	no	—
260	C₂²⋊Q₈	—	106	1	no	—
261	#121	—	53	2	no	—
262	#121	—	106	1	no	—
263	C₄×Q₈	—	53	2	no	—
264	C₄×Q₈	—	106	1	no	—
265	C₄×D₄	—	53	2	no	—
266	C₄×D₄	—	106	1	no	—
267	#117	—	106	1	no	—
268	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
269	#106	—	53	2	no	—
270	#106	—	53	2	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	53	2	no	—
273	C₄⋊C₈	—	53	2	no	—
274	C₄⋊C₈	—	106	2	no	—
275	C₄⋊C₈	—	106	2	no	—
276	C₄⋊C₈	D₂₁₂	1	2	yes	—
277	#103	—	53	2	no	—
278	#103	—	53	2	no	—
279	#102	—	53	2	no	—
280	#102	—	53	2	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	106	1	no	—
283	C₂²⋊C₈	—	106	1	no	—
284	C₂²⋊C₈	—	106	2	no	—
285	C₂²⋊C₈	—	106	2	no	—
286	C₂²⋊C₈	D₂₁₂	1	2	yes	—
287	#97	—	106	2	no	—
288	#97	—	106	1	no	—
289	#97	—	106	1	no	—
290	C₄×C₈	—	106	2	no	—
291	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
292	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
294	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
298	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
299	#41	—	424	2	no	—
300	C₂×Q₈	—	53	2	no	—
301	C₂×Q₈	—	106	1	no	—
302	C₂×Q₈	—	106	1	no	—
303	C₂×D₄	—	53	2	no	—
304	C₂×D₄	—	106	2	no	—
305	C₂×D₄	—	106	1	no	—
306	C₂×D₄	—	106	1	no	—
307	C₂²×C₄	—	106	2	no	—
308	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
310	C₂²×C₄	#11719945	1	1	yes	characteristic
311	SD₁₆	—	212	2	no	—
312	SD₁₆	—	212	2	no	—
313	M₁₆	—	424	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	106	2	no	—
319	C₂×C₈	—	106	2	no	—
320	C₂×C₈	—	424	2	no	—
321	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
325	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
326	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
327	C₄⋊C₄	—	53	2	no	—
328	C₄⋊C₄	—	53	2	no	—
329	C₄⋊C₄	—	106	1	no	—
330	C₄⋊C₄	—	106	1	no	—
331	C₄⋊C₄	—	106	1	no	—
332	C₄⋊C₄	—	106	2	no	—
333	C₄⋊C₄	—	106	1	no	—
334	C₄⋊C₄	—	106	2	no	—
335	C₂²⋊C₄	—	106	2	no	—
336	C₂²⋊C₄	—	106	2	no	—
337	C₂²⋊C₄	—	106	2	no	—
338	C₄²	—	106	2	no	—
339	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
340	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
341	C₄²	#11719945	1	1	yes	characteristic
342	C₄²	C₂³×D₅₃	1	1	yes	characteristic
343	C₂³	—	106	2	no	—
344	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
345	Q₈	—	106	2	no	—
346	Q₈	—	106	2	no	—
347	Q₈	—	424	1	no	—
348	Q₈	—	424	1	no	—
349	D₄	—	106	2	no	—
350	D₄	—	106	2	no	—
351	D₄	—	424	1	no	—
352	D₄	—	424	1	no	—
353	D₄	—	424	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	106	2	no	—
359	C₂×C₄	—	106	2	no	—
360	C₂×C₄	—	106	2	no	—
361	C₂×C₄	—	212	2	no	—
362	C₂×C₄	—	212	2	no	—
363	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
364	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
365	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
368	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	2	no	—
374	C₈	—	2	2	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	1	no	—
377	C₈	—	212	2	no	—
378	C₈	—	212	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	212	2	no	—
383	C₂²	—	212	2	no	—
384	C₂²	—	212	2	no	—
385	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	212	2	no	—
391	C₄	—	212	2	no	—
392	C₄	—	212	2	no	—
393	C₄	#16811389	1	1	yes	characteristic
394	C₄	#16811389	1	1	yes	characteristic
395	C₄	#16811435	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	424	2	no	—
398	C₂	#20283434	1	1	yes	characteristic
399	C₂	#20283576	1	1	yes	characteristic
400	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16811173	C₂²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#17990
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #20283809	C₂	semidirect
4. #20273595	C₂	semidirect
5. #20273574	C₂	semidirect
6. #20273560	C₂	semidirect
7. #20273553	C₂	semidirect
8. #20273478	C₂	semidirect
9. #20273189	C₂	semidirect
11. C₅₃×#1452	C₂	semidirect
206. #1452	D₅₃	semidirect
245. C₅₃	#17990	semidirect

Maximal subgroups

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

subgroup	index
#20285274	2
#20283809	2
#20283790	2
#20273595	2
#20273574	2
#20273560	2
#20273553	2
#20273478	2
#20273189	2
#20273185	2
C₅₃×#1452	2
#17990	53