Subgroups — #27353927

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283798	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283755	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719913	C₂	1	1	yes	maximal, characteristic
4	#20273616	C₂	1	4	yes	maximal, complemented
5	#20273604	C₂	1	4	yes	maximal, complemented
6	#20273109	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273105	C₂	1	1	yes	maximal, characteristic
8	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11719913	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
13	#16810753	—	2	2	no	—
14	#16810750	C₂²	1	4	yes	complemented
15	C₄×#11719913	C₄	1	2	yes	—
16	C₄×#11719913	C₄	1	2	yes	—
17	C₄×#11719913	C₂²	1	2	yes	—
18	C₄×#11719913	C₂²	1	2	yes	—
19	#16810702	—	2	2	no	—
20	#16810700	C₂²	1	4	yes	—
21	#16810677	—	2	2	no	—
22	#16810676	C₂²	1	1	yes	characteristic
23	#16810676	C₂²	1	1	yes	characteristic
24	C₂×#13133149	C₄	1	1	yes	characteristic
25	C₂×#13133142	C₄	1	1	yes	characteristic
26	C₂×#13133139	C₄	1	1	yes	characteristic
27	#16810585	—	2	2	no	—
28	#16810353	C₂²	1	4	yes	complemented
29	#16810350	C₂²	1	4	yes	—
30	#16810344	C₂²	1	2	yes	complemented
31	C₈×#88322	C₂²	1	2	yes	complemented
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₅₃×#431	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#335	C₂²	1	2	yes	complemented
36	C₅₃×#334	C₂²	1	2	yes	complemented
37	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×D₅₃	—	2	1	no	—
40	C₂×#11719913	—	2	4	no	—
41	C₂×#11719913	C₂×C₄	1	2	yes	—
42	C₂×#11719913	C₂×C₄	1	2	yes	—
43	C₂×#11719912	—	2	4	no	—
44	C₄×Dic₁₀₆	—	2	1	no	—
45	C₄×Dic₁₀₆	—	2	1	no	—
46	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₀₆	C₂³	1	2	yes	—
48	C₄×D₂₁₂	—	2	1	no	—
49	C₄×D₂₁₂	—	2	1	no	—
50	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
51	C₄×D₂₁₂	C₂³	1	2	yes	—
52	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719919	—	2	1	no	—
55	C₂×#11719921	—	2	1	no	—
56	C₂×C₄×Dic₅₃	—	2	1	no	—
57	#13133149	C₂×C₄	1	2	yes	—
58	#13133149	C₂×C₄	1	2	yes	—
59	C₄⋊#88322	—	2	1	no	—
60	C₄⋊#88322	D₄	1	2	yes	—
61	#13133146	—	2	4	no	—
62	#13133142	C₂×C₄	1	2	yes	—
63	#13133142	C₂×C₄	1	2	yes	—
64	#13133139	C₂×C₄	1	2	yes	—
65	#13133139	C₂×C₄	1	2	yes	—
66	C₂²⋊#88322	—	2	1	no	—
67	C₂²⋊#88322	D₄	1	2	yes	—
68	#13133129	—	2	4	no	—
69	#13133127	—	2	4	no	—
70	C₄×#88322	C₂³	1	2	yes	—
71	C₄×#88322	C₂³	1	2	yes	—
72	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
76	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
77	C₂²⋊C₈×C₅₃	—	2	1	no	—
78	C₂²⋊C₈×C₅₃	—	2	1	no	—
79	C₄×C₈×C₅₃	—	2	1	no	—
80	C₄×C₈×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₂³×D₅₃	—	2	1	no	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
89	C₂²×Dic₅₃	—	2	1	no	—
90	C₂²×Dic₅₃	—	2	1	no	—
91	C₂²×Dic₅₃	—	2	1	no	—
92	C₂×D₂₁₂	—	2	2	no	—
93	C₂×D₂₁₂	—	2	2	no	—
94	C₂×D₂₁₂	—	2	1	no	—
95	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₅₃	—	2	2	no	—
97	C₂×C₄×D₅₃	—	2	2	no	—
98	C₂×C₄×D₅₃	—	8	2	no	—
99	C₂×Dic₁₀₆	—	2	2	no	—
100	C₂×Dic₁₀₆	—	2	2	no	—
101	C₂×Dic₁₀₆	—	2	1	no	—
102	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	#41	1	1	yes	characteristic
114	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
115	#11719921	—	2	2	no	—
116	#11719921	—	2	2	no	—
117	C₄⋊Dic₅₃	—	2	1	no	—
118	C₄⋊Dic₅₃	—	2	1	no	—
119	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
120	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
121	#11719919	—	2	2	no	—
122	#11719919	—	2	2	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	C₂×D₄	1	4	yes	—
127	#11719913	—	2	4	no	—
128	#11719913	—	2	4	no	—
129	#11719912	—	8	4	no	—
130	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂²×D₅₃	—	2	2	no	—
132	C₂²×D₅₃	—	4	2	no	—
133	C₂²×D₅₃	—	8	1	no	—
134	C₂²×D₅₃	—	8	1	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₅₃	#131	1	2	yes	—
142	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₅₃	—	2	2	no	—
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	4	2	no	—
149	C₂×Dic₅₃	—	8	1	no	—
150	C₂×Dic₅₃	—	8	1	no	—
151	D₂₁₂	—	2	2	no	—
152	D₂₁₂	—	2	2	no	—
153	D₂₁₂	—	4	2	no	—
154	C₄×D₅₃	—	4	4	no	—
155	C₄×D₅₃	—	8	2	no	—
156	Dic₁₀₆	—	2	2	no	—
157	Dic₁₀₆	—	2	2	no	—
158	Dic₁₀₆	—	4	2	no	—
159	C₈×C₅₃	—	2	1	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	4	2	no	—
162	C₈×C₅₃	—	4	2	no	—
163	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
164	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
165	#88322	—	2	4	no	—
166	#88322	—	4	2	no	—
167	#17896	—	53	1	no	maximal, Sylow
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	#436	1	2	yes	—
171	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₅₃	—	4	2	no	—
173	C₂×D₅₃	—	4	2	no	—
174	C₂×D₅₃	—	4	2	no	—
175	C₂×D₅₃	—	8	4	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	#436	1	2	yes	—
180	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
182	C₄×C₅₃	#577	1	2	yes	—
183	Dic₅₃	—	4	2	no	—
184	Dic₅₃	—	4	2	no	—
185	Dic₅₃	—	4	2	no	—
186	#2983	—	53	1	no	—
187	#2975	—	53	1	no	—
188	C₂×C₄×SD₁₆	—	53	1	no	—
189	#1592	—	53	4	no	—
190	#1590	—	53	4	no	—
191	#1564	D₅₃	1	1	yes	characteristic, complemented
192	#1479	—	53	1	no	—
193	#1478	—	53	1	no	—
194	#1450	—	53	1	no	—
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
197	C₂×C₅₃	#2949	1	1	yes	characteristic
198	C₂×C₅₃	#2957	1	1	yes	characteristic
199	D₅₃	—	8	2	no	—
200	C₂²×SD₁₆	—	53	1	no	—
201	C₂×C₄×Q₈	—	53	1	no	—
202	C₂×C₄×D₄	—	53	1	no	—
203	#446	—	53	4	no	—
204	#446	—	106	2	no	—
205	C₄×SD₁₆	—	53	2	no	—
206	C₄×SD₁₆	—	53	2	no	—
207	C₄×SD₁₆	—	53	2	no	—
208	C₄×SD₁₆	—	53	2	no	—
209	#436	—	53	4	no	—
210	#436	—	106	2	no	—
211	#432	—	106	2	no	—
212	#431	—	53	1	no	—
213	#431	—	53	1	no	—
214	#431	C₂×D₅₃	1	1	yes	characteristic
215	#431	C₂×D₅₃	1	1	yes	characteristic
216	C₂×#106	—	53	1	no	—
217	C₂×#103	—	53	1	no	—
218	C₂×#102	—	53	1	no	—
219	#407	—	106	2	no	—
220	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
221	#335	C₂×D₅₃	1	2	yes	complemented
222	#334	C₂×D₅₃	1	2	yes	complemented
223	Q₈⋊C₈	—	53	4	no	—
224	D₄⋊C₈	—	53	4	no	—
225	#322	—	53	2	no	—
226	C₈²	—	53	2	no	—
227	C₅₃	#17896	1	1	yes	characteristic, complemented, Sylow
228	C₂²×Q₈	—	53	1	no	—
229	C₂²×D₄	—	53	1	no	—
230	C₂³×C₄	—	106	1	no	—
231	C₂×SD₁₆	—	53	2	no	—
232	C₂×SD₁₆	—	53	2	no	—
233	C₂×SD₁₆	—	106	4	no	—
234	C₂×M₁₆	—	106	4	no	—
235	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
236	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
237	C₄×Q₈	—	53	2	no	—
238	C₄×Q₈	—	53	2	no	—
239	C₄×Q₈	—	106	1	no	—
240	C₄×Q₈	—	106	1	no	—
241	C₄×D₄	—	53	2	no	—
242	C₄×D₄	—	53	2	no	—
243	C₄×D₄	—	106	1	no	—
244	C₄×D₄	—	106	1	no	—
245	C₂×C₄⋊C₄	—	53	1	no	—
246	C₂×C₄⋊C₄	—	53	1	no	—
247	C₂×C₄⋊C₄	—	106	1	no	—
248	C₂×C₂²⋊C₄	—	106	1	no	—
249	C₂×C₄²	—	106	1	no	—
250	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
251	#106	—	53	2	no	—
252	#106	—	53	2	no	—
253	C₄⋊C₈	—	53	2	no	—
254	C₄⋊C₈	—	106	4	no	—
255	C₄⋊C₈	—	106	1	no	—
256	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
257	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
258	#103	—	53	2	no	—
259	#103	—	53	2	no	—
260	#102	—	53	2	no	—
261	#102	—	53	2	no	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	2	1	no	—
264	C₂²⋊C₈	—	53	2	no	—
265	C₂²⋊C₈	—	106	4	no	—
266	C₂²⋊C₈	—	106	1	no	—
267	#97	—	106	4	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	—	53	2	no	—
271	C₄×C₈	—	53	2	no	—
272	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
273	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₂⁴	—	106	1	no	—
277	C₂×Q₈	—	53	2	no	—
278	C₂×Q₈	—	106	1	no	—
279	C₂×Q₈	—	106	2	no	—
280	C₂×Q₈	—	106	2	no	—
281	C₂×D₄	—	53	2	no	—
282	C₂×D₄	—	106	1	no	—
283	C₂×D₄	—	106	2	no	—
284	C₂×D₄	—	106	2	no	—
285	C₂²×C₄	—	106	1	no	—
286	C₂²×C₄	—	106	1	no	—
287	C₂²×C₄	—	106	1	no	—
288	C₂²×C₄	—	106	2	no	—
289	C₂²×C₄	—	106	2	no	—
290	C₂²×C₄	—	424	2	no	—
291	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
293	C₂²×C₄	#11719949	1	1	yes	characteristic
294	SD₁₆	—	106	4	no	—
295	SD₁₆	—	106	4	no	—
296	M₁₆	—	424	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	53	4	no	—
301	C₂×C₈	—	106	2	no	—
302	C₂×C₈	#11719912	1	4	yes	complemented
303	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
307	C₄⋊C₄	—	53	2	no	—
308	C₄⋊C₄	—	53	2	no	—
309	C₄⋊C₄	—	106	1	no	—
310	C₄⋊C₄	—	106	1	no	—
311	C₄⋊C₄	—	106	2	no	—
312	C₄⋊C₄	—	106	2	no	—
313	C₂²⋊C₄	—	106	2	no	—
314	C₂²⋊C₄	—	106	2	no	—
315	C₄²	—	106	2	no	—
316	C₄²	—	106	2	no	—
317	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
318	C₄²	D₄×D₅₃	1	1	yes	characteristic
319	C₄²	#11719949	1	1	yes	characteristic
320	C₄²	C₂³×D₅₃	1	1	yes	characteristic
321	C₂³	—	106	2	no	—
322	C₂³	—	424	1	no	—
323	C₂³	—	424	1	no	—
324	C₂³	—	212	2	no	—
325	C₂³	#13133179	1	1	yes	characteristic
326	Q₈	—	106	2	no	—
327	Q₈	—	106	2	no	—
328	Q₈	—	212	2	no	—
329	D₄	—	106	2	no	—
330	D₄	—	106	2	no	—
331	D₄	—	212	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	106	2	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	106	2	no	—
337	C₂×C₄	—	424	2	no	—
338	C₂×C₄	—	424	1	no	—
339	C₂×C₄	—	424	1	no	—
340	C₂×C₄	—	212	2	no	—
341	C₂×C₄	—	212	4	no	—
342	C₂×C₄	#13133179	1	1	yes	characteristic
343	C₂×C₄	#13133179	1	1	yes	characteristic
344	C₂×C₄	#13133179	1	1	yes	characteristic
345	C₂×C₄	C₂×#11719912	1	2	yes	—
346	C₂×C₄	#13133225	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	106	4	no	—
355	C₈	—	212	2	no	—
356	C₈	C₂×#11719912	1	2	yes	—
357	C₈	C₂×#11719912	1	2	yes	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	424	4	no	—
361	C₂²	—	212	2	no	—
362	C₂²	—	212	2	no	—
363	C₂²	—	212	2	no	—
364	C₂²	#16810700	1	2	yes	—
365	C₂²	C₂×#13133179	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	212	2	no	—
370	C₄	—	212	2	no	—
371	C₄	—	212	2	no	—
372	C₄	#16810700	1	2	yes	—
373	C₄	C₂²×#11719912	1	1	yes	characteristic
374	C₄	C₂×#13133225	1	1	yes	characteristic
375	C₄	#16811430	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	424	2	no	—
378	C₂	C₂×#16810700	1	1	yes	characteristic
379	C₂	#20283573	1	1	yes	characteristic
380	C₂	#20283631	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×#1564	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#17896
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #20283798	C₂	semidirect
4. #20273616	C₂	semidirect
5. #20273604	C₂	semidirect
6. #20273109	C₂	semidirect
8. #20272998	C₂	semidirect
9. C₅₃×#1564	C₂	semidirect
14. #16810750	C₂²	semidirect
28. #16810353	C₂²	semidirect
30. #16810344	C₂²	semidirect
31. C₈×#88322	C₂²	semidirect
35. C₅₃×#335	C₂²	semidirect
36. C₅₃×#334	C₂²	semidirect
106. C₂×C₈×C₅₃	M₁₆	semidirect
191. #1564	D₅₃	semidirect
221. #335	C₂×D₅₃	semidirect
222. #334	C₂×D₅₃	semidirect
227. C₅₃	#17896	semidirect
302. C₂×C₈	#11719912	semidirect

Maximal subgroups

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

subgroup	index
#20283798	2
#20283755	2
C₂×C₄×#11719913	2
#20273616	2
#20273604	2
#20273109	2
#20273105	2
#20272998	2
C₅₃×#1564	2
#17896	53