Subgroups — #27353433

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283842	C₂	1	2	yes	maximal
2	#20283681	C₂	1	1	yes	maximal, characteristic
3	#20273501	C₂	1	2	yes	maximal, complemented
4	#20273494	C₂	1	2	yes	maximal, complemented
5	#20273488	C₂	1	2	yes	maximal, complemented
6	#20273479	C₂	1	2	yes	maximal, complemented
7	#20273464	C₂	1	2	yes	maximal, complemented
8	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#11719945	C₂²	1	1	yes	characteristic
11	#16810753	C₄	1	1	yes	characteristic
12	#16810753	C₄	1	1	yes	characteristic
13	#16810751	—	2	2	no	—
14	#16810702	C₄	1	1	yes	characteristic
15	#16810702	C₄	1	1	yes	characteristic
16	#16810696	—	2	2	no	—
17	#16810694	C₄	1	2	yes	—
18	#16810694	C₄	1	2	yes	—
19	#16810691	—	2	2	no	—
20	#16810686	—	2	2	no	—
21	#16810679	C₂²	1	2	yes	—
22	#16810676	C₂²	1	1	yes	characteristic
23	#16810675	C₂²	1	2	yes	—
24	#16810674	C₂²	1	2	yes	—
25	#16810645	C₂²	1	2	yes	—
26	C₂×C₄⋊#88322	C₂²	1	2	yes	—
27	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
28	C₄×#11719912	C₂²	1	2	yes	—
29	#16810376	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	1	yes	characteristic
31	#16810376	C₂²	1	2	yes	—
32	#16810375	C₂²	1	2	yes	—
33	#16810375	C₂²	1	2	yes	—
34	#16810373	C₂²	1	2	yes	—
35	#16810373	C₂²	1	2	yes	—
36	C₅₃×#432	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#335	C₂²	1	1	yes	characteristic
39	C₅₃×#335	C₂²	1	1	yes	characteristic
40	C₅₃×#334	C₂²	1	2	yes	—
41	C₂×#11719945	Q₈	1	1	yes	characteristic
42	C₂×#11719912	—	2	2	no	—
43	C₂×#11719912	C₂×C₄	1	2	yes	—
44	C₂×#11719912	C₂×C₄	1	2	yes	—
45	C₂²×#88322	—	2	2	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
49	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₁₂	D₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₅₃	D₄	1	2	yes	—
53	#13133176	Q₈	1	1	yes	characteristic
54	#13133174	Q₈	1	2	yes	—
55	C₄²×D₅₃	C₂³	1	2	yes	—
56	C₄⋊#88322	—	2	2	no	—
57	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#88322	C₂³	1	2	yes	—
60	C₄⋊#88322	C₂³	1	2	yes	—
61	#13133146	—	2	2	no	—
62	#13133146	—	2	2	no	—
63	#13133146	C₂×C₄	1	2	yes	—
64	#13133146	C₂×C₄	1	2	yes	—
65	C₂²⋊#88322	C₂×C₄	1	2	yes	—
66	#13133129	—	2	2	no	—
67	#13133129	—	2	2	no	—
68	#13133129	C₂×C₄	1	2	yes	—
69	#13133129	C₂×C₄	1	2	yes	—
70	#13133127	—	2	2	no	—
71	#13133127	C₂×C₄	1	2	yes	—
72	#13133127	C₂×C₄	1	2	yes	—
73	C₄×#88322	C₂³	1	2	yes	—
74	C₈×Dic₅₃	—	2	2	no	—
75	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
79	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₅₃	—	2	2	no	—
81	#97×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	2	yes	—
90	#11719945	—	4	2	no	—
91	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
95	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	4	1	no	—
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
109	#11719921	C₄⋊C₄	1	2	yes	—
110	#11719921	C₄⋊C₄	1	2	yes	—
111	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
113	#11719919	C₄⋊C₄	1	2	yes	—
114	#11719919	C₄⋊C₄	1	2	yes	—
115	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
116	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
117	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	2	2	no	—
120	C₂×#88322	—	8	2	no	—
121	C₂×#88322	C₂²×C₄	1	2	yes	—
122	C₂×#88322	C₂²×C₄	1	2	yes	—
123	#11719912	—	2	4	no	—
124	#11719912	—	4	2	no	—
125	#11719912	—	8	2	no	—
126	C₈×D₅₃	—	8	2	no	—
127	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂²⋊D₅₃	—	2	4	no	—
141	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
144	D₂₁₂	—	4	1	no	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	2	2	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	2	2	no	—
149	C₄×D₅₃	—	4	2	no	—
150	Dic₁₀₆	—	4	1	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	1	no	—
154	C₈×C₅₃	—	4	2	no	—
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	#131	1	2	yes	—
157	#88322	—	2	2	no	—
158	#88322	—	2	2	no	—
159	#88322	—	4	2	no	—
160	#88322	—	4	2	no	—
161	#17840	—	53	1	no	maximal, Sylow
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	#430	1	2	yes	—
165	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂×D₅₃	—	2	2	no	—
167	C₂×D₅₃	—	2	2	no	—
168	C₂×D₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
174	C₄×C₅₃	#568	1	1	yes	characteristic
175	C₄×C₅₃	#576	1	1	yes	characteristic
176	C₄×C₅₃	#578	1	1	yes	characteristic
177	Dic₅₃	—	2	2	no	—
178	Dic₅₃	—	2	2	no	—
179	Dic₅₃	—	2	2	no	—
180	#2990	—	53	2	no	—
181	#2965	—	53	1	no	—
182	#1573	—	53	2	no	—
183	#1572	—	53	2	no	—
184	#1571	—	53	2	no	—
185	#1569	—	53	2	no	—
186	#1568	D₅₃	1	1	yes	characteristic, complemented
187	#1565	—	53	2	no	—
188	C₂×#335	—	53	1	no	—
189	C₂×C₅₃	—	2	2	no	—
190	C₂×C₅₃	#2906	1	1	yes	characteristic
191	C₂×C₅₃	#2913	1	1	yes	characteristic
192	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
193	D₅₃	—	4	2	no	—
194	C₄×#41	—	53	1	no	—
195	#446	—	53	1	no	—
196	#446	—	53	1	no	—
197	#446	—	106	2	no	—
198	#436	—	53	1	no	—
199	#436	—	53	1	no	—
200	#435	—	53	2	no	—
201	#435	—	53	2	no	—
202	#435	—	106	2	no	—
203	#435	—	106	2	no	—
204	C₈×D₄	—	106	2	no	—
205	#432	—	53	2	no	—
206	#432	C₂×D₅₃	1	2	yes	—
207	#431	—	53	1	no	—
208	#431	—	53	2	no	—
209	#431	—	53	2	no	—
210	#423	—	53	2	no	—
211	C₂×C₄⋊C₈	—	53	2	no	—
212	C₂×C₄⋊C₈	—	53	2	no	—
213	C₄×M₁₆	—	53	2	no	—
214	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
215	#335	—	53	2	no	—
216	#335	—	53	2	no	—
217	#335	—	53	1	no	—
218	#335	—	53	1	no	—
219	#335	—	53	2	no	—
220	#335	C₂×D₅₃	1	1	yes	characteristic
221	#335	C₂×D₅₃	1	1	yes	characteristic
222	#334	—	53	2	no	—
223	#334	—	53	2	no	—
224	#334	C₂×D₅₃	1	2	yes	—
225	C₅₃	#17840	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	53	1	no	—
227	C₂×M₁₆	—	53	2	no	—
228	C₂×M₁₆	—	53	2	no	—
229	C₂×M₁₆	—	106	2	no	—
230	C₂²×C₈	—	106	2	no	—
231	C₂²×C₈	—	106	2	no	—
232	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
234	C₄×Q₈	—	53	1	no	—
235	C₄×Q₈	—	53	1	no	—
236	C₄×D₄	—	53	2	no	—
237	C₄×D₄	—	53	2	no	—
238	C₄×D₄	—	53	1	no	—
239	C₄×D₄	—	53	1	no	—
240	#117	—	53	1	no	—
241	#117	—	53	2	no	—
242	C₂×C₄²	—	53	2	no	—
243	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
244	C₄⋊C₈	—	53	2	no	—
245	C₄⋊C₈	—	53	2	no	—
246	C₄⋊C₈	—	53	1	no	—
247	C₄⋊C₈	—	53	1	no	—
248	C₄⋊C₈	—	53	2	no	—
249	C₄⋊C₈	—	53	2	no	—
250	C₄⋊C₈	—	106	2	no	—
251	C₄⋊C₈	—	106	2	no	—
252	C₄⋊C₈	—	106	2	no	—
253	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
254	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	53	2	no	—
257	C₂²⋊C₈	—	53	2	no	—
258	C₂²⋊C₈	—	53	2	no	—
259	C₂²⋊C₈	—	106	2	no	—
260	C₂²⋊C₈	—	106	2	no	—
261	#97	—	2	2	no	—
262	#97	—	53	2	no	—
263	#97	—	53	2	no	—
264	#97	—	106	2	no	—
265	C₄×C₈	—	53	2	no	—
266	C₄×C₈	—	106	2	no	—
267	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
268	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
271	#41	—	212	2	no	—
272	C₂×Q₈	—	53	1	no	—
273	C₂×D₄	—	53	2	no	—
274	C₂×D₄	—	53	1	no	—
275	C₂²×C₄	—	53	2	no	—
276	C₂²×C₄	—	53	2	no	—
277	C₂²×C₄	—	53	2	no	—
278	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
280	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
281	M₁₆	—	106	4	no	—
282	M₁₆	—	424	2	no	—
283	M₁₆	—	212	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	53	2	no	—
289	C₂×C₈	—	53	2	no	—
290	C₂×C₈	—	106	2	no	—
291	C₂×C₈	—	106	2	no	—
292	C₂×C₈	—	424	2	no	—
293	C₂×C₈	—	424	2	no	—
294	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
298	C₄⋊C₄	—	53	2	no	—
299	C₄⋊C₄	—	53	2	no	—
300	C₄⋊C₄	—	53	1	no	—
301	C₄⋊C₄	—	53	1	no	—
302	C₂²⋊C₄	—	53	2	no	—
303	C₂²⋊C₄	—	53	2	no	—
304	C₂²⋊C₄	—	53	2	no	—
305	C₄²	—	53	2	no	—
306	C₄²	—	53	2	no	—
307	C₄²	—	53	2	no	—
308	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₄²	D₄×D₅₃	1	1	yes	characteristic
310	C₄²	Q₈×D₅₃	1	1	yes	characteristic
311	C₄²	C₂³×D₅₃	1	1	yes	characteristic
312	C₂³	—	53	2	no	—
313	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
314	Q₈	—	212	1	no	—
315	D₄	—	106	4	no	—
316	D₄	—	212	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	53	2	no	—
323	C₂×C₄	—	53	2	no	—
324	C₂×C₄	—	53	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	—	212	2	no	—
330	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
331	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
332	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	106	2	no	—
342	C₈	—	106	2	no	—
343	C₈	—	212	2	no	—
344	C₈	—	212	2	no	—
345	C₈	#13133225	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	106	2	no	—
349	C₂²	—	106	2	no	—
350	C₂²	—	106	2	no	—
351	C₂²	#16810670	1	2	yes	—
352	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	106	2	no	—
358	C₄	—	106	2	no	—
359	C₄	—	106	2	no	—
360	C₄	C₂×#13133225	1	1	yes	characteristic
361	C₄	#16811389	1	1	yes	characteristic
362	C₄	#16811425	1	1	yes	characteristic
363	C₄	#16811437	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	212	2	no	—
366	C₂	#20283284	1	1	yes	characteristic
367	C₂	#20283312	1	1	yes	characteristic
368	C₂	C₂×#16810670	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×#1568	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	Q₈
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#17840
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #20273501	C₂	semidirect
4. #20273494	C₂	semidirect
5. #20273488	C₂	semidirect
6. #20273479	C₂	semidirect
7. #20273464	C₂	semidirect
8. C₂×#16810376	C₂	semidirect
9. C₅₃×#1568	C₂	semidirect
186. #1568	D₅₃	semidirect
225. C₅₃	#17840	semidirect

Maximal subgroups

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

subgroup	index
#20283842	2
#20283681	2
#20273501	2
#20273494	2
#20273488	2
#20273479	2
#20273464	2
C₂×#16810376	2
C₅₃×#1568	2
#17840	53