Subgroups — #27273562

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810679	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810645	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719912	C₂	1	1	yes	maximal, characteristic
4	#20272550	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272549	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	—	2	1	no	—
9	C₂²×#11719912	C₄	1	2	yes	—
10	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	#16810679	—	2	2	no	—
12	#16810679	C₄	1	4	yes	—
13	#16810645	—	2	2	no	—
14	#16810645	C₄	1	4	yes	—
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×#13133146	—	2	1	no	—
18	C₂×#13133129	—	2	1	no	—
19	C₄×#11719912	—	2	2	no	—
20	C₄×#11719912	—	2	2	no	—
21	C₂×#13133127	—	2	1	no	—
22	C₂×#13133127	C₄	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
30	C₂×#11719912	—	2	4	no	—
31	C₂×#11719912	—	2	4	no	—
32	C₂×#11719912	—	2	4	no	—
33	C₂×#11719912	—	2	4	no	—
34	C₂×#11719912	—	4	2	no	—
35	C₂×#11719912	—	4	2	no	—
36	C₂²×#88322	—	2	1	no	—
37	C₂²×#88322	—	2	1	no	—
38	C₂²×#88322	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
42	C₄²×D₅₃	C₂×C₄	1	2	yes	—
43	C₄²×D₅₃	C₂×C₄	1	2	yes	—
44	C₄²×D₅₃	D₄	1	2	yes	—
45	C₄²×D₅₃	D₄	1	2	yes	—
46	C₄⋊#88322	—	2	1	no	—
47	C₄⋊#88322	—	2	1	no	—
48	C₄⋊#88322	C₂×C₄	1	2	yes	—
49	C₄⋊#88322	C₂×C₄	1	2	yes	—
50	#13133146	—	2	4	no	—
51	#13133129	—	2	4	no	—
52	#13133127	—	2	4	no	—
53	#13133127	—	2	4	no	—
54	C₄×#88322	—	2	1	no	—
55	C₄×#88322	—	2	1	no	—
56	C₂²×C₈×C₅₃	—	2	1	no	—
57	C₂²×C₈×C₅₃	—	2	1	no	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	1	no	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	—	2	1	no	—
65	C₄×C₈×C₅₃	—	2	1	no	—
66	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	2	no	—
74	C₂×C₄×D₅₃	—	2	2	no	—
75	C₂×C₄×D₅₃	—	2	1	no	—
76	C₂×C₄×D₅₃	—	2	1	no	—
77	C₂×C₄×D₅₃	—	2	1	no	—
78	C₂×C₄×D₅₃	—	2	1	no	—
79	C₂×C₄×D₅₃	—	4	2	no	—
80	C₂×C₄×D₅₃	C₄²	1	2	yes	—
81	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	4	1	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	4	1	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄×Dic₅₃	C₄²	1	2	yes	—
100	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
101	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
102	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
103	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
105	C₂×#88322	—	2	2	no	—
106	C₂×#88322	—	2	2	no	—
107	C₂×#88322	—	2	2	no	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	—	4	1	no	—
110	C₂×#88322	—	4	1	no	—
111	C₂×#88322	—	4	1	no	—
112	C₂×#88322	—	4	1	no	—
113	C₂×#88322	M₁₆	1	4	yes	complemented
114	#11719912	—	2	16	no	—
115	#11719912	—	4	8	no	—
116	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×D₅₃	—	2	2	no	—
118	C₂²×D₅₃	—	2	1	no	—
119	C₂²×D₅₃	—	2	1	no	—
120	C₂²×D₅₃	—	2	2	no	—
121	C₂²×D₅₃	#95	1	2	yes	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
135	C₂×Dic₅₃	—	2	2	no	—
136	C₂×Dic₅₃	—	2	1	no	—
137	C₂×Dic₅₃	—	2	1	no	—
138	C₂×Dic₅₃	—	2	2	no	—
139	C₂×Dic₅₃	#95	1	2	yes	—
140	C₂×Dic₅₃	#95	1	2	yes	—
141	C₂×Dic₅₃	#95	1	2	yes	—
142	C₄×D₅₃	—	2	2	no	—
143	C₄×D₅₃	—	2	2	no	—
144	C₄×D₅₃	—	2	4	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	C₄×D₅₃	—	4	2	no	—
151	C₈×C₅₃	—	2	4	no	—
152	C₈×C₅₃	—	4	2	no	—
153	C₈×C₅₃	—	4	2	no	—
154	#88322	—	2	4	no	—
155	#88322	—	4	2	no	—
156	#88322	—	4	2	no	—
157	#7876	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
159	C₂²×C₅₃	#423	1	2	yes	—
160	C₂²×C₅₃	#428	1	1	yes	characteristic
161	C₂²×C₅₃	#428	1	1	yes	characteristic
162	C₂²×C₅₃	#430	1	2	yes	—
163	C₂×D₅₃	—	2	2	no	—
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	—	2	2	no	—
166	C₂×D₅₃	—	4	2	no	—
167	C₂×D₅₃	—	4	2	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
171	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
172	C₄×C₅₃	#407	1	2	yes	—
173	C₄×C₅₃	#417	1	1	yes	characteristic
174	C₄×C₅₃	#417	1	1	yes	characteristic
175	C₄×C₅₃	#418	1	1	yes	characteristic
176	C₄×C₅₃	#419	1	1	yes	characteristic
177	Dic₅₃	—	2	2	no	—
178	Dic₅₃	—	2	2	no	—
179	Dic₅₃	—	2	2	no	—
180	C₂×#432	—	53	1	no	—
181	C₂×#423	—	53	1	no	—
182	C₂×C₄×M₁₆	—	53	1	no	—
183	#1277	—	53	1	no	—
184	#1277	—	53	1	no	—
185	#1277	—	53	1	no	—
186	#1277	D₅₃	1	1	yes	characteristic, complemented
187	C₂×C₅₃	#1569	1	2	yes	—
188	C₂×C₅₃	#1569	1	2	yes	—
189	C₂×C₅₃	#1728	1	1	yes	characteristic
190	C₂×C₅₃	#1736	1	1	yes	characteristic
191	C₂×C₅₃	#1738	1	1	yes	characteristic
192	D₅₃	—	4	2	no	—
193	C₂²×M₁₆	—	53	2	no	—
194	C₂²×M₁₆	—	106	1	no	—
195	C₂²×C₄²	—	53	1	no	—
196	#432	—	53	4	no	—
197	#432	—	106	2	no	—
198	#423	—	53	4	no	—
199	#423	—	106	2	no	—
200	C₂×C₄⋊C₈	—	53	1	no	—
201	C₂×C₄⋊C₈	—	53	1	no	—
202	C₂×C₄⋊C₈	—	106	1	no	—
203	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
205	C₂×C₂²⋊C₈	—	106	1	no	—
206	C₄×M₁₆	—	106	2	no	—
207	C₄×M₁₆	—	106	2	no	—
208	C₂×#97	—	53	2	no	—
209	C₂×#97	—	106	1	no	—
210	C₂×C₄×C₈	—	53	1	no	—
211	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
212	C₅₃	#7876	1	1	yes	characteristic, complemented, Sylow
213	C₂³×C₄	—	53	1	no	—
214	C₂³×C₄	—	53	1	no	—
215	C₂³×C₄	—	53	1	no	—
216	C₂×M₁₆	—	106	4	no	—
217	C₂×M₁₆	—	106	4	no	—
218	C₂×M₁₆	—	106	4	no	—
219	C₂×M₁₆	—	106	4	no	—
220	C₂×M₁₆	—	212	2	no	—
221	C₂×M₁₆	—	212	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	53	2	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	—	106	1	no	—
227	C₂²×C₈	C₄×D₅₃	1	2	yes	—
228	C₂×C₄²	—	53	1	no	—
229	C₂×C₄²	—	53	1	no	—
230	C₂×C₄²	—	53	1	no	—
231	C₂×C₄²	—	53	2	no	—
232	C₂×C₄²	—	53	2	no	—
233	C₂×C₄²	—	53	2	no	—
234	C₂×C₄²	—	53	2	no	—
235	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	53	2	no	—
239	C₄⋊C₈	—	53	2	no	—
240	C₄⋊C₈	—	106	4	no	—
241	C₄⋊C₈	—	106	1	no	—
242	C₄⋊C₈	—	106	1	no	—
243	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
244	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
245	C₂²⋊C₈	—	106	4	no	—
246	#97	—	106	4	no	—
247	#97	—	106	4	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	106	1	no	—
251	C₄×C₈	—	106	1	no	—
252	C₂⁴	—	53	1	no	—
253	C₂²×C₄	—	53	1	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	2	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	2	no	—
262	C₂²×C₄	—	106	2	no	—
263	C₂²×C₄	—	106	2	no	—
264	C₂²×C₄	—	106	1	no	—
265	C₂²×C₄	—	106	1	no	—
266	C₂²×C₄	—	106	1	no	—
267	C₂²×C₄	—	106	1	no	—
268	C₂²×C₄	—	212	2	no	—
269	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
271	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
272	M₁₆	—	106	16	no	—
273	M₁₆	—	212	8	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	53	4	no	—
283	C₂×C₈	—	106	2	no	—
284	C₂×C₈	—	106	2	no	—
285	C₂×C₈	—	106	2	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	212	1	no	—
288	C₂×C₈	—	212	1	no	—
289	C₂×C₈	—	212	1	no	—
290	C₂×C₈	—	212	1	no	—
291	C₂×C₈	#11719912	1	4	yes	complemented
292	C₄²	—	53	2	no	—
293	C₄²	—	53	2	no	—
294	C₄²	—	53	2	no	—
295	C₄²	—	53	2	no	—
296	C₄²	—	53	2	no	—
297	C₄²	—	53	2	no	—
298	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
300	C₄²	D₄×D₅₃	1	1	yes	characteristic
301	C₄²	D₄×D₅₃	1	1	yes	characteristic
302	C₂³	—	53	2	no	—
303	C₂³	—	106	2	no	—
304	C₂³	—	106	1	no	—
305	C₂³	—	106	1	no	—
306	C₂³	—	106	2	no	—
307	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	53	2	no	—
310	C₂×C₄	—	53	2	no	—
311	C₂×C₄	—	53	2	no	—
312	C₂×C₄	—	106	2	no	—
313	C₂×C₄	—	106	1	no	—
314	C₂×C₄	—	106	1	no	—
315	C₂×C₄	—	106	2	no	—
316	C₂×C₄	—	106	2	no	—
317	C₂×C₄	—	106	2	no	—
318	C₂×C₄	—	106	4	no	—
319	C₂×C₄	—	106	2	no	—
320	C₂×C₄	—	106	2	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	212	2	no	—
324	C₂×C₄	—	212	2	no	—
325	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
329	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
330	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
331	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
332	C₂×C₄	C₂×#11719912	1	2	yes	—
333	C₂×C₄	#13133241	1	1	yes	characteristic
334	C₂×C₄	#13133244	1	1	yes	characteristic
335	C₂×C₄	#13133246	1	1	yes	characteristic
336	C₂×C₄	#13133249	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	106	4	no	—
341	C₈	—	212	2	no	—
342	C₈	—	212	2	no	—
343	C₂²	—	106	2	no	—
344	C₂²	—	106	2	no	—
345	C₂²	—	106	2	no	—
346	C₂²	—	212	2	no	—
347	C₂²	—	212	2	no	—
348	C₂²	#95×D₅₃	1	1	yes	characteristic
349	C₂²	#16810645	1	2	yes	—
350	C₂²	#16810662	1	1	yes	characteristic
351	C₂²	#16810664	1	1	yes	characteristic
352	C₂²	#16810670	1	2	yes	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	106	2	no	—
356	C₄	—	106	2	no	—
357	C₄	—	106	2	no	—
358	C₄	C₄×#11719912	1	1	yes	characteristic
359	C₄	#16810575	1	1	yes	characteristic
360	C₄	#16810585	1	2	yes	—
361	C₄	#16810621	1	1	yes	characteristic
362	C₄	#16810623	1	1	yes	characteristic
363	C₄	#16810626	1	1	yes	characteristic
364	C₄	#16810630	1	1	yes	characteristic
365	C₂	—	212	2	no	—
366	C₂	#20273479	1	2	yes	—
367	C₂	#20273481	1	2	yes	—
368	C₂	#20274407	1	1	yes	characteristic
369	C₂	#20274433	1	1	yes	characteristic
370	C₂	#20274440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1277	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1277	D₅₃
hypocenter	C₅₃	#7876
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. #20272550	C₂	semidirect
5. #20272549	C₂	semidirect
6. #20272549	C₂	semidirect
7. C₅₃×#1277	C₂	semidirect
94. C₂×C₈×C₅₃	M₁₆	semidirect
113. C₂×#88322	M₁₆	semidirect
186. #1277	D₅₃	semidirect
212. C₅₃	#7876	semidirect
291. 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
C₂×#16810679	2
C₂×#16810645	2
C₂×C₄×#11719912	2
#20272550	2
#20272549	2
C₅₃×#1277	2
#7876	53