Subgroups — #27392467

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275439	C₂	1	1	yes	maximal, characteristic
2	#20275439	C₂	1	1	yes	maximal, characteristic
3	#20275274	C₂	1	2	yes	maximal
4	#20275274	C₂	1	2	yes	maximal
5	#20274835	C₂	1	1	yes	maximal, characteristic
6	#20274590	C₂	1	2	yes	maximal
7	#20274584	C₂	1	2	yes	maximal
8	#20274584	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic
10	#20274402	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2402	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133128	C₂²	1	2	yes	—
17	C₂×#13133128	C₂²	1	1	yes	characteristic
18	C₂×#13133128	C₂²	1	1	yes	characteristic
19	C₂×#13133128	C₂²	1	1	yes	characteristic
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	#16810378	C₂²	1	2	yes	—
25	#16810378	C₂²	1	2	yes	—
26	#16810378	C₂²	1	2	yes	—
27	#16810378	C₂²	1	2	yes	—
28	#16810378	C₂²	1	2	yes	—
29	#16810378	C₂²	1	2	yes	—
30	C₅₃×#400	C₄	1	2	yes	—
31	C₅₃×#400	C₄	1	2	yes	—
32	C₅₃×#398	C₄	1	2	yes	—
33	C₅₃×#398	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#376	C₂²	1	1	yes	characteristic
39	C₂²×#88322	C₂³	1	2	yes	—
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂²×#88322	C₂³	1	1	yes	characteristic
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	C₂²×#88322	C₂³	1	1	yes	characteristic
45	C₄⋊#88322	—	2	2	no	—
46	C₄⋊#88322	—	2	2	no	—
47	C₄⋊#88322	—	2	4	no	—
48	C₄⋊#88322	—	2	4	no	—
49	#13133128	—	2	2	no	—
50	#13133128	—	2	2	no	—
51	#13133128	—	2	2	no	—
52	#13133128	—	2	4	no	—
53	C₄×#88322	—	2	2	no	—
54	C₄×#88322	—	2	2	no	—
55	C₄×#88322	—	2	4	no	—
56	C₄×#88322	Q₈	1	4	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#95×C₅₃	C₂×C₄	1	2	yes	—
69	#95×C₅₃	C₂×C₄	1	2	yes	—
70	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
71	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
72	#95×C₅₃	C₂×C₄	1	2	yes	—
73	#95×C₅₃	C₂×C₄	1	2	yes	—
74	#95×C₅₃	C₂×C₄	1	2	yes	—
75	#95×C₅₃	C₂×C₄	1	2	yes	—
76	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
77	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	—	2	4	no	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	—	2	1	no	—
96	C₄²×C₅₃	—	2	1	no	—
97	C₄²×C₅₃	—	2	1	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₅₃	#41	1	2	yes	—
103	C₂×#88322	—	2	1	no	—
104	C₂×#88322	—	2	1	no	—
105	C₂×#88322	—	2	1	no	—
106	C₂×#88322	—	2	1	no	—
107	C₂×#88322	—	2	1	no	—
108	C₂×#88322	—	2	1	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	2	no	—
111	C₂×#88322	—	2	2	no	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	C₂×Q₈	1	2	yes	—
116	C₂×#88322	C₂×Q₈	1	2	yes	—
117	C₂×#88322	#41	1	2	yes	—
118	C₂×#88322	#41	1	2	yes	—
119	C₂×#88322	#41	1	2	yes	—
120	C₂×#88322	#41	1	2	yes	—
121	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₅₃	#131	1	4	yes	—
144	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
150	#88322	—	2	2	no	—
151	#88322	—	2	2	no	—
152	#88322	—	2	2	no	—
153	#88322	—	4	2	no	—
154	#88322	—	4	2	no	—
155	#88322	#125	1	4	yes	—
156	#21679	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
159	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
160	C₂²×C₅₃	C₂×#131	1	2	yes	—
161	C₂²×C₅₃	#568	1	2	yes	—
162	C₄×C₅₃	—	2	2	no	—
163	C₄×C₅₃	—	2	2	no	—
164	C₄×C₅₃	—	2	4	no	—
165	C₄×C₅₃	—	4	2	no	—
166	C₄×C₅₃	—	4	2	no	—
167	C₄×C₅₃	—	4	2	no	—
168	C₄×C₅₃	—	4	2	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
171	C₄×C₅₃	#529	1	1	yes	characteristic
172	C₄×C₅₃	#529	1	1	yes	characteristic
173	C₄×C₅₃	#533	1	1	yes	characteristic
174	#2402	D₅₃	1	1	yes	characteristic
175	#1940	—	53	1	no	—
176	#1940	—	53	1	no	—
177	#1916	—	53	2	no	—
178	#1916	—	53	2	no	—
179	#1832	—	53	1	no	—
180	#1771	—	53	2	no	—
181	#1770	—	53	2	no	—
182	#1770	—	53	1	no	—
183	C₈×C₄⋊C₄	—	53	1	no	—
184	#1726	—	53	1	no	—
185	C₂×C₅₃	C₄×#125	1	1	yes	characteristic
186	C₂×C₅₃	#2960	1	1	yes	characteristic
187	C₂×C₅₃	#2961	1	1	yes	characteristic
188	C₂×C₅₃	#2968	1	2	yes	—
189	C₂×C₅₃	#2969	1	2	yes	—
190	C₂×C₄⋊C₈	—	53	1	no	—
191	C₂×C₄⋊C₈	—	53	1	no	—
192	C₂×C₄⋊C₈	—	53	2	no	—
193	C₂×C₄⋊C₈	—	53	2	no	—
194	C₂×#97	—	53	1	no	—
195	C₂×#97	—	53	1	no	—
196	C₂×#97	—	53	1	no	—
197	C₂×#97	—	53	2	no	—
198	C₂×C₄×C₈	—	53	1	no	—
199	C₂×C₄×C₈	—	53	1	no	—
200	C₂×C₄×C₈	—	53	1	no	—
201	C₂×C₄×C₈	—	53	2	no	—
202	#400	Dic₅₃	1	2	yes	—
203	#400	Dic₅₃	1	2	yes	—
204	#398	Dic₅₃	1	2	yes	—
205	#398	Dic₅₃	1	2	yes	—
206	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
207	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
208	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
209	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
210	#376	C₂×D₅₃	1	1	yes	characteristic
211	#336	—	53	2	no	—
212	#336	—	53	2	no	—
213	#336	—	53	2	no	—
214	#336	—	53	2	no	—
215	#336	—	53	2	no	—
216	#336	—	53	2	no	—
217	C₅₃	#21679	1	1	yes	characteristic, complemented, Sylow
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	53	1	no	—
221	C₂²×C₈	—	53	1	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₂²×C₈	—	53	2	no	—
224	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
226	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
230	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
233	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
234	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
235	C₄⋊C₈	—	106	4	no	—
236	C₄⋊C₈	—	106	4	no	—
237	C₄⋊C₈	—	106	2	no	—
238	C₄⋊C₈	—	106	2	no	—
239	#97	—	106	2	no	—
240	#97	—	106	2	no	—
241	#97	—	106	4	no	—
242	#97	—	106	2	no	—
243	C₄×C₈	—	53	4	no	—
244	C₄×C₈	—	106	4	no	—
245	C₄×C₈	—	106	2	no	—
246	C₄×C₈	—	106	2	no	—
247	#95	C₂×Dic₅₃	1	2	yes	—
248	#95	C₂×Dic₅₃	1	2	yes	—
249	#95	C₂×Dic₅₃	1	1	yes	characteristic
250	#95	C₂×Dic₅₃	1	1	yes	characteristic
251	#95	C₂×Dic₅₃	1	2	yes	—
252	#95	C₂×Dic₅₃	1	2	yes	—
253	#95	C₂×Dic₅₃	1	2	yes	—
254	#95	C₂×Dic₅₃	1	2	yes	—
255	#95	C₂×Dic₅₃	1	1	yes	characteristic
256	#95	C₂×Dic₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
258	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
265	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
266	C₂×C₈	—	53	2	no	—
267	C₂×C₈	—	53	2	no	—
268	C₂×C₈	—	53	2	no	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	53	2	no	—
272	C₂×C₈	—	106	1	no	—
273	C₂×C₈	—	106	1	no	—
274	C₂×C₈	—	106	1	no	—
275	C₂×C₈	—	106	1	no	—
276	C₂×C₈	—	106	1	no	—
277	C₂×C₈	—	106	1	no	—
278	C₂×C₈	—	106	2	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	2	no	—
282	C₂×C₈	—	106	2	no	—
283	C₂×C₈	—	106	2	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	Q₈×D₅₃	1	2	yes	—
299	C₄²	#11719949	1	2	yes	—
300	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
321	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
322	C₂×C₄	#13133224	1	4	yes	—
323	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
324	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
325	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
326	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
327	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
329	C₈	—	53	4	no	—
330	C₈	—	106	2	no	—
331	C₈	—	106	2	no	—
332	C₈	—	106	2	no	—
333	C₈	—	212	2	no	—
334	C₈	—	212	2	no	—
335	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
336	C₂²	#41×Dic₅₃	1	1	yes	characteristic
337	C₂²	#41×Dic₅₃	1	1	yes	characteristic
338	C₂²	C₂×#13133224	1	2	yes	—
339	C₂²	#16811390	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	2	no	—
348	C₄	#125×D₅₃	1	1	yes	characteristic
349	C₄	#16811158	1	1	yes	characteristic
350	C₄	#16811158	1	1	yes	characteristic
351	C₄	#16811179	1	1	yes	characteristic
352	C₂	#125×Dic₅₃	1	1	yes	characteristic
353	C₂	#20283650	1	1	yes	characteristic
354	C₂	#20283656	1	1	yes	characteristic
355	C₂	#20283711	1	2	yes	—
356	C₂	#20283720	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2402	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2402	D₅₃
hypocenter	C₅₃	#21679
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2402

Complements

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

N	H ≅ G/N	type
217. C₅₃	#21679	semidirect

Maximal subgroups

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

subgroup	index
#20275439	2
#20275274	2
#20274835	2
#20274590	2
#20274584	2
C₄⋊C₄×#88322	2
#20274402	2
C₅₃×#2402	2
#21679	53