Subgroups — #27276868

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810365	C₂	1	2	yes	maximal
4	#20272596	C₂	1	2	yes	maximal
5	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×#13133128	C₂²	1	2	yes	—
9	C₂×#13133128	C₂²	1	2	yes	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	#16810365	—	2	4	no	—
13	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#389	C₄	1	1	yes	characteristic
15	C₅₃×#389	C₄	1	1	yes	characteristic
16	C₅₃×#387	—	2	1	no	—
17	C₅₃×#387	—	2	1	no	—
18	C₅₃×#382	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
21	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	4	no	—
28	#13133128	—	2	2	no	—
29	#13133128	—	2	2	no	—
30	#13133128	—	2	2	no	—
31	#13133128	—	2	2	no	—
32	#125×C₅₃	—	2	1	no	—
33	#125×C₅₃	—	2	1	no	—
34	#125×C₅₃	D₄	1	2	yes	—
35	#125×C₅₃	Q₈	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	#95×C₅₃	—	2	1	no	—
54	#95×C₅₃	—	2	1	no	—
55	#95×C₅₃	—	2	1	no	—
56	#95×C₅₃	—	2	1	no	—
57	#95×C₅₃	—	2	1	no	—
58	#95×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊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×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₄⋊C₄×C₅₃	—	4	1	no	—
78	C₄⋊C₄×C₅₃	—	4	1	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅₃	—	2	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	4	1	no	—
86	C₄²×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	—	4	2	no	—
89	C₄²×C₅₃	—	4	2	no	—
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	C₄²×C₅₃	#41	1	1	yes	characteristic
94	C₂×#88322	—	2	4	no	—
95	C₂×#88322	—	2	4	no	—
96	C₂×#88322	—	4	2	no	—
97	C₂×#88322	—	4	2	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	—	4	1	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	—	4	1	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	#101	1	2	yes	—
129	C₂×C₄×C₅₃	#104	1	2	yes	—
130	C₂×C₄×C₅₃	#104	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
137	#88322	—	4	4	no	—
138	#88322	—	4	4	no	—
139	#8582	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	#380	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
142	C₂²×C₅₃	#413	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
145	C₂²×C₅₃	#421	1	1	yes	characteristic
146	C₂²×C₅₃	#421	1	1	yes	characteristic
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	—	4	2	no	—
158	C₄×C₅₃	—	4	1	no	—
159	C₄×C₅₃	—	8	1	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×#125	D₅₃	1	1	yes	characteristic
162	#1767	—	53	1	no	—
163	#1767	—	53	1	no	—
164	C₂×#330	—	53	2	no	—
165	#1295	—	53	2	no	—
166	C₂×C₅₃	#1524	1	1	yes	characteristic
167	C₂×C₅₃	#1528	1	1	yes	characteristic
168	C₂×C₅₃	#1531	1	1	yes	characteristic
169	C₂×C₅₃	#1535	1	1	yes	characteristic
170	C₂×C₅₃	#1801	1	1	yes	characteristic
171	C₂×C₅₃	#1806	1	1	yes	characteristic
172	C₂×C₅₃	#1806	1	1	yes	characteristic
173	C₂×#125	C₂×D₅₃	1	1	yes	characteristic
174	C₂×C₄⋊C₈	—	106	1	no	—
175	C₂×C₄⋊C₈	—	106	1	no	—
176	C₂×#97	—	53	2	no	—
177	C₂×#97	—	53	2	no	—
178	#389	Dic₅₃	1	1	yes	characteristic
179	#389	Dic₅₃	1	1	yes	characteristic
180	#387	—	2	1	no	—
181	#387	—	2	1	no	—
182	#382	C₂×D₅₃	1	1	yes	characteristic
183	C₄×C₄⋊C₄	—	2	1	no	—
184	C₄×C₄⋊C₄	—	2	1	no	—
185	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
186	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
187	C₄³	C₂×D₅₃	1	1	yes	characteristic
188	#336	—	106	1	no	—
189	#336	—	106	1	no	—
190	#330	—	106	4	no	—
191	C₅₃	#8582	1	1	yes	characteristic, complemented, Sylow
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	—	106	2	no	—
194	#125	—	2	1	no	—
195	#125	—	2	1	no	—
196	#125	Dic₁₀₆	1	2	yes	—
197	#125	D₂₁₂	1	2	yes	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
215	C₄⋊C₈	—	106	4	no	—
216	C₄⋊C₈	—	106	4	no	—
217	#97	—	106	2	no	—
218	#97	—	106	2	no	—
219	#97	—	106	2	no	—
220	#97	—	106	2	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	#95	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×C₈	—	106	4	no	—
239	C₂×C₈	—	106	4	no	—
240	C₂×C₈	—	212	2	no	—
241	C₂×C₈	—	212	2	no	—
242	C₂×C₈	—	212	2	no	—
243	C₂×C₈	—	212	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
255	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
265	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
266	C₄²	#11719945	1	1	yes	characteristic
267	C₄²	#11719945	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	#13133136	1	2	yes	—
297	C₂×C₄	#13133144	1	2	yes	—
298	C₂×C₄	#13133144	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
300	C₂×C₄	#13133176	1	1	yes	characteristic
301	C₂×C₄	#13133191	1	1	yes	characteristic
302	C₂×C₄	#13133191	1	1	yes	characteristic
303	C₂×C₄	#13133196	1	1	yes	characteristic
304	C₂×C₄	#13133196	1	1	yes	characteristic
305	C₈	—	212	4	no	—
306	C₈	—	212	4	no	—
307	C₂²	#16810483	1	1	yes	characteristic
308	C₂²	C₂×#13133136	1	1	yes	characteristic
309	C₂²	#16810607	1	1	yes	characteristic
310	C₂²	C₂×#13133144	1	1	yes	characteristic
311	C₂²	C₂×#13133144	1	1	yes	characteristic
312	C₂²	#16810640	1	1	yes	characteristic
313	C₂²	#16810640	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₂	#20273303	1	1	yes	characteristic
329	C₂	#20273317	1	1	yes	characteristic
330	C₂	#20273330	1	1	yes	characteristic
331	C₂	#20273346	1	1	yes	characteristic
332	C₂	#20274711	1	1	yes	characteristic
333	C₂	#20274735	1	1	yes	characteristic
334	C₂	#20274735	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#8582
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
191. C₅₃	#8582	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₂×#16810365	2
#20272596	2
C₄×#125×C₅₃	2
#8582	53