Subgroups — #27276865

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	C₄×#13133209	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810364	C₂	1	2	yes	maximal
4	#20272597	C₂	1	2	yes	maximal
5	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×#13133128	C₂²	1	2	yes	—
9	#16810521	C₄	1	1	yes	characteristic
10	#16810521	C₄	1	1	yes	characteristic
11	#16810507	—	2	1	no	—
12	#16810507	—	2	1	no	—
13	#16810489	C₂²	1	1	yes	characteristic
14	C₄×#11719919	—	2	1	no	—
15	C₄×#11719919	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
17	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
18	#16810378	—	2	1	no	—
19	#16810364	—	2	4	no	—
20	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
21	C₂×#97×C₅₃	C₂²	1	2	yes	—
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#336	—	2	1	no	—
24	C₂²×#88322	—	2	2	no	—
25	#13133209	—	2	1	no	—
26	#13133209	—	2	1	no	—
27	#13133209	D₄	1	2	yes	—
28	#13133209	Q₈	1	2	yes	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719919	—	2	1	no	—
34	C₂×#11719919	—	2	1	no	—
35	C₂×#11719919	—	2	1	no	—
36	C₂×#11719919	—	2	1	no	—
37	C₂×C₄×Dic₅₃	—	2	1	no	—
38	C₂×C₄×Dic₅₃	—	2	1	no	—
39	C₄⋊#88322	—	2	4	no	—
40	#13133128	—	2	2	no	—
41	#13133128	—	2	2	no	—
42	#13133124	—	2	1	no	—
43	#13133124	—	2	1	no	—
44	#13133124	—	2	1	no	—
45	#13133124	—	2	1	no	—
46	#13133124	—	2	1	no	—
47	#13133124	—	2	1	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	4	no	—
57	#97×C₅₃	—	2	2	no	—
58	#97×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	—	2	1	no	—
67	C₂²×Dic₅₃	—	2	1	no	—
68	C₂²×Dic₅₃	—	2	1	no	—
69	C₂²×Dic₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	4	1	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	C₄²×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₅₃	#41	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₄⋊Dic₅₃	—	2	1	no	—
84	C₄⋊Dic₅₃	—	2	1	no	—
85	C₄⋊Dic₅₃	—	4	1	no	—
86	C₄⋊Dic₅₃	—	4	1	no	—
87	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
88	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
89	#11719919	—	2	2	no	—
90	#11719919	—	2	2	no	—
91	#11719919	—	2	2	no	—
92	#11719919	—	2	2	no	—
93	#11719919	—	4	2	no	—
94	#11719919	—	4	2	no	—
95	C₄×Dic₅₃	—	4	2	no	—
96	C₄×Dic₅₃	—	4	2	no	—
97	C₂×#88322	—	2	4	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	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	#101	1	2	yes	—
117	C₂×C₄×C₅₃	#104	1	2	yes	—
118	C₂×C₄×C₅₃	#104	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	4	1	no	—
128	C₂×Dic₅₃	—	4	1	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	—	4	1	no	—
134	C₂×Dic₅₃	—	4	1	no	—
135	C₂×Dic₅₃	—	4	1	no	—
136	C₂×Dic₅₃	—	4	1	no	—
137	C₈×C₅₃	—	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	1	no	—
157	Dic₅₃	—	4	2	no	—
158	Dic₅₃	—	4	2	no	—
159	Dic₅₃	—	8	1	no	—
160	Dic₅₃	—	8	1	no	—
161	C₄×#125	—	53	1	no	—
162	#1767	—	53	1	no	—
163	#1767	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
174	C₂×C₄⋊C₈	—	2	1	no	—
175	C₂×C₄⋊C₈	—	106	1	no	—
176	C₂×#97	—	53	2	no	—
177	C₂×#97	C₂×D₅₃	1	2	yes	—
178	#389	—	53	1	no	—
179	#389	—	53	1	no	—
180	#387	—	106	1	no	—
181	#387	—	106	1	no	—
182	#382	—	53	1	no	—
183	C₄×C₄⋊C₄	—	53	1	no	—
184	C₄×C₄⋊C₄	—	53	1	no	—
185	C₄×C₄⋊C₄	—	106	1	no	—
186	C₄×C₄⋊C₄	—	106	1	no	—
187	C₄³	C₂×D₅₃	1	1	yes	characteristic
188	#336	—	2	1	no	—
189	#336	—	106	1	no	—
190	#330	—	106	4	no	—
191	C₅₃	#8582	1	1	yes	characteristic, complemented, Sylow
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	106	2	no	—
194	#125	—	53	2	no	—
195	#125	—	53	2	no	—
196	#125	—	106	1	no	—
197	#125	—	106	1	no	—
198	C₂×C₄⋊C₄	—	53	1	no	—
199	C₂×C₄⋊C₄	—	53	1	no	—
200	C₂×C₄⋊C₄	—	53	1	no	—
201	C₂×C₄⋊C₄	—	53	1	no	—
202	C₂×C₄⋊C₄	—	106	1	no	—
203	C₂×C₄⋊C₄	—	106	1	no	—
204	C₂×C₄⋊C₄	—	106	1	no	—
205	C₂×C₄⋊C₄	—	106	1	no	—
206	C₂×C₄²	—	106	1	no	—
207	C₂×C₄²	—	106	1	no	—
208	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
209	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
210	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
211	C₂×C₄²	D₂₁₂	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	—	2	4	no	—
216	C₄⋊C₈	—	106	4	no	—
217	#97	—	2	2	no	—
218	#97	—	2	2	no	—
219	#97	—	106	2	no	—
220	#97	—	106	2	no	—
221	#95	—	106	1	no	—
222	#95	—	106	1	no	—
223	#95	—	106	1	no	—
224	#95	—	106	1	no	—
225	#95	—	106	1	no	—
226	#95	—	106	1	no	—
227	C₂²×C₄	—	106	1	no	—
228	C₂²×C₄	—	106	1	no	—
229	C₂²×C₄	—	106	1	no	—
230	C₂²×C₄	—	106	1	no	—
231	C₂²×C₄	#11719921	1	1	yes	characteristic
232	C₂²×C₄	#11719921	1	1	yes	characteristic
233	C₂²×C₄	#11719921	1	1	yes	characteristic
234	C₂²×C₄	#11719921	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	—	212	2	no	—
243	C₂×C₈	—	212	2	no	—
244	C₄⋊C₄	—	53	2	no	—
245	C₄⋊C₄	—	53	2	no	—
246	C₄⋊C₄	—	106	1	no	—
247	C₄⋊C₄	—	106	1	no	—
248	C₄⋊C₄	—	106	2	no	—
249	C₄⋊C₄	—	106	2	no	—
250	C₄⋊C₄	—	106	2	no	—
251	C₄⋊C₄	—	106	2	no	—
252	C₄⋊C₄	—	212	1	no	—
253	C₄⋊C₄	—	212	1	no	—
254	C₄⋊C₄	—	212	2	no	—
255	C₄⋊C₄	—	212	2	no	—
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	—	212	2	no	—
263	C₄²	—	212	2	no	—
264	C₄²	D₄×D₅₃	1	1	yes	characteristic
265	C₄²	#11719947	1	1	yes	characteristic
266	C₄²	Q₈×D₅₃	1	1	yes	characteristic
267	C₄²	#11719949	1	1	yes	characteristic
268	C₂³	C₂×#11719921	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	—	106	2	no	—
285	C₂×C₄	—	106	2	no	—
286	C₂×C₄	—	212	1	no	—
287	C₂×C₄	—	212	1	no	—
288	C₂×C₄	—	212	1	no	—
289	C₂×C₄	—	212	1	no	—
290	C₂×C₄	—	212	1	no	—
291	C₂×C₄	—	212	1	no	—
292	C₂×C₄	—	212	1	no	—
293	C₂×C₄	—	212	1	no	—
294	C₂×C₄	—	212	1	no	—
295	C₂×C₄	—	212	1	no	—
296	C₂×C₄	#13133135	1	2	yes	—
297	C₂×C₄	#13133143	1	2	yes	—
298	C₂×C₄	#13133145	1	2	yes	—
299	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
300	C₂×C₄	#13133175	1	1	yes	characteristic
301	C₂×C₄	#13133188	1	1	yes	characteristic
302	C₂×C₄	#13133192	1	1	yes	characteristic
303	C₂×C₄	#13133193	1	1	yes	characteristic
304	C₂×C₄	#13133197	1	1	yes	characteristic
305	C₈	—	4	4	no	—
306	C₈	—	212	4	no	—
307	C₂²	#16810480	1	1	yes	characteristic
308	C₂²	C₂×#13133135	1	1	yes	characteristic
309	C₂²	#16810605	1	1	yes	characteristic
310	C₂²	C₂×#13133143	1	1	yes	characteristic
311	C₂²	C₂×#13133145	1	1	yes	characteristic
312	C₂²	#16810639	1	1	yes	characteristic
313	C₂²	#16810641	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	1	no	—
324	C₄	—	424	1	no	—
325	C₄	—	424	1	no	—
326	C₄	—	212	2	no	—
327	C₄	—	212	2	no	—
328	C₂	#20273301	1	1	yes	characteristic
329	C₂	#20273316	1	1	yes	characteristic
330	C₂	#20273331	1	1	yes	characteristic
331	C₂	#20273347	1	1	yes	characteristic
332	C₂	#20274708	1	1	yes	characteristic
333	C₂	#20274732	1	1	yes	characteristic
334	C₂	#20274736	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1767	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1767	D₅₃
hypocenter	C₅₃	#8582
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

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
C₄×#13133209	2
#20274568	2
C₂×#16810364	2
#20272597	2
C₅₃×#1767	2
#8582	53