Subgroups — #27484784

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	#20276272	C₂	1	4	yes	maximal
3	#20275675	C₂	1	2	yes	maximal
4	#20275668	C₂	1	4	yes	maximal
5	#20274842	C₂	1	2	yes	maximal
6	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133217	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₆	—	2	1	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	#16810529	—	2	1	no	—
13	#16810521	—	2	1	no	—
14	#16810521	C₂²	1	2	yes	—
15	#16810493	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊Dic₅₃	—	2	1	no	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
18	#16810385	C₂²	1	8	yes	—
19	#16810385	C₂²	1	8	yes	—
20	#16810378	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#384	—	2	1	no	—
25	C₅₃×#384	—	2	1	no	—
26	C₅₃×#384	C₂²	1	2	yes	—
27	C₅₃×#384	C₂²	1	2	yes	—
28	C₄³×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₀₆	—	2	1	no	—
30	C₂²×#88322	C₂³	1	4	yes	—
31	#13133217	—	2	2	no	—
32	#13133217	Q₈	1	4	yes	complemented
33	C₄×Dic₁₀₆	—	4	4	no	—
34	C₂×C₄⋊Dic₅₃	—	2	1	no	—
35	C₂×C₄⋊Dic₅₃	—	2	1	no	—
36	C₂×C₄⋊Dic₅₃	—	2	1	no	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
39	C₂×#11719919	—	2	2	no	—
40	C₂×C₄×Dic₅₃	—	2	2	no	—
41	C₄⋊#88322	—	2	4	no	—
42	C₄×#88322	—	2	4	no	—
43	#13133124	—	2	2	no	—
44	#13133124	—	2	2	no	—
45	C₂²×Q₈×C₅₃	—	2	2	no	—
46	C₄⋊Q₈×C₅₃	—	2	8	no	—
47	C₄⋊Q₈×C₅₃	—	4	4	no	—
48	C₄⋊Q₈×C₅₃	—	4	4	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	—	4	4	no	—
63	C₂×Q₈×C₅₃	—	4	4	no	—
64	C₂×Q₈×C₅₃	—	4	4	no	—
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	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	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	2	2	no	—
74	C₂²×Dic₅₃	—	2	2	no	—
75	C₂×Dic₁₀₆	—	2	4	no	—
76	C₂×Dic₁₀₆	—	4	2	no	—
77	C₂×Dic₁₀₆	—	4	2	no	—
78	C₄⋊C₄×C₅₃	—	2	8	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₄⋊C₄×C₅₃	—	4	4	no	—
81	C₄⋊C₄×C₅₃	—	4	4	no	—
82	C₄⋊C₄×C₅₃	—	4	4	no	—
83	C₄⋊C₄×C₅₃	—	4	4	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	D₈	1	4	yes	—
88	C₄²×C₅₃	SD₁₆	1	4	yes	—
89	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₅₃	#41	1	2	yes	—
91	C₄²×C₅₃	#41	1	4	yes	—
92	C₄⋊Dic₅₃	—	2	4	no	—
93	C₄⋊Dic₅₃	—	2	2	no	—
94	C₄⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	4	2	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	C₄⋊Dic₅₃	C₂×Q₈	1	4	yes	—
98	#11719919	—	4	4	no	—
99	C₄×Dic₅₃	—	4	4	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	2	4	no	—
102	C₂×#88322	—	2	4	no	—
103	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₅₃	—	8	8	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	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	#124	1	2	yes	—
117	C₂×C₄×C₅₃	#124	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₅₃	#137	1	2	yes	—
125	C₂×C₄×C₅₃	#137	1	2	yes	—
126	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
130	C₂×Dic₅₃	—	2	4	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	4	2	no	—
136	Dic₁₀₆	—	4	8	no	—
137	#88322	—	4	4	no	—
138	#29219	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
141	C₂²×C₅₃	#536	1	1	yes	characteristic
142	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
143	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	8	4	no	—
150	C₄×C₅₃	#474	1	4	yes	—
151	C₄×C₅₃	#475	1	4	yes	—
152	C₄×C₅₃	#486	1	4	yes	—
153	C₄×C₅₃	#556	1	2	yes	—
154	C₄×C₅₃	#557	1	2	yes	—
155	Dic₅₃	—	4	4	no	—
156	Dic₅₃	—	8	2	no	—
157	#2868	D₅₃	1	1	yes	characteristic
158	C₄×C₄⋊Q₈	—	53	1	no	—
159	#2099	—	53	4	no	—
160	#1990	—	53	2	no	—
161	#1988	—	53	4	no	—
162	#1834	—	53	2	no	—
163	C₄×C₄⋊C₈	—	53	1	no	—
164	C₂×C₅₃	#2568	1	1	yes	characteristic
165	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
166	C₂×C₅₃	C₂×#475	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
168	C₂×C₅₃	#3139	1	1	yes	characteristic
169	C₂×C₅₃	#3226	1	1	yes	characteristic
170	C₂×C₅₃	#3229	1	1	yes	characteristic
171	C₂×C₄⋊Q₈	—	2	1	no	—
172	C₂×C₄⋊Q₈	—	2	1	no	—
173	C₂×C₄⋊Q₈	—	53	1	no	—
174	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
175	C₂×C₄×Q₈	—	106	1	no	—
176	C₂×C₄⋊C₈	—	53	2	no	—
177	C₂×C₄×C₈	—	53	2	no	—
178	#391	—	106	1	no	—
179	#389	—	53	2	no	—
180	#389	—	106	1	no	—
181	#384	—	2	1	no	—
182	#384	—	2	1	no	—
183	#384	—	53	1	no	—
184	#384	C₂×D₅₃	1	2	yes	—
185	#384	C₂×D₅₃	1	2	yes	—
186	C₄×C₄⋊C₄	—	53	2	no	—
187	C₄×C₄⋊C₄	—	106	1	no	—
188	C₄³	C₂×D₅₃	1	1	yes	characteristic
189	#340	—	53	8	no	—
190	#340	—	53	8	no	—
191	#336	—	53	2	no	—
192	C₅₃	#29219	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₈	—	2	2	no	—
194	C₂²×Q₈	—	106	1	no	—
195	C₂²×C₈	—	53	4	no	—
196	C₄⋊Q₈	—	2	8	no	—
197	C₄⋊Q₈	—	4	4	no	—
198	C₄⋊Q₈	—	4	4	no	—
199	C₄⋊Q₈	—	53	4	no	—
200	C₄⋊Q₈	—	106	2	no	—
201	C₄×Q₈	—	212	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	53	2	no	—
208	C₂×C₄⋊C₄	—	53	2	no	—
209	C₂×C₄⋊C₄	—	106	1	no	—
210	C₂×C₄⋊C₄	—	106	2	no	—
211	C₂×C₄⋊C₄	—	106	1	no	—
212	C₂×C₄⋊C₄	—	106	1	no	—
213	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
214	C₂×C₄²	—	106	2	no	—
215	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
222	C₄⋊C₈	—	106	4	no	—
223	C₄×C₈	—	106	4	no	—
224	#95	—	106	2	no	—
225	#95	—	106	2	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	106	4	no	—
230	C₂×Q₈	—	212	2	no	—
231	C₂×Q₈	—	212	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	106	2	no	—
234	C₂²×C₄	—	106	2	no	—
235	C₂²×C₄	C₂×C₂²⋊D₅₃	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	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	—	106	4	no	—
244	C₂×C₈	—	106	4	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	53	4	no	—
252	C₄⋊C₄	—	106	4	no	—
253	C₄⋊C₄	—	106	2	no	—
254	C₄⋊C₄	—	106	4	no	—
255	C₄⋊C₄	—	212	4	no	—
256	C₄⋊C₄	—	212	2	no	—
257	C₄⋊C₄	—	212	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	212	4	no	—
262	C₄²	#11719922	1	4	yes	—
263	C₄²	#11719923	1	4	yes	—
264	C₄²	#11719947	1	4	yes	—
265	C₄²	Q₈×D₅₃	1	2	yes	—
266	C₄²	#11719949	1	2	yes	—
267	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
268	Q₈	—	8	8	no	—
269	Q₈	—	212	8	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	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	106	4	no	—
280	C₂×C₄	—	212	2	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	—	212	2	no	—
283	C₂×C₄	—	212	2	no	—
284	C₂×C₄	—	212	2	no	—
285	C₂×C₄	#13133193	1	2	yes	—
286	C₂×C₄	#13133193	1	2	yes	—
287	C₂×C₄	#13133204	1	2	yes	—
288	C₂×C₄	#13133204	1	2	yes	—
289	C₂×C₄	C₂×#11719922	1	2	yes	—
290	C₂×C₄	C₂×#11719922	1	2	yes	—
291	C₂×C₄	C₂×#11719922	1	2	yes	—
292	C₂×C₄	C₂×#11719923	1	2	yes	—
293	C₂×C₄	C₂×#11719923	1	2	yes	—
294	C₂×C₄	C₂×#11719923	1	2	yes	—
295	C₂×C₄	#13133248	1	2	yes	—
296	C₂×C₄	#13133248	1	2	yes	—
297	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×#11719947	1	2	yes	—
299	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
300	C₂×C₄	#13133265	1	2	yes	—
301	C₈	—	212	4	no	—
302	C₂²	C₂×#13133193	1	1	yes	characteristic
303	C₂²	C₂×#13133204	1	1	yes	characteristic
304	C₂²	#16811197	1	1	yes	characteristic
305	C₂²	C₂²×#11719922	1	1	yes	characteristic
306	C₂²	C₂²×#11719923	1	1	yes	characteristic
307	C₂²	C₂×#13133248	1	1	yes	characteristic
308	C₂²	C₂×#13133248	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	8	4	no	—
313	C₄	—	424	2	no	—
314	C₄	—	212	4	no	—
315	C₄	#16810900	1	4	yes	—
316	C₄	#16810905	1	4	yes	—
317	C₄	#16810969	1	4	yes	—
318	C₄	#16811347	1	2	yes	—
319	C₄	#16811348	1	2	yes	—
320	C₂	#20279931	1	1	yes	characteristic
321	C₂	C₂×#16810900	1	1	yes	characteristic
322	C₂	C₂×#16810905	1	1	yes	characteristic
323	C₂	C₂×#16810969	1	1	yes	characteristic
324	C₂	#20285095	1	1	yes	characteristic
325	C₂	#20285836	1	1	yes	characteristic
326	C₂	#20285859	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2868	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2868	D₅₃
hypocenter	C₅₃	#29219
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
32. #13133217	Q₈	semidirect
192. C₅₃	#29219	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
#20276272	2
#20275675	2
#20275668	2
#20274842	2
C₄×C₄⋊#88322	2
C₅₃×#2868	2
#29219	53