Subgroups — #27276882

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 346 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	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810368	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	#16810529	—	2	1	no	—
12	#16810521	—	2	1	no	—
13	#16810521	C₄	1	2	yes	—
14	#16810493	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
17	#16810368	—	2	4	no	—
18	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₀₆	—	2	1	no	—
23	C₂²×#88322	—	2	2	no	—
24	#13133217	—	2	1	no	—
25	#13133217	—	2	1	no	—
26	#13133217	D₄	1	2	yes	—
27	#13133217	Q₈	1	2	yes	—
28	C₄×Dic₁₀₆	—	4	2	no	—
29	C₄×Dic₁₀₆	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	1	no	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂×#11719919	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	—	4	2	no	—
40	C₄⋊#88322	—	4	2	no	—
41	#13133124	—	2	2	no	—
42	#13133124	—	2	2	no	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	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	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	—	2	2	no	—
53	C₄⋊C₈×C₅₃	—	4	2	no	—
54	C₄⋊C₈×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	2	2	no	—
63	C₂²×Dic₅₃	—	2	2	no	—
64	C₂×Dic₁₀₆	—	2	2	no	—
65	C₂×Dic₁₀₆	—	2	2	no	—
66	C₂×Dic₁₀₆	—	4	1	no	—
67	C₂×Dic₁₀₆	—	4	1	no	—
68	C₂×Dic₁₀₆	—	4	1	no	—
69	C₂×Dic₁₀₆	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	2	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	1	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	D₈	1	2	yes	—
80	C₄²×C₅₃	SD₁₆	1	2	yes	—
81	C₄²×C₅₃	SD₁₆	1	2	yes	—
82	C₄²×C₅₃	Q₁₆	1	2	yes	—
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₅₃	#41	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	2	no	—
90	C₄⋊Dic₅₃	—	4	1	no	—
91	C₄⋊Dic₅₃	—	4	1	no	—
92	C₄⋊Dic₅₃	—	4	1	no	—
93	C₄⋊Dic₅₃	—	4	1	no	—
94	C₄⋊Dic₅₃	—	4	2	no	—
95	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
96	#11719919	—	4	4	no	—
97	C₄×Dic₅₃	—	4	4	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂×#88322	—	4	2	no	—
101	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#101	1	2	yes	—
112	C₂×C₄×C₅₃	#102	1	2	yes	—
113	C₂×C₄×C₅₃	#102	1	2	yes	—
114	C₂×C₄×C₅₃	#103	1	2	yes	—
115	C₂×C₄×C₅₃	#103	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
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	2	no	—
137	Dic₁₀₆	—	4	2	no	—
138	Dic₁₀₆	—	4	2	no	—
139	Dic₁₀₆	—	4	2	no	—
140	C₈×C₅₃	—	8	2	no	—
141	#88322	—	8	2	no	—
142	#8586	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	#380	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
145	C₂²×C₅₃	#413	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
148	C₂²×C₅₃	#417	1	1	yes	characteristic
149	C₂²×C₅₃	#417	1	1	yes	characteristic
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	4	1	no	—
155	C₄×C₅₃	#332	1	4	yes	complemented
156	C₄×C₅₃	#459	1	1	yes	characteristic
157	C₄×C₅₃	#460	1	1	yes	characteristic
158	C₄×C₅₃	#461	1	1	yes	characteristic
159	C₄×C₅₃	#462	1	1	yes	characteristic
160	C₄×C₅₃	#474	1	1	yes	characteristic
161	C₄×C₅₃	#475	1	1	yes	characteristic
162	C₄×C₅₃	#476	1	1	yes	characteristic
163	C₄×C₅₃	#477	1	1	yes	characteristic
164	Dic₅₃	—	4	4	no	—
165	Dic₅₃	—	8	2	no	—
166	C₄×C₄⋊Q₈	—	53	1	no	—
167	#1842	—	53	1	no	—
168	#1842	D₅₃	1	1	yes	characteristic
169	C₂×#332	—	53	2	no	—
170	#1296	—	53	2	no	—
171	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
172	C₂×C₅₃	#1547	1	1	yes	characteristic
173	C₂×C₅₃	#1550	1	1	yes	characteristic
174	C₂×C₅₃	#1561	1	1	yes	characteristic
175	C₂×C₅₃	#1801	1	1	yes	characteristic
176	C₂×C₅₃	#1802	1	1	yes	characteristic
177	C₂×C₅₃	#1803	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	—	53	1	no	—
179	C₂×C₄×Q₈	—	106	1	no	—
180	C₂×C₄⋊C₈	—	2	1	no	—
181	C₂×C₄⋊C₈	—	2	1	no	—
182	C₂×C₄⋊C₈	—	53	2	no	—
183	C₂×C₄⋊C₈	—	106	1	no	—
184	C₂×C₄⋊C₈	—	106	1	no	—
185	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
186	#391	—	106	1	no	—
187	#389	—	53	2	no	—
188	#389	—	106	1	no	—
189	#384	—	53	1	no	—
190	C₄×C₄⋊C₄	—	53	2	no	—
191	C₄×C₄⋊C₄	—	106	1	no	—
192	C₄³	C₂×D₅₃	1	1	yes	characteristic
193	#332	—	106	4	no	—
194	C₅₃	#8586	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	106	1	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	106	2	no	—
198	C₄⋊Q₈	—	53	2	no	—
199	C₄⋊Q₈	—	53	2	no	—
200	C₄⋊Q₈	—	106	1	no	—
201	C₄⋊Q₈	—	106	1	no	—
202	C₄×Q₈	—	212	2	no	—
203	C₄×Q₈	—	212	2	no	—
204	C₂×C₄⋊C₄	—	53	2	no	—
205	C₂×C₄⋊C₄	—	53	2	no	—
206	C₂×C₄⋊C₄	—	106	1	no	—
207	C₂×C₄⋊C₄	—	106	1	no	—
208	C₂×C₄⋊C₄	—	106	1	no	—
209	C₂×C₄⋊C₄	—	106	2	no	—
210	C₂×C₄²	—	106	2	no	—
211	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
214	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
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	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	4	2	no	—
221	C₄⋊C₈	—	4	2	no	—
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	2	no	—
224	C₄⋊C₈	—	212	2	no	—
225	C₄⋊C₈	—	212	2	no	—
226	#95	—	106	2	no	—
227	#95	—	106	2	no	—
228	C₂×Q₈	—	106	2	no	—
229	C₂×Q₈	—	106	2	no	—
230	C₂×Q₈	—	212	1	no	—
231	C₂×Q₈	—	212	1	no	—
232	C₂×Q₈	—	212	1	no	—
233	C₂×Q₈	—	212	1	no	—
234	C₂²×C₄	—	106	2	no	—
235	C₂²×C₄	—	106	2	no	—
236	C₂²×C₄	#11719921	1	1	yes	characteristic
237	C₂²×C₄	#11719921	1	1	yes	characteristic
238	C₂²×C₄	#11719921	1	1	yes	characteristic
239	C₂²×C₄	#11719921	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	212	2	no	—
247	C₂×C₈	—	212	2	no	—
248	C₂×C₈	—	212	2	no	—
249	C₄⋊C₄	—	53	4	no	—
250	C₄⋊C₄	—	106	2	no	—
251	C₄⋊C₄	—	106	2	no	—
252	C₄⋊C₄	—	106	2	no	—
253	C₄⋊C₄	—	212	2	no	—
254	C₄⋊C₄	—	212	4	no	—
255	C₄⋊C₄	—	212	1	no	—
256	C₄⋊C₄	—	212	1	no	—
257	C₄⋊C₄	—	212	1	no	—
258	C₄⋊C₄	—	212	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	212	4	no	—
266	C₄²	#11719913	1	2	yes	—
267	C₄²	D₄₂₄	1	2	yes	—
268	C₄²	#11719923	1	2	yes	—
269	C₄²	#11719925	1	2	yes	—
270	C₄²	D₄×D₅₃	1	1	yes	characteristic
271	C₄²	#11719947	1	1	yes	characteristic
272	C₄²	Q₈×D₅₃	1	1	yes	characteristic
273	C₄²	#11719949	1	1	yes	characteristic
274	C₂³	C₂×#11719921	1	1	yes	characteristic
275	Q₈	—	212	2	no	—
276	Q₈	—	212	2	no	—
277	Q₈	—	212	2	no	—
278	Q₈	—	212	2	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	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	106	4	no	—
289	C₂×C₄	—	212	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	#13133135	1	2	yes	—
295	C₂×C₄	#13133139	1	2	yes	—
296	C₂×C₄	#13133139	1	2	yes	—
297	C₂×C₄	#13133140	1	2	yes	—
298	C₂×C₄	#13133140	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	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
306	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
307	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
309	C₂×C₄	#13133243	1	1	yes	characteristic
310	C₂×C₄	#13133245	1	1	yes	characteristic
311	C₂×C₄	#13133248	1	1	yes	characteristic
312	C₂×C₄	#13133250	1	1	yes	characteristic
313	C₈	—	8	2	no	—
314	C₈	—	424	2	no	—
315	C₂²	#16810480	1	1	yes	characteristic
316	C₂²	C₂×#13133135	1	1	yes	characteristic
317	C₂²	#16810605	1	1	yes	characteristic
318	C₂²	C₂×#13133139	1	1	yes	characteristic
319	C₂²	C₂×#13133140	1	1	yes	characteristic
320	C₂²	#16810622	1	1	yes	characteristic
321	C₂²	#16810625	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	424	2	no	—
328	C₄	—	212	4	no	—
329	C₄	#16810368	1	4	yes	complemented
330	C₄	#16810819	1	1	yes	characteristic
331	C₄	#16810820	1	1	yes	characteristic
332	C₄	#16810829	1	1	yes	characteristic
333	C₄	#16810830	1	1	yes	characteristic
334	C₄	#16810904	1	1	yes	characteristic
335	C₄	#16810905	1	1	yes	characteristic
336	C₄	#16810914	1	1	yes	characteristic
337	C₄	#16810915	1	1	yes	characteristic
338	C₂	C₂×#16810368	1	1	yes	characteristic
339	C₂	#20273390	1	1	yes	characteristic
340	C₂	#20273406	1	1	yes	characteristic
341	C₂	#20273452	1	1	yes	characteristic
342	C₂	#20274708	1	1	yes	characteristic
343	C₂	#20274716	1	1	yes	characteristic
344	C₂	#20274717	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#8586
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
155. C₄×C₅₃	#332	semidirect
194. C₅₃	#8586	semidirect
329. C₄	#16810368	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
#20274867	2
C₂×#16810368	2
#20272599	2
C₅₃×#1842	2
#8586	53