Subgroups — #27354767

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285298	C₂	1	1	yes	maximal, characteristic
2	#20283811	C₂	1	2	yes	maximal
3	#20273625	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273610	C₂	1	2	yes	maximal, complemented
5	#20273534	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273533	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273519	C₂	1	2	yes	maximal, complemented
8	#20273475	C₂	1	1	yes	maximal, characteristic
9	#20273186	C₂	1	2	yes	maximal
10	#20273000	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811177	C₂²	1	2	yes	—
13	#16811027	C₄	1	1	yes	characteristic
14	#16811001	C₄	1	1	yes	characteristic
15	#16810972	C₄	1	2	yes	—
16	#16810865	C₄	1	1	yes	characteristic
17	#16810863	C₄	1	2	yes	—
18	#16810861	C₄	1	1	yes	characteristic
19	#16810753	C₂²	1	2	yes	—
20	#16810751	—	2	2	no	—
21	#16810750	—	2	2	no	—
22	Q₈×#88322	C₂²	1	2	yes	—
23	#16810747	C₂²	1	2	yes	—
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	C₄×#11719925	C₂²	1	2	yes	—
26	#16810685	—	2	2	no	—
27	#16810683	—	2	2	no	—
28	#16810679	C₂²	1	2	yes	—
29	#16810674	C₂²	1	1	yes	characteristic
30	#16810674	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810375	C₂²	1	2	yes	—
33	#16810374	C₂²	1	1	yes	characteristic
34	#16810355	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	C₂²	1	2	yes	—
37	#16810344	C₂²	1	1	yes	characteristic
38	C₈×#88322	C₂²	1	1	yes	characteristic
39	C₈×#88322	C₂²	1	1	yes	characteristic
40	C₅₃×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#334	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₅₃×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719925	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
49	#13133216	C₂×C₄	1	2	yes	—
50	#13133215	C₂×C₄	1	2	yes	—
51	#13133207	—	2	2	no	—
52	#13133197	C₂×C₄	1	2	yes	—
53	#13133195	—	2	2	no	—
54	#13133193	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₀₆	—	2	2	no	—
56	C₄×Dic₁₀₆	C₂³	1	2	yes	—
57	#13133174	—	2	2	no	—
58	C₄²×D₅₃	C₂³	1	1	yes	characteristic
59	#13133151	C₂×C₄	1	1	yes	characteristic
60	#13133150	C₂×C₄	1	2	yes	—
61	C₄⋊#88322	—	2	2	no	—
62	C₄⋊#88322	C₂³	1	2	yes	—
63	#13133146	—	2	2	no	—
64	#13133146	—	2	1	no	—
65	#13133146	—	2	2	no	—
66	#13133146	D₄	1	2	yes	—
67	#13133142	C₂×C₄	1	2	yes	—
68	#13133141	C₂×C₄	1	2	yes	—
69	#13133140	C₂×C₄	1	2	yes	—
70	#13133129	—	2	2	no	—
71	#13133129	—	2	1	no	—
72	#13133129	D₄	1	2	yes	—
73	#13133127	—	2	2	no	—
74	#13133127	—	2	2	no	—
75	C₄×#88322	—	2	2	no	—
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₈×Dic₅₃	—	2	2	no	—
80	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
84	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	—	2	2	no	—
86	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
87	#103×C₅₃	C₂×C₄	1	2	yes	—
88	#97×C₅₃	—	2	2	no	—
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
96	C₂×Dic₁₀₆	—	2	2	no	—
97	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
98	Q₁₆×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
108	#11719925	—	2	4	no	—
109	#11719921	—	2	2	no	—
110	#11719921	—	2	2	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
113	#11719919	—	2	2	no	—
114	#11719919	—	2	2	no	—
115	#11719919	—	2	2	no	—
116	#11719919	—	2	2	no	—
117	#11719919	—	2	2	no	—
118	C₄×Dic₅₃	—	2	2	no	—
119	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₅₃	#41	1	1	yes	characteristic
122	C₂×#88322	—	2	2	no	—
123	C₂×#88322	—	2	1	no	—
124	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
125	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
126	C₂×#88322	C₂×D₄	1	2	yes	—
127	Dic₂₁₂	—	2	2	no	—
128	C₈×D₅₃	—	2	2	no	—
129	C₈×D₅₃	—	2	2	no	—
130	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
131	Q₈×C₅₃	—	2	2	no	—
132	Q₈×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₅₃	—	2	2	no	—
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
143	C₄×D₅₃	—	2	1	no	—
144	C₄×D₅₃	—	2	1	no	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	4	1	no	—
147	Dic₁₀₆	—	2	2	no	—
148	Dic₁₀₆	—	2	2	no	—
149	Dic₁₀₆	—	8	2	no	—
150	C₈×C₅₃	—	2	1	no	—
151	C₈×C₅₃	—	2	2	no	—
152	C₈×C₅₃	—	4	1	no	—
153	C₈×C₅₃	—	4	2	no	—
154	C₈×C₅₃	#131	1	2	yes	—
155	#88322	—	2	1	no	—
156	#88322	—	2	2	no	—
157	#88322	—	4	1	no	—
158	#88322	—	4	2	no	—
159	#88322	#131	1	2	yes	—
160	#17972	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×D₅₃	—	2	1	no	—
163	C₂×D₅₃	—	2	1	no	—
164	C₂×D₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	4	2	no	—
167	C₄×C₅₃	—	4	2	no	—
168	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
169	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
170	C₄×C₅₃	#578	1	2	yes	—
171	Dic₅₃	—	2	1	no	—
172	Dic₅₃	—	2	1	no	—
173	Dic₅₃	—	2	1	no	—
174	Dic₅₃	—	4	2	no	—
175	Dic₅₃	—	4	2	no	—
176	#3170	—	53	1	no	—
177	#2985	—	53	2	no	—
178	#1593	—	53	1	no	—
179	#1593	D₅₃	1	1	yes	characteristic, complemented
180	#1591	—	53	2	no	—
181	#1579	—	53	1	no	—
182	#1579	—	53	1	no	—
183	C₈×Q₁₆	—	53	2	no	—
184	#1568	—	53	1	no	—
185	#1497	—	53	2	no	—
186	#1450	—	53	1	no	—
187	C₂×C₅₃	#2931	1	1	yes	characteristic
188	C₂×C₅₃	#2950	1	1	yes	characteristic
189	C₂×C₅₃	#2957	1	1	yes	characteristic
190	D₅₃	—	4	1	no	—
191	#533	—	53	2	no	—
192	#500	—	53	1	no	—
193	#494	—	53	1	no	—
194	#487	—	53	2	no	—
195	#467	—	53	1	no	—
196	#467	—	53	1	no	—
197	#467	—	53	2	no	—
198	#446	—	53	2	no	—
199	#446	—	106	2	no	—
200	#446	—	106	2	no	—
201	#446	C₂×D₅₃	1	2	yes	—
202	C₈×Q₈	—	53	2	no	—
203	C₈×Q₈	—	53	2	no	—
204	C₄×Q₁₆	—	53	2	no	—
205	C₄×Q₁₆	—	53	1	no	—
206	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
207	#433	—	106	2	no	—
208	#433	—	106	2	no	—
209	#432	—	53	2	no	—
210	#431	—	53	1	no	—
211	#431	—	53	1	no	—
212	C₂×C₄×C₈	—	53	1	no	—
213	#335	—	53	2	no	—
214	#334	—	53	1	no	—
215	#334	C₂×D₅₃	1	1	yes	characteristic
216	Q₈⋊C₈	—	53	2	no	—
217	Q₈⋊C₈	—	53	2	no	—
218	Q₈⋊C₈	—	53	2	no	—
219	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
220	#322	—	53	1	no	—
221	#322	C₂×D₅₃	1	1	yes	characteristic
222	C₈²	—	53	1	no	—
223	C₈²	—	53	1	no	—
224	C₅₃	#17972	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₁₆	—	53	2	no	—
226	C₂×Q₁₆	—	53	1	no	—
227	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
228	C₂²×C₈	—	53	1	no	—
229	C₂²×C₈	—	53	1	no	—
230	C₄⋊Q₈	—	53	2	no	—
231	C₄⋊Q₈	—	53	2	no	—
232	#125	—	106	2	no	—
233	C₂²⋊Q₈	—	53	2	no	—
234	C₂²⋊Q₈	—	53	2	no	—
235	C₂²⋊Q₈	—	106	2	no	—
236	C₄×Q₈	—	53	2	no	—
237	C₄×Q₈	—	106	2	no	—
238	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
239	#117	—	106	2	no	—
240	C₂×C₄²	—	53	1	no	—
241	#107	—	53	1	no	—
242	#107	—	53	2	no	—
243	#107	C₄×D₅₃	1	1	yes	characteristic
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	53	2	no	—
246	C₄⋊C₈	—	53	2	no	—
247	C₄⋊C₈	—	106	2	no	—
248	C₄⋊C₈	—	106	1	no	—
249	C₄⋊C₈	—	106	2	no	—
250	C₄⋊C₈	—	106	2	no	—
251	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
252	#103	—	53	2	no	—
253	#103	—	53	2	no	—
254	#103	—	53	2	no	—
255	#103	C₄×D₅₃	1	2	yes	—
256	C₂²⋊C₈	—	53	2	no	—
257	C₂²⋊C₈	—	106	1	no	—
258	C₂²⋊C₈	—	106	2	no	—
259	#97	—	2	2	no	—
260	#97	—	106	2	no	—
261	#97	—	106	2	no	—
262	C₄×C₈	—	53	1	no	—
263	C₄×C₈	—	53	1	no	—
264	C₄×C₈	—	53	1	no	—
265	C₄×C₈	—	53	1	no	—
266	C₄×C₈	—	53	1	no	—
267	C₄×C₈	—	106	2	no	—
268	C₄×C₈	—	106	2	no	—
269	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
272	C₂×Q₈	—	53	2	no	—
273	C₂×Q₈	—	106	2	no	—
274	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
275	C₂²×C₄	—	53	1	no	—
276	C₂²×C₄	—	53	1	no	—
277	C₂²×C₄	—	53	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	106	4	no	—
280	Q₁₆	—	106	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	53	1	no	—
284	C₂×C₈	—	53	1	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	106	2	no	—
288	C₂×C₈	—	106	2	no	—
289	C₂×C₈	—	106	1	no	—
290	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₅₃	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	53	2	no	—
295	C₄⋊C₄	—	106	2	no	—
296	C₄⋊C₄	—	106	2	no	—
297	C₄⋊C₄	—	106	2	no	—
298	C₄⋊C₄	—	106	2	no	—
299	C₄⋊C₄	—	106	2	no	—
300	C₄⋊C₄	—	106	2	no	—
301	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
302	C₂²⋊C₄	—	106	2	no	—
303	C₂²⋊C₄	—	106	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	53	1	no	—
306	C₄²	—	53	1	no	—
307	C₄²	—	53	1	no	—
308	C₄²	—	106	2	no	—
309	C₄²	C₂³×D₅₃	1	1	yes	characteristic
310	C₂³	—	53	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	106	2	no	—
314	Q₈	—	106	2	no	—
315	Q₈	—	424	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	106	1	no	—
322	C₂×C₄	—	106	1	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	212	1	no	—
327	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
329	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	2	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	53	2	no	—
335	C₈	—	106	1	no	—
336	C₈	—	106	2	no	—
337	C₈	—	212	1	no	—
338	C₈	—	212	2	no	—
339	C₈	#13133225	1	2	yes	—
340	C₂²	—	106	1	no	—
341	C₂²	—	106	1	no	—
342	C₂²	—	106	1	no	—
343	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	106	1	no	—
348	C₄	—	106	1	no	—
349	C₄	—	106	1	no	—
350	C₄	—	212	2	no	—
351	C₄	—	212	2	no	—
352	C₄	#131×D₅₃	1	1	yes	characteristic
353	C₄	C₂×#13133225	1	1	yes	characteristic
354	C₄	#16811439	1	2	yes	—
355	C₂	—	212	1	no	—
356	C₂	#20283424	1	1	yes	characteristic
357	C₂	#20283580	1	1	yes	characteristic
358	C₂	#20283629	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1593	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1593	D₅₃
hypocenter	C₅₃	#17972
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #20273625	C₂	semidirect
4. #20273610	C₂	semidirect
5. #20273534	C₂	semidirect
6. #20273533	C₂	semidirect
7. #20273519	C₂	semidirect
11. C₅₃×#1593	C₂	semidirect
179. #1593	D₅₃	semidirect
224. C₅₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#20285298	2
#20283811	2
#20273625	2
#20273610	2
#20273534	2
#20273533	2
#20273519	2
#20273475	2
#20273186	2
#20273000	2
C₅₃×#1593	2
#17972	53