Subgroups — #27354621

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 384 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	#20283646	C₂	1	2	yes	maximal
3	#20273613	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273610	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273521	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273519	C₂	1	2	yes	maximal, complemented
8	#20273472	C₂	1	1	yes	maximal, characteristic
9	#20273168	C₂	1	2	yes	maximal
10	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1591	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	Q₈×#88322	C₂²	1	2	yes	—
22	#16810747	—	2	2	no	—
23	#16810747	C₂²	1	2	yes	—
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	C₄×#11719925	C₂²	1	2	yes	—
26	#16810648	—	2	2	no	—
27	#16810645	C₂²	1	2	yes	—
28	#16810580	—	2	2	no	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810376	C₂²	1	1	yes	characteristic
33	#16810375	C₂²	1	2	yes	—
34	#16810355	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	C₂²	1	2	yes	—
37	C₈×#88322	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×#335	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719925	C₂×C₄	1	2	yes	—
47	C₂×#11719912	—	2	2	no	—
48	C₂×C₈×D₅₃	—	2	1	no	—
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
53	#13133216	C₂×C₄	1	2	yes	—
54	#13133215	C₂×C₄	1	2	yes	—
55	#13133207	—	2	2	no	—
56	#13133197	C₂×C₄	1	2	yes	—
57	#13133195	—	2	2	no	—
58	#13133193	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₀₆	—	2	2	no	—
60	C₄×Dic₁₀₆	C₂³	1	2	yes	—
61	#13133174	—	2	2	no	—
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	#13133151	C₂×C₄	1	1	yes	characteristic
64	#13133150	C₂×C₄	1	2	yes	—
65	C₄⋊#88322	—	2	2	no	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	#13133146	—	2	2	no	—
68	#13133146	—	2	2	no	—
69	#13133146	—	2	2	no	—
70	#13133142	C₂×C₄	1	2	yes	—
71	#13133141	C₂×C₄	1	2	yes	—
72	#13133140	C₂×C₄	1	2	yes	—
73	#13133127	—	2	2	no	—
74	C₄×#88322	—	2	2	no	—
75	C₄×#88322	C₂³	1	1	yes	characteristic
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₈×Dic₅₃	—	2	1	no	—
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	C₈×Dic₅₃	D₄	1	1	yes	characteristic
83	C₈×Dic₅₃	D₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
86	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	—	2	2	no	—
88	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
89	#103×C₅₃	C₂×C₄	1	2	yes	—
90	#97×C₅₃	—	2	2	no	—
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
98	C₂×Dic₁₀₆	—	2	2	no	—
99	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
100	Q₁₆×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
111	#11719925	—	2	4	no	—
112	#11719921	—	2	2	no	—
113	#11719921	—	2	2	no	—
114	C₄⋊Dic₅₃	—	2	2	no	—
115	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
116	#11719919	—	2	2	no	—
117	#11719919	—	2	2	no	—
118	#11719919	—	2	2	no	—
119	#11719919	—	2	2	no	—
120	#11719919	—	2	2	no	—
121	C₄×Dic₅₃	—	2	2	no	—
122	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₅₃	#41	1	1	yes	characteristic
125	C₂×#88322	—	2	1	no	—
126	C₂×#88322	—	2	2	no	—
127	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
128	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
129	C₂×#88322	C₂×D₄	1	1	yes	characteristic
130	C₂×#88322	C₂×D₄	1	1	yes	characteristic
131	Dic₂₁₂	—	2	2	no	—
132	#11719912	—	4	4	no	—
133	C₈×D₅₃	—	2	4	no	—
134	C₈×D₅₃	—	2	2	no	—
135	C₈×D₅₃	—	2	2	no	—
136	C₈×D₅₃	—	2	2	no	—
137	C₈×D₅₃	Q₁₆	1	4	yes	complemented
138	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₅₃	—	2	2	no	—
140	Q₈×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
150	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
151	C₄×D₅₃	—	2	1	no	—
152	C₄×D₅₃	—	2	1	no	—
153	C₄×D₅₃	—	2	2	no	—
154	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
155	C₄×D₅₃	#135	1	2	yes	—
156	Dic₁₀₆	—	2	2	no	—
157	Dic₁₀₆	—	2	2	no	—
158	Dic₁₀₆	—	4	4	no	—
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	4	2	no	—
163	C₈×C₅₃	#131	1	2	yes	—
164	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
165	#88322	—	2	2	no	—
166	#88322	—	2	1	no	—
167	#88322	—	2	1	no	—
168	#88322	—	4	2	no	—
169	#88322	#131	1	2	yes	—
170	#88322	C₂×Q₁₆	1	2	yes	—
171	#17958	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₅₃	—	2	1	no	—
174	C₂×D₅₃	—	2	1	no	—
175	C₂×D₅₃	C₄×Q₁₆	1	2	yes	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	4	2	no	—
178	C₄×C₅₃	—	4	2	no	—
179	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
180	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
181	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
182	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
183	Dic₅₃	—	2	1	no	—
184	Dic₅₃	—	4	2	no	—
185	Dic₅₃	—	4	2	no	—
186	Dic₅₃	C₄×Q₁₆	1	2	yes	—
187	Dic₅₃	#443	1	2	yes	—
188	#3170	—	53	1	no	—
189	#2960	—	53	2	no	—
190	#1591	—	53	2	no	—
191	#1591	—	53	1	no	—
192	#1591	D₅₃	1	1	yes	characteristic, complemented
193	C₈×Q₁₆	—	53	1	no	—
194	C₈×Q₁₆	—	53	1	no	—
195	C₈×Q₁₆	—	53	2	no	—
196	#1567	—	53	1	no	—
197	#1493	—	53	2	no	—
198	C₂×C₈²	—	53	1	no	—
199	C₂×C₅₃	#2931	1	1	yes	characteristic
200	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
202	D₅₃	—	2	2	no	—
203	#533	—	53	2	no	—
204	#500	—	53	1	no	—
205	#494	—	53	1	no	—
206	#487	—	53	2	no	—
207	#467	—	53	1	no	—
208	#467	—	53	1	no	—
209	#467	—	53	2	no	—
210	#446	—	53	2	no	—
211	#446	—	106	2	no	—
212	#446	C₂×D₅₃	1	2	yes	—
213	C₈×Q₈	—	53	2	no	—
214	C₈×Q₈	—	53	2	no	—
215	C₈×Q₈	—	106	2	no	—
216	C₄×Q₁₆	—	53	2	no	—
217	C₄×Q₁₆	—	53	1	no	—
218	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
219	#424	—	106	2	no	—
220	#423	—	53	2	no	—
221	#405	—	106	2	no	—
222	C₂×C₄×C₈	—	53	1	no	—
223	C₂×C₄×C₈	—	53	1	no	—
224	C₂×C₄×C₈	—	53	1	no	—
225	#335	—	53	2	no	—
226	#335	—	53	1	no	—
227	#335	C₂×D₅₃	1	1	yes	characteristic
228	Q₈⋊C₈	—	53	2	no	—
229	Q₈⋊C₈	—	53	2	no	—
230	Q₈⋊C₈	—	53	2	no	—
231	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
232	C₈²	—	53	1	no	—
233	C₈²	—	53	1	no	—
234	C₈²	—	53	1	no	—
235	C₈²	C₂×D₅₃	1	1	yes	characteristic
236	C₅₃	#17958	1	1	yes	characteristic, complemented, Sylow
237	C₂×Q₁₆	—	53	2	no	—
238	C₂×Q₁₆	—	53	1	no	—
239	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
240	C₂×M₁₆	—	106	2	no	—
241	C₂²×C₈	—	53	1	no	—
242	C₂²×C₈	—	53	1	no	—
243	C₂²×C₈	—	53	1	no	—
244	C₂²×C₈	—	53	1	no	—
245	C₂²×C₈	—	106	1	no	—
246	C₄⋊Q₈	—	53	2	no	—
247	C₄⋊Q₈	—	53	2	no	—
248	#125	—	106	2	no	—
249	C₂²⋊Q₈	—	53	2	no	—
250	C₂²⋊Q₈	—	53	2	no	—
251	C₂²⋊Q₈	—	106	2	no	—
252	C₄×Q₈	—	53	2	no	—
253	C₄×Q₈	—	106	2	no	—
254	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
255	#117	—	106	2	no	—
256	C₂×C₄²	—	53	1	no	—
257	#107	—	53	1	no	—
258	#107	—	53	2	no	—
259	#107	C₄×D₅₃	1	1	yes	characteristic
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	53	2	no	—
262	C₄⋊C₈	—	106	2	no	—
263	C₄⋊C₈	—	106	2	no	—
264	C₄⋊C₈	—	106	2	no	—
265	C₄⋊C₈	—	106	2	no	—
266	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
267	#103	—	53	2	no	—
268	#103	—	53	2	no	—
269	#103	—	53	2	no	—
270	#103	C₄×D₅₃	1	2	yes	—
271	#97	—	2	2	no	—
272	#97	—	106	2	no	—
273	C₄×C₈	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	—	53	1	no	—
278	C₄×C₈	—	53	1	no	—
279	C₄×C₈	—	53	1	no	—
280	C₄×C₈	—	106	2	no	—
281	C₄×C₈	—	106	1	no	—
282	C₄×C₈	—	106	2	no	—
283	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
286	C₂×Q₈	—	53	2	no	—
287	C₂×Q₈	—	106	2	no	—
288	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
289	C₂²×C₄	—	53	1	no	—
290	C₂²×C₄	—	53	1	no	—
291	C₂²×C₄	—	53	1	no	—
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	106	4	no	—
294	Q₁₆	—	106	2	no	—
295	M₁₆	—	212	4	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	53	1	no	—
299	C₂×C₈	—	53	1	no	—
300	C₂×C₈	—	53	4	no	—
301	C₂×C₈	—	53	1	no	—
302	C₂×C₈	—	53	1	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	—	106	4	no	—
306	C₂×C₈	—	106	1	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
312	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	53	2	no	—
315	C₄⋊C₄	—	106	2	no	—
316	C₄⋊C₄	—	106	2	no	—
317	C₄⋊C₄	—	106	2	no	—
318	C₄⋊C₄	—	106	2	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₄⋊C₄	—	106	2	no	—
321	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
322	C₂²⋊C₄	—	106	2	no	—
323	C₂²⋊C₄	—	106	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	53	1	no	—
326	C₄²	—	53	1	no	—
327	C₄²	—	53	1	no	—
328	C₄²	—	106	2	no	—
329	C₄²	C₂³×D₅₃	1	1	yes	characteristic
330	C₂³	—	53	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	106	2	no	—
334	Q₈	—	106	2	no	—
335	Q₈	—	212	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	53	1	no	—
339	C₂×C₄	—	53	1	no	—
340	C₂×C₄	—	53	2	no	—
341	C₂×C₄	—	53	2	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	106	2	no	—
344	C₂×C₄	—	106	1	no	—
345	C₂×C₄	—	106	1	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	—	106	2	no	—
348	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
350	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	4	2	no	—
355	C₈	—	53	2	no	—
356	C₈	—	53	2	no	—
357	C₈	—	106	2	no	—
358	C₈	—	106	1	no	—
359	C₈	—	106	1	no	—
360	C₈	—	212	2	no	—
361	C₈	#13133225	1	2	yes	—
362	C₈	Q₁₆×D₅₃	1	2	yes	—
363	C₂²	—	53	2	no	—
364	C₂²	—	106	1	no	—
365	C₂²	—	106	1	no	—
366	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	53	2	no	—
371	C₄	—	53	2	no	—
372	C₄	—	106	1	no	—
373	C₄	—	212	2	no	—
374	C₄	—	212	2	no	—
375	C₄	#131×D₅₃	1	1	yes	characteristic
376	C₄	C₂×#13133225	1	1	yes	characteristic
377	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
378	C₄	#135×D₅₃	1	1	yes	characteristic
379	C₂	—	106	2	no	—
380	C₂	#20283424	1	1	yes	characteristic
381	C₂	C₄×Q₁₆×D₅₃	1	1	yes	characteristic
382	C₂	#443×D₅₃	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×#1591	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1591	D₅₃
hypocenter	C₅₃	#17958
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #20273613	C₂	semidirect
4. #20273610	C₂	semidirect
5. Q₁₆×#88322	C₂	semidirect
6. #20273521	C₂	semidirect
7. #20273519	C₂	semidirect
11. C₅₃×#1591	C₂	semidirect
137. C₈×D₅₃	Q₁₆	semidirect
192. #1591	D₅₃	semidirect
236. C₅₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#20285298	2
#20283646	2
#20273613	2
#20273610	2
Q₁₆×#88322	2
#20273521	2
#20273519	2
#20273472	2
#20273168	2
C₈²×D₅₃	2
C₅₃×#1591	2
#17958	53