Subgroups — #27356139

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283875	C₂	1	2	yes	maximal, complemented
2	#20273618	C₂	1	2	yes	maximal, complemented
3	#20273566	C₂	1	2	yes	maximal, complemented
4	#20273545	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊#88322	C₂	1	2	yes	maximal, complemented
6	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273128	C₂	1	2	yes	maximal, complemented
8	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3168	C₂	1	1	yes	maximal, characteristic
10	#16810754	C₂²	1	2	yes	—
11	Q₈×#88322	C₂²	1	2	yes	—
12	#16810701	C₂²	1	2	yes	—
13	#16810695	—	2	2	no	—
14	D₄×#88322	—	2	2	no	—
15	D₄×#88322	C₂²	1	2	yes	—
16	#16810684	—	2	2	no	—
17	#16810676	C₂²	1	1	yes	characteristic
18	#16810650	—	2	2	no	—
19	C₂×C₄⋊#88322	C₂²	1	2	yes	—
20	C₄×#11719917	C₂²	1	1	yes	characteristic
21	#16810376	C₂²	1	1	yes	characteristic
22	#16810376	C₂²	1	2	yes	—
23	#16810376	C₂²	1	1	yes	characteristic
24	Q₈⋊#88322	C₂²	1	2	yes	—
25	Q₈⋊#88322	C₂²	1	2	yes	—
26	D₄⋊#88322	C₂²	1	2	yes	—
27	D₄⋊#88322	C₂²	1	2	yes	—
28	#16810345	C₂²	1	2	yes	—
29	#16810344	C₂²	1	2	yes	—
30	C₅₃×#529	C₂²	1	2	yes	—
31	C₅₃×#499	C₄	1	1	yes	characteristic
32	C₅₃×#495	C₄	1	1	yes	characteristic
33	C₅₃×#488	C₄	1	2	yes	—
34	C₅₃×#467	C₄	1	1	yes	characteristic
35	C₅₃×#466	C₄	1	1	yes	characteristic
36	C₅₃×#465	C₄	1	2	yes	—
37	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
39	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719917	—	2	2	no	—
42	C₂×#11719917	—	2	1	no	—
43	C₂²×#88322	—	2	2	no	—
44	C₂²×#88322	—	2	2	no	—
45	C₄⋊#88322	—	2	2	no	—
46	C₄⋊#88322	—	2	2	no	—
47	C₄⋊#88322	—	2	2	no	—
48	C₄⋊#88322	D₄	1	1	yes	characteristic
49	C₄⋊#88322	D₄	1	1	yes	characteristic
50	C₄⋊#88322	C₂³	1	2	yes	—
51	C₄⋊#88322	C₂³	1	2	yes	—
52	C₂²⋊#88322	—	2	2	no	—
53	C₂²⋊#88322	—	2	2	no	—
54	C₂²⋊#88322	D₄	1	2	yes	—
55	#13133128	—	2	1	no	—
56	#13133128	—	2	2	no	—
57	C₄×#88322	—	2	2	no	—
58	C₄×#88322	C₂³	1	2	yes	—
59	C₄×#88322	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#126×C₅₃	—	2	2	no	—
66	#125×C₅₃	C₂×C₄	1	2	yes	—
67	#124×C₅₃	C₂×C₄	1	2	yes	—
68	#123×C₅₃	—	2	2	no	—
69	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
70	#121×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
72	C₄×D₄×C₅₃	—	2	2	no	—
73	C₄×D₄×C₅₃	C₂³	1	2	yes	—
74	#117×C₅₃	—	2	2	no	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
77	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
78	#106×C₅₃	C₂×C₄	1	2	yes	—
79	#103×C₅₃	C₂×C₄	1	2	yes	—
80	#103×C₅₃	C₂×C₄	1	2	yes	—
81	#102×C₅₃	C₂×C₄	1	2	yes	—
82	#102×C₅₃	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
94	Q₁₆×C₅₃	—	4	1	no	—
95	SD₁₆×C₅₃	—	2	4	no	—
96	D₈×C₅₃	—	4	1	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	4	1	no	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
109	C₂²⋊C₄×C₅₃	—	2	2	no	—
110	C₂²⋊C₄×C₅₃	—	2	2	no	—
111	C₂²⋊C₄×C₅₃	—	2	2	no	—
112	C₂²⋊C₄×C₅₃	—	2	2	no	—
113	C₂²⋊C₄×C₅₃	—	2	2	no	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	#41	1	1	yes	characteristic
118	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
119	#11719917	—	4	2	no	—
120	#11719917	—	8	2	no	—
121	C₂×#88322	—	2	2	no	—
122	C₂×#88322	—	2	2	no	—
123	C₂×#88322	—	2	2	no	—
124	C₂×#88322	—	8	2	no	—
125	C₂×#88322	—	8	2	no	—
126	C₂×#88322	C₂×D₄	1	2	yes	—
127	C₂×#88322	C₂×D₄	1	2	yes	—
128	C₂³×C₅₃	—	2	2	no	—
129	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
130	Q₈×C₅₃	—	2	2	no	—
131	Q₈×C₅₃	—	2	2	no	—
132	D₄×C₅₃	—	2	2	no	—
133	D₄×C₅₃	—	2	2	no	—
134	D₄×C₅₃	—	4	4	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	1	no	—
154	C₈×C₅₃	#131	1	2	yes	—
155	#88322	—	2	2	no	—
156	#88322	—	2	2	no	—
157	#88322	—	4	2	no	—
158	#88322	—	4	2	no	—
159	#88322	—	4	2	no	—
160	#18092	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	4	2	no	—
164	C₂²×C₅₃	—	4	2	no	—
165	C₂²×C₅₃	—	4	2	no	—
166	C₂²×C₅₃	#444	1	2	yes	—
167	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	4	2	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
176	C₄×C₅₃	#568	1	1	yes	characteristic
177	C₄×C₅₃	#576	1	1	yes	characteristic
178	C₄×C₅₃	#578	1	1	yes	characteristic
179	#3168	D₅₃	1	1	yes	characteristic
180	#2994	—	53	2	no	—
181	#1592	—	53	2	no	—
182	#1584	—	53	2	no	—
183	#1581	—	53	2	no	—
184	SD₁₆⋊C₈	—	53	2	no	—
185	C₂×#335	—	53	1	no	—
186	#1483	—	53	2	no	—
187	#1451	—	53	1	no	—
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	—	8	2	no	—
190	C₂×C₅₃	#2935	1	1	yes	characteristic
191	C₂×C₅₃	#2951	1	1	yes	characteristic
192	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
193	#529	C₂×D₅₃	1	2	yes	—
194	#499	Dic₅₃	1	1	yes	characteristic
195	#495	Dic₅₃	1	1	yes	characteristic
196	#488	Dic₅₃	1	2	yes	—
197	#467	Dic₅₃	1	1	yes	characteristic
198	#466	Dic₅₃	1	1	yes	characteristic
199	#465	Dic₅₃	1	2	yes	—
200	#446	—	53	2	no	—
201	C₈×Q₈	—	53	2	no	—
202	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
203	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
204	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
205	#436	—	53	2	no	—
206	#435	—	106	2	no	—
207	C₈×D₄	—	53	2	no	—
208	C₈×D₄	—	106	2	no	—
209	#433	—	106	2	no	—
210	#431	—	53	1	no	—
211	#424	—	106	2	no	—
212	C₂×C₄⋊C₈	—	53	2	no	—
213	C₄×M₁₆	—	53	1	no	—
214	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
215	#335	—	53	1	no	—
216	#335	—	53	2	no	—
217	#335	—	53	1	no	—
218	Q₈⋊C₈	—	53	2	no	—
219	Q₈⋊C₈	—	53	2	no	—
220	D₄⋊C₈	—	53	2	no	—
221	D₄⋊C₈	—	53	2	no	—
222	#322	—	53	2	no	—
223	#322	—	53	2	no	—
224	C₅₃	#18092	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
226	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
227	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
228	C₂×M₁₆	—	106	2	no	—
229	C₂×M₁₆	—	106	1	no	—
230	C₂²×C₈	—	106	2	no	—
231	C₂²×C₈	—	106	2	no	—
232	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
233	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
234	#126	—	2	2	no	—
235	#125	C₂×Dic₅₃	1	2	yes	—
236	#124	C₂×Dic₅₃	1	2	yes	—
237	#123	—	2	2	no	—
238	C₂²⋊Q₈	C₂×Dic₅₃	1	2	yes	—
239	#121	C₂×Dic₅₃	1	2	yes	—
240	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	C₂²×D₅₃	1	2	yes	—
243	#117	—	2	2	no	—
244	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
245	#107	C₂×Dic₅₃	1	1	yes	characteristic
246	#107	C₂×Dic₅₃	1	1	yes	characteristic
247	#106	C₂×Dic₅₃	1	2	yes	—
248	C₄⋊C₈	—	53	1	no	—
249	C₄⋊C₈	—	53	1	no	—
250	C₄⋊C₈	—	53	2	no	—
251	C₄⋊C₈	—	53	2	no	—
252	C₄⋊C₈	—	106	2	no	—
253	C₄⋊C₈	—	106	2	no	—
254	C₄⋊C₈	—	106	2	no	—
255	#103	C₂×Dic₅₃	1	2	yes	—
256	#103	C₂×Dic₅₃	1	2	yes	—
257	#102	C₂×Dic₅₃	1	2	yes	—
258	#102	C₂×Dic₅₃	1	2	yes	—
259	C₂²⋊C₈	—	53	2	no	—
260	C₂²⋊C₈	—	106	2	no	—
261	C₂²⋊C₈	—	106	2	no	—
262	#97	—	106	2	no	—
263	#97	—	106	1	no	—
264	C₄×C₈	—	53	2	no	—
265	C₄×C₈	—	53	2	no	—
266	C₄×C₈	—	106	2	no	—
267	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
268	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
271	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
276	C₂²×C₄	#11719947	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
278	Q₁₆	—	4	1	no	—
279	SD₁₆	—	2	4	no	—
280	D₈	—	4	1	no	—
281	M₁₆	—	424	2	no	—
282	M₁₆	—	212	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	—	53	2	no	—
287	C₂×C₈	—	106	2	no	—
288	C₂×C₈	—	106	2	no	—
289	C₂×C₈	—	106	2	no	—
290	C₂×C₈	—	424	2	no	—
291	C₂×C₈	—	424	2	no	—
292	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
301	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	D₄×D₅₃	1	1	yes	characteristic
309	C₄²	#11719947	1	1	yes	characteristic
310	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
311	C₄²	C₂³×D₅₃	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
330	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	106	2	no	—
339	C₈	—	106	2	no	—
340	C₈	—	212	2	no	—
341	C₈	—	212	2	no	—
342	C₈	—	212	2	no	—
343	C₈	#13133224	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	#16810745	1	2	yes	—
350	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	C₂×#13133224	1	1	yes	characteristic
359	C₄	#16811390	1	1	yes	characteristic
360	C₄	#16811425	1	1	yes	characteristic
361	C₄	#16811437	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	8	2	no	—
364	C₂	#20283459	1	1	yes	characteristic
365	C₂	#20283591	1	1	yes	characteristic
366	C₂	C₂×#16810745	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×#3168	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#466	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3168	D₅₃
hypocenter	C₅₃	#18092
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #20283875	C₂	semidirect
2. #20273618	C₂	semidirect
3. #20273566	C₂	semidirect
4. #20273545	C₂	semidirect
5. SD₁₆⋊#88322	C₂	semidirect
6. C₂×#16810376	C₂	semidirect
7. #20273128	C₂	semidirect
8. #20273004	C₂	semidirect
224. C₅₃	#18092	semidirect

Maximal subgroups

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

subgroup	index
#20283875	2
#20273618	2
#20273566	2
#20273545	2
SD₁₆⋊#88322	2
C₂×#16810376	2
#20273128	2
#20273004	2
C₅₃×#3168	2
#18092	53