Subgroups — #27537665

Order 13568.

552 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
2	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276542	C₂	1	2	yes	maximal, complemented
4	#20276539	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276534	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276517	C₂	1	2	yes	maximal, complemented
8	#20276517	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133156	C₂	1	1	yes	maximal, characteristic, complemented
10	#346×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11719949	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	1	yes	characteristic
14	C₂×#13133156	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	Q₈×#88322	C₄	1	1	yes	characteristic
18	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
19	C₈×D₂₁₂	C₂²	1	2	yes	—
20	#16810688	C₄	1	2	yes	—
21	#16810688	C₄	1	1	yes	characteristic
22	#16810675	C₄	1	1	yes	characteristic
23	#16810675	C₄	1	2	yes	—
24	C₄×C₈×D₅₃	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊#11719908	C₂²	1	1	yes	characteristic
27	C₄⋊#11719908	C₂²	1	2	yes	—
28	#16810402	C₂²	1	2	yes	—
29	#16810402	C₂²	1	2	yes	—
30	#16810402	C₂²	1	2	yes	—
31	#16810399	C₂²	1	2	yes	—
32	#16810398	C₂²	1	2	yes	—
33	#16810398	C₂²	1	1	yes	characteristic
34	#16810398	C₂²	1	1	yes	characteristic
35	C₄×#11719908	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₅₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719949	C₈	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
46	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
47	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
48	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
49	#13133179	C₈	1	2	yes	—
50	#13133179	C₈	1	1	yes	characteristic
51	#13133175	C₈	1	2	yes	—
52	#13133175	C₈	1	1	yes	characteristic
53	C₄²×D₅₃	C₂×C₄	1	2	yes	—
54	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
55	#13133156	—	2	2	no	—
56	#13133156	—	2	2	no	—
57	C₂×#11719908	C₂³	1	2	yes	—
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₁₆×D₅₃	—	2	4	no	—
61	C₁₆×D₅₃	—	2	4	no	—
62	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#88322	C₂×C₄	1	2	yes	—
64	#13133129	C₂×C₄	1	2	yes	—
65	#13133129	C₂×C₄	1	2	yes	—
66	#13133129	C₂×C₄	1	2	yes	—
67	C₄×#88322	C₂×C₄	1	2	yes	—
68	C₄×#88322	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
70	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	2	yes	—
80	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
81	#11719949	—	2	4	no	—
82	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
83	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
94	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	#11719921	C₂×C₈	1	2	yes	—
98	#11719921	C₂×C₈	1	2	yes	—
99	#11719921	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
103	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
105	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×#88322	C₂²×C₄	1	2	yes	—
108	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
109	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	—	2	2	no	—
112	C₈×D₅₃	—	2	1	no	—
113	C₈×D₅₃	—	2	1	no	—
114	C₈×D₅₃	—	2	4	no	—
115	C₈×D₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	2	no	—
117	C₁₆×C₅₃	—	2	1	no	—
118	C₁₆×C₅₃	#41	1	2	yes	complemented
119	#11719908	—	2	2	no	—
120	#11719908	—	2	1	no	—
121	#11719908	#41	1	2	yes	complemented
122	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
124	Q₈×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
134	D₂₁₂	—	2	4	no	—
135	D₂₁₂	M₃₂	1	4	yes	complemented
136	C₄×D₅₃	—	2	4	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	2	1	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	1	no	—
141	C₄×D₅₃	—	2	2	no	—
142	C₄×D₅₃	—	4	2	no	—
143	C₄×D₅₃	—	4	1	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
148	#88322	—	2	2	no	—
149	#88322	—	2	1	no	—
150	#88322	#131	1	2	yes	—
151	#33484	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	2	1	no	—
156	C₂×D₅₃	—	2	1	no	—
157	C₂×D₅₃	—	2	1	no	—
158	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
163	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
165	Dic₅₃	—	2	1	no	—
166	Dic₅₃	—	2	1	no	—
167	Dic₅₃	—	2	2	no	—
168	Dic₅₃	—	2	1	no	—
169	C₈×#41	—	53	1	no	—
170	#2183	—	53	1	no	—
171	#2183	D₅₃	1	1	yes	characteristic, complemented
172	#2168	—	53	2	no	—
173	#2168	—	53	2	no	—
174	D₄×C₁₆	—	53	1	no	—
175	D₄×C₁₆	—	53	1	no	—
176	#2163	—	53	1	no	—
177	#2163	—	53	2	no	—
178	C₄×M₃₂	—	53	1	no	—
179	C₂×#346	—	53	2	no	—
180	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
181	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
182	C₂×C₅₃	#3407	1	1	yes	characteristic
183	D₅₃	—	2	4	no	—
184	D₅₃	—	4	1	no	—
185	C₄×#41	—	53	1	no	—
186	C₂×M₃₂	—	53	1	no	—
187	C₂×M₃₂	—	53	1	no	—
188	C₂²×C₁₆	—	53	2	no	—
189	C₂²×C₁₆	—	53	2	no	—
190	C₈×Q₈	—	53	1	no	—
191	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
192	C₈×D₄	—	53	2	no	—
193	C₈×D₄	—	53	1	no	—
194	C₈×D₄	—	53	1	no	—
195	C₈×D₄	—	53	2	no	—
196	#431	—	53	1	no	—
197	#431	—	53	2	no	—
198	C₂×C₄×C₈	—	53	2	no	—
199	C₂×C₄×C₈	—	53	1	no	—
200	C₄⋊C₁₆	—	53	1	no	—
201	C₄⋊C₁₆	—	53	2	no	—
202	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
204	C₂²⋊C₁₆	—	53	2	no	—
205	C₂²⋊C₁₆	—	53	2	no	—
206	C₂²⋊C₁₆	—	53	2	no	—
207	#346	—	53	1	no	—
208	#346	—	53	2	no	—
209	#346	—	53	1	no	—
210	#346	—	53	2	no	—
211	#346	C₂×D₅₃	1	2	yes	—
212	C₄×C₁₆	—	53	1	no	—
213	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
214	C₅₃	#33484	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	53	1	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₂²×C₈	—	53	1	no	—
218	C₂²×C₈	—	53	2	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₄×Q₈	—	53	1	no	—
221	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
222	C₄×D₄	—	53	1	no	—
223	C₄×D₄	—	53	2	no	—
224	C₄×D₄	—	53	1	no	—
225	C₄×D₄	—	53	2	no	—
226	#117	—	53	2	no	—
227	#117	—	53	1	no	—
228	C₂×C₄²	—	53	2	no	—
229	C₂×C₄²	—	53	1	no	—
230	M₃₂	—	106	2	no	—
231	M₃₂	—	106	2	no	—
232	C₂×C₁₆	—	53	1	no	—
233	C₂×C₁₆	—	53	1	no	—
234	C₂×C₁₆	—	53	2	no	—
235	C₂×C₁₆	—	106	4	no	—
236	C₂×C₁₆	—	106	4	no	—
237	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
239	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
240	C₄⋊C₈	—	53	1	no	—
241	C₄⋊C₈	—	53	2	no	—
242	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
243	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
244	C₂²⋊C₈	—	53	2	no	—
245	C₂²⋊C₈	—	53	2	no	—
246	C₂²⋊C₈	—	53	2	no	—
247	C₄×C₈	—	53	2	no	—
248	C₄×C₈	—	53	1	no	—
249	C₄×C₈	—	53	1	no	—
250	C₄×C₈	—	53	2	no	—
251	C₄×C₈	—	53	1	no	—
252	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₅₃	1	2	yes	—
254	#41	—	106	4	no	—
255	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
256	C₂×D₄	—	53	1	no	—
257	C₂×D₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂×C₈	—	53	1	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	1	no	—
267	C₂×C₈	—	106	4	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	—	106	2	no	—
270	C₂×C₈	—	106	1	no	—
271	C₂×C₈	—	106	2	no	—
272	C₂×C₈	—	106	1	no	—
273	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
275	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
276	C₄⋊C₄	—	53	2	no	—
277	C₄⋊C₄	—	53	1	no	—
278	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
279	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
280	C₂²⋊C₄	—	53	2	no	—
281	C₂²⋊C₄	—	53	2	no	—
282	C₂²⋊C₄	—	53	2	no	—
283	C₄²	—	53	2	no	—
284	C₄²	—	53	1	no	—
285	C₄²	—	53	2	no	—
286	C₄²	—	53	1	no	—
287	C₄²	—	53	1	no	—
288	C₄²	C₂×C₄×D₅₃	1	2	yes	—
289	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	53	2	no	—
293	C₁₆	—	106	2	no	—
294	C₁₆	—	106	1	no	—
295	C₁₆	#11719949	1	2	yes	complemented
296	C₂³	—	53	2	no	—
297	C₂³	—	53	1	no	—
298	Q₈	—	2	2	no	—
299	D₄	—	53	4	no	—
300	D₄	—	106	4	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	53	1	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	106	4	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	1	no	—
309	C₂×C₄	—	106	2	no	—
310	C₂×C₄	—	106	1	no	—
311	C₂×C₄	—	212	2	no	—
312	C₂×C₄	—	212	1	no	—
313	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
316	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	53	2	no	—
321	C₈	—	106	2	no	—
322	C₈	—	106	1	no	—
323	C₈	#41×D₅₃	1	1	yes	characteristic
324	C₈	C₂×#11719949	1	1	yes	characteristic
325	C₂²	—	53	4	no	—
326	C₂²	—	106	2	no	—
327	C₂²	—	106	2	no	—
328	C₂²	—	106	1	no	—
329	C₂²	—	106	1	no	—
330	C₂²	—	106	1	no	—
331	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	106	2	no	—
336	C₄	—	106	1	no	—
337	C₄	—	106	1	no	—
338	C₄	—	106	1	no	—
339	C₄	M₃₂×D₅₃	1	2	yes	—
340	C₄	C₄×#11719949	1	1	yes	characteristic
341	C₄	#131×D₅₃	1	1	yes	characteristic
342	C₂	—	106	4	no	—
343	C₂	—	212	1	no	—
344	C₂	C₈×#11719949	1	1	yes	characteristic
345	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
346	C₂	#20287816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2183	C₂
prime radical	C₂×#11719949	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2183	D₅₃
hypocenter	C₅₃	#33484
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #20276624	C₂	semidirect
3. #20276542	C₂	semidirect
4. #20276539	C₂	semidirect
5. C₁₆×D₂₁₂	C₂	semidirect
6. #20276534	C₂	semidirect
7. #20276517	C₂	semidirect
8. #20276517	C₂	semidirect
9. C₄×#13133156	C₂	semidirect
10. #346×D₅₃	C₂	semidirect
11. C₅₃×#2183	C₂	semidirect
118. C₁₆×C₅₃	#41	semidirect
121. #11719908	#41	semidirect
135. D₂₁₂	M₃₂	semidirect
171. #2183	D₅₃	semidirect
214. C₅₃	#33484	semidirect
295. C₁₆	#11719949	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719949	2
#20276624	2
#20276542	2
#20276539	2
C₁₆×D₂₁₂	2
#20276534	2
#20276517	2
C₄×#13133156	2
#346×D₅₃	2
C₅₃×#2183	2
#33484	53