Subgroups — #27283831

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810997	C₂	1	1	yes	maximal, characteristic
2	#20275677	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275521	C₂	1	1	yes	maximal, characteristic
4	#20275520	C₂	1	1	yes	maximal, characteristic
5	#20274871	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719924	—	2	2	no	—
9	C₂×#13133214	C₂²	1	1	yes	characteristic
10	#16810997	—	2	2	no	—
11	#16810997	C₄	1	4	yes	complemented
12	C₂×#13133148	—	2	2	no	—
13	C₂×#13133139	—	2	2	no	—
14	C₂×#13133137	—	2	2	no	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810493	C₂²	1	1	yes	characteristic
18	#16810385	—	2	2	no	—
19	C₈×#88322	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#384	C₂²	1	1	yes	characteristic
23	C₅₃×#340	—	2	2	no	—
24	C₂²×D₂₁₂	—	2	1	no	—
25	C₂²×D₂₁₂	—	2	1	no	—
26	C₂²×D₂₁₂	—	2	1	no	—
27	C₂×#11719924	—	2	8	no	—
28	C₂×#11719924	—	4	4	no	—
29	C₂×#11719924	—	4	4	no	—
30	C₂²×#88322	D₄	1	2	yes	—
31	C₂²×#88322	D₄	1	2	yes	—
32	#13133214	—	2	1	no	—
33	#13133214	—	2	1	no	—
34	#13133214	—	2	1	no	—
35	#13133214	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₅₃	—	2	1	no	—
37	C₂×C₄⋊Dic₅₃	—	2	1	no	—
38	C₂×C₄⋊Dic₅₃	—	2	1	no	—
39	#13133148	—	4	4	no	—
40	#13133139	—	4	4	no	—
41	#13133137	—	4	4	no	—
42	C₄×#88322	—	2	1	no	—
43	C₄×#88322	—	2	1	no	—
44	C₄×#88322	C₂×C₄	1	2	yes	—
45	C₄×#88322	D₄	1	2	yes	—
46	C₂²×Q₈×C₅₃	—	2	1	no	—
47	C₂²×C₈×C₅₃	D₄	1	2	yes	—
48	C₄⋊Q₈×C₅₃	—	2	1	no	—
49	C₄⋊Q₈×C₅₃	—	2	1	no	—
50	C₄⋊Q₈×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₂³×D₅₃	—	4	1	no	—
61	C₂×Q₈×C₅₃	—	2	2	no	—
62	C₂×Q₈×C₅₃	—	4	1	no	—
63	C₂×Q₈×C₅₃	—	4	2	no	—
64	C₂×Q₈×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	—	4	1	no	—
66	C₂²×C₄×C₅₃	—	4	1	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	4	1	no	—
71	C₂×D₂₁₂	—	2	2	no	—
72	C₂×D₂₁₂	—	2	2	no	—
73	C₂×D₂₁₂	—	2	2	no	—
74	C₂×D₂₁₂	—	4	2	no	—
75	C₂×D₂₁₂	—	4	2	no	—
76	C₂×D₂₁₂	—	4	2	no	—
77	C₂×D₂₁₂	—	4	2	no	—
78	C₂×D₂₁₂	—	4	2	no	—
79	C₂×D₂₁₂	—	4	2	no	—
80	C₂×D₂₁₂	—	4	1	no	—
81	C₂×D₂₁₂	—	4	1	no	—
82	C₂×D₂₁₂	—	4	1	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	4	1	no	—
89	C₄⋊C₄×C₅₃	—	4	1	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₃	#41	1	1	yes	characteristic
96	C₄²×C₅₃	#41	1	1	yes	characteristic
97	#11719924	—	4	16	no	—
98	C₄⋊Dic₅₃	—	4	2	no	—
99	C₄⋊Dic₅₃	—	4	2	no	—
100	C₄⋊Dic₅₃	—	4	2	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	C₂²⋊C₄	1	4	yes	—
104	C₂×#88322	D₈	1	4	yes	—
105	C₂×#88322	SD₁₆	1	4	yes	—
106	C₂×#88322	SD₁₆	1	4	yes	complemented
107	C₂³×C₅₃	#127	1	1	yes	characteristic
108	C₂²×D₅₃	—	4	2	no	—
109	C₂²×D₅₃	—	8	1	no	—
110	C₂²×D₅₃	—	8	1	no	—
111	C₂²×D₅₃	—	8	1	no	—
112	C₂²×D₅₃	—	8	1	no	—
113	C₂²×D₅₃	—	8	1	no	—
114	C₂²×D₅₃	—	8	1	no	—
115	Q₈×C₅₃	—	4	4	no	—
116	Q₈×C₅₃	—	8	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	8	1	no	—
119	C₂×C₄×C₅₃	—	8	1	no	—
120	C₂×C₄×C₅₃	—	8	1	no	—
121	C₂×C₄×C₅₃	—	8	1	no	—
122	C₂×C₄×C₅₃	—	8	1	no	—
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₅₃	#135	1	2	yes	—
134	C₂×C₄×C₅₃	#135	1	2	yes	—
135	C₂×Dic₅₃	—	8	1	no	—
136	C₂×Dic₅₃	—	8	1	no	—
137	C₂×Dic₅₃	—	8	1	no	—
138	D₂₁₂	—	4	4	no	—
139	D₂₁₂	—	4	4	no	—
140	D₂₁₂	—	4	4	no	—
141	D₂₁₂	—	8	2	no	—
142	D₂₁₂	—	8	2	no	—
143	D₂₁₂	—	8	2	no	—
144	C₈×C₅₃	—	2	4	no	—
145	#88322	—	2	4	no	—
146	#88322	#102	1	8	yes	complemented
147	#9820	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	#390	1	1	yes	characteristic
149	C₂²×C₅₃	#492	1	1	yes	characteristic
150	C₂²×C₅₃	#492	1	1	yes	characteristic
151	C₂²×C₅₃	#492	1	1	yes	characteristic
152	C₂²×C₅₃	#493	1	1	yes	characteristic
153	C₂²×C₅₃	#495	1	1	yes	characteristic
154	C₂²×C₅₃	#495	1	1	yes	characteristic
155	C₂×D₅₃	—	8	2	no	—
156	C₂×D₅₃	—	8	2	no	—
157	C₂×D₅₃	—	8	2	no	—
158	C₂×D₅₃	—	16	1	no	—
159	C₂×D₅₃	—	16	1	no	—
160	C₂×D₅₃	—	16	1	no	—
161	C₂×D₅₃	—	16	1	no	—
162	C₄×C₅₃	—	8	2	no	—
163	C₄×C₅₃	—	16	1	no	—
164	C₄×C₅₃	C₂×#102	1	2	yes	—
165	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
166	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
167	C₄×C₅₃	#465	1	2	yes	—
168	C₄×C₅₃	#465	1	2	yes	—
169	C₄×C₅₃	#486	1	2	yes	—
170	Dic₅₃	—	16	1	no	—
171	C₂×#492	—	53	1	no	—
172	#1990	—	53	1	no	—
173	#1990	D₅₃	1	1	yes	characteristic, complemented
174	#1957	—	53	1	no	—
175	#1957	—	53	1	no	—
176	#1844	—	53	1	no	—
177	C₂×C₈²	—	53	1	no	—
178	C₂×C₅₃	#1705	1	1	yes	characteristic
179	C₂×C₅₃	#1706	1	1	yes	characteristic
180	C₂×C₅₃	#1706	1	1	yes	characteristic
181	C₂×C₅₃	#1708	1	1	yes	characteristic
182	C₂×C₅₃	#1957	1	1	yes	characteristic
183	C₂×C₅₃	#1968	1	1	yes	characteristic
184	C₂×C₅₃	#1968	1	1	yes	characteristic
185	D₅₃	—	16	2	no	—
186	C₂²×SD₁₆	—	106	2	no	—
187	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
188	C₂×#127	—	53	1	no	—
189	#492	—	53	4	no	—
190	#492	—	106	2	no	—
191	C₂×#106	—	106	2	no	—
192	C₂×#102	—	106	2	no	—
193	C₂×#102	—	106	2	no	—
194	C₂×C₄×C₈	—	53	1	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
197	#384	—	53	1	no	—
198	#384	C₂×D₅₃	1	1	yes	characteristic
199	#340	—	2	2	no	—
200	#340	—	106	2	no	—
201	C₈²	—	106	2	no	—
202	C₅₃	#9820	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	106	1	no	—
205	C₂²×D₄	—	106	1	no	—
206	C₂²×D₄	—	106	1	no	—
207	C₂×SD₁₆	—	106	8	no	—
208	C₂×SD₁₆	—	212	4	no	—
209	C₂×SD₁₆	—	212	4	no	—
210	C₂²×C₈	—	53	2	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	C₄×D₅₃	1	2	yes	—
217	#127	—	53	2	no	—
218	#127	—	106	1	no	—
219	#127	—	106	1	no	—
220	#127	—	106	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	106	1	no	—
227	C₂×C₄⋊C₄	—	106	1	no	—
228	C₂×C₄⋊C₄	—	106	1	no	—
229	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
230	#106	—	212	4	no	—
231	#102	—	212	4	no	—
232	#102	—	212	4	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	53	2	no	—
236	C₄×C₈	—	53	2	no	—
237	C₄×C₈	—	106	1	no	—
238	C₄×C₈	—	106	1	no	—
239	C₂⁴	—	212	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×D₄	—	106	2	no	—
245	C₂×D₄	—	106	2	no	—
246	C₂×D₄	—	106	2	no	—
247	C₂×D₄	—	212	1	no	—
248	C₂×D₄	—	212	1	no	—
249	C₂×D₄	—	212	1	no	—
250	C₂×D₄	—	212	2	no	—
251	C₂×D₄	—	212	2	no	—
252	C₂×D₄	—	212	2	no	—
253	C₂×D₄	—	212	2	no	—
254	C₂×D₄	—	212	2	no	—
255	C₂×D₄	—	212	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	212	1	no	—
259	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
262	SD₁₆	—	212	16	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	53	4	no	—
266	C₂×C₈	—	53	4	no	—
267	C₂×C₈	—	53	4	no	—
268	C₂×C₈	—	53	4	no	—
269	C₂×C₈	—	106	2	no	—
270	C₂×C₈	—	106	2	no	—
271	C₂×C₈	#11719924	1	4	yes	complemented
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	212	2	no	—
280	C₄⋊C₄	—	212	2	no	—
281	C₄⋊C₄	—	212	2	no	—
282	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
283	C₄²	#11719945	1	1	yes	characteristic
284	C₄²	D₄×D₅₃	1	1	yes	characteristic
285	C₄²	#11719947	1	1	yes	characteristic
286	C₂³	—	424	1	no	—
287	C₂³	—	424	1	no	—
288	C₂³	—	424	1	no	—
289	C₂³	—	424	1	no	—
290	C₂³	—	424	1	no	—
291	C₂³	—	424	1	no	—
292	C₂³	—	212	2	no	—
293	C₂³	#13133213	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	424	2	no	—
297	D₄	—	424	2	no	—
298	D₄	—	424	2	no	—
299	D₄	—	212	4	no	—
300	D₄	—	212	4	no	—
301	D₄	—	212	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	424	1	no	—
309	C₂×C₄	—	424	1	no	—
310	C₂×C₄	—	424	1	no	—
311	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₄	#13133177	1	1	yes	characteristic
313	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
314	C₂×C₄	#13133189	1	1	yes	characteristic
315	C₂×C₄	#13133190	1	1	yes	characteristic
316	C₂×C₄	#13133203	1	1	yes	characteristic
317	C₂×C₄	D₈×D₅₃	1	2	yes	—
318	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
319	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
320	C₂×C₄	C₂×#11719924	1	2	yes	—
321	C₂×C₄	#13133237	1	2	yes	—
322	C₂×C₄	#13133240	1	2	yes	—
323	C₈	—	2	4	no	—
324	C₈	—	53	8	no	—
325	C₈	—	106	4	no	—
326	C₂²	—	848	1	no	—
327	C₂²	—	848	1	no	—
328	C₂²	—	848	1	no	—
329	C₂²	—	848	1	no	—
330	C₂²	—	424	2	no	—
331	C₂²	—	424	2	no	—
332	C₂²	—	424	2	no	—
333	C₂²	#16810523	1	1	yes	characteristic
334	C₂²	#16810994	1	1	yes	characteristic
335	C₂²	#16810994	1	1	yes	characteristic
336	C₂²	#16810997	1	1	yes	characteristic
337	C₂²	#16810998	1	1	yes	characteristic
338	C₂²	#16811005	1	1	yes	characteristic
339	C₂²	#16811008	1	1	yes	characteristic
340	C₄	—	8	2	no	—
341	C₄	—	16	1	no	—
342	C₄	—	848	1	no	—
343	C₄	#102×D₅₃	1	2	yes	—
344	C₄	#16810708	1	2	yes	—
345	C₄	C₄×#11719924	1	2	yes	—
346	C₄	#16810854	1	2	yes	—
347	C₄	#16810855	1	2	yes	—
348	C₄	#16810968	1	2	yes	—
349	C₂	—	848	2	no	—
350	C₂	#20274303	1	1	yes	characteristic
351	C₂	#20274306	1	1	yes	characteristic
352	C₂	#20274308	1	1	yes	characteristic
353	C₂	#20274318	1	1	yes	characteristic
354	C₂	#20275519	1	1	yes	characteristic
355	C₂	#20275566	1	1	yes	characteristic
356	C₂	#20275572	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1990	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#13133214	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1990	D₅₃
hypocenter	C₅₃	#9820
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. #20275677	C₂	semidirect
5. #20274871	C₂	semidirect
6. C₂×C₈×#88322	C₂	semidirect
7. C₅₃×#1990	C₂	semidirect
11. #16810997	C₄	semidirect
85. C₂×C₈×C₅₃	SD₁₆	semidirect
106. C₂×#88322	SD₁₆	semidirect
146. #88322	#102	semidirect
173. #1990	D₅₃	semidirect
202. C₅₃	#9820	semidirect
271. C₂×C₈	#11719924	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810997	2
#20275677	2
#20275521	2
#20275520	2
#20274871	2
C₂×C₈×#88322	2
C₅₃×#1990	2
#9820	53