Subgroups — #27274886

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133137	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810350	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272556	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	—	2	1	no	—
9	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
10	C₈×D₂₁₂	—	2	2	no	—
11	C₈×D₂₁₂	C₄	1	4	yes	complemented
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×#13133137	C₄	1	2	yes	—
14	C₂×#13133129	—	2	1	no	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810385	C₄	1	2	yes	—
18	#16810350	C₄	1	4	yes	complemented
19	#16810344	—	2	2	no	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₃	—	2	1	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#336	—	2	1	no	—
25	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	—	2	1	no	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅₃	—	2	4	no	—
31	C₂×C₈×D₅₃	—	4	2	no	—
32	C₂×C₈×D₅₃	—	4	2	no	—
33	C₄×D₂₁₂	—	2	1	no	—
34	C₄×D₂₁₂	—	2	1	no	—
35	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
36	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×#11719921	—	2	1	no	—
39	C₄⋊#88322	—	2	1	no	—
40	C₄⋊#88322	C₂×C₄	1	2	yes	—
41	#13133137	—	2	4	no	—
42	#13133129	—	2	4	no	—
43	C₄×#88322	—	2	1	no	—
44	C₄×#88322	—	2	1	no	—
45	C₄×#88322	—	2	1	no	—
46	C₄×#88322	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	—	2	1	no	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	—	2	1	no	—
54	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
55	#97×C₅₃	—	4	2	no	—
56	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₅₃	D₄	1	2	yes	—
58	#95×C₅₃	—	2	1	no	—
59	C₂³×D₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×Dic₅₃	—	2	1	no	—
65	C₂×D₂₁₂	—	2	1	no	—
66	C₂×D₂₁₂	—	2	2	no	—
67	C₂×D₂₁₂	C₄²	1	2	yes	—
68	C₂×D₂₁₂	M₁₆	1	4	yes	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	—	2	2	no	—
71	C₂×C₄×D₅₃	—	4	1	no	—
72	C₂×C₄×D₅₃	—	4	1	no	—
73	C₂×C₄×D₅₃	—	4	1	no	—
74	C₂×C₄×D₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	D₈	1	2	yes	—
83	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	#11719921	—	2	2	no	—
93	#11719921	—	2	2	no	—
94	C₄⋊Dic₅₃	—	2	1	no	—
95	C₄⋊Dic₅₃	C₄²	1	2	yes	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	—	4	2	no	—
102	C₂×#88322	—	4	1	no	—
103	C₂×#88322	—	4	1	no	—
104	C₈×D₅₃	—	4	8	no	—
105	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₅₃	—	2	2	no	—
107	C₂²×D₅₃	—	2	4	no	—
108	C₂²×D₅₃	—	4	1	no	—
109	C₂²×D₅₃	—	4	1	no	—
110	C₂²×D₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₅₃	#131	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	4	1	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	D₂₁₂	—	2	2	no	—
131	D₂₁₂	—	2	4	no	—
132	D₂₁₂	#97	1	4	yes	complemented
133	C₄×D₅₃	—	4	2	no	—
134	C₄×D₅₃	—	4	2	no	—
135	C₄×D₅₃	—	4	2	no	—
136	C₄×D₅₃	—	4	2	no	—
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	—	4	2	no	—
139	C₈×C₅₃	#102	1	4	yes	complemented
140	#88322	—	4	2	no	—
141	#88322	—	4	2	no	—
142	#88322	—	4	2	no	—
143	#8202	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₅₃	#435	1	2	yes	—
146	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
147	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
148	C₂²×C₅₃	#444	1	2	yes	—
149	C₂×D₅₃	—	2	4	no	—
150	C₂×D₅₃	—	4	4	no	—
151	C₂×D₅₃	—	4	4	no	—
152	C₂×D₅₃	—	4	2	no	—
153	C₂×D₅₃	—	4	2	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
158	C₄×C₅₃	#405	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
160	C₄×C₅₃	#416	1	1	yes	characteristic
161	C₄×C₅₃	#435	1	2	yes	—
162	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
163	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
164	Dic₅₃	—	4	2	no	—
165	C₂×C₈×D₄	—	53	1	no	—
166	#1770	D₅₃	1	1	yes	characteristic, complemented
167	C₄×#102	—	53	1	no	—
168	C₂×D₄⋊C₈	—	53	1	no	—
169	C₂×#322	—	53	1	no	—
170	#1279	—	53	1	no	—
171	#1277	—	53	1	no	—
172	C₂×C₅₃	#1581	1	2	yes	—
173	C₂×C₅₃	#1582	1	2	yes	—
174	C₂×C₅₃	#1752	1	1	yes	characteristic
175	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
176	C₂×C₅₃	#1765	1	1	yes	characteristic
177	D₅₃	—	4	4	no	—
178	C₂³×C₈	—	106	1	no	—
179	C₂×C₄×D₄	—	53	1	no	—
180	C₈×D₄	—	53	4	no	—
181	C₈×D₄	—	106	2	no	—
182	C₂×C₄⋊C₈	—	53	1	no	—
183	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
184	C₂×#102	—	53	2	no	—
185	C₂×C₂²⋊C₈	—	106	1	no	—
186	C₂×#97	—	2	1	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
191	#340	—	53	2	no	—
192	#336	—	2	1	no	—
193	D₄⋊C₈	—	53	4	no	—
194	#322	—	106	2	no	—
195	C₅₃	#8202	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	53	1	no	—
197	C₂³×C₄	—	106	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	53	2	no	—
200	C₂²×C₈	—	106	1	no	—
201	C₂²×C₈	—	106	1	no	—
202	C₂²×C₈	—	106	4	no	—
203	C₂²×C₈	—	212	2	no	—
204	C₂²×C₈	—	212	2	no	—
205	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄×D₄	—	53	2	no	—
208	C₄×D₄	—	53	2	no	—
209	C₄×D₄	—	106	1	no	—
210	C₄×D₄	—	106	1	no	—
211	C₂×C₄⋊C₄	—	53	1	no	—
212	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
213	C₂×C₂²⋊C₄	—	106	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	53	2	no	—
218	C₄⋊C₈	—	106	1	no	—
219	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
220	#102	—	106	4	no	—
221	C₂²⋊C₈	—	106	4	no	—
222	#97	—	4	2	no	—
223	C₄×C₈	—	53	2	no	—
224	C₄×C₈	—	106	1	no	—
225	C₄×C₈	—	106	1	no	—
226	C₄×C₈	—	106	1	no	—
227	C₄×C₈	C₄×D₅₃	1	2	yes	—
228	C₄×C₈	D₂₁₂	1	2	yes	—
229	#95	—	2	1	no	—
230	C₂⁴	—	106	1	no	—
231	C₂×D₄	—	53	2	no	—
232	C₂×D₄	—	53	4	no	—
233	C₂×D₄	—	106	1	no	—
234	C₂×D₄	—	106	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	106	1	no	—
237	C₂²×C₄	—	106	2	no	—
238	C₂²×C₄	—	106	2	no	—
239	C₂²×C₄	—	212	1	no	—
240	C₂²×C₄	—	212	1	no	—
241	C₂²×C₄	—	212	1	no	—
242	C₂²×C₄	—	212	1	no	—
243	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
244	C₂²×C₄	#11719945	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	106	2	no	—
252	C₂×C₈	—	106	2	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	—	212	2	no	—
257	C₂×C₈	—	212	1	no	—
258	C₂×C₈	—	212	1	no	—
259	C₂×C₈	—	212	8	no	—
260	C₂×C₈	#11719921	1	2	yes	—
261	C₂×C₈	#11719921	1	2	yes	—
262	C₂×C₈	#11719922	1	2	yes	—
263	C₂×C₈	#11719924	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	53	2	no	—
267	C₄⋊C₄	—	106	1	no	—
268	C₂²⋊C₄	—	106	2	no	—
269	C₂²⋊C₄	—	106	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
275	C₄²	#11719945	1	1	yes	characteristic
276	C₂³	—	106	4	no	—
277	C₂³	—	106	2	no	—
278	C₂³	—	212	1	no	—
279	C₂³	—	212	1	no	—
280	C₂³	—	212	2	no	—
281	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
282	D₄	—	53	4	no	—
283	D₄	—	106	2	no	—
284	D₄	—	106	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	212	1	no	—
291	C₂×C₄	—	212	1	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	—	212	2	no	—
295	C₂×C₄	—	212	2	no	—
296	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
298	C₂×C₄	#13133174	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
301	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
303	C₂×C₄	#13133225	1	2	yes	—
304	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
305	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
306	C₂×C₄	#13133237	1	1	yes	characteristic
307	C₂×C₄	#13133237	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	212	2	no	—
311	C₈	—	212	2	no	—
312	C₈	—	212	2	no	—
313	C₈	#13133137	1	4	yes	complemented
314	C₂²	—	106	4	no	—
315	C₂²	—	212	2	no	—
316	C₂²	—	212	2	no	—
317	C₂²	—	212	4	no	—
318	C₂²	—	212	4	no	—
319	C₂²	C₄×#11719921	1	1	yes	characteristic
320	C₂²	#16810692	1	2	yes	—
321	C₂²	C₄×#11719922	1	1	yes	characteristic
322	C₂²	C₄×#11719924	1	1	yes	characteristic
323	C₂²	#16810743	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	212	2	no	—
328	C₄	#97×D₅₃	1	1	yes	characteristic
329	C₄	#16810580	1	1	yes	characteristic
330	C₄	C₂×#13133137	1	1	yes	characteristic
331	C₄	#16810617	1	1	yes	characteristic
332	C₄	#16810696	1	2	yes	—
333	C₄	C₄×#11719922	1	1	yes	characteristic
334	C₄	C₄×#11719924	1	1	yes	characteristic
335	C₂	—	212	4	no	—
336	C₂	#20273542	1	2	yes	—
337	C₂	#20273549	1	2	yes	—
338	C₂	#20274492	1	1	yes	characteristic
339	C₂	C₄×#13133137	1	1	yes	characteristic
340	C₂	#20274561	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1770	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1770	D₅₃
hypocenter	C₅₃	#8202
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
4. C₂×#16810344	C₂	semidirect
5. #20272556	C₂	semidirect
6. #20272549	C₂	semidirect
7. C₅₃×#1770	C₂	semidirect
11. C₈×D₂₁₂	C₄	semidirect
18. #16810350	C₄	semidirect
132. D₂₁₂	#97	semidirect
139. C₈×C₅₃	#102	semidirect
166. #1770	D₅₃	semidirect
195. C₅₃	#8202	semidirect
313. C₈	#13133137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₁₂	2
C₄×#13133137	2
C₂×#16810350	2
C₂×#16810344	2
#20272556	2
#20272549	2
C₅₃×#1770	2
#8202	53