Subgroups — #27300975

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287817	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283411	C₂	1	1	yes	maximal, characteristic
3	#20276335	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276325	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276325	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272801	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#16811135	C₄	1	1	yes	characteristic
11	#16811041	—	2	2	no	—
12	#16811039	—	2	2	no	—
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	C₂×#13133157	C₂²	1	1	yes	characteristic
15	M₃₂×D₅₃	—	2	2	no	—
16	#16810696	—	2	2	no	—
17	C₈×D₂₁₂	—	2	2	no	—
18	#16810650	C₄	1	1	yes	characteristic
19	#16810591	C₄	1	1	yes	characteristic
20	#16810586	C₄	1	1	yes	characteristic
21	#16810404	—	2	2	no	—
22	#16810402	—	2	2	no	—
23	#16810402	—	2	2	no	—
24	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#405	C₂²	1	1	yes	characteristic
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	#13133223	—	2	1	no	—
30	#13133223	—	2	1	no	—
31	#13133223	—	2	1	no	—
32	#13133223	—	2	1	no	—
33	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
34	C₂×#11719912	D₄	1	2	yes	—
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	—	2	1	no	—
37	C₂×C₈×D₅₃	—	2	1	no	—
38	C₂×C₈×D₅₃	—	2	1	no	—
39	C₂×C₈×D₅₃	—	2	1	no	—
40	C₂×C₈×D₅₃	—	2	1	no	—
41	C₂×C₈×D₅₃	—	2	1	no	—
42	C₂×C₈×D₅₃	D₄	1	2	yes	—
43	#13133201	C₈	1	2	yes	—
44	#13133197	—	2	1	no	—
45	#13133192	—	2	1	no	—
46	#13133187	C₈	1	2	yes	—
47	C₄×D₂₁₂	—	2	2	no	—
48	#13133157	—	2	1	no	—
49	#13133157	—	2	1	no	—
50	#13133157	—	2	1	no	—
51	#13133157	—	2	1	no	—
52	#13133156	—	2	4	no	—
53	C₂×#11719908	—	2	1	no	—
54	C₂×#11719908	D₄	1	2	yes	complemented
55	C₁₆×D₅₃	—	2	4	no	—
56	C₄⋊#88322	—	2	2	no	—
57	#13133129	—	2	2	no	—
58	#13133129	—	2	2	no	—
59	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
62	M₃₂×C₅₃	—	2	1	no	—
63	M₃₂×C₅₃	—	2	1	no	—
64	M₃₂×C₅₃	—	2	1	no	—
65	M₃₂×C₅₃	—	2	1	no	—
66	C₂×C₁₆×C₅₃	—	2	1	no	—
67	C₂×C₁₆×C₅₃	D₄	1	2	yes	complemented
68	#97×C₅₃	—	2	1	no	—
69	C₄×C₈×C₅₃	—	2	1	no	—
70	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
73	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
74	#11719945	—	2	1	no	—
75	#11719945	—	2	1	no	—
76	#11719945	—	2	1	no	—
77	#11719945	—	2	1	no	—
78	C₂×D₂₁₂	—	2	2	no	—
79	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₅₃	—	2	1	no	—
81	C₂×C₄×D₅₃	—	2	1	no	—
82	C₂×C₄×D₅₃	—	2	1	no	—
83	C₂×C₄×D₅₃	—	2	1	no	—
84	C₂×C₄×D₅₃	—	2	1	no	—
85	C₂×C₄×D₅₃	—	2	1	no	—
86	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
87	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
88	M₁₆×C₅₃	—	2	1	no	—
89	M₁₆×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
99	C₄²×C₅₃	—	2	1	no	—
100	#11719921	—	2	2	no	—
101	#11719921	—	2	2	no	—
102	C₄⋊Dic₅₃	—	2	2	no	—
103	#11719917	—	2	1	no	—
104	#11719917	—	2	1	no	—
105	C₂×#88322	—	2	1	no	—
106	C₂×#88322	—	2	1	no	—
107	C₂×#88322	—	2	1	no	—
108	C₂×#88322	C₂²⋊C₄	1	2	yes	—
109	#11719912	—	2	2	no	—
110	#11719912	—	2	2	no	—
111	C₈×D₅₃	—	2	2	no	—
112	C₈×D₅₃	—	2	2	no	—
113	C₈×D₅₃	—	4	1	no	—
114	C₈×D₅₃	—	4	1	no	—
115	C₈×D₅₃	—	4	1	no	—
116	C₈×D₅₃	—	4	1	no	—
117	C₁₆×C₅₃	—	2	2	no	—
118	C₁₆×C₅₃	—	4	1	no	—
119	#11719908	—	2	2	no	—
120	#11719908	—	4	1	no	—
121	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₅₃	—	2	1	no	—
123	C₂²×D₅₃	—	2	1	no	—
124	C₂²×D₅₃	—	2	1	no	—
125	C₂²×D₅₃	—	4	2	no	—
126	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
134	C₂²⋊D₅₃	—	2	2	no	—
135	C₂²⋊D₅₃	—	2	2	no	—
136	C₂×Dic₅₃	—	2	1	no	—
137	C₂×Dic₅₃	—	2	1	no	—
138	C₂×Dic₅₃	—	2	1	no	—
139	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
140	D₂₁₂	—	2	1	no	—
141	D₂₁₂	—	2	1	no	—
142	D₂₁₂	—	4	4	no	—
143	C₄×D₅₃	—	2	2	no	—
144	C₄×D₅₃	—	2	2	no	—
145	C₄×D₅₃	—	4	1	no	—
146	C₄×D₅₃	—	4	1	no	—
147	C₄×D₅₃	—	4	1	no	—
148	C₄×D₅₃	—	4	1	no	—
149	Dic₁₀₆	—	2	1	no	—
150	Dic₁₀₆	—	2	1	no	—
151	C₈×C₅₃	—	4	1	no	—
152	C₈×C₅₃	#120	1	1	yes	characteristic
153	C₈×C₅₃	#121	1	1	yes	characteristic
154	C₈×C₅₃	#121	1	1	yes	characteristic
155	C₈×C₅₃	#123	1	1	yes	characteristic
156	#88322	—	2	2	no	—
157	#88322	—	4	1	no	—
158	#12981	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	4	1	no	—
160	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
162	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
163	C₂×D₅₃	—	2	2	no	—
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	—	2	2	no	—
166	C₂×D₅₃	—	2	2	no	—
167	C₂×D₅₃	—	4	1	no	—
168	C₂×D₅₃	—	4	1	no	—
169	C₂×D₅₃	—	4	1	no	—
170	C₂×D₅₃	—	4	1	no	—
171	C₂×D₅₃	—	4	2	no	—
172	C₄×C₅₃	—	4	2	no	—
173	C₄×C₅₃	#386	1	1	yes	characteristic
174	C₄×C₅₃	#408	1	1	yes	characteristic
175	C₄×C₅₃	#435	1	1	yes	characteristic
176	C₄×C₅₃	#435	1	1	yes	characteristic
177	Dic₅₃	—	2	2	no	—
178	Dic₅₃	—	4	1	no	—
179	#3407	—	53	1	no	—
180	#2929	—	53	1	no	—
181	#2115	—	53	1	no	—
182	#2112	—	53	1	no	—
183	#2112	—	53	1	no	—
184	#1378	—	53	1	no	—
185	#1378	D₅₃	1	1	yes	characteristic, complemented
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	2	1	no	—
188	C₂×C₅₃	—	2	1	no	—
189	C₂×C₅₃	#1869	1	1	yes	characteristic
190	D₅₃	—	4	2	no	—
191	D₅₃	—	4	2	no	—
192	C₂×#131	—	53	1	no	—
193	C₂³×C₈	—	53	1	no	—
194	#525	—	53	1	no	—
195	#504	—	106	2	no	—
196	#504	—	106	2	no	—
197	C₂×M₃₂	—	53	1	no	—
198	C₂×M₃₂	—	53	1	no	—
199	C₂×M₃₂	—	106	2	no	—
200	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
201	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
202	#435	—	106	2	no	—
203	C₈×D₄	—	106	2	no	—
204	#424	—	53	1	no	—
205	#408	—	53	1	no	—
206	#407	—	53	1	no	—
207	#405	C₂×D₅₃	1	1	yes	characteristic
208	#349	—	106	2	no	—
209	C₂²⋊C₁₆	—	106	2	no	—
210	C₂²⋊C₁₆	—	106	2	no	—
211	C₅₃	#12981	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	53	1	no	—
213	C₂³×C₄	—	53	1	no	—
214	#131	—	106	1	no	—
215	#131	—	106	1	no	—
216	#131	—	106	1	no	—
217	#131	—	106	1	no	—
218	C₂×M₁₆	—	53	1	no	—
219	C₂×M₁₆	—	53	2	no	—
220	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	—	53	1	no	—
223	C₂²×C₈	—	106	1	no	—
224	C₂²×C₈	—	106	1	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	—	106	1	no	—
227	C₂²×C₈	—	106	1	no	—
228	C₂²×C₈	—	106	1	no	—
229	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
230	#123	—	53	2	no	—
231	C₂²⋊Q₈	—	106	1	no	—
232	#121	—	106	1	no	—
233	#120	—	53	2	no	—
234	C₄×D₄	—	106	2	no	—
235	#117	C₄×D₅₃	1	1	yes	characteristic
236	M₃₂	—	2	1	no	—
237	M₃₂	—	2	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	2	1	no	—
240	M₃₂	—	106	4	no	—
241	M₃₂	—	106	1	no	—
242	M₃₂	—	106	1	no	—
243	M₃₂	—	106	1	no	—
244	M₃₂	—	106	1	no	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	53	2	no	—
247	C₂×C₁₆	—	106	4	no	—
248	C₂×C₁₆	—	106	1	no	—
249	C₂×C₁₆	D₂₁₂	1	2	yes	complemented
250	C₄⋊C₈	—	106	2	no	—
251	C₂²⋊C₈	—	106	2	no	—
252	C₂²⋊C₈	—	106	2	no	—
253	#97	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₂⁴	—	53	1	no	—
256	#41	—	106	1	no	—
257	#41	—	106	1	no	—
258	#41	—	106	1	no	—
259	#41	—	106	1	no	—
260	C₂×Q₈	—	53	1	no	—
261	C₂×D₄	—	53	2	no	—
262	C₂×D₄	—	53	1	no	—
263	C₂×D₄	—	106	2	no	—
264	C₂²×C₄	—	53	2	no	—
265	C₂²×C₄	—	53	1	no	—
266	C₂²×C₄	—	106	1	no	—
267	C₂²×C₄	—	106	1	no	—
268	C₂²×C₄	—	106	1	no	—
269	C₂²×C₄	—	106	1	no	—
270	C₂²×C₄	—	106	1	no	—
271	C₂²×C₄	—	106	1	no	—
272	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	106	1	no	—
276	M₁₆	—	106	1	no	—
277	M₁₆	—	106	2	no	—
278	M₁₆	—	106	2	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	53	2	no	—
281	C₂×C₈	—	106	2	no	—
282	C₂×C₈	—	106	2	no	—
283	C₂×C₈	—	106	1	no	—
284	C₂×C₈	—	106	1	no	—
285	C₂×C₈	—	106	1	no	—
286	C₂×C₈	—	212	1	no	—
287	C₂×C₈	—	212	1	no	—
288	C₂×C₈	—	212	1	no	—
289	C₂×C₈	—	212	1	no	—
290	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
291	C₂×C₈	#11719945	1	1	yes	characteristic
292	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
293	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
294	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
295	C₂×C₈	#11719949	1	1	yes	characteristic
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	106	2	no	—
298	C₂²⋊C₄	—	106	2	no	—
299	C₂²⋊C₄	—	106	2	no	—
300	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
301	C₄²	—	2	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	4	1	no	—
304	C₁₆	—	106	2	no	—
305	C₁₆	—	212	1	no	—
306	C₂³	—	53	2	no	—
307	C₂³	—	106	1	no	—
308	C₂³	—	106	1	no	—
309	C₂³	—	106	1	no	—
310	C₂³	—	212	2	no	—
311	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
312	Q₈	—	106	1	no	—
313	Q₈	—	106	1	no	—
314	D₄	—	106	2	no	—
315	D₄	—	106	2	no	—
316	D₄	—	106	1	no	—
317	D₄	—	106	1	no	—
318	D₄	—	212	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	1	no	—
324	C₂×C₄	—	106	1	no	—
325	C₂×C₄	—	106	1	no	—
326	C₂×C₄	—	212	1	no	—
327	C₂×C₄	—	212	1	no	—
328	C₂×C₄	—	212	1	no	—
329	C₂×C₄	—	212	1	no	—
330	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
331	C₂×C₄	#13133179	1	1	yes	characteristic
332	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
333	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
334	C₂×C₄	#13133223	1	1	yes	characteristic
335	C₂×C₄	#13133225	1	1	yes	characteristic
336	C₈	—	4	1	no	—
337	C₈	—	106	2	no	—
338	C₈	—	212	1	no	—
339	C₈	#13133187	1	1	yes	characteristic
340	C₈	#13133190	1	1	yes	characteristic
341	C₈	#13133192	1	1	yes	characteristic
342	C₈	#13133202	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	106	2	no	—
345	C₂²	—	106	2	no	—
346	C₂²	—	106	2	no	—
347	C₂²	—	106	2	no	—
348	C₂²	—	212	2	no	—
349	C₂²	—	212	1	no	—
350	C₂²	—	212	1	no	—
351	C₂²	—	212	1	no	—
352	C₂²	—	212	1	no	—
353	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
354	C₂²	#16810688	1	1	yes	characteristic
355	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
356	C₄	—	4	2	no	—
357	C₄	—	106	2	no	—
358	C₄	—	212	1	no	—
359	C₄	#16810501	1	1	yes	characteristic
360	C₄	#16810588	1	1	yes	characteristic
361	C₄	#16810693	1	1	yes	characteristic
362	C₄	#16810696	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	212	2	no	—
367	C₂	—	212	2	no	—
368	C₂	#20275006	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133187
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1378	C₂
socle	C₂×C₅₃	#1869
prime radical	#16811135	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1378	D₅₃
hypocenter	C₅₃	#12981
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
1. #20287817	C₂	semidirect
3. #20276335	C₂	semidirect
4. #20276325	C₂	semidirect
5. #20276325	C₂	semidirect
6. #20272801	C₂	semidirect
7. C₅₃×#1378	C₂	semidirect
54. C₂×#11719908	D₄	semidirect
67. C₂×C₁₆×C₅₃	D₄	semidirect
185. #1378	D₅₃	semidirect
211. C₅₃	#12981	semidirect
249. C₂×C₁₆	D₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
#20287817	2
#20283411	2
#20276335	2
#20276325	2
#20272801	2
C₅₃×#1378	2
#12981	53