Subgroups — #27388515

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283517	C₂	1	1	yes	maximal, characteristic
2	#20283357	C₂	1	2	yes	maximal, complemented
3	#20275608	C₂	1	1	yes	maximal, characteristic
4	#20275603	C₂	1	1	yes	maximal, characteristic
5	#20275468	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275463	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275426	C₂	1	2	yes	maximal, complemented
8	#20275419	C₂	1	2	yes	maximal, complemented
9	#20274621	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274526	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	C₄×#11719945	C₂²	1	2	yes	—
14	#16810736	C₄	1	2	yes	complemented
15	#16810731	C₄	1	2	yes	complemented
16	#16810725	C₄	1	4	yes	complemented
17	#16810682	—	2	4	no	—
18	#16810674	—	2	4	no	—
19	C₂×#13133153	C₂²	1	1	yes	characteristic
20	C₂×#13133153	C₂²	1	1	yes	characteristic
21	#16810661	C₂²	1	1	yes	characteristic
22	#16810649	C₂²	1	2	yes	—
23	#16810649	C₂²	1	2	yes	—
24	C₂×#13133145	C₂²	1	2	yes	—
25	C₂×#13133145	C₂²	1	2	yes	—
26	#16810620	C₂²	1	2	yes	—
27	#16810589	C₂²	1	2	yes	—
28	#16810589	C₂²	1	2	yes	—
29	C₄×#11719917	C₂²	1	2	yes	—
30	#16810388	C₂²	1	2	yes	—
31	#16810383	C₂²	1	2	yes	—
32	#16810383	C₂²	1	2	yes	—
33	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#376	C₂²	1	1	yes	characteristic
40	C₂×#11719945	C₂³	1	2	yes	—
41	#13133239	—	2	2	no	—
42	#13133239	—	2	2	no	—
43	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
44	C₂×#11719913	C₂×C₄	1	2	yes	—
45	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719917	C₂³	1	2	yes	—
47	C₂×#11719917	C₂³	1	2	yes	—
48	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
49	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
50	C₄×C₂²⋊D₅₃	—	2	4	no	—
51	#13133176	C₂³	1	2	yes	—
52	#13133174	—	2	2	no	—
53	C₄²×D₅₃	—	2	2	no	—
54	#13133153	—	2	2	no	—
55	#13133153	—	2	2	no	—
56	#13133151	C₂×C₄	1	2	yes	—
57	#13133149	C₂×C₄	1	2	yes	—
58	#13133146	—	2	4	no	—
59	#13133146	—	2	4	no	—
60	#13133145	—	2	4	no	—
61	#13133145	—	2	4	no	—
62	#13133142	C₂×C₄	1	4	yes	—
63	#13133139	C₂×C₄	1	4	yes	—
64	#13133129	—	2	4	no	—
65	#13133129	—	2	4	no	—
66	#13133128	—	2	2	no	—
67	C₄×#88322	—	2	2	no	—
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	#97×C₅₃	—	2	2	no	—
76	#97×C₅₃	—	2	2	no	—
77	#97×C₅₃	C₂×C₄	1	2	yes	—
78	#97×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	—	2	2	no	—
81	C₄×C₈×C₅₃	D₄	1	4	yes	—
82	#95×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	1	no	—
84	C₂²×C₄×C₅₃	—	2	1	no	—
85	C₂²×C₄×C₅₃	#41	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂×C₂²⋊D₅₃	—	2	2	no	—
88	#11719945	—	2	2	no	—
89	#11719945	—	2	2	no	—
90	#11719945	—	2	2	no	—
91	#11719945	—	2	2	no	—
92	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₅₃	—	2	4	no	—
94	C₂×C₄×D₅₃	—	2	2	no	—
95	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	—	2	1	no	—
97	C₂×C₈×C₅₃	—	2	1	no	—
98	C₂×C₈×C₅₃	—	2	1	no	—
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₅₃	#41	1	2	yes	—
108	C₂×C₈×C₅₃	#41	1	2	yes	—
109	C₄²×C₅₃	—	2	1	no	—
110	C₄²×C₅₃	—	2	1	no	—
111	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
112	C₄²×C₅₃	C₂×D₄	1	2	yes	—
113	C₄²×C₅₃	#41	1	2	yes	—
114	C₄²×C₅₃	#41	1	2	yes	—
115	C₂²⋊Dic₅₃	—	2	2	no	—
116	#11719921	—	2	4	no	—
117	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
118	#11719919	—	2	4	no	—
119	C₄×Dic₅₃	—	2	4	no	—
120	C₄×Dic₅₃	—	2	2	no	—
121	#11719917	—	2	2	no	—
122	#11719917	—	2	2	no	—
123	#11719917	—	2	2	no	—
124	#11719917	—	2	2	no	—
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	—	2	2	no	—
127	Dic₂₁₂	—	2	2	no	—
128	D₄₂₄	—	2	2	no	—
129	#11719913	—	2	4	no	—
130	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
131	C₂²×D₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	4	2	no	—
136	C₂×C₄×C₅₃	—	4	2	no	—
137	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
138	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₅₃	—	4	2	no	—
145	C₂²⋊D₅₃	—	4	2	no	—
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	—	2	4	no	—
148	D₂₁₂	—	2	2	no	—
149	D₂₁₂	—	2	2	no	—
150	C₄×D₅₃	—	4	8	no	—
151	C₄×D₅₃	—	4	2	no	—
152	C₄×D₅₃	—	4	2	no	—
153	Dic₁₀₆	—	2	2	no	—
154	Dic₁₀₆	—	2	2	no	—
155	C₈×C₅₃	—	2	2	no	—
156	C₈×C₅₃	—	2	2	no	—
157	C₈×C₅₃	—	2	2	no	—
158	C₈×C₅₃	#117	1	2	yes	—
159	C₈×C₅₃	#117	1	2	yes	—
160	#88322	—	4	2	no	—
161	#88322	—	4	2	no	—
162	#21296	—	53	1	no	maximal, Sylow
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₅₃	#529	1	1	yes	characteristic
166	C₂²×C₅₃	#529	1	1	yes	characteristic
167	C₂×D₅₃	—	4	4	no	—
168	C₂×D₅₃	—	4	2	no	—
169	C₄×C₅₃	—	2	4	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	—	4	2	no	—
172	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
173	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
174	C₄×C₅₃	#532	1	1	yes	characteristic
175	C₄×C₅₃	#532	1	1	yes	characteristic
176	Dic₅₃	—	4	4	no	—
177	Dic₅₃	—	4	2	no	—
178	#2943	—	53	1	no	—
179	#2922	—	53	2	no	—
180	#1975	—	53	1	no	—
181	#1974	—	53	1	no	—
182	#1946	—	53	1	no	—
183	#1945	—	53	1	no	—
184	#1938	—	53	2	no	—
185	#1937	—	53	2	no	—
186	#1778	—	53	1	no	—
187	#1759	—	53	2	no	—
188	#1726	D₅₃	1	1	yes	characteristic, complemented
189	C₂×C₅₃	—	2	1	no	—
190	C₂×C₅₃	—	2	1	no	—
191	C₂×C₅₃	#2293	1	1	yes	characteristic
192	C₂×C₅₃	#2955	1	2	yes	—
193	D₅₃	—	8	2	no	—
194	C₂×#135	—	53	1	no	—
195	C₄×#41	—	53	2	no	—
196	#442	—	53	2	no	—
197	#441	—	53	2	no	—
198	#440	—	53	4	no	—
199	#433	—	106	4	no	—
200	#431	—	106	4	no	—
201	C₂×#108	—	53	1	no	—
202	C₂×#108	—	53	1	no	—
203	#427	—	53	1	no	—
204	#424	—	53	2	no	—
205	#424	—	53	2	no	—
206	C₂×#104	—	53	2	no	—
207	C₂×#104	—	53	2	no	—
208	#416	—	53	2	no	—
209	#408	—	53	2	no	—
210	#408	—	53	2	no	—
211	C₄×M₁₆	—	53	2	no	—
212	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
213	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
214	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
217	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
218	#376	C₂×D₅₃	1	1	yes	characteristic
219	#341	—	53	2	no	—
220	#339	—	53	2	no	—
221	#339	—	53	2	no	—
222	C₅₃	#21296	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	—	53	2	no	—
224	#135	—	106	2	no	—
225	#135	—	106	2	no	—
226	C₂×Q₁₆	—	53	1	no	—
227	C₂×SD₁₆	—	53	2	no	—
228	C₂×D₈	—	53	1	no	—
229	C₂×M₁₆	—	53	2	no	—
230	C₂×M₁₆	—	53	2	no	—
231	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₄×Q₈	—	53	4	no	—
236	C₄×D₄	—	53	4	no	—
237	C₄×D₄	—	106	4	no	—
238	#117	—	53	2	no	—
239	#117	—	106	2	no	—
240	C₂×C₄²	—	106	2	no	—
241	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
244	#108	—	106	2	no	—
245	#108	—	106	2	no	—
246	#107	—	53	2	no	—
247	#106	—	53	2	no	—
248	C₄⋊C₈	—	106	4	no	—
249	C₄⋊C₈	—	106	4	no	—
250	#104	—	106	4	no	—
251	#104	—	106	4	no	—
252	#103	—	53	4	no	—
253	#102	—	53	4	no	—
254	C₂²⋊C₈	—	106	4	no	—
255	C₂²⋊C₈	—	106	4	no	—
256	#97	—	2	2	no	—
257	#97	—	2	2	no	—
258	#97	—	106	2	no	—
259	#97	C₄×D₅₃	1	2	yes	—
260	#97	C₄×D₅₃	1	2	yes	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	106	2	no	—
264	C₄×C₈	D₂₁₂	1	4	yes	—
265	#95	—	2	2	no	—
266	#41	—	106	2	no	—
267	#41	—	106	2	no	—
268	#41	—	106	2	no	—
269	#41	—	106	2	no	—
270	C₂×Q₈	—	53	2	no	—
271	C₂×D₄	—	53	2	no	—
272	C₂×D₄	—	106	2	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	106	4	no	—
276	C₂²×C₄	—	106	2	no	—
277	C₂²×C₄	#11719945	1	2	yes	—
278	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
279	Q₁₆	—	106	2	no	—
280	SD₁₆	—	106	4	no	—
281	D₈	—	106	2	no	—
282	M₁₆	—	106	2	no	—
283	M₁₆	—	106	2	no	—
284	M₁₆	—	106	2	no	—
285	M₁₆	—	106	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	106	2	no	—
293	C₂×C₈	—	106	2	no	—
294	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
297	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
298	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
299	C₂×C₈	#11719945	1	2	yes	—
300	C₂×C₈	#11719945	1	2	yes	—
301	C₄⋊C₄	—	53	4	no	—
302	C₄⋊C₄	—	106	4	no	—
303	C₂²⋊C₄	—	106	2	no	—
304	C₂²⋊C₄	—	106	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	106	4	no	—
308	C₄²	—	106	2	no	—
309	C₄²	C₂×C₄×D₅₃	1	2	yes	—
310	C₄²	C₂×D₂₁₂	1	2	yes	—
311	C₄²	#11719945	1	2	yes	—
312	C₄²	#11719945	1	2	yes	—
313	C₂³	—	106	2	no	—
314	C₂³	C₂×#11719945	1	1	yes	characteristic
315	Q₈	—	106	2	no	—
316	Q₈	—	106	2	no	—
317	D₄	—	106	2	no	—
318	D₄	—	106	2	no	—
319	D₄	—	212	2	no	—
320	D₄	—	212	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	106	4	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	212	8	no	—
329	C₂×C₄	—	212	2	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
332	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
334	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
335	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
336	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
337	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	212	2	no	—
342	C₈	—	212	2	no	—
343	C₈	#13133174	1	2	yes	—
344	C₈	#13133174	1	2	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	212	2	no	—
347	C₂²	—	212	4	no	—
348	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
349	C₂²	#16811157	1	1	yes	characteristic
350	C₂²	#16811157	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	212	2	no	—
355	C₄	—	212	4	no	—
356	C₄	C₂×#13133174	1	1	yes	characteristic
357	C₄	C₄×#11719945	1	1	yes	characteristic
358	C₄	#16811173	1	1	yes	characteristic
359	C₄	#16811173	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	424	2	no	—
363	C₂	#20277179	1	1	yes	characteristic
364	C₂	#20283619	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1726	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#21296
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. #20283357	C₂	semidirect
5. #20275468	C₂	semidirect
6. #20275463	C₂	semidirect
7. #20275426	C₂	semidirect
8. #20275419	C₂	semidirect
9. #20274621	C₂	semidirect
10. #20274526	C₂	semidirect
11. C₅₃×#1726	C₂	semidirect
14. #16810736	C₄	semidirect
15. #16810731	C₄	semidirect
16. #16810725	C₄	semidirect
188. #1726	D₅₃	semidirect
222. C₅₃	#21296	semidirect

Maximal subgroups

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

subgroup	index
#20283517	2
#20283357	2
#20275608	2
#20275603	2
#20275468	2
#20275463	2
#20275426	2
#20275419	2
#20274621	2
#20274526	2
C₅₃×#1726	2
#21296	53