Subgroups — #27354038

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284746	C₂	1	1	yes	maximal, characteristic
2	#20283428	C₂	1	1	yes	maximal, characteristic
3	#20273593	C₂	1	4	yes	maximal, complemented
4	#20273525	C₂	1	4	yes	maximal, complemented
5	#20273489	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273144	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273113	C₂	1	1	yes	maximal, characteristic
8	#20273007	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133223	—	2	1	no	—
11	#16811173	C₂²	1	1	yes	characteristic
12	#16810941	C₄	1	2	yes	—
13	#16810908	C₄	1	2	yes	—
14	#16810823	C₄	1	2	yes	—
15	#16810762	C₄	1	2	yes	—
16	C₄×#11719924	C₂²	1	4	yes	—
17	#16810702	—	2	1	no	—
18	#16810702	C₂²	1	2	yes	—
19	C₈×D₂₁₂	—	2	1	no	—
20	C₈×D₂₁₂	C₂²	1	2	yes	—
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810676	C₂²	1	1	yes	characteristic
23	#16810647	C₂²	1	1	yes	characteristic
24	#16810589	—	2	1	no	—
25	#16810375	C₂²	1	4	yes	—
26	Q₈⋊#88322	C₂²	1	4	yes	—
27	#16810350	C₂²	1	4	yes	—
28	#16810345	C₂²	1	2	yes	—
29	#16810345	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₅₃	—	2	1	no	—
32	C₈×Q₈×C₅₃	—	2	1	no	—
33	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
34	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₃	—	2	1	no	—
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#11719945	—	2	1	no	—
40	C₂×#11719924	C₂×C₄	1	4	yes	—
41	#13133223	—	4	4	no	—
42	C₂×#11719917	—	2	1	no	—
43	C₂×#11719912	—	2	2	no	—
44	C₂×C₈×D₅₃	—	2	2	no	—
45	#13133217	C₂×C₄	1	1	yes	characteristic
46	#13133214	C₂×C₄	1	1	yes	characteristic
47	#13133205	—	2	1	no	—
48	#13133191	—	2	1	no	—
49	#13133191	C₂×C₄	1	2	yes	—
50	C₄×D₂₁₂	—	2	1	no	—
51	C₄×D₂₁₂	C₂³	1	2	yes	—
52	#13133148	C₂×C₄	1	4	yes	—
53	C₄⋊#88322	—	2	1	no	—
54	C₄⋊#88322	—	2	1	no	—
55	C₄⋊#88322	C₂×C₄	1	2	yes	—
56	C₄⋊#88322	C₂³	1	2	yes	—
57	#13133141	C₂×C₄	1	4	yes	—
58	#13133137	C₂×C₄	1	4	yes	—
59	C₂²⋊#88322	—	2	1	no	—
60	C₂²⋊#88322	C₂×C₄	1	2	yes	—
61	#13133129	—	2	2	no	—
62	#13133129	—	2	2	no	—
63	C₄×#88322	C₂³	1	2	yes	—
64	C₄×#88322	C₂³	1	2	yes	—
65	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	—	2	1	no	—
67	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
69	C₄×Q₈×C₅₃	—	2	1	no	—
70	C₄×Q₈×C₅₃	—	2	1	no	—
71	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	—	2	1	no	—
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	—	2	2	no	—
78	C₄⋊C₈×C₅₃	—	2	2	no	—
79	C₄⋊C₈×C₅₃	—	2	1	no	—
80	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
81	C₄×C₈×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₅₃	—	2	4	no	—
88	C₂×Q₈×C₅₃	—	2	1	no	—
89	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	1	no	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₅₃	—	2	2	no	—
96	#11719945	—	4	2	no	—
97	#11719945	—	4	2	no	—
98	C₂×D₂₁₂	—	2	1	no	—
99	C₂×D₂₁₂	—	2	1	no	—
100	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₅₃	—	2	2	no	—
102	C₂×Dic₁₀₆	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	4	2	no	—
107	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₄⋊C₄×C₅₃	—	2	1	no	—
113	C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅₃	#41	1	1	yes	characteristic
120	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
121	#11719924	—	2	8	no	—
122	#11719921	—	2	2	no	—
123	#11719921	—	2	2	no	—
124	C₄⋊Dic₅₃	—	2	1	no	—
125	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
126	#11719917	—	4	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	C₂²×C₄	1	4	yes	—
130	#11719912	—	4	4	no	—
131	C₈×D₅₃	—	4	4	no	—
132	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₅₃	—	2	2	no	—
134	Q₈×C₅₃	—	2	4	no	—
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	#131	1	4	yes	complemented
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₅₃	#137	1	2	yes	—
148	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₅₃	—	4	4	no	—
152	C₂×Dic₅₃	—	2	2	no	—
153	D₂₁₂	—	2	4	no	—
154	D₂₁₂	—	4	2	no	—
155	D₂₁₂	—	4	2	no	—
156	C₄×D₅₃	—	4	2	no	—
157	C₄×D₅₃	—	4	2	no	—
158	Dic₁₀₆	—	4	2	no	—
159	C₈×C₅₃	—	2	1	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	—	4	2	no	—
164	#88322	—	2	4	no	—
165	#88322	—	4	2	no	—
166	#88322	—	4	2	no	—
167	#17908	—	53	1	no	maximal, Sylow
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	#440	1	2	yes	—
171	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₅₃	—	4	2	no	—
173	C₂×D₅₃	—	4	2	no	—
174	C₂×D₅₃	—	4	2	no	—
175	C₄×C₅₃	—	2	4	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	#440	1	2	yes	—
181	C₄×C₅₃	C₂×#131	1	2	yes	—
182	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
184	Dic₅₃	—	4	2	no	—
185	#3103	—	53	1	no	—
186	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
187	#2931	—	53	1	no	—
188	#1588	—	53	4	no	—
189	SD₁₆⋊C₈	—	53	4	no	—
190	#1571	—	53	1	no	—
191	#1487	—	53	1	no	—
192	#1480	—	53	1	no	—
193	#1452	—	53	1	no	—
194	C₂×C₅₃	—	2	2	no	—
195	C₂×C₅₃	#2929	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
197	C₂×C₅₃	#2955	1	1	yes	characteristic
198	D₅₃	—	8	2	no	—
199	C₂×#131	—	106	1	no	—
200	#532	—	53	1	no	—
201	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
202	#481	—	53	2	no	—
203	#475	—	53	2	no	—
204	#460	—	53	2	no	—
205	#448	—	53	2	no	—
206	C₈×Q₈	—	2	1	no	—
207	C₈×Q₈	—	2	1	no	—
208	C₈×Q₈	C₂×D₅₃	1	2	yes	—
209	C₈×Q₈	C₂×D₅₃	1	2	yes	—
210	C₄×SD₁₆	—	53	4	no	—
211	#436	—	53	2	no	—
212	#436	—	106	1	no	—
213	C₈×D₄	—	53	2	no	—
214	C₈×D₄	—	106	1	no	—
215	#431	—	53	1	no	—
216	#431	—	53	1	no	—
217	#423	—	53	1	no	—
218	C₂×C₄⋊C₈	—	2	1	no	—
219	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
220	#408	—	106	1	no	—
221	C₂×C₄×C₈	—	2	1	no	—
222	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
223	#335	—	53	4	no	—
224	Q₈⋊C₈	—	53	4	no	—
225	D₄⋊C₈	—	53	4	no	—
226	#322	—	53	2	no	—
227	#322	—	53	2	no	—
228	C₅₃	#17908	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	—	106	1	no	—
230	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
231	C₂×SD₁₆	—	53	4	no	—
232	#131	—	212	4	no	—
233	C₂×M₁₆	—	106	1	no	—
234	C₂×M₁₆	—	106	2	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	106	2	no	—
237	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₄⋊Q₈	—	53	1	no	—
240	#127	—	53	1	no	—
241	#124	—	106	1	no	—
242	#121	—	53	2	no	—
243	#121	—	106	1	no	—
244	C₄×Q₈	—	2	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₄×D₅₃	1	2	yes	—
247	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
248	C₄×D₄	—	53	2	no	—
249	C₄×D₄	—	106	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
254	#106	—	53	4	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	53	2	no	—
259	C₄⋊C₈	—	53	2	no	—
260	C₄⋊C₈	—	106	1	no	—
261	C₄⋊C₈	—	106	1	no	—
262	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
263	#103	—	53	4	no	—
264	#102	—	53	4	no	—
265	C₂²⋊C₈	—	53	2	no	—
266	C₂²⋊C₈	—	106	1	no	—
267	C₂²⋊C₈	—	106	2	no	—
268	C₂²⋊C₈	—	106	2	no	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	53	2	no	—
272	C₄×C₈	—	53	2	no	—
273	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
274	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
277	#41	—	212	2	no	—
278	#41	—	212	2	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	106	1	no	—
282	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
283	C₂×D₄	—	53	2	no	—
284	C₂×D₄	—	106	2	no	—
285	C₂×D₄	—	106	1	no	—
286	C₂×D₄	—	106	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	106	2	no	—
290	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂²×C₄	#11719945	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
293	SD₁₆	—	106	8	no	—
294	M₁₆	—	212	4	no	—
295	M₁₆	—	212	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	53	4	no	—
301	C₂×C₈	—	106	2	no	—
302	C₂×C₈	—	106	2	no	—
303	C₂×C₈	—	212	4	no	—
304	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	53	2	no	—
312	C₄⋊C₄	—	106	1	no	—
313	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
314	C₂²⋊C₄	—	106	2	no	—
315	C₂²⋊C₄	—	106	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
319	C₄²	#11719945	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
321	C₄²	C₂³×D₅₃	1	1	yes	characteristic
322	C₂³	—	106	2	no	—
323	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	212	2	no	—
327	Q₈	#13133223	1	4	yes	complemented
328	D₄	—	106	4	no	—
329	D₄	—	212	2	no	—
330	D₄	—	212	4	no	—
331	D₄	—	212	2	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	106	2	no	—
339	C₂×C₄	—	212	2	no	—
340	C₂×C₄	—	212	2	no	—
341	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
342	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
343	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
344	C₂×C₄	#13133242	1	2	yes	—
345	C₂×C₄	#13133248	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	4	2	no	—
354	C₈	—	106	4	no	—
355	C₈	—	212	2	no	—
356	C₈	—	212	2	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	212	2	no	—
360	C₂²	—	212	2	no	—
361	C₂²	—	212	2	no	—
362	C₂²	#16810721	1	2	yes	—
363	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
364	C₄	—	2	4	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	212	2	no	—
370	C₄	#16810721	1	2	yes	—
371	C₄	C₂×#13133223	1	2	yes	—
372	C₄	C₂×#13133242	1	1	yes	characteristic
373	C₄	C₂×#13133248	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	424	2	no	—
376	C₂	#20283409	1	1	yes	characteristic
377	C₂	C₂×#16810721	1	1	yes	characteristic
378	C₂	#20283616	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16811173	C₂²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#17908
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20273593	C₂	semidirect
4. #20273525	C₂	semidirect
5. #20273489	C₂	semidirect
6. #20273144	C₂	semidirect
8. #20273007	C₂	semidirect
9. C₂×C₈×Q₈×C₅₃	C₂	semidirect
136. Q₈×C₅₃	#131	semidirect
186. C₂×C₈×Q₈	D₅₃	semidirect
228. C₅₃	#17908	semidirect
327. Q₈	#13133223	semidirect

Maximal subgroups

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

subgroup	index
#20284746	2
#20283428	2
#20273593	2
#20273525	2
#20273489	2
#20273144	2
#20273113	2
#20273007	2
C₂×C₈×Q₈×C₅₃	2
#17908	53