Subgroups — #27355877

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285285	C₂	1	1	yes	maximal, characteristic
2	#20283713	C₂	1	2	yes	maximal, complemented
3	#20273611	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273599	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273557	C₂	1	2	yes	maximal, complemented
6	#20273550	C₂	1	2	yes	maximal, complemented
7	#20273521	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273509	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273121	C₂	1	2	yes	maximal, complemented
10	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811157	C₂²	1	2	yes	—
13	#16811025	C₄	1	1	yes	characteristic
14	#16811007	C₄	1	1	yes	characteristic
15	#16810978	C₄	1	2	yes	—
16	#16810864	C₄	1	1	yes	characteristic
17	#16810859	C₄	1	1	yes	characteristic
18	#16810852	C₄	1	2	yes	—
19	#16810750	C₂²	1	2	yes	—
20	#16810747	C₂²	1	2	yes	—
21	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
22	C₄×#11719913	C₂²	1	2	yes	—
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
24	#16810700	C₂²	1	2	yes	—
25	#16810698	—	2	2	no	—
26	#16810697	—	2	2	no	—
27	#16810688	C₂²	1	2	yes	—
28	#16810682	—	2	2	no	—
29	#16810676	C₂²	1	1	yes	characteristic
30	#16810577	—	2	2	no	—
31	C₄×#11719917	C₂²	1	1	yes	characteristic
32	#16810353	C₂²	1	2	yes	—
33	#16810353	C₂²	1	2	yes	—
34	#16810350	C₂²	1	2	yes	—
35	#16810350	C₂²	1	2	yes	—
36	#16810345	C₂²	1	1	yes	characteristic
37	#16810345	C₂²	1	1	yes	characteristic
38	C₈×#88322	C₂²	1	1	yes	characteristic
39	C₈×#88322	C₂²	1	1	yes	characteristic
40	C₅₃×#431	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#335	C₂²	1	1	yes	characteristic
43	C₅₃×#335	C₂²	1	1	yes	characteristic
44	C₅₃×#334	C₂²	1	2	yes	—
45	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719913	C₂×C₄	1	2	yes	—
47	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719917	D₄	1	1	yes	characteristic
49	C₂×#11719917	D₄	1	1	yes	characteristic
50	C₂×#11719912	—	2	2	no	—
51	C₂×C₈×D₅₃	—	2	2	no	—
52	#13133211	—	2	2	no	—
53	#13133209	C₂×C₄	1	2	yes	—
54	#13133205	C₂×C₄	1	2	yes	—
55	#13133200	—	2	2	no	—
56	#13133196	C₂×C₄	1	2	yes	—
57	#13133191	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₀₆	C₂³	1	2	yes	—
59	C₄×D₂₁₂	C₂³	1	2	yes	—
60	C₄×C₂²⋊D₅₃	—	2	2	no	—
61	#13133174	—	2	2	no	—
62	#13133151	C₂×C₄	1	1	yes	characteristic
63	#13133151	C₂×C₄	1	1	yes	characteristic
64	#13133149	C₂×C₄	1	2	yes	—
65	C₄⋊#88322	—	2	1	no	—
66	#13133146	—	2	2	no	—
67	#13133146	—	2	2	no	—
68	#13133142	C₂×C₄	1	2	yes	—
69	#13133142	C₂×C₄	1	2	yes	—
70	#13133139	C₂×C₄	1	2	yes	—
71	#13133139	C₂×C₄	1	2	yes	—
72	C₂²⋊#88322	—	2	1	no	—
73	#13133129	—	2	2	no	—
74	#13133129	—	2	2	no	—
75	#13133128	D₄	1	1	yes	characteristic
76	#13133128	D₄	1	1	yes	characteristic
77	#13133127	—	2	2	no	—
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	1	yes	characteristic
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₄×#88322	C₂³	1	1	yes	characteristic
82	C₈×Dic₅₃	—	2	2	no	—
83	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
87	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
88	C₂²⋊C₈×C₅₃	—	2	2	no	—
89	C₄×C₈×C₅₃	—	2	2	no	—
90	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
97	C₂×C₂²⋊D₅₃	—	2	2	no	—
98	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₅₃	—	2	2	no	—
100	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	4	1	no	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	#41	1	1	yes	characteristic
112	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₅₃	—	2	2	no	—
114	#11719921	—	2	2	no	—
115	#11719921	—	2	2	no	—
116	#11719921	—	2	2	no	—
117	#11719921	—	2	2	no	—
118	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
119	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
120	#11719919	—	2	2	no	—
121	#11719919	—	2	2	no	—
122	#11719919	—	2	2	no	—
123	#11719919	—	2	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	#11719917	—	2	2	no	—
126	#11719917	—	2	2	no	—
127	C₂×#88322	—	2	1	no	—
128	C₂×#88322	—	2	1	no	—
129	C₂×#88322	C₂×D₄	1	1	yes	characteristic
130	C₂×#88322	C₂×D₄	1	1	yes	characteristic
131	C₂×#88322	C₂×D₄	1	1	yes	characteristic
132	C₂×#88322	C₂×D₄	1	1	yes	characteristic
133	Dic₂₁₂	—	2	2	no	—
134	D₄₂₄	—	2	2	no	—
135	#11719913	—	4	2	no	—
136	#11719912	—	4	4	no	—
137	C₈×D₅₃	—	4	4	no	—
138	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
139	C₂²×D₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
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	—	8	2	no	—
152	C₂×Dic₅₃	—	2	2	no	—
153	C₂×Dic₅₃	—	2	2	no	—
154	C₂×Dic₅₃	—	2	2	no	—
155	D₂₁₂	—	2	2	no	—
156	D₂₁₂	—	2	2	no	—
157	C₄×D₅₃	—	4	2	no	—
158	C₄×D₅₃	—	4	2	no	—
159	Dic₁₀₆	—	2	2	no	—
160	Dic₁₀₆	—	2	2	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	4	2	no	—
165	C₈×C₅₃	—	4	2	no	—
166	C₈×C₅₃	#131	1	2	yes	—
167	#88322	—	2	1	no	—
168	#88322	—	2	1	no	—
169	#88322	—	4	1	no	—
170	#88322	—	4	1	no	—
171	#88322	#135	1	2	yes	—
172	#88322	#135	1	2	yes	—
173	#18069	—	53	1	no	maximal, Sylow
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	2	1	no	—
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×D₅₃	—	4	2	no	—
179	C₂×D₅₃	—	4	2	no	—
180	C₂×D₅₃	—	4	2	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
186	C₄×C₅₃	#568	1	1	yes	characteristic
187	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
188	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
189	Dic₅₃	—	4	2	no	—
190	Dic₅₃	—	4	2	no	—
191	Dic₅₃	—	4	2	no	—
192	#3168	—	53	1	no	—
193	#2969	—	53	2	no	—
194	#1591	—	53	1	no	—
195	#1589	—	53	1	no	—
196	#1583	—	53	2	no	—
197	#1582	—	53	2	no	—
198	C₈×Q₁₆	—	53	1	no	—
199	C₈×D₈	—	53	1	no	—
200	#1564	D₅₃	1	1	yes	characteristic, complemented
201	#1482	—	53	2	no	—
202	C₈×M₁₆	—	53	1	no	—
203	C₂×C₅₃	—	4	1	no	—
204	C₂×C₅₃	#2935	1	1	yes	characteristic
205	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
206	C₂×C₅₃	#2957	1	1	yes	characteristic
207	D₅₃	—	8	2	no	—
208	#529	—	53	2	no	—
209	#499	—	53	1	no	—
210	#495	—	53	1	no	—
211	#488	—	53	2	no	—
212	#467	—	53	1	no	—
213	#466	—	53	1	no	—
214	#465	—	53	2	no	—
215	#446	—	53	2	no	—
216	C₈×Q₈	—	53	2	no	—
217	C₄×Q₁₆	—	53	1	no	—
218	C₄×SD₁₆	—	53	2	no	—
219	C₄×D₈	—	53	1	no	—
220	#436	—	53	2	no	—
221	#436	—	106	2	no	—
222	#435	—	106	2	no	—
223	C₈×D₄	—	53	2	no	—
224	#433	—	106	2	no	—
225	#431	—	53	1	no	—
226	#431	C₂×D₅₃	1	2	yes	—
227	#405	—	106	2	no	—
228	C₄×M₁₆	—	53	1	no	—
229	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
230	#335	C₂×D₅₃	1	1	yes	characteristic
231	#335	C₂×D₅₃	1	1	yes	characteristic
232	#334	C₂×D₅₃	1	2	yes	—
233	Q₈⋊C₈	—	53	2	no	—
234	Q₈⋊C₈	—	53	2	no	—
235	D₄⋊C₈	—	53	2	no	—
236	D₄⋊C₈	—	53	2	no	—
237	#322	—	53	1	no	—
238	#322	—	53	1	no	—
239	C₈²	—	53	1	no	—
240	C₈²	—	53	1	no	—
241	C₅₃	#18069	1	1	yes	characteristic, complemented, Sylow
242	C₂×Q₁₆	—	53	1	no	—
243	C₂×SD₁₆	—	53	2	no	—
244	C₂×D₈	—	53	1	no	—
245	C₂×M₁₆	—	53	1	no	—
246	C₂×M₁₆	—	53	1	no	—
247	C₂×M₁₆	—	106	2	no	—
248	C₂²×C₈	—	106	2	no	—
249	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
250	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
251	#126	—	106	2	no	—
252	#125	—	53	2	no	—
253	#124	—	53	2	no	—
254	#123	—	106	2	no	—
255	C₂²⋊Q₈	—	53	2	no	—
256	#121	—	53	2	no	—
257	C₄×Q₈	—	53	2	no	—
258	C₄×D₄	—	53	2	no	—
259	C₄×D₄	—	106	2	no	—
260	#117	—	106	2	no	—
261	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
262	#107	—	53	1	no	—
263	#107	—	53	1	no	—
264	#106	—	53	2	no	—
265	C₄⋊C₈	—	106	2	no	—
266	C₄⋊C₈	—	106	2	no	—
267	C₄⋊C₈	—	106	1	no	—
268	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
269	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
270	#103	—	53	2	no	—
271	#103	—	53	2	no	—
272	#102	—	53	2	no	—
273	#102	—	53	2	no	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	106	2	no	—
276	C₂²⋊C₈	—	106	2	no	—
277	C₂²⋊C₈	—	106	1	no	—
278	#97	—	53	1	no	—
279	#97	—	53	1	no	—
280	#97	—	106	2	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	—	53	1	no	—
283	C₄×C₈	—	53	1	no	—
284	C₄×C₈	—	53	1	no	—
285	C₄×C₈	—	53	1	no	—
286	C₄×C₈	—	106	2	no	—
287	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
288	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
291	C₂×Q₈	—	53	2	no	—
292	C₂×D₄	—	53	2	no	—
293	C₂×D₄	—	106	2	no	—
294	C₂²×C₄	—	106	2	no	—
295	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
296	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
297	C₂²×C₄	#11719949	1	1	yes	characteristic
298	Q₁₆	—	106	2	no	—
299	SD₁₆	—	212	2	no	—
300	D₈	—	106	2	no	—
301	M₁₆	—	106	2	no	—
302	M₁₆	—	106	2	no	—
303	M₁₆	—	212	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	53	1	no	—
309	C₂×C₈	—	53	1	no	—
310	C₂×C₈	—	53	1	no	—
311	C₂×C₈	—	53	1	no	—
312	C₂×C₈	—	106	1	no	—
313	C₂×C₈	—	106	1	no	—
314	C₂×C₈	—	212	4	no	—
315	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
317	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
319	C₄⋊C₄	—	53	2	no	—
320	C₄⋊C₄	—	53	2	no	—
321	C₄⋊C₄	—	106	2	no	—
322	C₄⋊C₄	—	106	2	no	—
323	C₄⋊C₄	—	106	2	no	—
324	C₄⋊C₄	—	106	2	no	—
325	C₂²⋊C₄	—	106	2	no	—
326	C₂²⋊C₄	—	106	2	no	—
327	C₂²⋊C₄	—	106	2	no	—
328	C₂²⋊C₄	—	106	2	no	—
329	C₂²⋊C₄	—	106	2	no	—
330	C₄²	—	106	2	no	—
331	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
332	C₄²	D₄×D₅₃	1	1	yes	characteristic
333	C₄²	#11719949	1	1	yes	characteristic
334	C₄²	C₂³×D₅₃	1	1	yes	characteristic
335	C₂³	—	106	2	no	—
336	C₂³	#13133179	1	1	yes	characteristic
337	Q₈	—	106	2	no	—
338	Q₈	—	106	2	no	—
339	D₄	—	106	2	no	—
340	D₄	—	106	2	no	—
341	D₄	—	424	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	106	2	no	—
348	C₂×C₄	—	106	2	no	—
349	C₂×C₄	—	106	2	no	—
350	C₂×C₄	—	212	2	no	—
351	C₂×C₄	—	212	2	no	—
352	C₂×C₄	#13133179	1	1	yes	characteristic
353	C₂×C₄	#13133179	1	1	yes	characteristic
354	C₂×C₄	#13133179	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₈	—	53	2	no	—
364	C₈	—	53	2	no	—
365	C₈	—	106	1	no	—
366	C₈	—	106	1	no	—
367	C₈	—	212	1	no	—
368	C₈	—	212	1	no	—
369	C₈	#13133223	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	212	2	no	—
374	C₂²	—	212	2	no	—
375	C₂²	—	212	2	no	—
376	C₂²	C₂×#13133179	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	212	2	no	—
382	C₄	—	212	2	no	—
383	C₄	—	212	2	no	—
384	C₄	C₂×#13133223	1	1	yes	characteristic
385	C₄	#16811389	1	1	yes	characteristic
386	C₄	#135×D₅₃	1	1	yes	characteristic
387	C₄	#135×D₅₃	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	424	2	no	—
390	C₂	#20283458	1	1	yes	characteristic
391	C₂	#20283486	1	1	yes	characteristic
392	C₂	#20283631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810859	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#18069
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #20283713	C₂	semidirect
3. #20273611	C₂	semidirect
4. #20273599	C₂	semidirect
5. #20273557	C₂	semidirect
6. #20273550	C₂	semidirect
7. #20273521	C₂	semidirect
8. #20273509	C₂	semidirect
9. #20273121	C₂	semidirect
10. C₈×#11719917	C₂	semidirect
11. C₅₃×#1564	C₂	semidirect
200. #1564	D₅₃	semidirect
241. C₅₃	#18069	semidirect

Maximal subgroups

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

subgroup	index
#20285285	2
#20283713	2
#20273611	2
#20273599	2
#20273557	2
#20273550	2
#20273521	2
#20273509	2
#20273121	2
C₈×#11719917	2
C₅₃×#1564	2
#18069	53