Subgroups — #27354833

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285274	C₂	1	1	yes	maximal, characteristic
2	#20283649	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283428	C₂	1	1	yes	maximal, characteristic
4	#20273595	C₂	1	4	yes	maximal, complemented
5	#20273574	C₂	1	4	yes	maximal, complemented
6	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273171	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273167	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133223	—	2	1	no	—
11	#16811180	C₂²	1	1	yes	characteristic
12	#16811173	C₂²	1	1	yes	characteristic
13	#16811021	C₄	1	1	yes	characteristic
14	#16810996	C₄	1	1	yes	characteristic
15	#16810974	C₄	1	1	yes	characteristic
16	#16810970	C₄	1	1	yes	characteristic
17	#16810852	C₄	1	4	yes	—
18	#16810753	—	2	1	no	—
19	#16810753	C₂²	1	2	yes	—
20	C₈×Dic₁₀₆	—	2	1	no	—
21	C₈×Dic₁₀₆	C₂²	1	2	yes	—
22	C₄×#11719913	C₂²	1	4	yes	—
23	#16810702	—	2	1	no	—
24	#16810702	C₂²	1	2	yes	—
25	C₈×D₂₁₂	—	2	1	no	—
26	C₈×D₂₁₂	C₂²	1	2	yes	—
27	#16810649	—	2	1	no	—
28	#16810647	C₂²	1	1	yes	characteristic
29	#16810647	C₂²	1	1	yes	characteristic
30	#16810589	—	2	1	no	—
31	#16810578	—	2	1	no	—
32	#16810376	C₂²	1	4	yes	—
33	#16810355	C₂²	1	4	yes	—
34	#16810352	C₂²	1	4	yes	—
35	C₂×C₄×C₈×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	C₅₃×#322	C₂²	1	2	yes	—
39	C₅₃×#322	C₂²	1	2	yes	—
40	C₂×#11719945	—	2	1	no	—
41	C₂×#11719913	C₂×C₄	1	4	yes	—
42	#13133223	—	4	4	no	—
43	C₂×#11719917	—	2	1	no	—
44	C₂×#11719917	—	2	1	no	—
45	C₂×#11719912	—	2	2	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	#13133217	C₂×C₄	1	1	yes	characteristic
48	#13133217	C₂×C₄	1	1	yes	characteristic
49	#13133217	C₂×C₄	1	1	yes	characteristic
50	#13133214	C₂×C₄	1	1	yes	characteristic
51	#13133209	—	2	1	no	—
52	#13133205	—	2	1	no	—
53	#13133196	—	2	1	no	—
54	#13133196	C₂×C₄	1	2	yes	—
55	#13133191	—	2	1	no	—
56	#13133191	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₀₆	—	2	1	no	—
58	C₄×Dic₁₀₆	C₂³	1	2	yes	—
59	C₄×D₂₁₂	—	2	1	no	—
60	C₄×D₂₁₂	C₂³	1	2	yes	—
61	#13133176	—	2	1	no	—
62	#13133149	C₂×C₄	1	4	yes	—
63	C₄⋊#88322	—	2	1	no	—
64	C₄⋊#88322	—	2	1	no	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	#13133146	—	2	2	no	—
68	#13133146	—	2	2	no	—
69	#13133142	C₂×C₄	1	4	yes	—
70	#13133139	C₂×C₄	1	4	yes	—
71	#13133129	—	2	2	no	—
72	#13133129	—	2	2	no	—
73	#13133127	—	2	2	no	—
74	C₈×Dic₅₃	—	2	2	no	—
75	C₂²×C₈×C₅₃	—	2	1	no	—
76	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	C₄×C₈×C₅₃	C₂³	1	2	yes	—
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
92	C₂×C₂²⋊D₅₃	—	2	2	no	—
93	#11719945	—	4	2	no	—
94	#11719945	—	4	2	no	—
95	C₂×D₂₁₂	—	2	1	no	—
96	C₂×D₂₁₂	—	2	1	no	—
97	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₅₃	—	2	2	no	—
99	C₂×Dic₁₀₆	—	2	1	no	—
100	C₂×Dic₁₀₆	—	2	1	no	—
101	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₃	—	2	4	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	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	#41	1	1	yes	characteristic
116	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₂²⋊Dic₅₃	—	2	2	no	—
118	#11719921	—	2	2	no	—
119	#11719921	—	2	2	no	—
120	C₄⋊Dic₅₃	—	2	1	no	—
121	C₄⋊Dic₅₃	—	2	1	no	—
122	C₄⋊Dic₅₃	—	2	1	no	—
123	C₄⋊Dic₅₃	—	2	1	no	—
124	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
125	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
126	#11719919	—	2	2	no	—
127	#11719919	—	2	2	no	—
128	C₄×Dic₅₃	—	2	2	no	—
129	#11719917	—	4	2	no	—
130	#11719917	—	4	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	2	2	no	—
133	#11719913	—	4	4	no	—
134	#11719912	—	4	4	no	—
135	C₈×D₅₃	—	4	4	no	—
136	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₅₃	#137	1	2	yes	—
145	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₅₃	—	4	4	no	—
149	C₂×Dic₅₃	—	2	2	no	—
150	C₂×Dic₅₃	—	2	2	no	—
151	C₂×Dic₅₃	—	2	2	no	—
152	D₂₁₂	—	2	4	no	—
153	D₂₁₂	—	4	2	no	—
154	D₂₁₂	—	4	2	no	—
155	C₄×D₅₃	—	4	2	no	—
156	C₄×D₅₃	—	4	2	no	—
157	Dic₁₀₆	—	2	4	no	—
158	Dic₁₀₆	—	4	2	no	—
159	Dic₁₀₆	—	4	2	no	—
160	C₈×C₅₃	—	2	4	no	—
161	C₈×C₅₃	—	2	2	no	—
162	C₈×C₅₃	—	2	2	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	—	2	1	no	—
166	C₈×C₅₃	—	2	1	no	—
167	#88322	—	4	2	no	—
168	#88322	—	4	2	no	—
169	#17979	—	53	1	no	maximal, Sylow
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	#440	1	2	yes	—
173	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₅₃	—	4	2	no	—
175	C₂×D₅₃	—	4	2	no	—
176	C₂×D₅₃	—	4	2	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	#440	1	2	yes	—
180	C₄×C₅₃	#443	1	2	yes	—
181	C₄×C₅₃	#568	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	Dic₅₃	—	4	2	no	—
186	Dic₅₃	—	4	2	no	—
187	#3167	—	53	1	no	—
188	#2960	—	53	1	no	—
189	#2931	—	53	1	no	—
190	#1588	—	53	4	no	—
191	#1585	—	53	4	no	—
192	#1567	—	53	1	no	—
193	#1493	—	53	1	no	—
194	#1492	—	53	1	no	—
195	C₂×#322	D₅₃	1	1	yes	characteristic, complemented
196	C₂×C₅₃	—	2	2	no	—
197	C₂×C₅₃	#2932	1	1	yes	characteristic
198	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
200	D₅₃	—	8	2	no	—
201	C₂×#131	—	106	1	no	—
202	#533	—	53	1	no	—
203	#532	—	53	1	no	—
204	#498	—	53	1	no	—
205	#492	—	53	1	no	—
206	#487	—	53	1	no	—
207	#486	—	53	1	no	—
208	#465	—	53	4	no	—
209	#446	—	53	2	no	—
210	#446	—	106	1	no	—
211	C₈×Q₈	—	53	2	no	—
212	C₈×Q₈	—	106	1	no	—
213	C₄×SD₁₆	—	53	4	no	—
214	#436	—	53	2	no	—
215	#436	—	106	1	no	—
216	C₈×D₄	—	53	2	no	—
217	C₈×D₄	—	106	1	no	—
218	#424	—	106	1	no	—
219	#423	—	53	1	no	—
220	#423	—	53	1	no	—
221	#408	—	106	1	no	—
222	#405	—	106	1	no	—
223	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
224	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
226	#335	—	53	4	no	—
227	Q₈⋊C₈	—	53	4	no	—
228	D₄⋊C₈	—	53	4	no	—
229	#322	C₂×D₅₃	1	2	yes	—
230	#322	C₂×D₅₃	1	2	yes	—
231	C₅₃	#17979	1	1	yes	characteristic, complemented, Sylow
232	C₂×#41	—	106	1	no	—
233	C₂×SD₁₆	—	53	4	no	—
234	#131	—	212	4	no	—
235	C₂×M₁₆	—	106	1	no	—
236	C₂×M₁₆	—	106	1	no	—
237	C₂×M₁₆	—	106	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	106	2	no	—
240	C₂²×C₈	C₄×D₅₃	1	2	yes	—
241	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
242	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
243	C₄⋊Q₈	—	53	1	no	—
244	C₄⋊Q₈	—	53	1	no	—
245	C₄⋊Q₈	—	53	1	no	—
246	#127	—	53	1	no	—
247	#125	—	106	1	no	—
248	#124	—	106	1	no	—
249	C₂²⋊Q₈	—	53	2	no	—
250	C₂²⋊Q₈	—	106	1	no	—
251	#121	—	53	2	no	—
252	#121	—	106	1	no	—
253	C₄×Q₈	—	53	2	no	—
254	C₄×Q₈	—	106	1	no	—
255	C₄×D₄	—	53	2	no	—
256	C₄×D₄	—	106	1	no	—
257	#117	—	106	1	no	—
258	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
259	#106	—	53	4	no	—
260	C₄⋊C₈	—	53	2	no	—
261	C₄⋊C₈	—	53	2	no	—
262	C₄⋊C₈	—	106	2	no	—
263	C₄⋊C₈	—	106	2	no	—
264	C₄⋊C₈	—	106	1	no	—
265	C₄⋊C₈	—	106	1	no	—
266	#103	—	53	4	no	—
267	#102	—	53	4	no	—
268	C₂²⋊C₈	—	106	2	no	—
269	C₂²⋊C₈	—	106	2	no	—
270	#97	—	106	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	106	2	no	—
273	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
274	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
275	C₄×C₈	D₂₁₂	1	1	yes	characteristic
276	C₄×C₈	D₂₁₂	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₅₃	1	2	yes	—
278	C₄×C₈	C₂²×D₅₃	1	2	yes	—
279	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
281	#41	—	212	2	no	—
282	#41	—	212	2	no	—
283	C₂×Q₈	—	53	2	no	—
284	C₂×Q₈	—	106	1	no	—
285	C₂×Q₈	—	106	1	no	—
286	C₂×D₄	—	53	2	no	—
287	C₂×D₄	—	106	2	no	—
288	C₂×D₄	—	106	1	no	—
289	C₂×D₄	—	106	1	no	—
290	C₂²×C₄	—	106	2	no	—
291	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
293	C₂²×C₄	#11719945	1	1	yes	characteristic
294	SD₁₆	—	212	4	no	—
295	M₁₆	—	212	4	no	—
296	M₁₆	—	212	2	no	—
297	M₁₆	—	212	2	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	—	212	4	no	—
306	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
307	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
308	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
309	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
310	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
311	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
312	C₄⋊C₄	—	53	2	no	—
313	C₄⋊C₄	—	53	2	no	—
314	C₄⋊C₄	—	106	1	no	—
315	C₄⋊C₄	—	106	1	no	—
316	C₄⋊C₄	—	106	1	no	—
317	C₄⋊C₄	—	106	2	no	—
318	C₄⋊C₄	—	106	1	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₂²⋊C₄	—	106	2	no	—
321	C₂²⋊C₄	—	106	2	no	—
322	C₂²⋊C₄	—	106	2	no	—
323	C₄²	—	106	2	no	—
324	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
325	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
326	C₄²	#11719945	1	1	yes	characteristic
327	C₄²	C₂³×D₅₃	1	1	yes	characteristic
328	C₂³	—	106	2	no	—
329	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
330	Q₈	—	106	4	no	—
331	Q₈	—	212	2	no	—
332	Q₈	—	212	2	no	—
333	D₄	—	106	4	no	—
334	D₄	—	212	4	no	—
335	D₄	—	212	2	no	—
336	D₄	—	212	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	106	2	no	—
340	C₂×C₄	—	106	2	no	—
341	C₂×C₄	—	106	2	no	—
342	C₂×C₄	—	212	2	no	—
343	C₂×C₄	—	212	2	no	—
344	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
345	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
346	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
347	C₂×C₄	#13133245	1	2	yes	—
348	C₂×C₄	#13133250	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
351	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
352	C₈	—	2	4	no	—
353	C₈	—	2	2	no	—
354	C₈	—	2	2	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	212	2	no	—
360	C₈	—	212	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	212	2	no	—
364	C₂²	—	212	2	no	—
365	C₂²	—	212	2	no	—
366	C₂²	#16810725	1	2	yes	—
367	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	212	2	no	—
371	C₄	—	212	2	no	—
372	C₄	—	212	2	no	—
373	C₄	#16810725	1	2	yes	—
374	C₄	#16810741	1	2	yes	—
375	C₄	#16811389	1	2	yes	—
376	C₄	C₂×#13133245	1	1	yes	characteristic
377	C₄	C₂×#13133250	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	424	2	no	—
380	C₂	#20283434	1	1	yes	characteristic
381	C₂	C₂×#16810725	1	1	yes	characteristic
382	C₂	C₂×#16810741	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16811173	C₂²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#17979
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #20283649	C₂	semidirect
4. #20273595	C₂	semidirect
5. #20273574	C₂	semidirect
6. #20273474	C₂	semidirect
7. #20273171	C₂	semidirect
9. C₂×C₅₃×#322	C₂	semidirect
195. C₂×#322	D₅₃	semidirect
231. C₅₃	#17979	semidirect

Maximal subgroups

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

subgroup	index
#20285274	2
#20283649	2
#20283428	2
#20273595	2
#20273574	2
#20273474	2
#20273171	2
#20273167	2
C₂×C₅₃×#322	2
#17979	53