Subgroups — #27356141

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 368 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	#20283877	C₂	1	2	yes	maximal, complemented
3	#20273619	C₂	1	2	yes	maximal, complemented
4	#20273567	C₂	1	2	yes	maximal, complemented
5	#20273546	C₂	1	2	yes	maximal, complemented
6	#20273529	C₂	1	2	yes	maximal, complemented
7	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273129	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811157	C₂²	1	2	yes	—
11	#16811025	C₄	1	1	yes	characteristic
12	#16811007	C₄	1	1	yes	characteristic
13	#16810978	C₄	1	2	yes	—
14	#16810864	C₄	1	1	yes	characteristic
15	#16810859	C₄	1	1	yes	characteristic
16	#16810852	C₄	1	2	yes	—
17	#16810753	C₂²	1	2	yes	—
18	C₈×Dic₁₀₆	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	C₄×#11719913	C₂²	1	2	yes	—
21	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
22	#16810702	C₂²	1	2	yes	—
23	#16810691	—	2	2	no	—
24	C₈×D₂₁₂	C₂²	1	2	yes	—
25	#16810686	—	2	2	no	—
26	#16810683	—	2	2	no	—
27	#16810648	—	2	2	no	—
28	C₂×C₄⋊#88322	C₂²	1	2	yes	—
29	#16810376	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	2	yes	—
31	#16810376	C₂²	1	1	yes	characteristic
32	#16810355	C₂²	1	2	yes	—
33	#16810355	C₂²	1	2	yes	—
34	#16810352	C₂²	1	2	yes	—
35	#16810352	C₂²	1	2	yes	—
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#322	C₂²	1	2	yes	—
40	C₅₃×#322	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂×#11719913	C₂×C₄	1	2	yes	—
43	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
44	C₂×#11719912	—	2	2	no	—
45	C₂²×#88322	—	2	2	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	#13133211	—	2	2	no	—
48	#13133209	C₂×C₄	1	2	yes	—
49	#13133205	C₂×C₄	1	2	yes	—
50	#13133200	—	2	2	no	—
51	#13133196	C₂×C₄	1	2	yes	—
52	#13133191	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₀₆	C₂³	1	2	yes	—
54	C₄×D₂₁₂	C₂³	1	2	yes	—
55	C₄×C₂²⋊D₅₃	—	2	2	no	—
56	#13133174	—	2	2	no	—
57	#13133151	C₂×C₄	1	1	yes	characteristic
58	#13133151	C₂×C₄	1	1	yes	characteristic
59	#13133149	C₂×C₄	1	2	yes	—
60	C₄⋊#88322	—	2	2	no	—
61	C₄⋊#88322	C₂³	1	2	yes	—
62	C₄⋊#88322	C₂³	1	2	yes	—
63	#13133146	—	2	2	no	—
64	#13133146	—	2	2	no	—
65	#13133142	C₂×C₄	1	2	yes	—
66	#13133142	C₂×C₄	1	2	yes	—
67	#13133139	C₂×C₄	1	2	yes	—
68	#13133139	C₂×C₄	1	2	yes	—
69	#13133129	—	2	2	no	—
70	#13133129	—	2	2	no	—
71	#13133127	—	2	2	no	—
72	C₈×Dic₅₃	—	2	2	no	—
73	C₂×M₁₆×C₅₃	—	2	1	no	—
74	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
79	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
80	#97×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	C₂³	1	2	yes	—
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
90	C₂×C₂²⋊D₅₃	—	2	2	no	—
91	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₅₃	—	2	2	no	—
93	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
94	M₁₆×C₅₃	—	4	2	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	4	1	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
104	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	#41	1	1	yes	characteristic
107	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₅₃	—	2	2	no	—
109	#11719921	—	2	2	no	—
110	#11719921	—	2	2	no	—
111	#11719921	—	2	2	no	—
112	#11719921	—	2	2	no	—
113	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
114	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
115	#11719919	—	2	2	no	—
116	#11719919	—	2	2	no	—
117	#11719919	—	2	2	no	—
118	#11719919	—	2	2	no	—
119	C₄×Dic₅₃	—	2	2	no	—
120	C₂×#88322	—	2	2	no	—
121	C₂×#88322	—	2	2	no	—
122	C₂×#88322	—	8	2	no	—
123	Dic₂₁₂	—	4	1	no	—
124	D₄₂₄	—	4	1	no	—
125	#11719913	—	2	4	no	—
126	#11719912	—	8	2	no	—
127	C₈×D₅₃	—	8	2	no	—
128	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂²×D₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
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	C₂²⋊D₅₃	—	4	4	no	—
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	—	2	2	no	—
144	C₂×Dic₅₃	—	2	2	no	—
145	D₂₁₂	—	2	2	no	—
146	D₂₁₂	—	2	2	no	—
147	C₄×D₅₃	—	4	2	no	—
148	C₄×D₅₃	—	4	2	no	—
149	Dic₁₀₆	—	2	2	no	—
150	Dic₁₀₆	—	2	2	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	2	1	no	—
154	C₈×C₅₃	—	2	2	no	—
155	C₈×C₅₃	—	2	2	no	—
156	C₈×C₅₃	—	4	2	no	—
157	C₈×C₅₃	#131	1	2	yes	—
158	#88322	—	4	2	no	—
159	#88322	—	4	2	no	—
160	#18092	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	#444	1	2	yes	—
164	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₅₃	—	4	2	no	—
166	C₂×D₅₃	—	4	2	no	—
167	C₂×D₅₃	—	4	2	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
173	C₄×C₅₃	#568	1	1	yes	characteristic
174	C₄×C₅₃	#576	1	1	yes	characteristic
175	C₄×C₅₃	#578	1	1	yes	characteristic
176	Dic₅₃	—	4	2	no	—
177	Dic₅₃	—	4	2	no	—
178	Dic₅₃	—	4	2	no	—
179	#3168	—	53	1	no	—
180	#2994	—	53	2	no	—
181	#1592	—	53	2	no	—
182	#1584	—	53	2	no	—
183	#1581	—	53	2	no	—
184	SD₁₆⋊C₈	—	53	2	no	—
185	C₂×#335	—	53	1	no	—
186	#1483	—	53	2	no	—
187	#1451	D₅₃	1	1	yes	characteristic, complemented
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	#2935	1	1	yes	characteristic
190	C₂×C₅₃	#2951	1	1	yes	characteristic
191	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
192	D₅₃	—	8	2	no	—
193	#529	—	53	2	no	—
194	#499	—	53	1	no	—
195	#495	—	53	1	no	—
196	#488	—	53	2	no	—
197	#467	—	53	1	no	—
198	#466	—	53	1	no	—
199	#465	—	53	2	no	—
200	#446	—	53	2	no	—
201	C₈×Q₈	—	53	2	no	—
202	C₄×Q₁₆	—	53	1	no	—
203	C₄×SD₁₆	—	53	2	no	—
204	C₄×D₈	—	53	1	no	—
205	#436	—	53	2	no	—
206	#435	—	106	2	no	—
207	C₈×D₄	—	53	2	no	—
208	C₈×D₄	—	106	2	no	—
209	#433	—	106	2	no	—
210	#431	C₂×D₅₃	1	1	yes	characteristic
211	#424	—	106	2	no	—
212	C₂×C₄⋊C₈	—	53	2	no	—
213	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
215	#335	—	53	1	no	—
216	#335	—	53	2	no	—
217	#335	—	53	1	no	—
218	Q₈⋊C₈	—	53	2	no	—
219	Q₈⋊C₈	—	53	2	no	—
220	D₄⋊C₈	—	53	2	no	—
221	D₄⋊C₈	—	53	2	no	—
222	#322	C₂×D₅₃	1	2	yes	—
223	#322	C₂×D₅₃	1	2	yes	—
224	C₅₃	#18092	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₁₆	—	53	1	no	—
226	C₂×SD₁₆	—	53	2	no	—
227	C₂×D₈	—	53	1	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	106	2	no	—
230	C₂²×C₈	—	106	2	no	—
231	C₂²×C₈	—	106	2	no	—
232	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
234	#126	—	106	2	no	—
235	#125	—	53	2	no	—
236	#124	—	53	2	no	—
237	#123	—	106	2	no	—
238	C₂²⋊Q₈	—	53	2	no	—
239	#121	—	53	2	no	—
240	C₄×Q₈	—	53	2	no	—
241	C₄×D₄	—	53	2	no	—
242	C₄×D₄	—	106	2	no	—
243	#117	—	106	2	no	—
244	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
245	#107	—	53	1	no	—
246	#107	—	53	1	no	—
247	#106	—	53	2	no	—
248	C₄⋊C₈	—	53	2	no	—
249	C₄⋊C₈	—	53	2	no	—
250	C₄⋊C₈	—	106	2	no	—
251	C₄⋊C₈	—	106	2	no	—
252	C₄⋊C₈	—	106	2	no	—
253	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
254	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
255	#103	—	53	2	no	—
256	#103	—	53	2	no	—
257	#102	—	53	2	no	—
258	#102	—	53	2	no	—
259	C₂²⋊C₈	—	106	2	no	—
260	C₂²⋊C₈	—	106	2	no	—
261	C₂²⋊C₈	D₂₁₂	1	2	yes	—
262	#97	—	2	1	no	—
263	#97	—	106	2	no	—
264	C₄×C₈	—	106	2	no	—
265	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
266	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₅₃	1	2	yes	—
268	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₅₃	1	2	yes	—
271	C₂×Q₈	—	53	2	no	—
272	C₂×D₄	—	53	2	no	—
273	C₂×D₄	—	106	2	no	—
274	C₂²×C₄	—	106	2	no	—
275	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
277	C₂²×C₄	#11719945	1	1	yes	characteristic
278	Q₁₆	—	212	1	no	—
279	SD₁₆	—	106	4	no	—
280	D₈	—	212	1	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	424	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	106	2	no	—
288	C₂×C₈	—	424	2	no	—
289	C₂×C₈	—	424	2	no	—
290	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
294	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
295	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
296	C₄⋊C₄	—	53	2	no	—
297	C₄⋊C₄	—	53	2	no	—
298	C₄⋊C₄	—	106	2	no	—
299	C₄⋊C₄	—	106	2	no	—
300	C₄⋊C₄	—	106	2	no	—
301	C₄⋊C₄	—	106	2	no	—
302	C₂²⋊C₄	—	106	2	no	—
303	C₂²⋊C₄	—	106	2	no	—
304	C₂²⋊C₄	—	106	2	no	—
305	C₂²⋊C₄	—	106	2	no	—
306	C₂²⋊C₄	—	106	2	no	—
307	C₄²	—	106	2	no	—
308	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
310	C₄²	#11719945	1	1	yes	characteristic
311	C₄²	C₂³×D₅₃	1	1	yes	characteristic
312	C₂³	—	106	2	no	—
313	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
314	Q₈	—	106	2	no	—
315	Q₈	—	106	2	no	—
316	D₄	—	106	2	no	—
317	D₄	—	106	2	no	—
318	D₄	—	212	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	212	2	no	—
328	C₂×C₄	—	212	2	no	—
329	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
330	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
331	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
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	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	212	2	no	—
342	C₈	—	212	2	no	—
343	C₈	#13133225	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	212	2	no	—
347	C₂²	—	212	2	no	—
348	C₂²	—	212	2	no	—
349	C₂²	#16810746	1	2	yes	—
350	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	212	2	no	—
356	C₄	—	212	2	no	—
357	C₄	—	212	2	no	—
358	C₄	C₂×#13133225	1	1	yes	characteristic
359	C₄	#16811389	1	1	yes	characteristic
360	C₄	#16811428	1	1	yes	characteristic
361	C₄	#16811440	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	424	2	no	—
364	C₂	#20283461	1	1	yes	characteristic
365	C₂	#20283592	1	1	yes	characteristic
366	C₂	C₂×#16810746	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810859	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18092
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #20283877	C₂	semidirect
3. #20273619	C₂	semidirect
4. #20273567	C₂	semidirect
5. #20273546	C₂	semidirect
6. #20273529	C₂	semidirect
7. C₂×#16810376	C₂	semidirect
8. #20273129	C₂	semidirect
9. C₅₃×#1451	C₂	semidirect
187. #1451	D₅₃	semidirect
224. C₅₃	#18092	semidirect

Maximal subgroups

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

subgroup	index
#20285285	2
#20283877	2
#20273619	2
#20273567	2
#20273546	2
#20273529	2
C₂×#16810376	2
#20273129	2
C₅₃×#1451	2
#18092	53