Subgroups — #28327926

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	#20290938	C₂	1	2	yes	maximal, complemented
3	#20290937	C₂	1	4	yes	maximal, complemented
4	#20290936	C₂	1	2	yes	maximal, complemented
5	#20290932	C₂	1	4	yes	maximal, complemented
6	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133292	C₄	1	4	yes	complemented
8	D₅₃⋊M₃₂	C₄	1	4	yes	complemented
9	D₅₃⋊M₃₂	C₄	1	4	yes	complemented
10	D₅₃⋊M₃₂	C₂²	1	4	yes	—
11	D₅₃⋊M₃₂	C₂²	1	4	yes	—
12	C₂×#13133291	C₂²	1	4	yes	—
13	C₂×D₅₃⋊C₁₆	C₄	1	4	yes	complemented
14	#16811516	C₂²	1	8	yes	complemented
15	#16811515	C₂²	1	8	yes	complemented
16	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	#117×D₅₃	C₄	1	1	yes	characteristic
19	#16810581	C₂²	1	1	yes	characteristic
20	#16810580	C₄	1	2	yes	—
21	#16810579	C₂²	1	1	yes	characteristic
22	#16810578	C₄	1	1	yes	characteristic
23	#16810577	C₂²	1	2	yes	—
24	#97×D₅₃	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₄	1	2	yes	—
26	C₅₃×#405	C₄	1	1	yes	characteristic
27	#13133292	C₂×C₄	1	8	yes	—
28	#13133292	C₂×C₄	1	8	yes	complemented
29	#13133291	C₂×C₄	1	8	yes	complemented
30	#13133291	C₂³	1	8	yes	—
31	D₅₃⋊C₁₆	C₂×C₄	1	8	yes	—
32	D₅₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
33	C₂×#11719910	C₂×C₄	1	8	yes	complemented
34	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
36	C₂×#11719912	C₂×C₄	1	2	yes	—
37	M₁₆×D₅₃	—	2	2	no	—
38	M₁₆×D₅₃	—	2	2	no	—
39	C₂²×#88322	C₂³	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
49	#13133176	C₂×C₄	1	1	yes	characteristic
50	#13133175	—	2	1	no	—
51	#13133174	C₂×C₄	1	2	yes	—
52	#13133173	—	2	1	no	—
53	C₄⋊C₄×D₅₃	—	2	1	no	—
54	C₂²⋊C₄×D₅₃	—	2	1	no	—
55	C₄²×D₅₃	C₂×C₄	1	2	yes	—
56	#13133128	C₂³	1	2	yes	—
57	#13133127	C₂×C₄	1	2	yes	—
58	#13133127	C₂³	1	2	yes	—
59	C₄×#88322	C₂×C₄	1	2	yes	—
60	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
62	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#97×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
67	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
70	C₂×C₄×D₅₃	—	2	2	no	—
71	C₂×C₄×D₅₃	C₄²	1	2	yes	—
72	C₂×C₄×D₅₃	C₄²	1	2	yes	—
73	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
74	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
76	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
77	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
79	M₁₆×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	C₄²	1	2	yes	—
81	C₂×C₈×C₅₃	C₄²	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₂²⋊C₄×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²⋊Dic₅₃	—	2	1	no	—
89	#11719921	—	2	2	no	—
90	C₄⋊Dic₅₃	—	2	1	no	—
91	#11719919	—	2	2	no	—
92	C₄×Dic₅₃	—	2	2	no	—
93	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
94	#11719917	—	2	2	no	—
95	C₂×#88322	C₂²×C₄	1	2	yes	—
96	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
97	C₂×#88322	C₂²×C₄	1	2	yes	—
98	C₂×#88322	C₂²×C₄	1	2	yes	—
99	C₂×#88322	C₂⁴	1	1	yes	characteristic
100	#11719912	—	2	4	no	—
101	C₈×D₅₃	—	2	2	no	—
102	C₈×D₅₃	—	2	2	no	—
103	C₈×D₅₃	C₄²	1	2	yes	—
104	C₈×D₅₃	C₄²	1	1	yes	characteristic
105	C₈×D₅₃	C₄²	1	2	yes	—
106	C₈×D₅₃	C₄²	1	1	yes	characteristic
107	C₈×D₅₃	C₄²	1	1	yes	characteristic
108	C₈×D₅₃	C₄²	1	1	yes	characteristic
109	C₈×D₅₃	C₂²×C₄	1	2	yes	—
110	C₈×D₅₃	C₂²×C₄	1	2	yes	—
111	C₈×D₅₃	C₂²×C₄	1	2	yes	—
112	C₈×D₅₃	C₂²×C₄	1	2	yes	—
113	#11719910	C₂²×C₄	1	16	yes	—
114	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂²×D₅₃	—	4	2	no	—
116	C₂²×D₅₃	#97	1	2	yes	—
117	C₂²×D₅₃	#97	1	2	yes	—
118	C₂²×D₅₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂²×D₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	#97	1	2	yes	—
127	C₂×Dic₅₃	#97	1	2	yes	—
128	C₂×Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
130	C₄×D₅₃	—	4	4	no	—
131	C₄×D₅₃	#97	1	2	yes	—
132	C₄×D₅₃	#97	1	2	yes	—
133	C₄×D₅₃	#97	1	2	yes	—
134	C₄×D₅₃	#97	1	2	yes	—
135	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
136	C₄×D₅₃	C₂×C₄²	1	2	yes	—
137	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
138	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
139	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
140	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	C₂×C₄²	1	2	yes	—
143	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
144	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
145	#88322	—	2	2	no	—
146	#88322	C₂×C₄²	1	1	yes	characteristic
147	#88322	C₂×C₄²	1	1	yes	characteristic
148	#88322	C₂³×C₄	1	2	yes	—
149	C₂×#2110	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	4	1	no	—
151	C₂²×C₅₃	C₂×#97	1	2	yes	—
152	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	4	no	—
155	C₂×D₅₃	—	4	2	no	—
156	C₂×D₅₃	—	4	1	no	—
157	C₂×D₅₃	C₂×#97	1	1	yes	characteristic
158	C₂×D₅₃	C₂×#97	1	2	yes	—
159	C₂×D₅₃	C₂×#97	1	1	yes	characteristic
160	C₄×C₅₃	—	4	2	no	—
161	C₄×C₅₃	C₂×#97	1	2	yes	—
162	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
163	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
164	Dic₅₃	—	4	2	no	—
165	Dic₅₃	C₂×#97	1	1	yes	characteristic
166	Dic₅₃	C₂×#97	1	1	yes	characteristic
167	Dic₅₃	C₂×#97	1	2	yes	—
168	C₂²×M₃₂	—	53	2	no	—
169	C₂×#405	—	53	1	no	—
170	#2110	—	53	2	no	—
171	#2110	—	53	4	no	—
172	#2110	—	53	2	no	—
173	C₂×#347	—	53	4	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	—	2	2	no	—
176	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
177	D₅₃	—	2	1	no	—
178	D₅₃	—	2	2	no	—
179	D₅₃	#2110	1	2	yes	complemented
180	C₂²×M₁₆	—	53	1	no	—
181	C₂³×C₈	—	53	1	no	—
182	C₂×#117	—	53	1	no	—
183	C₂×M₃₂	—	53	4	no	—
184	C₂×M₃₂	—	53	4	no	—
185	C₂×M₃₂	—	53	4	no	—
186	C₂×M₃₂	—	53	4	no	—
187	C₂×M₃₂	—	53	4	no	—
188	C₂×M₃₂	—	53	4	no	—
189	C₂²×C₁₆	—	53	4	no	—
190	#405	—	53	1	no	—
191	#405	—	53	1	no	—
192	#405	—	53	1	no	—
193	#405	—	53	2	no	—
194	#405	—	53	2	no	—
195	#405	#6157	1	1	yes	characteristic
196	C₂×#97	—	53	2	no	—
197	C₂×C₄×C₈	—	53	2	no	—
198	#347	—	53	8	no	—
199	#347	—	53	8	no	—
200	C₅₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₄	—	53	1	no	—
202	C₂×M₁₆	—	53	1	no	—
203	C₂×M₁₆	—	53	2	no	—
204	C₂×M₁₆	—	106	2	no	—
205	C₂×M₁₆	—	106	2	no	—
206	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	53	2	no	—
209	C₂²×C₈	—	53	2	no	—
210	C₂²×C₈	—	53	2	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	1	no	—
214	C₂²×C₈	—	53	1	no	—
215	C₂²×C₈	—	53	1	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
218	#117	—	53	1	no	—
219	#117	—	53	2	no	—
220	#117	—	106	1	no	—
221	#117	—	106	1	no	—
222	#117	C₂×#6157	1	1	yes	characteristic
223	C₂×C₄⋊C₄	—	106	1	no	—
224	C₂×C₂²⋊C₄	—	106	1	no	—
225	C₂×C₄²	—	53	2	no	—
226	M₃₂	—	53	8	no	—
227	M₃₂	—	53	8	no	—
228	M₃₂	—	53	8	no	—
229	M₃₂	—	53	8	no	—
230	C₂×C₁₆	—	53	8	no	—
231	C₂×C₁₆	—	53	8	no	—
232	C₂×C₁₆	—	53	8	no	—
233	#97	—	53	2	no	—
234	#97	—	53	2	no	—
235	#97	—	53	2	no	—
236	#97	C₂×#6157	1	2	yes	—
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	53	2	no	—
239	C₄×C₈	—	53	2	no	—
240	C₄×C₈	C₂×#6157	1	2	yes	—
241	C₂⁴	—	53	1	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂²×C₄	—	53	2	no	—
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	—	53	1	no	—
251	C₂²×C₄	—	106	2	no	—
252	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
253	M₁₆	—	2	2	no	—
254	M₁₆	—	106	4	no	—
255	M₁₆	—	106	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	53	1	no	—
260	C₂×C₈	—	53	2	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	53	1	no	—
265	C₂×C₈	—	53	1	no	—
266	C₂×C₈	—	53	2	no	—
267	C₂×C₈	—	53	2	no	—
268	C₂×C₈	—	53	1	no	—
269	C₂×C₈	—	53	1	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	106	2	no	—
272	C₂×C₈	—	106	2	no	—
273	C₂×C₈	C₄×#6157	1	2	yes	—
274	C₂×C₈	C₄×#6157	1	2	yes	—
275	C₂×C₈	C₂²×#6157	1	2	yes	—
276	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
277	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	106	2	no	—
280	C₄⋊C₄	—	106	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	106	2	no	—
283	C₂²⋊C₄	—	106	1	no	—
284	C₄²	—	53	2	no	—
285	C₄²	—	106	2	no	—
286	C₄²	C₂²×#6157	1	2	yes	—
287	C₁₆	—	53	16	no	—
288	C₂³	—	53	2	no	—
289	C₂³	—	53	1	no	—
290	C₂³	—	53	2	no	—
291	C₂³	—	53	1	no	—
292	C₂³	—	212	2	no	—
293	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	53	1	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	53	2	no	—
306	C₂×C₄	—	53	1	no	—
307	C₂×C₄	—	53	1	no	—
308	C₂×C₄	—	53	2	no	—
309	C₂×C₄	—	106	2	no	—
310	C₂×C₄	—	212	4	no	—
311	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
312	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
313	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
314	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	53	1	no	—
317	C₈	—	53	1	no	—
318	C₈	—	53	2	no	—
319	C₈	—	106	2	no	—
320	C₈	C₂×C₄×#6157	1	2	yes	—
321	C₈	C₂×C₄×#6157	1	1	yes	characteristic
322	C₈	C₂×C₄×#6157	1	1	yes	characteristic
323	C₂²	—	4	1	no	—
324	C₂²	—	53	1	no	—
325	C₂²	—	53	1	no	—
326	C₂²	—	53	2	no	—
327	C₂²	—	106	2	no	—
328	C₂²	—	106	4	no	—
329	C₂²	—	212	1	no	—
330	C₂²	—	212	2	no	—
331	C₂²	#16811582	1	2	yes	—
332	C₂²	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
333	C₄	—	4	2	no	—
334	C₄	—	53	1	no	—
335	C₄	—	53	1	no	—
336	C₄	—	53	2	no	—
337	C₄	—	212	2	no	—
338	C₄	#16811582	1	2	yes	—
339	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
340	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	2	no	—
343	C₂	—	53	2	no	—
344	C₂	—	106	1	no	—
345	C₂	—	106	2	no	—
346	C₂	C₂×#16811582	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂×C₅₃	C₂²×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×#88322	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	C₂×#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20290938	C₂	semidirect
3. #20290937	C₂	semidirect
4. #20290936	C₂	semidirect
5. #20290932	C₂	semidirect
7. C₂×#13133292	C₄	semidirect
8. D₅₃⋊M₃₂	C₄	semidirect
9. D₅₃⋊M₃₂	C₄	semidirect
13. C₂×D₅₃⋊C₁₆	C₄	semidirect
14. #16811516	C₂²	semidirect
15. #16811515	C₂²	semidirect
28. #13133292	C₂×C₄	semidirect
29. #13133291	C₂×C₄	semidirect
32. D₅₃⋊C₁₆	C₂×C₄	semidirect
33. C₂×#11719910	C₂×C₄	semidirect
179. D₅₃	#2110	semidirect
200. C₅₃	C₂×#2110	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊M₃₂	2
#20290938	2
#20290937	2
#20290936	2
#20290932	2
#405×D₅₃	2
C₂×#2110	53