Subgroups — #27284088

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	2	yes	maximal
4	#20272556	C₂	1	2	yes	maximal
5	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133128	—	2	1	no	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#389	C₄	1	1	yes	characteristic
14	C₅₃×#389	C₄	1	1	yes	characteristic
15	C₅₃×#387	—	2	1	no	—
16	C₅₃×#387	—	2	1	no	—
17	C₅₃×#382	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	4	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	Q₈	1	4	yes	—
28	#13133128	—	4	2	no	—
29	C₄×#88322	—	2	4	no	—
30	#125×C₅₃	—	2	2	no	—
31	#125×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#95×C₅₃	—	2	1	no	—
50	#95×C₅₃	—	2	1	no	—
51	#95×C₅₃	—	2	1	no	—
52	#95×C₅₃	—	2	1	no	—
53	#95×C₅₃	—	2	1	no	—
54	#95×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	4	2	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₄⋊C₄×C₅₃	M₁₆	1	2	yes	—
75	C₄⋊C₄×C₅₃	M₁₆	1	2	yes	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	—	4	2	no	—
85	C₄²×C₅₃	—	4	1	no	—
86	C₄²×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
88	C₄²×C₅₃	#41	1	2	yes	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	4	2	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	—	4	1	no	—
121	C₂×C₄×C₅₃	#104	1	2	yes	—
122	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#131	1	2	yes	—
129	C₂×C₄×C₅₃	#135	1	2	yes	—
130	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
132	#88322	—	4	4	no	—
133	#88322	—	4	4	no	—
134	#9867	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	#391	1	1	yes	characteristic
136	C₂²×C₅₃	#407	1	1	yes	characteristic
137	C₂²×C₅₃	#408	1	1	yes	characteristic
138	C₂²×C₅₃	#416	1	1	yes	characteristic
139	C₂²×C₅₃	#417	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
141	C₂²×C₅₃	#421	1	1	yes	characteristic
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	2	no	—
145	C₄×C₅₃	—	4	1	no	—
146	C₄×C₅₃	—	4	2	no	—
147	C₄×C₅₃	—	4	1	no	—
148	C₄×C₅₃	—	4	2	no	—
149	C₄×C₅₃	—	4	1	no	—
150	C₄×C₅₃	—	8	1	no	—
151	C₄×C₅₃	—	8	1	no	—
152	C₄×C₅₃	#446	1	2	yes	—
153	C₄×C₅₃	#446	1	2	yes	—
154	C₄×C₅₃	#478	1	2	yes	—
155	C₄×C₅₃	#479	1	2	yes	—
156	C₄×#125	D₅₃	1	1	yes	characteristic
157	#1767	—	53	1	no	—
158	C₄×C₄⋊C₈	—	53	1	no	—
159	#1279	—	53	2	no	—
160	#1279	—	53	2	no	—
161	C₂×C₅₃	#1488	1	1	yes	characteristic
162	C₂×C₅₃	#1489	1	1	yes	characteristic
163	C₂×C₅₃	#1494	1	1	yes	characteristic
164	C₂×C₅₃	#1495	1	1	yes	characteristic
165	C₂×C₅₃	#1985	1	1	yes	characteristic
166	C₂×C₅₃	#1986	1	1	yes	characteristic
167	C₂×C₅₃	#1989	1	1	yes	characteristic
168	C₂×#125	Dic₅₃	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	53	2	no	—
170	C₂×C₄⋊C₈	—	106	1	no	—
171	C₂×#97	—	106	1	no	—
172	C₂×C₄×C₈	—	53	2	no	—
173	#389	Dic₅₃	1	1	yes	characteristic
174	#389	Dic₅₃	1	1	yes	characteristic
175	#387	—	2	1	no	—
176	#387	—	2	1	no	—
177	#382	Dic₅₃	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	2	1	no	—
179	C₄×C₄⋊C₄	—	2	1	no	—
180	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
181	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
182	C₄³	C₂×D₅₃	1	1	yes	characteristic
183	#336	—	106	1	no	—
184	#336	—	106	1	no	—
185	C₅₃	#9867	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₈	—	106	2	no	—
187	C₂²×C₈	—	106	2	no	—
188	#125	—	2	2	no	—
189	#125	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
201	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
207	C₄⋊C₈	—	53	4	no	—
208	C₄⋊C₈	—	106	4	no	—
209	C₄⋊C₈	—	106	2	no	—
210	#97	—	212	2	no	—
211	C₄×C₈	—	106	4	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	—	212	2	no	—
233	C₂×C₈	—	212	2	no	—
234	C₂×C₈	—	212	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	#11719917	1	2	yes	—
244	C₄⋊C₄	#11719917	1	2	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	1	no	—
256	C₄²	Q₈×D₅₃	1	2	yes	—
257	C₄²	#11719949	1	2	yes	—
258	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	#13133144	1	2	yes	—
285	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
286	C₂×C₄	#13133193	1	2	yes	—
287	C₂×C₄	#13133206	1	1	yes	characteristic
288	C₂×C₄	#13133215	1	1	yes	characteristic
289	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
291	C₂×C₄	#13133224	1	2	yes	—
292	C₂×C₄	#13133237	1	2	yes	—
293	C₂×C₄	#13133242	1	1	yes	characteristic
294	C₂×C₄	#13133247	1	1	yes	characteristic
295	C₈	—	212	4	no	—
296	C₈	—	212	4	no	—
297	C₂²	#16810530	1	1	yes	characteristic
298	C₂²	#16810587	1	1	yes	characteristic
299	C₂²	#16810590	1	1	yes	characteristic
300	C₂²	#16810619	1	1	yes	characteristic
301	C₂²	#16810624	1	1	yes	characteristic
302	C₂²	C₂×#13133144	1	1	yes	characteristic
303	C₂²	#16810640	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	#16810754	1	2	yes	—
315	C₄	#16810754	1	2	yes	—
316	C₄	#16810920	1	2	yes	—
317	C₄	#16810927	1	2	yes	—
318	C₂	#20273149	1	1	yes	characteristic
319	C₂	#20273154	1	1	yes	characteristic
320	C₂	#20273175	1	1	yes	characteristic
321	C₂	#20273180	1	1	yes	characteristic
322	C₂	#20275656	1	1	yes	characteristic
323	C₂	#20275660	1	1	yes	characteristic
324	C₂	#20275673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#9867
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
185. C₅₃	#9867	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₄×C₄⋊#88322	2
#20272556	2
C₄×#125×C₅₃	2
#9867	53