Subgroups — #27300325

Order 13568.

416 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283133	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810405	C₂	1	1	yes	maximal, characteristic
3	#20272802	C₂	1	2	yes	maximal, complemented
4	#20272800	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	1	yes	characteristic
7	C₂×#13133176	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16810581	—	2	1	no	—
10	#16810581	—	2	1	no	—
11	#16810578	C₄	1	1	yes	characteristic
12	#16810578	C₄	1	1	yes	characteristic
13	#16810578	C₂²	1	1	yes	characteristic
14	#16810578	C₂²	1	1	yes	characteristic
15	M₁₆×Dic₅₃	—	2	1	no	—
16	M₁₆×Dic₅₃	—	2	1	no	—
17	#16810405	—	2	1	no	—
18	#16810405	—	2	1	no	—
19	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#349	—	2	1	no	—
22	C₅₃×#349	—	2	1	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	1	no	—
26	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
27	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	—	2	1	no	—
29	#13133176	—	2	1	no	—
30	#13133176	—	2	1	no	—
31	#13133176	C₈	1	2	yes	—
32	#13133176	C₂×C₄	1	1	yes	characteristic
33	#13133176	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅₃	—	2	1	no	—
35	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
36	C₂×C₄×Dic₅₃	—	2	1	no	—
37	#13133157	—	2	2	no	—
38	#13133157	—	2	2	no	—
39	C₂×#11719908	—	2	2	no	—
40	#13133127	—	2	1	no	—
41	#13133127	—	2	1	no	—
42	#13133127	—	2	1	no	—
43	#13133127	—	2	1	no	—
44	C₈×Dic₅₃	—	2	1	no	—
45	C₈×Dic₅₃	—	2	1	no	—
46	C₈×Dic₅₃	—	2	1	no	—
47	C₈×Dic₅₃	—	2	1	no	—
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
52	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	M₃₂×C₅₃	—	2	2	no	—
58	M₃₂×C₅₃	—	2	2	no	—
59	C₂×C₁₆×C₅₃	—	2	2	no	—
60	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	—	2	2	no	—
72	M₁₆×C₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	2	yes	—
83	C₂×C₈×C₅₃	#41	1	2	yes	—
84	C₂²⋊Dic₅₃	—	2	2	no	—
85	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
86	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	1	no	—
89	C₄⋊Dic₅₃	—	2	1	no	—
90	C₄×Dic₅₃	—	2	2	no	—
91	C₄×Dic₅₃	—	2	1	no	—
92	C₄×Dic₅₃	—	2	1	no	—
93	#11719917	—	2	4	no	—
94	#11719917	—	2	1	no	—
95	#11719917	—	2	1	no	—
96	#11719917	M₁₆	1	2	yes	—
97	#11719917	M₁₆	1	2	yes	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	2	1	no	—
100	C₂×#88322	—	2	1	no	—
101	C₁₆×C₅₃	—	4	2	no	—
102	#11719908	—	4	2	no	—
103	C₂³×C₅₃	—	2	1	no	—
104	C₂³×C₅₃	—	2	1	no	—
105	C₂³×C₅₃	—	8	1	no	—
106	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	8	2	no	—
130	C₈×C₅₃	—	2	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	#123	1	2	yes	—
133	C₈×C₅₃	#123	1	2	yes	—
134	#88322	—	2	2	no	—
135	#88322	—	2	2	no	—
136	#12851	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	4	1	no	—
140	C₂²×C₅₃	—	4	1	no	—
141	C₂²×C₅₃	—	4	1	no	—
142	C₂²×C₅₃	—	8	1	no	—
143	C₂²×C₅₃	—	8	1	no	—
144	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₅₃	#431	1	1	yes	characteristic
146	C₂²×C₅₃	#431	1	1	yes	characteristic
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	#381	1	1	yes	characteristic
150	C₄×C₅₃	#407	1	1	yes	characteristic
151	C₄×C₅₃	#432	1	1	yes	characteristic
152	C₄×C₅₃	#432	1	1	yes	characteristic
153	Dic₅₃	—	4	2	no	—
154	Dic₅₃	—	4	2	no	—
155	#2873	—	53	1	no	—
156	C₂×#349	—	53	1	no	—
157	C₂×#349	D₅₃	1	1	yes	characteristic
158	#1378	—	53	2	no	—
159	#1378	—	53	2	no	—
160	C₂×C₅₃	—	2	1	no	—
161	C₂×C₅₃	—	2	1	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	—	4	1	no	—
164	C₂×C₅₃	—	4	1	no	—
165	C₂×C₅₃	#1817	1	1	yes	characteristic
166	C₂²×M₁₆	—	53	1	no	—
167	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
168	C₂×#117	—	53	1	no	—
169	C₂×M₃₂	—	53	2	no	—
170	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
171	#405	—	53	1	no	—
172	#405	—	53	1	no	—
173	#405	—	53	1	no	—
174	#405	—	53	1	no	—
175	#405	—	106	1	no	—
176	#405	—	106	1	no	—
177	C₄×M₁₆	—	106	1	no	—
178	C₄×M₁₆	—	106	1	no	—
179	#349	—	2	1	no	—
180	#349	—	2	1	no	—
181	#349	—	106	1	no	—
182	#349	—	106	1	no	—
183	C₅₃	#12851	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	53	1	no	—
188	C₂×M₁₆	—	53	1	no	—
189	C₂×M₁₆	—	106	2	no	—
190	C₂×M₁₆	—	106	2	no	—
191	C₂×M₁₆	—	106	1	no	—
192	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
193	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
194	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂²×C₈	—	106	1	no	—
197	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	#117	—	53	1	no	—
200	#117	—	53	1	no	—
201	#117	—	53	2	no	—
202	#117	—	106	1	no	—
203	#117	—	106	1	no	—
204	C₂×C₄⋊C₄	—	106	1	no	—
205	C₂×C₂²⋊C₄	—	53	2	no	—
206	C₂×C₄²	—	106	1	no	—
207	M₃₂	—	2	2	no	—
208	M₃₂	—	2	2	no	—
209	M₃₂	—	106	2	no	—
210	M₃₂	—	106	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	—	106	2	no	—
213	#97	—	106	1	no	—
214	#97	—	106	1	no	—
215	#97	—	106	1	no	—
216	#97	—	106	1	no	—
217	C₄×C₈	—	106	1	no	—
218	C₄×C₈	—	106	1	no	—
219	C₄×C₈	—	106	1	no	—
220	C₄×C₈	—	106	1	no	—
221	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	106	2	no	—
227	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
228	C₂²×C₄	#11719912	1	1	yes	characteristic
229	C₂²×C₄	#11719912	1	1	yes	characteristic
230	C₂²×C₄	#11719921	1	1	yes	characteristic
231	C₂²×C₄	#11719921	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	53	2	no	—
238	M₁₆	—	53	2	no	—
239	M₁₆	—	106	4	no	—
240	M₁₆	—	106	1	no	—
241	M₁₆	—	106	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	106	4	no	—
245	C₂×C₈	—	106	1	no	—
246	C₂×C₈	—	106	1	no	—
247	C₂×C₈	#11719921	1	1	yes	characteristic
248	C₂×C₈	#11719921	1	1	yes	characteristic
249	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
250	C₂×C₈	#11719947	1	2	yes	—
251	C₂×C₈	#11719947	1	2	yes	—
252	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
253	C₄⋊C₄	—	106	2	no	—
254	C₄⋊C₄	—	106	1	no	—
255	C₄⋊C₄	—	106	1	no	—
256	C₂²⋊C₄	—	53	2	no	—
257	C₂²⋊C₄	—	53	2	no	—
258	C₂²⋊C₄	—	106	2	no	—
259	C₄²	—	106	2	no	—
260	C₄²	—	106	1	no	—
261	C₄²	—	106	1	no	—
262	C₁₆	—	4	2	no	—
263	C₁₆	—	212	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	8	1	no	—
267	C₂³	#13133129	1	1	yes	characteristic
268	C₂³	#13133129	1	1	yes	characteristic
269	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
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	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	106	2	no	—
281	C₂×C₄	—	106	2	no	—
282	C₂×C₄	—	424	2	no	—
283	C₂×C₄	#13133129	1	1	yes	characteristic
284	C₂×C₄	#13133129	1	1	yes	characteristic
285	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
286	C₂×C₄	#13133173	1	1	yes	characteristic
287	C₂×C₄	#13133173	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	106	2	no	—
294	C₈	—	106	2	no	—
295	C₈	#13133198	1	2	yes	—
296	C₈	#13133201	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	C₂×#13133129	1	1	yes	characteristic
305	C₂²	#16810673	1	1	yes	characteristic
306	C₂²	#16810673	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	212	2	no	—
310	C₄	—	212	2	no	—
311	C₄	#16810484	1	1	yes	characteristic
312	C₄	#16810586	1	1	yes	characteristic
313	C₄	#16810677	1	1	yes	characteristic
314	C₄	#16810677	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	#20274781	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810484
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂×C₅₃	#1817
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	#12851
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. #20272802	C₂	semidirect
4. #20272800	C₂	semidirect
183. C₅₃	#12851	semidirect

Maximal subgroups

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

subgroup	index
#20283133	2
C₂×#16810405	2
#20272802	2
#20272800	2
C₂×C₅₃×#349	2
#12851	53