Subgroups — #27274839

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283676	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133145	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273095	C₂	1	1	yes	maximal, characteristic
5	#20272996	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272551	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11719945	C₂²	1	1	yes	characteristic
9	#16810750	C₄	1	2	yes	complemented
10	#16810700	C₄	1	2	yes	complemented
11	#16810691	—	2	2	no	—
12	#16810674	—	2	1	no	—
13	C₂×#13133145	C₄	1	2	yes	—
14	C₄×#11719917	C₂²	1	1	yes	characteristic
15	C₄×#11719912	—	2	1	no	—
16	C₂×#13133128	—	2	2	no	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	#16810383	C₄	1	2	yes	—
20	#16810353	C₄	1	2	yes	complemented
21	#16810350	C₄	1	2	yes	complemented
22	#16810345	—	2	1	no	—
23	C₈×#88322	—	2	1	no	—
24	C₅₃×#431	C₂²	1	1	yes	characteristic
25	C₅₃×#431	C₂²	1	1	yes	characteristic
26	C₄³×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂×#11719912	—	2	2	no	—
30	C₂²×#88322	—	2	1	no	—
31	C₂²×#88322	D₄	1	2	yes	—
32	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
34	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
35	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₅₃	—	2	1	no	—
37	C₄×C₂²⋊D₅₃	—	2	1	no	—
38	#13133176	C₂×C₄	1	1	yes	characteristic
39	#13133174	—	2	1	no	—
40	C₄²×D₅₃	—	2	1	no	—
41	#13133146	—	2	2	no	—
42	#13133145	—	2	4	no	—
43	#13133129	—	2	2	no	—
44	#13133128	—	2	8	no	—
45	#13133128	—	2	1	no	—
46	#13133127	—	2	2	no	—
47	C₄×#88322	—	2	1	no	—
48	C₄×#88322	—	2	1	no	—
49	C₄×#88322	C₂×C₄	1	2	yes	—
50	C₄×#88322	C₂×C₄	1	2	yes	—
51	C₄×#88322	D₄	1	1	yes	characteristic
52	C₄×#88322	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈×C₅₃	—	2	1	no	—
60	C₂²⋊C₈×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂×C₂²⋊D₅₃	—	2	1	no	—
68	#11719945	—	2	2	no	—
69	#11719945	—	2	2	no	—
70	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×C₄×D₅₃	—	2	1	no	—
74	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₂²⋊Dic₅₃	—	2	1	no	—
86	#11719921	—	2	1	no	—
87	#11719921	—	2	1	no	—
88	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
89	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
90	#11719919	—	2	1	no	—
91	#11719919	—	2	1	no	—
92	C₄×Dic₅₃	—	2	1	no	—
93	C₄×Dic₅₃	—	2	1	no	—
94	C₄×Dic₅₃	—	2	1	no	—
95	#11719917	—	2	2	no	—
96	#11719917	M₁₆	1	4	yes	complemented
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	C₂²⋊C₄	1	2	yes	—
102	C₂×#88322	C₂²⋊C₄	1	2	yes	—
103	#11719912	—	4	4	no	—
104	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₅₃	#131	1	2	yes	—
119	C₂²⋊D₅₃	—	4	2	no	—
120	C₂×Dic₅₃	—	2	1	no	—
121	C₂×Dic₅₃	—	2	1	no	—
122	C₂×Dic₅₃	—	2	1	no	—
123	D₂₁₂	—	2	1	no	—
124	D₂₁₂	—	2	1	no	—
125	C₄×D₅₃	—	4	2	no	—
126	C₄×D₅₃	—	4	2	no	—
127	C₄×D₅₃	—	4	2	no	—
128	Dic₁₀₆	—	2	1	no	—
129	Dic₁₀₆	—	2	1	no	—
130	C₈×C₅₃	—	4	2	no	—
131	C₈×C₅₃	—	4	2	no	—
132	#88322	—	2	4	no	—
133	#88322	—	2	2	no	—
134	#88322	—	2	4	no	—
135	#88322	#104	1	4	yes	complemented
136	#8196	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₅₃	#436	1	2	yes	—
141	C₂×D₅₃	—	4	1	no	—
142	C₂×D₅₃	—	4	1	no	—
143	C₂×D₅₃	—	4	1	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	4	4	no	—
147	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
148	C₄×C₅₃	#405	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#104	1	2	yes	—
150	C₄×C₅₃	#435	1	2	yes	—
151	C₄×C₅₃	#444	1	2	yes	—
152	Dic₅₃	—	4	1	no	—
153	Dic₅₃	—	4	1	no	—
154	Dic₅₃	—	4	1	no	—
155	#2965	—	53	1	no	—
156	C₄×#104	—	53	1	no	—
157	C₄×#97	—	53	1	no	—
158	#1476	—	53	1	no	—
159	C₈×M₁₆	—	53	1	no	—
160	#1278	—	53	1	no	—
161	#1276	D₅₃	1	1	yes	characteristic, complemented
162	C₂×C₅₃	—	2	2	no	—
163	C₂×C₅₃	#1753	1	1	yes	characteristic
164	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
165	C₂×C₅₃	#1765	1	1	yes	characteristic
166	D₅₃	—	8	1	no	—
167	C₄×#41	—	53	1	no	—
168	#446	—	53	2	no	—
169	#436	—	53	2	no	—
170	#435	—	106	2	no	—
171	#431	—	106	1	no	—
172	#431	C₂×D₅₃	1	1	yes	characteristic
173	#431	C₂×D₅₃	1	1	yes	characteristic
174	C₂×#104	—	53	2	no	—
175	C₄×M₁₆	—	53	1	no	—
176	C₄×M₁₆	—	106	1	no	—
177	C₂×#97	—	106	2	no	—
178	C₂×C₄×C₈	—	53	1	no	—
179	C₂×C₄×C₈	—	53	1	no	—
180	C₄³	C₂×D₅₃	1	1	yes	characteristic
181	#339	—	53	2	no	—
182	Q₈⋊C₈	—	53	2	no	—
183	D₄⋊C₈	—	53	2	no	—
184	#322	—	106	1	no	—
185	C₈²	—	106	1	no	—
186	C₅₃	#8196	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	53	1	no	—
188	C₂×M₁₆	—	53	2	no	—
189	C₂×M₁₆	—	106	2	no	—
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	106	1	no	—
192	C₄×Q₈	—	53	1	no	—
193	C₄×Q₈	—	53	1	no	—
194	C₄×D₄	—	53	1	no	—
195	C₄×D₄	—	53	1	no	—
196	C₄×D₄	—	106	1	no	—
197	C₄×D₄	—	106	1	no	—
198	#117	—	53	1	no	—
199	#117	—	106	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	106	1	no	—
203	C₂×C₄²	C₄×D₅₃	1	2	yes	—
204	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
205	C₄⋊C₈	—	106	2	no	—
206	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
207	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
208	#104	—	106	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	106	2	no	—
212	#97	—	106	8	no	—
213	#97	—	106	2	no	—
214	#97	—	106	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	53	2	no	—
218	C₄×C₈	—	53	2	no	—
219	C₄×C₈	—	53	1	no	—
220	C₄×C₈	—	53	1	no	—
221	C₄×C₈	—	106	1	no	—
222	C₄×C₈	—	106	1	no	—
223	#41	—	106	2	no	—
224	#41	—	106	2	no	—
225	C₂×Q₈	—	53	1	no	—
226	C₂×D₄	—	53	1	no	—
227	C₂×D₄	—	106	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	106	1	no	—
230	C₂²×C₄	—	106	1	no	—
231	C₂²×C₄	—	106	1	no	—
232	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
234	C₂²×C₄	#11719947	1	1	yes	characteristic
235	M₁₆	—	53	4	no	—
236	M₁₆	—	106	2	no	—
237	M₁₆	—	212	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	53	2	no	—
241	C₂×C₈	—	53	2	no	—
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	—	106	4	no	—
244	C₂×C₈	—	106	2	no	—
245	C₂×C₈	—	106	2	no	—
246	C₄⋊C₄	—	53	1	no	—
247	C₄⋊C₄	—	53	1	no	—
248	C₄⋊C₄	—	106	1	no	—
249	C₄⋊C₄	—	106	1	no	—
250	C₂²⋊C₄	—	106	1	no	—
251	C₂²⋊C₄	—	106	1	no	—
252	C₂²⋊C₄	—	106	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	2	no	—
257	C₄²	—	106	1	no	—
258	C₄²	—	106	1	no	—
259	C₄²	—	106	1	no	—
260	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
262	C₄²	D₄×D₅₃	1	1	yes	characteristic
263	C₄²	#11719949	1	1	yes	characteristic
264	C₂³	—	106	1	no	—
265	C₂³	#13133177	1	1	yes	characteristic
266	Q₈	—	106	1	no	—
267	Q₈	—	106	1	no	—
268	D₄	—	106	1	no	—
269	D₄	—	106	1	no	—
270	D₄	—	212	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	106	1	no	—
277	C₂×C₄	—	106	1	no	—
278	C₂×C₄	—	106	1	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	—	212	2	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
284	C₂×C₄	#13133175	1	1	yes	characteristic
285	C₂×C₄	#13133177	1	1	yes	characteristic
286	C₂×C₄	#13133179	1	1	yes	characteristic
287	C₂×C₄	#13133179	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
289	C₂×C₄	#13133223	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	53	4	no	—
293	C₈	—	106	4	no	—
294	C₈	—	106	2	no	—
295	C₈	—	106	4	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	212	1	no	—
299	C₂²	—	212	1	no	—
300	C₂²	—	212	1	no	—
301	C₂²	#16810470	1	1	yes	characteristic
302	C₂²	#16810698	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	212	1	no	—
307	C₄	—	212	1	no	—
308	C₄	—	212	1	no	—
309	C₄	#16810575	1	1	yes	characteristic
310	C₄	#16810577	1	1	yes	characteristic
311	C₄	#104×D₅₃	1	2	yes	—
312	C₄	#16810693	1	2	yes	—
313	C₄	#16810742	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	424	1	no	—
316	C₂	#20274497	1	1	yes	characteristic
317	C₂	#20274519	1	1	yes	characteristic
318	C₂	#20274556	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1276	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1276	D₅₃
hypocenter	C₅₃	#8196
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
3. C₄×#13133128	C₂	semidirect
5. #20272996	C₂	semidirect
6. #20272551	C₂	semidirect
7. C₅₃×#1276	C₂	semidirect
9. #16810750	C₄	semidirect
10. #16810700	C₄	semidirect
20. #16810353	C₄	semidirect
21. #16810350	C₄	semidirect
96. #11719917	M₁₆	semidirect
135. #88322	#104	semidirect
161. #1276	D₅₃	semidirect
186. C₅₃	#8196	semidirect

Maximal subgroups

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

subgroup	index
#20283676	2
C₄×#13133145	2
C₄×#13133128	2
#20273095	2
#20272996	2
#20272551	2
C₅₃×#1276	2
#8196	53