Subgroups — #27275186

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283426	C₂	1	1	yes	maximal, characteristic
2	#20274584	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273362	C₂	1	1	yes	maximal, characteristic
4	#20273209	C₂	1	1	yes	maximal, characteristic
5	#20272999	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272610	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133225	—	2	1	no	—
9	#16811173	C₂²	1	1	yes	characteristic
10	#16810700	—	2	1	no	—
11	#16810700	C₄	1	2	yes	complemented
12	#16810688	—	2	1	no	—
13	#16810688	C₄	1	2	yes	complemented
14	#16810676	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	#16810591	—	2	1	no	—
17	C₂×#13133128	—	2	1	no	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	#16810378	—	2	1	no	—
20	#16810368	C₄	1	2	yes	complemented
21	#16810366	C₄	1	2	yes	complemented
22	#16810356	C₄	1	4	yes	—
23	#16810345	—	2	1	no	—
24	C₈×#88322	—	2	1	no	—
25	C₅₃×#431	C₂²	1	1	yes	characteristic
26	C₅₃×#423	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#11719945	—	2	1	no	—
29	#13133225	—	4	2	no	—
30	#13133225	—	4	2	no	—
31	C₂×#11719912	—	2	2	no	—
32	C₂²×#88322	—	2	1	no	—
33	C₂²×#88322	D₄	1	1	yes	characteristic
34	C₂²×#88322	D₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	—	2	2	no	—
36	#13133217	C₂×C₄	1	1	yes	characteristic
37	#13133214	C₂×C₄	1	1	yes	characteristic
38	#13133205	—	2	1	no	—
39	#13133191	—	2	1	no	—
40	#13133191	C₂×C₄	1	2	yes	—
41	C₄×D₂₁₂	—	2	1	no	—
42	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
43	C₄⋊#88322	—	2	1	no	—
44	C₄⋊#88322	—	2	1	no	—
45	C₄⋊#88322	C₂×C₄	1	2	yes	—
46	C₂²⋊#88322	C₂×C₄	1	2	yes	—
47	#13133129	—	2	2	no	—
48	#13133129	—	2	2	no	—
49	#13133128	—	4	2	no	—
50	C₄×#88322	—	2	1	no	—
51	C₄×#88322	C₂×C₄	1	1	yes	characteristic
52	C₄×#88322	C₂×C₄	1	1	yes	characteristic
53	C₄×#88322	D₄	1	1	yes	characteristic
54	C₄×#88322	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄²×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₃	—	2	1	no	—
60	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₂²⋊C₈×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	1	no	—
64	#95×C₅₃	C₂×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₃	—	2	2	no	—
70	#11719945	—	4	1	no	—
71	#11719945	—	4	1	no	—
72	#11719945	—	4	1	no	—
73	#11719945	—	4	1	no	—
74	C₂×D₂₁₂	—	2	1	no	—
75	C₂×D₂₁₂	—	2	1	no	—
76	C₂×D₂₁₂	C₄²	1	2	yes	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×Dic₁₀₆	—	2	1	no	—
79	M₁₆×C₅₃	—	4	1	no	—
80	M₁₆×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	—	4	1	no	—
85	C₄²×C₅₃	—	4	1	no	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	#41	1	1	yes	characteristic
90	#11719921	—	2	2	no	—
91	#11719921	—	2	2	no	—
92	C₄⋊Dic₅₃	—	2	1	no	—
93	C₄⋊Dic₅₃	C₄²	1	2	yes	—
94	C₂×#88322	—	2	1	no	—
95	C₂×#88322	—	2	1	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	4	1	no	—
99	C₂×#88322	—	4	1	no	—
100	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#88322	D₈	1	2	yes	—
105	C₂×#88322	SD₁₆	1	2	yes	—
106	#11719912	—	4	4	no	—
107	C₈×D₅₃	—	4	4	no	—
108	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
125	C₂²⋊D₅₃	—	4	2	no	—
126	C₂²⋊D₅₃	—	4	2	no	—
127	C₂×Dic₅₃	—	2	2	no	—
128	D₂₁₂	—	4	2	no	—
129	D₂₁₂	—	4	1	no	—
130	D₂₁₂	—	4	1	no	—
131	D₂₁₂	—	4	2	no	—
132	C₄×D₅₃	—	4	2	no	—
133	C₄×D₅₃	—	4	2	no	—
134	Dic₁₀₆	—	4	1	no	—
135	Dic₁₀₆	—	4	1	no	—
136	C₈×C₅₃	—	4	2	no	—
137	C₈×C₅₃	—	4	2	no	—
138	#88322	—	2	1	no	—
139	#88322	—	2	1	no	—
140	#88322	—	4	2	no	—
141	#88322	—	4	2	no	—
142	#88322	#102	1	2	yes	—
143	#88322	#102	1	2	yes	—
144	#8255	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
149	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
150	C₂×D₅₃	—	4	2	no	—
151	C₂×D₅₃	—	4	2	no	—
152	C₂×D₅₃	—	4	2	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
158	C₄×C₅₃	#416	1	1	yes	characteristic
159	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
160	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
161	C₄×C₅₃	#444	1	2	yes	—
162	Dic₅₃	—	4	2	no	—
163	#2931	—	53	1	no	—
164	#1770	—	53	1	no	—
165	#1540	—	53	1	no	—
166	#1502	—	53	1	no	—
167	#1450	—	53	1	no	—
168	#1303	D₅₃	1	1	yes	characteristic, complemented
169	#1301	—	53	1	no	—
170	C₂×C₅₃	—	2	1	no	—
171	C₂×C₅₃	—	2	1	no	—
172	C₂×C₅₃	#1757	1	1	yes	characteristic
173	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
174	C₂×C₅₃	#1765	1	1	yes	characteristic
175	D₅₃	—	8	2	no	—
176	C₂×#131	—	106	1	no	—
177	#532	—	53	1	no	—
178	#436	—	53	2	no	—
179	#436	—	106	1	no	—
180	C₈×D₄	—	53	2	no	—
181	C₈×D₄	—	106	1	no	—
182	#431	—	53	1	no	—
183	#431	C₂×D₅₃	1	1	yes	characteristic
184	#423	C₂×D₅₃	1	1	yes	characteristic
185	C₂×C₄⋊C₈	—	53	1	no	—
186	#408	—	106	1	no	—
187	C₂×#97	—	106	1	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
190	#336	—	106	1	no	—
191	#332	—	53	2	no	—
192	#331	—	53	2	no	—
193	#327	—	53	4	no	—
194	#322	—	106	1	no	—
195	C₈²	—	106	1	no	—
196	C₅₃	#8255	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	106	1	no	—
198	#131	—	212	2	no	—
199	#131	—	212	2	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	106	2	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₂²×C₈	—	53	1	no	—
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	—	106	1	no	—
206	C₄⋊Q₈	—	53	1	no	—
207	#127	—	53	1	no	—
208	#124	—	106	1	no	—
209	#121	—	53	2	no	—
210	#121	—	106	1	no	—
211	C₄×D₄	—	53	2	no	—
212	C₄×D₄	—	106	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	53	2	no	—
218	C₄⋊C₈	—	106	1	no	—
219	C₄⋊C₈	—	106	1	no	—
220	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
221	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	53	2	no	—
224	C₂²⋊C₈	—	106	2	no	—
225	C₂²⋊C₈	—	106	2	no	—
226	#97	—	212	2	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	53	1	no	—
229	C₄×C₈	—	53	1	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	106	1	no	—
233	#95	C₄×D₅₃	1	2	yes	—
234	#41	—	212	1	no	—
235	#41	—	212	1	no	—
236	#41	—	212	1	no	—
237	#41	—	212	1	no	—
238	C₂×Q₈	—	106	1	no	—
239	C₂×D₄	—	53	2	no	—
240	C₂×D₄	—	106	2	no	—
241	C₂×D₄	—	106	1	no	—
242	C₂×D₄	—	106	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	106	2	no	—
245	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
247	C₂²×C₄	#11719947	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	212	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	106	1	no	—
260	C₂×C₈	—	106	1	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	212	4	no	—
264	C₂×C₈	—	212	1	no	—
265	C₂×C₈	—	212	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	53	2	no	—
268	C₄⋊C₄	—	106	1	no	—
269	C₂²⋊C₄	—	106	2	no	—
270	C₂²⋊C₄	—	106	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
275	C₄²	D₄×D₅₃	1	1	yes	characteristic
276	C₄²	#11719949	1	1	yes	characteristic
277	C₂³	—	106	2	no	—
278	C₂³	#13133177	1	1	yes	characteristic
279	Q₈	—	212	1	no	—
280	Q₈	—	212	1	no	—
281	D₄	—	212	2	no	—
282	D₄	—	212	2	no	—
283	D₄	—	212	2	no	—
284	D₄	—	212	1	no	—
285	D₄	—	212	1	no	—
286	D₄	—	212	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	—	212	2	no	—
295	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
297	C₂×C₄	#13133175	1	1	yes	characteristic
298	C₂×C₄	#13133177	1	1	yes	characteristic
299	C₂×C₄	#13133179	1	1	yes	characteristic
300	C₂×C₄	#13133179	1	1	yes	characteristic
301	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
303	C₂×C₄	#13133238	1	1	yes	characteristic
304	C₂×C₄	#13133240	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	53	2	no	—
308	C₈	—	53	2	no	—
309	C₈	—	106	1	no	—
310	C₈	—	106	1	no	—
311	C₈	—	212	2	no	—
312	C₈	—	212	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	212	2	no	—
316	C₂²	—	212	2	no	—
317	C₂²	—	212	2	no	—
318	C₂²	#16810470	1	1	yes	characteristic
319	C₂²	#16810703	1	1	yes	characteristic
320	C₂²	#16810708	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	212	2	no	—
326	C₄	#102×D₅₃	1	1	yes	characteristic
327	C₄	#16810618	1	1	yes	characteristic
328	C₄	#16810705	1	1	yes	characteristic
329	C₄	#16810710	1	1	yes	characteristic
330	C₄	#16810742	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	424	2	no	—
334	C₂	#20274514	1	1	yes	characteristic
335	C₂	#20274529	1	1	yes	characteristic
336	C₂	#20274556	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1303	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	#16811173	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1303	D₅₃
hypocenter	C₅₃	#8255
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
2. #20274584	C₂	semidirect
5. #20272999	C₂	semidirect
6. #20272610	C₂	semidirect
7. C₅₃×#1303	C₂	semidirect
11. #16810700	C₄	semidirect
13. #16810688	C₄	semidirect
20. #16810368	C₄	semidirect
21. #16810366	C₄	semidirect
168. #1303	D₅₃	semidirect
196. C₅₃	#8255	semidirect

Maximal subgroups

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

subgroup	index
#20283426	2
#20274584	2
#20273362	2
#20273209	2
#20272999	2
#20272610	2
C₅₃×#1303	2
#8255	53