Subgroups — #27274953

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810700	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133137	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810352	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	—	2	1	no	—
9	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
10	#16810700	—	2	2	no	—
11	#16810700	C₄	1	4	yes	complemented
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×#13133137	C₄	1	2	yes	—
14	C₂×#13133129	—	2	1	no	—
15	C₂×C₄×#88322	—	2	1	no	—
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	#16810385	C₄	1	2	yes	—
19	#16810378	—	2	1	no	—
20	#16810352	C₄	1	4	yes	complemented
21	#16810344	—	2	2	no	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	—	2	1	no	—
27	C₂×#11719912	—	2	4	no	—
28	C₂×#11719912	—	4	2	no	—
29	C₂×#11719912	—	4	2	no	—
30	C₂²×#88322	—	2	1	no	—
31	C₂²×#88322	C₂×C₄	1	2	yes	—
32	C₂²×#88322	D₄	1	1	yes	characteristic
33	C₂²×#88322	D₄	1	1	yes	characteristic
34	C₄×D₂₁₂	—	2	1	no	—
35	C₄×D₂₁₂	—	2	1	no	—
36	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
37	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719921	—	2	1	no	—
40	C₄⋊#88322	—	2	1	no	—
41	C₄⋊#88322	C₂×C₄	1	2	yes	—
42	#13133137	—	2	4	no	—
43	#13133129	—	2	4	no	—
44	C₄×#88322	—	2	1	no	—
45	C₄×#88322	—	2	1	no	—
46	C₄×#88322	—	4	2	no	—
47	C₄×#88322	C₂×C₄	1	2	yes	—
48	C₄×#88322	D₄	1	2	yes	—
49	C₂²×C₈×C₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	—	2	1	no	—
55	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	#95×C₅₃	—	2	1	no	—
59	C₂³×D₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×Dic₅₃	—	2	1	no	—
65	C₂×D₂₁₂	—	2	2	no	—
66	C₂×D₂₁₂	—	2	2	no	—
67	C₂×D₂₁₂	—	2	1	no	—
68	C₂×D₂₁₂	C₄²	1	2	yes	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	—	2	2	no	—
71	C₂×C₄×D₅₃	—	4	1	no	—
72	C₂×C₄×D₅₃	—	4	1	no	—
73	C₂×C₄×D₅₃	—	4	1	no	—
74	C₂×C₄×D₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	complemented
82	C₄²×C₅₃	—	4	1	no	—
83	C₄²×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	#41	1	1	yes	characteristic
88	#11719921	—	2	2	no	—
89	#11719921	—	2	2	no	—
90	C₄⋊Dic₅₃	—	2	1	no	—
91	C₄⋊Dic₅₃	C₄²	1	2	yes	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	—	2	1	no	—
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	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	4	1	no	—
101	C₂×#88322	—	4	1	no	—
102	C₂×#88322	C₂²⋊C₄	1	2	yes	—
103	C₂×#88322	C₂²⋊C₄	1	2	yes	—
104	#11719912	—	4	8	no	—
105	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₅₃	—	2	2	no	—
107	C₂²×D₅₃	—	4	2	no	—
108	C₂²×D₅₃	—	4	1	no	—
109	C₂²×D₅₃	—	4	1	no	—
110	C₂²×D₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	4	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×M₁₆	1	2	yes	—
122	C₂×C₄×C₅₃	#131	1	2	yes	—
123	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	4	1	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	D₂₁₂	—	2	2	no	—
131	D₂₁₂	—	2	2	no	—
132	D₂₁₂	—	4	2	no	—
133	C₄×D₅₃	—	4	2	no	—
134	C₄×D₅₃	—	4	2	no	—
135	C₄×D₅₃	—	4	2	no	—
136	C₄×D₅₃	—	4	2	no	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	—	4	2	no	—
139	#88322	—	2	2	no	—
140	#88322	—	2	2	no	—
141	#88322	—	4	2	no	—
142	#88322	—	4	2	no	—
143	#8212	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₅₃	#435	1	2	yes	—
146	C₂²×C₅₃	#440	1	1	yes	characteristic
147	C₂²×C₅₃	#442	1	1	yes	characteristic
148	C₂²×C₅₃	#443	1	2	yes	—
149	C₂×D₅₃	—	4	2	no	—
150	C₂×D₅₃	—	4	2	no	—
151	C₂×D₅₃	—	4	2	no	—
152	C₂×D₅₃	—	8	2	no	—
153	C₂×D₅₃	—	8	2	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	4	4	no	—
157	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
158	C₄×C₅₃	#405	1	1	yes	characteristic
159	C₄×C₅₃	#417	1	1	yes	characteristic
160	C₄×C₅₃	#418	1	1	yes	characteristic
161	C₄×C₅₃	#436	1	2	yes	—
162	C₄×C₅₃	#440	1	1	yes	characteristic
163	C₄×C₅₃	#442	1	1	yes	characteristic
164	Dic₅₃	—	4	2	no	—
165	C₂×#436	—	53	1	no	—
166	C₈×C₄⋊C₄	—	53	1	no	—
167	C₄×#102	—	53	1	no	—
168	C₂×D₄⋊C₈	—	53	1	no	—
169	C₂×#322	—	53	1	no	—
170	#1279	D₅₃	1	1	yes	characteristic, complemented
171	#1277	—	53	1	no	—
172	C₂×C₅₃	#1586	1	2	yes	—
173	C₂×C₅₃	#1588	1	2	yes	—
174	C₂×C₅₃	#1753	1	1	yes	characteristic
175	C₂×C₅₃	#1762	1	1	yes	characteristic
176	C₂×C₅₃	#1764	1	1	yes	characteristic
177	D₅₃	—	8	2	no	—
178	C₂²×M₁₆	—	106	1	no	—
179	C₂×C₄×D₄	—	53	1	no	—
180	#436	—	53	4	no	—
181	#436	—	106	2	no	—
182	C₂×C₄⋊C₈	—	53	1	no	—
183	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
184	C₂×#102	—	53	2	no	—
185	C₂×C₂²⋊C₈	—	106	1	no	—
186	C₂×C₄×C₈	—	53	1	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	106	1	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
191	#340	—	53	2	no	—
192	#336	—	106	1	no	—
193	D₄⋊C₈	—	53	4	no	—
194	#322	—	106	2	no	—
195	C₅₃	#8212	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	53	1	no	—
197	C₂³×C₄	—	106	1	no	—
198	C₂×M₁₆	—	106	4	no	—
199	C₂×M₁₆	—	212	2	no	—
200	C₂×M₁₆	—	212	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	53	1	no	—
204	C₂²×C₈	—	53	1	no	—
205	C₂²×C₈	—	53	2	no	—
206	C₂²×C₈	—	106	1	no	—
207	C₄×D₄	—	53	2	no	—
208	C₄×D₄	—	53	2	no	—
209	C₄×D₄	—	106	1	no	—
210	C₄×D₄	—	106	1	no	—
211	C₂×C₄⋊C₄	—	53	1	no	—
212	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
213	C₂×C₂²⋊C₄	—	106	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	C₄×D₅₃	1	2	yes	—
220	#102	—	106	4	no	—
221	C₂²⋊C₈	—	106	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	53	2	no	—
224	C₄×C₈	—	53	2	no	—
225	C₄×C₈	—	106	1	no	—
226	C₄×C₈	—	106	1	no	—
227	C₄×C₈	—	212	2	no	—
228	C₄×C₈	C₄×D₅₃	1	2	yes	—
229	#95	—	2	1	no	—
230	C₂⁴	—	106	1	no	—
231	C₂×D₄	—	53	2	no	—
232	C₂×D₄	—	106	1	no	—
233	C₂×D₄	—	106	2	no	—
234	C₂×D₄	—	106	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	106	1	no	—
237	C₂²×C₄	—	106	2	no	—
238	C₂²×C₄	—	106	2	no	—
239	C₂²×C₄	—	212	1	no	—
240	C₂²×C₄	—	212	1	no	—
241	C₂²×C₄	—	212	1	no	—
242	C₂²×C₄	—	212	1	no	—
243	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
245	C₂²×C₄	#11719947	1	1	yes	characteristic
246	M₁₆	—	212	8	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	106	1	no	—
255	C₂×C₈	—	106	1	no	—
256	C₂×C₈	—	106	1	no	—
257	C₂×C₈	—	106	1	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	212	1	no	—
263	C₂×C₈	—	212	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	53	2	no	—
266	C₄⋊C₄	—	106	1	no	—
267	C₄⋊C₄	#11719912	1	4	yes	complemented
268	C₂²⋊C₄	—	106	2	no	—
269	C₂²⋊C₄	—	106	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₄²	D₄×D₅₃	1	1	yes	characteristic
275	C₄²	#11719949	1	1	yes	characteristic
276	C₂³	—	106	2	no	—
277	C₂³	—	212	2	no	—
278	C₂³	—	212	1	no	—
279	C₂³	—	212	1	no	—
280	C₂³	—	212	2	no	—
281	C₂³	#13133177	1	1	yes	characteristic
282	D₄	—	106	2	no	—
283	D₄	—	106	2	no	—
284	D₄	—	212	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	212	1	no	—
291	C₂×C₄	—	212	1	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	—	212	2	no	—
295	C₂×C₄	—	212	2	no	—
296	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
298	C₂×C₄	#13133175	1	1	yes	characteristic
299	C₂×C₄	#13133177	1	1	yes	characteristic
300	C₂×C₄	#13133179	1	1	yes	characteristic
301	C₂×C₄	#13133179	1	1	yes	characteristic
302	C₂×C₄	C₂×#11719912	1	2	yes	—
303	C₂×C₄	#13133225	1	2	yes	—
304	C₂×C₄	#13133241	1	1	yes	characteristic
305	C₂×C₄	#13133244	1	1	yes	characteristic
306	C₂×C₄	#13133244	1	1	yes	characteristic
307	C₂×C₄	#13133249	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	106	2	no	—
311	C₈	—	106	2	no	—
312	C₈	—	212	2	no	—
313	C₈	—	212	2	no	—
314	C₂²	—	424	2	no	—
315	C₂²	—	424	2	no	—
316	C₂²	—	212	2	no	—
317	C₂²	—	212	2	no	—
318	C₂²	—	212	2	no	—
319	C₂²	#16810470	1	1	yes	characteristic
320	C₂²	#16810691	1	2	yes	—
321	C₂²	#16810726	1	1	yes	characteristic
322	C₂²	#16810732	1	1	yes	characteristic
323	C₂²	#16810737	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	4	no	—
327	C₄	—	212	2	no	—
328	C₄	C₄×#11719912	1	1	yes	characteristic
329	C₄	#16810580	1	1	yes	characteristic
330	C₄	#16810623	1	1	yes	characteristic
331	C₄	#16810626	1	1	yes	characteristic
332	C₄	#16810700	1	2	yes	—
333	C₄	#16810722	1	1	yes	characteristic
334	C₄	#16810734	1	1	yes	characteristic
335	C₂	—	424	2	no	—
336	C₂	#20273576	1	2	yes	—
337	C₂	#20273590	1	2	yes	—
338	C₂	#20274495	1	1	yes	characteristic
339	C₂	#20274539	1	1	yes	characteristic
340	C₂	#20274549	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1279	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1279	D₅₃
hypocenter	C₅₃	#8212
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×#88322	C₂	semidirect
5. C₂×#16810344	C₂	semidirect
6. #20272549	C₂	semidirect
7. C₅₃×#1279	C₂	semidirect
11. #16810700	C₄	semidirect
20. #16810352	C₄	semidirect
81. C₄⋊C₄×C₅₃	M₁₆	semidirect
170. #1279	D₅₃	semidirect
195. C₅₃	#8212	semidirect
267. C₄⋊C₄	#11719912	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810700	2
C₄⋊C₄×#88322	2
C₄×#13133137	2
C₂×#16810352	2
C₂×#16810344	2
#20272549	2
C₅₃×#1279	2
#8212	53