Subgroups — #27283340

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810971	C₂	1	1	yes	maximal, characteristic
2	#20275664	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275641	C₂	1	1	yes	maximal, characteristic
4	#20274580	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810373	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810366	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133214	C₂²	1	1	yes	characteristic
9	#16810971	—	2	2	no	—
10	#16810971	C₄	1	4	yes	complemented
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133139	—	2	2	no	—
13	C₂×#13133137	—	2	1	no	—
14	C₂×#13133137	—	2	1	no	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	—	2	1	no	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810385	—	2	2	no	—
19	#16810377	—	2	1	no	—
20	#16810373	—	2	2	no	—
21	#16810366	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#333	—	2	2	no	—
26	C₂²×D₂₁₂	—	2	1	no	—
27	C₂²×D₂₁₂	—	2	1	no	—
28	C₂²×D₂₁₂	—	2	1	no	—
29	C₂²×#88322	—	2	1	no	—
30	C₂²×#88322	D₄	1	1	yes	characteristic
31	C₂²×#88322	D₄	1	1	yes	characteristic
32	#13133214	—	2	1	no	—
33	#13133214	—	2	1	no	—
34	#13133214	—	2	1	no	—
35	#13133214	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₅₃	—	2	1	no	—
38	C₄⋊#88322	—	2	1	no	—
39	C₄⋊#88322	—	2	1	no	—
40	#13133139	—	4	4	no	—
41	#13133137	—	2	4	no	—
42	#13133137	—	2	4	no	—
43	C₄×#88322	C₂×C₄	1	2	yes	—
44	C₄×#88322	D₄	1	2	yes	—
45	C₈×Dic₅₃	—	4	2	no	—
46	#13133124	—	2	1	no	—
47	C₂²×Q₈×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	—	2	1	no	—
49	C₂²×C₈×C₅₃	—	2	1	no	—
50	C₄⋊Q₈×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	—	2	1	no	—
60	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
61	C₂³×D₅₃	—	4	1	no	—
62	C₂×Q₈×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	—	4	1	no	—
64	C₂×Q₈×C₅₃	—	8	2	no	—
65	C₂²×C₄×C₅₃	—	4	1	no	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	2	2	no	—
70	C₂×D₂₁₂	—	2	2	no	—
71	C₂×D₂₁₂	—	2	2	no	—
72	C₂×D₂₁₂	—	2	2	no	—
73	C₂×D₂₁₂	—	4	2	no	—
74	C₂×D₂₁₂	—	4	1	no	—
75	C₂×D₂₁₂	—	4	1	no	—
76	C₂×D₂₁₂	—	4	2	no	—
77	C₂×D₂₁₂	—	4	1	no	—
78	C₂×D₂₁₂	—	4	1	no	—
79	C₂×D₂₁₂	—	4	1	no	—
80	C₂×D₂₁₂	—	4	1	no	—
81	C₂×D₂₁₂	—	4	1	no	—
82	C₂×D₂₁₂	—	8	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	—	4	1	no	—
86	C₂×C₈×C₅₃	—	4	1	no	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	4	1	no	—
91	C₄⋊C₄×C₅₃	—	4	1	no	—
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₃	#41	1	1	yes	characteristic
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	SD₁₆	1	4	yes	—
98	C₄×Dic₅₃	—	4	1	no	—
99	C₄×Dic₅₃	—	4	1	no	—
100	C₂×#88322	—	2	1	no	—
101	C₂×#88322	—	2	1	no	—
102	C₂×#88322	—	2	1	no	—
103	C₂×#88322	—	2	1	no	—
104	C₂×#88322	—	4	1	no	—
105	C₂×#88322	—	4	2	no	—
106	C₂×#88322	C₂²⋊C₄	1	2	yes	—
107	C₂×#88322	C₂²⋊C₄	1	2	yes	—
108	C₂³×C₅₃	#121	1	1	yes	characteristic
109	C₂²×D₅₃	—	4	2	no	—
110	C₂²×D₅₃	—	8	1	no	—
111	C₂²×D₅₃	—	8	1	no	—
112	C₂²×D₅₃	—	8	2	no	—
113	C₂²×D₅₃	—	8	2	no	—
114	Q₈×C₅₃	—	8	2	no	—
115	Q₈×C₅₃	—	8	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#135	1	2	yes	—
129	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
131	C₂×Dic₅₃	—	2	4	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	D₂₁₂	—	4	2	no	—
135	D₂₁₂	—	4	2	no	—
136	D₂₁₂	—	4	4	no	—
137	D₂₁₂	—	8	2	no	—
138	D₂₁₂	—	8	2	no	—
139	D₂₁₂	—	8	1	no	—
140	D₂₁₂	—	8	1	no	—
141	D₂₁₂	—	8	1	no	—
142	D₂₁₂	—	8	1	no	—
143	C₈×C₅₃	—	4	2	no	—
144	C₈×C₅₃	—	8	1	no	—
145	#88322	—	2	2	no	—
146	#88322	—	2	2	no	—
147	#88322	—	8	1	no	—
148	#9735	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	#390	1	1	yes	characteristic
150	C₂²×C₅₃	#463	1	1	yes	characteristic
151	C₂²×C₅₃	#464	1	1	yes	characteristic
152	C₂²×C₅₃	#465	1	2	yes	—
153	C₂²×C₅₃	#472	1	2	yes	—
154	C₂×D₅₃	—	8	2	no	—
155	C₂×D₅₃	—	8	2	no	—
156	C₂×D₅₃	—	8	2	no	—
157	C₂×D₅₃	—	16	2	no	—
158	C₂×D₅₃	—	16	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	8	2	no	—
162	C₄×C₅₃	#416	1	1	yes	characteristic
163	C₄×C₅₃	#417	1	1	yes	characteristic
164	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
165	C₄×C₅₃	#463	1	1	yes	characteristic
166	C₄×C₅₃	#464	1	1	yes	characteristic
167	C₄×C₅₃	#492	1	1	yes	characteristic
168	C₄×C₅₃	#495	1	1	yes	characteristic
169	Dic₅₃	—	4	4	no	—
170	C₂×#486	—	53	1	no	—
171	#1987	—	53	1	no	—
172	#1982	—	53	1	no	—
173	C₈×C₄⋊C₄	—	53	1	no	—
174	C₂×#334	—	53	1	no	—
175	C₂×#333	D₅₃	1	1	yes	characteristic, complemented
176	C₂×#331	—	53	1	no	—
177	C₂×C₅₃	#1677	1	2	yes	—
178	C₂×C₅₃	#1678	1	2	yes	—
179	C₂×C₅₃	#1968	1	1	yes	characteristic
180	C₂×C₅₃	#1978	1	1	yes	characteristic
181	C₂×C₅₃	#1983	1	1	yes	characteristic
182	D₅₃	—	16	2	no	—
183	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
184	C₂×#127	—	53	1	no	—
185	#486	—	53	4	no	—
186	#486	—	106	2	no	—
187	C₂×C₄⋊C₈	—	53	1	no	—
188	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
190	C₂×#102	—	106	1	no	—
191	C₂×#102	—	106	1	no	—
192	C₂×#102	—	106	2	no	—
193	C₂×C₄×C₈	—	53	1	no	—
194	C₂×C₄×C₈	—	106	1	no	—
195	C₄×C₄⋊C₄	—	53	1	no	—
196	#340	—	106	2	no	—
197	#336	—	106	1	no	—
198	#334	—	106	2	no	—
199	#333	—	2	2	no	—
200	#331	—	106	2	no	—
201	C₅₃	#9735	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	106	1	no	—
204	C₂²×D₄	—	106	1	no	—
205	C₂²×D₄	—	106	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	106	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	C₄×D₅₃	1	2	yes	—
215	#127	—	53	2	no	—
216	#127	—	106	1	no	—
217	#127	—	106	1	no	—
218	#127	—	106	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	53	2	no	—
222	C₂×C₄²	—	106	1	no	—
223	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	106	1	no	—
228	C₄⋊C₈	—	106	1	no	—
229	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
230	#102	—	106	4	no	—
231	#102	—	106	4	no	—
232	#102	—	212	4	no	—
233	C₄×C₈	—	53	2	no	—
234	C₄×C₈	—	53	2	no	—
235	C₄×C₈	—	212	2	no	—
236	#95	—	106	1	no	—
237	C₂⁴	—	212	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	106	2	no	—
242	C₂×D₄	—	106	2	no	—
243	C₂×D₄	—	106	2	no	—
244	C₂×D₄	—	424	2	no	—
245	C₂×D₄	—	212	1	no	—
246	C₂×D₄	—	212	1	no	—
247	C₂×D₄	—	212	1	no	—
248	C₂×D₄	—	212	2	no	—
249	C₂×D₄	—	212	1	no	—
250	C₂×D₄	—	212	2	no	—
251	C₂×D₄	—	212	1	no	—
252	C₂×D₄	—	212	1	no	—
253	C₂×D₄	—	212	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	106	2	no	—
256	C₂²×C₄	#11719945	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	106	1	no	—
267	C₂×C₈	—	106	1	no	—
268	C₂×C₈	—	106	1	no	—
269	C₂×C₈	—	106	1	no	—
270	C₂×C₈	—	212	1	no	—
271	C₂×C₈	—	212	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	53	4	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄²	—	212	1	no	—
279	C₄²	—	212	1	no	—
280	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
281	C₄²	D₄×D₅₃	1	1	yes	characteristic
282	C₄²	#11719947	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₂³	—	424	1	no	—
285	C₂³	—	424	1	no	—
286	C₂³	—	424	2	no	—
287	C₂³	—	424	2	no	—
288	C₂³	—	212	2	no	—
289	C₂³	#13133190	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	424	2	no	—
293	D₄	—	424	2	no	—
294	D₄	—	424	1	no	—
295	D₄	—	424	1	no	—
296	D₄	—	424	1	no	—
297	D₄	—	424	1	no	—
298	D₄	—	212	2	no	—
299	D₄	—	212	2	no	—
300	D₄	—	212	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	106	4	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	—	212	2	no	—
307	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
308	C₂×C₄	#13133177	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
310	C₂×C₄	#13133189	1	1	yes	characteristic
311	C₂×C₄	#13133203	1	1	yes	characteristic
312	C₂×C₄	#13133213	1	1	yes	characteristic
313	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
314	C₂×C₄	#13133238	1	1	yes	characteristic
315	C₂×C₄	#13133240	1	2	yes	—
316	C₂×C₄	#13133240	1	1	yes	characteristic
317	C₂×C₄	#13133244	1	1	yes	characteristic
318	C₂×C₄	#13133249	1	1	yes	characteristic
319	C₈	—	4	2	no	—
320	C₈	—	8	1	no	—
321	C₈	—	106	2	no	—
322	C₈	—	106	2	no	—
323	C₈	—	424	1	no	—
324	C₂²	—	848	2	no	—
325	C₂²	—	848	2	no	—
326	C₂²	—	424	2	no	—
327	C₂²	—	424	2	no	—
328	C₂²	—	424	2	no	—
329	C₂²	#16810523	1	1	yes	characteristic
330	C₂²	#16810841	1	1	yes	characteristic
331	C₂²	#16810848	1	1	yes	characteristic
332	C₂²	#16810855	1	2	yes	—
333	C₂²	#16810892	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	212	4	no	—
338	C₄	#16810618	1	1	yes	characteristic
339	C₄	#16810623	1	1	yes	characteristic
340	C₄	#16810708	1	2	yes	—
341	C₄	#16810840	1	1	yes	characteristic
342	C₄	#16810847	1	1	yes	characteristic
343	C₄	#16810994	1	1	yes	characteristic
344	C₄	#16811008	1	1	yes	characteristic
345	C₂	—	848	2	no	—
346	C₂	#20274127	1	2	yes	—
347	C₂	#20274134	1	2	yes	—
348	C₂	#20275566	1	1	yes	characteristic
349	C₂	#20275622	1	1	yes	characteristic
350	C₂	#20275644	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133190
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#333	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂×#13133214	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#333	D₅₃
hypocenter	C₅₃	#9735
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #20275664	C₂	semidirect
4. #20274580	C₂	semidirect
5. C₂×#16810373	C₂	semidirect
7. C₂×C₅₃×#333	C₂	semidirect
10. #16810971	C₄	semidirect
175. C₂×#333	D₅₃	semidirect
201. C₅₃	#9735	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810971	2
#20275664	2
#20275641	2
#20274580	2
C₂×#16810373	2
C₂×#16810366	2
C₂×C₅₃×#333	2
#9735	53