Subgroups — #27368199

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20284846	C₂	1	2	yes	maximal
3	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
4	#20274270	C₂	1	4	yes	maximal, complemented
5	#20274234	C₂	1	2	yes	maximal, complemented
6	#20274210	C₂	1	2	yes	maximal, complemented
7	#20273416	C₂	1	2	yes	maximal
8	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	#16811028	C₂²	1	1	yes	characteristic
12	#16811003	C₂²	1	1	yes	characteristic
13	#16810974	C₂²	1	2	yes	—
14	#16810954	—	2	2	no	—
15	#16810909	C₂²	1	4	yes	—
16	#16810864	—	2	1	no	—
17	#16810864	—	2	1	no	—
18	#16810834	C₂²	1	4	yes	—
19	#16810781	—	2	2	no	—
20	#16810753	—	2	1	no	—
21	#16810753	—	2	1	no	—
22	#16810753	C₂²	1	2	yes	—
23	#16810753	C₂²	1	2	yes	—
24	#16810731	—	2	4	no	—
25	C₄×Dic₂₁₂	—	2	1	no	—
26	C₄×Dic₂₁₂	—	2	1	no	—
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×#13133146	—	2	1	no	—
29	C₂×#13133127	—	2	1	no	—
30	#16810370	C₂²	1	4	yes	—
31	#16810368	C₂²	1	4	yes	—
32	C₅₃×#432	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#335	C₂²	1	2	yes	—
35	C₅₃×#334	C₂²	1	2	yes	—
36	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
37	C₂×Dic₂₁₂	—	2	1	no	—
38	C₂×Dic₂₁₂	—	2	1	no	—
39	C₂×Dic₂₁₂	—	2	4	no	—
40	C₂²×#88322	—	2	1	no	—
41	#13133217	C₂³	1	2	yes	—
42	#13133217	C₂³	1	2	yes	—
43	#13133209	—	2	2	no	—
44	#13133196	—	2	2	no	—
45	#13133196	—	2	2	no	—
46	C₄×Dic₁₀₆	—	2	1	no	—
47	C₄×Dic₁₀₆	—	2	1	no	—
48	C₄×Dic₁₀₆	—	2	2	no	—
49	C₄×Dic₁₀₆	—	2	2	no	—
50	C₄×Dic₁₀₆	D₄	1	2	yes	—
51	C₄×Dic₁₀₆	C₂³	1	2	yes	—
52	#13133176	—	2	2	no	—
53	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
54	C₂×#11719919	—	2	1	no	—
55	C₂×C₄×Dic₅₃	—	2	1	no	—
56	#13133151	—	2	1	no	—
57	#13133151	—	2	1	no	—
58	#13133149	—	2	4	no	—
59	C₄⋊#88322	—	2	1	no	—
60	C₄⋊#88322	C₂³	1	2	yes	—
61	#13133146	—	2	2	no	—
62	#13133146	—	2	2	no	—
63	#13133142	—	2	2	no	—
64	#13133142	—	2	2	no	—
65	#13133142	—	2	4	no	—
66	#13133142	—	2	4	no	—
67	#13133127	—	2	2	no	—
68	#13133127	—	2	2	no	—
69	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
73	C₂²⋊C₈×C₅₃	—	2	2	no	—
74	#97×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	C₂²×Dic₅₃	—	2	1	no	—
83	C₂²×Dic₅₃	—	2	1	no	—
84	C₂×Dic₁₀₆	—	2	1	no	—
85	C₂×Dic₁₀₆	—	2	2	no	—
86	C₂×Dic₁₀₆	—	2	2	no	—
87	C₂×Dic₁₀₆	—	4	2	no	—
88	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₅₃	—	4	2	no	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	C₄⋊Dic₅₃	—	2	2	no	—
103	C₄⋊Dic₅₃	—	2	2	no	—
104	C₄⋊Dic₅₃	—	2	2	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
107	#11719919	—	2	2	no	—
108	#11719919	—	2	2	no	—
109	#11719919	—	4	2	no	—
110	#11719919	—	4	2	no	—
111	C₄×Dic₅₃	—	2	2	no	—
112	C₄×Dic₅₃	—	2	2	no	—
113	C₄×Dic₅₃	—	4	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	4	2	no	—
116	Dic₂₁₂	—	4	2	no	—
117	Dic₂₁₂	—	4	8	no	—
118	Dic₂₁₂	—	4	2	no	—
119	C₂³×C₅₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	4	2	no	—
136	Dic₁₀₆	—	2	4	no	—
137	Dic₁₀₆	—	4	2	no	—
138	Dic₁₀₆	—	4	2	no	—
139	Dic₁₀₆	—	4	2	no	—
140	Dic₁₀₆	—	4	4	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	4	4	no	—
145	C₈×C₅₃	#137	1	2	yes	—
146	#88322	—	4	2	no	—
147	#19290	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	#473	1	2	yes	—
151	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
157	C₄×C₅₃	#575	1	2	yes	—
158	C₄×C₅₃	#575	1	1	yes	characteristic
159	Dic₅₃	—	4	2	no	—
160	Dic₅₃	—	4	2	no	—
161	Dic₅₃	—	8	2	no	—
162	Dic₅₃	—	8	2	no	—
163	#3170	—	53	1	no	—
164	#3113	—	53	2	no	—
165	C₂×#446	—	53	1	no	—
166	#1700	—	53	4	no	—
167	#1694	—	53	2	no	—
168	#1690	—	53	2	no	—
169	#1568	D₅₃	1	1	yes	characteristic
170	#1552	—	53	2	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	#3037	1	1	yes	characteristic
173	C₂×C₅₃	#3054	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
175	#533	—	53	2	no	—
176	C₂×C₄×Q₈	—	53	1	no	—
177	#500	—	53	1	no	—
178	#494	—	53	1	no	—
179	#487	—	53	2	no	—
180	#483	—	106	2	no	—
181	#475	—	53	4	no	—
182	#467	—	106	1	no	—
183	#467	—	106	1	no	—
184	#462	—	53	4	no	—
185	#451	—	106	2	no	—
186	#446	—	53	2	no	—
187	#446	—	53	2	no	—
188	#446	—	106	1	no	—
189	#446	—	106	1	no	—
190	#441	—	106	4	no	—
191	C₄×Q₁₆	—	106	1	no	—
192	C₄×Q₁₆	—	106	1	no	—
193	#432	C₂×D₅₃	1	2	yes	—
194	C₂×C₄⋊C₈	—	53	1	no	—
195	C₂×C₄⋊C₈	—	106	1	no	—
196	C₂×#97	—	106	1	no	—
197	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
198	#335	C₂×D₅₃	1	2	yes	—
199	#334	C₂×D₅₃	1	2	yes	—
200	#332	—	53	4	no	—
201	#332	—	53	4	no	—
202	C₅₃	#19290	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	53	1	no	—
204	C₂×Q₁₆	—	106	4	no	—
205	C₂×Q₁₆	—	106	1	no	—
206	C₂×Q₁₆	—	106	1	no	—
207	C₂²×C₈	—	106	1	no	—
208	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
209	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
210	C₄⋊Q₈	—	53	2	no	—
211	C₄⋊Q₈	—	53	2	no	—
212	#125	—	106	2	no	—
213	C₂²⋊Q₈	—	106	2	no	—
214	C₂²⋊Q₈	—	106	2	no	—
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	—	53	2	no	—
217	C₄×Q₈	—	106	1	no	—
218	C₄×Q₈	—	106	1	no	—
219	C₄×Q₈	—	106	2	no	—
220	C₄×Q₈	—	106	2	no	—
221	#117	—	106	2	no	—
222	C₂×C₄⋊C₄	—	53	1	no	—
223	C₂×C₄⋊C₄	—	106	1	no	—
224	C₂×C₄²	—	106	1	no	—
225	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
226	#107	—	106	1	no	—
227	#107	—	106	1	no	—
228	#106	—	106	4	no	—
229	C₄⋊C₈	—	53	2	no	—
230	C₄⋊C₈	—	106	2	no	—
231	C₄⋊C₈	—	106	2	no	—
232	C₄⋊C₈	—	106	1	no	—
233	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
234	#103	—	106	4	no	—
235	#103	—	106	4	no	—
236	#103	—	106	2	no	—
237	#103	—	106	2	no	—
238	C₂²⋊C₈	—	2	2	no	—
239	#97	—	2	2	no	—
240	#97	—	106	2	no	—
241	#97	—	106	2	no	—
242	C₄×C₈	D₂₁₂	1	1	yes	characteristic
243	C₄×C₈	D₂₁₂	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
246	C₂×Q₈	—	53	2	no	—
247	C₂×Q₈	—	106	1	no	—
248	C₂×Q₈	—	106	2	no	—
249	C₂×Q₈	—	106	2	no	—
250	C₂×Q₈	—	212	2	no	—
251	C₂²×C₄	—	106	1	no	—
252	C₂²×C₄	—	106	1	no	—
253	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
255	C₂²×C₄	#11719949	1	1	yes	characteristic
256	Q₁₆	—	212	2	no	—
257	Q₁₆	—	212	8	no	—
258	Q₁₆	—	212	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	212	2	no	—
264	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
265	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
266	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
267	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
268	C₄⋊C₄	—	53	2	no	—
269	C₄⋊C₄	—	106	1	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	—	106	2	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	106	2	no	—
276	C₄⋊C₄	—	212	2	no	—
277	C₄⋊C₄	—	212	2	no	—
278	C₂²⋊C₄	—	212	2	no	—
279	C₄²	—	106	2	no	—
280	C₄²	—	106	2	no	—
281	C₄²	—	212	2	no	—
282	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
283	C₄²	D₄×D₅₃	1	1	yes	characteristic
284	C₄²	#11719949	1	1	yes	characteristic
285	C₄²	C₂³×D₅₃	1	1	yes	characteristic
286	C₂³	#13133192	1	1	yes	characteristic
287	Q₈	—	106	4	no	—
288	Q₈	—	212	2	no	—
289	Q₈	—	212	2	no	—
290	Q₈	—	212	4	no	—
291	Q₈	—	212	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	212	2	no	—
299	C₂×C₄	—	212	2	no	—
300	C₂×C₄	—	212	2	no	—
301	C₂×C₄	—	212	2	no	—
302	C₂×C₄	#13133192	1	1	yes	characteristic
303	C₂×C₄	#13133192	1	1	yes	characteristic
304	C₂×C₄	#13133192	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
307	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	4	no	—
312	C₈	—	212	2	no	—
313	C₈	#13133250	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	#16810899	1	2	yes	—
317	C₂²	C₂×#13133192	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	424	2	no	—
323	C₄	—	424	2	no	—
324	C₄	—	212	2	no	—
325	C₄	—	212	2	no	—
326	C₄	C₂×#13133250	1	1	yes	characteristic
327	C₄	#16811420	1	2	yes	—
328	C₄	#16811423	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	#20284189	1	1	yes	characteristic
331	C₂	#20284325	1	1	yes	characteristic
332	C₂	C₂×#16810899	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133192
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1568	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#19290
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
4. #20274270	C₂	semidirect
5. #20274234	C₂	semidirect
6. #20274210	C₂	semidirect
202. C₅₃	#19290	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20284846	2
C₂×#16810753	2
#20274270	2
#20274234	2
#20274210	2
#20273416	2
C₅₃×#1568	2
#19290	53