Subgroups — #27367356

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20286057	C₂	1	2	yes	maximal
2	#20283649	C₂	1	1	yes	maximal, characteristic
3	#20274218	C₂	1	4	yes	maximal, complemented
4	#20274116	C₂	1	2	yes	maximal, complemented
5	#20274106	C₂	1	2	yes	maximal, complemented
6	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
7	#20273405	C₂	1	2	yes	maximal
8	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	1	yes	characteristic
10	#16810918	—	2	2	no	—
11	#16810915	C₂²	1	4	yes	—
12	#16810862	—	2	4	no	—
13	#16810833	—	2	2	no	—
14	#16810830	C₂²	1	4	yes	—
15	#16810753	—	2	1	no	—
16	#16810753	C₂²	1	2	yes	—
17	C₈×Dic₁₀₆	—	2	1	no	—
18	C₈×Dic₁₀₆	C₂²	1	2	yes	—
19	#16810649	—	2	1	no	—
20	#16810647	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#88322	C₂²	1	2	yes	—
22	#16810578	—	2	1	no	—
23	#16810376	C₂²	1	2	yes	—
24	#16810376	C₂²	1	2	yes	—
25	#16810372	C₂²	1	4	yes	—
26	#16810369	C₂²	1	4	yes	—
27	C₅₃×#533	C₂²	1	2	yes	—
28	C₅₃×#500	C₂²	1	1	yes	characteristic
29	C₅₃×#500	C₂²	1	1	yes	characteristic
30	C₅₃×#499	—	2	1	no	—
31	C₅₃×#499	—	2	1	no	—
32	C₅₃×#498	C₂²	1	2	yes	—
33	C₅₃×#427	—	2	1	no	—
34	C₅₃×#427	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#11719925	—	2	4	no	—
37	C₂×#11719917	—	2	1	no	—
38	C₂²×#88322	—	2	2	no	—
39	#13133217	D₄	1	1	yes	characteristic
40	#13133217	D₄	1	1	yes	characteristic
41	#13133209	—	2	1	no	—
42	#13133196	—	2	1	no	—
43	#13133196	D₄	1	2	yes	—
44	C₄×Dic₁₀₆	—	2	1	no	—
45	C₄×Dic₁₀₆	C₂³	1	2	yes	—
46	#13133176	—	2	1	no	—
47	#13133150	—	2	4	no	—
48	C₄⋊#88322	—	2	1	no	—
49	C₄⋊#88322	—	2	2	no	—
50	C₄⋊#88322	C₂³	1	2	yes	—
51	C₄⋊#88322	C₂³	1	4	yes	—
52	#13133146	—	2	2	no	—
53	#13133146	—	2	2	no	—
54	#13133141	—	2	4	no	—
55	#13133140	—	2	4	no	—
56	#13133127	—	2	2	no	—
57	C₈×Dic₅₃	—	2	2	no	—
58	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	#125×C₅₃	—	2	2	no	—
63	C₂²⋊Q₈×C₅₃	—	2	2	no	—
64	C₂²⋊Q₈×C₅₃	—	2	2	no	—
65	C₄×Q₈×C₅₃	—	2	2	no	—
66	C₄×Q₈×C₅₃	—	2	2	no	—
67	#117×C₅₃	—	2	2	no	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#107×C₅₃	—	2	1	no	—
70	#107×C₅₃	—	2	1	no	—
71	#107×C₅₃	—	2	1	no	—
72	#107×C₅₃	—	2	1	no	—
73	#106×C₅₃	—	2	2	no	—
74	#106×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	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂×Dic₁₀₆	—	2	1	no	—
85	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₂²⋊C₄×C₅₃	—	4	2	no	—
99	C₄²×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	#41	1	1	yes	characteristic
103	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊Dic₅₃	—	2	2	no	—
105	#11719925	—	8	4	no	—
106	C₄⋊Dic₅₃	—	2	1	no	—
107	C₄⋊Dic₅₃	—	2	1	no	—
108	C₄⋊Dic₅₃	—	2	1	no	—
109	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
110	#11719919	—	2	2	no	—
111	#11719919	—	2	2	no	—
112	C₄×Dic₅₃	—	2	2	no	—
113	#11719917	—	4	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	2	4	no	—
116	C₂×#88322	—	8	2	no	—
117	C₂³×C₅₃	#121	1	1	yes	characteristic
118	Q₈×C₅₃	—	4	2	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	Q₈×C₅₃	—	8	2	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₅₃	#137	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₅₃	—	2	2	no	—
135	C₂×Dic₅₃	—	2	2	no	—
136	Dic₁₀₆	—	2	2	no	—
137	Dic₁₀₆	—	2	2	no	—
138	Dic₁₀₆	—	4	2	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
142	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
143	#88322	—	4	2	no	—
144	#88322	—	4	4	no	—
145	#19211	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	#462	1	2	yes	—
149	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	8	2	no	—
154	C₄×C₅₃	—	8	2	no	—
155	C₄×C₅₃	#462	1	2	yes	—
156	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
157	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
158	C₄×C₅₃	#576	1	1	yes	characteristic
159	C₄×C₅₃	#578	1	1	yes	characteristic
160	Dic₅₃	—	4	2	no	—
161	Dic₅₃	—	4	2	no	—
162	#3249	—	53	2	no	—
163	#3160	D₅₃	1	1	yes	characteristic
164	#2960	—	53	1	no	—
165	#1692	—	53	4	no	—
166	#1676	—	53	2	no	—
167	#1674	—	53	2	no	—
168	C₂×#335	—	53	1	no	—
169	#1550	—	53	2	no	—
170	C₂×C₅₃	—	2	2	no	—
171	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
172	C₂×C₅₃	#3061	1	1	yes	characteristic
173	C₂×C₅₃	#3067	1	1	yes	characteristic
174	#533	—	53	1	no	—
175	#533	C₂×D₅₃	1	2	yes	—
176	#500	C₂×D₅₃	1	1	yes	characteristic
177	#500	C₂×D₅₃	1	1	yes	characteristic
178	#499	—	2	1	no	—
179	#499	—	2	1	no	—
180	#498	C₂×D₅₃	1	2	yes	—
181	#477	—	53	4	no	—
182	#477	—	106	2	no	—
183	#467	—	106	4	no	—
184	#462	—	53	4	no	—
185	#462	—	106	2	no	—
186	#446	—	53	2	no	—
187	#446	—	106	1	no	—
188	C₈×Q₈	—	53	2	no	—
189	C₈×Q₈	—	106	1	no	—
190	#427	—	2	1	no	—
191	#427	—	2	1	no	—
192	#424	—	106	1	no	—
193	#423	—	53	1	no	—
194	C₂×C₄⋊C₈	—	53	2	no	—
195	#405	—	106	1	no	—
196	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
197	#335	—	53	2	no	—
198	#335	—	53	2	no	—
199	#333	—	53	4	no	—
200	#332	—	53	4	no	—
201	C₅₃	#19211	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	—	106	4	no	—
203	C₂×M₁₆	—	106	1	no	—
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄⋊Q₈	—	53	1	no	—
208	C₄⋊Q₈	—	53	1	no	—
209	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
210	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
211	#125	—	2	2	no	—
212	#125	—	106	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	53	2	no	—
216	C₂²⋊Q₈	—	106	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	53	2	no	—
220	C₄×Q₈	—	106	1	no	—
221	#117	—	2	2	no	—
222	#117	—	106	1	no	—
223	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	—	106	4	no	—
229	#106	—	2	2	no	—
230	#106	—	2	2	no	—
231	C₄⋊C₈	—	53	2	no	—
232	C₄⋊C₈	—	53	4	no	—
233	C₄⋊C₈	—	106	2	no	—
234	C₄⋊C₈	—	106	2	no	—
235	C₄⋊C₈	—	106	2	no	—
236	C₄⋊C₈	—	106	1	no	—
237	#103	—	106	4	no	—
238	#103	—	106	4	no	—
239	#97	—	106	2	no	—
240	C₄×C₈	—	106	2	no	—
241	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
242	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	53	2	no	—
248	C₂×Q₈	—	106	1	no	—
249	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
250	C₂²×C₄	#11719947	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
252	Q₁₆	—	424	4	no	—
253	M₁₆	—	212	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	106	4	no	—
256	C₂×C₈	—	106	2	no	—
257	C₂×C₈	—	424	2	no	—
258	C₂×C₈	#11719925	1	4	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	53	2	no	—
270	C₄⋊C₄	—	106	1	no	—
271	C₄⋊C₄	—	106	1	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	106	1	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	106	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	106	2	no	—
279	C₄²	D₄×D₅₃	1	1	yes	characteristic
280	C₄²	#11719947	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₄²	C₂³×D₅₃	1	1	yes	characteristic
283	C₂³	#13133188	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	106	2	no	—
288	Q₈	—	106	2	no	—
289	Q₈	—	212	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	#13133188	1	1	yes	characteristic
298	C₂×C₄	#13133188	1	1	yes	characteristic
299	C₂×C₄	#13133188	1	1	yes	characteristic
300	C₂×C₄	C₂×#11719925	1	2	yes	—
301	C₂×C₄	#13133248	1	2	yes	—
302	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	212	2	no	—
308	C₈	—	212	4	no	—
309	C₈	C₂×#11719925	1	2	yes	—
310	C₈	C₂×#11719925	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	#16810830	1	2	yes	—
314	C₂²	C₂×#13133188	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	212	2	no	—
321	C₄	—	212	2	no	—
322	C₄	#16810830	1	2	yes	—
323	C₄	C₂²×#11719925	1	1	yes	characteristic
324	C₄	C₂×#13133248	1	1	yes	characteristic
325	C₄	#16811425	1	1	yes	characteristic
326	C₄	#16811437	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	C₂×#16810830	1	1	yes	characteristic
329	C₂	#20284381	1	1	yes	characteristic
330	C₂	#20284431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#19211
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #20274218	C₂	semidirect
4. #20274116	C₂	semidirect
5. #20274106	C₂	semidirect
87. C₂×C₈×C₅₃	Q₁₆	semidirect
201. C₅₃	#19211	semidirect
258. C₂×C₈	#11719925	semidirect

Maximal subgroups

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

subgroup	index
#20286057	2
#20283649	2
#20274218	2
#20274116	2
#20274106	2
C₂×#16810376	2
#20273405	2
C₅₃×#3160	2
#19211	53