Subgroups — #27367352

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#20285229	C₂	1	1	yes	maximal, characteristic
3	#20274243	C₂	1	2	yes	maximal, complemented
4	#20274233	C₂	1	2	yes	maximal, complemented
5	#20274097	C₂	1	4	yes	maximal, complemented
6	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
7	#20273406	C₂	1	2	yes	maximal
8	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811028	C₂²	1	1	yes	characteristic
11	#16811028	C₂²	1	1	yes	characteristic
12	#16811025	—	2	1	no	—
13	#16811025	—	2	1	no	—
14	#16811021	C₂²	1	2	yes	—
15	#16810918	C₂²	1	4	yes	—
16	#16810915	—	2	2	no	—
17	#16810862	—	2	4	no	—
18	#16810833	C₂²	1	4	yes	—
19	#16810830	—	2	2	no	—
20	#16810661	—	2	1	no	—
21	#16810661	—	2	1	no	—
22	C₂×C₄⋊#88322	C₂²	1	2	yes	—
23	#16810374	C₂²	1	2	yes	—
24	#16810374	C₂²	1	2	yes	—
25	#16810371	C₂²	1	4	yes	—
26	#16810368	C₂²	1	4	yes	—
27	C₅₃×#533	C₂²	1	1	yes	characteristic
28	C₅₃×#446	—	2	1	no	—
29	C₅₃×#446	C₂²	1	2	yes	—
30	C₈×Q₈×C₅₃	—	2	1	no	—
31	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#424	—	2	1	no	—
33	C₅₃×#423	C₂²	1	1	yes	characteristic
34	C₅₃×#405	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#11719925	—	2	4	no	—
37	C₂²×#88322	—	2	2	no	—
38	#13133217	C₂³	1	2	yes	—
39	#13133217	C₂³	1	2	yes	—
40	#13133209	—	2	2	no	—
41	#13133196	—	2	2	no	—
42	#13133196	—	2	2	no	—
43	C₄×Dic₁₀₆	—	2	2	no	—
44	C₄×Dic₁₀₆	—	2	2	no	—
45	#13133176	—	2	2	no	—
46	#13133151	—	2	1	no	—
47	#13133151	—	2	1	no	—
48	#13133151	—	2	1	no	—
49	#13133151	—	2	1	no	—
50	#13133150	—	2	4	no	—
51	#13133149	—	2	2	no	—
52	#13133149	—	2	2	no	—
53	C₄⋊#88322	—	2	2	no	—
54	C₄⋊#88322	C₂³	1	4	yes	—
55	#13133141	—	2	4	no	—
56	#13133140	—	2	4	no	—
57	C₂×M₁₆×C₅₃	—	2	1	no	—
58	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
62	#125×C₅₃	—	2	1	no	—
63	C₂²⋊Q₈×C₅₃	—	2	1	no	—
64	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
65	C₄×Q₈×C₅₃	—	2	1	no	—
66	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
67	#117×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
73	#97×C₅₃	—	2	2	no	—
74	C₄×C₈×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	1	no	—
80	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
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	2	no	—
85	C₂×Dic₁₀₆	—	2	2	no	—
86	M₁₆×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
100	C₂²⋊C₄×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	#41	1	1	yes	characteristic
105	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊Dic₅₃	—	4	2	no	—
107	#11719925	—	8	4	no	—
108	C₄⋊Dic₅₃	—	2	2	no	—
109	C₄⋊Dic₅₃	—	2	2	no	—
110	C₄⋊Dic₅₃	—	2	2	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	#11719919	—	4	2	no	—
113	#11719919	—	4	2	no	—
114	C₄×Dic₅₃	—	4	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	—	2	4	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	4	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	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₅₃	#137	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	Dic₁₀₆	—	4	2	no	—
135	Dic₁₀₆	—	4	2	no	—
136	Dic₁₀₆	—	8	2	no	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	4	2	no	—
140	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
141	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
142	#88322	—	4	4	no	—
143	#19210	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	#462	1	2	yes	—
147	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	—	4	2	no	—
153	C₄×C₅₃	#462	1	2	yes	—
154	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
155	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
156	C₄×C₅₃	#577	1	2	yes	—
157	Dic₅₃	—	8	2	no	—
158	Dic₅₃	—	8	2	no	—
159	#3249	—	53	2	no	—
160	#3160	—	53	1	no	—
161	#2960	D₅₃	1	1	yes	characteristic
162	#1696	—	53	2	no	—
163	#1694	—	53	2	no	—
164	#1672	—	53	4	no	—
165	C₂×#334	—	53	1	no	—
166	#1550	—	53	2	no	—
167	C₂×C₅₃	—	2	2	no	—
168	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
169	C₂×C₅₃	#3060	1	1	yes	characteristic
170	C₂×C₅₃	#3067	1	1	yes	characteristic
171	#533	—	53	2	no	—
172	#533	C₂×D₅₃	1	1	yes	characteristic
173	#500	—	53	1	no	—
174	#500	—	53	1	no	—
175	#499	—	106	1	no	—
176	#499	—	106	1	no	—
177	#498	—	53	2	no	—
178	#477	—	53	4	no	—
179	#477	—	106	2	no	—
180	#467	—	106	4	no	—
181	#462	—	53	4	no	—
182	#462	—	106	2	no	—
183	#446	—	2	1	no	—
184	#446	C₂×D₅₃	1	2	yes	—
185	C₈×Q₈	—	2	1	no	—
186	C₈×Q₈	C₂×D₅₃	1	2	yes	—
187	#427	—	106	1	no	—
188	#427	—	106	1	no	—
189	#424	—	2	1	no	—
190	#423	C₂×D₅₃	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	53	2	no	—
192	#405	—	2	1	no	—
193	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
194	#334	—	53	2	no	—
195	#334	—	53	2	no	—
196	#333	—	53	4	no	—
197	#332	—	53	4	no	—
198	C₅₃	#19210	1	1	yes	characteristic, complemented, Sylow
199	C₂×Q₁₆	—	106	4	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂²×C₈	—	106	2	no	—
202	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	53	2	no	—
205	C₄⋊Q₈	—	53	2	no	—
206	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
207	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
208	#125	—	2	1	no	—
209	#125	—	106	2	no	—
210	C₂²⋊Q₈	—	2	1	no	—
211	C₂²⋊Q₈	—	106	2	no	—
212	C₂²⋊Q₈	—	106	2	no	—
213	C₂²⋊Q₈	D₂₁₂	1	2	yes	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	106	2	no	—
216	C₄×Q₈	—	106	2	no	—
217	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
218	#117	—	2	1	no	—
219	#117	—	106	2	no	—
220	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
221	#107	—	106	1	no	—
222	#107	—	106	1	no	—
223	#107	—	106	1	no	—
224	#107	—	106	1	no	—
225	#107	—	106	4	no	—
226	#106	—	106	2	no	—
227	#106	—	106	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	53	4	no	—
232	C₄⋊C₈	—	106	2	no	—
233	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
234	#103	—	106	4	no	—
235	#103	—	106	4	no	—
236	#97	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	106	2	no	—
244	C₂×Q₈	—	106	2	no	—
245	C₂×Q₈	C₂×D₂₁₂	1	2	yes	—
246	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
247	C₂²×C₄	#11719945	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
249	Q₁₆	—	424	4	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	106	4	no	—
254	C₂×C₈	—	424	2	no	—
255	C₂×C₈	#11719925	1	4	yes	complemented
256	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	106	2	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₄⋊C₄	—	106	2	no	—
269	C₄⋊C₄	—	212	2	no	—
270	C₄⋊C₄	—	212	2	no	—
271	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	212	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	212	2	no	—
276	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
277	C₄²	#11719945	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
279	C₄²	C₂³×D₅₃	1	1	yes	characteristic
280	C₂³	#13133191	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	424	2	no	—
284	Q₈	—	212	2	no	—
285	Q₈	—	212	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	#13133191	1	1	yes	characteristic
293	C₂×C₄	#13133191	1	1	yes	characteristic
294	C₂×C₄	#13133191	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719925	1	2	yes	—
296	C₂×C₄	#13133247	1	2	yes	—
297	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
299	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	212	4	no	—
304	C₈	C₂×#11719925	1	2	yes	—
305	C₈	C₂×#11719925	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#16810833	1	2	yes	—
309	C₂²	C₂×#13133191	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	424	2	no	—
316	C₄	—	424	2	no	—
317	C₄	#16810833	1	2	yes	—
318	C₄	C₂²×#11719925	1	1	yes	characteristic
319	C₄	C₂×#13133247	1	1	yes	characteristic
320	C₄	#16811435	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	C₂×#16810833	1	1	yes	characteristic
323	C₂	#20284377	1	1	yes	characteristic
324	C₂	#20284435	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#19210
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #20274243	C₂	semidirect
4. #20274233	C₂	semidirect
5. #20274097	C₂	semidirect
89. C₂×C₈×C₅₃	Q₁₆	semidirect
198. C₅₃	#19210	semidirect
255. 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
#20285229	2
#20274243	2
#20274233	2
#20274097	2
C₂×#16810374	2
#20273406	2
C₅₃×#2960	2
#19210	53