Subgroups — #27366457

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284750	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20274116	C₂	1	2	yes	maximal, complemented
4	#20274106	C₂	1	2	yes	maximal, complemented
5	#20274094	C₂	1	4	yes	maximal, complemented
6	#20273463	C₂	1	1	yes	maximal, characteristic
7	#20273372	C₂	1	2	yes	maximal
8	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#16810956	—	2	2	no	—
11	#16810915	C₂²	1	4	yes	—
12	#16810830	C₂²	1	4	yes	—
13	#16810778	—	2	2	no	—
14	C₈×Dic₁₀₆	—	2	1	no	—
15	C₈×Dic₁₀₆	—	2	1	no	—
16	C₈×Dic₁₀₆	C₂²	1	2	yes	—
17	C₈×Dic₁₀₆	C₂²	1	2	yes	—
18	C₄×#11719925	—	2	4	no	—
19	#16810676	C₂²	1	2	yes	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×#13133146	—	2	1	no	—
22	C₂×C₈×Dic₅₃	—	2	1	no	—
23	#16810376	C₂²	1	2	yes	—
24	#16810374	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	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
37	C₂×#11719925	—	2	4	no	—
38	C₂²×#88322	—	2	1	no	—
39	C₄×Dic₁₀₆	—	2	1	no	—
40	C₄×Dic₁₀₆	—	2	1	no	—
41	C₄×Dic₁₀₆	D₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
44	C₂×#11719919	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	#13133150	—	2	4	no	—
47	C₄⋊#88322	—	2	1	no	—
48	C₄⋊#88322	C₂³	1	4	yes	—
49	C₄⋊#88322	C₂³	1	2	yes	—
50	#13133146	—	2	2	no	—
51	#13133146	—	2	2	no	—
52	#13133141	—	2	4	no	—
53	#13133140	—	2	4	no	—
54	C₂²⋊#88322	—	2	2	no	—
55	C₄×#88322	—	2	2	no	—
56	C₈×Dic₅₃	—	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	C₂²×Dic₅₃	—	2	1	no	—
85	C₂²×Dic₅₃	—	2	1	no	—
86	C₂×Dic₁₀₆	—	2	1	no	—
87	C₂×Dic₁₀₆	—	2	4	no	—
88	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
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	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	4	2	no	—
100	C₄⋊C₄×C₅₃	—	4	2	no	—
101	C₂²⋊C₄×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	—	4	2	no	—
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₃	#41	1	1	yes	characteristic
106	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
107	#11719925	—	4	8	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	C₄×Dic₅₃	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	2	4	no	—
116	C₂×#88322	—	4	2	no	—
117	C₂³×C₅₃	#121	1	1	yes	characteristic
118	Q₈×C₅₃	—	4	4	no	—
119	Q₈×C₅₃	—	8	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	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	—	2	4	no	—
133	C₂×Dic₅₃	—	2	2	no	—
134	C₂×Dic₅₃	—	2	2	no	—
135	C₂×Dic₅₃	—	4	2	no	—
136	Dic₁₀₆	—	2	4	no	—
137	Dic₁₀₆	—	2	2	no	—
138	Dic₁₀₆	#135	1	4	yes	—
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	4	no	—
144	#88322	—	4	2	no	—
145	#19118	—	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×#135	1	2	yes	—
158	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
159	Dic₅₃	—	2	4	no	—
160	Dic₅₃	—	4	2	no	—
161	#3160	D₅₃	1	1	yes	characteristic
162	#3104	—	53	2	no	—
163	C₂×C₈×Q₈	—	53	1	no	—
164	#1676	—	53	2	no	—
165	#1674	—	53	2	no	—
166	#1672	—	53	4	no	—
167	#1564	—	53	1	no	—
168	#1542	—	53	2	no	—
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
171	C₂×C₅₃	#3050	1	1	yes	characteristic
172	C₂×C₅₃	#3067	1	1	yes	characteristic
173	#533	C₂×D₅₃	1	2	yes	—
174	C₂×C₄×Q₈	—	53	1	no	—
175	#500	C₂×D₅₃	1	1	yes	characteristic
176	#500	C₂×D₅₃	1	1	yes	characteristic
177	#499	—	2	1	no	—
178	#499	—	2	1	no	—
179	#498	C₂×D₅₃	1	2	yes	—
180	#484	—	106	2	no	—
181	#477	—	53	4	no	—
182	#462	—	53	4	no	—
183	#451	—	106	2	no	—
184	C₈×Q₈	—	53	2	no	—
185	C₈×Q₈	—	53	2	no	—
186	C₈×Q₈	—	106	1	no	—
187	C₈×Q₈	—	106	1	no	—
188	C₄×Q₁₆	—	106	4	no	—
189	#431	—	53	2	no	—
190	#427	—	2	1	no	—
191	#427	—	2	1	no	—
192	C₂×C₄⋊C₈	—	53	1	no	—
193	C₂×C₄⋊C₈	—	106	1	no	—
194	C₂×C₄×C₈	—	106	1	no	—
195	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
196	#335	—	53	2	no	—
197	#334	—	53	2	no	—
198	#333	—	53	4	no	—
199	#332	—	53	4	no	—
200	C₅₃	#19118	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	53	1	no	—
202	C₂×Q₁₆	—	106	4	no	—
203	C₂²×C₈	—	106	1	no	—
204	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
206	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
207	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
208	#125	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	53	2	no	—
214	C₄×Q₈	—	53	2	no	—
215	C₄×Q₈	—	106	1	no	—
216	C₄×Q₈	—	106	1	no	—
217	#117	—	2	2	no	—
218	C₂×C₄⋊C₄	—	53	1	no	—
219	C₂×C₄⋊C₄	—	106	1	no	—
220	C₂×C₄²	—	106	1	no	—
221	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#107	—	106	4	no	—
227	#106	—	2	2	no	—
228	#106	—	2	2	no	—
229	C₄⋊C₈	—	53	2	no	—
230	C₄⋊C₈	—	53	4	no	—
231	C₄⋊C₈	—	106	2	no	—
232	C₄⋊C₈	—	106	2	no	—
233	C₄⋊C₈	—	106	1	no	—
234	#103	—	106	4	no	—
235	#103	—	106	4	no	—
236	C₂²⋊C₈	—	106	2	no	—
237	C₄×C₈	—	106	2	no	—
238	C₄×C₈	—	106	2	no	—
239	C₄×C₈	—	106	2	no	—
240	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
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	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	53	2	no	—
247	C₂×Q₈	—	106	4	no	—
248	C₂×Q₈	—	106	1	no	—
249	C₂²×C₄	—	106	1	no	—
250	C₂²×C₄	—	106	1	no	—
251	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₄	#11719947	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
254	Q₁₆	—	212	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	106	4	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	212	2	no	—
259	C₂×C₈	#11719925	1	4	yes	complemented
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	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	53	2	no	—
271	C₄⋊C₄	—	106	1	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	106	2	no	—
277	C₄²	—	106	2	no	—
278	C₄²	D₄×D₅₃	1	1	yes	characteristic
279	C₄²	#11719947	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
281	C₄²	C₂³×D₅₃	1	1	yes	characteristic
282	C₂³	#13133188	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	53	4	no	—
286	Q₈	—	106	4	no	—
287	Q₈	—	106	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	106	4	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	106	2	no	—
295	C₂×C₄	—	212	2	no	—
296	C₂×C₄	#13133188	1	1	yes	characteristic
297	C₂×C₄	#13133188	1	1	yes	characteristic
298	C₂×C₄	#13133188	1	1	yes	characteristic
299	C₂×C₄	C₂×#11719925	1	2	yes	—
300	C₂×C₄	#13133248	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	212	4	no	—
307	C₈	—	212	2	no	—
308	C₈	C₂×#11719925	1	2	yes	—
309	C₈	C₂×#11719925	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#16810830	1	2	yes	—
313	C₂²	C₂×#13133188	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	106	4	no	—
320	C₄	—	212	2	no	—
321	C₄	#16810830	1	2	yes	—
322	C₄	C₂²×#11719925	1	1	yes	characteristic
323	C₄	#135×D₅₃	1	2	yes	—
324	C₄	C₂×#13133248	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×#16810830	1	1	yes	characteristic
327	C₂	#20284292	1	1	yes	characteristic
328	C₂	#20284431	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#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	#19118
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #20274116	C₂	semidirect
4. #20274106	C₂	semidirect
5. #20274094	C₂	semidirect
90. C₂×C₈×C₅₃	Q₁₆	semidirect
200. C₅₃	#19118	semidirect
259. 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
#20284750	2
C₂×C₈×Dic₁₀₆	2
#20274116	2
#20274106	2
#20274094	2
#20273463	2
#20273372	2
C₅₃×#3160	2
#19118	53