Subgroups — #27372027

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285836	C₂	1	2	yes	maximal, complemented
2	#20285294	C₂	1	1	yes	maximal, characteristic
3	#20285172	C₂	1	2	yes	maximal, complemented
4	#20274385	C₂	1	4	yes	maximal, complemented
5	#20274332	C₂	1	2	yes	maximal, complemented
6	#20274308	C₂	1	2	yes	maximal, complemented
7	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133248	—	2	4	no	—
10	#16811173	C₂²	1	2	yes	—
11	#16811028	C₂²	1	1	yes	characteristic
12	#16811026	C₂²	1	4	yes	—
13	#16811015	—	2	4	no	—
14	#16811002	C₂²	1	4	yes	—
15	#16811000	C₂²	1	1	yes	characteristic
16	#16810997	C₂²	1	4	yes	—
17	#16810971	C₂²	1	4	yes	—
18	#16810970	C₂²	1	2	yes	—
19	#16810936	—	2	4	no	—
20	#16810900	—	2	4	no	—
21	#16810859	—	2	2	no	—
22	C₄×D₄₂₄	—	2	2	no	—
23	#16810676	C₂²	1	1	yes	characteristic
24	C₄×#11719917	C₂²	1	1	yes	characteristic
25	#16810345	C₂²	1	2	yes	—
26	C₈×#88322	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#500	—	2	2	no	—
29	C₅₃×#500	C₂²	1	2	yes	—
30	C₅₃×#500	C₂²	1	2	yes	—
31	C₂×#107×C₅₃	—	2	2	no	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#11719945	—	2	2	no	—
34	#13133248	—	4	16	no	—
35	C₂×#11719925	—	2	8	no	—
36	C₂×#11719924	—	2	8	no	—
37	C₂×D₄₂₄	—	2	2	no	—
38	C₂×#11719917	D₄	1	2	yes	—
39	#13133217	C₂³	1	2	yes	—
40	#13133214	C₂³	1	2	yes	—
41	#13133205	—	2	2	no	—
42	#13133191	—	2	2	no	—
43	#13133191	—	2	2	no	—
44	C₄×D₂₁₂	—	2	2	no	—
45	C₄×D₂₁₂	—	2	2	no	—
46	#13133151	—	2	2	no	—
47	#13133150	—	2	8	no	—
48	C₄⋊#88322	D₄	1	2	yes	—
49	#13133139	—	2	4	no	—
50	#13133137	—	2	8	no	—
51	C₂²⋊#88322	D₄	1	2	yes	—
52	#13133128	D₄	1	2	yes	—
53	C₄×#88322	C₂³	1	2	yes	—
54	C₄×#88322	C₂³	1	2	yes	—
55	C₂²×Q₈×C₅₃	—	2	2	no	—
56	C₂²×C₈×C₅₃	D₄	1	2	yes	—
57	C₄⋊Q₈×C₅₃	—	2	2	no	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	—	2	2	no	—
60	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	—	2	4	no	—
65	#107×C₅₃	—	2	4	no	—
66	C₄×C₈×C₅₃	D₄	1	2	yes	—
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₅₃	—	2	4	no	—
70	C₂×Q₈×C₅₃	—	4	2	no	—
71	C₂×Q₈×C₅₃	—	4	8	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₃	—	4	2	no	—
77	#11719945	—	4	8	no	—
78	C₂×D₂₁₂	—	2	2	no	—
79	C₂×D₂₁₂	—	2	2	no	—
80	C₂×D₂₁₂	—	2	2	no	—
81	C₂×C₄×D₅₃	—	4	2	no	—
82	C₂×Dic₁₀₆	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	2	4	no	—
88	C₄⋊C₄×C₅₃	—	4	2	no	—
89	C₄⋊C₄×C₅₃	—	4	2	no	—
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
94	#11719925	—	4	16	no	—
95	#11719924	—	4	16	no	—
96	#11719921	—	4	2	no	—
97	#11719921	—	4	2	no	—
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	—	2	2	no	—
100	#11719917	D₈	1	8	yes	complemented
101	C₂×#88322	C₂×D₄	1	4	yes	—
102	C₂×#88322	C₂×D₄	1	4	yes	—
103	D₄₂₄	—	4	4	no	—
104	C₂³×C₅₃	#127	1	1	yes	characteristic
105	C₂²×D₅₃	—	4	2	no	—
106	Q₈×C₅₃	—	4	8	no	—
107	Q₈×C₅₃	—	8	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂²⋊D₅₃	—	8	4	no	—
120	C₂×Dic₅₃	—	4	2	no	—
121	D₂₁₂	—	4	4	no	—
122	D₂₁₂	—	4	4	no	—
123	D₂₁₂	—	4	4	no	—
124	C₄×D₅₃	—	8	4	no	—
125	Dic₁₀₆	—	4	4	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	#137	1	4	yes	complemented
128	#88322	—	2	4	no	—
129	#88322	C₂×D₈	1	8	yes	—
130	#19630	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	#493	1	2	yes	—
134	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
135	C₂×D₅₃	—	8	2	no	—
136	C₂×D₅₃	—	8	2	no	—
137	C₂×D₅₃	—	8	2	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	8	4	no	—
141	C₄×C₅₃	#493	1	2	yes	—
142	C₄×C₅₃	C₂²×D₈	1	2	yes	—
143	C₄×C₅₃	C₂×#137	1	2	yes	—
144	C₄×C₅₃	#578	1	2	yes	—
145	Dic₅₃	—	8	2	no	—
146	#3226	—	53	2	no	—
147	#3169	—	53	1	no	—
148	C₂×#500	D₅₃	1	1	yes	characteristic, complemented
149	#3152	—	53	2	no	—
150	#1722	—	53	4	no	—
151	#1713	—	53	2	no	—
152	#1706	—	53	2	no	—
153	C₈×M₁₆	—	53	1	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
156	C₂×C₅₃	#3152	1	1	yes	characteristic
157	C₂×C₅₃	#3156	1	1	yes	characteristic
158	D₅₃	—	16	2	no	—
159	C₂×#137	—	106	4	no	—
160	#532	—	53	2	no	—
161	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
162	#500	—	2	2	no	—
163	#500	—	53	4	no	—
164	#500	—	53	1	no	—
165	#500	C₂×D₅₃	1	2	yes	—
166	#500	C₂×D₅₃	1	2	yes	—
167	#497	—	106	4	no	—
168	#494	—	53	4	no	—
169	#493	—	53	1	no	—
170	#492	—	53	4	no	—
171	#486	—	53	4	no	—
172	#486	—	53	2	no	—
173	#480	—	106	4	no	—
174	#474	—	106	4	no	—
175	#466	—	106	2	no	—
176	C₄×D₈	—	106	2	no	—
177	#431	—	53	1	no	—
178	C₂×#107	—	2	2	no	—
179	C₄×M₁₆	—	53	1	no	—
180	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
181	#322	—	53	2	no	—
182	C₈²	—	53	2	no	—
183	C₅₃	#19630	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	106	2	no	—
185	C₂²×Q₈	—	2	2	no	—
186	#137	—	212	16	no	—
187	C₂×Q₁₆	—	106	8	no	—
188	C₂×SD₁₆	—	106	8	no	—
189	C₂×D₈	—	106	2	no	—
190	C₂×M₁₆	—	53	2	no	—
191	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	53	2	no	—
196	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
197	#127	—	53	2	no	—
198	#124	—	106	2	no	—
199	#121	—	106	2	no	—
200	#121	—	106	2	no	—
201	C₄×D₄	—	106	2	no	—
202	C₄×D₄	—	106	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	#107	—	106	8	no	—
209	#107	—	106	2	no	—
210	C₄⋊C₈	—	53	2	no	—
211	#102	—	106	8	no	—
212	#102	—	106	4	no	—
213	C₂²⋊C₈	—	53	2	no	—
214	#97	—	53	2	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	—	53	2	no	—
217	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
220	#41	—	212	8	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	106	2	no	—
225	C₂×D₄	—	106	2	no	—
226	C₂×D₄	—	106	2	no	—
227	C₂×D₄	—	106	2	no	—
228	C₂×D₄	—	212	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	212	2	no	—
231	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
234	Q₁₆	—	212	16	no	—
235	SD₁₆	—	212	16	no	—
236	D₈	—	212	4	no	—
237	M₁₆	—	53	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	53	4	no	—
241	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	106	2	no	—
248	C₄⋊C₄	—	106	2	no	—
249	C₂²⋊C₄	—	212	2	no	—
250	C₂²⋊C₄	—	212	2	no	—
251	C₄²	D₄×D₅₃	1	1	yes	characteristic
252	C₄²	D₄×D₅₃	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
254	C₄²	C₂³×D₅₃	1	1	yes	characteristic
255	C₂³	—	212	2	no	—
256	C₂³	#13133213	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	Q₈	—	8	4	no	—
259	Q₈	—	212	4	no	—
260	D₄	—	424	4	no	—
261	D₄	—	212	4	no	—
262	D₄	—	212	4	no	—
263	D₄	—	212	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	424	4	no	—
269	C₂×C₄	—	212	2	no	—
270	C₂×C₄	#13133213	1	1	yes	characteristic
271	C₂×C₄	#13133213	1	1	yes	characteristic
272	C₂×C₄	#13133213	1	1	yes	characteristic
273	C₂×C₄	D₈×D₅₃	1	4	yes	—
274	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	53	8	no	—
279	C₈	—	106	4	no	—
280	C₈	#13133248	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	424	2	no	—
284	C₂²	—	424	2	no	—
285	C₂²	—	424	2	no	—
286	C₂²	#16810998	1	2	yes	—
287	C₂²	C₂×#13133213	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	4	no	—
291	C₄	—	424	2	no	—
292	C₄	#16810998	1	2	yes	—
293	C₄	C₂×D₈×D₅₃	1	2	yes	—
294	C₄	C₂×#13133248	1	2	yes	—
295	C₄	#16811436	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	848	2	no	—
298	C₂	C₂×#16810998	1	1	yes	characteristic
299	C₂	#20285172	1	1	yes	characteristic
300	C₂	#20285204	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	#20285294	C₂
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#19630
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
1. #20285836	C₂	semidirect
3. #20285172	C₂	semidirect
4. #20274385	C₂	semidirect
5. #20274332	C₂	semidirect
6. #20274308	C₂	semidirect
7. C₈×#11719917	C₂	semidirect
8. C₂×C₅₃×#500	C₂	semidirect
100. #11719917	D₈	semidirect
127. C₈×C₅₃	#137	semidirect
148. C₂×#500	D₅₃	semidirect
183. C₅₃	#19630	semidirect
280. C₈	#13133248	semidirect

Maximal subgroups

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

subgroup	index
#20285836	2
#20285294	2
#20285172	2
#20274385	2
#20274332	2
#20274308	2
C₈×#11719917	2
C₂×C₅₃×#500	2
#19630	53