Subgroups — #27300646

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810446	C₂	1	2	yes	maximal, complemented
3	#20274740	C₂	1	1	yes	maximal, characteristic
4	#20272814	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	—	2	1	no	—
7	C₂×#13133153	—	2	4	no	—
8	C₂×#13133153	—	2	1	no	—
9	C₂×#13133153	C₂²	1	2	yes	—
10	C₂×#13133149	C₂²	1	1	yes	characteristic
11	C₂×#13133149	C₂²	1	1	yes	characteristic
12	#16810483	—	2	1	no	—
13	#16810446	—	2	4	no	—
14	#16810386	—	2	1	no	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	complemented
18	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	complemented
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	4	2	no	—
22	C₂×#11719917	—	4	1	no	—
23	C₂²×#88322	—	4	1	no	—
24	C₂×C₄⋊Dic₅₃	—	2	1	no	—
25	C₂×C₄⋊Dic₅₃	—	2	1	no	—
26	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
27	#13133153	—	2	4	no	—
28	#13133153	—	2	2	no	—
29	#13133153	D₄	1	4	yes	—
30	#13133149	—	2	2	no	—
31	#13133149	—	2	2	no	—
32	#13133124	—	4	1	no	—
33	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	2	no	—
43	M₃₂×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
45	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	1	no	—
57	C₂²×Dic₅₃	—	4	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₅₃	#41	1	2	yes	—
70	C₂²⋊Dic₅₃	—	8	2	no	—
71	C₄⋊Dic₅₃	—	4	1	no	—
72	C₄⋊Dic₅₃	—	4	1	no	—
73	C₄⋊Dic₅₃	—	4	1	no	—
74	C₄⋊Dic₅₃	—	4	1	no	—
75	#11719917	—	2	4	no	—
76	#11719917	—	4	4	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	4	4	no	—
79	C₂×#88322	—	4	2	no	—
80	C₁₆×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	—	4	1	no	—
85	C₂³×C₅₃	#106	1	1	yes	characteristic
86	C₂³×C₅₃	#106	1	1	yes	characteristic
87	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
106	C₂×Dic₅₃	—	8	1	no	—
107	C₂×Dic₅₃	—	8	1	no	—
108	C₂×Dic₅₃	—	8	1	no	—
109	C₂×Dic₅₃	—	8	1	no	—
110	C₂×Dic₅₃	—	8	1	no	—
111	C₂×Dic₅₃	—	8	1	no	—
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	C₄×D₄	1	2	yes	—
115	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	#88322	—	4	4	no	—
117	#12928	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	4	2	no	—
127	C₂²×C₅₃	#365	1	2	yes	complemented
128	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
129	C₂²×C₅₃	#448	1	1	yes	characteristic
130	C₂²×C₅₃	#481	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	#385	1	1	yes	characteristic
134	C₄×C₅₃	#428	1	1	yes	characteristic
135	C₄×C₅₃	#450	1	1	yes	characteristic
136	C₄×C₅₃	#483	1	1	yes	characteristic
137	Dic₅₃	—	16	1	no	—
138	Dic₅₃	—	16	1	no	—
139	C₂²×#108	—	53	1	no	—
140	C₂×#365	—	53	2	no	—
141	#2112	D₅₃	1	1	yes	characteristic
142	#1807	—	53	1	no	—
143	#1383	—	53	2	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	—	4	1	no	—
148	C₂×C₅₃	#1853	1	1	yes	characteristic
149	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
150	C₂×C₅₃	#2155	1	1	yes	characteristic
151	C₂²×M₁₆	—	106	1	no	—
152	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
153	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
154	C₂×#108	—	53	2	no	—
155	C₂×#108	—	106	4	no	—
156	C₂×#108	—	106	1	no	—
157	C₂×#106	—	53	1	no	—
158	C₂×#106	—	53	1	no	—
159	#380	—	106	1	no	—
160	#365	—	106	4	no	—
161	C₂²⋊C₁₆	Dic₅₃	1	2	yes	complemented
162	C₂²⋊C₁₆	Dic₅₃	1	2	yes	complemented
163	#340	—	106	1	no	—
164	C₅₃	#12928	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
166	C₂×M₁₆	—	106	4	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	—	212	2	no	—
169	C₂×M₁₆	—	212	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	212	1	no	—
173	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
174	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
175	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
176	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
179	C₂×C₄⋊C₄	—	106	1	no	—
180	C₂×C₄⋊C₄	—	106	1	no	—
181	C₂×C₂²⋊C₄	—	212	1	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	2	2	no	—
184	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
185	#108	—	53	4	no	—
186	#108	—	106	4	no	—
187	#108	—	106	2	no	—
188	#106	—	106	2	no	—
189	#106	—	106	2	no	—
190	#95	—	212	1	no	—
191	C₂⁴	#11719913	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	212	1	no	—
197	C₂²×C₄	—	212	1	no	—
198	C₂²×C₄	#11719913	1	1	yes	characteristic
199	C₂²×C₄	#11719913	1	1	yes	characteristic
200	C₂²×C₄	#11719913	1	1	yes	characteristic
201	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
204	M₁₆	—	106	4	no	—
205	M₁₆	—	212	4	no	—
206	M₁₆	—	212	2	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	212	4	no	—
214	C₂×C₈	—	212	2	no	—
215	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
216	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
218	C₂×C₈	D₄×D₅₃	1	2	yes	—
219	C₂×C₈	#11719947	1	2	yes	—
220	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
221	C₄⋊C₄	—	212	1	no	—
222	C₄⋊C₄	—	212	1	no	—
223	C₄⋊C₄	—	212	1	no	—
224	C₄⋊C₄	—	212	1	no	—
225	C₂²⋊C₄	—	424	2	no	—
226	C₁₆	—	2	4	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	#13133149	1	1	yes	characteristic
232	C₂³	#13133149	1	1	yes	characteristic
233	C₂³	C₂×#11719913	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	424	1	no	—
245	C₂×C₄	—	424	1	no	—
246	C₂×C₄	—	424	1	no	—
247	C₂×C₄	—	424	1	no	—
248	C₂×C₄	—	424	1	no	—
249	C₂×C₄	—	424	1	no	—
250	C₂×C₄	#13133149	1	1	yes	characteristic
251	C₂×C₄	#13133149	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	#13133187	1	1	yes	characteristic
254	C₂×C₄	#13133201	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
256	C₂×C₄	#13133245	1	1	yes	characteristic
257	C₂×C₄	#13133250	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	212	4	no	—
261	C₈	D₄×Dic₅₃	1	2	yes	—
262	C₈	#13133194	1	2	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	#16810446	1	2	yes	complemented
273	C₂²	C₂×#13133149	1	1	yes	characteristic
274	C₂²	#16810764	1	1	yes	characteristic
275	C₂²	#16810942	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	848	1	no	—
279	C₄	—	848	1	no	—
280	C₄	#16810498	1	1	yes	characteristic
281	C₄	#16810665	1	1	yes	characteristic
282	C₄	#16810776	1	1	yes	characteristic
283	C₄	#16810954	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	—	4	1	no	—
288	C₂	#20274910	1	1	yes	characteristic
289	C₂	C₂×#16810446	1	1	yes	characteristic
290	C₂	#20276488	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133149
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2112	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2112	D₅₃
hypocenter	C₅₃	#12928
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. C₂×#16810446	C₂	semidirect
4. #20272814	C₂	semidirect
17. C₅₃×C₂²⋊C₁₆	C₄	semidirect
18. C₅₃×C₂²⋊C₁₆	C₄	semidirect
127. C₂²×C₅₃	#365	semidirect
161. C₂²⋊C₁₆	Dic₅₃	semidirect
162. C₂²⋊C₁₆	Dic₅₃	semidirect
164. C₅₃	#12928	semidirect
272. C₂²	#16810446	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810446	2
#20274740	2
#20272814	2
C₅₃×#2112	2
#12928	53