Subgroups — #27275106

Order 13568.

436 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283428	C₂	1	1	yes	maximal, characteristic
2	#20273375	C₂	1	1	yes	maximal, characteristic
3	#20273306	C₂	1	1	yes	maximal, characteristic
4	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272606	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272605	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₄	1	2	yes	—
9	#16811173	C₂²	1	1	yes	characteristic
10	#16810702	—	2	1	no	—
11	#16810702	—	2	1	no	—
12	C₈×D₂₁₂	—	2	1	no	—
13	C₈×D₂₁₂	—	2	1	no	—
14	#16810676	C₂²	1	1	yes	characteristic
15	#16810647	C₂²	1	1	yes	characteristic
16	#16810589	C₄	1	2	yes	—
17	C₄×#11719917	C₂²	1	1	yes	characteristic
18	#16810371	C₄	1	2	yes	—
19	#16810366	C₄	1	2	yes	—
20	#16810362	C₄	1	4	yes	—
21	#16810345	—	2	1	no	—
22	#16810344	—	2	1	no	—
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#322	—	2	1	no	—
27	C₅₃×#322	—	2	1	no	—
28	C₂×#11719945	C₂×C₄	1	2	yes	—
29	#13133223	—	4	4	no	—
30	C₂×#11719917	—	2	1	no	—
31	C₂×#11719917	C₂×C₄	1	2	yes	—
32	C₂×#11719912	—	2	2	no	—
33	C₂×C₈×D₅₃	—	2	2	no	—
34	#13133217	C₂×C₄	1	1	yes	characteristic
35	#13133214	C₂×C₄	1	1	yes	characteristic
36	#13133205	C₂×C₄	1	2	yes	—
37	#13133191	—	2	1	no	—
38	#13133191	—	2	1	no	—
39	C₄×D₂₁₂	—	2	1	no	—
40	C₄×D₂₁₂	—	2	1	no	—
41	C₄⋊#88322	—	2	1	no	—
42	C₄⋊#88322	—	2	1	no	—
43	C₄⋊#88322	C₂×C₄	1	2	yes	—
44	C₂²⋊#88322	—	2	1	no	—
45	#13133129	—	2	2	no	—
46	#13133129	—	2	2	no	—
47	#13133128	C₂×C₄	1	2	yes	—
48	C₄×#88322	—	2	1	no	—
49	C₄×#88322	—	2	1	no	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₅₃	—	2	1	no	—
56	#97×C₅₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₃	—	2	2	no	—
67	#11719945	—	4	2	no	—
68	#11719945	—	4	2	no	—
69	C₂×D₂₁₂	—	2	1	no	—
70	C₂×D₂₁₂	—	2	1	no	—
71	C₂×D₂₁₂	C₄²	1	2	yes	—
72	C₂×C₄×D₅₃	—	2	2	no	—
73	C₂×Dic₁₀₆	C₄²	1	2	yes	—
74	M₁₆×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	#11719921	—	2	2	no	—
88	#11719921	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	1	no	—
90	C₄⋊Dic₅₃	—	2	1	no	—
91	#11719917	—	4	2	no	—
92	#11719917	—	4	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	#11719912	—	4	4	no	—
97	C₈×D₅₃	—	4	4	no	—
98	C₂³×C₅₃	#117	1	1	yes	characteristic
99	C₂²×D₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂²⋊D₅₃	—	4	4	no	—
111	C₂×Dic₅₃	—	2	2	no	—
112	D₂₁₂	—	4	2	no	—
113	D₂₁₂	—	4	2	no	—
114	D₂₁₂	—	4	2	no	—
115	C₄×D₅₃	—	4	2	no	—
116	C₄×D₅₃	—	4	2	no	—
117	Dic₁₀₆	—	4	2	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	4	2	no	—
122	C₈×C₅₃	—	4	2	no	—
123	C₈×C₅₃	#100	1	2	yes	—
124	#88322	—	4	2	no	—
125	#88322	—	4	2	no	—
126	#88322	—	4	2	no	—
127	#8243	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×D₅₃	—	4	2	no	—
133	C₂×D₅₃	—	4	2	no	—
134	C₂×D₅₃	—	4	2	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
139	C₄×C₅₃	#413	1	1	yes	characteristic
140	C₄×C₅₃	#444	1	2	yes	—
141	C₄×C₅₃	#444	1	2	yes	—
142	Dic₅₃	—	4	2	no	—
143	#2931	—	53	1	no	—
144	#1543	—	53	1	no	—
145	#1525	—	53	1	no	—
146	#1451	—	53	1	no	—
147	#1451	D₅₃	1	1	yes	characteristic, complemented
148	#1299	—	53	1	no	—
149	#1299	—	53	1	no	—
150	C₂×C₅₃	—	4	1	no	—
151	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
152	C₂×C₅₃	#1765	1	1	yes	characteristic
153	C₂×C₅₃	#1765	1	1	yes	characteristic
154	D₅₃	—	8	2	no	—
155	C₂×#131	—	53	2	no	—
156	#532	—	53	1	no	—
157	#436	—	106	1	no	—
158	#436	—	106	1	no	—
159	C₈×D₄	—	106	1	no	—
160	C₈×D₄	—	106	1	no	—
161	#431	—	53	1	no	—
162	#431	C₂×D₅₃	1	1	yes	characteristic
163	#423	—	53	1	no	—
164	#408	—	53	2	no	—
165	C₄×M₁₆	—	53	1	no	—
166	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
168	#333	—	53	2	no	—
169	#331	—	53	2	no	—
170	#329	—	53	4	no	—
171	#322	—	2	1	no	—
172	#322	—	2	1	no	—
173	#322	—	106	1	no	—
174	#322	—	106	1	no	—
175	C₅₃	#8243	1	1	yes	characteristic, complemented, Sylow
176	C₂×#41	—	53	2	no	—
177	#131	—	212	4	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	53	2	no	—
180	C₂×M₁₆	—	106	1	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
185	C₄⋊Q₈	—	53	1	no	—
186	#127	—	53	1	no	—
187	#124	—	53	2	no	—
188	#121	—	106	1	no	—
189	#121	—	106	1	no	—
190	C₄×D₄	—	106	1	no	—
191	C₄×D₄	—	106	1	no	—
192	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
193	C₄⋊C₈	—	53	2	no	—
194	C₄⋊C₈	—	106	1	no	—
195	C₄⋊C₈	—	106	1	no	—
196	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	106	1	no	—
199	C₂²⋊C₈	—	106	2	no	—
200	C₂²⋊C₈	—	106	2	no	—
201	#97	—	53	2	no	—
202	#97	C₄×D₅₃	1	2	yes	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	106	1	no	—
206	C₄×C₈	—	106	1	no	—
207	C₄×C₈	D₂₁₂	1	1	yes	characteristic
208	C₄×C₈	D₂₁₂	1	1	yes	characteristic
209	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
211	#41	—	212	2	no	—
212	#41	—	212	2	no	—
213	C₂×Q₈	—	53	2	no	—
214	C₂×D₄	—	53	2	no	—
215	C₂×D₄	—	106	2	no	—
216	C₂×D₄	—	106	1	no	—
217	C₂×D₄	—	106	1	no	—
218	C₂²×C₄	—	106	2	no	—
219	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₂²×C₄	#11719945	1	1	yes	characteristic
221	C₂²×C₄	#11719945	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	M₁₆	—	212	4	no	—
224	M₁₆	—	212	2	no	—
225	M₁₆	—	212	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	212	4	no	—
234	C₂×C₈	#11719921	1	1	yes	characteristic
235	C₂×C₈	#11719921	1	1	yes	characteristic
236	C₂×C₈	#11719921	1	1	yes	characteristic
237	C₂×C₈	#11719921	1	1	yes	characteristic
238	C₄⋊C₄	—	106	1	no	—
239	C₄⋊C₄	—	106	1	no	—
240	C₂²⋊C₄	—	106	2	no	—
241	C₂²⋊C₄	—	106	2	no	—
242	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
244	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂³	—	106	2	no	—
247	C₂³	#13133174	1	1	yes	characteristic
248	Q₈	—	212	2	no	—
249	D₄	—	212	2	no	—
250	D₄	—	212	4	no	—
251	D₄	—	212	2	no	—
252	D₄	—	212	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	212	2	no	—
259	C₂×C₄	—	212	2	no	—
260	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
265	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	212	2	no	—
272	C₈	—	212	2	no	—
273	C₈	—	212	2	no	—
274	C₈	#13133133	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	212	2	no	—
279	C₂²	—	212	2	no	—
280	C₂²	—	212	2	no	—
281	C₂²	C₄×#11719921	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	212	2	no	—
286	C₄	C₂×#13133133	1	1	yes	characteristic
287	C₄	#16810605	1	1	yes	characteristic
288	C₄	#16810743	1	2	yes	—
289	C₄	#16810746	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	424	2	no	—
292	C₂	C₄×#13133133	1	1	yes	characteristic
293	C₂	#20274559	1	1	yes	characteristic
294	C₂	#20274561	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	#16811173	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#8243
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
4. #20273004	C₂	semidirect
5. #20272606	C₂	semidirect
6. #20272605	C₂	semidirect
7. C₅₃×#1451	C₂	semidirect
147. #1451	D₅₃	semidirect
175. C₅₃	#8243	semidirect

Maximal subgroups

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

subgroup	index
#20283428	2
#20273375	2
#20273306	2
#20273004	2
#20272606	2
#20272605	2
C₅₃×#1451	2
#8243	53