Subgroups — #27355918

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285294	C₂	1	1	yes	maximal, characteristic
2	#20283461	C₂	1	2	yes	maximal, complemented
3	#20273630	C₂	1	2	yes	maximal, complemented
4	#20273581	C₂	1	4	yes	maximal, complemented
5	#20273540	C₂	1	2	yes	maximal, complemented
6	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273115	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133225	—	2	2	no	—
10	#16811173	C₂²	1	2	yes	—
11	#16811028	C₄	1	1	yes	characteristic
12	#16811000	C₄	1	1	yes	characteristic
13	#16810970	C₄	1	2	yes	—
14	#16810859	C₄	1	2	yes	—
15	#16810859	C₄	1	2	yes	—
16	C₄×D₄₂₄	C₂²	1	2	yes	—
17	C₄×D₄₂₄	C₂²	1	2	yes	—
18	#16810702	C₂²	1	4	yes	—
19	#16810696	—	2	4	no	—
20	C₈×D₂₁₂	C₂²	1	4	yes	—
21	C₂×C₄⋊#88322	C₂²	1	2	yes	—
22	#16810591	—	2	2	no	—
23	#16810374	C₂²	1	1	yes	characteristic
24	#16810374	C₂²	1	2	yes	—
25	#16810374	C₂²	1	1	yes	characteristic
26	#16810352	C₂²	1	4	yes	—
27	#16810352	C₂²	1	4	yes	—
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₂×#11719945	—	2	2	no	—
34	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
35	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
36	#13133225	—	4	8	no	—
37	C₂×#11719912	—	2	4	no	—
38	C₂²×#88322	—	2	2	no	—
39	C₂×C₈×D₅₃	—	2	4	no	—
40	#13133217	C₂×C₄	1	2	yes	—
41	#13133214	C₂×C₄	1	2	yes	—
42	#13133205	—	2	2	no	—
43	#13133191	—	2	2	no	—
44	#13133191	C₂×C₄	1	4	yes	—
45	C₄×D₂₁₂	—	2	2	no	—
46	C₄×D₂₁₂	C₂³	1	4	yes	—
47	#13133151	C₂×C₄	1	2	yes	—
48	#13133151	C₂×C₄	1	2	yes	—
49	C₄⋊#88322	—	2	2	no	—
50	C₄⋊#88322	C₂³	1	2	yes	—
51	C₄⋊#88322	C₂³	1	2	yes	—
52	#13133139	C₂×C₄	1	4	yes	—
53	#13133139	C₂×C₄	1	4	yes	—
54	#13133129	—	2	4	no	—
55	#13133129	—	2	4	no	—
56	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	1	no	—
61	C₂²⋊C₈×C₅₃	—	2	1	no	—
62	#97×C₅₃	D₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₅₃	—	2	4	no	—
73	#11719945	—	4	4	no	—
74	#11719945	—	4	4	no	—
75	C₂×D₂₁₂	—	2	2	no	—
76	C₂×D₂₁₂	—	2	2	no	—
77	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₅₃	—	2	4	no	—
79	C₂×Dic₁₀₆	—	2	2	no	—
80	M₁₆×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	#11719921	—	2	4	no	—
94	#11719921	—	2	4	no	—
95	C₄⋊Dic₅₃	—	2	2	no	—
96	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	4	4	no	—
100	D₄₂₄	—	2	4	no	—
101	D₄₂₄	—	4	2	no	—
102	#11719912	—	4	8	no	—
103	C₈×D₅₃	—	4	8	no	—
104	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
116	C₂²⋊D₅₃	—	4	8	no	—
117	C₂×Dic₅₃	—	2	4	no	—
118	D₂₁₂	—	2	4	no	—
119	D₂₁₂	—	2	4	no	—
120	D₂₁₂	—	4	4	no	—
121	D₂₁₂	—	8	2	no	—
122	C₄×D₅₃	—	4	4	no	—
123	C₄×D₅₃	—	4	4	no	—
124	Dic₁₀₆	—	4	4	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	—	2	1	no	—
128	C₈×C₅₃	—	2	4	no	—
129	C₈×C₅₃	—	4	2	no	—
130	C₈×C₅₃	#131	1	2	yes	complemented
131	#88322	—	4	2	no	—
132	#88322	—	4	2	no	—
133	#18073	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	#442	1	2	yes	—
137	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×D₅₃	—	4	4	no	—
139	C₂×D₅₃	—	4	4	no	—
140	C₂×D₅₃	—	4	4	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	#442	1	2	yes	—
145	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
146	C₄×C₅₃	#568	1	1	yes	characteristic
147	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
148	Dic₅₃	—	4	4	no	—
149	#3169	—	53	1	no	—
150	#2935	—	53	2	no	—
151	#1594	—	53	2	no	—
152	#1586	—	53	4	no	—
153	#1580	—	53	2	no	—
154	C₂×#334	—	53	1	no	—
155	#1480	—	53	2	no	—
156	#1451	D₅₃	1	1	yes	characteristic, complemented
157	C₂×C₅₃	—	2	2	no	—
158	C₂×C₅₃	#2935	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
160	C₂×C₅₃	#2957	1	1	yes	characteristic
161	D₅₃	—	8	4	no	—
162	C₂×#131	—	106	2	no	—
163	#532	—	53	2	no	—
164	#500	—	53	1	no	—
165	#493	—	53	1	no	—
166	#486	—	53	2	no	—
167	#466	—	53	2	no	—
168	#466	—	53	2	no	—
169	C₄×D₈	—	53	2	no	—
170	C₄×D₈	—	53	2	no	—
171	#436	—	53	4	no	—
172	#435	—	106	4	no	—
173	C₈×D₄	—	53	4	no	—
174	#431	C₂×D₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₈	—	53	2	no	—
176	#408	—	106	2	no	—
177	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
179	#334	—	53	1	no	—
180	#334	—	53	2	no	—
181	#334	—	53	1	no	—
182	D₄⋊C₈	—	53	4	no	—
183	D₄⋊C₈	—	53	4	no	—
184	#322	C₂×D₅₃	1	2	yes	—
185	#322	C₂×D₅₃	1	2	yes	—
186	C₅₃	#18073	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	106	2	no	—
188	C₂×D₈	—	53	2	no	—
189	C₂×D₈	—	53	2	no	—
190	#131	—	212	8	no	—
191	C₂×M₁₆	—	106	4	no	—
192	C₂×M₁₆	D₂₁₂	1	2	yes	—
193	C₂²×C₈	—	106	2	no	—
194	C₂²×C₈	—	106	4	no	—
195	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	53	2	no	—
198	#127	—	53	2	no	—
199	#124	—	106	2	no	—
200	#121	—	53	4	no	—
201	#121	—	106	2	no	—
202	C₄×D₄	—	53	4	no	—
203	C₄×D₄	—	106	2	no	—
204	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
205	#107	—	53	2	no	—
206	#107	—	53	2	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	53	2	no	—
209	C₄⋊C₈	—	53	2	no	—
210	C₄⋊C₈	—	106	2	no	—
211	#102	—	53	4	no	—
212	#102	—	53	4	no	—
213	C₂²⋊C₈	—	2	1	no	—
214	C₂²⋊C₈	—	106	4	no	—
215	C₂²⋊C₈	—	106	4	no	—
216	#97	D₂₁₂	1	2	yes	—
217	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
218	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅₃	1	2	yes	—
220	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₅₃	1	2	yes	—
223	#41	—	212	4	no	—
224	#41	—	212	4	no	—
225	C₂×Q₈	—	106	2	no	—
226	C₂×D₄	—	53	4	no	—
227	C₂×D₄	—	106	4	no	—
228	C₂×D₄	—	106	2	no	—
229	C₂×D₄	—	106	2	no	—
230	C₂²×C₄	—	106	4	no	—
231	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
233	C₂²×C₄	#11719945	1	1	yes	characteristic
234	D₈	—	106	4	no	—
235	D₈	—	212	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	212	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	106	2	no	—
242	C₂×C₈	—	106	2	no	—
243	C₂×C₈	—	212	8	no	—
244	C₂×C₈	—	212	4	no	—
245	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
250	C₄⋊C₄	—	53	4	no	—
251	C₄⋊C₄	—	106	2	no	—
252	C₂²⋊C₄	—	106	4	no	—
253	C₂²⋊C₄	—	106	4	no	—
254	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
256	C₄²	#11719945	1	1	yes	characteristic
257	C₄²	C₂³×D₅₃	1	1	yes	characteristic
258	C₂³	—	106	4	no	—
259	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
260	Q₈	—	212	4	no	—
261	D₄	—	106	4	no	—
262	D₄	—	106	4	no	—
263	D₄	—	424	2	no	—
264	D₄	—	212	8	no	—
265	D₄	—	212	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	106	4	no	—
270	C₂×C₄	—	212	4	no	—
271	C₂×C₄	—	212	4	no	—
272	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
273	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
274	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
275	C₂×C₄	#13133245	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
278	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	212	2	no	—
285	C₈	—	212	2	no	—
286	C₈	#13133225	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	212	4	no	—
290	C₂²	—	212	4	no	—
291	C₂²	—	212	4	no	—
292	C₂²	#16810736	1	2	yes	—
293	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	212	4	no	—
298	C₄	#16810736	1	2	yes	—
299	C₄	C₂×#13133225	1	1	yes	characteristic
300	C₄	#16811389	1	1	yes	characteristic
301	C₄	C₂×#13133245	1	2	yes	—
302	C₂	—	2	2	no	—
303	C₂	—	424	4	no	—
304	C₂	#20283461	1	1	yes	characteristic
305	C₂	C₂×#16810736	1	1	yes	characteristic
306	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285294	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18073
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #20283461	C₂	semidirect
3. #20273630	C₂	semidirect
4. #20273581	C₂	semidirect
5. #20273540	C₂	semidirect
6. C₂×#16810374	C₂	semidirect
7. #20273115	C₂	semidirect
8. C₅₃×#1451	C₂	semidirect
130. C₈×C₅₃	#131	semidirect
156. #1451	D₅₃	semidirect
186. C₅₃	#18073	semidirect
286. C₈	#13133225	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20283461	2
#20273630	2
#20273581	2
#20273540	2
C₂×#16810374	2
#20273115	2
C₅₃×#1451	2
#18073	53