Subgroups — #27538527

Order 13568.

710 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	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284292	C₂	1	2	yes	maximal, complemented
3	#20276808	C₂	1	4	yes	maximal, complemented
4	#20276803	C₂	1	2	yes	maximal, complemented
5	#20276798	C₂	1	2	yes	maximal, complemented
6	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276422	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#16811130	—	2	2	no	—
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₅₃	—	2	4	no	—
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	#16810861	C₂²	1	4	yes	—
16	#16810856	C₂²	1	4	yes	—
17	#16810849	—	2	4	no	—
18	#16810661	C₂²	1	2	yes	—
19	#16810619	—	2	2	no	—
20	#16810451	C₂²	1	2	yes	complemented
21	#16810449	C₂²	1	2	yes	complemented
22	#16810427	C₂²	1	4	yes	complemented
23	#16810424	C₂²	1	4	yes	complemented
24	C₂×#135×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#508	—	2	4	no	—
26	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	complemented
28	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
31	C₂²×#88322	D₄	1	1	yes	characteristic
32	C₂×C₈×D₅₃	—	2	2	no	—
33	C₂×C₈×D₅₃	—	2	2	no	—
34	C₂×C₈×D₅₃	D₄	1	2	yes	—
35	C₂×C₈×D₅₃	C₂³	1	2	yes	—
36	#13133198	—	4	2	no	—
37	#13133193	—	2	4	no	—
38	#13133188	—	2	4	no	—
39	#13133185	—	4	2	no	—
40	C₄×C₂²⋊D₅₃	—	4	4	no	—
41	#13133176	—	2	2	no	—
42	C₂×#11719908	—	2	2	no	—
43	C₂×#11719908	C₂³	1	2	yes	—
44	C₁₆×D₅₃	—	2	4	no	—
45	C₁₆×D₅₃	—	2	4	no	—
46	#13133151	C₂³	1	4	yes	—
47	#13133149	—	2	2	no	—
48	#13133141	—	2	4	no	—
49	#13133138	—	2	4	no	—
50	C₂×#41×C₅₃	—	2	2	no	—
51	#135×C₅₃	—	2	4	no	—
52	#135×C₅₃	—	2	4	no	—
53	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
54	C₂×SD₁₆×C₅₃	—	2	2	no	—
55	C₂×D₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	Q₃₂×C₅₃	—	2	2	no	—
58	SD₃₂×C₅₃	—	2	4	no	—
59	D₁₆×C₅₃	—	2	2	no	—
60	C₂×C₁₆×C₅₃	D₄	1	4	yes	complemented
61	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
63	C₂³×D₅₃	D₈	1	1	yes	characteristic
64	#41×C₅₃	—	4	4	no	—
65	#41×C₅₃	—	4	4	no	—
66	C₂×Q₈×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₅₃	—	4	4	no	—
72	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	2	no	—
74	C₂×C₄×D₅₃	—	2	2	no	—
75	C₂×C₄×D₅₃	D₈	1	2	yes	—
76	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
77	Q₁₆×C₅₃	—	2	2	no	—
78	Q₁₆×C₅₃	—	2	2	no	—
79	SD₁₆×C₅₃	—	4	4	no	—
80	D₈×C₅₃	—	2	2	no	—
81	D₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	#41	1	2	yes	—
85	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₅₃	—	4	2	no	—
87	C₂²⋊Dic₅₃	—	4	4	no	—
88	#11719921	—	4	4	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	#11719919	—	4	4	no	—
91	C₄×Dic₅₃	—	4	2	no	—
92	C₂×#88322	—	2	2	no	—
93	C₂×#88322	C₂×D₄	1	2	yes	—
94	C₈×D₅₃	—	2	4	no	—
95	C₈×D₅₃	—	2	2	no	—
96	C₈×D₅₃	—	2	2	no	—
97	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
98	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
99	#11719908	—	2	2	no	—
100	C₂³×C₅₃	—	4	2	no	—
101	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₅₃	—	2	2	no	—
103	C₂²×D₅₃	—	2	4	no	—
104	C₂²×D₅₃	C₂×D₈	1	2	yes	—
105	Q₈×C₅₃	—	4	2	no	—
106	Q₈×C₅₃	—	4	2	no	—
107	D₄×C₅₃	—	4	2	no	—
108	D₄×C₅₃	—	4	2	no	—
109	D₄×C₅₃	—	8	4	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	#121	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#135	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²⋊D₅₃	—	8	8	no	—
117	C₂×Dic₅₃	—	2	2	no	—
118	C₂×Dic₅₃	—	4	4	no	—
119	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
120	C₄×D₅₃	—	2	2	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	—	2	4	no	—
123	C₈×C₅₃	#121	1	2	yes	—
124	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
126	#88322	—	2	2	no	—
127	#33600	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	8	4	no	—
130	C₂²×C₅₃	—	8	2	no	—
131	C₂²×C₅₃	#466	1	2	yes	—
132	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
133	C₂×D₅₃	—	2	8	no	—
134	C₂×D₅₃	—	2	2	no	—
135	C₂×D₅₃	—	2	2	no	—
136	C₂×D₅₃	#508	1	4	yes	complemented
137	C₄×C₅₃	—	8	2	no	—
138	C₄×C₅₃	#466	1	2	yes	—
139	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
141	Dic₅₃	—	2	2	no	—
142	Dic₅₃	—	8	4	no	—
143	C₂×#508	D₅₃	1	1	yes	characteristic, complemented
144	C₂³×C₁₆	—	53	1	no	—
145	#3050	—	53	2	no	—
146	#2217	—	53	4	no	—
147	#2216	—	53	2	no	—
148	#2215	—	53	2	no	—
149	#2158	—	53	1	no	—
150	#2138	—	53	2	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	—	16	2	no	—
153	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
154	C₂×C₅₃	C₂×#508	1	2	yes	—
155	D₅₃	—	2	4	no	—
156	C₂×#135	C₂×D₅₃	1	2	yes	—
157	C₂³×C₈	—	53	1	no	—
158	#525	—	106	2	no	—
159	#508	—	2	4	no	—
160	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
161	C₂×SD₃₂	C₂×D₅₃	1	2	yes	complemented
162	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
163	C₂²×C₁₆	—	53	1	no	—
164	C₂²×C₁₆	—	53	2	no	—
165	C₂²×C₁₆	—	53	2	no	—
166	C₂²×C₁₆	—	106	4	no	—
167	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
168	#467	—	53	4	no	—
169	#466	—	53	4	no	—
170	#465	—	106	4	no	—
171	#427	—	53	2	no	—
172	#416	—	106	2	no	—
173	#367	—	53	2	no	—
174	#366	—	53	2	no	—
175	#358	—	53	4	no	—
176	#357	—	53	4	no	—
177	C₅₃	#33600	1	1	yes	characteristic, complemented, Sylow
178	C₂×#41	—	2	2	no	—
179	C₂³×C₄	—	53	1	no	—
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂²×D₅₃	1	2	yes	—
185	C₂²×C₈	—	53	1	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	—	53	2	no	—
188	C₂²×C₈	—	106	2	no	—
189	C₂²×C₈	—	106	2	no	—
190	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	#123	—	212	2	no	—
192	C₂²⋊Q₈	—	106	4	no	—
193	#121	—	106	4	no	—
194	#120	—	212	2	no	—
195	C₄×D₄	—	212	4	no	—
196	#117	—	106	2	no	—
197	Q₃₂	—	2	2	no	—
198	SD₃₂	—	2	4	no	—
199	D₁₆	—	2	2	no	—
200	C₂×C₁₆	—	53	2	no	—
201	C₂×C₁₆	—	106	2	no	—
202	C₂×C₁₆	—	106	4	no	—
203	C₂×C₁₆	—	106	4	no	—
204	C₂×C₁₆	C₂²⋊D₅₃	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
207	#107	—	53	4	no	—
208	#106	—	106	2	no	—
209	#103	—	106	4	no	—
210	#102	—	106	4	no	—
211	C₂⁴	—	53	1	no	—
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	212	4	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	53	1	no	—
220	C₂²×C₄	—	53	2	no	—
221	C₂²×C₄	—	53	2	no	—
222	C₂²×C₄	—	106	2	no	—
223	C₂²×C₄	—	106	2	no	—
224	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	D₈	—	2	2	no	—
229	D₈	—	2	2	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	2	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
236	C₂×C₈	#11719947	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
238	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
239	C₄⋊C₄	—	106	4	no	—
240	C₄⋊C₄	—	212	4	no	—
241	C₂²⋊C₄	—	212	4	no	—
242	C₂²⋊C₄	—	212	4	no	—
243	C₂²⋊C₄	—	212	2	no	—
244	C₄²	—	212	2	no	—
245	C₁₆	—	106	2	no	—
246	C₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
247	C₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
248	C₂³	—	4	2	no	—
249	C₂³	—	53	2	no	—
250	C₂³	—	106	4	no	—
251	C₂³	—	106	2	no	—
252	C₂³	D₈×D₅₃	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	424	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	53	2	no	—
262	C₂×C₄	—	106	2	no	—
263	C₂×C₄	—	106	4	no	—
264	C₂×C₄	—	106	2	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	212	4	no	—
267	C₂×C₄	#13133188	1	2	yes	—
268	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
269	C₂×C₄	#13133236	1	2	yes	—
270	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
271	C₈	—	106	2	no	—
272	C₈	#13133188	1	2	yes	—
273	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₈	C₂×#11719947	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	8	4	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	53	4	no	—
279	C₂²	—	106	8	no	—
280	C₂²	—	106	2	no	—
281	C₂²	—	106	2	no	—
282	C₂²	#16810856	1	2	yes	—
283	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
284	C₄	—	8	2	no	—
285	C₄	—	106	2	no	—
286	C₄	—	424	4	no	—
287	C₄	#16810856	1	2	yes	—
288	C₄	C₂×#13133188	1	1	yes	characteristic
289	C₄	C₂×#13133236	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	16	2	no	—
292	C₂	—	106	4	no	—
293	C₂	C₂×#16810856	1	1	yes	characteristic
294	C₂	#508×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#508	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	#33600
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₅₃	C₂	semidirect
2. #20284292	C₂	semidirect
3. #20276808	C₂	semidirect
4. #20276803	C₂	semidirect
5. #20276798	C₂	semidirect
6. #20276499	C₂	semidirect
7. #20276422	C₂	semidirect
8. C₂×C₅₃×#508	C₂	semidirect
20. #16810451	C₂²	semidirect
21. #16810449	C₂²	semidirect
22. #16810427	C₂²	semidirect
23. #16810424	C₂²	semidirect
26. C₂×Q₃₂×C₅₃	C₂²	semidirect
27. C₂×SD₃₂×C₅₃	C₂²	semidirect
28. C₂×D₁₆×C₅₃	C₂²	semidirect
60. C₂×C₁₆×C₅₃	D₄	semidirect
136. C₂×D₅₃	#508	semidirect
143. C₂×#508	D₅₃	semidirect
160. C₂×Q₃₂	C₂×D₅₃	semidirect
161. C₂×SD₃₂	C₂×D₅₃	semidirect
162. C₂×D₁₆	C₂×D₅₃	semidirect
177. C₅₃	#33600	semidirect
204. C₂×C₁₆	C₂²⋊D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₃	2
#20284292	2
#20276808	2
#20276803	2
#20276798	2
#20276499	2
#20276422	2
C₂×C₅₃×#508	2
#33600	53