Subgroups — #27539603

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285248	C₂	1	2	yes	maximal
2	#20277000	C₂	1	2	yes	maximal, complemented
3	#20276998	C₂	1	2	yes	maximal, complemented
4	#20276996	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276994	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276993	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276502	C₂	1	2	yes	maximal
8	#20276498	C₂	1	1	yes	maximal, characteristic
9	#367×D₅₃	C₂	1	1	yes	maximal, characteristic
10	#20276310	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811250	—	2	2	no	—
13	C₂×#13133156	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	#16811031	—	2	2	no	—
16	#16811029	—	2	2	no	—
17	#16811027	C₂²	1	2	yes	—
18	#16811025	C₂²	1	2	yes	—
19	#16811024	C₂²	1	2	yes	—
20	#16810664	—	2	2	no	—
21	#16810660	C₂²	1	2	yes	—
22	#107×D₅₃	C₂²	1	2	yes	—
23	#16810577	C₂²	1	1	yes	characteristic
24	#16810451	C₂²	1	1	yes	characteristic
25	#16810451	C₂²	1	1	yes	characteristic
26	#16810450	C₂²	1	2	yes	—
27	#16810450	C₂²	1	2	yes	—
28	#16810449	C₂²	1	2	yes	—
29	#16810448	C₂²	1	2	yes	—
30	#16810448	C₂²	1	2	yes	—
31	#16810399	C₂²	1	1	yes	characteristic
32	#16810398	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#500	C₂²	1	2	yes	—
35	C₅₃×#367	C₂²	1	1	yes	characteristic
36	C₅₃×#367	C₂²	1	1	yes	characteristic
37	C₅₃×#367	C₂²	1	2	yes	—
38	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#11719912	D₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
41	#13133216	—	2	2	no	—
42	#13133215	—	4	2	no	—
43	#13133209	—	2	2	no	—
44	#13133208	—	2	2	no	—
45	#13133207	—	4	2	no	—
46	#13133195	—	4	2	no	—
47	#13133193	—	4	2	no	—
48	#13133175	—	2	2	no	—
49	#13133174	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₅₃	—	2	2	no	—
51	#13133156	Q₈	1	4	yes	complemented
52	C₂×#11719908	C₂³	1	1	yes	characteristic
53	C₂×#11719908	C₂³	1	1	yes	characteristic
54	C₁₆×D₅₃	Q₈	1	4	yes	complemented
55	#13133151	C₂³	1	2	yes	—
56	#13133151	C₂³	1	2	yes	—
57	#13133150	—	2	2	no	—
58	#13133150	—	2	2	no	—
59	#13133149	—	2	2	no	—
60	#13133148	—	2	2	no	—
61	#13133148	—	2	2	no	—
62	#13133128	D₄	1	1	yes	characteristic
63	#13133127	D₄	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₃	—	2	2	no	—
66	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	#107×C₅₃	—	2	2	no	—
69	#107×C₅₃	C₂³	1	2	yes	—
70	#107×C₅₃	C₂³	1	2	yes	—
71	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₅₃	—	4	2	no	—
73	C₂×C₄×D₅₃	—	4	2	no	—
74	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₁₀₆	—	4	2	no	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	#11719921	—	4	2	no	—
83	#11719921	D₈	1	2	yes	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	—	2	2	no	—
86	C₄⋊Dic₅₃	—	4	2	no	—
87	#11719919	—	4	2	no	—
88	#11719919	—	4	2	no	—
89	#11719919	—	4	2	no	—
90	#11719919	—	4	2	no	—
91	#11719919	—	4	2	no	—
92	#11719919	D₈	1	2	yes	—
93	C₄×Dic₅₃	—	4	2	no	—
94	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×#88322	C₂×D₄	1	1	yes	characteristic
96	C₂×#88322	C₂×D₄	1	1	yes	characteristic
97	#11719912	Q₁₆	1	4	yes	—
98	C₈×D₅₃	C₂×Q₈	1	2	yes	—
99	C₈×D₅₃	C₂×Q₈	1	2	yes	—
100	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
101	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
102	#11719908	C₂×Q₈	1	2	yes	—
103	#11719908	C₂×Q₈	1	2	yes	—
104	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₅₃	—	16	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₅₃	—	4	2	no	—
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
113	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
114	C₄×D₅₃	—	16	2	no	—
115	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
116	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
117	Dic₁₀₆	—	16	2	no	—
118	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
119	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	#88322	C₄⋊Q₈	1	2	yes	—
122	#88322	C₂×Q₁₆	1	2	yes	—
123	#33730	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	#500	1	2	yes	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	8	2	no	—
129	C₄×C₅₃	—	8	2	no	—
130	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	8	2	no	—
134	Dic₅₃	—	8	2	no	—
135	Dic₅₃	#500	1	2	yes	—
136	#3164	—	53	2	no	—
137	#2255	—	53	2	no	—
138	#2255	—	53	2	no	—
139	#2254	—	53	1	no	—
140	#2254	D₅₃	1	1	yes	characteristic, complemented
141	#2253	—	53	1	no	—
142	#2253	—	53	1	no	—
143	#2159	—	53	2	no	—
144	#2158	—	53	1	no	—
145	C₂×#367	—	53	1	no	—
146	#2108	—	53	1	no	—
147	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
148	C₂×C₅₃	#3416	1	2	yes	—
149	D₅₃	—	2	2	no	—
150	#544	—	106	2	no	—
151	C₂×M₃₂	—	53	1	no	—
152	C₂²×C₁₆	—	53	1	no	—
153	#501	—	106	2	no	—
154	#501	—	106	2	no	—
155	#500	—	53	2	no	—
156	#500	C₂×D₅₃	1	2	yes	—
157	#499	—	53	2	no	—
158	#499	—	53	2	no	—
159	#428	—	106	2	no	—
160	#427	—	53	2	no	—
161	C₂×#107	—	53	2	no	—
162	#405	—	53	1	no	—
163	#367	—	53	1	no	—
164	#367	—	53	1	no	—
165	#367	—	53	2	no	—
166	#367	—	53	2	no	—
167	#367	C₂×D₅₃	1	1	yes	characteristic
168	#367	C₂×D₅₃	1	1	yes	characteristic
169	#367	C₂×D₅₃	1	2	yes	—
170	#366	—	53	2	no	—
171	#366	—	53	2	no	—
172	#366	—	53	2	no	—
173	#346	—	53	1	no	—
174	#346	—	53	1	no	—
175	C₄×C₁₆	—	53	1	no	—
176	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
177	C₅₃	#33730	1	1	yes	characteristic, complemented, Sylow
178	C₂×M₁₆	—	53	1	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	106	2	no	—
182	C₄⋊Q₈	—	212	2	no	—
183	#125	—	106	2	no	—
184	#125	—	106	2	no	—
185	#125	—	212	2	no	—
186	C₂²⋊Q₈	—	212	2	no	—
187	C₂²⋊Q₈	—	212	2	no	—
188	#117	—	53	1	no	—
189	#117	—	106	2	no	—
190	C₂×C₄⋊C₄	—	106	2	no	—
191	M₃₂	—	53	4	no	—
192	C₂×C₁₆	—	53	1	no	—
193	C₂×C₁₆	—	53	1	no	—
194	C₂×C₁₆	—	53	4	no	—
195	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
196	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
197	#107	—	2	2	no	—
198	#107	—	53	2	no	—
199	#107	—	53	2	no	—
200	#107	—	106	2	no	—
201	#107	—	106	2	no	—
202	#107	C₂²×D₅₃	1	2	yes	—
203	#107	C₂²×D₅₃	1	2	yes	—
204	#106	—	106	2	no	—
205	#106	—	106	2	no	—
206	#106	—	106	2	no	—
207	#97	—	53	1	no	—
208	#97	—	53	1	no	—
209	C₄×C₈	—	53	1	no	—
210	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	212	2	no	—
213	C₂²×C₄	—	53	1	no	—
214	C₂²×C₄	—	212	2	no	—
215	M₁₆	—	53	4	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	2	no	—
219	C₂×C₈	—	53	2	no	—
220	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	53	2	no	—
226	C₄⋊C₄	—	106	2	no	—
227	C₄⋊C₄	—	106	2	no	—
228	C₄⋊C₄	—	212	2	no	—
229	C₄⋊C₄	—	212	2	no	—
230	C₄⋊C₄	—	212	2	no	—
231	C₄⋊C₄	—	212	2	no	—
232	C₄⋊C₄	—	212	2	no	—
233	C₄⋊C₄	—	212	2	no	—
234	C₂²⋊C₄	—	53	2	no	—
235	C₂²⋊C₄	—	212	2	no	—
236	C₄²	—	53	1	no	—
237	C₄²	—	212	2	no	—
238	C₄²	D₄×D₅₃	1	1	yes	characteristic
239	C₁₆	—	53	2	no	—
240	C₁₆	—	53	2	no	—
241	C₁₆	Q₈×D₅₃	1	2	yes	—
242	C₁₆	Q₈×D₅₃	1	2	yes	—
243	C₂³	—	53	1	no	—
244	Q₈	—	16	2	no	—
245	Q₈	—	848	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	53	1	no	—
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	53	2	no	—
251	C₂×C₄	—	53	2	no	—
252	C₂×C₄	—	848	2	no	—
253	C₂×C₄	—	212	2	no	—
254	C₂×C₄	—	212	2	no	—
255	C₂×C₄	D₈×D₅₃	1	2	yes	—
256	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
257	C₈	—	53	2	no	—
258	C₈	—	53	2	no	—
259	C₈	Q₁₆×D₅₃	1	2	yes	—
260	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
262	C₂²	—	53	2	no	—
263	C₂²	—	106	2	no	—
264	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	53	2	no	—
269	C₄	—	106	2	no	—
270	C₄	—	424	2	no	—
271	C₄	—	424	2	no	—
272	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
273	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
274	C₂	—	106	2	no	—
275	C₂	#500×D₅₃	1	1	yes	characteristic
276	C₂	#20287857	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2254	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2254	D₅₃
hypocenter	C₅₃	#33730
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
2. #20277000	C₂	semidirect
3. #20276998	C₂	semidirect
4. #20276996	C₂	semidirect
5. #20276994	C₂	semidirect
6. #20276993	C₂	semidirect
11. C₅₃×#2254	C₂	semidirect
51. #13133156	Q₈	semidirect
54. C₁₆×D₅₃	Q₈	semidirect
140. #2254	D₅₃	semidirect
177. C₅₃	#33730	semidirect

Maximal subgroups

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

subgroup	index
#20285248	2
#20277000	2
#20276998	2
#20276996	2
#20276994	2
#20276993	2
#20276502	2
#20276498	2
#367×D₅₃	2
#20276310	2
C₅₃×#2254	2
#33730	53