Subgroups — #27299332

Order 13568.

430 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	C₂×#16810739	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810429	C₂	1	1	yes	maximal, characteristic
3	#20276423	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810398	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274597	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272812	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	C₂²	1	1	yes	characteristic
9	C₂×#13133225	—	2	1	no	—
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	#16810739	—	2	2	no	—
12	#16810739	C₄	1	4	yes	complemented
13	#16810661	C₂²	1	1	yes	characteristic
14	C₂×#13133143	—	2	1	no	—
15	#16810581	—	2	1	no	—
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810429	C₄	1	4	yes	complemented
18	#16810428	C₄	1	2	yes	complemented
19	#16810425	C₄	1	2	yes	complemented
20	#16810398	—	2	2	no	—
21	#16810384	—	2	1	no	—
22	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719945	—	2	1	no	—
26	#13133239	—	2	1	no	—
27	#13133239	—	2	1	no	—
28	#13133239	C₂×C₄	1	2	yes	—
29	#13133239	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719913	—	2	1	no	—
32	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
33	#13133225	—	2	4	no	—
34	#13133225	—	4	2	no	—
35	C₂×#11719912	—	4	1	no	—
36	C₂²×#88322	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	—	4	1	no	—
38	#13133176	—	2	1	no	—
39	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	C₂×C₄	1	2	yes	—
43	#13133151	C₂×C₄	1	2	yes	—
44	#13133149	—	2	1	no	—
45	#13133143	—	2	4	no	—
46	#13133127	—	4	1	no	—
47	C₈×Dic₅₃	—	4	1	no	—
48	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₅₃	D₄	1	2	yes	—
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂×M₁₆×C₅₃	—	2	1	no	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₅₃	—	2	1	no	—
54	C₂×C₁₆×C₅₃	—	2	1	no	—
55	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
56	#108×C₅₃	—	2	1	no	—
57	#108×C₅₃	—	2	1	no	—
58	#108×C₅₃	C₂×C₄	1	2	yes	—
59	#108×C₅₃	C₂×C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	—	4	1	no	—
62	C₂²×Dic₅₃	—	2	1	no	—
63	#11719945	—	2	2	no	—
64	#11719945	—	2	2	no	—
65	#11719945	—	4	1	no	—
66	#11719945	—	4	1	no	—
67	C₂×D₂₁₂	—	2	1	no	—
68	C₂×C₄×D₅₃	—	4	1	no	—
69	C₂×Dic₁₀₆	—	2	1	no	—
70	M₁₆×C₅₃	—	2	2	no	—
71	M₁₆×C₅₃	—	2	2	no	—
72	M₁₆×C₅₃	—	4	1	no	—
73	M₁₆×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₂²⋊Dic₅₃	—	4	1	no	—
83	C₄⋊Dic₅₃	—	2	2	no	—
84	C₄×Dic₅₃	—	2	1	no	—
85	C₄×Dic₅₃	—	4	1	no	—
86	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
87	C₂×#88322	C₂²⋊C₄	1	2	yes	—
88	C₂×#88322	D₈	1	1	yes	characteristic
89	C₂×#88322	D₈	1	1	yes	characteristic
90	C₂×#88322	SD₁₆	1	2	yes	—
91	Dic₂₁₂	—	2	1	no	—
92	Dic₂₁₂	C₄²	1	2	yes	—
93	D₄₂₄	—	2	1	no	—
94	D₄₂₄	C₄²	1	2	yes	—
95	#11719913	—	2	2	no	—
96	#11719913	—	2	2	no	—
97	#11719912	—	4	4	no	—
98	C₈×D₅₃	—	4	4	no	—
99	C₁₆×C₅₃	—	2	2	no	—
100	#11719908	—	2	2	no	—
101	C₂³×C₅₃	#135	1	1	yes	characteristic
102	C₂²×D₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
109	C₂²⋊D₅₃	—	4	2	no	—
110	C₂²⋊D₅₃	—	4	2	no	—
111	C₂×Dic₅₃	—	2	2	no	—
112	C₂×Dic₅₃	—	4	2	no	—
113	C₂×Dic₅₃	—	4	1	no	—
114	C₂×Dic₅₃	—	4	2	no	—
115	D₂₁₂	—	2	2	no	—
116	D₂₁₂	—	4	1	no	—
117	C₄×D₅₃	—	4	2	no	—
118	C₄×D₅₃	—	4	2	no	—
119	Dic₁₀₆	—	2	2	no	—
120	Dic₁₀₆	—	4	1	no	—
121	C₈×C₅₃	—	4	2	no	—
122	C₈×C₅₃	—	4	2	no	—
123	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₅₃	#117	1	1	yes	characteristic
125	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
127	#88322	#102	1	2	yes	—
128	#88322	#102	1	2	yes	—
129	#12642	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	#416	1	1	yes	characteristic
132	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
133	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
134	C₂×D₅₃	—	4	2	no	—
135	C₂×D₅₃	—	8	2	no	—
136	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
138	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
139	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
140	Dic₅₃	—	4	2	no	—
141	Dic₅₃	—	4	2	no	—
142	Dic₅₃	—	8	2	no	—
143	C₂×#443	—	53	1	no	—
144	C₂×#359	—	53	1	no	—
145	#2138	—	53	1	no	—
146	C₂×#346	—	53	1	no	—
147	#1772	—	53	1	no	—
148	#1382	—	53	1	no	—
149	#1381	D₅₃	1	1	yes	characteristic, complemented
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
153	C₂×C₅₃	#2179	1	2	yes	—
154	D₅₃	—	8	2	no	—
155	C₂×#135	—	53	1	no	—
156	C₂×#131	—	106	1	no	—
157	C₂²×C₁₆	—	53	1	no	—
158	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
159	#443	—	53	4	no	—
160	#443	—	106	2	no	—
161	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
162	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
163	#427	—	53	1	no	—
164	C₂×#104	—	106	1	no	—
165	#405	—	106	1	no	—
166	C₂×C₄×C₈	—	53	1	no	—
167	#359	—	53	4	no	—
168	#358	—	53	2	no	—
169	#357	—	53	2	no	—
170	#346	—	106	2	no	—
171	#339	—	106	1	no	—
172	C₅₃	#12642	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	—	106	1	no	—
174	#135	—	53	2	no	—
175	#135	—	53	2	no	—
176	#135	—	106	1	no	—
177	#135	—	106	1	no	—
178	C₂×Q₁₆	—	53	1	no	—
179	C₂×SD₁₆	—	106	1	no	—
180	C₂×D₈	—	53	1	no	—
181	#131	—	106	4	no	—
182	#131	—	212	2	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	212	1	no	—
186	C₂²×C₈	—	53	1	no	—
187	C₂²×C₈	—	212	1	no	—
188	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	#117	—	106	1	no	—
190	C₂×C₄²	—	53	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	53	2	no	—
194	C₂×C₁₆	—	106	1	no	—
195	C₂×C₁₆	—	106	1	no	—
196	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
197	#108	—	2	1	no	—
198	#108	—	2	1	no	—
199	#108	C₄×D₅₃	1	2	yes	—
200	#108	C₄×D₅₃	1	2	yes	—
201	#107	—	53	2	no	—
202	#106	—	106	1	no	—
203	#104	—	106	4	no	—
204	#97	—	212	1	no	—
205	C₄×C₈	—	53	2	no	—
206	C₄×C₈	—	53	2	no	—
207	C₄×C₈	—	212	1	no	—
208	#41	—	106	2	no	—
209	#41	—	106	2	no	—
210	#41	—	212	1	no	—
211	#41	—	212	1	no	—
212	C₂×Q₈	—	106	1	no	—
213	C₂×D₄	—	106	1	no	—
214	C₂×D₄	—	212	1	no	—
215	C₂²×C₄	—	106	1	no	—
216	C₂²×C₄	—	212	1	no	—
217	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
218	Q₁₆	—	53	2	no	—
219	Q₁₆	—	106	1	no	—
220	SD₁₆	—	106	2	no	—
221	SD₁₆	—	106	2	no	—
222	D₈	—	53	2	no	—
223	D₈	—	106	1	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	4	1	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	212	4	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	53	2	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	2	no	—
235	C₂×C₈	—	212	4	no	—
236	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
240	C₂×C₈	#11719947	1	1	yes	characteristic
241	C₂×C₈	#11719949	1	1	yes	characteristic
242	C₄⋊C₄	—	106	2	no	—
243	C₂²⋊C₄	—	212	1	no	—
244	C₄²	—	53	2	no	—
245	C₄²	—	106	1	no	—
246	C₄²	—	212	1	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	106	2	no	—
249	C₂³	—	212	1	no	—
250	C₂³	#13133240	1	1	yes	characteristic
251	Q₈	—	106	2	no	—
252	Q₈	—	212	1	no	—
253	D₄	—	106	2	no	—
254	D₄	—	212	2	no	—
255	D₄	—	212	2	no	—
256	D₄	—	212	1	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	212	2	no	—
259	C₂×C₄	—	212	2	no	—
260	C₂×C₄	—	212	1	no	—
261	C₂×C₄	—	212	2	no	—
262	C₂×C₄	—	212	2	no	—
263	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄	#13133177	1	1	yes	characteristic
265	C₂×C₄	#13133179	1	1	yes	characteristic
266	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
267	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
268	C₂×C₄	#13133238	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	53	2	no	—
272	C₈	—	53	2	no	—
273	C₈	C₄²×D₅₃	1	1	yes	characteristic
274	C₈	#13133175	1	1	yes	characteristic
275	C₈	#13133177	1	1	yes	characteristic
276	C₈	#13133179	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	424	2	no	—
279	C₂²	—	212	2	no	—
280	C₂²	#16810618	1	1	yes	characteristic
281	C₂²	#16810705	1	1	yes	characteristic
282	C₂²	#16810708	1	1	yes	characteristic
283	C₄	—	424	2	no	—
284	C₄	—	212	2	no	—
285	C₄	—	212	2	no	—
286	C₄	#16810470	1	1	yes	characteristic
287	C₄	#102×D₅₃	1	1	yes	characteristic
288	C₄	#16810703	1	1	yes	characteristic
289	C₄	#16810710	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	424	2	no	—
293	C₂	#20274529	1	1	yes	characteristic
294	C₂	#20276600	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1381	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1381	D₅₃
hypocenter	C₅₃	#12642
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

Complements

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

N	H ≅ G/N	type
4. C₂×#16810398	C₂	semidirect
5. #20274597	C₂	semidirect
6. #20272812	C₂	semidirect
7. C₅₃×#1381	C₂	semidirect
12. #16810739	C₄	semidirect
17. #16810429	C₄	semidirect
18. #16810428	C₄	semidirect
19. #16810425	C₄	semidirect
149. #1381	D₅₃	semidirect
172. C₅₃	#12642	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810739	2
C₂×#16810429	2
#20276423	2
C₂×#16810398	2
#20274597	2
#20272812	2
C₅₃×#1381	2
#12642	53