Subgroups — #27537504

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287851	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810739	C₂	1	2	yes	maximal, complemented
3	#20276600	C₂	1	4	yes	maximal, complemented
4	#20276595	C₂	1	4	yes	maximal, complemented
5	#20276442	C₂	1	2	yes	maximal, complemented
6	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	C₂²	1	2	yes	—
9	C₂×#13133225	—	2	2	no	—
10	#16811066	C₄	1	4	yes	complemented
11	#16811061	C₄	1	4	yes	complemented
12	#16811055	C₂²	1	4	yes	—
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	#16810739	—	2	2	no	—
15	#16810739	—	2	2	no	—
16	#16810739	C₂²	1	4	yes	complemented
17	#16810739	C₂²	1	4	yes	complemented
18	C₂×#13133143	—	2	2	no	—
19	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	#16810429	C₂²	1	8	yes	complemented
21	#16810398	C₂²	1	2	yes	complemented
22	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#368	C₂²	1	4	yes	complemented
26	C₂×#11719945	—	2	2	no	—
27	#13133239	—	2	2	no	—
28	#13133239	—	2	2	no	—
29	#13133239	C₂×C₄	1	4	yes	—
30	#13133239	C₂³	1	4	yes	—
31	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
32	C₂×#11719913	—	2	2	no	—
33	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
34	#13133225	—	2	4	no	—
35	#13133225	—	2	4	no	—
36	#13133225	—	4	2	no	—
37	#13133225	—	4	2	no	—
38	C₂×#11719912	—	4	2	no	—
39	C₂²×#88322	D₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	—	4	2	no	—
41	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
42	Dic₄₂₄	C₂×C₄	1	4	yes	complemented
43	#13133162	C₂×C₄	1	8	yes	complemented
44	D₈₄₈	C₂×C₄	1	4	yes	complemented
45	#13133157	—	2	2	no	—
46	C₂×#11719908	C₂³	1	2	yes	—
47	#13133143	—	2	4	no	—
48	#13133143	—	2	4	no	—
49	C₈×Dic₅₃	D₄	1	1	yes	characteristic
50	C₈×Dic₅₃	D₄	1	1	yes	characteristic
51	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
52	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	—	2	2	no	—
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
56	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
57	#108×C₅₃	—	2	2	no	—
58	#108×C₅₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	—	4	2	no	—
61	C₂²×Dic₅₃	—	2	1	no	—
62	#11719945	—	2	4	no	—
63	#11719945	—	2	4	no	—
64	#11719945	—	4	2	no	—
65	#11719945	—	4	2	no	—
66	C₂×D₂₁₂	—	2	2	no	—
67	C₂×C₄×D₅₃	—	4	2	no	—
68	C₂×Dic₁₀₆	—	2	2	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	2	yes	—
75	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₄×Dic₅₃	—	2	1	no	—
77	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×#88322	D₈	1	2	yes	—
80	C₂×#88322	D₈	1	2	yes	—
81	C₂×#88322	C₂×D₄	1	1	yes	characteristic
82	C₂×#88322	C₂×D₄	1	1	yes	characteristic
83	Dic₂₁₂	—	2	2	no	—
84	Dic₂₁₂	C₂²×C₄	1	4	yes	—
85	D₄₂₄	—	2	2	no	—
86	D₄₂₄	C₂²×C₄	1	4	yes	—
87	#11719913	—	2	4	no	—
88	#11719913	—	2	4	no	—
89	#11719912	—	4	4	no	—
90	#11719912	—	4	4	no	—
91	C₈×D₅₃	—	4	4	no	—
92	C₈×D₅₃	—	4	4	no	—
93	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
94	#11719908	—	2	2	no	—
95	C₂³×C₅₃	#135	1	1	yes	characteristic
96	C₂²×D₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²⋊D₅₃	—	4	4	no	—
102	C₂²⋊D₅₃	—	4	4	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	4	2	no	—
106	C₂×Dic₅₃	—	4	2	no	—
107	D₂₁₂	—	2	4	no	—
108	D₂₁₂	—	4	2	no	—
109	C₄×D₅₃	—	4	4	no	—
110	C₄×D₅₃	—	4	4	no	—
111	Dic₁₀₆	—	2	4	no	—
112	Dic₁₀₆	—	4	2	no	—
113	C₈×C₅₃	—	4	4	no	—
114	C₈×C₅₃	C₄×D₄	1	2	yes	—
115	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
117	#88322	C₂×D₈	1	1	yes	characteristic
118	#88322	C₂×D₈	1	2	yes	—
119	#88322	C₂×D₈	1	1	yes	characteristic
120	#33463	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	4	1	no	—
122	C₂²×C₅₃	C₄×D₈	1	2	yes	—
123	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
124	C₂×D₅₃	—	4	4	no	—
125	C₂×D₅₃	—	8	4	no	—
126	C₄×C₅₃	C₄×D₈	1	2	yes	—
127	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
129	Dic₅₃	—	2	2	no	—
130	Dic₅₃	—	4	1	no	—
131	Dic₅₃	—	4	4	no	—
132	#3414	—	53	1	no	—
133	C₂×#443	—	53	2	no	—
134	#2179	—	53	4	no	—
135	#2178	—	53	4	no	—
136	#2161	D₅₃	1	1	yes	characteristic, complemented
137	#2143	—	53	2	no	—
138	C₄×M₃₂	—	53	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	2	2	no	—
141	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
142	D₅₃	—	8	4	no	—
143	C₂×#135	—	53	2	no	—
144	C₂×#131	—	106	2	no	—
145	#510	—	53	4	no	—
146	#509	—	53	4	no	—
147	#508	—	53	4	no	—
148	C₂×M₃₂	—	53	1	no	—
149	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
150	#443	—	53	4	no	—
151	#443	—	53	4	no	—
152	#443	—	106	2	no	—
153	#443	—	106	2	no	—
154	C₂×#108	C₂×D₅₃	1	2	yes	—
155	C₂×#104	—	106	2	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	#368	C₂×D₅₃	1	4	yes	complemented
158	#359	—	53	8	no	—
159	#346	—	53	2	no	—
160	C₄×C₁₆	—	53	2	no	—
161	C₅₃	#33463	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	106	2	no	—
163	#135	—	53	4	no	—
164	#135	—	53	4	no	—
165	#135	—	106	2	no	—
166	#135	—	106	2	no	—
167	C₂×Q₁₆	—	53	2	no	—
168	C₂×SD₁₆	—	106	2	no	—
169	C₂×D₈	—	53	2	no	—
170	#131	—	106	4	no	—
171	#131	—	106	4	no	—
172	#131	—	212	2	no	—
173	#131	—	212	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	212	2	no	—
176	C₂²×C₈	—	53	1	no	—
177	C₂²×C₈	—	212	2	no	—
178	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
179	C₂×C₄²	—	53	1	no	—
180	Q₃₂	—	53	4	no	—
181	SD₃₂	—	53	8	no	—
182	D₁₆	—	53	4	no	—
183	M₃₂	—	106	2	no	—
184	M₃₂	C₄×D₅₃	1	4	yes	—
185	C₂×C₁₆	—	53	2	no	—
186	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
187	#108	—	2	2	no	—
188	#108	C₂²×D₅₃	1	4	yes	—
189	#104	—	106	4	no	—
190	#104	—	106	4	no	—
191	C₄×C₈	—	53	1	no	—
192	C₄×C₈	—	53	1	no	—
193	C₄×C₈	—	53	1	no	—
194	C₄×C₈	—	53	1	no	—
195	#41	—	106	4	no	—
196	#41	—	106	4	no	—
197	#41	—	212	2	no	—
198	#41	—	212	2	no	—
199	C₂×Q₈	—	106	2	no	—
200	C₂×D₄	—	106	2	no	—
201	C₂×D₄	—	212	2	no	—
202	C₂²×C₄	—	106	1	no	—
203	C₂²×C₄	—	212	2	no	—
204	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
205	Q₁₆	—	53	4	no	—
206	Q₁₆	—	106	2	no	—
207	SD₁₆	—	106	4	no	—
208	SD₁₆	—	106	4	no	—
209	D₈	—	53	4	no	—
210	D₈	—	106	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	212	4	no	—
214	M₁₆	—	212	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	53	1	no	—
220	C₂×C₈	—	212	4	no	—
221	C₂×C₈	—	212	4	no	—
222	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
223	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
224	C₂×C₈	#11719949	1	2	yes	—
225	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
226	C₄²	—	53	1	no	—
227	C₄²	—	53	1	no	—
228	C₄²	—	106	1	no	—
229	C₁₆	—	106	2	no	—
230	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
231	C₂³	—	212	2	no	—
232	C₂³	#13133240	1	1	yes	characteristic
233	Q₈	—	106	4	no	—
234	Q₈	—	212	2	no	—
235	D₄	—	106	4	no	—
236	D₄	—	212	4	no	—
237	D₄	—	212	4	no	—
238	D₄	—	212	2	no	—
239	C₂×C₄	—	106	1	no	—
240	C₂×C₄	—	106	1	no	—
241	C₂×C₄	—	212	2	no	—
242	C₂×C₄	—	212	2	no	—
243	C₂×C₄	—	212	4	no	—
244	C₂×C₄	—	212	4	no	—
245	C₂×C₄	#13133179	1	2	yes	—
246	C₂×C₄	D₈×D₅₃	1	2	yes	—
247	C₂×C₄	#13133240	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	53	1	no	—
251	C₈	—	53	2	no	—
252	C₈	—	53	1	no	—
253	C₈	#13133179	1	2	yes	—
254	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
255	C₈	C₂×#11719949	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	424	4	no	—
258	C₂²	—	212	4	no	—
259	C₂²	#16810705	1	2	yes	—
260	C₂²	C₂×#13133240	1	1	yes	characteristic
261	C₄	—	106	2	no	—
262	C₄	—	212	1	no	—
263	C₄	—	212	4	no	—
264	C₄	#16810705	1	2	yes	—
265	C₄	C₂×#13133179	1	1	yes	characteristic
266	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	424	4	no	—
270	C₂	C₂×#16810705	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2161	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	#20287851	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2161	D₅₃
hypocenter	C₅₃	#33463
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

Complements

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

N	H ≅ G/N	type
2. C₂×#16810739	C₂	semidirect
3. #20276600	C₂	semidirect
4. #20276595	C₂	semidirect
5. #20276442	C₂	semidirect
6. M₃₂×Dic₅₃	C₂	semidirect
7. C₅₃×#2161	C₂	semidirect
10. #16811066	C₄	semidirect
11. #16811061	C₄	semidirect
16. #16810739	C₂²	semidirect
17. #16810739	C₂²	semidirect
20. #16810429	C₂²	semidirect
21. #16810398	C₂²	semidirect
22. C₁₆×Dic₅₃	C₂²	semidirect
25. C₅₃×#368	C₂²	semidirect
42. Dic₄₂₄	C₂×C₄	semidirect
43. #13133162	C₂×C₄	semidirect
44. D₈₄₈	C₂×C₄	semidirect
136. #2161	D₅₃	semidirect
157. #368	C₂×D₅₃	semidirect
161. C₅₃	#33463	semidirect

Maximal subgroups

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

subgroup	index
#20287851	2
C₂×#16810739	2
#20276600	2
#20276595	2
#20276442	2
M₃₂×Dic₅₃	2
C₅₃×#2161	2
#33463	53