Subgroups — #27537519

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283632	C₂	1	2	yes	maximal, complemented
2	#20276601	C₂	1	4	yes	maximal, complemented
3	#20276596	C₂	1	4	yes	maximal, complemented
4	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276443	C₂	1	2	yes	maximal, complemented
6	#20276311	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic
8	#16811390	—	2	2	no	—
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	#16810745	—	2	4	no	—
11	#16810740	C₂²	1	8	yes	complemented
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810640	—	2	2	no	—
14	#16810578	C₂²	1	1	yes	characteristic
15	#16810453	C₂²	1	2	yes	complemented
16	#16810453	C₂²	1	2	yes	complemented
17	#16810430	C₂²	1	8	yes	complemented
18	#16810398	C₂²	1	2	yes	complemented
19	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
20	C₂×#135×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#508	C₄	1	2	yes	complemented
22	C₅₃×#508	C₄	1	2	yes	complemented
23	C₅₃×#508	C₂²	1	2	yes	—
24	C₅₃×#508	C₂²	1	2	yes	—
25	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
26	C₂×SD₃₂×C₅₃	C₄	1	2	yes	complemented
27	C₂×D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
28	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
29	#13133224	—	2	8	no	—
30	#13133224	—	4	4	no	—
31	C₂×#11719917	—	2	2	no	—
32	C₂×#11719917	—	4	4	no	—
33	C₂×#11719917	D₄	1	1	yes	characteristic
34	#13133176	D₄	1	1	yes	characteristic
35	#13133157	—	2	1	no	—
36	#13133157	—	2	1	no	—
37	C₂×#11719908	C₂³	1	2	yes	—
38	#13133153	—	2	2	no	—
39	#13133153	C₂³	1	4	yes	—
40	#13133144	—	2	8	no	—
41	#13133127	D₄	1	2	yes	—
42	C₈×Dic₅₃	C₂³	1	2	yes	—
43	C₂×#41×C₅₃	—	2	2	no	—
44	#135×C₅₃	—	2	2	no	—
45	#135×C₅₃	—	2	2	no	—
46	#135×C₅₃	C₂×C₄	1	4	yes	—
47	#135×C₅₃	C₂³	1	4	yes	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
53	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
54	SD₃₂×C₅₃	C₂×C₄	1	4	yes	—
55	SD₃₂×C₅₃	C₂×C₄	1	4	yes	—
56	D₁₆×C₅₃	C₂×C₄	1	2	yes	—
57	D₁₆×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
64	#41×C₅₃	—	2	4	no	—
65	#41×C₅₃	—	2	4	no	—
66	#41×C₅₃	—	4	2	no	—
67	#41×C₅₃	—	4	2	no	—
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×D₄×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₅₃	—	2	2	no	—
74	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
75	SD₁₆×C₅₃	—	2	4	no	—
76	SD₁₆×C₅₃	—	2	4	no	—
77	D₈×C₅₃	—	2	2	no	—
78	D₈×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂²⋊Dic₅₃	—	2	1	no	—
86	C₄⋊Dic₅₃	—	2	1	no	—
87	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
88	#11719917	—	2	4	no	—
89	#11719917	—	4	2	no	—
90	#11719917	—	4	8	no	—
91	#11719917	D₈	1	2	yes	—
92	#11719917	D₈	1	2	yes	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	8	no	—
95	C₂×#88322	C₂×D₄	1	2	yes	—
96	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	#11719908	—	2	2	no	—
101	C₂³×C₅₃	—	4	2	no	—
102	C₂³×C₅₃	#135	1	1	yes	characteristic
103	Q₈×C₅₃	—	2	4	no	—
104	Q₈×C₅₃	—	4	2	no	—
105	D₄×C₅₃	—	2	4	no	—
106	D₄×C₅₃	—	4	4	no	—
107	D₄×C₅₃	—	4	4	no	—
108	D₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₅₃	—	2	2	no	—
119	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
123	#88322	—	4	4	no	—
124	#88322	C₂×D₈	1	4	yes	—
125	#33465	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	16	2	no	—
129	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
130	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
134	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
135	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
137	Dic₅₃	—	4	2	no	—
138	C₂×#508	D₅₃	1	1	yes	characteristic
139	#2957	—	53	2	no	—
140	#2179	—	53	4	no	—
141	#2178	—	53	4	no	—
142	C₂×#368	—	53	1	no	—
143	#2143	—	53	2	no	—
144	#2108	—	53	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	8	4	no	—
149	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
150	C₂×#135	C₂×D₅₃	1	2	yes	—
151	#568	—	106	2	no	—
152	#508	Dic₅₃	1	2	yes	complemented
153	#508	Dic₅₃	1	2	yes	complemented
154	#508	C₂×D₅₃	1	2	yes	—
155	#508	C₂×D₅₃	1	2	yes	—
156	C₂×Q₃₂	Dic₅₃	1	1	yes	characteristic, complemented
157	C₂×SD₃₂	Dic₅₃	1	2	yes	complemented
158	C₂×D₁₆	Dic₅₃	1	1	yes	characteristic, complemented
159	C₂×M₃₂	—	53	1	no	—
160	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
161	#444	—	106	4	no	—
162	#443	—	53	8	no	—
163	C₂×#108	—	53	2	no	—
164	#421	—	106	2	no	—
165	#405	—	53	1	no	—
166	#368	—	53	2	no	—
167	#368	—	53	2	no	—
168	#359	—	53	8	no	—
169	#346	—	53	2	no	—
170	C₄×C₁₆	—	53	2	no	—
171	C₅₃	#33465	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	—	2	2	no	—
175	#135	C₂×Dic₅₃	1	4	yes	—
176	#135	C₂²×D₅₃	1	4	yes	—
177	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
180	#131	—	106	8	no	—
181	#131	—	212	4	no	—
182	C₂×M₁₆	—	53	1	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	—	212	4	no	—
185	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
186	#117	—	53	1	no	—
187	Q₃₂	C₂×Dic₅₃	1	2	yes	—
188	Q₃₂	C₂×Dic₅₃	1	2	yes	—
189	SD₃₂	C₂×Dic₅₃	1	4	yes	—
190	SD₃₂	C₂×Dic₅₃	1	4	yes	—
191	D₁₆	C₂×Dic₅₃	1	2	yes	—
192	D₁₆	C₂×Dic₅₃	1	2	yes	—
193	M₃₂	—	106	1	no	—
194	M₃₂	—	106	1	no	—
195	C₂×C₁₆	—	53	2	no	—
196	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
197	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
198	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
199	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
200	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
202	#108	—	53	4	no	—
203	#108	—	106	2	no	—
204	#104	—	106	8	no	—
205	#97	—	53	2	no	—
206	C₄×C₈	—	53	2	no	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×Dic₅₃	1	4	yes	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×Dic₅₃	1	4	yes	—
222	M₁₆	—	53	2	no	—
223	M₁₆	—	53	2	no	—
224	M₁₆	—	106	4	no	—
225	M₁₆	—	212	8	no	—
226	M₁₆	—	212	2	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	212	8	no	—
229	C₂×C₈	—	212	2	no	—
230	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
231	C₂×C₈	#11719947	1	1	yes	characteristic
232	C₂×C₈	#11719947	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
236	C₄⋊C₄	—	106	1	no	—
237	C₂²⋊C₄	—	106	1	no	—
238	C₄²	—	53	2	no	—
239	C₁₆	—	106	2	no	—
240	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
241	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
242	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
243	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	#13133236	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	106	2	no	—
256	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
258	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
259	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
260	C₂×C₄	#13133236	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
262	C₈	—	53	4	no	—
263	C₈	—	212	4	no	—
264	C₈	D₄×Dic₅₃	1	1	yes	characteristic
265	C₈	D₄×Dic₅₃	1	1	yes	characteristic
266	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
267	C₈	C₂×#11719947	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
272	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
273	C₂²	C₂×#13133236	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	212	2	no	—
276	C₄	D₈×Dic₅₃	1	1	yes	characteristic
277	C₄	D₈×Dic₅₃	1	1	yes	characteristic
278	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
279	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#508	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	#33465
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #20283632	C₂	semidirect
2. #20276601	C₂	semidirect
3. #20276596	C₂	semidirect
4. C₂×#16810453	C₂	semidirect
5. #20276443	C₂	semidirect
6. #20276311	C₂	semidirect
11. #16810740	C₂²	semidirect
15. #16810453	C₂²	semidirect
16. #16810453	C₂²	semidirect
17. #16810430	C₂²	semidirect
18. #16810398	C₂²	semidirect
19. C₁₆×Dic₅₃	C₂²	semidirect
21. C₅₃×#508	C₄	semidirect
22. C₅₃×#508	C₄	semidirect
25. C₂×Q₃₂×C₅₃	C₄	semidirect
26. C₂×SD₃₂×C₅₃	C₄	semidirect
27. C₂×D₁₆×C₅₃	C₄	semidirect
152. #508	Dic₅₃	semidirect
153. #508	Dic₅₃	semidirect
156. C₂×Q₃₂	Dic₅₃	semidirect
157. C₂×SD₃₂	Dic₅₃	semidirect
158. C₂×D₁₆	Dic₅₃	semidirect
171. C₅₃	#33465	semidirect

Maximal subgroups

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

subgroup	index
#20283632	2
#20276601	2
#20276596	2
C₂×#16810453	2
#20276443	2
#20276311	2
C₂×C₅₃×#508	2
#33465	53