Subgroups — #27537505

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	C₂×#16810740	C₂	1	2	yes	maximal, complemented
2	#20276601	C₂	1	4	yes	maximal, complemented
3	#20276596	C₂	1	4	yes	maximal, complemented
4	#20276510	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276443	C₂	1	2	yes	maximal, complemented
6	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3414	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133224	—	2	2	no	—
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	#16810740	—	2	2	no	—
11	#16810740	—	2	2	no	—
12	#16810740	C₂²	1	4	yes	complemented
13	#16810740	C₂²	1	4	yes	complemented
14	C₂×#13133153	C₂²	1	2	yes	—
15	C₂×#13133144	—	2	2	no	—
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810453	C₂²	1	4	yes	complemented
18	#16810430	C₂²	1	8	yes	complemented
19	#16810398	C₂²	1	2	yes	complemented
20	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
21	C₂×#135×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#510	C₄	1	4	yes	complemented
23	C₅₃×#509	C₄	1	4	yes	complemented
24	C₅₃×#508	C₂²	1	4	yes	—
25	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
26	#13133224	—	2	4	no	—
27	#13133224	—	2	4	no	—
28	#13133224	—	4	2	no	—
29	#13133224	—	4	2	no	—
30	C₂×#11719917	—	2	2	no	—
31	C₂×#11719917	—	4	2	no	—
32	C₂²×#88322	—	4	2	no	—
33	C₂²×#88322	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	#13133157	—	2	2	no	—
36	C₂×#11719908	C₂³	1	2	yes	—
37	#13133153	—	2	2	no	—
38	#13133153	C₂³	1	4	yes	—
39	#13133144	—	2	4	no	—
40	#13133144	—	2	4	no	—
41	C₈×Dic₅₃	D₄	1	1	yes	characteristic
42	C₈×Dic₅₃	D₄	1	1	yes	characteristic
43	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
44	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
45	C₂×#41×C₅₃	—	2	2	no	—
46	#135×C₅₃	—	2	2	no	—
47	#135×C₅₃	—	2	2	no	—
48	#135×C₅₃	C₂×C₄	1	4	yes	—
49	#135×C₅₃	C₂³	1	4	yes	—
50	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×SD₁₆×C₅₃	—	2	2	no	—
52	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
55	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
56	D₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
57	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
59	#41×C₅₃	—	2	4	no	—
60	#41×C₅₃	—	2	4	no	—
61	#41×C₅₃	—	4	2	no	—
62	#41×C₅₃	—	4	2	no	—
63	C₂×Q₈×C₅₃	—	2	2	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	—	4	2	no	—
66	C₂²×C₄×C₅₃	—	4	2	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	2	1	no	—
69	Q₁₆×C₅₃	—	2	2	no	—
70	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
71	SD₁₆×C₅₃	—	2	4	no	—
72	SD₁₆×C₅₃	—	2	4	no	—
73	D₈×C₅₃	—	2	2	no	—
74	D₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₄×Dic₅₃	—	2	1	no	—
80	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
82	#11719917	—	2	4	no	—
83	#11719917	—	4	2	no	—
84	#11719917	—	4	4	no	—
85	#11719917	—	4	4	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	4	no	—
88	C₂×#88322	—	4	4	no	—
89	C₂×#88322	D₈	1	2	yes	—
90	C₂×#88322	D₈	1	2	yes	—
91	C₂×#88322	C₂×D₄	1	1	yes	characteristic
92	C₂×#88322	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
94	#11719908	—	2	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	#135	1	1	yes	characteristic
97	Q₈×C₅₃	—	2	4	no	—
98	Q₈×C₅₃	—	4	2	no	—
99	D₄×C₅₃	—	2	4	no	—
100	D₄×C₅₃	—	4	4	no	—
101	D₄×C₅₃	—	4	4	no	—
102	D₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₅₃	—	2	1	no	—
111	C₂×Dic₅₃	—	2	1	no	—
112	C₂×Dic₅₃	—	4	2	no	—
113	C₈×C₅₃	C₄×D₄	1	2	yes	—
114	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
116	#88322	—	4	4	no	—
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	—	4	4	no	—
123	C₂²×C₅₃	—	8	4	no	—
124	C₂²×C₅₃	C₄×D₈	1	2	yes	—
125	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
126	C₄×C₅₃	—	4	4	no	—
127	C₄×C₅₃	C₄×D₈	1	2	yes	—
128	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
130	Dic₅₃	—	2	2	no	—
131	Dic₅₃	—	4	1	no	—
132	#3414	D₅₃	1	1	yes	characteristic
133	C₂×#443	—	53	2	no	—
134	#2179	—	53	4	no	—
135	#2178	—	53	4	no	—
136	#2161	—	53	1	no	—
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	—	8	4	no	—
142	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
143	C₂×#135	C₂×D₅₃	1	2	yes	—
144	C₂×#131	—	106	2	no	—
145	#510	Dic₅₃	1	4	yes	complemented
146	#509	Dic₅₃	1	4	yes	complemented
147	#508	C₂×D₅₃	1	4	yes	—
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	—	53	2	no	—
155	C₂×#104	—	106	2	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	#368	—	53	4	no	—
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	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	C₂×Dic₅₃	1	4	yes	—
166	#135	C₂²×D₅₃	1	4	yes	—
167	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
170	#131	—	106	4	no	—
171	#131	—	106	4	no	—
172	#131	—	212	2	no	—
173	#131	—	212	2	no	—
174	C₂×M₁₆	—	106	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	C₂×Dic₅₃	1	4	yes	complemented
181	SD₃₂	C₂×Dic₅₃	1	8	yes	complemented
182	D₁₆	C₂×Dic₅₃	1	4	yes	complemented
183	M₃₂	—	106	2	no	—
184	M₃₂	C₂×Dic₅₃	1	4	yes	—
185	C₂×C₁₆	—	53	2	no	—
186	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
187	#108	—	53	4	no	—
188	#108	—	106	2	no	—
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	—	2	4	no	—
196	#41	—	2	4	no	—
197	#41	—	4	2	no	—
198	#41	—	4	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	106	1	no	—
204	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
205	Q₁₆	—	2	2	no	—
206	Q₁₆	C₂²×Dic₅₃	1	4	yes	—
207	SD₁₆	—	2	4	no	—
208	SD₁₆	—	2	4	no	—
209	D₈	—	2	2	no	—
210	D₈	C₂²×Dic₅₃	1	4	yes	—
211	M₁₆	—	106	4	no	—
212	M₁₆	—	212	4	no	—
213	M₁₆	—	212	4	no	—
214	M₁₆	—	212	2	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	212	4	no	—
220	C₂×C₈	—	212	4	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
223	C₂×C₈	#11719947	1	2	yes	—
224	C₂×C₈	C₂²×Dic₅₃	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	C₂²×Dic₅₃	1	4	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	#13133236	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	2	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	106	1	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	—	212	2	no	—
245	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
246	C₂×C₄	D₈×D₅₃	1	2	yes	—
247	C₂×C₄	#13133236	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
249	C₈	—	53	1	no	—
250	C₈	—	53	2	no	—
251	C₈	—	53	1	no	—
252	C₈	—	212	4	no	—
253	C₈	D₄×Dic₅₃	1	2	yes	—
254	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
255	C₈	C₂×#11719947	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	D₈×Dic₅₃	1	2	yes	—
260	C₂²	C₂×#13133236	1	1	yes	characteristic
261	C₄	—	4	4	no	—
262	C₄	—	106	2	no	—
263	C₄	—	212	1	no	—
264	C₄	D₈×Dic₅₃	1	2	yes	—
265	C₄	C₂×D₄×Dic₅₃	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	—	8	4	no	—
270	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	C₅₃×#3414	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#3414	D₅₃
hypocenter	C₅₃	#33463
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

Complements

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

N	H ≅ G/N	type
1. C₂×#16810740	C₂	semidirect
2. #20276601	C₂	semidirect
3. #20276596	C₂	semidirect
4. #20276510	C₂	semidirect
5. #20276443	C₂	semidirect
6. M₃₂×Dic₅₃	C₂	semidirect
12. #16810740	C₂²	semidirect
13. #16810740	C₂²	semidirect
17. #16810453	C₂²	semidirect
18. #16810430	C₂²	semidirect
19. #16810398	C₂²	semidirect
20. C₁₆×Dic₅₃	C₂²	semidirect
22. C₅₃×#510	C₄	semidirect
23. C₅₃×#509	C₄	semidirect
54. Q₃₂×C₅₃	C₂×C₄	semidirect
55. SD₃₂×C₅₃	C₂×C₄	semidirect
56. D₁₆×C₅₃	C₂×C₄	semidirect
145. #510	Dic₅₃	semidirect
146. #509	Dic₅₃	semidirect
161. C₅₃	#33463	semidirect
180. Q₃₂	C₂×Dic₅₃	semidirect
181. SD₃₂	C₂×Dic₅₃	semidirect
182. D₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810740	2
#20276601	2
#20276596	2
#20276510	2
#20276443	2
M₃₂×Dic₅₃	2
C₅₃×#3414	2
#33463	53