Subgroups — #27537129

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133156	C₂	1	1	yes	maximal, characteristic
2	#427×D₅₃	C₂	1	2	yes	maximal
3	#20276502	C₂	1	4	yes	maximal, complemented
4	#20276500	C₂	1	4	yes	maximal, complemented
5	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276497	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	#117×D₅₃	—	2	2	no	—
10	C₂×#13133156	C₄	1	8	yes	complemented
11	C₂×#13133156	C₂²	1	2	yes	—
12	C₂×#13133156	C₂²	1	2	yes	—
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	#16810661	C₂²	1	2	yes	—
15	#16810660	C₂²	1	4	yes	—
16	#16810659	—	2	2	no	—
17	#16810658	—	2	2	no	—
18	#107×D₅₃	C₂²	1	4	yes	—
19	#106×D₅₃	—	2	2	no	—
20	#16810451	C₂²	1	2	yes	complemented
21	#16810450	C₂²	1	4	yes	complemented
22	#16810449	C₂²	1	2	yes	complemented
23	#16810448	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#427	C₂²	1	2	yes	—
26	C₅₃×#367	C₂²	1	2	yes	complemented
27	C₅₃×#366	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
29	C₂²×#88322	D₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₅₃	D₄	1	2	yes	—
32	C₂×C₈×D₅₃	Q₈	1	4	yes	—
33	C₂×C₈×D₅₃	C₂³	1	2	yes	—
34	#13133176	—	2	2	no	—
35	#13133175	—	2	4	no	—
36	#13133174	—	4	2	no	—
37	#13133173	—	4	2	no	—
38	C₄⋊C₄×D₅₃	—	2	4	no	—
39	C₂²⋊C₄×D₅₃	—	4	2	no	—
40	C₄²×D₅₃	—	4	2	no	—
41	#13133156	C₂×C₄	1	8	yes	complemented
42	#13133156	C₂×C₄	1	8	yes	complemented
43	C₂×#11719908	C₂×C₄	1	4	yes	complemented
44	C₂×#11719908	C₂³	1	2	yes	—
45	#13133151	C₂³	1	4	yes	—
46	#13133150	—	2	4	no	—
47	#13133149	—	2	2	no	—
48	#13133148	—	2	4	no	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	#117×C₅₃	—	2	2	no	—
51	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
54	#107×C₅₃	C₂³	1	4	yes	—
55	#106×C₅₃	—	2	2	no	—
56	C₂³×D₅₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
59	C₂×C₄×D₅₃	—	4	4	no	—
60	C₂×C₄×D₅₃	C₄⋊C₄	1	4	yes	—
61	C₂×C₄×D₅₃	D₈	1	2	yes	—
62	C₂×C₄×D₅₃	Q₁₆	1	4	yes	—
63	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₂²⋊C₄×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	—	4	2	no	—
71	C₂²⋊Dic₅₃	—	4	2	no	—
72	#11719921	—	4	4	no	—
73	C₄⋊Dic₅₃	—	2	4	no	—
74	#11719919	—	4	4	no	—
75	C₄×Dic₅₃	—	4	2	no	—
76	C₄×Dic₅₃	—	4	4	no	—
77	C₂×#88322	C₄⋊C₄	1	4	yes	—
78	C₂×#88322	C₂×D₄	1	2	yes	—
79	C₈×D₅₃	C₄⋊C₄	1	8	yes	—
80	C₈×D₅₃	C₂²×C₄	1	4	yes	—
81	C₈×D₅₃	C₂×Q₈	1	4	yes	—
82	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
83	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
84	#11719908	C₂²×C₄	1	4	yes	—
85	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₅₃	—	4	2	no	—
87	C₂²×D₅₃	#107	1	4	yes	—
88	C₂²×D₅₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₅₃	—	4	4	no	—
95	C₂×Dic₅₃	#107	1	4	yes	—
96	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
97	C₄×D₅₃	—	8	8	no	—
98	C₄×D₅₃	#107	1	8	yes	—
99	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
101	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
104	#88322	C₂×C₄⋊C₄	1	4	yes	—
105	#33409	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	C₂×#107	1	2	yes	—
108	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
109	C₂×D₅₃	—	2	4	no	—
110	C₂×D₅₃	—	4	4	no	—
111	C₂×D₅₃	C₂×#107	1	4	yes	—
112	C₄×C₅₃	—	8	4	no	—
113	C₄×C₅₃	C₂×#107	1	2	yes	—
114	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
116	Dic₅₃	—	8	4	no	—
117	Dic₅₃	C₂×#107	1	4	yes	—
118	C₂²×M₃₂	—	53	1	no	—
119	C₂×#427	—	53	2	no	—
120	#2159	—	53	4	no	—
121	#2159	—	53	4	no	—
122	#2158	—	53	1	no	—
123	#2158	—	53	2	no	—
124	#2158	D₅₃	1	1	yes	characteristic, complemented
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
127	C₂×C₅₃	#3414	1	2	yes	—
128	D₅₃	—	2	4	no	—
129	C₂³×C₈	—	53	1	no	—
130	C₂×#117	—	106	2	no	—
131	C₂×M₃₂	—	53	8	no	—
132	C₂×M₃₂	—	53	2	no	—
133	C₂×M₃₂	—	53	2	no	—
134	C₂²×C₁₆	—	53	1	no	—
135	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
136	#427	—	53	2	no	—
137	#427	—	53	4	no	—
138	#427	—	106	2	no	—
139	#427	—	106	2	no	—
140	#427	C₂×D₅₃	1	2	yes	—
141	C₂×#107	—	53	4	no	—
142	C₂×#106	—	106	2	no	—
143	#367	—	53	2	no	—
144	#367	—	53	4	no	—
145	#367	C₂×D₅₃	1	2	yes	complemented
146	#366	—	53	2	no	—
147	#366	—	53	4	no	—
148	#366	C₂×D₅₃	1	2	yes	complemented
149	C₅₃	#33409	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	—	53	1	no	—
151	C₂²×C₈	—	53	1	no	—
152	C₂²×C₈	—	53	4	no	—
153	C₂²×C₈	—	53	2	no	—
154	C₂²×C₈	—	53	4	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
157	#117	—	2	2	no	—
158	#117	—	106	4	no	—
159	#117	—	106	2	no	—
160	#117	—	212	2	no	—
161	#117	—	212	2	no	—
162	C₂×C₄⋊C₄	—	106	4	no	—
163	C₂×C₂²⋊C₄	—	212	2	no	—
164	C₂×C₄²	—	212	2	no	—
165	M₃₂	—	53	8	no	—
166	M₃₂	—	53	8	no	—
167	C₂×C₁₆	—	53	4	no	—
168	C₂×C₁₆	—	53	2	no	—
169	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
170	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
172	#107	—	53	4	no	—
173	#107	—	106	4	no	—
174	#107	C₂²×D₅₃	1	4	yes	—
175	#106	—	2	2	no	—
176	#106	—	106	4	no	—
177	#106	—	106	2	no	—
178	C₂⁴	—	53	1	no	—
179	C₂²×C₄	—	53	1	no	—
180	C₂²×C₄	—	53	4	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	4	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	—	212	4	no	—
185	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
186	C₂×C₈	—	53	4	no	—
187	C₂×C₈	—	53	2	no	—
188	C₂×C₈	—	53	8	no	—
189	C₂×C₈	—	53	4	no	—
190	C₂×C₈	—	53	4	no	—
191	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
192	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
193	C₂×C₈	Q₈×D₅₃	1	2	yes	—
194	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	106	4	no	—
197	C₄⋊C₄	—	212	4	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	212	4	no	—
200	C₂²⋊C₄	—	212	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	212	4	no	—
203	C₄²	—	212	2	no	—
204	C₁₆	—	53	4	no	—
205	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
206	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
207	C₂³	—	53	2	no	—
208	C₂³	—	53	4	no	—
209	C₂³	—	212	2	no	—
210	C₂³	D₈×D₅₃	1	1	yes	characteristic
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	53	2	no	—
213	C₂×C₄	—	53	4	no	—
214	C₂×C₄	—	53	4	no	—
215	C₂×C₄	—	53	4	no	—
216	C₂×C₄	—	53	8	no	—
217	C₂×C₄	—	424	8	no	—
218	C₂×C₄	—	212	4	no	—
219	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
220	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
221	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
223	C₈	—	53	4	no	—
224	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
225	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
226	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	53	4	no	—
229	C₂²	—	106	4	no	—
230	C₂²	—	212	4	no	—
231	C₂²	#107×D₅₃	1	2	yes	—
232	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
233	C₄	—	8	4	no	—
234	C₄	—	53	4	no	—
235	C₄	—	424	4	no	—
236	C₄	#107×D₅₃	1	2	yes	—
237	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
238	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	—	106	4	no	—
241	C₂	C₂×#107×D₅₃	1	1	yes	characteristic
242	C₂	#20287848	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2158	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2158	D₅₃
hypocenter	C₅₃	#33409
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #20276502	C₂	semidirect
4. #20276500	C₂	semidirect
5. #20276499	C₂	semidirect
6. #20276497	C₂	semidirect
7. C₅₃×#2158	C₂	semidirect
10. C₂×#13133156	C₄	semidirect
20. #16810451	C₂²	semidirect
21. #16810450	C₂²	semidirect
22. #16810449	C₂²	semidirect
23. #16810448	C₂²	semidirect
26. C₅₃×#367	C₂²	semidirect
27. C₅₃×#366	C₂²	semidirect
41. #13133156	C₂×C₄	semidirect
42. #13133156	C₂×C₄	semidirect
43. C₂×#11719908	C₂×C₄	semidirect
51. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
124. #2158	D₅₃	semidirect
145. #367	C₂×D₅₃	semidirect
148. #366	C₂×D₅₃	semidirect
149. C₅₃	#33409	semidirect
169. C₂×C₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133156	2
#427×D₅₃	2
#20276502	2
#20276500	2
#20276499	2
#20276497	2
C₅₃×#2158	2
#33409	53