Subgroups — #27537454

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283567	C₂	1	2	yes	maximal, complemented
2	#20276591	C₂	1	4	yes	maximal, complemented
3	D₁₆×Dic₅₃	C₂	1	4	yes	maximal, complemented
4	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276431	C₂	1	2	yes	maximal, complemented
6	#20276311	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811098	—	2	2	no	—
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	#16810735	—	2	4	no	—
11	D₈×Dic₅₃	C₂²	1	8	yes	complemented
12	#16810661	C₂²	1	2	yes	—
13	#16810628	—	2	2	no	—
14	#16810578	C₂²	1	1	yes	characteristic
15	#16810451	C₂²	1	2	yes	complemented
16	#16810449	C₂²	1	2	yes	complemented
17	#16810424	C₂²	1	8	yes	complemented
18	#16810398	C₂²	1	2	yes	complemented
19	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
20	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
21	C₂×D₁₆×C₅₃	C₄	1	8	yes	complemented
22	C₂×D₁₆×C₅₃	C₂²	1	2	yes	—
23	C₂×D₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719917	D₄	1	1	yes	characteristic
26	D₄×Dic₅₃	—	2	8	no	—
27	D₄×Dic₅₃	—	4	4	no	—
28	#13133176	—	2	2	no	—
29	#13133176	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₅₃	—	4	4	no	—
31	#13133157	—	2	2	no	—
32	C₂×#11719908	C₂³	1	2	yes	—
33	#13133151	C₂³	1	4	yes	—
34	#13133149	—	2	2	no	—
35	#13133138	—	2	8	no	—
36	#13133127	D₄	1	2	yes	—
37	C₈×Dic₅₃	C₂³	1	2	yes	—
38	C₂²×D₄×C₅₃	—	2	2	no	—
39	C₂×D₈×C₅₃	—	2	2	no	—
40	C₂×D₈×C₅₃	—	2	4	no	—
41	C₂×D₈×C₅₃	C₂×C₄	1	8	yes	complemented
42	C₂×D₈×C₅₃	C₂³	1	4	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	D₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
45	D₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
46	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
49	C₂⁴×C₅₃	—	4	2	no	—
50	C₂×D₄×C₅₃	—	2	8	no	—
51	C₂×D₄×C₅₃	—	2	4	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	4	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	4	8	no	—
56	D₈×C₅₃	—	2	8	no	—
57	D₈×C₅₃	—	2	4	no	—
58	D₈×C₅₃	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	#41	1	2	yes	—
62	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂²⋊Dic₅₃	—	4	2	no	—
64	C₂²⋊Dic₅₃	—	4	8	no	—
65	C₂²⋊Dic₅₃	D₈	1	2	yes	—
66	C₄⋊Dic₅₃	—	2	4	no	—
67	C₄⋊Dic₅₃	D₈	1	2	yes	—
68	C₄×Dic₅₃	—	4	2	no	—
69	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
70	#11719917	—	2	2	no	—
71	C₂×#88322	C₂×D₄	1	2	yes	—
72	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
73	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
74	#11719908	—	2	2	no	—
75	C₂³×C₅₃	—	4	4	no	—
76	C₂³×C₅₃	—	4	8	no	—
77	C₂³×C₅₃	—	16	4	no	—
78	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
79	D₄×C₅₃	—	2	8	no	—
80	D₄×C₅₃	—	4	4	no	—
81	D₄×C₅₃	—	4	8	no	—
82	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#135	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₅₃	—	4	4	no	—
87	C₂×Dic₅₃	—	16	8	no	—
88	C₂×Dic₅₃	C₂×D₈	1	4	yes	—
89	C₈×C₅₃	C₄×D₄	1	2	yes	—
90	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
92	#88322	—	2	2	no	—
93	#33456	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	4	8	no	—
96	C₂²×C₅₃	—	8	8	no	—
97	C₂²×C₅₃	—	16	8	no	—
98	C₂²×C₅₃	C₄×D₈	1	2	yes	—
99	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₅₃	C₄×D₈	1	2	yes	—
101	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
103	Dic₅₃	—	2	4	no	—
104	Dic₅₃	—	8	4	no	—
105	C₂²×D₁₆	D₅₃	1	1	yes	characteristic
106	#2948	—	53	2	no	—
107	#2177	—	53	4	no	—
108	C₄×D₁₆	—	53	4	no	—
109	#2158	—	53	1	no	—
110	#2140	—	53	2	no	—
111	#2108	—	53	1	no	—
112	C₂×C₅₃	—	2	2	no	—
113	C₂×C₅₃	—	8	8	no	—
114	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
115	C₂×C₅₃	#3415	1	2	yes	—
116	C₂²×D₈	C₂×D₅₃	1	2	yes	—
117	#518	—	106	2	no	—
118	C₂×D₁₆	Dic₅₃	1	8	yes	complemented
119	C₂×D₁₆	C₂×D₅₃	1	2	yes	—
120	C₂×D₁₆	C₂×D₅₃	1	2	yes	—
121	C₂×M₃₂	—	53	1	no	—
122	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
123	#442	—	106	4	no	—
124	C₄×D₈	—	53	8	no	—
125	#427	—	53	2	no	—
126	#418	—	106	2	no	—
127	#405	—	53	1	no	—
128	#367	—	53	2	no	—
129	#366	—	53	2	no	—
130	#357	—	53	8	no	—
131	#346	—	53	2	no	—
132	C₄×C₁₆	—	53	2	no	—
133	C₅₃	#33456	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	—	2	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	C₂×Dic₅₃	1	8	yes	complemented
138	C₂×D₈	C₂²×D₅₃	1	4	yes	—
139	C₂×M₁₆	—	53	1	no	—
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₄×D₄	—	106	8	no	—
142	C₄×D₄	—	212	4	no	—
143	#117	—	53	1	no	—
144	#117	—	106	2	no	—
145	C₂×C₂²⋊C₄	—	212	4	no	—
146	D₁₆	C₂×Dic₅₃	1	8	yes	complemented
147	D₁₆	C₂×Dic₅₃	1	8	yes	complemented
148	M₃₂	—	106	2	no	—
149	C₂×C₁₆	—	53	2	no	—
150	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
153	#107	—	53	4	no	—
154	#106	—	106	2	no	—
155	#102	—	106	8	no	—
156	#97	—	53	2	no	—
157	C₄×C₈	—	53	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	212	8	no	—
164	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
165	D₈	—	2	8	no	—
166	D₈	—	2	4	no	—
167	D₈	C₂²×Dic₅₃	1	8	yes	—
168	M₁₆	—	106	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
171	C₂×C₈	#11719947	1	2	yes	—
172	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
173	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
174	C₄⋊C₄	—	53	2	no	—
175	C₄⋊C₄	—	106	4	no	—
176	C₂²⋊C₄	—	53	2	no	—
177	C₂²⋊C₄	—	212	8	no	—
178	C₂²⋊C₄	—	212	2	no	—
179	C₄²	—	53	2	no	—
180	C₄²	—	212	2	no	—
181	C₁₆	—	106	2	no	—
182	C₁₆	C₂²×Dic₅₃	1	2	yes	—
183	C₁₆	C₂²×Dic₅₃	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	D₈×D₅₃	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	4	no	—
190	D₄	—	4	8	no	—
191	C₂×C₄	—	53	4	no	—
192	C₂×C₄	—	848	8	no	—
193	C₂×C₄	—	212	4	no	—
194	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
195	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
196	C₂×C₄	#13133236	1	2	yes	—
197	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
198	C₈	—	106	2	no	—
199	C₈	D₄×Dic₅₃	1	2	yes	—
200	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
201	C₈	C₂×#11719947	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	D₈×Dic₅₃	1	2	yes	—
207	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
208	C₄	—	106	4	no	—
209	C₄	—	424	4	no	—
210	C₄	D₈×Dic₅₃	1	2	yes	—
211	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
212	C₄	C₂×#13133236	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic
216	C₂	#20287852	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂²×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×D₁₆	D₅₃
hypocenter	C₅₃	#33456
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20283567	C₂	semidirect
2. #20276591	C₂	semidirect
3. D₁₆×Dic₅₃	C₂	semidirect
4. #20276499	C₂	semidirect
5. #20276431	C₂	semidirect
6. #20276311	C₂	semidirect
11. D₈×Dic₅₃	C₂²	semidirect
15. #16810451	C₂²	semidirect
16. #16810449	C₂²	semidirect
17. #16810424	C₂²	semidirect
18. #16810398	C₂²	semidirect
19. C₁₆×Dic₅₃	C₂²	semidirect
21. C₂×D₁₆×C₅₃	C₄	semidirect
41. C₂×D₈×C₅₃	C₂×C₄	semidirect
44. D₁₆×C₅₃	C₂×C₄	semidirect
45. D₁₆×C₅₃	C₂×C₄	semidirect
46. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
118. C₂×D₁₆	Dic₅₃	semidirect
133. C₅₃	#33456	semidirect
137. C₂×D₈	C₂×Dic₅₃	semidirect
146. D₁₆	C₂×Dic₅₃	semidirect
147. D₁₆	C₂×Dic₅₃	semidirect
150. C₂×C₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283567	2
#20276591	2
D₁₆×Dic₅₃	2
#20276499	2
#20276431	2
#20276311	2
C₂²×D₁₆×C₅₃	2
#33456	53