Subgroups — #20283142

Order 6784.

556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133223	C₂	1	1	yes	maximal, characteristic
3	#16811091	C₂	1	1	yes	maximal, characteristic
4	#16810580	C₂	1	2	yes	maximal, complemented
5	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810577	C₂	1	2	yes	maximal, complemented
7	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11719912	C₂	1	2	yes	maximal, complemented
9	C₂×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133225	C₄	1	8	yes	complemented
14	#13133223	C₄	1	8	yes	complemented
15	C₂×#11719917	C₂²	1	1	yes	characteristic
16	C₂×#11719912	C₂²	1	2	yes	—
17	C₂×#11719912	C₂²	1	2	yes	—
18	C₂²×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	2	yes	—
20	C₂×C₈×D₅₃	C₂²	1	2	yes	—
21	#13133182	C₄	1	2	yes	—
22	#13133179	C₄	1	2	yes	—
23	#13133177	C₄	1	4	yes	—
24	#13133174	C₂²	1	2	yes	—
25	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
26	C₄²×D₅₃	C₂²	1	2	yes	—
27	#13133128	C₂²	1	2	yes	—
28	#13133127	C₂²	1	1	yes	characteristic
29	#13133127	C₂²	1	2	yes	—
30	#13133127	C₂²	1	1	yes	characteristic
31	#13133127	C₂²	1	1	yes	characteristic
32	#13133127	C₂²	1	1	yes	characteristic
33	C₄×#88322	C₂²	1	2	yes	—
34	C₈×Dic₅₃	C₂²	1	2	yes	—
35	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	#117×C₅₃	C₂²	1	1	yes	characteristic
38	#97×C₅₃	C₂²	1	2	yes	—
39	C₄×C₈×C₅₃	C₂²	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
42	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
43	#11719945	C₂×C₄	1	4	yes	—
44	#11719945	C₂×C₄	1	4	yes	—
45	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₅₃	C₂³	1	2	yes	—
48	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
49	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₄²×C₅₃	C₂³	1	2	yes	—
57	#11719921	C₂×C₄	1	4	yes	—
58	#11719919	C₂×C₄	1	4	yes	—
59	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
60	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
62	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
63	#11719917	C₂×C₄	1	4	yes	complemented
64	C₂×#88322	C₂×C₄	1	4	yes	complemented
65	C₂×#88322	C₂³	1	2	yes	—
66	C₂×#88322	C₂³	1	1	yes	characteristic
67	C₂×#88322	C₂³	1	1	yes	characteristic
68	#11719912	C₂×C₄	1	8	yes	complemented
69	#11719912	C₂×C₄	1	8	yes	complemented
70	C₈×D₅₃	C₂×C₄	1	8	yes	complemented
71	C₈×D₅₃	C₂×C₄	1	8	yes	complemented
72	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊D₅₃	C₄²	1	8	yes	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
84	D₂₁₂	C₄²	1	4	yes	—
85	C₄×D₅₃	—	4	4	no	—
86	C₄×D₅₃	C₂²×C₄	1	4	yes	—
87	C₄×D₅₃	C₂²×C₄	1	4	yes	—
88	Dic₁₀₆	C₄²	1	4	yes	—
89	C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₈×C₅₃	C₂²×C₄	1	4	yes	—
91	C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	#88322	C₂²×C₄	1	2	yes	—
93	#88322	C₂²×C₄	1	4	yes	—
94	#88322	C₂²×C₄	1	2	yes	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
97	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₅₃	—	2	4	no	—
99	C₂×D₅₃	C₂×C₄²	1	4	yes	—
100	C₄×C₅₃	—	2	4	no	—
101	C₄×C₅₃	C₂×C₄²	1	2	yes	—
102	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	Dic₅₃	—	2	2	no	—
105	Dic₅₃	—	2	2	no	—
106	Dic₅₃	C₂×C₄²	1	4	yes	—
107	#2875	—	53	1	no	maximal, Sylow
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
110	C₂×C₅₃	#568	1	2	yes	—
111	D₅₃	—	2	4	no	—
112	C₂×#131	—	53	1	no	—
113	C₂×#131	—	53	1	no	—
114	C₄×#41	—	53	1	no	—
115	#405	—	53	1	no	—
116	#405	—	53	2	no	—
117	#405	—	53	2	no	—
118	#405	D₅₃	1	1	yes	characteristic, complemented
119	C₄×M₁₆	—	53	1	no	—
120	C₄×M₁₆	—	53	2	no	—
121	C₂×#97	—	53	1	no	—
122	C₂×#97	—	53	2	no	—
123	C₅₃	#2875	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	53	1	no	—
125	#131	—	53	8	no	—
126	#131	—	53	8	no	—
127	C₂×M₁₆	—	53	1	no	—
128	C₂×M₁₆	—	53	2	no	—
129	C₂×M₁₆	—	53	2	no	—
130	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	2	no	—
133	C₂²×C₈	—	53	2	no	—
134	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₄×Q₈	—	53	2	no	—
136	C₄×D₄	—	53	2	no	—
137	C₄×D₄	—	53	4	no	—
138	#117	—	53	2	no	—
139	#117	C₂×D₅₃	1	1	yes	characteristic
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₄²	—	53	2	no	—
142	#97	—	53	1	no	—
143	#97	—	53	2	no	—
144	#97	—	53	1	no	—
145	#97	—	53	1	no	—
146	#97	—	53	1	no	—
147	#97	—	53	2	no	—
148	#97	C₂×D₅₃	1	2	yes	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	C₂×D₅₃	1	2	yes	—
152	#41	—	53	4	no	—
153	#41	—	53	4	no	—
154	C₂×Q₈	—	53	1	no	—
155	C₂×D₄	—	53	2	no	—
156	C₂×D₄	—	53	1	no	—
157	C₂²×C₄	—	53	2	no	—
158	C₂²×C₄	—	53	2	no	—
159	C₂²×C₄	—	53	4	no	—
160	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
161	M₁₆	—	53	4	no	—
162	M₁₆	—	53	8	no	—
163	M₁₆	—	53	8	no	—
164	M₁₆	C₄×D₅₃	1	4	yes	complemented
165	C₂×C₈	—	53	4	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	53	8	no	—
170	C₂×C₈	—	53	8	no	—
171	C₂×C₈	C₄×D₅₃	1	4	yes	complemented
172	C₂×C₈	C₂²×D₅₃	1	2	yes	—
173	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	C₄⋊C₄	—	53	4	no	—
176	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
177	C₂²⋊C₄	—	53	4	no	—
178	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
179	C₄²	—	53	4	no	—
180	C₄²	—	53	2	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	C₂²×D₅₃	1	2	yes	—
184	C₂³	—	53	2	no	—
185	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
186	Q₈	—	53	4	no	—
187	D₄	—	53	8	no	—
188	D₄	—	53	4	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	4	no	—
191	C₂×C₄	—	53	4	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	212	4	no	—
195	C₂×C₄	—	212	2	no	—
196	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
197	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
200	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
201	C₈	—	53	2	no	—
202	C₈	—	53	4	no	—
203	C₈	—	53	2	no	—
204	C₈	C₂×C₄×D₅₃	1	2	yes	—
205	C₈	C₂×C₄×D₅₃	1	4	yes	—
206	C₈	C₂×C₄×D₅₃	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	53	4	no	—
209	C₂²	—	106	4	no	—
210	C₂²	C₄²×D₅₃	1	2	yes	—
211	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	53	4	no	—
214	C₄	—	106	2	no	—
215	C₄	—	106	2	no	—
216	C₄	C₄²×D₅₃	1	2	yes	—
217	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
218	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	106	4	no	—
221	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
222	C₂	#16811388	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2875
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #16810580	C₂	semidirect
5. #16810579	C₂	semidirect
6. #16810577	C₂	semidirect
7. M₁₆×Dic₅₃	C₂	semidirect
8. C₄×#11719912	C₂	semidirect
9. C₂×#13133127	C₂	semidirect
10. #97×D₅₃	C₂	semidirect
11. C₅₃×#405	C₂	semidirect
13. #13133225	C₄	semidirect
14. #13133223	C₄	semidirect
49. M₁₆×C₅₃	C₂×C₄	semidirect
50. C₂×C₈×C₅₃	C₂×C₄	semidirect
63. #11719917	C₂×C₄	semidirect
64. C₂×#88322	C₂×C₄	semidirect
68. #11719912	C₂×C₄	semidirect
69. #11719912	C₂×C₄	semidirect
70. C₈×D₅₃	C₂×C₄	semidirect
71. C₈×D₅₃	C₂×C₄	semidirect
118. #405	D₅₃	semidirect
123. C₅₃	#2875	semidirect
164. M₁₆	C₄×D₅₃	semidirect
171. C₂×C₈	C₄×D₅₃	semidirect