Subgroups — #20288508

Order 6784.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133264	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133263	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811285	C₂	1	4	yes	maximal, complemented
4	#16811268	C₂	1	12	yes	maximal, complemented
5	#16811199	C₂	1	3	yes	maximal, complemented
6	#16811192	C₂	1	3	yes	maximal, complemented
7	#16811173	C₂	1	3	yes	maximal, complemented
8	C₂×#13133205	C₂	1	3	yes	maximal, complemented
9	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133264	—	2	8	no	—
11	#13133263	—	2	8	no	—
12	C₂×#11719949	C₂²	1	2	yes	—
13	C₂×#11719945	C₂²	1	3	yes	—
14	C₂×#11719945	C₂²	1	3	yes	complemented
15	C₂×#11719947	C₂²	1	6	yes	complemented
16	C₂²×Dic₁₀₆	C₂²	1	3	yes	complemented
17	C₂×Q₈×D₅₃	C₂²	1	2	yes	complemented
18	C₂²×D₂₁₂	C₂²	1	3	yes	—
19	C₂×D₄×D₅₃	C₂²	1	6	yes	—
20	#13133217	C₂²	1	3	yes	complemented
21	#13133214	C₂²	1	3	yes	complemented
22	#13133205	C₂²	1	12	yes	complemented
23	#13133205	C₂²	1	3	yes	complemented
24	#13133205	C₂²	1	6	yes	complemented
25	#13133205	C₂²	1	1	yes	characteristic, complemented
26	#13133201	C₂²	1	12	yes	complemented
27	#13133197	C₂²	1	12	yes	complemented
28	#13133192	C₂²	1	12	yes	complemented
29	#13133191	C₂²	1	12	yes	complemented
30	#13133187	C₂²	1	12	yes	complemented
31	C₄×D₂₁₂	C₂²	1	12	yes	complemented
32	C₂×#11719921	C₂²	1	12	yes	complemented
33	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
35	C₄×D₄×C₅₃	C₂²	1	6	yes	complemented
36	#117×C₅₃	C₂²	1	3	yes	complemented
37	C₂×C₄²×C₅₃	C₂²	1	3	yes	complemented
38	C₂³×D₅₃	C₂³	1	6	yes	—
39	#41×C₅₃	D₄	1	8	yes	complemented
40	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₅₃	C₂³	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
43	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
44	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
45	C₂×C₂²⋊D₅₃	C₂³	1	6	yes	—
46	C₂²×Dic₅₃	C₂³	1	6	yes	—
47	#11719949	—	2	8	no	—
48	Q₈×D₅₃	—	2	8	no	—
49	#11719947	—	2	24	no	—
50	D₄×D₅₃	—	2	24	no	—
51	#11719945	—	2	12	no	—
52	#11719945	—	2	12	no	—
53	C₂×D₂₁₂	—	2	12	no	—
54	C₂×D₂₁₂	C₂³	1	6	yes	—
55	C₂×D₂₁₂	C₂³	1	3	yes	—
56	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
57	C₂×D₂₁₂	C₂³	1	12	yes	—
58	C₂×C₄×D₅₃	C₂³	1	6	yes	—
59	C₂×C₄×D₅₃	C₂³	1	6	yes	—
60	C₂×Dic₁₀₆	—	2	12	no	—
61	C₂×Dic₁₀₆	C₂³	1	6	yes	—
62	C₂×Dic₁₀₆	C₂³	1	3	yes	—
63	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
65	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
66	C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄²×C₅₃	C₂³	1	3	yes	—
68	C₄²×C₅₃	C₂³	1	6	yes	—
69	#11719921	C₂³	1	12	yes	—
70	#11719921	C₂³	1	24	yes	—
71	#11719921	C₂³	1	4	yes	—
72	C₄⋊Dic₅₃	C₂³	1	12	yes	—
73	C₂³×C₅₃	C₂⁴	1	3	yes	—
74	C₂²×D₅₃	—	2	12	no	—
75	C₂²×D₅₃	—	4	12	no	—
76	C₂²×D₅₃	C₂⁴	1	2	yes	—
77	C₂²×D₅₃	C₂⁴	1	6	yes	—
78	Q₈×C₅₃	C₂×D₄	1	4	yes	—
79	D₄×C₅₃	C₂×D₄	1	12	yes	—
80	C₂×C₄×C₅₃	—	4	6	no	—
81	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
82	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
83	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
84	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
86	C₂²⋊D₅₃	—	2	12	no	—
87	C₂²⋊D₅₃	—	2	12	no	—
88	C₂×Dic₅₃	—	2	12	no	—
89	C₂×Dic₅₃	C₂⁴	1	6	yes	—
90	C₂×Dic₅₃	C₂⁴	1	2	yes	—
91	D₂₁₂	—	2	6	no	—
92	D₂₁₂	—	2	12	no	—
93	D₂₁₂	—	2	2	no	—
94	D₂₁₂	—	4	12	no	—
95	C₄×D₅₃	—	2	12	no	—
96	C₄×D₅₃	—	2	12	no	—
97	Dic₁₀₆	—	2	6	no	—
98	Dic₁₀₆	—	2	12	no	—
99	Dic₁₀₆	—	2	2	no	—
100	C₂²×C₅₃	—	2	6	no	—
101	C₂²×C₅₃	C₂²×D₄	1	6	yes	—
102	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
103	C₂×D₅₃	—	2	2	no	—
104	C₂×D₅₃	—	2	6	no	—
105	C₂×D₅₃	—	2	4	no	—
106	C₂×D₅₃	—	2	12	no	—
107	C₂×D₅₃	—	8	12	no	—
108	C₄×C₅₃	—	2	6	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	C₂²×D₄	1	2	yes	—
111	C₄×C₅₃	C₂²×D₄	1	6	yes	—
112	Dic₅₃	—	2	6	no	—
113	Dic₅₃	—	2	2	no	—
114	#3500	—	53	1	no	maximal, Sylow
115	C₂×C₅₃	—	2	6	no	—
116	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
117	C₂×C₅₃	#585	1	2	yes	—
118	D₅₃	—	4	2	no	—
119	D₅₃	—	4	6	no	—
120	C₂×2₋¹⁺⁴	—	53	1	no	—
121	C₂×H(5,𝔽₂)	—	53	1	no	—
122	#548	—	53	4	no	—
123	#546	—	53	12	no	—
124	#536	—	53	3	no	—
125	#535	—	53	3	no	—
126	#532	—	53	3	no	—
127	C₂×#124	—	53	3	no	—
128	C₄×#41	D₅₃	1	1	yes	characteristic, complemented
129	C₅₃	#3500	1	1	yes	characteristic, complemented, Sylow
130	2₋¹⁺⁴	—	106	8	no	—
131	H(5,𝔽₂)	—	106	8	no	—
132	C₂×#41	—	53	2	no	—
133	C₂×#41	—	53	6	no	—
134	C₂×#41	—	53	3	no	—
135	C₂×#41	—	53	3	no	—
136	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
137	C₂²×Q₈	—	53	2	no	—
138	C₂²×Q₈	—	53	3	no	—
139	C₂²×D₄	—	53	6	no	—
140	C₂²×D₄	—	53	3	no	—
141	C₄⋊Q₈	—	53	3	no	—
142	#127	—	53	3	no	—
143	#124	—	53	3	no	—
144	#124	—	53	6	no	—
145	#124	—	53	12	no	—
146	#124	—	53	1	no	—
147	#123	—	53	12	no	—
148	C₂²⋊Q₈	—	53	12	no	—
149	#121	—	53	12	no	—
150	#121	—	53	12	no	—
151	#120	—	53	12	no	—
152	C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
153	C₄×D₄	—	53	12	no	—
154	C₄×D₄	C₂×D₅₃	1	6	yes	complemented
155	#117	C₂×D₅₃	1	3	yes	complemented
156	C₂×C₂²⋊C₄	—	53	12	no	—
157	C₂×C₄²	C₂×D₅₃	1	3	yes	complemented
158	C₂⁴	—	53	6	no	—
159	#41	—	106	8	no	—
160	#41	—	106	24	no	—
161	#41	—	106	12	no	—
162	#41	—	106	12	no	—
163	#41	D₂₁₂	1	8	yes	complemented
164	C₂×Q₈	—	53	6	no	—
165	C₂×Q₈	—	53	3	no	—
166	C₂×Q₈	—	53	1	no	—
167	C₂×Q₈	—	106	8	no	—
168	C₂×Q₈	—	106	12	no	—
169	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
170	C₂×D₄	—	53	6	no	—
171	C₂×D₄	—	53	6	no	—
172	C₂×D₄	—	53	6	no	—
173	C₂×D₄	—	53	3	no	—
174	C₂×D₄	—	53	1	no	—
175	C₂×D₄	—	53	12	no	—
176	C₂×D₄	—	106	24	no	—
177	C₂×D₄	—	106	12	no	—
178	C₂×D₄	C₂²×D₅₃	1	3	yes	—
179	C₂²×C₄	—	53	6	no	—
180	C₂²×C₄	—	53	6	no	—
181	C₂²×C₄	—	53	6	no	—
182	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
183	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
184	C₄⋊C₄	—	53	12	no	—
185	C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
186	C₂²⋊C₄	—	53	12	no	—
187	C₂²⋊C₄	—	53	24	no	—
188	C₂²⋊C₄	—	53	4	no	—
189	C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
190	C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	C₄²	C₂²×D₅₃	1	3	yes	—
192	C₄²	C₂²×D₅₃	1	6	yes	—
193	C₂³	—	53	6	no	—
194	C₂³	—	53	2	no	—
195	C₂³	—	106	12	no	—
196	C₂³	—	212	12	no	—
197	C₂³	C₂³×D₅₃	1	3	yes	—
198	Q₈	—	106	12	no	—
199	Q₈	—	106	6	no	—
200	Q₈	—	106	2	no	—
201	Q₈	C₂×D₂₁₂	1	4	yes	—
202	D₄	—	106	12	no	—
203	D₄	—	106	12	no	—
204	D₄	—	106	12	no	—
205	D₄	—	106	6	no	—
206	D₄	—	106	2	no	—
207	D₄	—	212	12	no	—
208	D₄	C₂×D₂₁₂	1	12	yes	—
209	C₂×C₄	—	4	6	no	—
210	C₂×C₄	—	53	6	no	—
211	C₂×C₄	—	53	2	no	—
212	C₂×C₄	—	106	12	no	—
213	C₂×C₄	—	106	12	no	—
214	C₂×C₄	—	106	12	no	—
215	C₂×C₄	C₂×D₂₁₂	1	12	yes	—
216	C₂×C₄	C₂³×D₅₃	1	2	yes	—
217	C₂×C₄	C₂³×D₅₃	1	6	yes	—
218	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₅₃	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	106	2	no	—
222	C₂²	—	106	6	no	—
223	C₂²	—	106	4	no	—
224	C₂²	—	106	12	no	—
225	C₂²	—	424	12	no	—
226	C₂²	C₂²×D₂₁₂	1	6	yes	—
227	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	6	no	—
230	C₄	—	106	2	no	—
231	C₄	—	106	6	no	—
232	C₄	C₂²×D₂₁₂	1	6	yes	—
233	C₄	C₂²×D₂₁₂	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	212	6	no	—
236	C₂	—	212	2	no	—
237	C₂	C₂³×D₂₁₂	1	1	yes	characteristic
238	C₂	#16811459	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₄×#41×C₅₃	C₂
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	#3500
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#13133264	C₂	semidirect
2. C₂×#13133263	C₂	semidirect
3. #16811285	C₂	semidirect
4. #16811268	C₂	semidirect
5. #16811199	C₂	semidirect
6. #16811192	C₂	semidirect
7. #16811173	C₂	semidirect
8. C₂×#13133205	C₂	semidirect
9. C₄×#41×C₅₃	C₂	semidirect
14. C₂×#11719945	C₂²	semidirect
15. C₂×#11719947	C₂²	semidirect
16. C₂²×Dic₁₀₆	C₂²	semidirect
17. C₂×Q₈×D₅₃	C₂²	semidirect
20. #13133217	C₂²	semidirect
21. #13133214	C₂²	semidirect
22. #13133205	C₂²	semidirect
23. #13133205	C₂²	semidirect
24. #13133205	C₂²	semidirect
25. #13133205	C₂²	semidirect
26. #13133201	C₂²	semidirect
27. #13133197	C₂²	semidirect
28. #13133192	C₂²	semidirect
29. #13133191	C₂²	semidirect
30. #13133187	C₂²	semidirect
31. C₄×D₂₁₂	C₂²	semidirect
32. C₂×#11719921	C₂²	semidirect
34. C₄×Q₈×C₅₃	C₂²	semidirect
35. C₄×D₄×C₅₃	C₂²	semidirect
36. #117×C₅₃	C₂²	semidirect
37. C₂×C₄²×C₅₃	C₂²	semidirect
39. #41×C₅₃	D₄	semidirect
128. C₄×#41	D₅₃	semidirect
129. C₅₃	#3500	semidirect
152. C₄×Q₈	C₂×D₅₃	semidirect
154. C₄×D₄	C₂×D₅₃	semidirect
155. #117	C₂×D₅₃	semidirect
157. C₂×C₄²	C₂×D₅₃	semidirect
163. #41	D₂₁₂	semidirect