Subgroups — #20283545

Order 6784.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133245	C₂	1	1	yes	maximal, characteristic
2	#16811104	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811097	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810734	C₂	1	2	yes	maximal, complemented
5	#16810722	C₂	1	2	yes	maximal, complemented
6	#16810710	C₂	1	2	yes	maximal, complemented
7	#16810705	C₂	1	2	yes	maximal, complemented
8	#16810617	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133137	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810581	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
14	#13133245	C₄	1	8	yes	complemented
15	C₂×#11719913	C₂²	1	2	yes	—
16	C₂×D₄₂₄	C₂²	1	2	yes	—
17	C₂²×#88322	C₂²	1	1	yes	characteristic
18	#13133182	C₂²	1	2	yes	complemented
19	C₄×D₂₁₂	—	2	1	no	—
20	C₄×D₂₁₂	—	2	1	no	—
21	#13133179	C₂²	1	2	yes	—
22	#13133179	C₂²	1	2	yes	—
23	#13133179	C₂²	1	2	yes	complemented
24	C₄×C₂²⋊D₅₃	—	2	2	no	—
25	#13133176	C₂²	1	1	yes	characteristic
26	#13133175	—	2	1	no	—
27	C₄⋊C₄×D₅₃	—	2	1	no	—
28	C₂²⋊C₄×D₅₃	—	2	2	no	—
29	#13133140	C₂²	1	1	yes	characteristic, complemented
30	#13133140	C₂²	1	1	yes	characteristic, complemented
31	#13133137	C₂²	1	2	yes	—
32	#13133137	C₂²	1	2	yes	—
33	#13133137	C₂²	1	1	yes	characteristic, complemented
34	#13133137	C₂²	1	1	yes	characteristic, complemented
35	#13133127	C₂²	1	2	yes	complemented
36	C₈×Dic₅₃	C₂²	1	2	yes	complemented
37	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	#117×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	#107×C₅₃	C₂²	1	2	yes	complemented
41	#106×C₅₃	C₂²	1	2	yes	complemented
42	C₂³×D₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₅₃	—	2	1	no	—
46	#11719945	—	2	2	no	—
47	#11719945	C₂×C₄	1	4	yes	complemented
48	C₂×D₂₁₂	—	2	1	no	—
49	C₂×D₂₁₂	—	2	2	no	—
50	C₂×D₂₁₂	C₂×C₄	1	4	yes	complemented
51	C₂×D₂₁₂	C₂³	1	2	yes	—
52	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×C₄×D₅₃	—	2	2	no	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
58	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
59	C₂×C₈×C₅₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₂²⋊Dic₅₃	D₄	1	2	yes	—
67	#11719921	—	2	2	no	—
68	#11719921	—	2	2	no	—
69	#11719921	—	2	2	no	—
70	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
71	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
72	#11719919	—	2	2	no	—
73	C₄×Dic₅₃	—	2	2	no	—
74	C₄×Dic₅₃	C₂³	1	2	yes	—
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	C₂³	1	1	yes	characteristic
77	C₂×#88322	C₂³	1	1	yes	characteristic
78	D₄₂₄	C₂×C₄	1	8	yes	complemented
79	#11719913	C₂×C₄	1	8	yes	complemented
80	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₅₃	—	2	1	no	—
82	C₂²×D₅₃	—	2	2	no	—
83	C₂²×D₅₃	—	2	4	no	—
84	C₂²×D₅₃	—	8	1	no	—
85	C₂²×D₅₃	—	8	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	8	1	no	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#41	1	2	yes	—
93	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₅₃	—	2	4	no	—
95	C₂×Dic₅₃	—	2	1	no	—
96	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
97	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
98	D₂₁₂	—	2	4	no	—
99	D₂₁₂	—	2	2	no	—
100	D₂₁₂	—	2	1	no	—
101	D₂₁₂	C₂²×C₄	1	4	yes	—
102	D₂₁₂	C₂²×C₄	1	2	yes	—
103	C₄×D₅₃	—	2	4	no	—
104	C₄×D₅₃	—	8	2	no	—
105	C₄×D₅₃	—	8	2	no	—
106	C₄×D₅₃	—	8	2	no	—
107	Dic₁₀₆	—	2	1	no	—
108	Dic₁₀₆	C₂²×C₄	1	2	yes	—
109	C₈×C₅₃	C₂²×C₄	1	4	yes	—
110	#88322	—	2	1	no	—
111	#88322	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	C₄×D₄	1	2	yes	—
115	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₅₃	—	2	2	no	—
117	C₂×D₅₃	—	2	4	no	—
118	C₂×D₅₃	—	4	1	no	—
119	C₂×D₅₃	—	4	2	no	—
120	C₂×D₅₃	—	4	2	no	—
121	C₂×D₅₃	—	4	1	no	—
122	C₂×D₅₃	—	8	4	no	—
123	C₄×C₅₃	—	4	1	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	C₄×D₄	1	2	yes	—
127	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	Dic₅₃	—	2	2	no	—
130	Dic₅₃	—	2	2	no	—
131	Dic₅₃	—	2	2	no	—
132	#2946	—	53	1	no	maximal, Sylow
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₅₃	#576	1	1	yes	characteristic
136	C₂×C₅₃	#577	1	1	yes	characteristic
137	D₅₃	—	4	2	no	—
138	D₅₃	—	4	4	no	—
139	C₂×AGL(1,ℤ/8)	—	53	1	no	—
140	#520	—	53	1	no	—
141	#518	—	53	1	no	—
142	#442	—	53	2	no	—
143	#440	—	53	2	no	—
144	C₄×SD₁₆	—	53	2	no	—
145	C₄×D₈	—	53	2	no	—
146	#428	D₅₃	1	1	yes	characteristic, complemented
147	#416	—	53	1	no	—
148	C₂×#102	—	53	1	no	—
149	#405	—	53	1	no	—
150	C₅₃	#2946	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	53	1	no	—
152	C₂²×D₄	—	53	1	no	—
153	AGL(1,ℤ/8)	—	53	8	no	—
154	C₂×SD₁₆	—	53	2	no	—
155	C₂×D₈	—	53	2	no	—
156	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
157	C₂²×C₈	—	53	1	no	—
158	C₄×Q₈	—	53	2	no	—
159	C₄×D₄	—	53	2	no	—
160	C₄×D₄	—	53	2	no	—
161	C₄×D₄	—	53	2	no	—
162	C₄×D₄	—	106	1	no	—
163	C₄×D₄	—	106	1	no	—
164	C₄×D₄	—	106	2	no	—
165	#117	—	53	1	no	—
166	#117	—	106	1	no	—
167	#117	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	106	1	no	—
169	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	106	2	no	—
171	#107	C₂×D₅₃	1	2	yes	complemented
172	#106	C₂×D₅₃	1	2	yes	complemented
173	#103	—	53	1	no	—
174	#103	—	53	1	no	—
175	#102	—	53	2	no	—
176	#102	—	53	2	no	—
177	#102	—	53	1	no	—
178	#102	—	53	1	no	—
179	#97	—	53	2	no	—
180	C₄×C₈	—	53	2	no	—
181	C₂⁴	—	106	1	no	—
182	#41	—	53	4	no	—
183	#41	—	106	2	no	—
184	C₂×Q₈	—	53	1	no	—
185	C₂×D₄	—	53	4	no	—
186	C₂×D₄	—	53	1	no	—
187	C₂×D₄	—	53	2	no	—
188	C₂×D₄	—	106	1	no	—
189	C₂×D₄	—	106	1	no	—
190	C₂×D₄	—	106	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	106	2	no	—
193	C₂²×C₄	—	106	2	no	—
194	C₂²×C₄	—	106	2	no	—
195	C₂²×C₄	—	106	1	no	—
196	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
197	SD₁₆	—	53	8	no	—
198	D₈	—	53	8	no	—
199	M₁₆	C₄×D₅₃	1	4	yes	complemented
200	C₂×C₈	—	53	1	no	—
201	C₂×C₈	—	53	1	no	—
202	C₂×C₈	—	106	2	no	—
203	C₂×C₈	C₂²×D₅₃	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	53	1	no	—
206	C₄⋊C₄	—	53	1	no	—
207	C₄⋊C₄	—	106	2	no	—
208	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
209	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	53	2	no	—
213	C₂²⋊C₄	—	106	2	no	—
214	C₂²⋊C₄	—	106	2	no	—
215	C₂²⋊C₄	—	106	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	53	2	no	—
218	C₄²	—	106	2	no	—
219	C₂³	—	106	4	no	—
220	C₂³	—	106	1	no	—
221	C₂³	—	106	2	no	—
222	C₂³	—	424	1	no	—
223	C₂³	—	424	1	no	—
224	C₂³	D₄×D₅₃	1	1	yes	characteristic
225	Q₈	—	53	2	no	—
226	Q₈	—	106	1	no	—
227	D₄	—	53	2	no	—
228	D₄	—	53	4	no	—
229	D₄	—	106	4	no	—
230	D₄	—	106	4	no	—
231	D₄	—	106	2	no	—
232	D₄	—	106	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	53	2	no	—
238	C₂×C₄	—	53	2	no	—
239	C₂×C₄	—	106	1	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	—	424	2	no	—
242	C₂×C₄	—	424	2	no	—
243	C₂×C₄	—	424	2	no	—
244	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
245	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
246	C₂×C₄	#11719949	1	2	yes	—
247	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
248	C₈	—	106	1	no	—
249	C₈	—	106	1	no	—
250	C₈	C₂×C₄×D₅₃	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	106	2	no	—
254	C₂²	—	106	4	no	—
255	C₂²	—	424	4	no	—
256	C₂²	—	212	1	no	—
257	C₂²	—	212	2	no	—
258	C₂²	—	212	2	no	—
259	C₂²	—	212	1	no	—
260	C₂²	#13133179	1	2	yes	—
261	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	106	2	no	—
266	C₄	—	106	2	no	—
267	C₄	—	106	2	no	—
268	C₄	#13133179	1	2	yes	—
269	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
270	C₄	C₂×#11719949	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	212	4	no	—
273	C₂	—	212	2	no	—
274	C₂	C₂×#13133179	1	1	yes	characteristic
275	C₂	#16811427	1	1	yes	characteristic
276	C₂	#16811432	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#428	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#428	D₅₃
hypocenter	C₅₃	#2946
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
2. #16811104	C₂	semidirect
3. #16811097	C₂	semidirect
4. #16810734	C₂	semidirect
5. #16810722	C₂	semidirect
6. #16810710	C₂	semidirect
7. #16810705	C₂	semidirect
8. #16810617	C₂	semidirect
9. C₂×#13133137	C₂	semidirect
10. #16810581	C₂	semidirect
11. C₅₃×#428	C₂	semidirect
14. #13133245	C₄	semidirect
18. #13133182	C₂²	semidirect
23. #13133179	C₂²	semidirect
29. #13133140	C₂²	semidirect
30. #13133140	C₂²	semidirect
33. #13133137	C₂²	semidirect
34. #13133137	C₂²	semidirect
35. #13133127	C₂²	semidirect
36. C₈×Dic₅₃	C₂²	semidirect
40. #107×C₅₃	C₂²	semidirect
41. #106×C₅₃	C₂²	semidirect
47. #11719945	C₂×C₄	semidirect
50. C₂×D₂₁₂	C₂×C₄	semidirect
58. M₁₆×C₅₃	C₂×C₄	semidirect
78. D₄₂₄	C₂×C₄	semidirect
79. #11719913	C₂×C₄	semidirect
146. #428	D₅₃	semidirect
150. C₅₃	#2946	semidirect
171. #107	C₂×D₅₃	semidirect
172. #106	C₂×D₅₃	semidirect
199. M₁₆	C₄×D₅₃	semidirect