Subgroups — #20288488

Order 6784.

1748 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133263	C₂	1	2	yes	maximal, complemented
2	#16811294	C₂	1	4	yes	maximal, complemented
3	D₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
4	#16811257	C₂	1	8	yes	maximal, complemented
5	#16811191	C₂	1	4	yes	maximal, complemented
6	#16811187	C₂	1	2	yes	maximal, complemented
7	#16811174	C₂	1	2	yes	maximal, complemented
8	C₂×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
9	#127×D₅₃	C₂	1	2	yes	maximal, complemented
10	C₄×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	#13133263	—	2	16	no	—
13	C₂×#11719949	C₂²	1	4	yes	complemented
14	C₂×#11719945	C₂²	1	1	yes	characteristic
15	C₂×#11719945	C₂²	1	4	yes	complemented
16	C₂²×D₂₁₂	C₂²	1	2	yes	complemented
17	C₂²×D₂₁₂	C₂²	1	4	yes	complemented
18	C₂²×D₂₁₂	C₂²	1	4	yes	complemented
19	C₂×D₄×D₅₃	C₂²	1	4	yes	complemented
20	C₂×D₄×D₅₃	C₂²	1	8	yes	complemented
21	C₂×D₄×D₅₃	C₂²	1	8	yes	complemented
22	#13133214	C₂²	1	4	yes	complemented
23	#13133214	C₂²	1	2	yes	complemented
24	#13133214	C₂²	1	2	yes	complemented
25	#13133213	C₂²	1	8	yes	complemented
26	#13133213	C₂²	1	4	yes	complemented
27	#13133206	C₂²	1	4	yes	complemented
28	#13133192	C₂²	1	8	yes	complemented
29	#13133191	C₂²	1	8	yes	complemented
30	#13133190	C₂²	1	16	yes	complemented
31	#13133187	C₂²	1	16	yes	complemented
32	#13133185	C₂²	1	8	yes	complemented
33	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
34	C₄×D₂₁₂	C₂²	1	2	yes	complemented
35	#13133179	C₂²	1	8	yes	complemented
36	C₄×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
37	#13133176	C₂²	1	1	yes	characteristic, complemented
38	#13133174	C₂²	1	2	yes	complemented
39	C₄²×D₅₃	C₂²	1	2	yes	complemented
40	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
41	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	#124×C₅₃	C₂²	1	2	yes	complemented
44	#121×C₅₃	C₂²	1	4	yes	complemented
45	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
46	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂³×D₅₃	C₂³	1	4	yes	—
48	C₂³×D₅₃	C₂³	1	8	yes	—
49	#41×C₅₃	—	2	8	no	—
50	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
51	C₂×D₄×C₅₃	C₂³	1	4	yes	—
52	C₂×D₄×C₅₃	C₂³	1	4	yes	—
53	C₂×D₄×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₅₃	C₂³	1	16	yes	—
59	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
60	#11719949	—	2	16	no	—
61	D₄×D₅₃	—	2	32	no	—
62	D₄×D₅₃	—	2	16	no	—
63	D₄×D₅₃	—	4	16	no	—
64	#11719945	—	2	16	no	—
65	#11719945	D₄	1	8	yes	complemented
66	C₂×D₂₁₂	—	2	8	no	—
67	C₂×D₂₁₂	—	2	16	no	—
68	C₂×D₂₁₂	—	4	8	no	—
69	C₂×D₂₁₂	C₂³	1	8	yes	—
70	C₂×D₂₁₂	C₂³	1	8	yes	—
71	C₂×D₂₁₂	C₂³	1	4	yes	—
72	C₂×D₂₁₂	C₂³	1	4	yes	—
73	C₂×D₂₁₂	C₂³	1	8	yes	—
74	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₂³	1	4	yes	—
76	C₂×C₄×D₅₃	C₂³	1	8	yes	—
77	C₂×C₄×D₅₃	C₂³	1	2	yes	—
78	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
81	C₄²×C₅₃	C₂³	1	2	yes	—
82	C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
85	#11719921	C₂³	1	4	yes	—
86	#11719921	C₂³	1	16	yes	—
87	C₄⋊Dic₅₃	C₂³	1	2	yes	—
88	#11719919	C₂³	1	4	yes	—
89	C₄×Dic₅₃	C₂³	1	2	yes	—
90	C₄×Dic₅₃	C₂³	1	4	yes	—
91	C₂³×C₅₃	C₂⁴	1	4	yes	—
92	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂²×D₅₃	—	2	8	no	—
94	C₂²×D₅₃	—	2	16	no	—
95	C₂²×D₅₃	—	4	8	no	—
96	C₂²×D₅₃	—	4	16	no	—
97	C₂²×D₅₃	C₂⁴	1	8	yes	—
98	C₂²×D₅₃	C₂⁴	1	2	yes	—
99	Q₈×C₅₃	—	2	4	no	—
100	D₄×C₅₃	—	2	8	no	—
101	D₄×C₅₃	—	2	4	no	—
102	D₄×C₅₃	—	2	8	no	—
103	C₂×C₄×C₅₃	—	2	8	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	complemented
106	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
107	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
111	C₂²⋊D₅₃	—	2	16	no	—
112	C₂²⋊D₅₃	—	4	16	no	—
113	C₂²⋊D₅₃	C₂×D₄	1	8	yes	complemented
114	C₂×Dic₅₃	C₂⁴	1	4	yes	—
115	C₂×Dic₅₃	C₂⁴	1	2	yes	—
116	D₂₁₂	—	2	16	no	—
117	D₂₁₂	—	2	8	no	—
118	D₂₁₂	—	2	8	no	—
119	D₂₁₂	—	2	16	no	—
120	D₂₁₂	—	4	8	no	—
121	D₂₁₂	C₂×D₄	1	4	yes	complemented
122	C₄×D₅₃	—	2	8	no	—
123	C₄×D₅₃	—	2	16	no	—
124	C₄×D₅₃	C₂×D₄	1	8	yes	complemented
125	Dic₁₀₆	C₂×D₄	1	4	yes	complemented
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	8	no	—
129	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
130	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
131	C₂×D₅₃	—	2	4	no	—
132	C₂×D₅₃	—	2	8	no	—
133	C₂×D₅₃	—	2	16	no	—
134	C₂×D₅₃	—	8	16	no	—
135	C₂×D₅₃	—	8	8	no	—
136	C₂×D₅₃	C₂²×D₄	1	4	yes	—
137	C₄×C₅₃	—	2	4	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	C₂²×D₄	1	2	yes	—
140	C₄×C₅₃	C₂²×D₄	1	2	yes	—
141	C₄×C₅₃	H(5,𝔽₂)	1	4	yes	complemented
142	Dic₅₃	—	2	4	no	—
143	Dic₅₃	C₂²×D₄	1	4	yes	—
144	#3498	—	53	1	no	maximal, Sylow
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	—	4	4	no	—
147	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
148	C₂×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
149	D₅₃	—	2	4	no	—
150	D₅₃	—	4	8	no	—
151	C₂×H(5,𝔽₂)	—	53	2	no	—
152	#550	—	53	4	no	—
153	D₄²	—	53	8	no	—
154	D₄²	—	53	4	no	—
155	#535	—	53	2	no	—
156	#535	—	53	4	no	—
157	#532	—	53	2	no	—
158	#532	D₅₃	1	1	yes	characteristic, complemented
159	C₂×#127	—	53	2	no	—
160	C₂×#127	—	53	1	no	—
161	C₄×#41	—	53	1	no	—
162	C₅₃	#3498	1	1	yes	characteristic, complemented, Sylow
163	H(5,𝔽₂)	—	106	16	no	—
164	C₂×#41	—	53	4	no	—
165	C₂×#41	—	53	1	no	—
166	C₂×#41	—	53	4	no	—
167	C₂×#41	C₂×D₅₃	1	2	yes	complemented
168	C₂²×D₄	—	53	4	no	—
169	C₂²×D₄	—	53	8	no	—
170	C₂²×D₄	—	53	8	no	—
171	C₂²×D₄	—	53	2	no	—
172	C₂²×D₄	—	53	4	no	—
173	C₂²×D₄	—	53	4	no	—
174	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
175	#127	—	53	4	no	—
176	#127	—	53	8	no	—
177	#127	—	53	2	no	—
178	#127	—	53	2	no	—
179	#127	—	53	4	no	—
180	#127	C₂×D₅₃	1	1	yes	characteristic, complemented
181	#124	—	53	4	no	—
182	#124	C₂×D₅₃	1	2	yes	complemented
183	#121	—	53	8	no	—
184	#121	—	53	8	no	—
185	#121	—	53	16	no	—
186	#121	C₂×D₅₃	1	4	yes	complemented
187	#120	—	53	8	no	—
188	#120	—	53	16	no	—
189	C₄×Q₈	—	53	2	no	—
190	C₄×D₄	—	53	4	no	—
191	C₄×D₄	—	53	8	no	—
192	C₄×D₄	—	53	2	no	—
193	C₄×D₄	C₂×D₅₃	1	4	yes	complemented
194	#117	—	53	1	no	—
195	#117	—	53	2	no	—
196	C₂×C₄²	—	53	2	no	—
197	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
198	C₂⁴	—	53	8	no	—
199	C₂⁴	—	53	4	no	—
200	#41	—	2	8	no	—
201	#41	—	53	8	no	—
202	#41	—	106	16	no	—
203	#41	—	106	16	no	—
204	C₂×Q₈	—	53	1	no	—
205	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
206	C₂×D₄	—	53	2	no	—
207	C₂×D₄	—	53	8	no	—
208	C₂×D₄	—	53	16	no	—
209	C₂×D₄	—	53	1	no	—
210	C₂×D₄	—	53	8	no	—
211	C₂×D₄	—	53	8	no	—
212	C₂×D₄	—	53	4	no	—
213	C₂×D₄	—	53	4	no	—
214	C₂×D₄	—	53	8	no	—
215	C₂×D₄	—	106	16	no	—
216	C₂×D₄	—	106	16	no	—
217	C₂×D₄	—	106	32	no	—
218	C₂×D₄	—	106	8	no	—
219	C₂×D₄	—	212	8	no	—
220	C₂×D₄	—	212	16	no	—
221	C₂×D₄	C₂²×D₅₃	1	4	yes	—
222	C₂×D₄	C₂²×D₅₃	1	4	yes	—
223	C₂×D₄	C₂²×D₅₃	1	2	yes	—
224	C₂²×C₄	—	53	2	no	—
225	C₂²×C₄	—	53	4	no	—
226	C₂²×C₄	—	53	8	no	—
227	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
228	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
229	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
230	C₄⋊C₄	—	53	4	no	—
231	C₄⋊C₄	—	53	2	no	—
232	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
233	C₂²⋊C₄	—	53	2	no	—
234	C₂²⋊C₄	—	53	4	no	—
235	C₂²⋊C₄	—	53	16	no	—
236	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
237	C₄²	—	53	2	no	—
238	C₄²	—	53	4	no	—
239	C₄²	C₂²×D₅₃	1	1	yes	characteristic
240	C₄²	C₂²×D₅₃	1	1	yes	characteristic
241	C₄²	C₂²×D₅₃	1	2	yes	—
242	C₂³	—	53	2	no	—
243	C₂³	—	53	8	no	—
244	C₂³	—	106	16	no	—
245	C₂³	—	106	8	no	—
246	C₂³	—	212	8	no	—
247	C₂³	—	212	16	no	—
248	C₂³	C₂³×D₅₃	1	4	yes	—
249	C₂³	C₂³×D₅₃	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	53	4	no	—
252	D₄	—	2	8	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	8	no	—
255	D₄	—	53	8	no	—
256	D₄	—	53	4	no	—
257	D₄	—	106	16	no	—
258	D₄	—	106	16	no	—
259	D₄	—	106	8	no	—
260	D₄	—	106	8	no	—
261	D₄	—	106	16	no	—
262	D₄	—	212	8	no	—
263	D₄	—	212	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	53	2	no	—
267	C₂×C₄	—	53	8	no	—
268	C₂×C₄	—	53	4	no	—
269	C₂×C₄	—	106	16	no	—
270	C₂×C₄	—	106	8	no	—
271	C₂×C₄	D₄×D₅₃	1	4	yes	complemented
272	C₂×C₄	C₂³×D₅₃	1	2	yes	—
273	C₂×C₄	C₂³×D₅₃	1	4	yes	—
274	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₅₃	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	53	4	no	—
281	C₂²	—	106	4	no	—
282	C₂²	—	106	8	no	—
283	C₂²	—	106	16	no	—
284	C₂²	—	424	8	no	—
285	C₂²	—	424	16	no	—
286	C₂²	C₂×D₄×D₅₃	1	2	yes	—
287	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	53	4	no	—
291	C₄	—	106	4	no	—
292	C₄	C₂×D₄×D₅₃	1	2	yes	—
293	C₄	C₂×D₄×D₅₃	1	2	yes	—
294	C₄	#13133263	1	4	yes	complemented
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	106	4	no	—
298	C₂	—	212	8	no	—
299	C₂	C₂²×D₄×D₅₃	1	1	yes	characteristic
300	C₂	C₂×#13133263	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#532	C₂
hypercenter	#532	D₅₃
hypocenter	C₅₃	#3498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#13133263	C₂	semidirect
2. #16811294	C₂	semidirect
3. D₄×D₂₁₂	C₂	semidirect
4. #16811257	C₂	semidirect
5. #16811191	C₂	semidirect
6. #16811187	C₂	semidirect
7. #16811174	C₂	semidirect
8. C₂×#13133214	C₂	semidirect
9. #127×D₅₃	C₂	semidirect
10. C₄×#11719945	C₂	semidirect
11. C₅₃×#532	C₂	semidirect
13. C₂×#11719949	C₂²	semidirect
15. C₂×#11719945	C₂²	semidirect
16. C₂²×D₂₁₂	C₂²	semidirect
17. C₂²×D₂₁₂	C₂²	semidirect
18. C₂²×D₂₁₂	C₂²	semidirect
19. C₂×D₄×D₅₃	C₂²	semidirect
20. C₂×D₄×D₅₃	C₂²	semidirect
21. C₂×D₄×D₅₃	C₂²	semidirect
22. #13133214	C₂²	semidirect
23. #13133214	C₂²	semidirect
24. #13133214	C₂²	semidirect
25. #13133213	C₂²	semidirect
26. #13133213	C₂²	semidirect
27. #13133206	C₂²	semidirect
28. #13133192	C₂²	semidirect
29. #13133191	C₂²	semidirect
30. #13133190	C₂²	semidirect
31. #13133187	C₂²	semidirect
32. #13133185	C₂²	semidirect
33. C₄×Dic₁₀₆	C₂²	semidirect
34. C₄×D₂₁₂	C₂²	semidirect
35. #13133179	C₂²	semidirect
36. C₄×C₂²⋊D₅₃	C₂²	semidirect
37. #13133176	C₂²	semidirect
38. #13133174	C₂²	semidirect
39. C₄²×D₅₃	C₂²	semidirect
40. C₂×#41×C₅₃	C₂²	semidirect
41. C₄⋊Q₈×C₅₃	C₂²	semidirect
42. #127×C₅₃	C₂²	semidirect
43. #124×C₅₃	C₂²	semidirect
44. #121×C₅₃	C₂²	semidirect
45. C₄×D₄×C₅₃	C₂²	semidirect
46. C₂×C₄²×C₅₃	C₂²	semidirect
65. #11719945	D₄	semidirect
105. C₂×C₄×C₅₃	C₂×D₄	semidirect
113. C₂²⋊D₅₃	C₂×D₄	semidirect
121. D₂₁₂	C₂×D₄	semidirect
124. C₄×D₅₃	C₂×D₄	semidirect
125. Dic₁₀₆	C₂×D₄	semidirect
141. C₄×C₅₃	H(5,𝔽₂)	semidirect
158. #532	D₅₃	semidirect
162. C₅₃	#3498	semidirect
167. C₂×#41	C₂×D₅₃	semidirect
174. C₄⋊Q₈	C₂×D₅₃	semidirect
180. #127	C₂×D₅₃	semidirect
182. #124	C₂×D₅₃	semidirect
186. #121	C₂×D₅₃	semidirect
193. C₄×D₄	C₂×D₅₃	semidirect
197. C₂×C₄²	C₂×D₅₃	semidirect
271. C₂×C₄	D₄×D₅₃	semidirect
294. C₄	#13133263	semidirect