Subgroups — #35227742

Order 20864.

1660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104781	C₂	1	1	yes	maximal, characteristic
2	#23782147	C₂	1	2	yes	maximal, complemented
3	#23782146	C₂	1	3	yes	maximal, complemented
4	#23782139	C₂	1	3	yes	maximal, complemented
5	#23782137	C₂	1	6	yes	maximal, complemented
6	#41×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#14579700	C₂	1	2	yes	maximal, complemented
8	#23782133	C₂	1	3	yes	maximal, complemented
9	C₂×#19104697	C₂	1	3	yes	maximal, complemented
10	C₄×D₄×D₁₆₃	C₂	1	6	yes	maximal, complemented
11	#520×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19104781	C₄	1	16	yes	complemented
13	C₂×#14579700	C₂²	1	2	yes	—
14	C₂×#14579699	C₂²	1	3	yes	—
15	C₂²×D₆₅₂	C₂²	1	3	yes	—
16	C₂×D₄×D₁₆₃	C₂²	1	6	yes	—
17	C₂²×C₄×D₁₆₃	C₂²	1	6	yes	complemented
18	Q₈×Dic₁₆₃	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₁₆₃	C₂²	1	1	yes	characteristic, complemented
20	#19104700	C₂²	1	3	yes	complemented
21	#19104700	C₂²	1	1	yes	characteristic, complemented
22	C₄×D₆₅₂	C₂²	1	6	yes	complemented
23	C₄×D₆₅₂	C₂²	1	6	yes	complemented
24	D₄×Dic₁₆₃	C₂²	1	3	yes	complemented
25	D₄×Dic₁₆₃	C₂²	1	3	yes	complemented
26	#19104697	C₂²	1	6	yes	complemented
27	#19104697	C₂²	1	6	yes	complemented
28	#19104697	C₂²	1	6	yes	complemented
29	#19104697	C₂²	1	3	yes	complemented
30	#19104697	C₂²	1	3	yes	complemented
31	C₄×C₂²⋊D₁₆₃	C₂²	1	12	yes	complemented
32	(C₂²⋊D₁₆₃)⋊C₄	C₂²	1	6	yes	complemented
33	(C₂²⋊D₁₆₃)⋊C₄	C₂²	1	6	yes	complemented
34	#19104694	C₂²	1	3	yes	complemented
35	#19104693	C₂²	1	6	yes	complemented
36	#19104692	C₂²	1	6	yes	complemented
37	C₄⋊C₄×D₁₆₃	C₂²	1	6	yes	complemented
38	C₂×(C₂×D₁₆₃)⋊C₄	C₂²	1	6	yes	complemented
39	C₂²⋊C₄×D₁₆₃	C₂²	1	12	yes	complemented
40	C₂×C₄×Dic₁₆₃	C₂²	1	3	yes	complemented
41	C₄²×D₁₆₃	C₂²	1	6	yes	complemented
42	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₁₆₃	C₂²	1	2	yes	complemented
44	C₄×D₄×C₁₆₃	C₂²	1	6	yes	complemented
45	#117×C₁₆₃	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	3	yes	complemented
47	C₂³×D₁₆₃	C₂³	1	6	yes	—
48	#14579700	C₂×C₄	1	16	yes	complemented
49	#14579699	C₂×C₄	1	24	yes	complemented
50	C₂×Dic₃₂₆	C₂³	1	1	yes	characteristic
51	C₂×D₆₅₂	C₂×C₄	1	24	yes	complemented
52	C₂×D₆₅₂	C₂³	1	6	yes	—
53	C₂×D₆₅₂	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₁₆₃	C₂³	1	6	yes	—
55	D₄×D₁₆₃	C₂×C₄	1	48	yes	complemented
56	C₂²×Dic₁₆₃	C₂³	1	3	yes	—
57	C₂²×Dic₁₆₃	C₂³	1	3	yes	—
58	C₂×C₄×D₁₆₃	—	4	12	no	—
59	C₂×C₄×D₁₆₃	C₂³	1	6	yes	—
60	C₂×C₄×D₁₆₃	C₂³	1	12	yes	—
61	C₂×C₄×D₁₆₃	C₂³	1	6	yes	—
62	C₂×C₄×D₁₆₃	C₂³	1	6	yes	—
63	C₄⋊Dic₁₆₃	C₂³	1	3	yes	—
64	C₄⋊Dic₁₆₃	C₂³	1	3	yes	—
65	Dic₁₆₃⋊C₄	C₂³	1	6	yes	—
66	Dic₁₆₃⋊C₄	C₂³	1	2	yes	—
67	C₂²⋊Dic₁₆₃	C₂³	1	6	yes	—
68	(C₂×D₁₆₃)⋊C₄	C₂³	1	6	yes	—
69	(C₂×D₁₆₃)⋊C₄	C₂³	1	12	yes	—
70	(C₂×D₁₆₃)⋊C₄	C₂³	1	6	yes	—
71	C₄×Dic₁₆₃	C₂³	1	2	yes	—
72	C₄×Dic₁₆₃	C₂³	1	3	yes	—
73	C₄×Dic₁₆₃	C₂³	1	6	yes	—
74	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
75	C₄×Dic₁₆₃	C₂³	1	3	yes	—
76	C₄×Dic₁₆₃	C₂³	1	3	yes	—
77	#41×C₁₆₃	C₂×C₄	1	8	yes	complemented
78	C₂×Q₈×C₁₆₃	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₁₆₃	C₂³	1	3	yes	—
80	C₂²×C₄×C₁₆₃	C₂³	1	6	yes	—
81	C₂²×C₄×C₁₆₃	C₂³	1	3	yes	—
82	C₄⋊C₄×C₁₆₃	C₂³	1	3	yes	—
83	C₄⋊C₄×C₁₆₃	C₂³	1	6	yes	—
84	C₄⋊C₄×C₁₆₃	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₆₃	C₂³	1	6	yes	—
86	C₄²×C₁₆₃	C₂³	1	6	yes	—
87	C₂²×D₁₆₃	—	4	6	no	—
88	C₂²×D₁₆₃	—	4	6	no	—
89	C₂²×D₁₆₃	C₂²×C₄	1	24	yes	complemented
90	C₂²×D₁₆₃	C₂⁴	1	6	yes	—
91	Dic₃₂₆	C₂²×C₄	1	4	yes	complemented
92	D₆₅₂	C₂²×C₄	1	24	yes	complemented
93	D₆₅₂	C₂²×C₄	1	12	yes	complemented
94	C₂²⋊D₁₆₃	C₂²×C₄	1	24	yes	complemented
95	C₂×Dic₁₆₃	—	2	6	no	—
96	C₂×Dic₁₆₃	—	4	3	no	—
97	C₂×Dic₁₆₃	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₁₆₃	C₂⁴	1	3	yes	—
99	C₂×Dic₁₆₃	C₂⁴	1	3	yes	—
100	C₂×Dic₁₆₃	C₂⁴	1	2	yes	—
101	C₂×Dic₁₆₃	C₂⁴	1	1	yes	characteristic
102	C₄×D₁₆₃	—	2	12	no	—
103	C₄×D₁₆₃	—	4	12	no	—
104	C₄×D₁₆₃	—	4	6	no	—
105	C₄×D₁₆₃	C₂²×C₄	1	24	yes	complemented
106	C₂³×C₁₆₃	C₂⁴	1	3	yes	—
107	Q₈×C₁₆₃	C₂²×C₄	1	4	yes	complemented
108	D₄×C₁₆₃	C₂²×C₄	1	12	yes	complemented
109	C₂×C₄×C₁₆₃	—	4	6	no	—
110	C₂×C₄×C₁₆₃	C₂²×C₄	1	12	yes	complemented
111	C₂×C₄×C₁₆₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₆₃	C₂⁴	1	6	yes	—
113	C₂×C₄×C₁₆₃	C₂⁴	1	3	yes	—
114	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
115	C₂²×C₁₆₃	—	2	3	no	—
116	C₂²×C₁₆₃	—	2	3	no	—
117	C₂²×C₁₆₃	C₂³×C₄	1	6	yes	—
118	C₂²×C₁₆₃	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₆₃	—	2	6	no	—
120	C₂×D₁₆₃	—	2	6	no	—
121	C₂×D₁₆₃	—	4	24	no	—
122	C₂×D₁₆₃	C₂³×C₄	1	12	yes	—
123	C₄×C₁₆₃	—	2	6	no	—
124	C₄×C₁₆₃	—	2	2	no	—
125	C₄×C₁₆₃	C₂³×C₄	1	2	yes	—
126	C₄×C₁₆₃	C₂³×C₄	1	6	yes	—
127	Dic₁₆₃	—	2	3	no	—
128	Dic₁₆₃	—	2	3	no	—
129	Dic₁₆₃	—	2	1	no	—
130	Dic₁₆₃	C₂³×C₄	1	4	yes	—
131	Dic₁₆₃	H(5,𝔽₂)	1	2	yes	complemented
132	C₂×C₁₆₃	—	2	6	no	—
133	C₂×C₁₆₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₂×C₁₆₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₁₆₃	#585	1	1	yes	characteristic
136	D₁₆₃	—	2	12	no	—
137	C₁₆₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
138	C₄×H(5,𝔽₂)	—	163	1	no	maximal, Sylow
139	C₂×H(5,𝔽₂)	—	163	1	no	—
140	#520	—	163	2	no	—
141	#520	—	163	3	no	—
142	#520	D₁₆₃	1	1	yes	characteristic, complemented
143	#518	—	163	6	no	—
144	#518	—	163	3	no	—
145	C₄×#41	—	163	1	no	—
146	C₄×#41	—	163	2	no	—
147	C₄×#41	—	163	3	no	—
148	C₂×C₄×D₄	—	163	6	no	—
149	C₂×C₄×D₄	—	163	3	no	—
150	H(5,𝔽₂)	—	163	16	no	—
151	C₂×#41	—	163	2	no	—
152	C₂×#41	—	163	3	no	—
153	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
154	C₂²×D₄	—	163	6	no	—
155	C₂²×D₄	—	163	3	no	—
156	C₂³×C₄	—	163	6	no	—
157	C₄×Q₈	—	163	1	no	—
158	C₄×Q₈	—	163	1	no	—
159	C₄×Q₈	—	163	1	no	—
160	C₄×Q₈	—	163	3	no	—
161	C₄×Q₈	C₂×D₁₆₃	1	2	yes	complemented
162	C₄×D₄	—	163	3	no	—
163	C₄×D₄	—	163	3	no	—
164	C₄×D₄	—	163	12	no	—
165	C₄×D₄	—	163	6	no	—
166	C₄×D₄	—	163	3	no	—
167	C₄×D₄	—	163	6	no	—
168	C₄×D₄	—	163	6	no	—
169	C₄×D₄	—	163	3	no	—
170	C₄×D₄	—	163	6	no	—
171	C₄×D₄	—	163	6	no	—
172	C₄×D₄	—	163	6	no	—
173	C₄×D₄	—	163	6	no	—
174	C₄×D₄	C₂×D₁₆₃	1	6	yes	complemented
175	#117	—	163	3	no	—
176	#117	—	163	6	no	—
177	#117	—	163	6	no	—
178	#117	C₂×D₁₆₃	1	3	yes	complemented
179	C₂×C₄⋊C₄	—	163	6	no	—
180	C₂×C₄⋊C₄	C₂×D₁₆₃	1	3	yes	complemented
181	C₂×C₂²⋊C₄	—	163	6	no	—
182	C₂×C₂²⋊C₄	—	163	12	no	—
183	C₂×C₄²	—	163	3	no	—
184	C₂×C₄²	—	163	6	no	—
185	C₂⁴	—	163	6	no	—
186	#41	—	163	16	no	—
187	#41	—	163	24	no	—
188	#41	C₄×D₁₆₃	1	8	yes	complemented
189	C₂×Q₈	—	163	1	no	—
190	C₂×Q₈	C₂²×D₁₆₃	1	1	yes	characteristic
191	C₂×D₄	—	163	6	no	—
192	C₂×D₄	—	163	48	no	—
193	C₂×D₄	—	163	24	no	—
194	C₂×D₄	—	163	3	no	—
195	C₂×D₄	—	163	6	no	—
196	C₂×D₄	C₂²×D₁₆₃	1	3	yes	—
197	C₂²×C₄	—	652	12	no	—
198	C₂²×C₄	—	163	3	no	—
199	C₂²×C₄	—	163	6	no	—
200	C₂²×C₄	—	163	3	no	—
201	C₂²×C₄	—	163	6	no	—
202	C₂²×C₄	—	163	12	no	—
203	C₂²×C₄	—	163	6	no	—
204	C₂²×C₄	C₂²×D₁₆₃	1	6	yes	—
205	C₂²×C₄	C₂²×D₁₆₃	1	3	yes	—
206	C₄⋊C₄	—	163	6	no	—
207	C₄⋊C₄	—	163	3	no	—
208	C₄⋊C₄	—	163	2	no	—
209	C₄⋊C₄	—	163	3	no	—
210	C₄⋊C₄	C₂²×D₁₆₃	1	6	yes	—
211	C₄⋊C₄	C₂²×D₁₆₃	1	3	yes	—
212	C₄⋊C₄	C₂²×D₁₆₃	1	1	yes	characteristic
213	C₂²⋊C₄	—	163	6	no	—
214	C₂²⋊C₄	—	163	6	no	—
215	C₂²⋊C₄	—	163	12	no	—
216	C₂²⋊C₄	—	163	6	no	—
217	C₂²⋊C₄	C₂²×D₁₆₃	1	6	yes	—
218	C₄²	—	163	1	no	—
219	C₄²	—	163	3	no	—
220	C₄²	—	163	2	no	—
221	C₄²	—	163	3	no	—
222	C₄²	—	163	3	no	—
223	C₄²	—	163	6	no	—
224	C₄²	C₂²×D₁₆₃	1	6	yes	—
225	C₂³	—	652	6	no	—
226	C₂³	—	652	6	no	—
227	C₂³	—	163	6	no	—
228	C₂³	—	163	24	no	—
229	C₂³	C₂³×D₁₆₃	1	3	yes	—
230	Q₈	—	163	4	no	—
231	Q₈	C₂×C₄×D₁₆₃	1	4	yes	complemented
232	D₄	—	163	24	no	—
233	D₄	—	163	24	no	—
234	D₄	—	163	12	no	—
235	D₄	C₂×C₄×D₁₆₃	1	12	yes	complemented
236	C₂×C₄	—	4	6	no	—
237	C₂×C₄	—	326	12	no	—
238	C₂×C₄	—	326	6	no	—
239	C₂×C₄	—	652	12	no	—
240	C₂×C₄	—	652	6	no	—
241	C₂×C₄	—	652	3	no	—
242	C₂×C₄	—	163	3	no	—
243	C₂×C₄	—	163	1	no	—
244	C₂×C₄	—	163	3	no	—
245	C₂×C₄	—	163	1	no	—
246	C₂×C₄	—	163	2	no	—
247	C₂×C₄	—	163	24	no	—
248	C₂×C₄	C₂×C₄×D₁₆₃	1	12	yes	complemented
249	C₂×C₄	C₂³×D₁₆₃	1	2	yes	—
250	C₂×C₄	C₂³×D₁₆₃	1	6	yes	—
251	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₆₃	1	3	yes	—
253	C₂²	—	2	3	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	326	6	no	—
256	C₂²	—	326	6	no	—
257	C₂²	—	652	24	no	—
258	C₂²	—	163	12	no	—
259	C₂²	C₂²×C₄×D₁₆₃	1	6	yes	—
260	C₂²	C₂⁴×D₁₆₃	1	1	yes	characteristic
261	C₄	—	2	6	no	—
262	C₄	—	2	2	no	—
263	C₄	—	326	3	no	—
264	C₄	—	326	3	no	—
265	C₄	—	326	1	no	—
266	C₄	—	163	4	no	—
267	C₄	—	163	2	no	—
268	C₄	C₂²×C₄×D₁₆₃	1	6	yes	—
269	C₄	C₂²×C₄×D₁₆₃	1	2	yes	—
270	C₂	—	2	6	no	—
271	C₂	—	326	12	no	—
272	C₂	C₂³×C₄×D₁₆₃	1	1	yes	characteristic
273	C₂	H(5,𝔽₂)×D₁₆₃	1	1	yes	characteristic
274	C₂	#23782502	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₆₃
commutator	C₂×C₁₆₃	C₂⁴×C₄
Fitting	#520×C₁₆₃	C₂
socle, cosocle	C₂²×C₁₆₃	C₂⁵
prime radical	C₂×#19104781	C₂
hypercenter	#520	D₁₆₃
hypocenter	C₁₆₃	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #23782147	C₂	semidirect
3. #23782146	C₂	semidirect
4. #23782139	C₂	semidirect
5. #23782137	C₂	semidirect
6. #41×Dic₁₆₃	C₂	semidirect
7. C₄×#14579700	C₂	semidirect
8. #23782133	C₂	semidirect
9. C₂×#19104697	C₂	semidirect
10. C₄×D₄×D₁₆₃	C₂	semidirect
11. #520×C₁₆₃	C₂	semidirect
12. #19104781	C₄	semidirect
17. C₂²×C₄×D₁₆₃	C₂²	semidirect
18. Q₈×Dic₁₆₃	C₂²	semidirect
19. Q₈×Dic₁₆₃	C₂²	semidirect
20. #19104700	C₂²	semidirect
21. #19104700	C₂²	semidirect
22. C₄×D₆₅₂	C₂²	semidirect
23. C₄×D₆₅₂	C₂²	semidirect
24. D₄×Dic₁₆₃	C₂²	semidirect
25. D₄×Dic₁₆₃	C₂²	semidirect
26. #19104697	C₂²	semidirect
27. #19104697	C₂²	semidirect
28. #19104697	C₂²	semidirect
29. #19104697	C₂²	semidirect
30. #19104697	C₂²	semidirect
31. C₄×C₂²⋊D₁₆₃	C₂²	semidirect
32. (C₂²⋊D₁₆₃)⋊C₄	C₂²	semidirect
33. (C₂²⋊D₁₆₃)⋊C₄	C₂²	semidirect
34. #19104694	C₂²	semidirect
35. #19104693	C₂²	semidirect
36. #19104692	C₂²	semidirect
37. C₄⋊C₄×D₁₆₃	C₂²	semidirect
38. C₂×(C₂×D₁₆₃)⋊C₄	C₂²	semidirect
39. C₂²⋊C₄×D₁₆₃	C₂²	semidirect
40. C₂×C₄×Dic₁₆₃	C₂²	semidirect
41. C₄²×D₁₆₃	C₂²	semidirect
43. C₄×Q₈×C₁₆₃	C₂²	semidirect
44. C₄×D₄×C₁₆₃	C₂²	semidirect
45. #117×C₁₆₃	C₂²	semidirect
46. C₂×C₄⋊C₄×C₁₆₃	C₂²	semidirect
48. #14579700	C₂×C₄	semidirect
49. #14579699	C₂×C₄	semidirect
51. C₂×D₆₅₂	C₂×C₄	semidirect
55. D₄×D₁₆₃	C₂×C₄	semidirect
77. #41×C₁₆₃	C₂×C₄	semidirect
89. C₂²×D₁₆₃	C₂²×C₄	semidirect
91. Dic₃₂₆	C₂²×C₄	semidirect
92. D₆₅₂	C₂²×C₄	semidirect
93. D₆₅₂	C₂²×C₄	semidirect
94. C₂²⋊D₁₆₃	C₂²×C₄	semidirect
105. C₄×D₁₆₃	C₂²×C₄	semidirect
107. Q₈×C₁₆₃	C₂²×C₄	semidirect
108. D₄×C₁₆₃	C₂²×C₄	semidirect
110. C₂×C₄×C₁₆₃	C₂²×C₄	semidirect
131. Dic₁₆₃	H(5,𝔽₂)	semidirect
137. C₁₆₃	C₄×H(5,𝔽₂)	semidirect
142. #520	D₁₆₃	semidirect
161. C₄×Q₈	C₂×D₁₆₃	semidirect
174. C₄×D₄	C₂×D₁₆₃	semidirect
178. #117	C₂×D₁₆₃	semidirect
180. C₂×C₄⋊C₄	C₂×D₁₆₃	semidirect
188. #41	C₄×D₁₆₃	semidirect
231. Q₈	C₂×C₄×D₁₆₃	semidirect
235. D₄	C₂×C₄×D₁₆₃	semidirect
248. C₂×C₄	C₂×C₄×D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104781	2
#23782147	2
#23782146	2
#23782139	2
#23782137	2
#41×Dic₁₆₃	2
C₄×#14579700	2
#23782133	2
C₂×#19104697	2
C₄×D₄×D₁₆₃	2
#520×C₁₆₃	2
C₄×H(5,𝔽₂)	163