Subgroups — #35228172

Order 20864.

1900 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104780	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#23782310	C₂	1	8	yes	maximal, complemented
4	D₄×D₆₅₂	C₂	1	8	yes	maximal, complemented
5	#23782234	C₂	1	2	yes	maximal, complemented
6	#23782227	C₂	1	4	yes	maximal, complemented
7	#23782177	C₂	1	2	yes	maximal, complemented
8	C₂×#19104705	C₂	1	4	yes	maximal, complemented
9	#518×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₆₃	C₂²	1	2	yes	complemented
11	#19104780	—	2	8	no	—
12	C₂×#14579699	C₂²	1	2	yes	complemented
13	C₂×#14579698	C₂²	1	4	yes	complemented
14	C₂²×D₆₅₂	C₂²	1	1	yes	characteristic, complemented
15	C₂²×D₆₅₂	C₂²	1	8	yes	complemented
16	C₂²×C₂²⋊D₁₆₃	C₂²	1	2	yes	complemented
17	C₂²×C₂²⋊D₁₆₃	C₂²	1	4	yes	complemented
18	C₂×D₄×D₁₆₃	—	2	8	no	—
19	C₂×D₄×D₁₆₃	C₂²	1	8	yes	complemented
20	C₂×D₄×D₁₆₃	C₂²	1	4	yes	complemented
21	C₂²×C₄×D₁₆₃	C₂²	1	1	yes	characteristic, complemented
22	#19104732	C₂²	1	4	yes	complemented
23	#19104723	C₂²	1	4	yes	complemented
24	#19104719	C₂²	1	4	yes	complemented
25	#19104719	C₂²	1	8	yes	complemented
26	#19104715	C₂²	1	4	yes	complemented
27	#19104710	C₂²	1	4	yes	complemented
28	#19104710	C₂²	1	8	yes	complemented
29	#19104709	C₂²	1	16	yes	complemented
30	#19104705	C₂²	1	4	yes	complemented
31	#19104705	C₂²	1	8	yes	complemented
32	#19104705	C₂²	1	16	yes	complemented
33	#19104705	C₂²	1	8	yes	complemented
34	C₄×D₆₅₂	C₂²	1	8	yes	complemented
35	C₂×(C₂×D₁₆₃)⋊C₄	C₂²	1	8	yes	complemented
36	C₂²×D₄×C₁₆₃	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₁₆₃	C₂²	1	8	yes	complemented
38	#117×C₁₆₃	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₁₆₃	C₂²	1	4	yes	complemented
40	C₂³×D₁₆₃	—	2	8	no	—
41	C₂³×D₁₆₃	—	4	8	no	—
42	C₂³×D₁₆₃	C₂³	1	1	yes	characteristic
43	C₂³×D₁₆₃	C₂³	1	8	yes	—
44	C₂³×D₁₆₃	C₂³	1	2	yes	—
45	C₂³×D₁₆₃	C₂³	1	4	yes	complemented
46	#14579699	—	2	8	no	—
47	#14579698	—	2	16	no	—
48	C₂×Dic₃₂₆	C₂³	1	2	yes	complemented
49	C₂×D₆₅₂	—	2	4	no	—
50	C₂×D₆₅₂	—	4	16	no	—
51	C₂×D₆₅₂	C₂³	1	4	yes	—
52	C₂×D₆₅₂	C₂³	1	8	yes	—
53	C₂×D₆₅₂	C₂³	1	2	yes	complemented
54	C₂×D₆₅₂	C₂³	1	16	yes	complemented
55	C₂×C₂²⋊D₁₆₃	—	2	8	no	—
56	C₂×C₂²⋊D₁₆₃	—	2	16	no	—
57	C₂×C₂²⋊D₁₆₃	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₁₆₃	C₂³	1	4	yes	—
59	C₂×C₂²⋊D₁₆₃	C₂³	1	8	yes	complemented
60	D₄×D₁₆₃	—	2	32	no	—
61	D₄×D₁₆₃	—	2	16	no	—
62	C₂²×Dic₁₆₃	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₆₃	C₂³	1	4	yes	complemented
64	C₂×C₄×D₁₆₃	—	2	4	no	—
65	C₂×C₄×D₁₆₃	C₂³	1	4	yes	—
66	C₂×C₄×D₁₆₃	C₂³	1	4	yes	complemented
67	C₄⋊Dic₁₆₃	C₂³	1	8	yes	complemented
68	(C₂×D₁₆₃)⋊C₄	C₂³	1	8	yes	—
69	(C₂×D₁₆₃)⋊C₄	C₂³	1	16	yes	complemented
70	(C₂×D₁₆₃)⋊C₄	C₂³	1	8	yes	complemented
71	C₂⁴×C₁₆₃	C₂³	1	2	yes	—
72	C₂×D₄×C₁₆₃	D₄	1	8	yes	complemented
73	C₂×D₄×C₁₆₃	C₂³	1	4	yes	—
74	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₆₃	C₂³	1	8	yes	complemented
76	C₄⋊C₄×C₁₆₃	C₂³	1	4	yes	complemented
77	C₂²⋊C₄×C₁₆₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₁₆₃	C₂³	1	8	yes	complemented
79	C₄²×C₁₆₃	C₂³	1	4	yes	complemented
80	C₂²×D₁₆₃	—	2	2	no	—
81	C₂²×D₁₆₃	—	2	4	no	—
82	C₂²×D₁₆₃	—	2	16	no	—
83	C₂²×D₁₆₃	—	2	4	no	—
84	C₂²×D₁₆₃	—	2	8	no	—
85	C₂²×D₁₆₃	—	2	16	no	—
86	C₂²×D₁₆₃	—	4	8	no	—
87	C₂²×D₁₆₃	—	4	4	no	—
88	C₂²×D₁₆₃	—	4	8	no	—
89	C₂²×D₁₆₃	—	4	32	no	—
90	C₂²×D₁₆₃	C₂⁴	1	4	yes	—
91	C₂²×D₁₆₃	C₂⁴	1	4	yes	—
92	C₂²×D₁₆₃	C₂⁴	1	2	yes	—
93	Dic₃₂₆	—	2	4	no	—
94	D₆₅₂	—	2	16	no	—
95	D₆₅₂	—	2	8	no	—
96	D₆₅₂	—	2	4	no	—
97	D₆₅₂	—	4	16	no	—
98	C₂²⋊D₁₆₃	—	2	16	no	—
99	C₂²⋊D₁₆₃	—	2	16	no	—
100	C₂²⋊D₁₆₃	—	2	8	no	—
101	C₂×Dic₁₆₃	—	2	2	no	—
102	C₂×Dic₁₆₃	—	2	8	no	—
103	C₂×Dic₁₆₃	C₂⁴	1	4	yes	—
104	C₂×Dic₁₆₃	C₂⁴	1	2	yes	—
105	C₄×D₁₆₃	—	2	8	no	—
106	C₄×D₁₆₃	—	2	8	no	—
107	C₂³×C₁₆₃	—	4	4	no	—
108	C₂³×C₁₆₃	C₂×D₄	1	8	yes	complemented
109	C₂³×C₁₆₃	C₂⁴	1	1	yes	characteristic
110	C₂³×C₁₆₃	C₂⁴	1	4	yes	—
111	D₄×C₁₆₃	C₂×D₄	1	16	yes	complemented
112	C₂×C₄×C₁₆₃	—	4	8	no	—
113	C₂×C₄×C₁₆₃	C₂×D₄	1	4	yes	complemented
114	C₂×C₄×C₁₆₃	C₂⁴	1	8	yes	—
115	C₂×C₄×C₁₆₃	C₂⁴	1	2	yes	—
116	C₂²×C₁₆₃	—	2	8	no	—
117	C₂²×C₁₆₃	—	2	2	no	—
118	C₂²×C₁₆₃	—	4	8	no	—
119	C₂²×C₁₆₃	C₂²×D₄	1	2	yes	—
120	C₂²×C₁₆₃	C₂²×D₄	1	8	yes	—
121	C₂²×C₁₆₃	C₂⁵	1	1	yes	characteristic
122	C₂×D₁₆₃	—	2	8	no	—
123	C₂×D₁₆₃	—	2	2	no	—
124	C₂×D₁₆₃	—	2	2	no	—
125	C₂×D₁₆₃	—	2	4	no	—
126	C₂×D₁₆₃	—	2	4	no	—
127	C₂×D₁₆₃	—	2	8	no	—
128	C₂×D₁₆₃	—	2	8	no	—
129	C₂×D₁₆₃	—	4	32	no	—
130	C₂×D₁₆₃	—	4	8	no	—
131	C₂×D₁₆₃	—	8	8	no	—
132	C₂×D₁₆₃	H(5,𝔽₂)	1	4	yes	complemented
133	C₄×C₁₆₃	—	2	8	no	—
134	C₄×C₁₆₃	C₂²×D₄	1	4	yes	—
135	Dic₁₆₃	—	2	2	no	—
136	Dic₁₆₃	—	2	4	no	—
137	C₂×C₁₆₃	—	2	2	no	—
138	C₂×C₁₆₃	—	2	8	no	—
139	C₂×C₁₆₃	C₂³×D₄	1	1	yes	characteristic
140	C₂×C₁₆₃	C₂×H(5,𝔽₂)	1	2	yes	—
141	D₁₆₃	—	2	4	no	—
142	D₁₆₃	—	4	4	no	—
143	D₁₆₃	—	4	4	no	—
144	C₁₆₃	#3484	1	1	yes	characteristic, complemented, Sylow
145	#3484	—	163	1	no	maximal, Sylow
146	C₂×H(5,𝔽₂)	—	163	1	no	—
147	C₂³×D₄	—	163	1	no	—
148	#546	—	163	8	no	—
149	D₄²	—	163	8	no	—
150	#535	—	163	2	no	—
151	#534	—	163	4	no	—
152	#525	—	163	2	no	—
153	C₂×#120	—	163	4	no	—
154	#518	D₁₆₃	1	1	yes	characteristic, complemented
155	C₂⁵	—	163	2	no	—
156	H(5,𝔽₂)	—	326	8	no	—
157	C₂×#41	—	163	4	no	—
158	C₂×#41	—	163	2	no	—
159	C₂²×D₄	—	326	8	no	—
160	C₂²×D₄	—	163	4	no	—
161	C₂²×D₄	—	163	2	no	—
162	C₂²×D₄	—	163	4	no	—
163	C₂²×D₄	—	163	8	no	—
164	C₂²×D₄	—	163	1	no	—
165	C₂²×D₄	—	163	8	no	—
166	C₂²×D₄	C₂×D₁₆₃	1	1	yes	characteristic
167	C₂³×C₄	—	163	1	no	—
168	#127	—	163	4	no	—
169	#124	—	163	4	no	—
170	#123	—	163	4	no	—
171	#123	—	163	8	no	—
172	C₂²⋊Q₈	—	163	4	no	—
173	#121	—	163	16	no	—
174	#121	—	163	4	no	—
175	#121	—	163	8	no	—
176	#120	—	163	4	no	—
177	#120	—	163	8	no	—
178	#120	—	163	16	no	—
179	#120	—	163	8	no	—
180	C₄×D₄	—	163	8	no	—
181	C₄×D₄	C₂×D₁₆₃	1	8	yes	complemented
182	#117	C₂×D₁₆₃	1	2	yes	complemented
183	C₂×C₂²⋊C₄	—	163	8	no	—
184	C₂×C₂²⋊C₄	C₂×D₁₆₃	1	4	yes	complemented
185	C₂⁴	—	326	8	no	—
186	C₂⁴	—	652	8	no	—
187	C₂⁴	—	163	1	no	—
188	C₂⁴	—	163	8	no	—
189	C₂⁴	—	163	2	no	—
190	C₂⁴	—	163	4	no	—
191	C₂⁴	C₂²×D₁₆₃	1	2	yes	—
192	#41	—	326	16	no	—
193	#41	—	326	8	no	—
194	C₂×Q₈	—	163	2	no	—
195	C₂×D₄	—	326	8	no	—
196	C₂×D₄	—	326	16	no	—
197	C₂×D₄	—	326	32	no	—
198	C₂×D₄	—	326	16	no	—
199	C₂×D₄	—	326	4	no	—
200	C₂×D₄	—	652	16	no	—
201	C₂×D₄	—	163	4	no	—
202	C₂×D₄	—	163	8	no	—
203	C₂×D₄	—	163	8	no	—
204	C₂×D₄	—	163	8	no	—
205	C₂×D₄	—	163	4	no	—
206	C₂×D₄	—	163	2	no	—
207	C₂×D₄	—	163	16	no	—
208	C₂×D₄	D₆₅₂	1	8	yes	complemented
209	C₂×D₄	C₂²×D₁₆₃	1	4	yes	—
210	C₂²×C₄	—	326	4	no	—
211	C₂²×C₄	—	163	4	no	—
212	C₂²×C₄	—	163	1	no	—
213	C₂²×C₄	—	163	4	no	—
214	C₂²×C₄	—	163	4	no	—
215	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₁₆₃	1	8	yes	complemented
217	C₄⋊C₄	—	163	8	no	—
218	C₄⋊C₄	C₂²×D₁₆₃	1	4	yes	complemented
219	C₂²⋊C₄	—	163	8	no	—
220	C₂²⋊C₄	—	163	16	no	—
221	C₂²⋊C₄	—	163	8	no	—
222	C₂²⋊C₄	C₂²×D₁₆₃	1	4	yes	—
223	C₂²⋊C₄	C₂²×D₁₆₃	1	8	yes	complemented
224	C₄²	C₂²×D₁₆₃	1	4	yes	complemented
225	C₂³	—	4	4	no	—
226	C₂³	—	326	4	no	—
227	C₂³	—	326	16	no	—
228	C₂³	—	326	8	no	—
229	C₂³	—	326	2	no	—
230	C₂³	—	326	16	no	—
231	C₂³	—	326	4	no	—
232	C₂³	—	652	8	no	—
233	C₂³	—	652	8	no	—
234	C₂³	—	652	4	no	—
235	C₂³	—	652	32	no	—
236	C₂³	—	163	4	no	—
237	C₂³	—	163	4	no	—
238	C₂³	—	163	2	no	—
239	C₂³	C₂×D₆₅₂	1	8	yes	complemented
240	C₂³	C₂³×D₁₆₃	1	1	yes	characteristic
241	C₂³	C₂³×D₁₆₃	1	4	yes	—
242	Q₈	—	326	4	no	—
243	D₄	—	326	16	no	—
244	D₄	—	326	16	no	—
245	D₄	—	326	8	no	—
246	D₄	—	326	16	no	—
247	D₄	—	326	8	no	—
248	D₄	—	326	4	no	—
249	D₄	—	652	16	no	—
250	D₄	C₂×D₆₅₂	1	16	yes	complemented
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	326	2	no	—
253	C₂×C₄	—	326	8	no	—
254	C₂×C₄	—	326	8	no	—
255	C₂×C₄	—	326	8	no	—
256	C₂×C₄	—	163	4	no	—
257	C₂×C₄	—	163	2	no	—
258	C₂×C₄	C₂×D₆₅₂	1	4	yes	complemented
259	C₂×C₄	C₂³×D₁₆₃	1	8	yes	—
260	C₂×C₄	C₂³×D₁₆₃	1	2	yes	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	8	no	—
264	C₂²	—	1304	8	no	—
265	C₂²	—	326	8	no	—
266	C₂²	—	326	2	no	—
267	C₂²	—	326	2	no	—
268	C₂²	—	326	4	no	—
269	C₂²	—	326	4	no	—
270	C₂²	—	326	8	no	—
271	C₂²	—	326	8	no	—
272	C₂²	—	652	32	no	—
273	C₂²	—	652	8	no	—
274	C₂²	—	163	4	no	—
275	C₂²	C₂²×D₆₅₂	1	2	yes	—
276	C₂²	C₂²×D₆₅₂	1	8	yes	—
277	C₂²	C₂⁴×D₁₆₃	1	1	yes	characteristic
278	C₄	—	2	8	no	—
279	C₄	—	326	2	no	—
280	C₄	—	326	4	no	—
281	C₄	C₂²×D₆₅₂	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	8	no	—
284	C₂	—	326	4	no	—
285	C₂	—	652	4	no	—
286	C₂	—	652	4	no	—
287	C₂	C₂³×D₆₅₂	1	1	yes	characteristic
288	C₂	H(5,𝔽₂)×D₁₆₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₆₃
commutator, socle, cosocle	C₂²×C₁₆₃	C₂⁵
Fitting	#518×C₁₆₃	C₂
hypercenter	#518	D₁₆₃
hypocenter	C₁₆₃	#3484
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. C₂×#19104780	C₂	semidirect
2. C₂²×D₄×D₁₆₃	C₂	semidirect
3. #23782310	C₂	semidirect
4. D₄×D₆₅₂	C₂	semidirect
5. #23782234	C₂	semidirect
6. #23782227	C₂	semidirect
7. #23782177	C₂	semidirect
8. C₂×#19104705	C₂	semidirect
9. #518×C₁₆₃	C₂	semidirect
10. C₂⁴×D₁₆₃	C₂²	semidirect
12. C₂×#14579699	C₂²	semidirect
13. C₂×#14579698	C₂²	semidirect
14. C₂²×D₆₅₂	C₂²	semidirect
15. C₂²×D₆₅₂	C₂²	semidirect
16. C₂²×C₂²⋊D₁₆₃	C₂²	semidirect
17. C₂²×C₂²⋊D₁₆₃	C₂²	semidirect
19. C₂×D₄×D₁₆₃	C₂²	semidirect
20. C₂×D₄×D₁₆₃	C₂²	semidirect
21. C₂²×C₄×D₁₆₃	C₂²	semidirect
22. #19104732	C₂²	semidirect
23. #19104723	C₂²	semidirect
24. #19104719	C₂²	semidirect
25. #19104719	C₂²	semidirect
26. #19104715	C₂²	semidirect
27. #19104710	C₂²	semidirect
28. #19104710	C₂²	semidirect
29. #19104709	C₂²	semidirect
30. #19104705	C₂²	semidirect
31. #19104705	C₂²	semidirect
32. #19104705	C₂²	semidirect
33. #19104705	C₂²	semidirect
34. C₄×D₆₅₂	C₂²	semidirect
35. C₂×(C₂×D₁₆₃)⋊C₄	C₂²	semidirect
37. C₄×D₄×C₁₆₃	C₂²	semidirect
38. #117×C₁₆₃	C₂²	semidirect
39. C₂×C₂²⋊C₄×C₁₆₃	C₂²	semidirect
45. C₂³×D₁₆₃	C₂³	semidirect
48. C₂×Dic₃₂₆	C₂³	semidirect
53. C₂×D₆₅₂	C₂³	semidirect
54. C₂×D₆₅₂	C₂³	semidirect
59. C₂×C₂²⋊D₁₆₃	C₂³	semidirect
63. C₂²×Dic₁₆₃	C₂³	semidirect
66. C₂×C₄×D₁₆₃	C₂³	semidirect
67. C₄⋊Dic₁₆₃	C₂³	semidirect
69. (C₂×D₁₆₃)⋊C₄	C₂³	semidirect
70. (C₂×D₁₆₃)⋊C₄	C₂³	semidirect
72. C₂×D₄×C₁₆₃	D₄	semidirect
75. C₂²×C₄×C₁₆₃	C₂³	semidirect
76. C₄⋊C₄×C₁₆₃	C₂³	semidirect
78. C₂²⋊C₄×C₁₆₃	C₂³	semidirect
79. C₄²×C₁₆₃	C₂³	semidirect
108. C₂³×C₁₆₃	C₂×D₄	semidirect
111. D₄×C₁₆₃	C₂×D₄	semidirect
113. C₂×C₄×C₁₆₃	C₂×D₄	semidirect
132. C₂×D₁₆₃	H(5,𝔽₂)	semidirect
144. C₁₆₃	#3484	semidirect
154. #518	D₁₆₃	semidirect
181. C₄×D₄	C₂×D₁₆₃	semidirect
182. #117	C₂×D₁₆₃	semidirect
184. C₂×C₂²⋊C₄	C₂×D₁₆₃	semidirect
208. C₂×D₄	D₆₅₂	semidirect
216. C₂²×C₄	C₂²×D₁₆₃	semidirect
218. C₄⋊C₄	C₂²×D₁₆₃	semidirect
223. C₂²⋊C₄	C₂²×D₁₆₃	semidirect
224. C₄²	C₂²×D₁₆₃	semidirect
239. C₂³	C₂×D₆₅₂	semidirect
250. D₄	C₂×D₆₅₂	semidirect
258. C₂×C₄	C₂×D₆₅₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104780	2
C₂²×D₄×D₁₆₃	2
#23782310	2
D₄×D₆₅₂	2
#23782234	2
#23782227	2
#23782177	2
C₂×#19104705	2
#518×C₁₆₃	2
#3484	163