Subgroups — #35223368

Order 20864.

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×#19104761	C₂	1	1	yes	maximal, characteristic
2	#23782145	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781775	C₂	1	2	yes	maximal, complemented
4	#23781765	C₂	1	2	yes	maximal, complemented
5	C₄×#14579687	C₂	1	2	yes	maximal, complemented
6	C₄×#14579684	C₂	1	2	yes	maximal, complemented
7	#23781708	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781659	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19104656	C₂	1	1	yes	maximal, characteristic, complemented
10	#23781621	C₂	1	1	yes	maximal, characteristic, complemented
11	#518×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	#19104761	C₄	1	8	yes	complemented
14	C₂×#14579687	C₂²	1	2	yes	—
15	C₂×#14579684	C₂²	1	2	yes	—
16	C₂×#14579683	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₃₂₆	C₂²	1	2	yes	complemented
19	C₄×D₆₅₂	C₂²	1	2	yes	complemented
20	(C₂²⋊D₁₆₃)⋊C₄	—	2	2	no	—
21	#19104693	—	2	1	no	—
22	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₆₃	—	2	1	no	—
24	#19104668	C₂²	1	2	yes	complemented
25	#19104666	C₂²	1	2	yes	complemented
26	#19104658	C₂²	1	1	yes	characteristic, complemented
27	#19104658	C₂²	1	1	yes	characteristic, complemented
28	#19104656	C₂²	1	2	yes	—
29	#19104656	C₂²	1	2	yes	—
30	#19104655	C₂²	1	1	yes	characteristic, complemented
31	#19104655	C₂²	1	1	yes	characteristic, complemented
32	#19104646	C₂²	1	2	yes	complemented
33	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₆₃	—	2	1	no	—
36	C₄×D₄×C₁₆₃	—	2	1	no	—
37	C₄×D₄×C₁₆₃	C₂²	1	2	yes	—
38	C₄×D₄×C₁₆₃	C₂²	1	2	yes	—
39	#117×C₁₆₃	C₂²	1	1	yes	characteristic
40	#117×C₁₆₃	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₆₃	—	2	2	no	—
42	#14579699	—	2	2	no	—
43	#14579699	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₃₂₆	C₂³	1	1	yes	characteristic
45	C₂×D₆₅₂	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₆₃	—	2	1	no	—
47	C₂²×Dic₁₆₃	—	2	1	no	—
48	C₂×C₄×D₁₆₃	—	2	1	no	—
49	C₂×C₄×D₁₆₃	—	2	2	no	—
50	#14579687	C₂×C₄	1	8	yes	complemented
51	#14579684	C₂×C₄	1	8	yes	complemented
52	#14579683	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆₃⋊C₈	—	2	2	no	—
54	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
57	C₄⋊Dic₁₆₃	—	2	1	no	—
58	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
59	Dic₁₆₃⋊C₄	—	2	2	no	—
60	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
61	C₄×Dic₁₆₃	—	2	2	no	—
62	C₂⁴×C₁₆₃	—	2	1	no	—
63	C₂×D₄×C₁₆₃	—	2	1	no	—
64	C₂×D₄×C₁₆₃	—	2	2	no	—
65	C₂×D₄×C₁₆₃	C₂×C₄	1	4	yes	complemented
66	C₂×D₄×C₁₆₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₆₃	—	2	2	no	—
68	C₂²×C₄×C₁₆₃	—	2	2	no	—
69	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₆₃	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₆₃	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₆₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₆₃	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
75	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
76	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
77	C₂²⋊C₄×C₁₆₃	D₄	1	2	yes	—
78	C₄²×C₁₆₃	—	2	1	no	—
79	C₄²×C₁₆₃	C₂³	1	2	yes	—
80	C₂²×D₁₆₃	—	2	1	no	—
81	Dic₃₂₆	—	2	1	no	—
82	Dic₃₂₆	C₂²×C₄	1	2	yes	—
83	D₆₅₂	—	2	1	no	—
84	D₆₅₂	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₆₃	—	2	4	no	—
86	C₂×Dic₁₆₃	—	2	1	no	—
87	C₂×Dic₁₆₃	—	2	2	no	—
88	C₂×Dic₁₆₃	—	8	1	no	—
89	C₄×D₁₆₃	—	2	4	no	—
90	C₄×D₁₆₃	—	8	2	no	—
91	C₁₆₃⋊C₈	—	2	1	no	—
92	C₁₆₃⋊C₈	—	2	1	no	—
93	C₁₆₃⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂³×C₁₆₃	—	2	2	no	—
95	C₂³×C₁₆₃	—	2	4	no	—
96	C₂³×C₁₆₃	—	8	1	no	—
97	C₂³×C₁₆₃	—	8	1	no	—
98	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
99	D₄×C₁₆₃	—	2	4	no	—
100	D₄×C₁₆₃	—	2	2	no	—
101	D₄×C₁₆₃	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₁₆₃	—	2	1	no	—
103	C₂×C₄×C₁₆₃	—	2	1	no	—
104	C₂×C₄×C₁₆₃	—	8	2	no	—
105	C₂×C₄×C₁₆₃	—	8	2	no	—
106	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₆₃	#41	1	2	yes	—
111	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
112	C₂²×C₁₆₃	—	2	1	no	—
113	C₂²×C₁₆₃	—	2	1	no	—
114	C₂²×C₁₆₃	—	2	4	no	—
115	C₂²×C₁₆₃	—	4	2	no	—
116	C₂²×C₁₆₃	—	4	2	no	—
117	C₂²×C₁₆₃	—	8	4	no	—
118	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
119	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₆₃	—	2	2	no	—
121	C₂×D₁₆₃	—	4	1	no	—
122	C₂×D₁₆₃	—	4	1	no	—
123	C₄×C₁₆₃	—	2	2	no	—
124	C₄×C₁₆₃	—	2	2	no	—
125	C₄×C₁₆₃	—	4	1	no	—
126	C₄×C₁₆₃	—	4	1	no	—
127	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
128	C₄×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
130	Dic₁₆₃	—	2	2	no	—
131	Dic₁₆₃	—	4	2	no	—
132	C₂×C₁₆₃	—	2	2	no	—
133	C₂×C₁₆₃	—	4	4	no	—
134	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₆₃	#576	1	1	yes	characteristic
136	C₂×C₁₆₃	#577	1	1	yes	characteristic
137	D₁₆₃	—	4	2	no	—
138	C₁₆₃	#2946	1	1	yes	characteristic, complemented, Sylow
139	#2946	—	163	1	no	maximal, Sylow
140	C₂×AGL(1,ℤ/8)	—	163	1	no	—
141	#520	—	163	1	no	—
142	#518	D₁₆₃	1	1	yes	characteristic, complemented
143	#442	—	163	2	no	—
144	#440	—	163	2	no	—
145	C₄×SD₁₆	—	163	2	no	—
146	C₄×D₈	—	163	2	no	—
147	#428	—	163	1	no	—
148	#416	—	163	1	no	—
149	C₂×#102	—	163	1	no	—
150	#405	—	163	1	no	—
151	C₂×#41	—	163	1	no	—
152	C₂²×D₄	C₂×D₁₆₃	1	1	yes	characteristic
153	AGL(1,ℤ/8)	—	163	8	no	—
154	C₂×SD₁₆	—	163	2	no	—
155	C₂×D₈	—	163	2	no	—
156	C₂×M₁₆	—	163	1	no	—
157	C₂²×C₈	—	163	1	no	—
158	C₄×Q₈	—	163	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	326	2	no	—
162	C₄×D₄	—	163	2	no	—
163	C₄×D₄	C₂×D₁₆₃	1	2	yes	—
164	C₄×D₄	C₂×D₁₆₃	1	2	yes	—
165	#117	—	326	1	no	—
166	#117	C₂×D₁₆₃	1	1	yes	characteristic
167	#117	C₂×D₁₆₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	326	1	no	—
169	C₂×C₄⋊C₄	—	163	1	no	—
170	C₂×C₂²⋊C₄	—	2	2	no	—
171	#107	—	163	2	no	—
172	#106	—	163	2	no	—
173	#103	—	163	1	no	—
174	#103	—	163	1	no	—
175	#102	—	163	2	no	—
176	#102	—	163	2	no	—
177	#102	—	163	1	no	—
178	#102	—	163	1	no	—
179	#97	—	163	2	no	—
180	C₄×C₈	—	163	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	326	2	no	—
183	#41	—	163	4	no	—
184	C₂×Q₈	—	163	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	326	1	no	—
188	C₂×D₄	—	163	1	no	—
189	C₂×D₄	C₄×D₁₆₃	1	4	yes	complemented
190	C₂×D₄	C₂²×D₁₆₃	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	326	1	no	—
194	C₂²×C₄	—	326	2	no	—
195	C₂²×C₄	—	326	1	no	—
196	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
197	SD₁₆	—	163	8	no	—
198	D₈	—	163	8	no	—
199	M₁₆	—	163	4	no	—
200	C₂×C₈	—	326	2	no	—
201	C₂×C₈	—	163	1	no	—
202	C₂×C₈	—	163	1	no	—
203	C₂×C₈	—	163	2	no	—
204	C₄⋊C₄	—	326	1	no	—
205	C₄⋊C₄	—	326	2	no	—
206	C₄⋊C₄	—	163	2	no	—
207	C₄⋊C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₁₆₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₆₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	326	2	no	—
215	C₂²⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	326	2	no	—
218	C₄²	C₂²×D₁₆₃	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	326	1	no	—
224	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
225	Q₈	—	326	1	no	—
226	Q₈	—	163	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	326	4	no	—
230	D₄	—	326	1	no	—
231	D₄	—	163	2	no	—
232	D₄	C₂×C₄×D₁₆₃	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	1304	2	no	—
238	C₂×C₄	—	1304	1	no	—
239	C₂×C₄	—	326	1	no	—
240	C₂×C₄	—	326	4	no	—
241	C₂×C₄	—	326	2	no	—
242	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
246	C₂×C₄	#14579699	1	2	yes	—
247	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
248	C₈	—	326	1	no	—
249	C₈	—	326	1	no	—
250	C₈	—	163	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	326	2	no	—
258	C₂²	—	652	1	no	—
259	C₂²	—	652	1	no	—
260	C₂²	C₄×C₂²⋊D₁₆₃	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₁₆₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	326	2	no	—
267	C₄	—	652	2	no	—
268	C₄	C₄×C₂²⋊D₁₆₃	1	2	yes	—
269	C₄	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
270	C₄	C₂×#14579699	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	652	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₁₆₃	1	1	yes	characteristic
275	C₂	#23782468	1	1	yes	characteristic
276	C₂	#23782476	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₆₃
commutator	C₄×C₁₆₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	#518×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	C₂×#19104761	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#518	D₁₆₃
hypocenter	C₁₆₃	#2946
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #23782145	C₂	semidirect
3. #23781775	C₂	semidirect
4. #23781765	C₂	semidirect
5. C₄×#14579687	C₂	semidirect
6. C₄×#14579684	C₂	semidirect
7. #23781708	C₂	semidirect
8. #23781659	C₂	semidirect
9. C₂×#19104656	C₂	semidirect
10. #23781621	C₂	semidirect
11. #518×C₁₆₃	C₂	semidirect
13. #19104761	C₄	semidirect
18. C₄×Dic₃₂₆	C₂²	semidirect
19. C₄×D₆₅₂	C₂²	semidirect
24. #19104668	C₂²	semidirect
25. #19104666	C₂²	semidirect
26. #19104658	C₂²	semidirect
27. #19104658	C₂²	semidirect
30. #19104655	C₂²	semidirect
31. #19104655	C₂²	semidirect
32. #19104646	C₂²	semidirect
33. C₄×C₁₆₃⋊C₈	C₂²	semidirect
43. #14579699	C₂×C₄	semidirect
50. #14579687	C₂×C₄	semidirect
51. #14579684	C₂×C₄	semidirect
52. #14579683	C₂×C₄	semidirect
65. C₂×D₄×C₁₆₃	C₂×C₄	semidirect
138. C₁₆₃	#2946	semidirect
142. #518	D₁₆₃	semidirect
189. C₂×D₄	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×#19104761	2
#23782145	2
#23781775	2
#23781765	2
C₄×#14579687	2
C₄×#14579684	2
#23781708	2
#23781659	2
C₂×#19104656	2
#23781621	2
#518×C₁₆₃	2
#2946	163