Subgroups — #35223415

Order 20864.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104755	C₂	1	1	yes	maximal, characteristic
2	#23782148	C₂	1	1	yes	maximal, characteristic, complemented
3	#23782146	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781774	C₂	1	2	yes	maximal, complemented
5	#23781769	C₂	1	2	yes	maximal, complemented
6	#23781756	C₂	1	2	yes	maximal, complemented
7	#23781750	C₂	1	2	yes	maximal, complemented
8	C₂×#19104666	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781667	C₂	1	1	yes	maximal, characteristic, complemented
10	#23781623	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	#19104755	C₄	1	8	yes	complemented
14	C₂×#14579689	C₂²	1	1	yes	characteristic
15	C₂×#14579687	C₂²	1	1	yes	characteristic
16	C₂×#14579686	C₂²	1	1	yes	characteristic
17	C₂×#14579684	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₆₃	C₂²	1	2	yes	—
20	#19104700	C₂²	1	2	yes	—
21	D₄×Dic₁₆₃	—	2	2	no	—
22	D₄×Dic₁₆₃	C₂²	1	2	yes	—
23	#19104697	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₁₆₃	—	2	2	no	—
25	#19104694	—	2	1	no	—
26	#19104694	C₂²	1	1	yes	characteristic
27	#19104693	—	2	1	no	—
28	C₂×C₄⋊Dic₁₆₃	—	2	1	no	—
29	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₆₃	—	2	1	no	—
31	#19104666	C₂²	1	2	yes	—
32	#19104666	C₂²	1	2	yes	—
33	#19104660	C₂²	1	2	yes	—
34	#19104657	C₂²	1	2	yes	—
35	#19104645	C₂²	1	2	yes	—
36	C₈×Dic₁₆₃	C₂²	1	2	yes	—
37	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
38	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
40	#103×C₁₆₃	C₂²	1	2	yes	—
41	#102×C₁₆₃	C₂²	1	2	yes	—
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	2	no	—
48	C₂²×Dic₁₆₃	—	2	1	no	—
49	C₂×C₄×D₁₆₃	—	2	2	no	—
50	C₂×C₄×D₁₆₃	—	2	1	no	—
51	#14579689	C₂×C₄	1	4	yes	complemented
52	#14579687	C₂×C₄	1	4	yes	complemented
53	#14579686	C₂×C₄	1	4	yes	complemented
54	#14579684	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₆₃	—	2	2	no	—
59	C₄⋊Dic₁₆₃	—	2	1	no	—
60	C₄⋊Dic₁₆₃	D₄	1	1	yes	characteristic
61	C₄⋊Dic₁₆₃	D₄	1	1	yes	characteristic
62	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
63	Dic₁₆₃⋊C₄	—	2	2	no	—
64	C₂²⋊Dic₁₆₃	—	2	2	no	—
65	C₂²⋊Dic₁₆₃	D₄	1	2	yes	—
66	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
67	C₄×Dic₁₆₃	—	2	2	no	—
68	C₄×Dic₁₆₃	—	2	2	no	—
69	C₄×Dic₁₆₃	C₂³	1	2	yes	—
70	#41×C₁₆₃	—	2	2	no	—
71	#41×C₁₆₃	C₂×C₄	1	4	yes	complemented
72	C₂×Q₈×C₁₆₃	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₁₆₃	—	2	1	no	—
74	C₂×D₄×C₁₆₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₆₃	—	2	1	no	—
76	C₂²×C₄×C₁₆₃	—	2	1	no	—
77	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
78	M₁₆×C₁₆₃	—	2	2	no	—
79	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₆₃	—	2	1	no	—
81	C₄⋊C₄×C₁₆₃	C₂³	1	2	yes	—
82	C₂²×D₁₆₃	—	2	1	no	—
83	Dic₃₂₆	—	2	1	no	—
84	Dic₃₂₆	C₂²×C₄	1	2	yes	—
85	D₆₅₂	—	2	1	no	—
86	D₆₅₂	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₆₃	—	2	4	no	—
88	C₂×Dic₁₆₃	—	2	1	no	—
89	C₂×Dic₁₆₃	—	2	2	no	—
90	C₂×Dic₁₆₃	—	8	2	no	—
91	C₂×Dic₁₆₃	—	8	1	no	—
92	C₂×Dic₁₆₃	C₂×D₄	1	2	yes	—
93	C₂×Dic₁₆₃	C₂×D₄	1	2	yes	—
94	C₄×D₁₆₃	—	2	4	no	—
95	C₄×D₁₆₃	—	8	2	no	—
96	C₁₆₃⋊C₈	C₂²×C₄	1	2	yes	—
97	C₁₆₃⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂³×C₁₆₃	—	2	1	no	—
99	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₁₆₃	—	2	1	no	—
101	Q₈×C₁₆₃	C₂²×C₄	1	2	yes	—
102	D₄×C₁₆₃	—	2	4	no	—
103	D₄×C₁₆₃	—	2	1	no	—
104	D₄×C₁₆₃	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₁₆₃	—	2	1	no	—
106	C₂×C₄×C₁₆₃	—	2	4	no	—
107	C₂×C₄×C₁₆₃	—	2	2	no	—
108	C₂×C₄×C₁₆₃	—	8	1	no	—
109	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₆₃	#41	1	2	yes	—
112	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
113	C₈×C₁₆₃	—	2	2	no	—
114	C₂²×C₁₆₃	—	2	1	no	—
115	C₂²×C₁₆₃	—	2	1	no	—
116	C₂²×C₁₆₃	—	2	2	no	—
117	C₂²×C₁₆₃	—	4	1	no	—
118	C₂²×C₁₆₃	—	4	1	no	—
119	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
120	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₆₃	—	2	2	no	—
122	C₂×D₁₆₃	—	4	1	no	—
123	C₂×D₁₆₃	—	4	1	no	—
124	C₄×C₁₆₃	—	2	2	no	—
125	C₄×C₁₆₃	—	4	2	no	—
126	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
127	C₄×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
129	Dic₁₆₃	—	2	2	no	—
130	Dic₁₆₃	—	2	2	no	—
131	Dic₁₆₃	—	2	2	no	—
132	Dic₁₆₃	—	4	2	no	—
133	C₂×C₁₆₃	—	2	2	no	—
134	C₂×C₁₆₃	—	4	2	no	—
135	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₆₃	#577	1	1	yes	characteristic
137	C₂×C₁₆₃	#577	1	1	yes	characteristic
138	D₁₆₃	—	4	2	no	—
139	C₁₆₃	#2952	1	1	yes	characteristic, complemented, Sylow
140	#2952	—	163	1	no	maximal, Sylow
141	C₂×#135	—	163	1	no	—
142	#520	—	163	1	no	—
143	#520	—	163	1	no	—
144	#442	—	163	2	no	—
145	#441	—	163	2	no	—
146	C₄×SD₁₆	—	163	2	no	—
147	C₄×SD₁₆	—	163	2	no	—
148	C₂×#106	—	163	1	no	—
149	#417	—	163	1	no	—
150	#417	D₁₆₃	1	1	yes	characteristic, complemented
151	#405	—	163	1	no	—
152	C₂×#41	—	163	1	no	—
153	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
154	#135	—	163	8	no	—
155	C₂×Q₁₆	—	163	1	no	—
156	C₂×SD₁₆	—	163	1	no	—
157	C₂×SD₁₆	—	163	1	no	—
158	C₂×D₈	—	163	1	no	—
159	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
160	C₂²×C₈	—	163	1	no	—
161	C₄×Q₈	—	163	2	no	—
162	C₄×Q₈	—	163	2	no	—
163	C₄×D₄	—	326	2	no	—
164	C₄×D₄	—	326	2	no	—
165	C₄×D₄	—	163	2	no	—
166	C₄×D₄	—	163	2	no	—
167	#117	—	326	1	no	—
168	#117	—	326	1	no	—
169	#117	—	163	1	no	—
170	C₂×C₄⋊C₄	—	326	1	no	—
171	C₂×C₄⋊C₄	—	326	1	no	—
172	C₂×C₄⋊C₄	—	163	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
174	#106	—	163	2	no	—
175	#106	—	163	2	no	—
176	#103	—	163	2	no	—
177	#103	C₂×D₁₆₃	1	2	yes	—
178	#102	—	163	2	no	—
179	#102	C₂×D₁₆₃	1	2	yes	—
180	#97	—	163	2	no	—
181	C₄×C₈	—	163	2	no	—
182	#41	—	2	2	no	—
183	#41	—	326	2	no	—
184	#41	—	163	4	no	—
185	#41	C₄×D₁₆₃	1	4	yes	complemented
186	C₂×Q₈	—	163	1	no	—
187	C₂×Q₈	C₂²×D₁₆₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	326	1	no	—
190	C₂×D₄	—	163	1	no	—
191	C₂×D₄	C₂²×D₁₆₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	326	2	no	—
195	C₂²×C₄	—	326	1	no	—
196	C₂²×C₄	—	326	2	no	—
197	C₂²×C₄	—	326	1	no	—
198	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
199	Q₁₆	—	163	4	no	—
200	SD₁₆	—	163	4	no	—
201	SD₁₆	—	163	4	no	—
202	D₈	—	163	4	no	—
203	M₁₆	—	2	2	no	—
204	C₂×C₈	—	163	4	no	—
205	C₂×C₈	—	163	1	no	—
206	C₂×C₈	—	163	1	no	—
207	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	326	2	no	—
210	C₄⋊C₄	—	326	2	no	—
211	C₄⋊C₄	—	326	1	no	—
212	C₄⋊C₄	—	163	2	no	—
213	C₄⋊C₄	—	163	1	no	—
214	C₄⋊C₄	—	163	1	no	—
215	C₄⋊C₄	C₂²×D₁₆₃	1	2	yes	—
216	C₂²⋊C₄	—	326	2	no	—
217	C₂²⋊C₄	—	326	2	no	—
218	C₂²⋊C₄	—	163	2	no	—
219	C₄²	—	326	2	no	—
220	C₄²	—	326	2	no	—
221	C₄²	—	163	2	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	326	1	no	—
224	C₂³	D₄×D₁₆₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	326	1	no	—
227	Q₈	—	163	2	no	—
228	Q₈	C₂×C₄×D₁₆₃	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	326	4	no	—
232	D₄	—	326	1	no	—
233	D₄	—	163	2	no	—
234	D₄	C₂×C₄×D₁₆₃	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	1304	2	no	—
240	C₂×C₄	—	1304	2	no	—
241	C₂×C₄	—	1304	1	no	—
242	C₂×C₄	—	326	1	no	—
243	C₂×C₄	—	326	4	no	—
244	C₂×C₄	—	326	2	no	—
245	C₂×C₄	—	163	2	no	—
246	C₂×C₄	—	163	2	no	—
247	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
248	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
249	C₂×C₄	#14579698	1	2	yes	—
250	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	163	2	no	—
253	C₈	—	163	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	326	2	no	—
260	C₂²	—	652	1	no	—
261	C₂²	—	652	1	no	—
262	C₂²	(C₂²⋊D₁₆₃)⋊C₄	1	2	yes	—
263	C₂²	C₂×D₄×D₁₆₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	326	2	no	—
267	C₄	—	326	2	no	—
268	C₄	—	326	2	no	—
269	C₄	—	652	2	no	—
270	C₄	(C₂²⋊D₁₆₃)⋊C₄	1	2	yes	—
271	C₄	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
272	C₄	C₂×#14579698	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	652	2	no	—
276	C₂	C₂×(C₂²⋊D₁₆₃)⋊C₄	1	1	yes	characteristic
277	C₂	#23782474	1	1	yes	characteristic
278	C₂	#23782475	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₆₃
commutator	C₄×C₁₆₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	#417×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	C₂×#19104755	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#417	D₁₆₃
hypocenter	C₁₆₃	#2952
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #23782148	C₂	semidirect
3. #23782146	C₂	semidirect
4. #23781774	C₂	semidirect
5. #23781769	C₂	semidirect
6. #23781756	C₂	semidirect
7. #23781750	C₂	semidirect
8. C₂×#19104666	C₂	semidirect
9. #23781667	C₂	semidirect
10. #23781623	C₂	semidirect
11. #417×C₁₆₃	C₂	semidirect
13. #19104755	C₄	semidirect
43. #14579699	C₂×C₄	semidirect
51. #14579689	C₂×C₄	semidirect
52. #14579687	C₂×C₄	semidirect
53. #14579686	C₂×C₄	semidirect
54. #14579684	C₂×C₄	semidirect
55. C₂×C₁₆₃⋊C₈	C₂×C₄	semidirect
71. #41×C₁₆₃	C₂×C₄	semidirect
139. C₁₆₃	#2952	semidirect
150. #417	D₁₆₃	semidirect
185. #41	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×#19104755	2
#23782148	2
#23782146	2
#23781774	2
#23781769	2
#23781756	2
#23781750	2
C₂×#19104666	2
#23781667	2
#23781623	2
#417×C₁₆₃	2
#2952	163