Subgroups — #35223304

Order 20864.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#41×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#23782133	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781757	C₂	1	2	yes	maximal, complemented
5	#23781756	C₂	1	2	yes	maximal, complemented
6	#23781750	C₂	1	2	yes	maximal, complemented
7	#23781745	C₂	1	2	yes	maximal, complemented
8	#23781701	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781662	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×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	(C₂²⋊D₁₆₃)⋊C₄	—	2	2	no	—
25	#19104694	—	2	1	no	—
26	#19104694	C₂²	1	1	yes	characteristic
27	#19104692	—	2	1	no	—
28	C₂×C₄×Dic₁₆₃	—	2	1	no	—
29	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
30	C₄²×D₁₆₃	—	2	1	no	—
31	#19104668	C₂²	1	2	yes	—
32	#19104666	C₂²	1	2	yes	—
33	#19104660	C₂²	1	2	yes	—
34	#19104657	C₂²	1	2	yes	—
35	C₈×Dic₁₆₃	C₂²	1	2	yes	—
36	C₈×Dic₁₆₃	C₂²	1	2	yes	—
37	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
39	#117×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	D₄	1	2	yes	—
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	C₂³	1	2	yes	—
60	Dic₁₆₃⋊C₄	—	2	2	no	—
61	C₂²⋊Dic₁₆₃	—	2	2	no	—
62	C₂²⋊Dic₁₆₃	—	2	1	no	—
63	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
64	C₄×Dic₁₆₃	—	2	2	no	—
65	C₄×Dic₁₆₃	—	2	2	no	—
66	C₄×Dic₁₆₃	—	2	1	no	—
67	C₄×Dic₁₆₃	D₄	1	1	yes	characteristic
68	C₄×Dic₁₆₃	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₆₃	—	2	2	no	—
78	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₆₃	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
81	C₄²×C₁₆₃	—	2	1	no	—
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	4	no	—
90	C₂×Dic₁₆₃	—	2	2	no	—
91	C₂×Dic₁₆₃	—	4	4	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	—	4	4	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	C₂²×C₄	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	C₈×C₁₆₃	—	2	2	no	—
113	C₂²×C₁₆₃	—	2	2	no	—
114	C₂²×C₁₆₃	—	2	2	no	—
115	C₂²×C₁₆₃	—	4	2	no	—
116	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
117	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₆₃	—	2	2	no	—
119	C₂×D₁₆₃	—	4	2	no	—
120	C₄×C₁₆₃	—	2	2	no	—
121	C₄×C₁₆₃	—	4	2	no	—
122	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
123	C₄×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
125	Dic₁₆₃	—	2	2	no	—
126	Dic₁₆₃	—	2	2	no	—
127	Dic₁₆₃	—	4	2	no	—
128	Dic₁₆₃	#135	1	4	yes	complemented
129	C₂×C₁₆₃	—	2	2	no	—
130	C₂×C₁₆₃	—	4	2	no	—
131	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₆₃	C₂×#135	1	2	yes	—
133	D₁₆₃	—	4	2	no	—
134	C₁₆₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
135	C₄×#135	—	163	1	no	maximal, Sylow
136	C₂×#135	—	163	1	no	—
137	C₄×#41	—	163	1	no	—
138	C₄×#41	—	163	1	no	—
139	C₄×Q₁₆	—	163	2	no	—
140	C₄×SD₁₆	—	163	2	no	—
141	C₄×SD₁₆	—	163	2	no	—
142	C₄×D₈	—	163	2	no	—
143	#427	—	163	1	no	—
144	#416	—	163	1	no	—
145	#416	D₁₆₃	1	1	yes	characteristic, complemented
146	C₂×C₄×C₈	—	163	1	no	—
147	C₂×#41	—	163	1	no	—
148	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
149	#135	—	163	8	no	—
150	C₂×Q₁₆	—	163	1	no	—
151	C₂×SD₁₆	—	163	1	no	—
152	C₂×SD₁₆	—	163	1	no	—
153	C₂×D₈	—	163	1	no	—
154	C₂²×C₈	—	163	1	no	—
155	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
156	C₄×Q₈	—	163	2	no	—
157	C₄×Q₈	—	163	2	no	—
158	C₄×D₄	—	326	2	no	—
159	C₄×D₄	—	326	2	no	—
160	C₄×D₄	—	163	2	no	—
161	C₄×D₄	—	163	2	no	—
162	#117	—	326	1	no	—
163	#117	—	326	1	no	—
164	#117	—	163	1	no	—
165	#117	C₂×D₁₆₃	1	1	yes	characteristic
166	C₂×C₄²	—	326	1	no	—
167	C₂×C₄²	—	326	1	no	—
168	C₂×C₄²	—	163	1	no	—
169	#107	—	163	2	no	—
170	#106	—	163	2	no	—
171	#103	—	163	2	no	—
172	#103	C₂×D₁₆₃	1	2	yes	—
173	#102	—	163	2	no	—
174	#102	C₂×D₁₆₃	1	2	yes	—
175	C₄×C₈	—	163	2	no	—
176	C₄×C₈	—	163	2	no	—
177	#41	—	2	2	no	—
178	#41	—	326	2	no	—
179	#41	—	163	4	no	—
180	#41	C₄×D₁₆₃	1	4	yes	complemented
181	C₂×Q₈	—	163	1	no	—
182	C₂×Q₈	C₂²×D₁₆₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	326	1	no	—
185	C₂×D₄	—	163	1	no	—
186	C₂×D₄	C₂²×D₁₆₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	326	2	no	—
189	C₂²×C₄	—	326	1	no	—
190	C₂²×C₄	—	326	2	no	—
191	C₂²×C₄	—	163	2	no	—
192	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
193	Q₁₆	—	163	4	no	—
194	SD₁₆	—	163	4	no	—
195	SD₁₆	—	163	4	no	—
196	D₈	—	163	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	163	4	no	—
199	C₂×C₈	—	163	1	no	—
200	C₂×C₈	—	163	1	no	—
201	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
202	C₄⋊C₄	—	326	2	no	—
203	C₄⋊C₄	—	326	2	no	—
204	C₄⋊C₄	—	163	2	no	—
205	C₄⋊C₄	C₂²×D₁₆₃	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	326	2	no	—
208	C₂²⋊C₄	—	326	1	no	—
209	C₂²⋊C₄	—	326	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	326	2	no	—
212	C₄²	—	326	2	no	—
213	C₄²	—	326	1	no	—
214	C₄²	—	163	1	no	—
215	C₄²	—	163	1	no	—
216	C₄²	—	163	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	326	1	no	—
219	C₂³	D₄×D₁₆₃	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	326	1	no	—
222	Q₈	—	163	2	no	—
223	Q₈	C₂×C₄×D₁₆₃	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	326	4	no	—
227	D₄	—	326	1	no	—
228	D₄	—	163	2	no	—
229	D₄	C₂×C₄×D₁₆₃	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	326	1	no	—
234	C₂×C₄	—	326	4	no	—
235	C₂×C₄	—	326	4	no	—
236	C₂×C₄	—	326	2	no	—
237	C₂×C₄	—	652	4	no	—
238	C₂×C₄	—	652	4	no	—
239	C₂×C₄	—	163	2	no	—
240	C₂×C₄	—	163	2	no	—
241	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
242	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
243	C₂×C₄	#14579698	1	2	yes	—
244	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	163	2	no	—
247	C₈	—	163	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	326	2	no	—
252	C₂²	—	652	2	no	—
253	C₂²	(C₂²⋊D₁₆₃)⋊C₄	1	2	yes	—
254	C₂²	C₂×D₄×D₁₆₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	326	2	no	—
258	C₄	—	326	2	no	—
259	C₄	—	652	2	no	—
260	C₄	—	163	4	no	—
261	C₄	(C₂²⋊D₁₆₃)⋊C₄	1	2	yes	—
262	C₄	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
263	C₄	C₂×#14579698	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	652	2	no	—
267	C₂	C₂×(C₂²⋊D₁₆₃)⋊C₄	1	1	yes	characteristic
268	C₂	#135×D₁₆₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₆₃
commutator	C₄×C₁₆₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	#416×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	C₂×#19104755	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#416	D₁₆₃
hypocenter	C₁₆₃	C₄×#135
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₁₆₃	C₂	semidirect
3. #23782133	C₂	semidirect
4. #23781757	C₂	semidirect
5. #23781756	C₂	semidirect
6. #23781750	C₂	semidirect
7. #23781745	C₂	semidirect
8. #23781701	C₂	semidirect
9. #23781662	C₂	semidirect
10. C₂×C₈×Dic₁₆₃	C₂	semidirect
11. #416×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
128. Dic₁₆₃	#135	semidirect
134. C₁₆₃	C₄×#135	semidirect
145. #416	D₁₆₃	semidirect
180. #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
#41×Dic₁₆₃	2
#23782133	2
#23781757	2
#23781756	2
#23781750	2
#23781745	2
#23781701	2
#23781662	2
C₂×C₈×Dic₁₆₃	2
#416×C₁₆₃	2
C₄×#135	163