Subgroups — #35223245

Order 20864.

500 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×#19104743	C₂	1	2	yes	maximal, complemented
2	#23782212	C₂	1	1	yes	maximal, characteristic
3	#23781744	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781741	C₂	1	2	yes	maximal, complemented
5	D₄×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781730	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₁₆₃)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#23781687	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781630	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	2	yes	—
13	#19104743	—	2	8	no	—
14	C₂×#14579682	C₂²	1	2	yes	—
15	C₂×#14579682	C₂²	1	2	yes	—
16	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
18	C₂×C₈×D₁₆₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₆₃	C₂²	1	1	yes	characteristic
20	#19104734	C₄	1	1	yes	characteristic
21	#19104731	C₄	1	1	yes	characteristic
22	#19104721	C₄	1	2	yes	—
23	#19104710	C₄	1	1	yes	characteristic
24	#19104708	C₄	1	2	yes	—
25	#19104706	C₄	1	1	yes	characteristic
26	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₆₃	C₂²	1	2	yes	—
28	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
29	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
30	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
31	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
32	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
34	C₄×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₆₃	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₆₃	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
38	C₄×D₄×C₁₆₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₆₃	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₁₆₃	C₂²	1	2	yes	—
41	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
42	#14579699	—	2	4	no	—
43	#14579699	—	2	4	no	—
44	C₂×Dic₃₂₆	C₂×C₄	1	2	yes	—
45	C₂×D₆₅₂	C₂×C₄	1	2	yes	—
46	C₂×D₆₅₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₆₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₆₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
53	C₂×C₄×D₁₆₃	C₂³	1	1	yes	characteristic
54	#14579682	—	2	4	no	—
55	#14579682	—	2	4	no	—
56	C₂×C₁₆₃⋊C₈	—	2	4	no	—
57	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
59	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
60	C₈×D₁₆₃	—	2	2	no	—
61	C₈×D₁₆₃	—	2	4	no	—
62	C₈×D₁₆₃	D₄	1	4	yes	complemented
63	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
64	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
66	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
67	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
73	M₁₆×C₁₆₃	—	2	4	no	—
74	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
76	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
79	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
80	C₂²×D₁₆₃	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
82	Dic₃₂₆	—	2	4	no	—
83	D₆₅₂	—	2	4	no	—
84	D₆₅₂	—	2	2	no	—
85	C₂²⋊D₁₆₃	—	2	4	no	—
86	C₂²⋊D₁₆₃	—	2	4	no	—
87	C₂²⋊D₁₆₃	—	2	4	no	—
88	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₁₆₃	—	2	2	no	—
93	C₄×D₁₆₃	—	2	2	no	—
94	C₄×D₁₆₃	—	2	2	no	—
95	C₄×D₁₆₃	—	2	2	no	—
96	C₄×D₁₆₃	C₂×D₄	1	2	yes	—
97	C₄×D₁₆₃	#41	1	2	yes	—
98	C₁₆₃⋊C₈	—	2	2	no	—
99	C₁₆₃⋊C₈	—	2	1	no	—
100	C₁₆₃⋊C₈	C₂×D₄	1	2	yes	—
101	C₂³×C₁₆₃	C₂²×C₄	1	2	yes	—
102	D₄×C₁₆₃	—	2	2	no	—
103	C₂×C₄×C₁₆₃	—	2	2	no	—
104	C₂×C₄×C₁₆₃	—	2	2	no	—
105	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
109	C₈×C₁₆₃	—	2	2	no	—
110	C₈×C₁₆₃	—	2	1	no	—
111	C₈×C₁₆₃	C₂×D₄	1	2	yes	—
112	C₂²×C₁₆₃	—	2	2	no	—
113	C₂²×C₁₆₃	—	2	2	no	—
114	C₂²×C₁₆₃	—	2	2	no	—
115	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₆₃	—	2	1	no	—
117	C₂×D₁₆₃	—	2	1	no	—
118	C₂×D₁₆₃	—	2	2	no	—
119	C₂×D₁₆₃	—	2	2	no	—
120	C₂×D₁₆₃	—	2	2	no	—
121	C₂×D₁₆₃	C₄×D₄	1	2	yes	—
122	C₄×C₁₆₃	—	2	2	no	—
123	C₄×C₁₆₃	—	2	1	no	—
124	C₄×C₁₆₃	#131	1	2	yes	complemented
125	C₄×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
127	Dic₁₆₃	—	2	2	no	—
128	Dic₁₆₃	—	2	1	no	—
129	Dic₁₆₃	C₄×D₄	1	2	yes	—
130	Dic₁₆₃	#131	1	2	yes	complemented
131	C₂×C₁₆₃	—	4	2	no	—
132	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
134	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
135	D₁₆₃	—	2	2	no	—
136	D₁₆₃	—	4	2	no	—
137	C₁₆₃	#2931	1	1	yes	characteristic, complemented, Sylow
138	#2931	—	163	1	no	maximal, Sylow
139	C₂×#131	—	163	2	no	—
140	#532	—	163	1	no	—
141	#436	—	163	2	no	—
142	#436	—	163	1	no	—
143	#436	D₁₆₃	1	1	yes	characteristic, complemented
144	C₈×D₄	—	163	1	no	—
145	C₈×D₄	—	163	1	no	—
146	C₈×D₄	—	163	2	no	—
147	#423	—	163	1	no	—
148	#408	—	163	2	no	—
149	C₂×C₄×C₈	—	163	1	no	—
150	C₂×#41	—	163	2	no	—
151	#131	—	326	8	no	—
152	C₂×M₁₆	—	163	2	no	—
153	C₂×M₁₆	—	163	2	no	—
154	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
155	C₂²×C₈	—	163	2	no	—
156	C₂²×C₈	—	163	2	no	—
157	C₂²×C₈	—	163	1	no	—
158	C₂²×C₈	—	163	1	no	—
159	C₄⋊Q₈	—	163	1	no	—
160	#127	—	163	1	no	—
161	#124	—	163	2	no	—
162	#121	—	163	1	no	—
163	#121	—	163	1	no	—
164	#121	—	163	2	no	—
165	C₄×D₄	—	163	2	no	—
166	C₄×D₄	—	163	1	no	—
167	C₄×D₄	C₂×D₁₆₃	1	1	yes	characteristic
168	C₂×C₄²	—	163	1	no	—
169	C₄⋊C₈	—	163	2	no	—
170	C₄⋊C₈	—	163	1	no	—
171	C₄⋊C₈	C₂×D₁₆₃	1	1	yes	characteristic
172	C₂²⋊C₈	—	163	2	no	—
173	C₂²⋊C₈	—	163	2	no	—
174	C₂²⋊C₈	—	163	2	no	—
175	C₂²⋊C₈	C₂×D₁₆₃	1	2	yes	—
176	C₄×C₈	—	163	1	no	—
177	C₄×C₈	—	163	1	no	—
178	C₄×C₈	—	163	1	no	—
179	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
180	#41	—	326	4	no	—
181	#41	—	326	4	no	—
182	C₂×Q₈	—	163	2	no	—
183	C₂×D₄	—	163	2	no	—
184	C₂×D₄	—	163	2	no	—
185	C₂×D₄	—	163	2	no	—
186	C₂×D₄	—	163	1	no	—
187	C₂×D₄	—	163	2	no	—
188	C₂×D₄	C₄×D₁₆₃	1	1	yes	characteristic
189	C₂²×C₄	—	163	1	no	—
190	C₂²×C₄	—	163	1	no	—
191	C₂²×C₄	—	163	2	no	—
192	C₂²×C₄	—	163	1	no	—
193	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	326	4	no	—
196	M₁₆	—	326	4	no	—
197	C₂×C₈	—	326	2	no	—
198	C₂×C₈	—	326	4	no	—
199	C₂×C₈	—	326	4	no	—
200	C₂×C₈	—	163	1	no	—
201	C₂×C₈	—	163	2	no	—
202	C₂×C₈	—	163	1	no	—
203	C₂×C₈	—	163	4	no	—
204	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
206	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
207	C₄⋊C₄	—	163	2	no	—
208	C₄⋊C₄	—	163	1	no	—
209	C₄⋊C₄	C₄×D₁₆₃	1	1	yes	characteristic
210	C₂²⋊C₄	—	163	2	no	—
211	C₂²⋊C₄	—	163	2	no	—
212	C₂²⋊C₄	—	163	2	no	—
213	C₂²⋊C₄	C₄×D₁₆₃	1	2	yes	—
214	C₄²	—	163	1	no	—
215	C₄²	—	163	1	no	—
216	C₄²	—	163	1	no	—
217	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
218	C₂³	—	163	2	no	—
219	C₂³	—	163	1	no	—
220	C₂³	C₂×C₄×D₁₆₃	1	2	yes	—
221	Q₈	—	326	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	326	4	no	—
224	D₄	—	326	4	no	—
225	D₄	—	326	4	no	—
226	D₄	—	326	4	no	—
227	D₄	—	326	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	326	2	no	—
231	C₂×C₄	—	326	2	no	—
232	C₂×C₄	—	326	2	no	—
233	C₂×C₄	—	326	2	no	—
234	C₂×C₄	—	163	1	no	—
235	C₂×C₄	—	163	2	no	—
236	C₂×C₄	—	163	1	no	—
237	C₂×C₄	—	163	1	no	—
238	C₂×C₄	—	163	2	no	—
239	C₂×C₄	—	163	2	no	—
240	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	326	2	no	—
247	C₈	—	326	1	no	—
248	C₈	—	163	2	no	—
249	C₈	D₄×D₁₆₃	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	326	1	no	—
254	C₂²	—	326	1	no	—
255	C₂²	—	326	2	no	—
256	C₂²	—	326	2	no	—
257	C₂²	—	326	2	no	—
258	C₂²	—	163	2	no	—
259	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	326	1	no	—
263	C₄	—	326	2	no	—
264	C₄	—	163	2	no	—
265	C₄	—	163	2	no	—
266	C₄	#19104743	1	2	yes	complemented
267	C₄	C₂×D₄×D₁₆₃	1	1	yes	characteristic
268	C₄	#41×D₁₆₃	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	326	2	no	—
271	C₂	—	652	2	no	—
272	C₂	C₄×D₄×D₁₆₃	1	1	yes	characteristic
273	C₂	#131×D₁₆₃	1	1	yes	characteristic
274	C₂	C₂×#19104743	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#436×C₁₆₃	C₂
prime radical	#23782212	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#436	D₁₆₃
hypocenter	C₁₆₃	#2931
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#19104743	C₂	semidirect
3. #23781744	C₂	semidirect
4. #23781741	C₂	semidirect
5. D₄×C₁₆₃⋊C₈	C₂	semidirect
6. #23781730	C₂	semidirect
7. (C₂²⋊D₁₆₃)⋊C₈	C₂	semidirect
8. #23781687	C₂	semidirect
9. #23781630	C₂	semidirect
10. C₄×C₈×D₁₆₃	C₂	semidirect
11. #436×C₁₆₃	C₂	semidirect
62. C₈×D₁₆₃	D₄	semidirect
124. C₄×C₁₆₃	#131	semidirect
130. Dic₁₆₃	#131	semidirect
137. C₁₆₃	#2931	semidirect
143. #436	D₁₆₃	semidirect
266. C₄	#19104743	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104743	2
#23782212	2
#23781744	2
#23781741	2
D₄×C₁₆₃⋊C₈	2
#23781730	2
(C₂²⋊D₁₆₃)⋊C₈	2
#23781687	2
#23781630	2
C₄×C₈×D₁₆₃	2
#436×C₁₆₃	2
#2931	163