Subgroups — #35223228

Order 20864.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#23782172	C₂	1	1	yes	maximal, characteristic
3	#23781734	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₁₆₃	C₂	1	4	yes	maximal, complemented
5	#23781693	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781632	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781628	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781620	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×#14579683	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₆₃	—	2	4	no	—
14	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
15	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
16	#19104716	C₄	1	2	yes	—
17	#19104711	C₄	1	2	yes	—
18	#19104706	C₄	1	2	yes	—
19	#19104703	C₄	1	2	yes	—
20	C₄×C₂²⋊D₁₆₃	C₂²	1	4	yes	—
21	#19104694	C₂²	1	1	yes	characteristic
22	Dic₁₆₃⋊C₈	C₂²	1	4	yes	complemented
23	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	4	yes	complemented
24	(C₂×D₁₆₃)⋊C₈	C₂²	1	4	yes	—
25	#19104645	C₂²	1	2	yes	complemented
26	C₈×Dic₁₆₃	C₂²	1	2	yes	complemented
27	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
28	#131×C₁₆₃	—	2	4	no	—
29	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	complemented
31	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	complemented
33	C₂³×D₁₆₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	4	yes	—
35	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₆₃	—	2	2	no	—
37	C₂×C₄×D₁₆₃	—	2	2	no	—
38	C₂×C₄×D₁₆₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
40	#14579683	—	2	2	no	—
41	C₂×C₁₆₃⋊C₈	—	2	2	no	—
42	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
43	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
44	C₈×D₁₆₃	—	2	4	no	—
45	C₈×D₁₆₃	—	2	4	no	—
46	C₄⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
47	Dic₁₆₃⋊C₄	C₂×C₄	1	4	yes	—
48	C₂²⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
49	C₂²⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
50	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	4	yes	—
51	C₄×Dic₁₆₃	C₂³	1	2	yes	—
52	#41×C₁₆₃	—	2	2	no	—
53	#41×C₁₆₃	—	2	2	no	—
54	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
55	C₂×D₄×C₁₆₃	C₂×C₄	1	2	yes	—
56	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
59	M₁₆×C₁₆₃	—	2	2	no	—
60	M₁₆×C₁₆₃	—	2	4	no	—
61	C₂×C₈×C₁₆₃	—	2	4	no	—
62	C₂×C₈×C₁₆₃	D₄	1	4	yes	complemented
63	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
64	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
66	C₂²×D₁₆₃	—	2	2	no	—
67	C₂²×D₁₆₃	—	2	4	no	—
68	C₂²×D₁₆₃	C₂²×C₄	1	2	yes	—
69	C₂²⋊D₁₆₃	—	2	8	no	—
70	C₂×Dic₁₆₃	—	2	2	no	—
71	C₂×Dic₁₆₃	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
73	C₄×D₁₆₃	—	2	4	no	—
74	C₄×D₁₆₃	—	2	2	no	—
75	C₄×D₁₆₃	—	2	2	no	—
76	C₁₆₃⋊C₈	—	2	2	no	—
77	C₁₆₃⋊C₈	—	2	2	no	—
78	C₂³×C₁₆₃	C₂²×C₄	1	2	yes	—
79	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
80	Q₈×C₁₆₃	—	2	2	no	—
81	D₄×C₁₆₃	—	2	4	no	—
82	D₄×C₁₆₃	—	2	2	no	—
83	C₂×C₄×C₁₆₃	—	2	2	no	—
84	C₂×C₄×C₁₆₃	—	2	2	no	—
85	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₆₃	#41	1	2	yes	—
89	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₆₃	—	2	2	no	—
91	C₈×C₁₆₃	C₂×D₄	1	2	yes	—
92	C₈×C₁₆₃	C₂×D₄	1	2	yes	—
93	C₂²×C₁₆₃	—	2	2	no	—
94	C₂²×C₁₆₃	—	2	2	no	—
95	C₂²×C₁₆₃	—	2	4	no	—
96	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
97	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₆₃	—	2	8	no	—
99	C₂×D₁₆₃	—	2	2	no	—
100	C₂×D₁₆₃	—	2	2	no	—
101	C₂×D₁₆₃	#131	1	4	yes	complemented
102	C₄×C₁₆₃	—	2	2	no	—
103	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
104	C₄×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
106	Dic₁₆₃	—	2	2	no	—
107	Dic₁₆₃	—	2	4	no	—
108	C₂×C₁₆₃	—	2	2	no	—
109	C₂×C₁₆₃	—	4	2	no	—
110	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₁₆₃	C₂×#131	1	2	yes	—
112	D₁₆₃	—	2	4	no	—
113	C₁₆₃	#2929	1	1	yes	characteristic, complemented, Sylow
114	#2929	—	163	1	no	maximal, Sylow
115	C₂×#131	D₁₆₃	1	1	yes	characteristic, complemented
116	C₂³×C₈	—	163	1	no	—
117	#525	—	163	1	no	—
118	#435	—	163	4	no	—
119	C₈×D₄	—	163	4	no	—
120	#424	—	163	1	no	—
121	#408	—	163	1	no	—
122	#407	—	163	1	no	—
123	#405	—	163	1	no	—
124	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
125	C₂³×C₄	—	163	1	no	—
126	#131	—	2	4	no	—
127	C₂×M₁₆	—	163	1	no	—
128	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
129	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	complemented
130	C₂²×C₈	—	326	4	no	—
131	C₂²×C₈	—	163	1	no	—
132	C₂²×C₈	—	163	2	no	—
133	C₂²×C₈	—	163	2	no	—
134	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₁₆₃	1	2	yes	complemented
136	#123	—	163	2	no	—
137	C₂²⋊Q₈	—	163	2	no	—
138	#121	—	163	2	no	—
139	#120	—	163	2	no	—
140	C₄×D₄	—	163	4	no	—
141	#117	—	163	1	no	—
142	C₄⋊C₈	—	163	4	no	—
143	C₂²⋊C₈	—	163	4	no	—
144	C₂²⋊C₈	—	163	4	no	—
145	#97	—	163	2	no	—
146	C₄×C₈	—	163	2	no	—
147	C₂⁴	—	163	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₁₆₃	1	1	yes	characteristic
151	C₂×D₄	—	163	4	no	—
152	C₂×D₄	C₄×D₁₆₃	1	2	yes	—
153	C₂×D₄	C₄×D₁₆₃	1	1	yes	characteristic
154	C₂²×C₄	—	326	2	no	—
155	C₂²×C₄	—	326	2	no	—
156	C₂²×C₄	—	163	1	no	—
157	C₂²×C₄	—	163	2	no	—
158	C₂²×C₄	—	163	2	no	—
159	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	326	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	326	2	no	—
166	C₂×C₈	—	326	4	no	—
167	C₂×C₈	—	326	4	no	—
168	C₂×C₈	—	163	2	no	—
169	C₂×C₈	—	163	2	no	—
170	C₂×C₈	C₂²⋊D₁₆₃	1	4	yes	complemented
171	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
172	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
174	C₄⋊C₄	—	163	2	no	—
175	C₄⋊C₄	—	163	4	no	—
176	C₂²⋊C₄	—	163	2	no	—
177	C₂²⋊C₄	—	163	4	no	—
178	C₂²⋊C₄	—	163	4	no	—
179	C₄²	—	163	2	no	—
180	C₂³	—	326	4	no	—
181	C₂³	—	326	2	no	—
182	C₂³	—	163	2	no	—
183	C₂³	C₂×C₄×D₁₆₃	1	2	yes	—
184	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	326	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	326	2	no	—
192	C₂×C₄	—	326	2	no	—
193	C₂×C₄	—	326	2	no	—
194	C₂×C₄	—	326	4	no	—
195	C₂×C₄	—	163	4	no	—
196	C₂×C₄	—	163	2	no	—
197	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
200	C₂×C₄	#14579699	1	2	yes	—
201	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	326	2	no	—
204	C₈	—	326	2	no	—
205	C₈	C₂×C₂²⋊D₁₆₃	1	2	yes	—
206	C₈	C₂×C₂²⋊D₁₆₃	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	326	8	no	—
211	C₂²	—	326	2	no	—
212	C₂²	—	326	2	no	—
213	C₂²	—	163	4	no	—
214	C₂²	C₄×C₂²⋊D₁₆₃	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	326	4	no	—
218	C₄	—	326	2	no	—
219	C₄	C₄×C₂²⋊D₁₆₃	1	2	yes	—
220	C₄	C₂²×C₂²⋊D₁₆₃	1	1	yes	characteristic
221	C₄	C₂×#14579699	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	326	4	no	—
225	C₂	C₂×C₄×C₂²⋊D₁₆₃	1	1	yes	characteristic
226	C₂	#131×D₁₆₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	C₂×#131×C₁₆₃	C₂
prime radical	#23782172	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	C₂×#131	D₁₆₃
hypocenter	C₁₆₃	#2929
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₁₆₃	C₂	semidirect
3. #23781734	C₂	semidirect
4. C₈×C₂²⋊D₁₆₃	C₂	semidirect
5. #23781693	C₂	semidirect
6. #23781632	C₂	semidirect
7. #23781628	C₂	semidirect
8. #23781620	C₂	semidirect
9. C₂×#131×C₁₆₃	C₂	semidirect
22. Dic₁₆₃⋊C₈	C₂²	semidirect
23. C₂²⋊(C₁₆₃⋊C₈)	C₂²	semidirect
25. #19104645	C₂²	semidirect
26. C₈×Dic₁₆₃	C₂²	semidirect
30. C₂×M₁₆×C₁₆₃	C₂²	semidirect
32. C₂²×C₈×C₁₆₃	C₂²	semidirect
62. C₂×C₈×C₁₆₃	D₄	semidirect
101. C₂×D₁₆₃	#131	semidirect
113. C₁₆₃	#2929	semidirect
115. C₂×#131	D₁₆₃	semidirect
129. C₂×M₁₆	C₂×D₁₆₃	semidirect
135. C₂²×C₈	C₂×D₁₆₃	semidirect
170. C₂×C₈	C₂²⋊D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₁₆₃	2
#23782172	2
#23781734	2
C₈×C₂²⋊D₁₆₃	2
#23781693	2
#23781632	2
#23781628	2
#23781620	2
C₂×#131×C₁₆₃	2
#2929	163