Subgroups — #35223232

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	C₂×#19104741	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#23782177	C₂	1	1	yes	maximal, characteristic
4	#23781738	C₂	1	4	yes	maximal, complemented
5	C₈×D₆₅₂	C₂	1	4	yes	maximal, complemented
6	#23781692	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781633	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781628	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579699	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₆₃	C₂²	1	1	yes	characteristic
12	#19104741	—	2	4	no	—
13	C₂×#14579683	C₂²	1	1	yes	characteristic
14	C₂×#14579682	C₂²	1	2	yes	complemented
15	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₆₃	—	2	4	no	—
17	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
18	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
19	C₂×C₈×D₁₆₃	C₂²	1	2	yes	complemented
20	#19104719	C₄	1	2	yes	—
21	#19104715	C₄	1	2	yes	—
22	#19104710	C₄	1	2	yes	—
23	#19104705	C₄	1	2	yes	—
24	C₄×D₆₅₂	C₂²	1	4	yes	—
25	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	4	yes	complemented
26	(C₂×D₁₆₃)⋊C₈	C₂²	1	4	yes	—
27	(C₂×D₁₆₃)⋊C₈	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
30	#117×C₁₆₃	C₂²	1	1	yes	characteristic
31	#97×C₁₆₃	C₂²	1	2	yes	complemented
32	C₄×C₈×C₁₆₃	C₂²	1	2	yes	complemented
33	C₂³×D₁₆₃	C₂×C₄	1	1	yes	characteristic
34	#14579699	—	2	2	no	—
35	#14579699	—	2	2	no	—
36	C₂×Dic₃₂₆	C₂×C₄	1	1	yes	characteristic
37	C₂×D₆₅₂	C₂×C₄	1	1	yes	characteristic
38	C₂×D₆₅₂	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₆₃	—	2	2	no	—
42	C₂×C₄×D₁₆₃	—	2	2	no	—
43	C₂×C₄×D₁₆₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
45	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
46	#14579683	—	2	2	no	—
47	#14579682	—	2	4	no	—
48	C₂×C₁₆₃⋊C₈	—	2	2	no	—
49	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
50	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
51	C₈×D₁₆₃	—	2	4	no	—
52	C₈×D₁₆₃	—	2	4	no	—
53	C₈×D₁₆₃	—	2	4	no	—
54	C₄⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
55	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	4	yes	—
56	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₆₃	—	2	2	no	—
59	C₂×C₈×C₁₆₃	D₄	1	4	yes	complemented
60	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
61	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
65	C₄²×C₁₆₃	C₂³	1	2	yes	—
66	C₂²×D₁₆₃	—	2	2	no	—
67	C₂²×D₁₆₃	—	2	4	no	—
68	C₂²×D₁₆₃	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₆₃	C₂²×C₄	1	2	yes	—
70	Dic₃₂₆	—	2	2	no	—
71	D₆₅₂	—	2	8	no	—
72	D₆₅₂	—	2	2	no	—
73	C₂²⋊D₁₆₃	—	2	4	no	—
74	C₂×Dic₁₆₃	—	2	2	no	—
75	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
77	C₄×D₁₆₃	—	2	4	no	—
78	C₄×D₁₆₃	—	2	2	no	—
79	C₄×D₁₆₃	—	2	2	no	—
80	C₄×D₁₆₃	—	2	2	no	—
81	C₄×D₁₆₃	—	2	2	no	—
82	C₁₆₃⋊C₈	—	2	2	no	—
83	C₁₆₃⋊C₈	—	2	2	no	—
84	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₆₃	C₂²×C₄	1	4	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	C₄×D₄	1	2	yes	—
95	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₆₃	—	2	8	no	—
97	C₂×D₁₆₃	—	2	2	no	—
98	C₂×D₁₆₃	—	2	2	no	—
99	C₂×D₁₆₃	—	2	2	no	—
100	C₂×D₁₆₃	—	2	4	no	—
101	C₂×D₁₆₃	#131	1	4	yes	complemented
102	C₄×C₁₆₃	—	2	4	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	2	no	—
108	C₂×C₁₆₃	—	2	2	no	—
109	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂×C₁₆₃	C₂×#131	1	2	yes	—
111	D₁₆₃	—	2	4	no	—
112	D₁₆₃	—	4	2	no	—
113	C₁₆₃	#2929	1	1	yes	characteristic, complemented, Sylow
114	#2929	—	163	1	no	maximal, Sylow
115	C₂×#131	—	163	1	no	—
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	D₁₆₃	1	1	yes	characteristic, complemented
124	C₂×#41	—	163	1	no	—
125	C₂³×C₄	—	163	1	no	—
126	#131	—	326	4	no	—
127	C₂×M₁₆	—	163	1	no	—
128	C₂×M₁₆	—	163	2	no	—
129	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
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	—	163	2	no	—
135	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
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	C₂×D₁₆₃	1	1	yes	characteristic
142	C₄⋊C₈	—	163	4	no	—
143	C₂²⋊C₈	—	163	4	no	—
144	C₂²⋊C₈	—	163	4	no	—
145	#97	C₂×D₁₆₃	1	2	yes	complemented
146	C₄×C₈	C₂×D₁₆₃	1	2	yes	complemented
147	C₂⁴	—	163	1	no	—
148	#41	—	326	2	no	—
149	#41	—	326	2	no	—
150	C₂×Q₈	—	163	1	no	—
151	C₂×D₄	—	163	2	no	—
152	C₂×D₄	—	163	4	no	—
153	C₂×D₄	—	163	1	no	—
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	—	163	2	no	—
160	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	326	2	no	—
163	M₁₆	—	326	4	no	—
164	C₂×C₈	—	326	2	no	—
165	C₂×C₈	—	326	4	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	D₆₅₂	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	4	no	—
175	C₄⋊C₄	C₄×D₁₆₃	1	2	yes	—
176	C₂²⋊C₄	—	163	4	no	—
177	C₂²⋊C₄	—	163	4	no	—
178	C₂²⋊C₄	C₄×D₁₆₃	1	2	yes	—
179	C₄²	C₂²×D₁₆₃	1	2	yes	—
180	C₂³	—	326	4	no	—
181	C₂³	—	326	2	no	—
182	C₂³	—	163	2	no	—
183	C₂³	—	163	2	no	—
184	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
185	Q₈	—	326	2	no	—
186	D₄	—	326	8	no	—
187	D₄	—	326	4	no	—
188	D₄	—	326	2	no	—
189	C₂×C₄	—	326	2	no	—
190	C₂×C₄	—	326	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	2	no	—
196	C₂×C₄	—	163	2	no	—
197	C₂×C₄	C₂×C₄×D₁₆₃	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₆₅₂	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×D₆₅₂	1	2	yes	—
206	C₈	C₂×D₆₅₂	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	326	8	no	—
209	C₂²	—	326	2	no	—
210	C₂²	—	326	2	no	—
211	C₂²	—	326	4	no	—
212	C₂²	—	326	2	no	—
213	C₂²	—	163	4	no	—
214	C₂²	C₄×D₆₅₂	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	326	2	no	—
218	C₄	—	326	2	no	—
219	C₄	C₄×D₆₅₂	1	2	yes	—
220	C₄	C₂²×D₆₅₂	1	1	yes	characteristic
221	C₄	C₂×#14579699	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	326	4	no	—
224	C₂	—	652	2	no	—
225	C₂	C₂×C₄×D₆₅₂	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	#405×C₁₆₃	C₂
prime radical	#23782177	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#405	D₁₆₃
hypocenter	C₁₆₃	#2929
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#19104741	C₂	semidirect
2. C₂²×C₈×D₁₆₃	C₂	semidirect
4. #23781738	C₂	semidirect
5. C₈×D₆₅₂	C₂	semidirect
6. #23781692	C₂	semidirect
7. #23781633	C₂	semidirect
8. #23781628	C₂	semidirect
9. #405×C₁₆₃	C₂	semidirect
14. C₂×#14579682	C₂²	semidirect
19. C₂×C₈×D₁₆₃	C₂²	semidirect
25. C₄⋊(C₁₆₃⋊C₈)	C₂²	semidirect
27. (C₂×D₁₆₃)⋊C₈	C₂²	semidirect
31. #97×C₁₆₃	C₂²	semidirect
32. C₄×C₈×C₁₆₃	C₂²	semidirect
59. C₂×C₈×C₁₆₃	D₄	semidirect
101. C₂×D₁₆₃	#131	semidirect
113. C₁₆₃	#2929	semidirect
123. #405	D₁₆₃	semidirect
145. #97	C₂×D₁₆₃	semidirect
146. C₄×C₈	C₂×D₁₆₃	semidirect
170. C₂×C₈	D₆₅₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104741	2
C₂²×C₈×D₁₆₃	2
#23782177	2
#23781738	2
C₈×D₆₅₂	2
#23781692	2
#23781633	2
#23781628	2
#405×C₁₆₃	2
#2929	163