Subgroups — #35223632

Order 20864.

322 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	#23782219	C₂	1	1	yes	maximal, characteristic
2	#23781795	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781793	C₂	1	2	yes	maximal, complemented
4	#23781792	C₂	1	2	yes	maximal, complemented
5	Q₈×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781789	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781727	C₂	1	2	yes	maximal
8	#23781725	C₂	1	2	yes	maximal
9	#23781721	C₂	1	1	yes	maximal, characteristic
10	#23781716	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19104734	C₄	1	1	yes	characteristic
13	#19104733	C₄	1	1	yes	characteristic
14	#19104725	C₄	1	2	yes	—
15	#19104715	C₄	1	1	yes	characteristic
16	#19104713	C₄	1	2	yes	—
17	#19104711	C₄	1	1	yes	characteristic
18	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
19	C₄×Dic₃₂₆	C₂²	1	2	yes	—
20	#19104692	C₂²	1	2	yes	—
21	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
22	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
24	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
27	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₁₆₃)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₆₃)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
31	#19104645	C₂²	1	2	yes	—
32	#19104645	C₂²	1	2	yes	—
33	C₄×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
34	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
35	C₈×Dic₁₆₃	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₆₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₆₃	C₂²	1	2	yes	—
39	#97×C₁₆₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₂₆	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₃₂₆	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₆₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₆₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₆₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
49	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
51	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₆₃	C₂³	1	2	yes	—
61	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₄²×C₁₆₃	C₂³	1	2	yes	—
70	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
71	Dic₃₂₆	—	2	2	no	—
72	Dic₃₂₆	—	4	2	no	—
73	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
78	C₄×D₁₆₃	—	2	2	no	—
79	C₄×D₁₆₃	—	2	1	no	—
80	C₄×D₁₆₃	—	2	1	no	—
81	C₄×D₁₆₃	—	4	1	no	—
82	C₁₆₃⋊C₈	—	2	1	no	—
83	C₁₆₃⋊C₈	—	2	1	no	—
84	C₁₆₃⋊C₈	—	2	2	no	—
85	Q₈×C₁₆₃	—	2	2	no	—
86	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₆₃	—	2	1	no	—
92	C₈×C₁₆₃	—	2	2	no	—
93	C₈×C₁₆₃	—	2	1	no	—
94	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₁₆₃	—	2	1	no	—
96	C₂×D₁₆₃	—	2	1	no	—
97	C₂×D₁₆₃	—	2	1	no	—
98	C₄×C₁₆₃	—	2	2	no	—
99	C₄×C₁₆₃	—	2	1	no	—
100	C₄×C₁₆₃	—	2	2	no	—
101	C₄×C₁₆₃	#131	1	2	yes	—
102	C₄×C₁₆₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₁₆₃	2₋¹⁺⁴	1	1	yes	characteristic
104	Dic₁₆₃	—	2	1	no	—
105	Dic₁₆₃	—	2	1	no	—
106	Dic₁₆₃	—	2	2	no	—
107	Dic₁₆₃	—	2	1	no	—
108	Dic₁₆₃	—	2	2	no	—
109	C₂×C₁₆₃	#519	1	1	yes	characteristic
110	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
111	C₂×C₁₆₃	#568	1	1	yes	characteristic
112	D₁₆₃	—	4	1	no	—
113	C₁₆₃	#2985	1	1	yes	characteristic, complemented, Sylow
114	#2985	—	163	1	no	maximal, Sylow
115	#533	—	163	1	no	—
116	#446	—	163	2	no	—
117	#446	—	163	2	no	—
118	#446	—	163	1	no	—
119	#446	D₁₆₃	1	1	yes	characteristic, complemented
120	C₈×Q₈	—	163	1	no	—
121	C₈×Q₈	—	163	1	no	—
122	#433	—	163	2	no	—
123	#433	—	163	2	no	—
124	#432	—	163	1	no	—
125	#431	—	163	1	no	—
126	C₄⋊Q₈	—	163	1	no	—
127	C₄⋊Q₈	—	163	1	no	—
128	#125	—	163	2	no	—
129	C₂²⋊Q₈	—	163	1	no	—
130	C₂²⋊Q₈	—	163	2	no	—
131	C₂²⋊Q₈	—	163	1	no	—
132	C₄×Q₈	—	163	1	no	—
133	C₄×Q₈	—	163	2	no	—
134	C₄×Q₈	C₂×D₁₆₃	1	1	yes	characteristic
135	#117	—	163	2	no	—
136	C₂×C₄²	—	163	1	no	—
137	C₄⋊C₈	—	163	1	no	—
138	C₄⋊C₈	—	163	1	no	—
139	C₄⋊C₈	—	163	2	no	—
140	C₄⋊C₈	—	163	2	no	—
141	C₄⋊C₈	—	163	1	no	—
142	C₄⋊C₈	—	163	2	no	—
143	C₄⋊C₈	C₂×D₁₆₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₁₆₃	1	2	yes	—
145	C₂²⋊C₈	—	163	1	no	—
146	C₂²⋊C₈	—	163	1	no	—
147	C₂²⋊C₈	—	163	2	no	—
148	#97	—	163	2	no	—
149	#97	—	163	2	no	—
150	#97	C₂×D₁₆₃	1	2	yes	—
151	C₄×C₈	—	163	2	no	—
152	C₄×C₈	—	163	1	no	—
153	C₄×C₈	—	163	2	no	—
154	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
155	C₂×Q₈	—	163	1	no	—
156	C₂×Q₈	—	163	2	no	—
157	C₂×Q₈	C₄×D₁₆₃	1	1	yes	characteristic
158	C₂²×C₄	—	163	1	no	—
159	C₂²×C₄	—	163	1	no	—
160	C₂²×C₄	—	163	1	no	—
161	C₂×C₈	—	163	1	no	—
162	C₂×C₈	—	163	1	no	—
163	C₂×C₈	—	163	2	no	—
164	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
165	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
166	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
167	C₄⋊C₄	—	163	2	no	—
168	C₄⋊C₄	—	163	2	no	—
169	C₄⋊C₄	—	163	2	no	—
170	C₄⋊C₄	—	163	2	no	—
171	C₄⋊C₄	—	163	2	no	—
172	C₄⋊C₄	—	163	2	no	—
173	C₄⋊C₄	—	163	1	no	—
174	C₄⋊C₄	C₄×D₁₆₃	1	2	yes	—
175	C₄⋊C₄	C₄×D₁₆₃	1	1	yes	characteristic
176	C₂²⋊C₄	—	163	2	no	—
177	C₂²⋊C₄	—	163	2	no	—
178	C₄²	—	163	1	no	—
179	C₄²	—	163	1	no	—
180	C₄²	—	163	2	no	—
181	C₄²	—	163	1	no	—
182	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
183	C₄²	C₂²×D₁₆₃	1	2	yes	—
184	C₂³	—	163	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	326	2	no	—
187	Q₈	—	652	2	no	—
188	C₂×C₄	—	326	1	no	—
189	C₂×C₄	—	326	1	no	—
190	C₂×C₄	—	326	2	no	—
191	C₂×C₄	—	652	1	no	—
192	C₂×C₄	—	163	1	no	—
193	C₂×C₄	—	163	1	no	—
194	C₂×C₄	—	163	1	no	—
195	C₂×C₄	—	163	2	no	—
196	C₂×C₄	—	163	2	no	—
197	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	326	1	no	—
206	C₈	—	326	1	no	—
207	C₈	—	326	2	no	—
208	C₂²	—	326	1	no	—
209	C₂²	—	326	1	no	—
210	C₂²	—	326	1	no	—
211	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	326	1	no	—
216	C₄	—	326	2	no	—
217	C₄	—	326	1	no	—
218	C₄	—	326	2	no	—
219	C₄	—	326	1	no	—
220	C₄	#19104743	1	2	yes	—
221	C₄	#19104782	1	1	yes	characteristic
222	C₄	#19104783	1	1	yes	characteristic
223	C₂	—	652	1	no	—
224	C₂	#23782141	1	1	yes	characteristic
225	C₂	C₂×#19104743	1	1	yes	characteristic
226	C₂	#23782430	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#446×C₁₆₃	C₂
prime radical	C₂²×D₁₆₃	C₂²×C₄
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#446	D₁₆₃
hypocenter	C₁₆₃	#2985
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #23781795	C₂	semidirect
3. #23781793	C₂	semidirect
4. #23781792	C₂	semidirect
5. Q₈×C₁₆₃⋊C₈	C₂	semidirect
6. #23781789	C₂	semidirect
11. #446×C₁₆₃	C₂	semidirect
113. C₁₆₃	#2985	semidirect
119. #446	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782219	2
#23781795	2
#23781793	2
#23781792	2
Q₈×C₁₆₃⋊C₈	2
#23781789	2
#23781727	2
#23781725	2
#23781721	2
#23781716	2
#446×C₁₆₃	2
#2985	163