Subgroups — #35223699

Order 20864.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782202	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781742	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781738	C₂	1	2	yes	maximal, complemented
5	C₈×D₆₅₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781730	C₂	1	2	yes	maximal, complemented
7	#23781727	C₂	1	2	yes	maximal, complemented
8	#23781716	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781690	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₈×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#446×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579682	C₂²	1	2	yes	—
13	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
14	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
15	#19104730	C₄	1	2	yes	—
16	#19104726	C₄	1	1	yes	characteristic
17	#19104724	C₄	1	1	yes	characteristic
18	#19104720	C₄	1	2	yes	—
19	#19104711	C₄	1	1	yes	characteristic
20	#19104710	C₄	1	1	yes	characteristic
21	C₄×D₆₅₂	C₂²	1	2	yes	—
22	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
23	#19104692	C₂²	1	2	yes	—
24	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
27	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
28	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
29	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₆₃)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₆₃)⋊C₈	C₂²	1	1	yes	characteristic
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	C₄×C₁₆₃⋊C₈	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×D₆₅₂	C₂×C₄	1	2	yes	—
42	C₂×D₆₅₂	C₂×C₄	1	1	yes	characteristic
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×C₄×D₁₆₃	C₂³	1	2	yes	—
47	#14579682	—	4	2	no	—
48	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
50	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
51	C₈×D₁₆₃	—	4	2	no	—
52	C₈×D₁₆₃	—	4	2	no	—
53	C₄⋊Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
55	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
61	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
71	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₆₃	C₂³	1	2	yes	—
73	C₂²×D₁₆₃	C₂²×C₄	1	2	yes	—
74	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
75	D₆₅₂	—	2	4	no	—
76	D₆₅₂	—	2	2	no	—
77	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
81	C₄×D₁₆₃	—	2	2	no	—
82	C₄×D₁₆₃	—	2	2	no	—
83	C₄×D₁₆₃	—	2	1	no	—
84	C₄×D₁₆₃	—	2	2	no	—
85	C₄×D₁₆₃	—	2	1	no	—
86	C₄×D₁₆₃	—	2	2	no	—
87	C₁₆₃⋊C₈	—	2	2	no	—
88	C₁₆₃⋊C₈	—	2	1	no	—
89	C₁₆₃⋊C₈	—	2	1	no	—
90	Q₈×C₁₆₃	—	2	2	no	—
91	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₆₃	—	2	1	no	—
97	C₈×C₁₆₃	—	2	2	no	—
98	C₈×C₁₆₃	—	2	1	no	—
99	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₁₆₃	—	2	1	no	—
101	C₂×D₁₆₃	—	2	1	no	—
102	C₂×D₁₆₃	—	2	2	no	—
103	C₂×D₁₆₃	—	2	2	no	—
104	C₂×D₁₆₃	—	2	2	no	—
105	C₂×D₁₆₃	#131	1	2	yes	—
106	C₄×C₁₆₃	—	2	2	no	—
107	C₄×C₁₆₃	—	2	1	no	—
108	C₄×C₁₆₃	—	2	2	no	—
109	C₄×C₁₆₃	#131	1	2	yes	—
110	C₄×C₁₆₃	H(5,𝔽₂)	1	1	yes	characteristic
111	C₄×C₁₆₃	2₋¹⁺⁴	1	1	yes	characteristic
112	Dic₁₆₃	—	2	1	no	—
113	Dic₁₆₃	—	2	2	no	—
114	Dic₁₆₃	—	2	1	no	—
115	Dic₁₆₃	—	2	1	no	—
116	C₂×C₁₆₃	#520	1	1	yes	characteristic
117	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
118	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
119	D₁₆₃	—	2	2	no	—
120	D₁₆₃	—	4	2	no	—
121	C₁₆₃	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	163	1	no	maximal, Sylow
123	#529	—	163	1	no	—
124	#446	D₁₆₃	1	1	yes	characteristic, complemented
125	C₈×Q₈	—	163	1	no	—
126	#436	—	163	1	no	—
127	#435	—	163	2	no	—
128	C₈×D₄	—	163	2	no	—
129	C₈×D₄	—	163	1	no	—
130	#433	—	163	2	no	—
131	#431	—	163	1	no	—
132	#424	—	163	2	no	—
133	C₂×C₄⋊C₈	—	163	1	no	—
134	C₂×M₁₆	—	163	2	no	—
135	C₂²×C₈	—	163	2	no	—
136	C₂²×C₈	—	163	2	no	—
137	#126	—	163	2	no	—
138	#125	—	163	1	no	—
139	#124	—	163	1	no	—
140	#123	—	163	2	no	—
141	C₂²⋊Q₈	—	163	1	no	—
142	#121	—	163	1	no	—
143	C₄×Q₈	C₂×D₁₆₃	1	1	yes	characteristic
144	C₄×D₄	—	163	2	no	—
145	C₄×D₄	—	163	1	no	—
146	#117	—	163	2	no	—
147	C₂×C₄²	—	163	1	no	—
148	C₄⋊C₈	—	163	2	no	—
149	C₄⋊C₈	—	163	1	no	—
150	C₄⋊C₈	—	163	1	no	—
151	C₄⋊C₈	—	163	1	no	—
152	C₄⋊C₈	—	163	2	no	—
153	C₄⋊C₈	C₂×D₁₆₃	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₁₆₃	1	2	yes	—
155	C₂²⋊C₈	—	163	1	no	—
156	C₂²⋊C₈	—	163	1	no	—
157	C₂²⋊C₈	—	163	2	no	—
158	C₂²⋊C₈	—	163	2	no	—
159	#97	C₂×D₁₆₃	1	2	yes	—
160	C₄×C₈	—	163	1	no	—
161	C₄×C₈	—	163	2	no	—
162	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
163	C₂×Q₈	C₄×D₁₆₃	1	1	yes	characteristic
164	C₂×D₄	—	163	2	no	—
165	C₂×D₄	—	163	1	no	—
166	C₂²×C₄	—	163	1	no	—
167	C₂²×C₄	—	163	1	no	—
168	C₂²×C₄	—	163	1	no	—
169	C₂²×C₄	—	163	2	no	—
170	M₁₆	—	652	2	no	—
171	C₂×C₈	—	652	2	no	—
172	C₂×C₈	—	652	2	no	—
173	C₂×C₈	—	163	1	no	—
174	C₂×C₈	—	163	2	no	—
175	C₂×C₈	—	163	1	no	—
176	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
179	C₄⋊C₄	—	163	2	no	—
180	C₄⋊C₄	—	163	2	no	—
181	C₄⋊C₄	—	163	1	no	—
182	C₄⋊C₄	—	163	2	no	—
183	C₄⋊C₄	C₄×D₁₆₃	1	1	yes	characteristic
184	C₄⋊C₄	C₄×D₁₆₃	1	2	yes	—
185	C₂²⋊C₄	—	163	2	no	—
186	C₂²⋊C₄	—	163	2	no	—
187	C₂²⋊C₄	—	163	2	no	—
188	C₂²⋊C₄	—	163	2	no	—
189	C₂²⋊C₄	—	163	2	no	—
190	C₄²	—	163	1	no	—
191	C₄²	—	163	1	no	—
192	C₄²	—	163	1	no	—
193	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
194	C₄²	C₂²×D₁₆₃	1	2	yes	—
195	C₂³	—	163	2	no	—
196	C₂³	—	163	1	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	326	4	no	—
199	D₄	—	326	2	no	—
200	C₂×C₄	—	326	2	no	—
201	C₂×C₄	—	326	1	no	—
202	C₂×C₄	—	326	2	no	—
203	C₂×C₄	—	326	1	no	—
204	C₂×C₄	—	326	2	no	—
205	C₂×C₄	—	326	2	no	—
206	C₂×C₄	—	163	1	no	—
207	C₂×C₄	—	163	1	no	—
208	C₂×C₄	—	163	1	no	—
209	C₂×C₄	—	163	2	no	—
210	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	326	2	no	—
219	C₈	—	326	1	no	—
220	C₈	—	326	1	no	—
221	C₂²	—	326	1	no	—
222	C₂²	—	326	1	no	—
223	C₂²	—	326	2	no	—
224	C₂²	—	326	2	no	—
225	C₂²	—	326	2	no	—
226	C₂²	—	163	2	no	—
227	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	326	1	no	—
232	C₄	—	326	2	no	—
233	C₄	—	326	1	no	—
234	C₄	—	326	1	no	—
235	C₄	#19104743	1	2	yes	—
236	C₄	#19104781	1	1	yes	characteristic
237	C₄	#19104783	1	1	yes	characteristic
238	C₂	—	326	2	no	—
239	C₂	—	652	2	no	—
240	C₂	#23782147	1	1	yes	characteristic
241	C₂	#131×D₁₆₃	1	1	yes	characteristic
242	C₂	C₂×#19104743	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	#19104710	C₄
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#446	D₁₆₃
hypocenter	C₁₆₃	#2994
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. Q₈×C₁₆₃⋊C₈	C₂	semidirect
3. #23781742	C₂	semidirect
4. #23781738	C₂	semidirect
5. C₈×D₆₅₂	C₂	semidirect
6. #23781730	C₂	semidirect
7. #23781727	C₂	semidirect
8. #23781716	C₂	semidirect
9. #23781690	C₂	semidirect
10. C₄⋊C₈×D₁₆₃	C₂	semidirect
11. #446×C₁₆₃	C₂	semidirect
121. C₁₆₃	#2994	semidirect
124. #446	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782202	2
Q₈×C₁₆₃⋊C₈	2
#23781742	2
#23781738	2
C₈×D₆₅₂	2
#23781730	2
#23781727	2
#23781716	2
#23781690	2
C₄⋊C₈×D₁₆₃	2
#446×C₁₆₃	2
#2994	163