Subgroups — #35223276

Order 20864.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104742	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19104741	C₂	1	1	yes	maximal, characteristic, complemented
3	#23782211	C₂	1	1	yes	maximal, characteristic
4	#23781740	C₂	1	2	yes	maximal, complemented
5	#23781739	C₂	1	2	yes	maximal, complemented
6	#23781733	C₂	1	2	yes	maximal, complemented
7	(C₂²⋊D₁₆₃)⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×Dic₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781633	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#408×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	#19104742	—	2	4	no	—
14	#19104741	—	2	4	no	—
15	C₂×#14579683	C₂²	1	2	yes	—
16	C₂×#14579683	C₂²	1	1	yes	characteristic
17	C₂×#14579682	C₂²	1	2	yes	—
18	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
20	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
21	#19104733	C₄	1	1	yes	characteristic
22	#19104731	C₄	1	1	yes	characteristic
23	#19104724	C₄	1	1	yes	characteristic
24	#19104722	C₄	1	1	yes	characteristic
25	#19104707	C₄	1	2	yes	—
26	#19104707	C₄	1	2	yes	—
27	C₄×C₂²⋊D₁₆₃	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₁₆₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
30	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
31	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
32	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
33	Dic₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
34	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
36	#19104645	C₂²	1	2	yes	—
37	C₈×Dic₁₆₃	C₂²	1	2	yes	—
38	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
40	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₆₃	C₂²	1	2	yes	—
42	C₂²⋊C₈×C₁₆₃	C₂²	1	2	yes	—
43	#14579699	—	2	2	no	—
44	#14579699	—	2	2	no	—
45	C₂×Dic₃₂₆	C₂×C₄	1	1	yes	characteristic
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	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
53	#14579683	—	2	4	no	—
54	#14579683	D₄	1	4	yes	complemented
55	#14579682	—	2	4	no	—
56	C₂×C₁₆₃⋊C₈	—	2	4	no	—
57	C₂×C₁₆₃⋊C₈	—	2	2	no	—
58	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
59	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
60	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
61	C₈×D₁₆₃	—	2	4	no	—
62	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
64	C₂²⋊Dic₁₆₃	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	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
72	#41×C₁₆₃	—	2	2	no	—
73	#41×C₁₆₃	—	2	2	no	—
74	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
75	C₂×D₄×C₁₆₃	C₂×C₄	1	2	yes	—
76	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₆₃	—	2	2	no	—
80	C₂×C₈×C₁₆₃	—	2	2	no	—
81	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
84	C₂²×D₁₆₃	C₂²×C₄	1	2	yes	—
85	Dic₃₂₆	—	2	2	no	—
86	D₆₅₂	—	2	2	no	—
87	C₂²⋊D₁₆₃	—	2	4	no	—
88	C₂²⋊D₁₆₃	—	2	4	no	—
89	C₂²⋊D₁₆₃	—	4	2	no	—
90	C₂×Dic₁₆₃	—	2	2	no	—
91	C₂×Dic₁₆₃	—	4	1	no	—
92	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×D₁₆₃	—	2	2	no	—
98	C₄×D₁₆₃	—	2	2	no	—
99	C₁₆₃⋊C₈	—	2	1	no	—
100	C₁₆₃⋊C₈	—	2	1	no	—
101	C₁₆₃⋊C₈	C₂×D₄	1	4	yes	—
102	C₂³×C₁₆₃	C₂²×C₄	1	2	yes	—
103	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
104	Q₈×C₁₆₃	—	2	2	no	—
105	D₄×C₁₆₃	—	2	4	no	—
106	D₄×C₁₆₃	—	2	2	no	—
107	C₂×C₄×C₁₆₃	—	2	2	no	—
108	C₂×C₄×C₁₆₃	—	2	2	no	—
109	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₆₃	#41	1	2	yes	—
113	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
114	C₈×C₁₆₃	—	2	1	no	—
115	C₈×C₁₆₃	—	2	1	no	—
116	C₈×C₁₆₃	—	2	2	no	—
117	C₂²×C₁₆₃	—	2	1	no	—
118	C₂²×C₁₆₃	—	2	1	no	—
119	C₂²×C₁₆₃	—	2	2	no	—
120	C₂²×C₁₆₃	—	2	2	no	—
121	C₂²×C₁₆₃	—	2	2	no	—
122	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
123	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₁₆₃	—	2	2	no	—
125	C₂×D₁₆₃	—	2	2	no	—
126	C₂×D₁₆₃	—	2	2	no	—
127	C₄×C₁₆₃	—	2	2	no	—
128	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
129	C₄×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
131	Dic₁₆₃	—	2	2	no	—
132	Dic₁₆₃	—	2	1	no	—
133	Dic₁₆₃	—	2	1	no	—
134	Dic₁₆₃	—	2	1	no	—
135	Dic₁₆₃	#131	1	2	yes	complemented
136	C₂×C₁₆₃	—	2	2	no	—
137	C₂×C₁₆₃	—	4	2	no	—
138	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₁₆₃	C₂×#131	1	1	yes	characteristic
140	C₂×C₁₆₃	#568	1	1	yes	characteristic
141	D₁₆₃	—	4	2	no	—
142	C₁₆₃	#2935	1	1	yes	characteristic, complemented, Sylow
143	#2935	—	163	1	no	maximal, Sylow
144	C₂×#131	—	163	1	no	—
145	C₂×#131	—	163	1	no	—
146	#532	—	163	1	no	—
147	#436	—	163	2	no	—
148	#435	—	163	2	no	—
149	#435	—	163	2	no	—
150	C₈×D₄	—	163	2	no	—
151	C₂×C₄⋊C₈	—	163	1	no	—
152	#408	—	163	1	no	—
153	#408	D₁₆₃	1	1	yes	characteristic, complemented
154	C₄×M₁₆	—	163	1	no	—
155	C₂×#41	—	163	1	no	—
156	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
157	#131	—	326	4	no	—
158	#131	—	326	4	no	—
159	C₂×M₁₆	—	163	2	no	—
160	C₂×M₁₆	—	163	1	no	—
161	C₂×M₁₆	—	163	2	no	—
162	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
163	C₂²×C₈	—	163	1	no	—
164	C₂²×C₈	—	163	2	no	—
165	C₂²×C₈	—	163	2	no	—
166	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
167	C₄⋊Q₈	—	163	1	no	—
168	#127	—	163	1	no	—
169	#124	—	163	1	no	—
170	#124	—	163	1	no	—
171	#121	—	163	2	no	—
172	#121	—	163	2	no	—
173	C₄×D₄	—	163	2	no	—
174	C₄×D₄	—	163	2	no	—
175	C₂×C₄²	—	163	1	no	—
176	C₄⋊C₈	—	163	1	no	—
177	C₄⋊C₈	—	163	1	no	—
178	C₄⋊C₈	—	163	1	no	—
179	C₄⋊C₈	—	163	1	no	—
180	C₂²⋊C₈	—	163	2	no	—
181	C₂²⋊C₈	—	163	2	no	—
182	C₂²⋊C₈	C₂×D₁₆₃	1	2	yes	—
183	C₂²⋊C₈	C₂×D₁₆₃	1	2	yes	—
184	#97	—	163	2	no	—
185	C₄×C₈	—	163	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	326	2	no	—
189	#41	—	326	2	no	—
190	C₂×Q₈	—	163	1	no	—
191	C₂×Q₈	C₄×D₁₆₃	1	1	yes	characteristic
192	C₂×D₄	—	163	2	no	—
193	C₂×D₄	—	163	2	no	—
194	C₂×D₄	—	163	2	no	—
195	C₂×D₄	—	163	1	no	—
196	C₂×D₄	C₄×D₁₆₃	1	2	yes	—
197	C₂×D₄	C₄×D₁₆₃	1	1	yes	characteristic
198	C₂²×C₄	—	163	1	no	—
199	C₂²×C₄	—	163	2	no	—
200	C₂²×C₄	—	163	1	no	—
201	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	326	4	no	—
205	M₁₆	—	326	4	no	—
206	M₁₆	—	163	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	326	4	no	—
209	C₂×C₈	—	326	2	no	—
210	C₂×C₈	—	326	4	no	—
211	C₂×C₈	—	163	1	no	—
212	C₂×C₈	—	163	1	no	—
213	C₂×C₈	—	163	2	no	—
214	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
215	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
217	C₄⋊C₄	—	163	2	no	—
218	C₄⋊C₄	—	163	2	no	—
219	C₂²⋊C₄	—	163	2	no	—
220	C₂²⋊C₄	—	163	2	no	—
221	C₂²⋊C₄	—	163	2	no	—
222	C₂²⋊C₄	—	163	2	no	—
223	C₄²	—	163	1	no	—
224	C₄²	—	163	1	no	—
225	C₄²	—	163	1	no	—
226	C₄²	—	163	1	no	—
227	C₂³	—	163	2	no	—
228	C₂³	C₂×C₄×D₁₆₃	1	2	yes	—
229	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	326	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	326	4	no	—
235	D₄	—	326	4	no	—
236	D₄	—	326	2	no	—
237	D₄	—	652	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	326	2	no	—
241	C₂×C₄	—	326	2	no	—
242	C₂×C₄	—	326	2	no	—
243	C₂×C₄	—	652	1	no	—
244	C₂×C₄	—	163	1	no	—
245	C₂×C₄	—	163	1	no	—
246	C₂×C₄	—	163	1	no	—
247	C₂×C₄	—	163	2	no	—
248	C₂×C₄	—	163	1	no	—
249	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
251	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
252	C₂×C₄	#14579698	1	2	yes	—
253	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	2	no	—
257	C₈	—	326	1	no	—
258	C₈	—	326	1	no	—
259	C₈	—	163	4	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	326	2	no	—
266	C₂²	—	326	2	no	—
267	C₂²	—	326	2	no	—
268	C₂²	(C₂²⋊D₁₆₃)⋊C₄	1	2	yes	—
269	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	326	1	no	—
272	C₄	—	326	1	no	—
273	C₄	—	326	1	no	—
274	C₄	—	326	2	no	—
275	C₄	—	163	2	no	—
276	C₄	(C₂²⋊D₁₆₃)⋊C₄	1	2	yes	—
277	C₄	C₂×D₄×D₁₆₃	1	1	yes	characteristic
278	C₄	C₂×#14579698	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	652	2	no	—
282	C₂	C₂×(C₂²⋊D₁₆₃)⋊C₄	1	1	yes	characteristic
283	C₂	#131×D₁₆₃	1	1	yes	characteristic
284	C₂	#23782430	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#408×C₁₆₃	C₂
prime radical	#23782211	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#408	D₁₆₃
hypocenter	C₁₆₃	#2935
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#19104742	C₂	semidirect
2. C₂×#19104741	C₂	semidirect
4. #23781740	C₂	semidirect
5. #23781739	C₂	semidirect
6. #23781733	C₂	semidirect
7. (C₂²⋊D₁₆₃)⋊C₈	C₂	semidirect
8. C₂×Dic₁₆₃⋊C₈	C₂	semidirect
9. #23781633	C₂	semidirect
10. M₁₆×Dic₁₆₃	C₂	semidirect
11. #408×C₁₆₃	C₂	semidirect
54. #14579683	D₄	semidirect
135. Dic₁₆₃	#131	semidirect
142. C₁₆₃	#2935	semidirect
153. #408	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104742	2
C₂×#19104741	2
#23782211	2
#23781740	2
#23781739	2
#23781733	2
(C₂²⋊D₁₆₃)⋊C₈	2
C₂×Dic₁₆₃⋊C₈	2
#23781633	2
M₁₆×Dic₁₆₃	2
#408×C₁₆₃	2
#2935	163