Subgroups — #35224445

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104694	C₂	1	1	yes	maximal, characteristic, complemented
2	#23782003	C₂	1	2	yes	maximal, complemented
3	#23781998	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781993	C₂	1	1	yes	maximal, characteristic, complemented
5	#23781986	C₂	1	2	yes	maximal, complemented
6	#23781981	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781976	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781701	C₂	1	2	yes	maximal, complemented
9	#23781661	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₁₆₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₁₆₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
13	#19104694	—	2	1	no	—
14	#19104694	—	2	1	no	—
15	#19104694	C₂²	1	2	yes	—
16	#19104694	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₆₃	—	2	1	no	—
20	C₂×C₄×Dic₁₆₃	—	2	1	no	—
21	#19104668	C₂²	1	2	yes	complemented
22	#19104668	C₂²	1	2	yes	complemented
23	#19104666	C₂²	1	2	yes	complemented
24	#19104666	C₂²	1	2	yes	complemented
25	#19104659	C₂²	1	2	yes	complemented
26	#19104659	C₂²	1	2	yes	complemented
27	#19104656	C₂²	1	2	yes	complemented
28	#19104656	C₂²	1	2	yes	complemented
29	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
32	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
34	#123×C₁₆₃	—	2	1	no	—
35	C₂²⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
37	#121×C₁₆₃	C₂²	1	1	yes	characteristic
38	#121×C₁₆₃	C₂²	1	1	yes	characteristic
39	#120×C₁₆₃	—	2	1	no	—
40	C₄×D₄×C₁₆₃	—	2	1	no	—
41	C₄×D₄×C₁₆₃	—	2	1	no	—
42	#117×C₁₆₃	C₂²	1	1	yes	characteristic
43	C₂²×Dic₁₆₃	—	2	1	no	—
44	C₂²×Dic₁₆₃	—	2	1	no	—
45	C₂×C₁₆₃⋊C₈	—	2	4	no	—
46	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
47	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₁₆₃	—	2	1	no	—
49	C₄⋊Dic₁₆₃	—	2	1	no	—
50	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
51	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
52	C₂²⋊Dic₁₆₃	—	2	2	no	—
53	C₂²⋊Dic₁₆₃	—	2	2	no	—
54	C₄×Dic₁₆₃	—	2	2	no	—
55	C₄×Dic₁₆₃	—	2	2	no	—
56	C₂⁴×C₁₆₃	D₄	1	1	yes	characteristic
57	#41×C₁₆₃	—	2	4	no	—
58	C₂×Q₈×C₁₆₃	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₆₃	—	2	1	no	—
60	C₂×D₄×C₁₆₃	—	2	1	no	—
61	C₂×D₄×C₁₆₃	—	2	1	no	—
62	C₂×D₄×C₁₆₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₆₃	—	2	4	no	—
64	C₂²×C₄×C₁₆₃	—	2	1	no	—
65	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₆₃	—	2	1	no	—
72	C₄⋊C₄×C₁₆₃	—	2	1	no	—
73	C₄⋊C₄×C₁₆₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₆₃	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
76	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
77	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
78	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
79	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
80	C₄²×C₁₆₃	—	2	1	no	—
81	C₂×Dic₁₆₃	—	2	2	no	—
82	C₂×Dic₁₆₃	—	2	2	no	—
83	C₂×Dic₁₆₃	—	4	2	no	—
84	C₂×Dic₁₆₃	—	4	2	no	—
85	C₁₆₃⋊C₈	—	2	2	no	—
86	C₁₆₃⋊C₈	—	2	2	no	—
87	C₂³×C₁₆₃	—	2	2	no	—
88	C₂³×C₁₆₃	—	2	2	no	—
89	C₂³×C₁₆₃	—	2	1	no	—
90	C₂³×C₁₆₃	—	4	1	no	—
91	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
93	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₁₆₃	—	2	2	no	—
95	D₄×C₁₆₃	—	2	2	no	—
96	D₄×C₁₆₃	—	4	2	no	—
97	D₄×C₁₆₃	—	4	2	no	—
98	D₄×C₁₆₃	—	4	2	no	—
99	C₂×C₄×C₁₆₃	—	2	2	no	—
100	C₂×C₄×C₁₆₃	—	2	2	no	—
101	C₂×C₄×C₁₆₃	—	2	2	no	—
102	C₂×C₄×C₁₆₃	—	2	2	no	—
103	C₂×C₄×C₁₆₃	—	2	2	no	—
104	C₂×C₄×C₁₆₃	—	2	1	no	—
105	C₂×C₄×C₁₆₃	—	2	1	no	—
106	C₂×C₄×C₁₆₃	—	2	1	no	—
107	C₂×C₄×C₁₆₃	—	4	2	no	—
108	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₆₃	#41	1	4	yes	—
112	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
113	C₂²×C₁₆₃	—	2	1	no	—
114	C₂²×C₁₆₃	—	2	1	no	—
115	C₂²×C₁₆₃	—	2	1	no	—
116	C₂²×C₁₆₃	—	2	1	no	—
117	C₂²×C₁₆₃	—	2	1	no	—
118	C₂²×C₁₆₃	—	2	1	no	—
119	C₂²×C₁₆₃	—	2	1	no	—
120	C₂²×C₁₆₃	—	4	4	no	—
121	C₂²×C₁₆₃	—	4	1	no	—
122	C₂²×C₁₆₃	—	4	1	no	—
123	C₂²×C₁₆₃	—	4	1	no	—
124	C₂²×C₁₆₃	#123	1	2	yes	—
125	C₂²×C₁₆₃	#135	1	2	yes	—
126	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₆₃	—	2	1	no	—
128	C₄×C₁₆₃	—	2	1	no	—
129	C₄×C₁₆₃	—	4	1	no	—
130	C₄×C₁₆₃	—	4	1	no	—
131	C₄×C₁₆₃	—	4	1	no	—
132	C₄×C₁₆₃	#123	1	2	yes	—
133	C₄×C₁₆₃	C₂×#41	1	2	yes	—
134	Dic₁₆₃	—	4	2	no	—
135	Dic₁₆₃	—	4	2	no	—
136	C₂×C₁₆₃	—	2	2	no	—
137	C₂×C₁₆₃	—	2	2	no	—
138	C₂×C₁₆₃	—	4	1	no	—
139	C₂×C₁₆₃	—	8	1	no	—
140	C₂×C₁₆₃	C₂×#123	1	1	yes	characteristic
141	C₂×C₁₆₃	C₂×#135	1	1	yes	characteristic
142	C₂×C₁₆₃	#575	1	1	yes	characteristic
143	C₁₆₃	#3091	1	1	yes	characteristic, complemented, Sylow
144	#3091	—	163	1	no	maximal, Sylow
145	#525	D₁₆₃	1	1	yes	characteristic
146	C₂×#117	—	163	1	no	—
147	#485	—	163	2	no	—
148	#484	—	163	1	no	—
149	#483	—	163	1	no	—
150	#482	—	163	2	no	—
151	#481	—	163	1	no	—
152	#480	—	163	1	no	—
153	#427	—	163	2	no	—
154	#416	—	163	2	no	—
155	C₂×C₂²⋊C₈	—	163	1	no	—
156	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
157	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
158	C₂²×C₈	—	163	2	no	—
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
161	C₂²⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
162	#121	C₂×D₁₆₃	1	1	yes	characteristic
163	#121	C₂×D₁₆₃	1	1	yes	characteristic
164	#120	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	#117	—	326	1	no	—
168	#117	—	326	1	no	—
169	#117	—	163	2	no	—
170	#117	—	163	2	no	—
171	#117	C₂×D₁₆₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	163	1	no	—
173	C₂×C₄⋊C₄	—	163	1	no	—
174	C₂×C₂²⋊C₄	—	326	1	no	—
175	C₂×C₄²	—	326	1	no	—
176	#107	—	163	2	no	—
177	#107	—	163	2	no	—
178	#106	—	163	2	no	—
179	#106	—	163	2	no	—
180	#103	—	163	2	no	—
181	#103	—	163	2	no	—
182	#102	—	163	2	no	—
183	#102	—	163	2	no	—
184	C₂²⋊C₈	—	163	1	no	—
185	C₂²⋊C₈	—	163	1	no	—
186	C₂²⋊C₈	—	163	2	no	—
187	C₂⁴	C₂²⋊D₁₆₃	1	1	yes	characteristic
188	#41	—	2	4	no	—
189	C₂×Q₈	C₂²×D₁₆₃	1	1	yes	characteristic
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂²×D₁₆₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	326	1	no	—
197	C₂²×C₄	—	326	1	no	—
198	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
204	C₂×C₈	—	326	4	no	—
205	C₂×C₈	—	163	2	no	—
206	C₂×C₈	—	163	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	326	1	no	—
210	C₄⋊C₄	—	326	1	no	—
211	C₄⋊C₄	—	163	2	no	—
212	C₄⋊C₄	—	163	2	no	—
213	C₄⋊C₄	C₂²×D₁₆₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂²×D₁₆₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	326	2	no	—
221	C₂²⋊C₄	—	326	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	326	2	no	—
224	C₄²	—	326	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
230	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
231	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	326	2	no	—
247	C₂×C₄	—	326	2	no	—
248	C₂×C₄	—	652	2	no	—
249	C₂×C₄	—	652	2	no	—
250	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
253	C₂×C₄	#14579698	1	4	yes	—
254	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
255	C₈	—	326	2	no	—
256	C₈	—	326	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#19104716	1	2	yes	—
269	C₂²	#19104755	1	2	yes	—
270	C₂²	C₂²×C₂²⋊D₁₆₃	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	652	2	no	—
277	C₄	—	652	2	no	—
278	C₄	#19104716	1	2	yes	—
279	C₄	C₂×#14579698	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×#19104716	1	1	yes	characteristic
285	C₂	C₂×#19104755	1	1	yes	characteristic
286	C₂	#23782463	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₆₃
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting, prime radical	#525×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
hypercenter	#525	D₁₆₃
hypocenter	C₁₆₃	#3091
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#19104694	C₂	semidirect
2. #23782003	C₂	semidirect
3. #23781998	C₂	semidirect
4. #23781993	C₂	semidirect
5. #23781986	C₂	semidirect
6. #23781981	C₂	semidirect
7. #23781976	C₂	semidirect
8. #23781701	C₂	semidirect
9. #23781661	C₂	semidirect
10. C₂×C₂²⋊(C₁₆₃⋊C₈)	C₂	semidirect
21. #19104668	C₂²	semidirect
22. #19104668	C₂²	semidirect
23. #19104666	C₂²	semidirect
24. #19104666	C₂²	semidirect
25. #19104659	C₂²	semidirect
26. #19104659	C₂²	semidirect
27. #19104656	C₂²	semidirect
28. #19104656	C₂²	semidirect
143. C₁₆₃	#3091	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104694	2
#23782003	2
#23781998	2
#23781993	2
#23781986	2
#23781981	2
#23781976	2
#23781701	2
#23781661	2
C₂×C₂²⋊(C₁₆₃⋊C₈)	2
#525×C₁₆₃	2
#3091	163