Subgroups — #35227928

Order 20864.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782289	C₂	1	4	yes	maximal, complemented
2	#23782278	C₂	1	2	yes	maximal, complemented
3	#23782271	C₂	1	2	yes	maximal, complemented
4	#23782264	C₂	1	4	yes	maximal, complemented
5	#23782255	C₂	1	4	yes	maximal, complemented
6	#23782215	C₂	1	1	yes	maximal, characteristic, complemented
7	#23782202	C₂	1	8	yes	maximal, complemented
8	#23782177	C₂	1	2	yes	maximal, complemented
9	#23782133	C₂	1	2	yes	maximal, complemented
10	C₂×#19104692	C₂	1	1	yes	maximal, characteristic, complemented
11	#533×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₆₃	C₂²	1	1	yes	characteristic
14	#19104735	C₂²	1	1	yes	characteristic, complemented
15	#19104732	C₂²	1	1	yes	characteristic, complemented
16	#19104730	C₂²	1	8	yes	—
17	#19104730	C₂²	1	8	yes	complemented
18	#19104726	C₂²	1	8	yes	complemented
19	#19104724	C₂²	1	8	yes	—
20	#19104723	C₂²	1	2	yes	complemented
21	#19104720	C₂²	1	8	yes	—
22	#19104720	C₂²	1	8	yes	complemented
23	#19104719	C₂²	1	4	yes	—
24	#19104715	C₂²	1	4	yes	—
25	#19104711	C₂²	1	8	yes	complemented
26	#19104710	C₂²	1	4	yes	—
27	#19104710	C₂²	1	8	yes	—
28	#19104709	C₂²	1	4	yes	—
29	#19104705	C₂²	1	4	yes	—
30	#19104700	C₂²	1	4	yes	complemented
31	C₄×D₆₅₂	C₂²	1	8	yes	—
32	C₄×D₆₅₂	C₂²	1	4	yes	complemented
33	#19104697	C₂²	1	4	yes	complemented
34	(C₂²⋊D₁₆₃)⋊C₄	C₂²	1	8	yes	—
35	#19104692	C₂²	1	4	yes	—
36	#19104692	C₂²	1	2	yes	—
37	#19104692	C₂²	1	2	yes	—
38	#19104692	C₂²	1	4	yes	complemented
39	C₂×Dic₁₆₃⋊C₄	C₂²	1	2	yes	—
40	C₂×(C₂×D₁₆₃)⋊C₄	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
42	C₄²×D₁₆₃	C₂²	1	4	yes	complemented
43	C₄⋊Q₈×C₁₆₃	C₂²	1	2	yes	complemented
44	#125×C₁₆₃	C₂²	1	2	yes	complemented
45	C₂²⋊Q₈×C₁₆₃	C₂²	1	4	yes	—
46	C₄×Q₈×C₁₆₃	C₂²	1	4	yes	complemented
47	#117×C₁₆₃	C₂²	1	2	yes	—
48	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
49	C₂³×D₁₆₃	C₂³	1	1	yes	characteristic
50	#14579699	—	2	8	no	—
51	C₂×Dic₃₂₆	C₂³	1	2	yes	—
52	C₂×D₆₅₂	C₂³	1	2	yes	—
53	C₂×D₆₅₂	C₂³	1	8	yes	—
54	C₂×D₆₅₂	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₁₆₃	C₂³	1	4	yes	—
56	C₂²×Dic₁₆₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₆₃	C₂³	1	2	yes	—
58	C₂×C₄×D₁₆₃	—	2	4	no	—
59	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
60	C₂×C₄×D₁₆₃	C₂³	1	2	yes	—
61	C₂×C₄×D₁₆₃	C₂³	1	4	yes	—
62	C₂×C₄×D₁₆₃	C₂³	1	8	yes	—
63	C₄⋊Dic₁₆₃	C₂³	1	8	yes	—
64	C₄⋊Dic₁₆₃	C₂³	1	4	yes	—
65	Dic₁₆₃⋊C₄	C₂³	1	4	yes	—
66	Dic₁₆₃⋊C₄	C₂³	1	4	yes	—
67	Dic₁₆₃⋊C₄	C₂³	1	8	yes	—
68	(C₂×D₁₆₃)⋊C₄	C₂³	1	4	yes	—
69	(C₂×D₁₆₃)⋊C₄	C₂³	1	4	yes	—
70	(C₂×D₁₆₃)⋊C₄	C₂³	1	8	yes	—
71	(C₂×D₁₆₃)⋊C₄	C₂³	1	8	yes	—
72	(C₂×D₁₆₃)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₁₆₃)⋊C₄	C₂³	1	8	yes	—
74	C₄×Dic₁₆₃	C₂³	1	2	yes	—
75	C₄×Dic₁₆₃	C₂³	1	8	yes	—
76	C₄×Dic₁₆₃	C₂³	1	2	yes	—
77	C₂×Q₈×C₁₆₃	C₂³	1	4	yes	—
78	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₆₃	C₂³	1	4	yes	—
81	C₄⋊C₄×C₁₆₃	C₂³	1	8	yes	—
82	C₄⋊C₄×C₁₆₃	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₁₆₃	C₂³	1	4	yes	—
84	C₄²×C₁₆₃	C₂³	1	2	yes	—
85	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
86	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
87	C₄²×C₁₆₃	C₂³	1	4	yes	—
88	C₂²×D₁₆₃	—	2	2	no	—
89	C₂²×D₁₆₃	—	4	2	no	—
90	C₂²×D₁₆₃	C₂⁴	1	2	yes	—
91	C₂²×D₁₆₃	C₂⁴	1	4	yes	—
92	Dic₃₂₆	—	2	4	no	—
93	D₆₅₂	—	2	4	no	—
94	D₆₅₂	—	4	4	no	—
95	D₆₅₂	—	4	8	no	—
96	C₂²⋊D₁₆₃	—	2	8	no	—
97	C₂×Dic₁₆₃	—	2	2	no	—
98	C₂×Dic₁₆₃	—	4	2	no	—
99	C₂×Dic₁₆₃	C₂⁴	1	4	yes	—
100	C₂×Dic₁₆₃	C₂⁴	1	4	yes	—
101	C₂×Dic₁₆₃	C₂⁴	1	2	yes	—
102	C₄×D₁₆₃	—	2	4	no	—
103	C₄×D₁₆₃	—	2	4	no	—
104	C₄×D₁₆₃	—	2	8	no	—
105	C₄×D₁₆₃	—	4	8	no	—
106	C₂³×C₁₆₃	C₂⁴	1	1	yes	characteristic
107	Q₈×C₁₆₃	—	4	4	no	—
108	C₂×C₄×C₁₆₃	—	4	2	no	—
109	C₂×C₄×C₁₆₃	#41	1	4	yes	—
110	C₂×C₄×C₁₆₃	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₆₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₆₃	C₂⁴	1	8	yes	—
113	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
115	C₂²×C₁₆₃	—	2	2	no	—
116	C₂²×C₁₆₃	C₂×#41	1	2	yes	—
117	C₂²×C₁₆₃	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₆₃	—	2	2	no	—
119	C₂×D₁₆₃	—	2	4	no	—
120	C₂×D₁₆₃	—	2	8	no	—
121	C₂×D₁₆₃	—	2	4	no	—
122	C₂×D₁₆₃	—	4	4	no	—
123	C₄×C₁₆₃	—	2	2	no	—
124	C₄×C₁₆₃	—	2	2	no	—
125	C₄×C₁₆₃	—	2	8	no	—
126	C₄×C₁₆₃	C₂×#41	1	2	yes	—
127	C₄×C₁₆₃	C₂×#41	1	2	yes	—
128	Dic₁₆₃	—	2	2	no	—
129	Dic₁₆₃	—	2	4	no	—
130	Dic₁₆₃	—	2	4	no	—
131	C₂×C₁₆₃	—	2	2	no	—
132	C₂×C₁₆₃	C₂²×#41	1	1	yes	characteristic
133	C₂×C₁₆₃	#585	1	2	yes	—
134	D₁₆₃	—	4	4	no	—
135	D₁₆₃	—	4	2	no	—
136	C₁₆₃	#3455	1	1	yes	characteristic, complemented, Sylow
137	#3455	—	163	1	no	maximal, Sylow
138	#542	—	163	4	no	—
139	#541	—	163	2	no	—
140	#540	—	163	2	no	—
141	#539	—	163	4	no	—
142	#538	—	163	4	no	—
143	#533	D₁₆₃	1	1	yes	characteristic, complemented
144	#532	—	163	1	no	—
145	#529	—	163	8	no	—
146	#525	—	163	2	no	—
147	C₄×#41	—	163	2	no	—
148	C₂×#117	—	163	1	no	—
149	C₂×#41	—	163	2	no	—
150	C₂³×C₄	—	163	1	no	—
151	C₄⋊Q₈	—	163	1	no	—
152	C₄⋊Q₈	C₂×D₁₆₃	1	2	yes	complemented
153	#127	—	163	1	no	—
154	#126	—	163	8	no	—
155	#126	—	163	8	no	—
156	#125	—	163	8	no	—
157	#125	C₂×D₁₆₃	1	2	yes	complemented
158	#124	—	163	8	no	—
159	#124	—	163	2	no	—
160	#123	—	163	8	no	—
161	#123	—	163	8	no	—
162	#123	—	163	4	no	—
163	C₂²⋊Q₈	—	163	8	no	—
164	C₂²⋊Q₈	—	163	4	no	—
165	C₂²⋊Q₈	C₂×D₁₆₃	1	4	yes	—
166	#121	—	163	4	no	—
167	#121	—	163	4	no	—
168	#121	—	163	8	no	—
169	#120	—	163	4	no	—
170	C₄×Q₈	—	163	4	no	—
171	C₄×Q₈	C₂×D₁₆₃	1	4	yes	complemented
172	C₄×D₄	—	163	4	no	—
173	C₄×D₄	—	163	8	no	—
174	C₄×D₄	—	163	8	no	—
175	C₄×D₄	—	163	4	no	—
176	#117	—	163	4	no	—
177	#117	—	163	2	no	—
178	#117	—	163	2	no	—
179	#117	—	163	4	no	—
180	#117	C₂×D₁₆₃	1	2	yes	—
181	C₂×C₄⋊C₄	—	163	2	no	—
182	C₂×C₂²⋊C₄	—	163	2	no	—
183	C₂×C₄²	—	163	1	no	—
184	C₂×C₄²	—	163	4	no	—
185	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
186	C₂⁴	—	163	1	no	—
187	#41	—	326	8	no	—
188	C₂×Q₈	—	163	2	no	—
189	C₂×Q₈	C₂²×D₁₆₃	1	4	yes	—
190	C₂×D₄	—	163	4	no	—
191	C₂×D₄	—	163	2	no	—
192	C₂×D₄	—	163	8	no	—
193	C₂×D₄	—	163	4	no	—
194	C₂²×C₄	—	326	4	no	—
195	C₂²×C₄	—	163	1	no	—
196	C₂²×C₄	—	163	2	no	—
197	C₂²×C₄	—	163	4	no	—
198	C₂²×C₄	—	163	2	no	—
199	C₂²×C₄	—	163	2	no	—
200	C₂²×C₄	—	163	8	no	—
201	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
203	C₄⋊C₄	—	163	4	no	—
204	C₄⋊C₄	—	163	4	no	—
205	C₄⋊C₄	—	163	8	no	—
206	C₄⋊C₄	—	163	4	no	—
207	C₄⋊C₄	—	163	8	no	—
208	C₄⋊C₄	C₂²×D₁₆₃	1	4	yes	—
209	C₄⋊C₄	C₂²×D₁₆₃	1	8	yes	—
210	C₄⋊C₄	C₂²×D₁₆₃	1	4	yes	—
211	C₂²⋊C₄	—	163	4	no	—
212	C₂²⋊C₄	—	163	4	no	—
213	C₂²⋊C₄	—	163	8	no	—
214	C₂²⋊C₄	—	163	8	no	—
215	C₂²⋊C₄	—	163	8	no	—
216	C₂²⋊C₄	—	163	8	no	—
217	C₂²⋊C₄	C₂²×D₁₆₃	1	4	yes	—
218	C₄²	—	163	2	no	—
219	C₄²	—	163	8	no	—
220	C₄²	—	163	2	no	—
221	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
222	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
223	C₄²	C₂²×D₁₆₃	1	4	yes	—
224	C₄²	C₂²×D₁₆₃	1	2	yes	—
225	C₂³	—	326	2	no	—
226	C₂³	—	652	2	no	—
227	C₂³	—	163	2	no	—
228	C₂³	—	163	4	no	—
229	C₂³	C₂³×D₁₆₃	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	326	4	no	—
232	D₄	—	326	8	no	—
233	D₄	—	326	4	no	—
234	D₄	—	652	4	no	—
235	D₄	—	652	8	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	326	2	no	—
238	C₂×C₄	—	326	4	no	—
239	C₂×C₄	—	326	4	no	—
240	C₂×C₄	—	326	8	no	—
241	C₂×C₄	—	652	8	no	—
242	C₂×C₄	—	652	2	no	—
243	C₂×C₄	—	163	4	no	—
244	C₂×C₄	—	163	4	no	—
245	C₂×C₄	—	163	2	no	—
246	C₂×C₄	#14579700	1	4	yes	—
247	C₂×C₄	C₂³×D₁₆₃	1	8	yes	—
248	C₂×C₄	C₂³×D₁₆₃	1	2	yes	—
249	C₂×C₄	C₂³×D₁₆₃	1	2	yes	—
250	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	326	2	no	—
254	C₂²	—	326	4	no	—
255	C₂²	—	326	8	no	—
256	C₂²	—	326	4	no	—
257	C₂²	—	652	4	no	—
258	C₂²	C₂×#14579700	1	2	yes	—
259	C₂²	C₂⁴×D₁₆₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	326	2	no	—
264	C₄	—	326	4	no	—
265	C₄	—	326	4	no	—
266	C₄	C₂×#14579700	1	2	yes	—
267	C₄	C₂×#14579700	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	652	2	no	—
270	C₂	—	652	4	no	—
271	C₂	C₂²×#14579700	1	1	yes	characteristic
272	C₂	#23782501	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₆₃
commutator, socle, cosocle	C₂²×C₁₆₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₆₃
Fitting	#533×C₁₆₃	C₂
hypercenter	#533	D₁₆₃
hypocenter	C₁₆₃	#3455
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #23782289	C₂	semidirect
2. #23782278	C₂	semidirect
3. #23782271	C₂	semidirect
4. #23782264	C₂	semidirect
5. #23782255	C₂	semidirect
6. #23782215	C₂	semidirect
7. #23782202	C₂	semidirect
8. #23782177	C₂	semidirect
9. #23782133	C₂	semidirect
10. C₂×#19104692	C₂	semidirect
11. #533×C₁₆₃	C₂	semidirect
14. #19104735	C₂²	semidirect
15. #19104732	C₂²	semidirect
17. #19104730	C₂²	semidirect
18. #19104726	C₂²	semidirect
20. #19104723	C₂²	semidirect
22. #19104720	C₂²	semidirect
25. #19104711	C₂²	semidirect
30. #19104700	C₂²	semidirect
32. C₄×D₆₅₂	C₂²	semidirect
33. #19104697	C₂²	semidirect
38. #19104692	C₂²	semidirect
42. C₄²×D₁₆₃	C₂²	semidirect
43. C₄⋊Q₈×C₁₆₃	C₂²	semidirect
44. #125×C₁₆₃	C₂²	semidirect
46. C₄×Q₈×C₁₆₃	C₂²	semidirect
136. C₁₆₃	#3455	semidirect
143. #533	D₁₆₃	semidirect
152. C₄⋊Q₈	C₂×D₁₆₃	semidirect
157. #125	C₂×D₁₆₃	semidirect
171. C₄×Q₈	C₂×D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782289	2
#23782278	2
#23782271	2
#23782264	2
#23782255	2
#23782215	2
#23782202	2
#23782177	2
#23782133	2
C₂×#19104692	2
#533×C₁₆₃	2
#3455	163