Subgroups — #1845×D₈₉ #36554827

Order 22784.

1358 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107×D₈₉	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#384×D₈₉	C₂	1	2	yes	maximal
4	#26020412	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020411	C₂	1	2	yes	maximal, complemented
6	#26020410	C₂	1	4	yes	maximal, complemented
7	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×D₈₉	—	2	4	no	—
10	C₂×C₄⋊C₄×D₈₉	C₂²	1	4	yes	—
11	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
12	C₂×#15667946	C₂²	1	4	yes	—
13	C₂×#15667945	C₂²	1	8	yes	—
14	#107×D₈₉	—	2	16	no	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
17	C₄×C₈×D₈₉	C₄	1	8	yes	complemented
18	#19151468	C₂²	1	2	yes	—
19	#19151467	—	2	4	no	—
20	#19151467	C₂²	1	4	yes	—
21	C₂×#107×C₈₉	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#384×C₈₉	C₂²	1	2	yes	—
24	C₂²×C₄×D₈₉	—	2	4	no	—
25	C₂²×C₄×D₈₉	D₄	1	2	yes	—
26	C₂²×C₄×D₈₉	C₂³	1	1	yes	characteristic
27	C₂²×#10623494	C₂³	1	2	yes	—
28	C₂×C₈×D₈₉	C₂×C₄	1	8	yes	—
29	C₂×C₈×D₈₉	D₄	1	8	yes	—
30	C₂×C₈×D₈₉	Q₈	1	8	yes	—
31	C₂×C₄⋊Dic₈₉	—	2	4	no	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
33	C₂×#13083553	—	2	4	no	—
34	C₂×#13083553	—	2	8	no	—
35	C₄⋊C₄×D₈₉	—	2	16	no	—
36	C₄⋊C₄×D₈₉	—	4	16	no	—
37	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
38	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
39	C₄²×D₈₉	C₂×C₄	1	4	yes	—
40	C₄²×D₈₉	Q₈	1	4	yes	—
41	#15667946	—	2	8	no	—
42	#15667945	—	2	16	no	—
43	C₄×#10623494	C₂×C₄	1	4	yes	complemented
44	C₈×Dic₈₉	C₂×C₄	1	8	yes	complemented
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
47	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	#107×C₈₉	—	2	8	no	—
50	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	complemented
51	C₂³×D₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₈₉	—	2	4	no	—
53	C₂²×Dic₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×D₈₉	—	4	8	no	—
56	C₂×C₄×D₈₉	—	4	16	no	—
57	C₂×C₄×D₈₉	C₄⋊C₄	1	8	yes	—
58	C₂×C₄×D₈₉	D₈	1	8	yes	—
59	C₂×C₄×D₈₉	Q₁₆	1	8	yes	—
60	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×D₈₉	C₂×D₄	1	4	yes	—
62	C₂×C₄×D₈₉	C₂×Q₈	1	4	yes	—
63	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
64	C₂×#10623494	C₂²×C₄	1	4	yes	—
65	C₂×#10623494	C₂×D₄	1	4	yes	—
66	C₂×#10623494	C₂×Q₈	1	4	yes	—
67	C₈×D₈₉	C₄⋊C₄	1	32	yes	complemented
68	C₄⋊Dic₈₉	—	2	8	no	—
69	C₄⋊Dic₈₉	—	4	8	no	—
70	#13083553	—	4	8	no	—
71	#13083553	—	4	16	no	—
72	C₄×Dic₈₉	C₄⋊C₄	1	8	yes	—
73	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
74	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₈₉	—	2	4	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
81	C₄⋊C₄×C₈₉	—	2	8	no	—
82	C₄⋊C₄×C₈₉	—	4	8	no	—
83	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
84	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
85	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
86	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×D₈₉	#127	1	2	yes	—
88	C₂²×D₈₉	C₄⋊Q₈	1	2	yes	—
89	C₂²×D₈₉	C₂×D₈	1	4	yes	—
90	C₂²×D₈₉	C₂×Q₁₆	1	4	yes	—
91	C₂×C₄×C₈₉	—	4	4	no	—
92	C₂×C₄×C₈₉	—	4	8	no	—
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
95	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
98	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
99	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	4	8	no	—
102	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×Dic₈₉	#127	1	2	yes	—
105	C₂×Dic₈₉	C₄⋊Q₈	1	2	yes	—
106	C₂×Dic₈₉	C₂×D₈	1	4	yes	—
107	C₂×Dic₈₉	C₂×Q₁₆	1	4	yes	—
108	C₄×D₈₉	—	8	16	no	—
109	C₄×D₈₉	#107	1	32	yes	complemented
110	C₄×D₈₉	C₂×C₄⋊C₄	1	8	yes	—
111	C₄×D₈₉	C₄⋊Q₈	1	8	yes	—
112	C₈×C₈₉	C₂×C₄⋊C₄	1	8	yes	complemented
113	#10623494	C₂×C₄⋊C₄	1	8	yes	complemented
114	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
116	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₈₉	C₂²×D₈	1	2	yes	—
118	C₂²×C₈₉	C₂²×Q₁₆	1	2	yes	—
119	C₂×D₈₉	#384	1	4	yes	—
120	C₂×D₈₉	C₂×#107	1	8	yes	—
121	C₂×D₈₉	#493	1	4	yes	—
122	C₂×D₈₉	#494	1	4	yes	—
123	C₂×D₈₉	#500	1	8	yes	—
124	C₄×C₈₉	—	8	4	no	—
125	C₄×C₈₉	C₂×#107	1	8	yes	complemented
126	C₄×C₈₉	C₂²×C₄⋊C₄	1	2	yes	—
127	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
128	Dic₈₉	—	8	4	no	—
129	Dic₈₉	#384	1	4	yes	—
130	Dic₈₉	C₂×#107	1	8	yes	complemented
131	C₂×#1845	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	C₂×#384	1	1	yes	characteristic
133	C₂×C₈₉	C₂²×#107	1	2	yes	—
134	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
135	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
136	C₂×C₈₉	C₂×#500	1	2	yes	—
137	D₈₉	#1845	1	8	yes	direct product factor
138	C₂²×#107	—	89	4	no	—
139	C₂²×C₄×C₈	—	89	1	no	—
140	C₂×#384	—	89	2	no	—
141	#1845	—	89	4	no	—
142	#1845	—	89	1	no	—
143	#1845	—	89	2	no	—
144	#1845	D₈₉	1	1	yes	characteristic, direct product factor
145	C₈₉	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₈	—	89	2	no	—
147	C₂²×C₄⋊C₄	—	89	4	no	—
148	C₂²×C₄⋊C₄	—	178	4	no	—
149	C₂²×C₄²	—	89	1	no	—
150	C₂×#107	—	89	4	no	—
151	C₂×#107	—	89	8	no	—
152	C₂×#107	—	178	16	no	—
153	C₂×#107	C₂×D₈₉	1	4	yes	—
154	C₂×C₄×C₈	—	89	2	no	—
155	C₂×C₄×C₈	—	89	1	no	—
156	C₂×C₄×C₈	—	89	8	no	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	#384	—	89	4	no	—
159	#384	—	89	2	no	—
160	#384	—	178	4	no	—
161	#384	C₂×D₈₉	1	2	yes	—
162	C₂³×C₄	—	89	2	no	—
163	C₂³×C₄	—	89	1	no	—
164	C₂³×C₄	—	178	4	no	—
165	C₂²×C₈	—	89	2	no	—
166	C₂²×C₈	—	89	8	no	—
167	C₂²×C₈	—	89	8	no	—
168	C₂²×C₈	—	89	8	no	—
169	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	89	4	no	—
172	C₂×C₄⋊C₄	—	356	16	no	—
173	C₂×C₄⋊C₄	—	178	16	no	—
174	C₂×C₄⋊C₄	—	178	4	no	—
175	C₂×C₄⋊C₄	—	178	8	no	—
176	C₂×C₄⋊C₄	—	178	4	no	—
177	C₂×C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
178	C₂×C₄²	—	89	2	no	—
179	C₂×C₄²	—	89	1	no	—
180	C₂×C₄²	—	89	4	no	—
181	C₂×C₄²	—	89	4	no	—
182	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
183	#107	—	2	8	no	—
184	#107	—	178	16	no	—
185	#107	—	178	8	no	—
186	C₄×C₈	—	89	8	no	—
187	C₄×C₈	—	89	4	no	—
188	C₄×C₈	C₄×D₈₉	1	4	yes	complemented
189	C₂⁴	—	89	1	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	89	1	no	—
192	C₂²×C₄	—	89	2	no	—
193	C₂²×C₄	—	89	8	no	—
194	C₂²×C₄	—	89	8	no	—
195	C₂²×C₄	—	89	2	no	—
196	C₂²×C₄	—	89	4	no	—
197	C₂²×C₄	—	89	2	no	—
198	C₂²×C₄	—	89	4	no	—
199	C₂²×C₄	—	89	8	no	—
200	C₂²×C₄	—	356	16	no	—
201	C₂²×C₄	—	356	8	no	—
202	C₂²×C₄	—	178	4	no	—
203	C₂²×C₄	D₄×D₈₉	1	2	yes	—
204	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
205	C₂×C₈	—	89	4	no	—
206	C₂×C₈	—	89	4	no	—
207	C₂×C₈	—	89	4	no	—
208	C₂×C₈	—	89	32	no	—
209	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
210	C₂×C₈	D₄×D₈₉	1	4	yes	—
211	C₂×C₈	Q₈×D₈₉	1	4	yes	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	4	8	no	—
214	C₄⋊C₄	—	356	16	no	—
215	C₄⋊C₄	—	356	8	no	—
216	C₄⋊C₄	—	356	8	no	—
217	C₄⋊C₄	—	178	8	no	—
218	C₄²	—	89	2	no	—
219	C₄²	—	89	2	no	—
220	C₄²	—	89	8	no	—
221	C₄²	C₂×C₄×D₈₉	1	2	yes	—
222	C₄²	Q₈×D₈₉	1	2	yes	—
223	C₂³	—	89	2	no	—
224	C₂³	—	89	2	no	—
225	C₂³	—	89	2	no	—
226	C₂³	—	89	4	no	—
227	C₂³	—	89	4	no	—
228	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	89	2	no	—
232	C₂×C₄	—	89	2	no	—
233	C₂×C₄	—	89	4	no	—
234	C₂×C₄	—	89	2	no	—
235	C₂×C₄	—	89	4	no	—
236	C₂×C₄	—	89	4	no	—
237	C₂×C₄	—	89	8	no	—
238	C₂×C₄	—	89	32	no	—
239	C₂×C₄	—	89	8	no	—
240	C₂×C₄	—	356	4	no	—
241	C₂×C₄	—	356	8	no	—
242	C₂×C₄	—	712	16	no	—
243	C₂×C₄	C₄⋊C₄×D₈₉	1	4	yes	—
244	C₂×C₄	D₈×D₈₉	1	4	yes	—
245	C₂×C₄	Q₁₆×D₈₉	1	4	yes	—
246	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
248	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₈×D₈₉	1	2	yes	—
250	C₈	—	89	8	no	—
251	C₈	C₄⋊C₄×D₈₉	1	8	yes	complemented
252	C₂²	—	89	8	no	—
253	C₂²	—	89	4	no	—
254	C₂²	—	89	4	no	—
255	C₂²	—	89	8	no	—
256	C₂²	—	89	4	no	—
257	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
258	C₂²	#127×D₈₉	1	1	yes	characteristic
259	C₂²	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
260	C₂²	C₂×D₈×D₈₉	1	2	yes	—
261	C₂²	C₂×Q₁₆×D₈₉	1	2	yes	—
262	C₄	—	8	4	no	—
263	C₄	—	89	8	no	—
264	C₄	—	89	4	no	—
265	C₄	—	712	4	no	—
266	C₄	#107×D₈₉	1	8	yes	complemented
267	C₄	C₂×C₄⋊C₄×D₈₉	1	2	yes	—
268	C₄	C₄⋊Q₈×D₈₉	1	2	yes	—
269	C₂	—	89	8	no	—
270	C₂	#384×D₈₉	1	1	yes	characteristic
271	C₂	C₂×#107×D₈₉	1	2	yes	—
272	C₂	#493×D₈₉	1	1	yes	characteristic
273	C₂	#494×D₈₉	1	1	yes	characteristic
274	C₂	#500×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1845	C₂
socle	C₂³×C₈₉	C₂²×D₄
prime radical	C₂³×D₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1845	D₈₉
hypocenter	C₈₉	C₂×#1845
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
4. #26020412	C₂	semidirect
5. #26020411	C₂	semidirect
6. #26020410	C₂	semidirect
7. C₈₉×#1845	C₂	semidirect
17. C₄×C₈×D₈₉	C₄	semidirect
43. C₄×#10623494	C₂×C₄	semidirect
44. C₈×Dic₈₉	C₂×C₄	semidirect
50. C₄×C₈×C₈₉	C₂×C₄	semidirect
67. C₈×D₈₉	C₄⋊C₄	semidirect
109. C₄×D₈₉	#107	semidirect
112. C₈×C₈₉	C₂×C₄⋊C₄	semidirect
113. #10623494	C₂×C₄⋊C₄	semidirect
125. C₄×C₈₉	C₂×#107	semidirect
130. Dic₈₉	C₂×#107	semidirect
137. D₈₉	#1845	direct (N × H)
144. #1845	D₈₉	direct (N × H)
145. C₈₉	C₂×#1845	semidirect
188. C₄×C₈	C₄×D₈₉	semidirect
251. C₈	C₄⋊C₄×D₈₉	semidirect
266. C₄	#107×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107×D₈₉	2
C₂×C₄×C₈×D₈₉	2
#384×D₈₉	2
#26020412	2
#26020411	2
#26020410	2
C₈₉×#1845	2
C₂×#1845	89

Subgroups — #1845×D89 #36554827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1845×D₈₉ #36554827