Subgroups — C₈₉×#8645 #36386493

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1803	C₂	1	2	yes	maximal
4	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1296	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₈₉	—	2	2	no	—
8	C₈×Q₈×C₈₉	C₄	1	4	yes	complemented
9	C₂×#107×C₈₉	—	2	1	no	—
10	C₂×#107×C₈₉	—	2	1	no	—
11	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#103×C₈₉	—	2	2	no	—
15	C₂×C₄×C₈×C₈₉	—	2	1	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#384×C₈₉	C₂²	1	2	yes	—
18	#340×C₈₉	—	2	2	no	—
19	#335×C₈₉	—	2	2	no	—
20	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₉	—	2	1	no	—
22	C₂²×C₈×C₈₉	—	2	2	no	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₈₉	—	2	1	no	—
26	C₄×Q₈×C₈₉	—	2	1	no	—
27	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₄×Q₈×C₈₉	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	—	2	1	no	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	#107×C₈₉	—	4	2	no	—
37	#107×C₈₉	—	4	2	no	—
38	C₄⋊C₈×C₈₉	—	2	4	no	—
39	C₄⋊C₈×C₈₉	—	2	1	no	—
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	C₄⋊C₈×C₈₉	—	2	2	no	—
42	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
43	#103×C₈₉	—	4	4	no	—
44	C₄×C₈×C₈₉	—	2	4	no	—
45	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₈₉	D₄	1	2	yes	—
47	C₂×Q₈×C₈₉	—	2	1	no	—
48	C₂×Q₈×C₈₉	—	2	1	no	—
49	C₂×Q₈×C₈₉	—	2	1	no	—
50	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
51	C₂×Q₈×C₈₉	D₈	1	2	yes	—
52	C₂×Q₈×C₈₉	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₈₉	—	2	1	no	—
54	C₂²×C₄×C₈₉	—	2	1	no	—
55	C₂²×C₄×C₈₉	—	4	2	no	—
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
59	C₂×C₈×C₈₉	—	2	1	no	—
60	C₂×C₈×C₈₉	—	2	1	no	—
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	—	4	1	no	—
63	C₂×C₈×C₈₉	—	4	1	no	—
64	C₂×C₈×C₈₉	—	4	2	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₂×C₈×C₈₉	—	4	2	no	—
67	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
70	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
71	C₄⋊C₄×C₈₉	—	2	2	no	—
72	C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₄⋊C₄×C₈₉	—	2	2	no	—
74	C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₄⋊C₄×C₈₉	—	4	2	no	—
76	C₄⋊C₄×C₈₉	—	4	2	no	—
77	C₄⋊C₄×C₈₉	—	4	2	no	—
78	C₄⋊C₄×C₈₉	—	8	2	no	—
79	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
80	C₄²×C₈₉	—	2	2	no	—
81	C₄²×C₈₉	—	2	2	no	—
82	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₈₉	#41	1	1	yes	characteristic
86	C₂³×C₈₉	#121	1	1	yes	characteristic
87	Q₈×C₈₉	—	2	2	no	—
88	Q₈×C₈₉	—	2	2	no	—
89	Q₈×C₈₉	—	2	2	no	—
90	Q₈×C₈₉	#107	1	4	yes	complemented
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	4	1	no	—
98	C₂×C₄×C₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	—	8	2	no	—
100	C₂×C₄×C₈₉	—	8	2	no	—
101	C₂×C₄×C₈₉	—	8	2	no	—
102	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
114	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
116	C₈×C₈₉	—	2	2	no	—
117	C₈×C₈₉	—	4	2	no	—
118	C₈×C₈₉	—	8	2	no	—
119	C₈×C₈₉	#103	1	4	yes	complemented
120	C₂²×C₈₉	#380	1	1	yes	characteristic
121	C₂²×C₈₉	#460	1	1	yes	characteristic
122	C₂²×C₈₉	#462	1	1	yes	characteristic
123	C₂²×C₈₉	#466	1	1	yes	characteristic
124	C₂²×C₈₉	#467	1	1	yes	characteristic
125	C₂²×C₈₉	#472	1	1	yes	characteristic
126	C₂²×C₈₉	#473	1	1	yes	characteristic
127	C₄×C₈₉	—	2	4	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	4	2	no	—
131	C₄×C₈₉	—	16	2	no	—
132	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
133	C₄×C₈₉	#417	1	1	yes	characteristic
134	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
135	C₄×C₈₉	#427	1	1	yes	characteristic
136	C₄×C₈₉	#466	1	1	yes	characteristic
137	C₄×C₈₉	#467	1	1	yes	characteristic
138	C₄×C₈₉	#475	1	1	yes	characteristic
139	C₄×C₈₉	#477	1	1	yes	characteristic
140	#8645	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₈₉	#1668	1	1	yes	characteristic
142	C₂×C₈₉	#1670	1	1	yes	characteristic
143	C₂×C₈₉	#1674	1	1	yes	characteristic
144	C₂×C₈₉	#1676	1	1	yes	characteristic
145	C₂×C₈₉	#1803	1	1	yes	characteristic
146	C₂×C₈₉	#1808	1	1	yes	characteristic
147	C₂×C₈₉	#1816	1	1	yes	characteristic
148	C₂×C₈×Q₈	C₂×C₈₉	1	1	yes	characteristic
149	#1845	C₂×C₈₉	1	1	yes	characteristic
150	#1803	C₂×C₈₉	1	2	yes	—
151	C₂×#335	C₂×C₈₉	1	1	yes	characteristic
152	#1296	C₂×C₈₉	1	2	yes	—
153	C₈₉	#8645	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
155	C₈×Q₈	—	2	2	no	—
156	C₈×Q₈	C₄×C₈₉	1	4	yes	complemented
157	C₂×#107	—	2	1	no	—
158	C₂×#107	—	2	1	no	—
159	C₂×C₄⋊C₈	—	2	1	no	—
160	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
162	C₂×#103	—	2	2	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
165	#384	C₂²×C₈₉	1	2	yes	—
166	#340	—	2	2	no	—
167	#335	—	2	2	no	—
168	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
173	C₄×Q₈	—	2	1	no	—
174	C₄×Q₈	—	2	1	no	—
175	C₄×Q₈	C₂×C₄×C₈₉	1	2	yes	—
176	C₄×Q₈	Q₈×C₈₉	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
184	#107	—	4	2	no	—
185	#107	—	4	2	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	C₂×C₄×C₈₉	1	2	yes	—
191	#103	—	4	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
194	C₄×C₈	D₄×C₈₉	1	2	yes	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	C₄⋊C₄×C₈₉	1	2	yes	—
199	C₂×Q₈	D₈×C₈₉	1	2	yes	—
200	C₂×Q₈	Q₁₆×C₈₉	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
206	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
216	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
217	C₂×C₈	SD₁₆×C₈₉	1	2	yes	—
218	C₂×C₈	Q₁₆×C₈₉	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	8	2	no	—
227	C₄⋊C₄	C₄⋊C₄×C₈₉	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
232	C₄²	C₂×Q₈×C₈₉	1	1	yes	characteristic
233	C₄²	#41×C₈₉	1	1	yes	characteristic
234	C₂³	#121×C₈₉	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	2	no	—
238	Q₈	#107×C₈₉	1	4	yes	complemented
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	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
252	C₂×C₄	#117×C₈₉	1	1	yes	characteristic
253	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
256	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
257	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₁₆×C₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂×Q₁₆×C₈₉	1	1	yes	characteristic
260	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
261	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
262	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
263	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	8	2	no	—
267	C₈	#103×C₈₉	1	4	yes	complemented
268	C₂²	#380×C₈₉	1	1	yes	characteristic
269	C₂²	#460×C₈₉	1	1	yes	characteristic
270	C₂²	#462×C₈₉	1	1	yes	characteristic
271	C₂²	#466×C₈₉	1	1	yes	characteristic
272	C₂²	#467×C₈₉	1	1	yes	characteristic
273	C₂²	#472×C₈₉	1	1	yes	characteristic
274	C₂²	#473×C₈₉	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	16	2	no	—
280	C₄	C₂×#103×C₈₉	1	1	yes	characteristic
281	C₄	#417×C₈₉	1	1	yes	characteristic
282	C₄	C₂×#107×C₈₉	1	1	yes	characteristic
283	C₄	#427×C₈₉	1	1	yes	characteristic
284	C₄	#466×C₈₉	1	1	yes	characteristic
285	C₄	#467×C₈₉	1	1	yes	characteristic
286	C₄	#475×C₈₉	1	1	yes	characteristic
287	C₄	#477×C₈₉	1	1	yes	characteristic
288	C₂	C₈₉×#1668	1	1	yes	characteristic
289	C₂	C₈₉×#1670	1	1	yes	characteristic
290	C₂	C₈₉×#1674	1	1	yes	characteristic
291	C₂	C₈₉×#1676	1	1	yes	characteristic
292	C₂	C₈₉×#1803	1	1	yes	characteristic
293	C₂	C₈₉×#1808	1	1	yes	characteristic
294	C₂	C₈₉×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	#121
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₈₉	C₄	semidirect
90. Q₈×C₈₉	#107	semidirect
119. C₈×C₈₉	#103	semidirect
140. #8645	C₈₉	direct (N × H)
153. C₈₉	#8645	direct (N × H)
156. C₈×Q₈	C₄×C₈₉	semidirect
238. Q₈	#107×C₈₉	semidirect
267. C₈	#103×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₈₉	2
C₈₉×#1845	2
C₈₉×#1803	2
C₂×#335×C₈₉	2
C₈₉×#1296	2
#8645	89

Subgroups — C89×#8645 #36386493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#8645 #36386493