Subgroups — C₈₉×#8643 #36386491

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#1844	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1803	C₂	1	2	yes	maximal
4	C₂×#334×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×#106×C₈₉	—	2	1	no	—
10	C₂×#106×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	#334×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	#106×C₈₉	—	4	2	no	—
37	#106×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	2	no	—
49	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
50	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	—
51	C₂²×C₄×C₈₉	—	2	1	no	—
52	C₂²×C₄×C₈₉	—	2	1	no	—
53	C₂²×C₄×C₈₉	—	4	2	no	—
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	—	2	1	no	—
59	C₂×C₈×C₈₉	—	2	2	no	—
60	C₂×C₈×C₈₉	—	4	1	no	—
61	C₂×C₈×C₈₉	—	4	1	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₂×C₈×C₈₉	—	4	2	no	—
64	C₂×C₈×C₈₉	—	4	2	no	—
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₈₉	—	2	2	no	—
70	C₄⋊C₄×C₈₉	—	2	1	no	—
71	C₄⋊C₄×C₈₉	—	2	2	no	—
72	C₄⋊C₄×C₈₉	—	4	2	no	—
73	C₄⋊C₄×C₈₉	—	4	2	no	—
74	C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₄⋊C₄×C₈₉	—	4	2	no	—
76	C₄⋊C₄×C₈₉	—	8	2	no	—
77	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₈₉	#41	1	1	yes	characteristic
84	C₂³×C₈₉	#121	1	1	yes	characteristic
85	Q₈×C₈₉	—	2	2	no	—
86	Q₈×C₈₉	—	2	4	no	—
87	Q₈×C₈₉	#106	1	4	yes	complemented
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	—	8	2	no	—
94	C₂×C₄×C₈₉	—	8	2	no	—
95	C₂×C₄×C₈₉	—	8	2	no	—
96	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#135	1	2	yes	—
106	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
108	C₈×C₈₉	—	2	2	no	—
109	C₈×C₈₉	—	4	2	no	—
110	C₈×C₈₉	—	8	2	no	—
111	C₈×C₈₉	#103	1	4	yes	complemented
112	C₂²×C₈₉	#380	1	1	yes	characteristic
113	C₂²×C₈₉	#460	1	1	yes	characteristic
114	C₂²×C₈₉	#462	1	1	yes	characteristic
115	C₂²×C₈₉	#465	1	2	yes	—
116	C₂²×C₈₉	#471	1	2	yes	—
117	C₄×C₈₉	—	2	4	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	—	2	1	no	—
120	C₄×C₈₉	—	4	2	no	—
121	C₄×C₈₉	—	16	2	no	—
122	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
123	C₄×C₈₉	#417	1	1	yes	characteristic
124	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
125	C₄×C₈₉	#427	1	1	yes	characteristic
126	C₄×C₈₉	#465	1	2	yes	—
127	C₄×C₈₉	#475	1	1	yes	characteristic
128	C₄×C₈₉	#477	1	1	yes	characteristic
129	#8643	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₈₉	#1666	1	2	yes	—
131	C₂×C₈₉	#1672	1	2	yes	—
132	C₂×C₈₉	#1803	1	1	yes	characteristic
133	C₂×C₈₉	#1807	1	1	yes	characteristic
134	C₂×C₈₉	#1815	1	1	yes	characteristic
135	C₂×C₈×Q₈	C₂×C₈₉	1	1	yes	characteristic
136	#1844	C₂×C₈₉	1	1	yes	characteristic
137	#1803	C₂×C₈₉	1	2	yes	—
138	C₂×#334	C₂×C₈₉	1	1	yes	characteristic
139	#1296	C₂×C₈₉	1	2	yes	—
140	C₈₉	#8643	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
142	C₈×Q₈	—	2	2	no	—
143	C₈×Q₈	C₄×C₈₉	1	4	yes	complemented
144	C₂×#106	—	2	1	no	—
145	C₂×#106	—	2	1	no	—
146	C₂×C₄⋊C₈	—	2	1	no	—
147	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
148	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
149	C₂×#103	—	2	2	no	—
150	C₂×C₄×C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
152	#384	C₂²×C₈₉	1	2	yes	—
153	#340	—	2	2	no	—
154	#334	—	2	2	no	—
155	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	C₂×C₄×C₈₉	1	2	yes	—
163	C₄×Q₈	Q₈×C₈₉	1	2	yes	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
171	#106	—	4	2	no	—
172	#106	—	4	2	no	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂×C₄×C₈₉	1	2	yes	—
178	#103	—	4	4	no	—
179	C₄×C₈	—	2	4	no	—
180	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
181	C₄×C₈	D₄×C₈₉	1	2	yes	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₄⋊C₄×C₈₉	1	2	yes	—
185	C₂×Q₈	SD₁₆×C₈₉	1	4	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
191	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
201	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
202	C₂×C₈	SD₁₆×C₈₉	1	2	yes	—
203	C₂×C₈	Q₁₆×C₈₉	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	8	2	no	—
212	C₄⋊C₄	C₄⋊C₄×C₈₉	1	2	yes	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
217	C₄²	C₂×Q₈×C₈₉	1	1	yes	characteristic
218	C₄²	#41×C₈₉	1	1	yes	characteristic
219	C₂³	#121×C₈₉	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	2	4	no	—
222	Q₈	#106×C₈₉	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
233	C₂×C₄	#117×C₈₉	1	1	yes	characteristic
234	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
235	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
236	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
237	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
238	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
239	C₂×C₄	C₂×Q₁₆×C₈₉	1	1	yes	characteristic
240	C₂×C₄	#135×C₈₉	1	2	yes	—
241	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
242	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	8	2	no	—
246	C₈	#103×C₈₉	1	4	yes	complemented
247	C₂²	#380×C₈₉	1	1	yes	characteristic
248	C₂²	#460×C₈₉	1	1	yes	characteristic
249	C₂²	#462×C₈₉	1	1	yes	characteristic
250	C₂²	#465×C₈₉	1	2	yes	—
251	C₂²	#471×C₈₉	1	2	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	16	2	no	—
257	C₄	C₂×#103×C₈₉	1	1	yes	characteristic
258	C₄	#417×C₈₉	1	1	yes	characteristic
259	C₄	C₂×#106×C₈₉	1	1	yes	characteristic
260	C₄	#427×C₈₉	1	1	yes	characteristic
261	C₄	#465×C₈₉	1	2	yes	—
262	C₄	#475×C₈₉	1	1	yes	characteristic
263	C₄	#477×C₈₉	1	1	yes	characteristic
264	C₂	C₈₉×#1666	1	2	yes	—
265	C₂	C₈₉×#1672	1	2	yes	—
266	C₂	C₈₉×#1803	1	1	yes	characteristic
267	C₂	C₈₉×#1807	1	1	yes	characteristic
268	C₂	C₈₉×#1815	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
87. Q₈×C₈₉	#106	semidirect
111. C₈×C₈₉	#103	semidirect
129. #8643	C₈₉	direct (N × H)
140. C₈₉	#8643	direct (N × H)
143. C₈×Q₈	C₄×C₈₉	semidirect
222. Q₈	#106×C₈₉	semidirect
246. 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×#1844	2
C₈₉×#1803	2
C₂×#334×C₈₉	2
C₈₉×#1296	2
#8643	89

Subgroups — C89×#8643 #36386491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#8643 #36386491