Subgroups — C₈₉×#33858 #36411706

Order 22784.

668 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3415	C₂	1	2	yes	maximal, complemented
2	C₈₉×#3292	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2221	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2213	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2192	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	#576×C₈₉	—	2	2	no	—
10	#576×C₈₉	C₂²	1	4	yes	complemented
11	#509×C₈₉	—	2	4	no	—
12	#509×C₈₉	C₂²	1	4	yes	complemented
13	#508×C₈₉	—	2	2	no	—
14	#508×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×D₁₆×C₈₉	—	2	2	no	—
16	C₂×D₁₆×C₈₉	C₂²	1	2	yes	complemented
17	C₂×D₁₆×C₈₉	C₂²	1	2	yes	complemented
18	#504×C₈₉	C₂²	1	2	yes	complemented
19	#493×C₈₉	C₂²	1	2	yes	complemented
20	#472×C₈₉	—	2	2	no	—
21	#472×C₈₉	C₂²	1	4	yes	complemented
22	#453×C₈₉	—	2	4	no	—
23	#443×C₈₉	C₂²	1	1	yes	characteristic
24	#443×C₈₉	C₂²	1	2	yes	complemented
25	#368×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	#364×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	#361×C₈₉	C₂²	1	4	yes	complemented
28	#359×C₈₉	C₂²	1	2	yes	complemented
29	#350×C₈₉	C₂²	1	2	yes	complemented
30	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
31	H(5,𝔽₂)×C₈₉	—	2	4	no	—
32	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
33	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
34	AGL(1,ℤ/8)×C₈₉	—	4	4	no	—
35	AGL(1,ℤ/8)×C₈₉	—	4	4	no	—
36	#135×C₈₉	—	2	4	no	—
37	#135×C₈₉	—	4	2	no	—
38	#135×C₈₉	D₄	1	1	yes	characteristic
39	#135×C₈₉	C₂³	1	2	yes	—
40	C₂×D₈×C₈₉	—	2	4	no	—
41	C₂×D₈×C₈₉	—	2	2	no	—
42	C₂×D₈×C₈₉	—	4	2	no	—
43	C₂×D₈×C₈₉	C₂³	1	4	yes	—
44	#131×C₈₉	—	2	2	no	—
45	#131×C₈₉	C₂³	1	2	yes	—
46	#127×C₈₉	—	2	2	no	—
47	Q₃₂×C₈₉	D₄	1	1	yes	characteristic, complemented
48	SD₃₂×C₈₉	—	2	4	no	—
49	SD₃₂×C₈₉	D₄	1	2	yes	complemented
50	D₁₆×C₈₉	—	2	4	no	—
51	D₁₆×C₈₉	—	2	2	no	—
52	D₁₆×C₈₉	—	2	2	no	—
53	D₁₆×C₈₉	—	4	2	no	—
54	D₁₆×C₈₉	D₄	1	1	yes	characteristic, complemented
55	M₃₂×C₈₉	—	2	2	no	—
56	M₃₂×C₈₉	C₂³	1	2	yes	—
57	C₂×C₁₆×C₈₉	—	2	2	no	—
58	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
60	#108×C₈₉	D₄	1	1	yes	characteristic
61	#108×C₈₉	C₂³	1	2	yes	—
62	#104×C₈₉	—	2	2	no	—
63	#104×C₈₉	D₄	1	2	yes	—
64	#100×C₈₉	—	2	4	no	—
65	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	#41×C₈₉	—	2	2	no	—
67	#41×C₈₉	—	4	4	no	—
68	#41×C₈₉	—	4	4	no	—
69	#41×C₈₉	C₂×D₄	1	2	yes	—
70	C₂×D₄×C₈₉	—	2	4	no	—
71	C₂×D₄×C₈₉	—	4	4	no	—
72	C₂×D₄×C₈₉	—	4	4	no	—
73	C₂×D₄×C₈₉	—	8	2	no	—
74	C₂×D₄×C₈₉	—	8	4	no	—
75	Q₁₆×C₈₉	D₈	1	1	yes	characteristic
76	Q₁₆×C₈₉	C₂×D₄	1	2	yes	—
77	SD₁₆×C₈₉	—	2	2	no	—
78	SD₁₆×C₈₉	—	4	4	no	—
79	SD₁₆×C₈₉	—	4	4	no	—
80	SD₁₆×C₈₉	D₈	1	2	yes	—
81	D₈×C₈₉	—	2	4	no	—
82	D₈×C₈₉	—	2	4	no	—
83	D₈×C₈₉	—	4	4	no	—
84	D₈×C₈₉	—	4	4	no	—
85	D₈×C₈₉	—	4	2	no	—
86	D₈×C₈₉	—	4	2	no	—
87	D₈×C₈₉	D₈	1	1	yes	characteristic
88	D₈×C₈₉	C₂×D₄	1	2	yes	—
89	M₁₆×C₈₉	—	2	2	no	—
90	M₁₆×C₈₉	—	2	2	no	—
91	M₁₆×C₈₉	—	4	2	no	—
92	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	4	2	no	—
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₁₆×C₈₉	—	2	2	no	—
99	C₁₆×C₈₉	—	2	1	no	—
100	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₈₉	—	4	4	no	—
103	C₂³×C₈₉	—	8	4	no	—
104	Q₈×C₈₉	—	2	2	no	—
105	Q₈×C₈₉	C₂×D₈	1	2	yes	—
106	D₄×C₈₉	—	2	2	no	—
107	D₄×C₈₉	—	2	2	no	—
108	D₄×C₈₉	—	4	2	no	—
109	D₄×C₈₉	—	4	4	no	—
110	D₄×C₈₉	—	4	4	no	—
111	D₄×C₈₉	—	8	4	no	—
112	D₄×C₈₉	—	8	4	no	—
113	D₄×C₈₉	—	16	4	no	—
114	D₄×C₈₉	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	8	4	no	—
119	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₈₉	—	2	2	no	—
121	C₈×C₈₉	—	4	2	no	—
122	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₈₉	—	2	2	no	—
127	C₂²×C₈₉	—	4	2	no	—
128	C₂²×C₈₉	—	8	4	no	—
129	C₂²×C₈₉	—	16	4	no	—
130	C₂²×C₈₉	—	32	4	no	—
131	C₂²×C₈₉	#576	1	1	yes	characteristic
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	—	2	2	no	—
134	C₄×C₈₉	—	4	2	no	—
135	C₄×C₈₉	D₄²	1	1	yes	characteristic
136	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
137	#33858	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₈₉	—	2	1	no	—
139	C₂×C₈₉	—	4	2	no	—
140	C₂×C₈₉	—	8	2	no	—
141	C₂×C₈₉	—	16	4	no	—
142	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
143	#3415	C₂×C₈₉	1	2	yes	complemented
144	#3292	C₂×C₈₉	1	2	yes	complemented
145	#2246	C₂×C₈₉	1	1	yes	characteristic, complemented
146	#2221	C₂×C₈₉	1	2	yes	complemented
147	#2213	C₂×C₈₉	1	2	yes	complemented
148	#2192	C₂×C₈₉	1	4	yes	complemented
149	#2178	C₂×C₈₉	1	1	yes	characteristic, complemented
150	#2170	C₂×C₈₉	1	1	yes	characteristic, complemented
151	C₈₉	#33858	1	1	yes	characteristic, direct product factor, Sylow
152	#576	—	2	2	no	—
153	#576	C₂²×C₈₉	1	4	yes	complemented
154	#509	—	2	4	no	—
155	#509	C₂²×C₈₉	1	4	yes	complemented
156	#508	—	2	2	no	—
157	#508	C₂²×C₈₉	1	1	yes	characteristic
158	C₂×D₁₆	—	2	2	no	—
159	C₂×D₁₆	C₂²×C₈₉	1	2	yes	complemented
160	C₂×D₁₆	C₂²×C₈₉	1	2	yes	complemented
161	#504	C₂²×C₈₉	1	2	yes	complemented
162	#493	C₂²×C₈₉	1	2	yes	complemented
163	#472	—	2	2	no	—
164	#472	C₂²×C₈₉	1	4	yes	complemented
165	#453	—	2	4	no	—
166	#443	C₂²×C₈₉	1	1	yes	characteristic
167	#443	C₂²×C₈₉	1	2	yes	complemented
168	#368	C₂²×C₈₉	1	1	yes	characteristic, complemented
169	#364	C₂²×C₈₉	1	1	yes	characteristic, complemented
170	#361	C₂²×C₈₉	1	4	yes	complemented
171	#359	C₂²×C₈₉	1	2	yes	complemented
172	#350	C₂²×C₈₉	1	2	yes	complemented
173	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic, complemented
174	H(5,𝔽₂)	—	2	4	no	—
175	AGL(1,ℤ/8)	—	2	4	no	—
176	AGL(1,ℤ/8)	—	2	4	no	—
177	AGL(1,ℤ/8)	—	4	4	no	—
178	AGL(1,ℤ/8)	—	4	4	no	—
179	#135	—	2	4	no	—
180	#135	—	4	2	no	—
181	#135	D₄×C₈₉	1	1	yes	characteristic
182	#135	C₂³×C₈₉	1	2	yes	—
183	C₂×D₈	—	2	4	no	—
184	C₂×D₈	—	2	2	no	—
185	C₂×D₈	—	4	2	no	—
186	C₂×D₈	C₂³×C₈₉	1	4	yes	—
187	#131	—	2	2	no	—
188	#131	C₂³×C₈₉	1	2	yes	—
189	#127	—	2	2	no	—
190	Q₃₂	D₄×C₈₉	1	1	yes	characteristic, complemented
191	SD₃₂	—	2	4	no	—
192	SD₃₂	D₄×C₈₉	1	2	yes	complemented
193	D₁₆	—	2	4	no	—
194	D₁₆	—	2	2	no	—
195	D₁₆	—	2	2	no	—
196	D₁₆	—	4	2	no	—
197	D₁₆	D₄×C₈₉	1	1	yes	characteristic, complemented
198	M₃₂	—	2	2	no	—
199	M₃₂	C₂³×C₈₉	1	2	yes	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
202	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
203	#108	D₄×C₈₉	1	1	yes	characteristic
204	#108	C₂³×C₈₉	1	2	yes	—
205	#104	—	2	2	no	—
206	#104	D₄×C₈₉	1	2	yes	—
207	#100	—	2	4	no	—
208	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
209	#41	—	2	2	no	—
210	#41	—	4	4	no	—
211	#41	—	4	4	no	—
212	#41	C₂×D₄×C₈₉	1	2	yes	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	8	2	no	—
217	C₂×D₄	—	8	4	no	—
218	Q₁₆	D₈×C₈₉	1	1	yes	characteristic
219	Q₁₆	C₂×D₄×C₈₉	1	2	yes	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	4	4	no	—
222	SD₁₆	—	4	4	no	—
223	SD₁₆	D₈×C₈₉	1	2	yes	—
224	D₈	—	2	4	no	—
225	D₈	—	2	4	no	—
226	D₈	—	4	4	no	—
227	D₈	—	4	4	no	—
228	D₈	—	4	2	no	—
229	D₈	—	4	2	no	—
230	D₈	D₈×C₈₉	1	1	yes	characteristic
231	D₈	C₂×D₄×C₈₉	1	2	yes	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	C₂×D₄×C₈₉	1	2	yes	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
239	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	1	no	—
243	C₁₆	C₂×D₄×C₈₉	1	1	yes	characteristic
244	C₁₆	C₂×D₄×C₈₉	1	1	yes	characteristic
245	C₂³	—	4	4	no	—
246	C₂³	—	8	4	no	—
247	Q₈	—	2	2	no	—
248	Q₈	C₂×D₈×C₈₉	1	2	yes	—
249	D₄	—	2	2	no	—
250	D₄	—	2	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	8	4	no	—
255	D₄	—	8	4	no	—
256	D₄	—	16	4	no	—
257	D₄	C₂×D₈×C₈₉	1	2	yes	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	C₂×D₈×C₈₉	1	1	yes	characteristic
266	C₈	C₂×D₈×C₈₉	1	1	yes	characteristic
267	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
268	C₈	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	4	no	—
273	C₂²	—	32	4	no	—
274	C₂²	#576×C₈₉	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	D₄²×C₈₉	1	1	yes	characteristic
279	C₄	C₂²×D₈×C₈₉	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	4	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	16	4	no	—
284	C₂	D₄×D₈×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂×C₄×C₈₉ ◁ #443×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3415	C₂	semidirect
2. C₈₉×#3292	C₂	semidirect
3. C₈₉×#2246	C₂	semidirect
4. C₈₉×#2221	C₂	semidirect
5. C₈₉×#2213	C₂	semidirect
6. C₈₉×#2192	C₂	semidirect
7. C₈₉×#2178	C₂	semidirect
8. C₈₉×#2170	C₂	semidirect
10. #576×C₈₉	C₂²	semidirect
12. #509×C₈₉	C₂²	semidirect
16. C₂×D₁₆×C₈₉	C₂²	semidirect
17. C₂×D₁₆×C₈₉	C₂²	semidirect
18. #504×C₈₉	C₂²	semidirect
19. #493×C₈₉	C₂²	semidirect
21. #472×C₈₉	C₂²	semidirect
24. #443×C₈₉	C₂²	semidirect
25. #368×C₈₉	C₂²	semidirect
26. #364×C₈₉	C₂²	semidirect
27. #361×C₈₉	C₂²	semidirect
28. #359×C₈₉	C₂²	semidirect
29. #350×C₈₉	C₂²	semidirect
30. C₄×C₁₆×C₈₉	C₂²	semidirect
47. Q₃₂×C₈₉	D₄	semidirect
49. SD₃₂×C₈₉	D₄	semidirect
54. D₁₆×C₈₉	D₄	semidirect
137. #33858	C₈₉	direct (N × H)
143. #3415	C₂×C₈₉	semidirect
144. #3292	C₂×C₈₉	semidirect
145. #2246	C₂×C₈₉	semidirect
146. #2221	C₂×C₈₉	semidirect
147. #2213	C₂×C₈₉	semidirect
148. #2192	C₂×C₈₉	semidirect
149. #2178	C₂×C₈₉	semidirect
150. #2170	C₂×C₈₉	semidirect
151. C₈₉	#33858	direct (N × H)
153. #576	C₂²×C₈₉	semidirect
155. #509	C₂²×C₈₉	semidirect
159. C₂×D₁₆	C₂²×C₈₉	semidirect
160. C₂×D₁₆	C₂²×C₈₉	semidirect
161. #504	C₂²×C₈₉	semidirect
162. #493	C₂²×C₈₉	semidirect
164. #472	C₂²×C₈₉	semidirect
167. #443	C₂²×C₈₉	semidirect
168. #368	C₂²×C₈₉	semidirect
169. #364	C₂²×C₈₉	semidirect
170. #361	C₂²×C₈₉	semidirect
171. #359	C₂²×C₈₉	semidirect
172. #350	C₂²×C₈₉	semidirect
173. C₄×C₁₆	C₂²×C₈₉	semidirect
190. Q₃₂	D₄×C₈₉	semidirect
192. SD₃₂	D₄×C₈₉	semidirect
197. D₁₆	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3415	2
C₈₉×#3292	2
C₈₉×#2246	2
C₈₉×#2221	2
C₈₉×#2213	2
C₈₉×#2192	2
C₈₉×#2178	2
C₈₉×#2170	2
#33858	89

Subgroups — C89×#33858 #36411706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33858 #36411706