Subgroups — C₈₉×#35390 #36413238

Order 22784.

5610 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×#534×C₈₉	C₂	1	3	yes	maximal, complemented
2	C₂²×#120×C₈₉	C₂	1	3	yes	maximal, complemented
3	C₈₉×#2539	C₂	1	12	yes	maximal, complemented
4	C₈₉×#2433	C₂	1	12	yes	maximal, complemented
5	C₈₉×#2273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁶×C₈₉	C₂²	1	1	yes	characteristic
7	C₂³×D₄×C₈₉	C₂²	1	3	yes	—
8	C₂³×D₄×C₈₉	C₂²	1	6	yes	complemented
9	#534×C₈₉	—	2	24	no	—
10	C₂×#127×C₈₉	C₂²	1	8	yes	complemented
11	C₂×#123×C₈₉	C₂²	1	12	yes	complemented
12	C₂×#121×C₈₉	C₂²	1	12	yes	complemented
13	C₂×#120×C₈₉	—	2	24	no	—
14	C₂×#120×C₈₉	C₂²	1	24	yes	complemented
15	C₂×#120×C₈₉	C₂²	1	12	yes	complemented
16	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
17	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	6	yes	complemented
18	#399×C₈₉	C₂²	1	8	yes	complemented
19	#392×C₈₉	C₂²	1	24	yes	complemented
20	#390×C₈₉	C₂²	1	24	yes	complemented
21	C₄×C₂²⋊C₄×C₈₉	C₂²	1	12	yes	complemented
22	C₂⁵×C₈₉	—	2	12	no	—
23	C₂⁵×C₈₉	—	4	8	no	—
24	C₂⁵×C₈₉	C₂³	1	6	yes	—
25	C₂⁵×C₈₉	C₂³	1	3	yes	—
26	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
27	C₂²×D₄×C₈₉	—	2	12	no	—
28	C₂²×D₄×C₈₉	—	2	24	no	—
29	C₂²×D₄×C₈₉	—	2	24	no	—
30	C₂²×D₄×C₈₉	—	4	24	no	—
31	C₂²×D₄×C₈₉	C₂³	1	12	yes	—
32	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
33	C₂²×D₄×C₈₉	C₂³	1	12	yes	complemented
34	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
36	#127×C₈₉	—	4	16	no	—
37	#123×C₈₉	—	2	48	no	—
38	#121×C₈₉	—	2	48	no	—
39	#120×C₈₉	—	2	96	no	—
40	#120×C₈₉	—	2	48	no	—
41	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	complemented
42	C₂×C₂²⋊C₄×C₈₉	—	2	24	no	—
43	C₂×C₂²⋊C₄×C₈₉	D₄	1	24	yes	complemented
44	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
45	C₂×C₂²⋊C₄×C₈₉	C₂³	1	12	yes	—
46	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	complemented
47	C₂×C₄²×C₈₉	C₂³	1	8	yes	complemented
48	#95×C₈₉	C₂³	1	8	yes	complemented
49	C₂⁴×C₈₉	—	2	12	no	—
50	C₂⁴×C₈₉	—	2	24	no	—
51	C₂⁴×C₈₉	—	2	24	no	—
52	C₂⁴×C₈₉	—	2	6	no	—
53	C₂⁴×C₈₉	—	2	12	no	—
54	C₂⁴×C₈₉	—	4	4	no	—
55	C₂⁴×C₈₉	—	4	12	no	—
56	C₂⁴×C₈₉	—	4	12	no	—
57	C₂⁴×C₈₉	—	4	96	no	—
58	C₂⁴×C₈₉	—	4	16	no	—
59	C₂⁴×C₈₉	C₂×D₄	1	12	yes	—
60	C₂⁴×C₈₉	C₂⁴	1	6	yes	—
61	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
62	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
63	C₂×D₄×C₈₉	—	2	24	no	—
64	C₂×D₄×C₈₉	—	2	24	no	—
65	C₂×D₄×C₈₉	—	2	24	no	—
66	C₂×D₄×C₈₉	—	2	48	no	—
67	C₂×D₄×C₈₉	—	2	48	no	—
68	C₂×D₄×C₈₉	—	2	24	no	—
69	C₂×D₄×C₈₉	—	4	24	no	—
70	C₂×D₄×C₈₉	—	4	24	no	—
71	C₂×D₄×C₈₉	—	4	96	no	—
72	C₂×D₄×C₈₉	—	4	48	no	—
73	C₂²×C₄×C₈₉	—	2	6	no	—
74	C₂²×C₄×C₈₉	—	2	12	no	—
75	C₂²×C₄×C₈₉	—	4	12	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	complemented
77	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
79	C₄⋊C₄×C₈₉	—	2	24	no	—
80	C₂²⋊C₄×C₈₉	—	2	48	no	—
81	C₂²⋊C₄×C₈₉	—	2	48	no	—
82	C₂²⋊C₄×C₈₉	C₂×D₄	1	48	yes	complemented
83	C₄²×C₈₉	—	4	8	no	—
84	C₂³×C₈₉	—	2	12	no	—
85	C₂³×C₈₉	—	2	48	no	—
86	C₂³×C₈₉	—	2	12	no	—
87	C₂³×C₈₉	—	2	12	no	—
88	C₂³×C₈₉	—	2	12	no	—
89	C₂³×C₈₉	—	2	6	no	—
90	C₂³×C₈₉	—	2	24	no	—
91	C₂³×C₈₉	—	4	24	no	—
92	C₂³×C₈₉	—	4	96	no	—
93	C₂³×C₈₉	—	4	24	no	—
94	C₂³×C₈₉	—	4	24	no	—
95	C₂³×C₈₉	—	4	128	no	—
96	C₂³×C₈₉	—	4	24	no	—
97	C₂³×C₈₉	—	4	12	no	—
98	C₂³×C₈₉	—	8	24	no	—
99	C₂³×C₈₉	#127	1	16	yes	complemented
100	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
101	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
102	C₂³×C₈₉	H(5,𝔽₂)	1	8	yes	—
103	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
104	D₄×C₈₉	—	4	96	no	—
105	D₄×C₈₉	—	4	48	no	—
106	D₄×C₈₉	—	8	24	no	—
107	C₂×C₄×C₈₉	—	2	24	no	—
108	C₂×C₄×C₈₉	—	2	6	no	—
109	C₂×C₄×C₈₉	—	2	12	no	—
110	C₂×C₄×C₈₉	—	4	24	no	—
111	C₂×C₄×C₈₉	—	4	12	no	—
112	C₂×C₄×C₈₉	C₂²×D₄	1	24	yes	—
113	C₂²×C₈₉	—	2	12	no	—
114	C₂²×C₈₉	—	2	24	no	—
115	C₂²×C₈₉	—	2	24	no	—
116	C₂²×C₈₉	—	4	16	no	—
117	C₂²×C₈₉	—	4	96	no	—
118	C₂²×C₈₉	—	4	4	no	—
119	C₂²×C₈₉	—	4	12	no	—
120	C₂²×C₈₉	—	4	6	no	—
121	C₂²×C₈₉	—	4	12	no	—
122	C₂²×C₈₉	—	4	24	no	—
123	C₂²×C₈₉	—	8	24	no	—
124	C₂²×C₈₉	C₂×#127	1	12	yes	—
125	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
126	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
127	C₄×C₈₉	—	2	24	no	—
128	C₄×C₈₉	—	4	6	no	—
129	#35390	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₈₉	—	2	12	no	—
131	C₂×C₈₉	—	4	8	no	—
132	C₂×C₈₉	—	8	6	no	—
133	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
134	C₂×C₈₉	#3484	1	6	yes	—
135	C₂×#534	C₂×C₈₉	1	3	yes	complemented
136	C₂²×#120	C₂×C₈₉	1	3	yes	complemented
137	#2539	C₂×C₈₉	1	12	yes	complemented
138	#2433	C₂×C₈₉	1	12	yes	complemented
139	#2273	C₂×C₈₉	1	1	yes	characteristic, complemented
140	C₈₉	#35390	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁶	C₂²×C₈₉	1	1	yes	characteristic
142	C₂³×D₄	C₂²×C₈₉	1	3	yes	—
143	C₂³×D₄	C₂²×C₈₉	1	6	yes	complemented
144	#534	—	2	24	no	—
145	C₂×#127	C₂²×C₈₉	1	8	yes	complemented
146	C₂×#123	C₂²×C₈₉	1	12	yes	complemented
147	C₂×#121	C₂²×C₈₉	1	12	yes	complemented
148	C₂×#120	—	2	24	no	—
149	C₂×#120	C₂²×C₈₉	1	24	yes	complemented
150	C₂×#120	C₂²×C₈₉	1	12	yes	complemented
151	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
152	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	6	yes	complemented
153	#399	C₂²×C₈₉	1	8	yes	complemented
154	#392	C₂²×C₈₉	1	24	yes	complemented
155	#390	C₂²×C₈₉	1	24	yes	complemented
156	C₄×C₂²⋊C₄	C₂²×C₈₉	1	12	yes	complemented
157	C₂⁵	—	2	12	no	—
158	C₂⁵	—	4	8	no	—
159	C₂⁵	C₂³×C₈₉	1	6	yes	—
160	C₂⁵	C₂³×C₈₉	1	3	yes	—
161	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
162	C₂²×D₄	—	2	12	no	—
163	C₂²×D₄	—	2	24	no	—
164	C₂²×D₄	—	2	24	no	—
165	C₂²×D₄	—	4	24	no	—
166	C₂²×D₄	C₂³×C₈₉	1	12	yes	—
167	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
168	C₂²×D₄	C₂³×C₈₉	1	12	yes	complemented
169	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
170	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
171	#127	—	4	16	no	—
172	#123	—	2	48	no	—
173	#121	—	2	48	no	—
174	#120	—	2	96	no	—
175	#120	—	2	48	no	—
176	C₂×C₄⋊C₄	C₂³×C₈₉	1	12	yes	complemented
177	C₂×C₂²⋊C₄	—	2	24	no	—
178	C₂×C₂²⋊C₄	D₄×C₈₉	1	24	yes	complemented
179	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
180	C₂×C₂²⋊C₄	C₂³×C₈₉	1	12	yes	—
181	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	complemented
182	C₂×C₄²	C₂³×C₈₉	1	8	yes	complemented
183	#95	C₂³×C₈₉	1	8	yes	complemented
184	C₂⁴	—	2	12	no	—
185	C₂⁴	—	2	24	no	—
186	C₂⁴	—	2	24	no	—
187	C₂⁴	—	2	6	no	—
188	C₂⁴	—	2	12	no	—
189	C₂⁴	—	4	4	no	—
190	C₂⁴	—	4	12	no	—
191	C₂⁴	—	4	12	no	—
192	C₂⁴	—	4	96	no	—
193	C₂⁴	—	4	16	no	—
194	C₂⁴	C₂×D₄×C₈₉	1	12	yes	—
195	C₂⁴	C₂⁴×C₈₉	1	6	yes	—
196	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
197	C₂⁴	C₂⁴×C₈₉	1	3	yes	—
198	C₂×D₄	—	2	24	no	—
199	C₂×D₄	—	2	24	no	—
200	C₂×D₄	—	2	24	no	—
201	C₂×D₄	—	2	48	no	—
202	C₂×D₄	—	2	48	no	—
203	C₂×D₄	—	2	24	no	—
204	C₂×D₄	—	4	24	no	—
205	C₂×D₄	—	4	24	no	—
206	C₂×D₄	—	4	96	no	—
207	C₂×D₄	—	4	48	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	—	4	12	no	—
211	C₂²×C₄	C₂×D₄×C₈₉	1	24	yes	complemented
212	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
213	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
214	C₄⋊C₄	—	2	24	no	—
215	C₂²⋊C₄	—	2	48	no	—
216	C₂²⋊C₄	—	2	48	no	—
217	C₂²⋊C₄	C₂×D₄×C₈₉	1	48	yes	complemented
218	C₄²	—	4	8	no	—
219	C₂³	—	2	12	no	—
220	C₂³	—	2	48	no	—
221	C₂³	—	2	12	no	—
222	C₂³	—	2	12	no	—
223	C₂³	—	2	12	no	—
224	C₂³	—	2	6	no	—
225	C₂³	—	2	24	no	—
226	C₂³	—	4	24	no	—
227	C₂³	—	4	96	no	—
228	C₂³	—	4	24	no	—
229	C₂³	—	4	24	no	—
230	C₂³	—	4	128	no	—
231	C₂³	—	4	24	no	—
232	C₂³	—	4	12	no	—
233	C₂³	—	8	24	no	—
234	C₂³	#127×C₈₉	1	16	yes	complemented
235	C₂³	C₂²×D₄×C₈₉	1	12	yes	—
236	C₂³	C₂²×D₄×C₈₉	1	6	yes	—
237	C₂³	H(5,𝔽₂)×C₈₉	1	8	yes	—
238	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
239	D₄	—	4	96	no	—
240	D₄	—	4	48	no	—
241	D₄	—	8	24	no	—
242	C₂×C₄	—	2	24	no	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	4	24	no	—
246	C₂×C₄	—	4	12	no	—
247	C₂×C₄	C₂²×D₄×C₈₉	1	24	yes	—
248	C₂²	—	2	12	no	—
249	C₂²	—	2	24	no	—
250	C₂²	—	2	24	no	—
251	C₂²	—	4	16	no	—
252	C₂²	—	4	96	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	12	no	—
255	C₂²	—	4	6	no	—
256	C₂²	—	4	12	no	—
257	C₂²	—	4	24	no	—
258	C₂²	—	8	24	no	—
259	C₂²	C₂×#127×C₈₉	1	12	yes	—
260	C₂²	C₂³×D₄×C₈₉	1	3	yes	—
261	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
262	C₄	—	2	24	no	—
263	C₄	—	4	6	no	—
264	C₂	—	2	12	no	—
265	C₂	—	4	8	no	—
266	C₂	—	8	6	no	—
267	C₂	C₂²×#127×C₈₉	1	1	yes	characteristic
268	C₂	C₈₉×#3484	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#534×C₈₉	C₂	semidirect
2. C₂²×#120×C₈₉	C₂	semidirect
3. C₈₉×#2539	C₂	semidirect
4. C₈₉×#2433	C₂	semidirect
5. C₈₉×#2273	C₂	semidirect
8. C₂³×D₄×C₈₉	C₂²	semidirect
10. C₂×#127×C₈₉	C₂²	semidirect
11. C₂×#123×C₈₉	C₂²	semidirect
12. C₂×#121×C₈₉	C₂²	semidirect
14. C₂×#120×C₈₉	C₂²	semidirect
15. C₂×#120×C₈₉	C₂²	semidirect
17. C₂²×C₂²⋊C₄×C₈₉	C₂²	semidirect
18. #399×C₈₉	C₂²	semidirect
19. #392×C₈₉	C₂²	semidirect
20. #390×C₈₉	C₂²	semidirect
21. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
33. C₂²×D₄×C₈₉	C₂³	semidirect
41. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
43. C₂×C₂²⋊C₄×C₈₉	D₄	semidirect
46. C₂×C₂²⋊C₄×C₈₉	C₂³	semidirect
47. C₂×C₄²×C₈₉	C₂³	semidirect
48. #95×C₈₉	C₂³	semidirect
76. C₂²×C₄×C₈₉	C₂×D₄	semidirect
82. C₂²⋊C₄×C₈₉	C₂×D₄	semidirect
99. C₂³×C₈₉	#127	semidirect
129. #35390	C₈₉	direct (N × H)
135. C₂×#534	C₂×C₈₉	semidirect
136. C₂²×#120	C₂×C₈₉	semidirect
137. #2539	C₂×C₈₉	semidirect
138. #2433	C₂×C₈₉	semidirect
139. #2273	C₂×C₈₉	semidirect
140. C₈₉	#35390	direct (N × H)
143. C₂³×D₄	C₂²×C₈₉	semidirect
145. C₂×#127	C₂²×C₈₉	semidirect
146. C₂×#123	C₂²×C₈₉	semidirect
147. C₂×#121	C₂²×C₈₉	semidirect
149. C₂×#120	C₂²×C₈₉	semidirect
150. C₂×#120	C₂²×C₈₉	semidirect
152. C₂²×C₂²⋊C₄	C₂²×C₈₉	semidirect
153. #399	C₂²×C₈₉	semidirect
154. #392	C₂²×C₈₉	semidirect
155. #390	C₂²×C₈₉	semidirect
156. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
168. C₂²×D₄	C₂³×C₈₉	semidirect
176. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
178. C₂×C₂²⋊C₄	D₄×C₈₉	semidirect
181. C₂×C₂²⋊C₄	C₂³×C₈₉	semidirect
182. C₂×C₄²	C₂³×C₈₉	semidirect
183. #95	C₂³×C₈₉	semidirect
211. C₂²×C₄	C₂×D₄×C₈₉	semidirect
217. C₂²⋊C₄	C₂×D₄×C₈₉	semidirect
234. C₂³	#127×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#534×C₈₉	2
C₂²×#120×C₈₉	2
C₈₉×#2539	2
C₈₉×#2433	2
C₈₉×#2273	2
#35390	89

Subgroups — C89×#35390 #36413238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35390 #36413238