Subgroups — C₂²×#578×C₈₉ #36440934

Order 22784.

10774 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₂×#578×C₈₉	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#137×C₈₉	C₂	1	6	yes	maximal, complemented
4	C₂²×#135×C₈₉	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₁₆×C₈₉	C₂	1	3	yes	maximal, complemented
6	C₂²×#131×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×2₋¹⁺⁴×C₈₉	—	2	24	no	—
8	C₂×2₋¹⁺⁴×C₈₉	C₂²	1	48	yes	complemented
9	C₂²×#41×C₈₉	—	2	6	no	—
10	C₂²×#41×C₈₉	C₂²	1	6	yes	—
11	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₈₉	—	2	2	no	—
13	C₂³×Q₈×C₈₉	C₂²	1	6	yes	—
14	#578×C₈₉	C₂²	1	256	yes	direct product factor
15	C₂×#137×C₈₉	C₂²	1	144	yes	complemented
16	C₂×#135×C₈₉	C₂²	1	72	yes	complemented
17	C₂²×Q₁₆×C₈₉	C₂²	1	3	yes	—
18	C₂²×Q₁₆×C₈₉	C₂²	1	6	yes	—
19	C₂²×Q₁₆×C₈₉	C₂²	1	72	yes	complemented
20	C₂²×SD₁₆×C₈₉	C₂²	1	6	yes	—
21	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#131×C₈₉	C₂²	1	24	yes	complemented
23	C₂²×M₁₆×C₈₉	C₂²	1	3	yes	—
24	C₂³×C₈×C₈₉	C₂²	1	3	yes	—
25	2₋¹⁺⁴×C₈₉	—	2	192	no	—
26	2₋¹⁺⁴×C₈₉	C₂³	1	128	yes	complemented
27	C₂×#41×C₈₉	—	2	144	no	—
28	C₂×#41×C₈₉	—	2	36	no	—
29	C₂×#41×C₈₉	D₄	1	12	yes	—
30	C₂×#41×C₈₉	C₂³	1	72	yes	—
31	C₂×#41×C₈₉	C₂³	1	12	yes	—
32	C₂²×Q₈×C₈₉	—	2	48	no	—
33	C₂²×Q₈×C₈₉	—	2	6	no	—
34	C₂²×Q₈×C₈₉	—	2	36	no	—
35	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₈₉	C₂³	1	6	yes	—
37	C₂²×Q₈×C₈₉	C₂³	1	72	yes	—
38	C₂²×D₄×C₈₉	—	2	6	no	—
39	C₂²×D₄×C₈₉	D₄	1	3	yes	—
40	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
41	C₂³×C₄×C₈₉	—	2	6	no	—
42	C₂³×C₄×C₈₉	—	2	6	no	—
43	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
44	#137×C₈₉	C₂³	1	384	yes	complemented
45	#135×C₈₉	C₂³	1	192	yes	complemented
46	C₂×Q₁₆×C₈₉	C₂³	1	36	yes	—
47	C₂×Q₁₆×C₈₉	C₂³	1	72	yes	—
48	C₂×Q₁₆×C₈₉	C₂³	1	192	yes	complemented
49	C₂×SD₁₆×C₈₉	C₂³	1	72	yes	—
50	C₂×D₈×C₈₉	C₂³	1	12	yes	—
51	#131×C₈₉	C₂³	1	64	yes	complemented
52	C₂×M₁₆×C₈₉	C₂³	1	36	yes	—
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	36	yes	—
56	C₂⁴×C₈₉	—	2	2	no	—
57	C₂⁴×C₈₉	C₂×D₄	1	3	yes	—
58	#41×C₈₉	—	2	384	no	—
59	#41×C₈₉	—	2	144	no	—
60	#41×C₈₉	C₂×D₄	1	48	yes	—
61	#41×C₈₉	C₂⁴	1	96	yes	—
62	#41×C₈₉	C₂⁴	1	16	yes	—
63	C₂×Q₈×C₈₉	—	2	128	no	—
64	C₂×Q₈×C₈₉	—	2	72	no	—
65	C₂×Q₈×C₈₉	—	2	18	no	—
66	C₂×Q₈×C₈₉	—	2	144	no	—
67	C₂×Q₈×C₈₉	C₂×D₄	1	12	yes	—
68	C₂×Q₈×C₈₉	C₂⁴	1	36	yes	—
69	C₂×Q₈×C₈₉	C₂⁴	1	96	yes	—
70	C₂×D₄×C₈₉	—	2	72	no	—
71	C₂×D₄×C₈₉	—	2	6	no	—
72	C₂×D₄×C₈₉	C₂×D₄	1	36	yes	—
73	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
74	C₂²×C₄×C₈₉	—	2	72	no	—
75	C₂²×C₄×C₈₉	—	2	6	no	—
76	C₂²×C₄×C₈₉	—	2	72	no	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	18	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₈₉	C₂⁴	1	18	yes	—
80	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₈₉	C₂⁴	1	48	yes	—
82	Q₁₆×C₈₉	C₂⁴	1	96	yes	—
83	SD₁₆×C₈₉	C₂⁴	1	96	yes	—
84	D₈×C₈₉	C₂⁴	1	16	yes	—
85	M₁₆×C₈₉	C₂⁴	1	48	yes	—
86	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
87	C₂×C₈×C₈₉	C₂⁴	1	18	yes	—
88	C₂×C₈×C₈₉	C₂⁴	1	48	yes	—
89	C₂³×C₈₉	—	2	12	no	—
90	C₂³×C₈₉	—	2	12	no	—
91	C₂³×C₈₉	—	4	8	no	—
92	C₂³×C₈₉	C₂²×D₄	1	18	yes	—
93	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₈₉	—	2	96	no	—
95	Q₈×C₈₉	—	2	36	no	—
96	Q₈×C₈₉	C₂²×D₄	1	16	yes	—
97	Q₈×C₈₉	C₂⁵	1	24	yes	—
98	D₄×C₈₉	—	2	96	no	—
99	D₄×C₈₉	—	2	12	no	—
100	D₄×C₈₉	C₂²×D₄	1	48	yes	—
101	D₄×C₈₉	C₂⁵	1	8	yes	—
102	C₂×C₄×C₈₉	—	2	36	no	—
103	C₂×C₄×C₈₉	—	2	96	no	—
104	C₂×C₄×C₈₉	—	2	96	no	—
105	C₂×C₄×C₈₉	C₂²×D₄	1	18	yes	—
106	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
107	C₂×C₄×C₈₉	C₂²×D₄	1	36	yes	—
108	C₂×C₄×C₈₉	C₂⁵	1	3	yes	—
109	C₂×C₄×C₈₉	C₂⁵	1	12	yes	—
110	C₈×C₈₉	C₂⁵	1	4	yes	—
111	C₈×C₈₉	C₂⁵	1	12	yes	—
112	C₂²×C₈₉	—	2	36	no	—
113	C₂²×C₈₉	—	2	8	no	—
114	C₂²×C₈₉	—	4	24	no	—
115	C₂²×C₈₉	#578	1	4	yes	direct product factor
116	C₂²×C₈₉	C₂³×D₄	1	12	yes	—
117	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
118	C₄×C₈₉	—	2	24	no	—
119	C₄×C₈₉	C₂³×D₄	1	12	yes	—
120	C₄×C₈₉	C₂³×D₄	1	3	yes	—
121	C₄×C₈₉	C₂⁶	1	1	yes	characteristic
122	C₂²×#578	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₈₉	—	2	12	no	—
124	C₂×C₈₉	—	4	8	no	—
125	C₂×C₈₉	C₂×#578	1	6	yes	direct product factor
126	C₂×C₈₉	C₂⁴×D₄	1	1	yes	characteristic
127	C₂²×2₋¹⁺⁴	C₂×C₈₉	1	2	yes	complemented
128	C₂×#578	C₂×C₈₉	1	48	yes	direct product factor
129	C₂²×#137	C₂×C₈₉	1	6	yes	complemented
130	C₂²×#135	C₂×C₈₉	1	3	yes	complemented
131	C₂³×Q₁₆	C₂×C₈₉	1	3	yes	complemented
132	C₂²×#131	C₂×C₈₉	1	1	yes	characteristic, complemented
133	C₈₉	C₂²×#578	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×2₋¹⁺⁴	—	2	24	no	—
135	C₂×2₋¹⁺⁴	C₂²×C₈₉	1	48	yes	complemented
136	C₂²×#41	—	2	6	no	—
137	C₂²×#41	C₂²×C₈₉	1	6	yes	—
138	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
139	C₂³×Q₈	—	2	2	no	—
140	C₂³×Q₈	C₂²×C₈₉	1	6	yes	—
141	#578	C₂²×C₈₉	1	256	yes	direct product factor
142	C₂×#137	C₂²×C₈₉	1	144	yes	complemented
143	C₂×#135	C₂²×C₈₉	1	72	yes	complemented
144	C₂²×Q₁₆	C₂²×C₈₉	1	3	yes	—
145	C₂²×Q₁₆	C₂²×C₈₉	1	6	yes	—
146	C₂²×Q₁₆	C₂²×C₈₉	1	72	yes	complemented
147	C₂²×SD₁₆	C₂²×C₈₉	1	6	yes	—
148	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
149	C₂×#131	C₂²×C₈₉	1	24	yes	complemented
150	C₂²×M₁₆	C₂²×C₈₉	1	3	yes	—
151	C₂³×C₈	C₂²×C₈₉	1	3	yes	—
152	2₋¹⁺⁴	—	2	192	no	—
153	2₋¹⁺⁴	C₂³×C₈₉	1	128	yes	complemented
154	C₂×#41	—	2	144	no	—
155	C₂×#41	—	2	36	no	—
156	C₂×#41	D₄×C₈₉	1	12	yes	—
157	C₂×#41	C₂³×C₈₉	1	72	yes	—
158	C₂×#41	C₂³×C₈₉	1	12	yes	—
159	C₂²×Q₈	—	2	48	no	—
160	C₂²×Q₈	—	2	6	no	—
161	C₂²×Q₈	—	2	36	no	—
162	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
163	C₂²×Q₈	C₂³×C₈₉	1	6	yes	—
164	C₂²×Q₈	C₂³×C₈₉	1	72	yes	—
165	C₂²×D₄	—	2	6	no	—
166	C₂²×D₄	D₄×C₈₉	1	3	yes	—
167	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
168	C₂³×C₄	—	2	6	no	—
169	C₂³×C₄	—	2	6	no	—
170	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
171	#137	C₂³×C₈₉	1	384	yes	complemented
172	#135	C₂³×C₈₉	1	192	yes	complemented
173	C₂×Q₁₆	C₂³×C₈₉	1	36	yes	—
174	C₂×Q₁₆	C₂³×C₈₉	1	72	yes	—
175	C₂×Q₁₆	C₂³×C₈₉	1	192	yes	complemented
176	C₂×SD₁₆	C₂³×C₈₉	1	72	yes	—
177	C₂×D₈	C₂³×C₈₉	1	12	yes	—
178	#131	C₂³×C₈₉	1	64	yes	complemented
179	C₂×M₁₆	C₂³×C₈₉	1	36	yes	—
180	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
182	C₂²×C₈	C₂³×C₈₉	1	36	yes	—
183	C₂⁴	—	2	2	no	—
184	C₂⁴	C₂×D₄×C₈₉	1	3	yes	—
185	#41	—	2	384	no	—
186	#41	—	2	144	no	—
187	#41	C₂×D₄×C₈₉	1	48	yes	—
188	#41	C₂⁴×C₈₉	1	96	yes	—
189	#41	C₂⁴×C₈₉	1	16	yes	—
190	C₂×Q₈	—	2	128	no	—
191	C₂×Q₈	—	2	72	no	—
192	C₂×Q₈	—	2	18	no	—
193	C₂×Q₈	—	2	144	no	—
194	C₂×Q₈	C₂×D₄×C₈₉	1	12	yes	—
195	C₂×Q₈	C₂⁴×C₈₉	1	36	yes	—
196	C₂×Q₈	C₂⁴×C₈₉	1	96	yes	—
197	C₂×D₄	—	2	72	no	—
198	C₂×D₄	—	2	6	no	—
199	C₂×D₄	C₂×D₄×C₈₉	1	36	yes	—
200	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
201	C₂²×C₄	—	2	6	no	—
202	C₂²×C₄	—	2	72	no	—
203	C₂²×C₄	—	2	72	no	—
204	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
205	C₂²×C₄	C₂×D₄×C₈₉	1	18	yes	—
206	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₈₉	1	18	yes	—
208	Q₁₆	C₂⁴×C₈₉	1	48	yes	—
209	Q₁₆	C₂⁴×C₈₉	1	96	yes	—
210	SD₁₆	C₂⁴×C₈₉	1	96	yes	—
211	D₈	C₂⁴×C₈₉	1	16	yes	—
212	M₁₆	C₂⁴×C₈₉	1	48	yes	—
213	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
214	C₂×C₈	C₂⁴×C₈₉	1	18	yes	—
215	C₂×C₈	C₂⁴×C₈₉	1	48	yes	—
216	C₂³	—	2	12	no	—
217	C₂³	—	2	12	no	—
218	C₂³	—	4	8	no	—
219	C₂³	C₂²×D₄×C₈₉	1	18	yes	—
220	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
221	Q₈	—	2	96	no	—
222	Q₈	—	2	36	no	—
223	Q₈	C₂²×D₄×C₈₉	1	16	yes	—
224	Q₈	C₂⁵×C₈₉	1	24	yes	—
225	D₄	—	2	96	no	—
226	D₄	—	2	12	no	—
227	D₄	C₂²×D₄×C₈₉	1	48	yes	—
228	D₄	C₂⁵×C₈₉	1	8	yes	—
229	C₂×C₄	—	2	36	no	—
230	C₂×C₄	—	2	96	no	—
231	C₂×C₄	—	2	96	no	—
232	C₂×C₄	C₂²×D₄×C₈₉	1	18	yes	—
233	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
234	C₂×C₄	C₂²×D₄×C₈₉	1	36	yes	—
235	C₂×C₄	C₂⁵×C₈₉	1	3	yes	—
236	C₂×C₄	C₂⁵×C₈₉	1	12	yes	—
237	C₈	C₂⁵×C₈₉	1	4	yes	—
238	C₈	C₂⁵×C₈₉	1	12	yes	—
239	C₂²	—	2	36	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	4	24	no	—
242	C₂²	#578×C₈₉	1	4	yes	direct product factor
243	C₂²	C₂³×D₄×C₈₉	1	12	yes	—
244	C₂²	C₂³×D₄×C₈₉	1	3	yes	—
245	C₄	—	2	24	no	—
246	C₄	C₂³×D₄×C₈₉	1	12	yes	—
247	C₄	C₂³×D₄×C₈₉	1	3	yes	—
248	C₄	C₂⁶×C₈₉	1	1	yes	characteristic
249	C₂	—	2	12	no	—
250	C₂	—	4	8	no	—
251	C₂	C₂×#578×C₈₉	1	6	yes	direct product factor
252	C₂	C₂⁴×D₄×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₈₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴×C₈₉	C₂	semidirect
2. C₂×#578×C₈₉	C₂	direct (N × H)
3. C₂²×#137×C₈₉	C₂	semidirect
4. C₂²×#135×C₈₉	C₂	semidirect
5. C₂³×Q₁₆×C₈₉	C₂	semidirect
6. C₂²×#131×C₈₉	C₂	semidirect
8. C₂×2₋¹⁺⁴×C₈₉	C₂²	semidirect
14. #578×C₈₉	C₂²	direct (N × H)
15. C₂×#137×C₈₉	C₂²	semidirect
16. C₂×#135×C₈₉	C₂²	semidirect
19. C₂²×Q₁₆×C₈₉	C₂²	semidirect
22. C₂×#131×C₈₉	C₂²	semidirect
26. 2₋¹⁺⁴×C₈₉	C₂³	semidirect
44. #137×C₈₉	C₂³	semidirect
45. #135×C₈₉	C₂³	semidirect
48. C₂×Q₁₆×C₈₉	C₂³	semidirect
51. #131×C₈₉	C₂³	semidirect
115. C₂²×C₈₉	#578	direct (N × H)
122. C₂²×#578	C₈₉	direct (N × H)
125. C₂×C₈₉	C₂×#578	direct (N × H)
127. C₂²×2₋¹⁺⁴	C₂×C₈₉	semidirect
128. C₂×#578	C₂×C₈₉	direct (N × H)
129. C₂²×#137	C₂×C₈₉	semidirect
130. C₂²×#135	C₂×C₈₉	semidirect
131. C₂³×Q₁₆	C₂×C₈₉	semidirect
132. C₂²×#131	C₂×C₈₉	semidirect
133. C₈₉	C₂²×#578	direct (N × H)
135. C₂×2₋¹⁺⁴	C₂²×C₈₉	semidirect
141. #578	C₂²×C₈₉	direct (N × H)
142. C₂×#137	C₂²×C₈₉	semidirect
143. C₂×#135	C₂²×C₈₉	semidirect
146. C₂²×Q₁₆	C₂²×C₈₉	semidirect
149. C₂×#131	C₂²×C₈₉	semidirect
153. 2₋¹⁺⁴	C₂³×C₈₉	semidirect
171. #137	C₂³×C₈₉	semidirect
172. #135	C₂³×C₈₉	semidirect
175. C₂×Q₁₆	C₂³×C₈₉	semidirect
178. #131	C₂³×C₈₉	semidirect
242. C₂²	#578×C₈₉	direct (N × H)
251. C₂	C₂×#578×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₈₉	2
C₂×#578×C₈₉	2
C₂²×#137×C₈₉	2
C₂²×#135×C₈₉	2
C₂³×Q₁₆×C₈₉	2
C₂²×#131×C₈₉	2
C₂²×#578	89

Subgroups — C22×#578×C89 #36440934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#578×C₈₉ #36440934