Subgroups — C₂×C₄×C₁₆×D₈₉ #36707049

Order 22784.

1022 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₉	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₈₉	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×D₈₉	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₈₉	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₈₉	C₄	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	2	yes	—
11	C₂×C₁₆×D₈₉	C₄	1	16	yes	direct product factor
12	C₂×C₁₆×D₈₉	C₂²	1	8	yes	—
13	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₈₉	C₄	1	4	yes	—
17	C₄×C₈×D₈₉	C₂²	1	4	yes	—
18	C₄×#13083549	C₂²	1	4	yes	complemented
19	C₁₆×Dic₈₉	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₈₉	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₈₉	C₈	1	2	yes	—
24	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₈₉	C₈	1	8	yes	—
28	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₈₉	C₂×C₄	1	8	yes	—
30	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₈₉	C₂³	1	2	yes	—
32	C₂×C₄×Dic₈₉	C₈	1	2	yes	—
33	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₈₉	C₈	1	4	yes	—
35	C₄²×D₈₉	C₂×C₄	1	4	yes	—
36	C₂×#13083549	C₂×C₄	1	8	yes	complemented
37	C₂×#13083549	C₂³	1	4	yes	—
38	C₁₆×D₈₉	C₂×C₄	1	32	yes	direct product factor
39	C₄×#10623494	C₂×C₄	1	4	yes	—
40	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
41	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
42	C₈×Dic₈₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₈₉	C₂×C₄	1	8	yes	complemented
47	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
48	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
53	C₂×C₄×D₈₉	C₁₆	1	16	yes	direct product factor
54	C₂×C₄×D₈₉	C₄²	1	4	yes	—
55	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
57	C₂×C₄×D₈₉	C₂×C₈	1	8	yes	—
58	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
59	C₂×#10623494	C₄²	1	4	yes	—
60	C₂×#10623494	C₂×C₈	1	4	yes	—
61	C₂×#10623494	C₂²×C₄	1	2	yes	—
62	C₂×#10623494	C₂²×C₄	1	2	yes	—
63	C₈×D₈₉	C₄²	1	8	yes	—
64	C₈×D₈₉	C₂×C₈	1	16	yes	—
65	C₈×D₈₉	C₂²×C₄	1	8	yes	—
66	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
67	C₄×Dic₈₉	C₂×C₈	1	8	yes	—
68	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
69	#13083549	C₂²×C₄	1	8	yes	complemented
70	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
77	C₄²×C₈₉	C₂×C₈	1	2	yes	—
78	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
79	C₁₆×C₈₉	C₂²×C₄	1	8	yes	complemented
80	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
81	C₂²×D₈₉	C₄×C₈	1	4	yes	—
82	C₂²×D₈₉	C₂×C₁₆	1	8	yes	—
83	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₈₉	C₂×C₁₆	1	8	yes	complemented
85	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
86	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₈₉	C₄×C₈	1	4	yes	—
91	C₂×Dic₈₉	C₂×C₁₆	1	8	yes	complemented
92	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
93	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
94	C₄×D₈₉	C₄×C₈	1	8	yes	—
95	C₄×D₈₉	C₂×C₁₆	1	32	yes	direct product factor
96	C₄×D₈₉	C₂×C₄²	1	4	yes	—
97	C₄×D₈₉	C₂²×C₈	1	4	yes	—
98	C₈×C₈₉	C₂×C₄²	1	2	yes	—
99	C₈×C₈₉	C₂²×C₈	1	4	yes	—
100	C₈×C₈₉	C₂³×C₄	1	2	yes	—
101	#10623494	C₂×C₄²	1	4	yes	—
102	#10623494	C₂²×C₈	1	4	yes	—
103	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
104	C₂²×C₈₉	C₂²×C₁₆	1	4	yes	—
105	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
106	C₂×D₈₉	C₄×C₁₆	1	16	yes	direct product factor
107	C₂×D₈₉	C₂×C₄×C₈	1	4	yes	—
108	C₂×D₈₉	C₂²×C₁₆	1	8	yes	—
109	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
110	C₄×C₈₉	C₂²×C₁₆	1	8	yes	complemented
111	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
113	Dic₈₉	C₂×C₄×C₈	1	4	yes	—
114	Dic₈₉	C₂²×C₁₆	1	8	yes	complemented
115	C₂²×C₄×C₁₆	—	89	1	no	maximal, Sylow
116	C₂×C₈₉	C₂×C₄×C₁₆	1	4	yes	complemented
117	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
118	C₂×C₈₉	C₂³×C₁₆	1	2	yes	—
119	D₈₉	C₂×C₄×C₁₆	1	8	yes	direct product factor
120	C₂³×C₁₆	—	89	2	no	—
121	C₂²×C₄×C₈	—	89	1	no	—
122	C₂×C₄×C₁₆	—	89	2	no	—
123	C₂×C₄×C₁₆	—	89	1	no	—
124	C₂×C₄×C₁₆	—	89	8	no	—
125	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, direct product factor
126	C₈₉	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₈	—	89	1	no	—
128	C₂³×C₈	—	89	1	no	—
129	C₂²×C₄²	—	89	1	no	—
130	C₂²×C₁₆	—	89	2	no	—
131	C₂²×C₁₆	—	89	16	no	—
132	C₂²×C₁₆	—	89	8	no	—
133	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
134	C₂×C₄×C₈	—	89	1	no	—
135	C₂×C₄×C₈	—	89	1	no	—
136	C₂×C₄×C₈	—	89	1	no	—
137	C₂×C₄×C₈	—	89	4	no	—
138	C₂×C₄×C₈	—	89	4	no	—
139	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
140	C₄×C₁₆	—	89	8	no	—
141	C₄×C₁₆	—	89	4	no	—
142	C₄×C₁₆	C₂×D₈₉	1	4	yes	direct product factor
143	C₂³×C₄	—	89	2	no	—
144	C₂³×C₄	—	89	1	no	—
145	C₂²×C₈	—	89	1	no	—
146	C₂²×C₈	—	89	1	no	—
147	C₂²×C₈	—	89	8	no	—
148	C₂²×C₈	—	89	4	no	—
149	C₂²×C₈	—	89	2	no	—
150	C₂²×C₈	—	89	8	no	—
151	C₂²×C₈	—	89	2	no	—
152	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
154	C₂×C₄²	—	89	2	no	—
155	C₂×C₄²	—	89	1	no	—
156	C₂×C₄²	—	89	4	no	—
157	C₂×C₄²	—	89	4	no	—
158	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
159	C₂×C₁₆	—	89	8	no	—
160	C₂×C₁₆	—	89	4	no	—
161	C₂×C₁₆	—	89	32	no	—
162	C₂×C₁₆	C₄×D₈₉	1	8	yes	direct product factor
163	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
164	C₄×C₈	—	89	4	no	—
165	C₄×C₈	—	89	2	no	—
166	C₄×C₈	—	89	2	no	—
167	C₄×C₈	—	89	4	no	—
168	C₄×C₈	C₄×D₈₉	1	2	yes	—
169	C₄×C₈	C₂²×D₈₉	1	2	yes	—
170	C₂⁴	—	89	1	no	—
171	C₂²×C₄	—	89	2	no	—
172	C₂²×C₄	—	89	1	no	—
173	C₂²×C₄	—	89	8	no	—
174	C₂²×C₄	—	89	2	no	—
175	C₂²×C₄	—	89	4	no	—
176	C₂²×C₄	—	89	2	no	—
177	C₂²×C₄	—	89	16	no	—
178	C₂²×C₄	—	89	4	no	—
179	C₂²×C₄	C₈×D₈₉	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
181	C₂×C₈	—	89	4	no	—
182	C₂×C₈	—	89	2	no	—
183	C₂×C₈	—	89	2	no	—
184	C₂×C₈	—	89	4	no	—
185	C₂×C₈	—	89	16	no	—
186	C₂×C₈	—	89	8	no	—
187	C₂×C₈	—	89	8	no	—
188	C₂×C₈	C₈×D₈₉	1	4	yes	—
189	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
191	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
192	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
193	C₄²	—	89	8	no	—
194	C₄²	—	89	2	no	—
195	C₄²	—	89	2	no	—
196	C₄²	C₈×D₈₉	1	2	yes	—
197	C₄²	C₂×C₄×D₈₉	1	2	yes	—
198	C₁₆	—	89	8	no	—
199	C₁₆	C₂×C₄×D₈₉	1	8	yes	direct product factor
200	C₂³	—	89	8	no	—
201	C₂³	—	89	4	no	—
202	C₂³	—	89	2	no	—
203	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
204	C₂×C₄	—	89	8	no	—
205	C₂×C₄	—	89	4	no	—
206	C₂×C₄	—	89	2	no	—
207	C₂×C₄	—	89	4	no	—
208	C₂×C₄	—	89	32	no	—
209	C₂×C₄	—	89	4	no	—
210	C₂×C₄	—	89	8	no	—
211	C₂×C₄	—	89	4	no	—
212	C₂×C₄	C₁₆×D₈₉	1	8	yes	direct product factor
213	C₂×C₄	C₄²×D₈₉	1	2	yes	—
214	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
215	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
216	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
217	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
218	C₈	—	89	4	no	—
219	C₈	—	89	4	no	—
220	C₈	C₄²×D₈₉	1	2	yes	—
221	C₈	C₂×C₈×D₈₉	1	4	yes	—
222	C₈	C₂²×C₄×D₈₉	1	2	yes	—
223	C₂²	—	89	4	no	—
224	C₂²	—	89	8	no	—
225	C₂²	—	89	16	no	—
226	C₂²	C₄×C₈×D₈₉	1	2	yes	—
227	C₂²	C₂×C₁₆×D₈₉	1	4	yes	—
228	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
229	C₄	—	89	4	no	—
230	C₄	—	89	8	no	—
231	C₄	C₄×C₈×D₈₉	1	2	yes	—
232	C₄	C₂×C₁₆×D₈₉	1	8	yes	direct product factor
233	C₄	C₂×C₄²×D₈₉	1	1	yes	characteristic
234	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
235	C₂	—	89	8	no	—
236	C₂	C₄×C₁₆×D₈₉	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₈×D₈₉	1	1	yes	characteristic
238	C₂	C₂²×C₁₆×D₈₉	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₈₉
commutator, hypocenter	C₈₉	C₂²×C₄×C₁₆
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
LCS	G ▷ C₈₉
UCS	C₁ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#13083549	C₂	semidirect
4. C₂×C₁₆×Dic₈₉	C₂	semidirect
5. C₄×C₁₆×D₈₉	C₂	direct (N × H)
6. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
11. C₂×C₁₆×D₈₉	C₄	direct (N × H)
18. C₄×#13083549	C₂²	semidirect
19. C₁₆×Dic₈₉	C₂²	semidirect
22. C₄×C₁₆×C₈₉	C₂²	semidirect
36. C₂×#13083549	C₂×C₄	semidirect
38. C₁₆×D₈₉	C₂×C₄	direct (N × H)
46. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
53. C₂×C₄×D₈₉	C₁₆	direct (N × H)
69. #13083549	C₂²×C₄	semidirect
79. C₁₆×C₈₉	C₂²×C₄	semidirect
84. C₂×C₄×C₈₉	C₂×C₁₆	semidirect
91. C₂×Dic₈₉	C₂×C₁₆	semidirect
95. C₄×D₈₉	C₂×C₁₆	direct (N × H)
106. C₂×D₈₉	C₄×C₁₆	direct (N × H)
110. C₄×C₈₉	C₂²×C₁₆	semidirect
114. Dic₈₉	C₂²×C₁₆	semidirect
116. C₂×C₈₉	C₂×C₄×C₁₆	semidirect
119. D₈₉	C₂×C₄×C₁₆	direct (N × H)
125. C₂×C₄×C₁₆	D₈₉	direct (N × H)
126. C₈₉	C₂²×C₄×C₁₆	semidirect
142. C₄×C₁₆	C₂×D₈₉	direct (N × H)
162. C₂×C₁₆	C₄×D₈₉	direct (N × H)
199. C₁₆	C₂×C₄×D₈₉	direct (N × H)
212. C₂×C₄	C₁₆×D₈₉	direct (N × H)
232. C₄	C₂×C₁₆×D₈₉	direct (N × H)
236. C₂	C₄×C₁₆×D₈₉	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×C₁₆×D₈₉	2
C₂×C₄×C₈×D₈₉	2
C₂×C₄×#13083549	2
C₂×C₁₆×Dic₈₉	2
C₄×C₁₆×D₈₉	2
C₂×C₄×C₁₆×C₈₉	2
C₂²×C₄×C₁₆	89

Subgroups — C2×C4×C16×D89 #36707049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₆×D₈₉ #36707049