Subgroups — #2252×D₈₉ #36709902

Order 22784.

558 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₈₉	C₂	1	2	yes	maximal
2	#26022527	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022526	C₂	1	4	yes	maximal, complemented
4	#26022525	C₂	1	2	yes	maximal, complemented
5	#366×D₈₉	C₂	1	4	yes	maximal
6	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊Q₈×D₈₉	—	2	2	no	—
9	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
10	#19152003	C₂²	1	2	yes	—
11	#19152002	—	2	2	no	—
12	#19152002	C₂²	1	4	yes	—
13	#19152001	—	2	2	no	—
14	#107×D₈₉	—	2	2	no	—
15	#107×D₈₉	C₂²	1	4	yes	—
16	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
17	#19151424	C₂²	1	4	yes	—
18	#19151423	C₂²	1	8	yes	—
19	C₄×#13083549	C₂²	1	1	yes	characteristic
20	C₁₆×Dic₈₉	C₂²	1	2	yes	—
21	#500×C₈₉	C₂²	1	2	yes	—
22	#366×C₈₉	C₂²	1	4	yes	—
23	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×Q₈×D₈₉	—	4	2	no	—
25	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
26	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
27	#15668012	—	2	2	no	—
28	#15668011	—	2	4	no	—
29	#15668011	—	4	2	no	—
30	#15668010	—	4	2	no	—
31	C₄⋊C₄×D₈₉	—	2	4	no	—
32	C₄⋊C₄×D₈₉	—	4	2	no	—
33	C₄²×D₈₉	D₄	1	1	yes	characteristic
34	C₂×#13083549	C₂³	1	2	yes	—
35	C₁₆×D₈₉	Q₈	1	8	yes	complemented
36	#15667946	—	2	2	no	—
37	#15667946	C₂³	1	4	yes	—
38	#15667945	—	2	4	no	—
39	#15667945	—	2	4	no	—
40	C₄×#10623494	D₄	1	1	yes	characteristic
41	C₈×Dic₈₉	D₄	1	1	yes	characteristic
42	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₈₉	—	2	2	no	—
44	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
45	#107×C₈₉	—	2	2	no	—
46	#107×C₈₉	C₂³	1	4	yes	—
47	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂×Dic₁₇₈	—	4	4	no	—
50	Q₈×D₈₉	—	8	8	no	—
51	C₂×C₄×D₈₉	—	4	4	no	—
52	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
53	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
54	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
56	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
57	C₈×D₈₉	Q₁₆	1	4	yes	—
58	C₈×D₈₉	C₂×Q₈	1	4	yes	—
59	C₄⋊Dic₈₉	—	2	4	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	#13083553	—	4	4	no	—
62	#13083553	—	4	4	no	—
63	C₄×Dic₈₉	D₈	1	1	yes	characteristic
64	C₄×Dic₈₉	D₈	1	1	yes	characteristic
65	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
66	#13083549	C₂×Q₈	1	4	yes	complemented
67	C₂×Q₈×C₈₉	—	4	2	no	—
68	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₈₉	—	2	4	no	—
71	C₄⋊C₄×C₈₉	—	4	2	no	—
72	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₈₉	C₂×Q₈	1	4	yes	complemented
74	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₈₉	—	8	4	no	—
76	C₂×C₄×C₈₉	—	4	4	no	—
77	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₈₉	—	4	4	no	—
81	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
82	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
83	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
84	C₄×D₈₉	—	8	8	no	—
85	C₄×D₈₉	D₁₆	1	4	yes	—
86	C₄×D₈₉	Q₃₂	1	4	yes	—
87	C₄×D₈₉	C₄⋊Q₈	1	2	yes	—
88	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
89	Dic₁₇₈	—	8	4	no	—
90	Dic₁₇₈	—	8	8	no	—
91	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
92	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
93	#10623494	C₄⋊Q₈	1	2	yes	—
94	#10623494	C₂×Q₁₆	1	2	yes	—
95	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
96	C₂×D₈₉	#500	1	2	yes	—
97	C₂×D₈₉	C₂×D₁₆	1	2	yes	—
98	C₂×D₈₉	C₂×Q₃₂	1	2	yes	—
99	C₄×C₈₉	—	8	4	no	—
100	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
101	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
102	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₈₉	—	8	4	no	—
105	Dic₈₉	#500	1	2	yes	—
106	Dic₈₉	C₂×D₁₆	1	2	yes	—
107	Dic₈₉	C₂×Q₃₂	1	2	yes	—
108	C₂×#2252	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
110	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
111	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
112	D₈₉	#2252	1	4	yes	direct product factor
113	C₂×#500	—	89	2	no	—
114	#2252	—	89	2	no	—
115	#2252	—	89	4	no	—
116	#2252	—	89	1	no	—
117	#2252	D₈₉	1	1	yes	characteristic, direct product factor
118	C₂×#366	—	89	4	no	—
119	C₂×C₄×C₁₆	—	89	1	no	—
120	C₈₉	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄⋊Q₈	—	178	2	no	—
122	C₂²×C₁₆	—	89	2	no	—
123	#500	—	89	4	no	—
124	#500	—	89	2	no	—
125	#500	—	178	2	no	—
126	#500	—	178	2	no	—
127	#500	C₂×D₈₉	1	2	yes	—
128	C₂×#107	—	89	4	no	—
129	C₂×#107	—	178	2	no	—
130	C₂×C₄×C₈	—	89	1	no	—
131	#366	—	89	4	no	—
132	#366	—	89	8	no	—
133	#366	C₂×D₈₉	1	4	yes	—
134	C₄×C₁₆	—	89	2	no	—
135	C₄×C₁₆	—	89	1	no	—
136	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
137	C₂²×Q₈	—	356	2	no	—
138	C₂²×C₈	—	89	1	no	—
139	C₂²×C₈	—	89	1	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	356	2	no	—
142	C₄⋊Q₈	—	356	2	no	—
143	C₄⋊Q₈	—	178	4	no	—
144	C₄⋊Q₈	—	178	2	no	—
145	C₂×C₄⋊C₄	—	356	2	no	—
146	C₂×C₄⋊C₄	—	178	4	no	—
147	C₂×C₄²	—	89	1	no	—
148	C₂×C₁₆	—	89	2	no	—
149	C₂×C₁₆	—	89	8	no	—
150	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	89	4	no	—
153	#107	—	178	4	no	—
154	#107	—	178	4	no	—
155	#107	—	178	2	no	—
156	#107	C₂²×D₈₉	1	4	yes	—
157	C₄×C₈	—	89	1	no	—
158	C₄×C₈	—	89	1	no	—
159	C₄×C₈	—	89	1	no	—
160	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	356	4	no	—
163	C₂×Q₈	—	356	2	no	—
164	C₂×Q₈	—	712	8	no	—
165	C₂²×C₄	—	89	1	no	—
166	C₂²×C₄	—	89	1	no	—
167	C₂²×C₄	—	89	1	no	—
168	C₂²×C₄	—	356	4	no	—
169	C₂×C₈	—	89	1	no	—
170	C₂×C₈	—	89	1	no	—
171	C₂×C₈	—	89	4	no	—
172	C₂×C₈	—	89	4	no	—
173	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
174	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	356	4	no	—
178	C₄⋊C₄	—	356	4	no	—
179	C₄⋊C₄	—	356	2	no	—
180	C₄⋊C₄	—	178	4	no	—
181	C₄²	—	89	1	no	—
182	C₄²	—	89	1	no	—
183	C₄²	—	89	1	no	—
184	C₄²	D₄×D₈₉	1	1	yes	characteristic
185	C₁₆	—	89	4	no	—
186	C₁₆	Q₈×D₈₉	1	4	yes	complemented
187	C₂³	—	89	1	no	—
188	Q₈	—	8	4	no	—
189	Q₈	—	712	8	no	—
190	Q₈	—	712	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	89	1	no	—
193	C₂×C₄	—	89	1	no	—
194	C₂×C₄	—	89	1	no	—
195	C₂×C₄	—	89	2	no	—
196	C₂×C₄	—	89	4	no	—
197	C₂×C₄	—	89	4	no	—
198	C₂×C₄	—	89	2	no	—
199	C₂×C₄	—	356	4	no	—
200	C₂×C₄	—	712	8	no	—
201	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
202	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
204	C₈	—	89	2	no	—
205	C₈	—	89	2	no	—
206	C₈	Q₁₆×D₈₉	1	2	yes	—
207	C₈	C₂×Q₈×D₈₉	1	2	yes	—
208	C₂²	—	89	2	no	—
209	C₂²	—	89	2	no	—
210	C₂²	—	89	2	no	—
211	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
212	C₄	—	8	4	no	—
213	C₄	—	89	2	no	—
214	C₄	—	89	2	no	—
215	C₄	—	89	2	no	—
216	C₄	—	712	4	no	—
217	C₄	D₁₆×D₈₉	1	2	yes	—
218	C₄	Q₃₂×D₈₉	1	2	yes	—
219	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
220	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
221	C₂	—	89	4	no	—
222	C₂	#500×D₈₉	1	1	yes	characteristic
223	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic
224	C₂	C₂×Q₃₂×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2252	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2252	D₈₉
hypocenter	C₈₉	C₂×#2252
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #26022527	C₂	semidirect
3. #26022526	C₂	semidirect
4. #26022525	C₂	semidirect
7. C₈₉×#2252	C₂	semidirect
35. C₁₆×D₈₉	Q₈	semidirect
66. #13083549	C₂×Q₈	semidirect
73. C₁₆×C₈₉	C₂×Q₈	semidirect
112. D₈₉	#2252	direct (N × H)
117. #2252	D₈₉	direct (N × H)
120. C₈₉	C₂×#2252	semidirect
186. C₁₆	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#500×D₈₉	2
#26022527	2
#26022526	2
#26022525	2
#366×D₈₉	2
C₄×C₁₆×D₈₉	2
C₈₉×#2252	2
C₂×#2252	89

Subgroups — #2252×D89 #36709902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2252×D₈₉ #36709902