Subgroups — #2253×D₈₉ #36709906

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×D₈₉	C₂	1	2	yes	maximal
2	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022530	C₂	1	2	yes	maximal, complemented
4	#26022529	C₂	1	2	yes	maximal, complemented
5	#26022528	C₂	1	2	yes	maximal, complemented
6	#367×D₈₉	C₂	1	2	yes	maximal
7	#366×D₈₉	C₂	1	2	yes	maximal
8	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
10	#125×D₈₉	—	2	2	no	—
11	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
13	#19152000	C₂²	1	2	yes	—
14	#19151999	C₂²	1	4	yes	—
15	#19151998	—	2	2	no	—
16	#19151997	—	2	2	no	—
17	#107×D₈₉	C₂²	1	4	yes	—
18	#106×D₈₉	—	2	2	no	—
19	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#19151426	C₂²	1	2	yes	—
21	#19151425	C₂²	1	4	yes	—
22	#19151424	C₂²	1	2	yes	—
23	#19151423	C₂²	1	4	yes	—
24	C₄×#13083549	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
27	#499×C₈₉	C₂²	1	2	yes	—
28	#367×C₈₉	C₂²	1	2	yes	—
29	#366×C₈₉	C₂²	1	2	yes	—
30	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
32	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
33	#15668004	—	2	2	no	—
34	#15668003	—	2	4	no	—
35	#15668002	—	4	4	no	—
36	C₄⋊C₄×D₈₉	—	2	4	no	—
37	C₄⋊C₄×D₈₉	—	4	4	no	—
38	C₄²×D₈₉	D₄	1	1	yes	characteristic
39	C₂×#13083549	C₂³	1	1	yes	characteristic
40	C₂×#13083549	C₂³	1	1	yes	characteristic
41	C₁₆×D₈₉	Q₈	1	4	yes	—
42	C₁₆×D₈₉	Q₈	1	4	yes	—
43	#15667946	C₂³	1	4	yes	—
44	#15667945	—	2	4	no	—
45	#15667944	—	2	2	no	—
46	#15667943	—	2	4	no	—
47	C₄×#10623494	D₄	1	1	yes	characteristic
48	C₈×Dic₈₉	D₄	1	1	yes	characteristic
49	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
50	#125×C₈₉	—	2	2	no	—
51	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
53	#107×C₈₉	C₂³	1	4	yes	—
54	#106×C₈₉	—	2	2	no	—
55	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₈₉	—	4	4	no	—
57	C₂×C₄×D₈₉	D₈	1	2	yes	—
58	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
60	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
61	C₈×D₈₉	Q₁₆	1	4	yes	—
62	C₈×D₈₉	C₂×Q₈	1	2	yes	—
63	C₈×D₈₉	C₂×Q₈	1	2	yes	—
64	C₄⋊Dic₈₉	—	2	4	no	—
65	C₄⋊Dic₈₉	—	4	4	no	—
66	#13083553	—	4	4	no	—
67	#13083553	—	4	4	no	—
68	#13083553	—	4	4	no	—
69	C₄×Dic₈₉	D₈	1	2	yes	—
70	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
71	#13083549	C₂×Q₈	1	2	yes	—
72	#13083549	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₈₉	—	2	4	no	—
76	C₄⋊C₄×C₈₉	—	4	4	no	—
77	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
79	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
80	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	—	4	4	no	—
82	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₈₉	—	4	4	no	—
85	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
86	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
87	C₄×D₈₉	—	2	4	no	—
88	C₄×D₈₉	—	8	8	no	—
89	C₄×D₈₉	C₄⋊Q₈	1	2	yes	—
90	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
91	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
92	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
94	#10623494	C₄⋊Q₈	1	2	yes	—
95	#10623494	C₂×Q₁₆	1	2	yes	—
96	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₈₉	#500	1	2	yes	—
98	C₂×D₈₉	#508	1	4	yes	—
99	C₄×C₈₉	—	2	2	no	—
100	C₄×C₈₉	—	8	4	no	—
101	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₈₉	—	2	2	no	—
104	Dic₈₉	—	8	4	no	—
105	Dic₈₉	#500	1	2	yes	—
106	C₂×#2253	—	89	1	no	maximal, Sylow
107	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
108	C₂×C₈₉	C₂×#508	1	2	yes	—
109	D₈₉	#2253	1	4	yes	direct product factor
110	C₂×#499	—	89	2	no	—
111	#2253	—	89	2	no	—
112	#2253	—	89	2	no	—
113	#2253	—	89	2	no	—
114	#2253	—	89	1	no	—
115	#2253	D₈₉	1	1	yes	characteristic, direct product factor
116	C₂×#367	—	89	2	no	—
117	C₂×#366	—	89	2	no	—
118	C₂×C₄×C₁₆	—	89	1	no	—
119	C₈₉	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
120	C₂×#125	—	178	2	no	—
121	C₂²×C₁₆	—	89	1	no	—
122	C₂²×C₁₆	—	89	1	no	—
123	#499	—	89	4	no	—
124	#499	—	89	2	no	—
125	#499	—	178	2	no	—
126	#499	—	178	2	no	—
127	#499	C₂×D₈₉	1	2	yes	—
128	C₂×#107	—	89	4	no	—
129	C₂×#106	—	178	2	no	—
130	C₂×C₄×C₈	—	89	1	no	—
131	#367	—	89	2	no	—
132	#367	—	89	4	no	—
133	#367	C₂×D₈₉	1	2	yes	—
134	#366	—	89	2	no	—
135	#366	—	89	4	no	—
136	#366	C₂×D₈₉	1	2	yes	—
137	C₄×C₁₆	—	89	1	no	—
138	C₄×C₁₆	—	89	1	no	—
139	C₄×C₁₆	—	89	1	no	—
140	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
141	C₂²×C₈	—	89	1	no	—
142	C₂²×C₈	—	89	1	no	—
143	#125	—	2	2	no	—
144	#125	—	356	4	no	—
145	#125	—	178	4	no	—
146	#125	—	178	2	no	—
147	C₂×C₄⋊C₄	—	356	4	no	—
148	C₂×C₄⋊C₄	—	178	4	no	—
149	C₂×C₄²	—	89	1	no	—
150	C₂×C₁₆	—	89	1	no	—
151	C₂×C₁₆	—	89	1	no	—
152	C₂×C₁₆	—	89	4	no	—
153	C₂×C₁₆	—	89	4	no	—
154	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
156	#107	—	89	4	no	—
157	#107	—	178	4	no	—
158	#107	C₂²×D₈₉	1	4	yes	—
159	#106	—	2	2	no	—
160	#106	—	178	4	no	—
161	#106	—	178	2	no	—
162	C₄×C₈	—	89	1	no	—
163	C₄×C₈	—	89	1	no	—
164	C₄×C₈	—	89	1	no	—
165	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
166	C₂²×C₄	—	89	1	no	—
167	C₂²×C₄	—	89	2	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	2	no	—
173	C₂×C₈	—	89	2	no	—
174	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	356	4	no	—
179	C₄⋊C₄	—	356	4	no	—
180	C₄⋊C₄	—	356	4	no	—
181	C₄⋊C₄	—	356	4	no	—
182	C₄⋊C₄	—	178	4	no	—
183	C₄²	—	89	2	no	—
184	C₄²	—	89	1	no	—
185	C₄²	D₄×D₈₉	1	1	yes	characteristic
186	C₁₆	—	89	2	no	—
187	C₁₆	—	89	2	no	—
188	C₁₆	Q₈×D₈₉	1	2	yes	—
189	C₁₆	Q₈×D₈₉	1	2	yes	—
190	C₂³	—	89	1	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	89	1	no	—
193	C₂×C₄	—	89	2	no	—
194	C₂×C₄	—	89	2	no	—
195	C₂×C₄	—	89	2	no	—
196	C₂×C₄	—	356	4	no	—
197	C₂×C₄	—	178	4	no	—
198	C₂×C₄	—	712	8	no	—
199	C₂×C₄	D₈×D₈₉	1	2	yes	—
200	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
201	C₈	—	89	2	no	—
202	C₈	—	89	2	no	—
203	C₈	Q₁₆×D₈₉	1	2	yes	—
204	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
205	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
206	C₂²	—	89	2	no	—
207	C₂²	—	89	4	no	—
208	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	89	2	no	—
212	C₄	—	178	2	no	—
213	C₄	—	712	4	no	—
214	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
215	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
216	C₂	—	89	4	no	—
217	C₂	#500×D₈₉	1	1	yes	characteristic
218	C₂	#508×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2253	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2253	D₈₉
hypocenter	C₈₉	C₂×#2253
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #26022531	C₂	semidirect
3. #26022530	C₂	semidirect
4. #26022529	C₂	semidirect
5. #26022528	C₂	semidirect
9. C₈₉×#2253	C₂	semidirect
109. D₈₉	#2253	direct (N × H)
115. #2253	D₈₉	direct (N × H)
119. C₈₉	C₂×#2253	semidirect

Maximal subgroups

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

subgroup	index
#499×D₈₉	2
#26022531	2
#26022530	2
#26022529	2
#26022528	2
#367×D₈₉	2
#366×D₈₉	2
C₄×C₁₆×D₈₉	2
C₈₉×#2253	2
C₂×#2253	89

Subgroups — #2253×D89 #36709906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2253×D₈₉ #36709906