Subgroups — #1747×D₈₉ #36551929

Order 22784.

930 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#405×D₈₉	C₂	1	3	yes	maximal
2	#26020008	C₂	1	1	yes	maximal, characteristic, complemented
3	#26020007	C₂	1	3	yes	maximal, complemented
4	#26020006	C₂	1	3	yes	maximal, complemented
5	#338×D₈₉	C₂	1	1	yes	maximal, characteristic
6	H(3,ℤ/4)×D₈₉	C₂	1	3	yes	maximal
7	C₈₉×#1747	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	C₂²	1	3	yes	—
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	#117×D₈₉	C₂²	1	3	yes	—
11	#19151556	C₂²	1	3	yes	—
12	#19151555	C₄	1	6	yes	complemented
13	#19151554	C₂²	1	3	yes	—
14	#19151553	C₂²	1	3	yes	—
15	#19151552	C₄	1	6	yes	complemented
16	#97×D₈₉	C₄	1	6	yes	complemented
17	C₄×C₈×D₈₉	C₄	1	6	yes	complemented
18	#19151356	C₂²	1	3	yes	—
19	#19151355	C₂²	1	3	yes	—
20	#19151354	C₂²	1	6	yes	—
21	#405×C₈₉	C₂²	1	3	yes	—
22	#338×C₈₉	C₂²	1	1	yes	characteristic
23	H(3,ℤ/4)×C₈₉	C₂²	1	3	yes	—
24	C₂²×C₄×D₈₉	C₂³	1	1	yes	characteristic
25	C₂×#13083558	C₂³	1	3	yes	—
26	C₂×#13083557	C₂×C₄	1	6	yes	—
27	M₁₆×D₈₉	—	2	6	no	—
28	M₁₆×D₈₉	—	2	6	no	—
29	C₂²×#10623494	C₂³	1	1	yes	characteristic
30	C₂×C₈×D₈₉	C₂×C₄	1	6	yes	—
31	C₂×C₈×D₈₉	C₂×C₄	1	3	yes	—
32	C₂×C₈×D₈₉	C₂×C₄	1	3	yes	—
33	C₂×C₈×D₈₉	C₂×C₄	1	3	yes	—
34	C₂×C₈×D₈₉	C₂×C₄	1	3	yes	—
35	#15667971	C₂³	1	3	yes	—
36	#15667970	—	2	3	no	—
37	#15667969	C₂×C₄	1	6	yes	—
38	#15667968	—	2	3	no	—
39	C₄⋊C₄×D₈₉	—	2	3	no	—
40	C₂²⋊C₄×D₈₉	—	2	3	no	—
41	C₄²×D₈₉	C₂×C₄	1	6	yes	—
42	#15667923	C₂×C₄	1	6	yes	complemented
43	#15667922	C₂×C₄	1	6	yes	complemented
44	#15667922	C₂×C₄	1	6	yes	complemented
45	C₄×#10623494	C₂×C₄	1	6	yes	complemented
46	C₈×Dic₈₉	C₂×C₄	1	6	yes	complemented
47	C₈×Dic₈₉	C₂×C₄	1	6	yes	complemented
48	C₂×M₁₆×C₈₉	C₂³	1	3	yes	—
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	#117×C₈₉	C₂³	1	3	yes	—
51	#97×C₈₉	C₂×C₄	1	6	yes	complemented
52	C₄×C₈×C₈₉	C₂×C₄	1	6	yes	complemented
53	C₂³×D₈₉	#41	1	1	yes	characteristic
54	C₂²×Dic₈₉	#41	1	1	yes	characteristic
55	C₂×C₄×D₈₉	—	2	6	no	—
56	C₂×C₄×D₈₉	C₂²×C₄	1	3	yes	—
57	C₂×C₄×D₈₉	C₂²×C₄	1	3	yes	—
58	C₂×C₄×D₈₉	#41	1	3	yes	—
59	C₂×C₄×D₈₉	#41	1	3	yes	—
60	#13083558	—	2	6	no	—
61	#13083557	—	2	12	no	—
62	C₂×#10623494	C₂²×C₄	1	6	yes	—
63	C₂×#10623494	C₂²×C₄	1	3	yes	—
64	C₂×#10623494	C₂²×C₄	1	3	yes	—
65	C₈×D₈₉	—	2	6	no	—
66	C₈×D₈₉	—	2	6	no	—
67	C₈×D₈₉	C₄²	1	3	yes	—
68	C₈×D₈₉	C₄²	1	3	yes	—
69	C₈×D₈₉	C₄²	1	6	yes	—
70	C₈×D₈₉	C₄²	1	1	yes	characteristic
71	C₈×D₈₉	C₄²	1	3	yes	—
72	C₄⋊Dic₈₉	—	2	3	no	—
73	#13083553	—	2	6	no	—
74	C₂²⋊Dic₈₉	—	2	3	no	—
75	#13083551	—	2	6	no	—
76	C₄×Dic₈₉	—	2	6	no	—
77	C₄×Dic₈₉	C₂²×C₄	1	6	yes	—
78	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₈₉	—	2	6	no	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
83	C₄⋊C₄×C₈₉	—	2	3	no	—
84	C₂²⋊C₄×C₈₉	—	2	3	no	—
85	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
86	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
87	C₂²×D₈₉	—	4	2	no	—
88	C₂²×D₈₉	#117	1	3	yes	—
89	C₂²×D₈₉	#117	1	3	yes	—
90	C₂×C₄×C₈₉	—	2	6	no	—
91	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
92	C₂×C₄×C₈₉	C₂×#41	1	3	yes	—
93	C₂×Dic₈₉	—	2	6	no	—
94	C₂×Dic₈₉	#117	1	3	yes	—
95	C₂×Dic₈₉	#117	1	3	yes	—
96	C₄×D₈₉	—	4	12	no	—
97	C₄×D₈₉	C₂×C₄²	1	3	yes	—
98	C₄×D₈₉	C₂×C₄²	1	1	yes	characteristic
99	C₄×D₈₉	#117	1	3	yes	—
100	C₄×D₈₉	#117	1	3	yes	—
101	C₄×D₈₉	#117	1	6	yes	—
102	C₈×C₈₉	—	2	6	no	—
103	C₈×C₈₉	C₂×C₄²	1	3	yes	—
104	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
105	#10623494	—	2	6	no	—
106	#10623494	C₂×C₄²	1	1	yes	characteristic
107	#10623494	C₂×C₄²	1	3	yes	—
108	C₂²×C₈₉	—	4	1	no	—
109	C₂²×C₈₉	C₂×#117	1	3	yes	—
110	C₂×D₈₉	—	2	6	no	—
111	C₂×D₈₉	—	4	3	no	—
112	C₂×D₈₉	#376	1	3	yes	—
113	C₂×D₈₉	#376	1	1	yes	characteristic
114	C₄×C₈₉	—	4	6	no	—
115	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
116	C₄×C₈₉	C₂×#117	1	3	yes	—
117	Dic₈₉	—	4	6	no	—
118	Dic₈₉	#376	1	1	yes	characteristic
119	Dic₈₉	#376	1	3	yes	—
120	C₂×#1747	—	89	1	no	maximal, Sylow
121	C₂×C₈₉	—	2	3	no	—
122	C₂×C₈₉	C₂×#376	1	1	yes	characteristic
123	D₈₉	—	2	3	no	—
124	D₈₉	#1747	1	2	yes	direct product factor
125	C₂×#405	—	89	3	no	—
126	#1747	—	89	3	no	—
127	#1747	—	89	1	no	—
128	#1747	—	89	3	no	—
129	#1747	D₈₉	1	1	yes	characteristic, direct product factor
130	C₂×#338	—	89	1	no	—
131	C₂×H(3,ℤ/4)	—	89	3	no	—
132	C₈₉	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
133	C₂²×M₁₆	—	89	3	no	—
134	C₂³×C₈	—	89	1	no	—
135	C₂×#117	—	89	3	no	—
136	#405	—	89	3	no	—
137	#405	—	89	3	no	—
138	#405	—	89	3	no	—
139	#405	—	89	6	no	—
140	#405	—	89	6	no	—
141	#405	C₂×D₈₉	1	3	yes	—
142	C₂×#97	—	89	6	no	—
143	C₂×C₄×C₈	—	89	6	no	—
144	#338	—	89	3	no	—
145	#338	C₂×D₈₉	1	1	yes	characteristic
146	H(3,ℤ/4)	—	89	3	no	—
147	H(3,ℤ/4)	—	89	6	no	—
148	H(3,ℤ/4)	C₂×D₈₉	1	3	yes	—
149	C₂³×C₄	—	89	1	no	—
150	C₂×M₁₆	—	89	3	no	—
151	C₂×M₁₆	—	89	6	no	—
152	C₂×M₁₆	—	178	6	no	—
153	C₂×M₁₆	—	178	6	no	—
154	C₂×M₁₆	C₂²×D₈₉	1	3	yes	—
155	C₂²×C₈	—	89	1	no	—
156	C₂²×C₈	—	89	3	no	—
157	C₂²×C₈	—	89	3	no	—
158	C₂²×C₈	—	89	3	no	—
159	C₂²×C₈	—	89	3	no	—
160	C₂²×C₈	—	89	6	no	—
161	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
162	#117	—	89	3	no	—
163	#117	—	89	6	no	—
164	#117	—	178	3	no	—
165	#117	—	178	3	no	—
166	#117	C₂²×D₈₉	1	3	yes	—
167	C₂×C₄⋊C₄	—	178	3	no	—
168	C₂×C₂²⋊C₄	—	178	3	no	—
169	C₂×C₄²	—	89	6	no	—
170	#97	—	89	6	no	—
171	#97	—	89	6	no	—
172	#97	—	89	6	no	—
173	#97	C₄×D₈₉	1	6	yes	complemented
174	C₄×C₈	—	89	6	no	—
175	C₄×C₈	—	89	6	no	—
176	C₄×C₈	—	89	6	no	—
177	C₄×C₈	C₄×D₈₉	1	6	yes	complemented
178	C₂⁴	—	89	1	no	—
179	C₂²×C₄	—	89	1	no	—
180	C₂²×C₄	—	89	3	no	—
181	C₂²×C₄	—	89	3	no	—
182	C₂²×C₄	—	89	3	no	—
183	C₂²×C₄	—	89	3	no	—
184	C₂²×C₄	—	178	6	no	—
185	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
186	M₁₆	—	2	6	no	—
187	M₁₆	—	178	12	no	—
188	M₁₆	—	178	6	no	—
189	C₂×C₈	—	89	3	no	—
190	C₂×C₈	—	89	3	no	—
191	C₂×C₈	—	89	3	no	—
192	C₂×C₈	—	89	3	no	—
193	C₂×C₈	—	89	6	no	—
194	C₂×C₈	—	89	1	no	—
195	C₂×C₈	—	89	3	no	—
196	C₂×C₈	—	89	6	no	—
197	C₂×C₈	—	178	6	no	—
198	C₂×C₈	—	178	6	no	—
199	C₂×C₈	C₂×C₄×D₈₉	1	6	yes	—
200	C₂×C₈	C₂×C₄×D₈₉	1	3	yes	—
201	C₂×C₈	C₂×C₄×D₈₉	1	3	yes	—
202	C₄⋊C₄	—	2	3	no	—
203	C₄⋊C₄	—	178	6	no	—
204	C₄⋊C₄	—	178	3	no	—
205	C₂²⋊C₄	—	2	3	no	—
206	C₂²⋊C₄	—	178	6	no	—
207	C₂²⋊C₄	—	178	3	no	—
208	C₄²	—	89	6	no	—
209	C₄²	—	178	6	no	—
210	C₄²	C₂×C₄×D₈₉	1	6	yes	—
211	C₂³	—	89	3	no	—
212	C₂³	—	89	3	no	—
213	C₂³	—	356	2	no	—
214	C₂³	#41×D₈₉	1	1	yes	characteristic
215	C₂×C₄	—	2	6	no	—
216	C₂×C₄	—	89	3	no	—
217	C₂×C₄	—	89	3	no	—
218	C₂×C₄	—	89	1	no	—
219	C₂×C₄	—	89	3	no	—
220	C₂×C₄	—	89	3	no	—
221	C₂×C₄	—	89	3	no	—
222	C₂×C₄	—	89	6	no	—
223	C₂×C₄	—	356	12	no	—
224	C₂×C₄	—	178	6	no	—
225	C₂×C₄	C₂²×C₄×D₈₉	1	3	yes	—
226	C₂×C₄	#41×D₈₉	1	3	yes	—
227	C₈	—	2	6	no	—
228	C₈	—	89	1	no	—
229	C₈	—	89	3	no	—
230	C₈	—	178	6	no	—
231	C₈	C₄²×D₈₉	1	3	yes	—
232	C₈	C₄²×D₈₉	1	1	yes	characteristic
233	C₂²	—	4	1	no	—
234	C₂²	—	89	1	no	—
235	C₂²	—	89	3	no	—
236	C₂²	—	356	3	no	—
237	C₂²	—	178	6	no	—
238	C₂²	#117×D₈₉	1	3	yes	—
239	C₄	—	4	6	no	—
240	C₄	—	89	1	no	—
241	C₄	—	89	3	no	—
242	C₄	—	356	6	no	—
243	C₄	C₂×C₄²×D₈₉	1	1	yes	characteristic
244	C₄	#117×D₈₉	1	3	yes	—
245	C₂	—	2	3	no	—
246	C₂	—	89	2	no	—
247	C₂	—	178	3	no	—
248	C₂	#376×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₈₉
commutator	C₄×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1747	C₂
socle	C₂×C₈₉	C₂×#376
prime radical	C₂³×D₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1747	D₈₉
hypocenter	C₈₉	C₂×#1747
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
2. #26020008	C₂	semidirect
3. #26020007	C₂	semidirect
4. #26020006	C₂	semidirect
7. C₈₉×#1747	C₂	semidirect
12. #19151555	C₄	semidirect
15. #19151552	C₄	semidirect
16. #97×D₈₉	C₄	semidirect
17. C₄×C₈×D₈₉	C₄	semidirect
42. #15667923	C₂×C₄	semidirect
43. #15667922	C₂×C₄	semidirect
44. #15667922	C₂×C₄	semidirect
45. C₄×#10623494	C₂×C₄	semidirect
46. C₈×Dic₈₉	C₂×C₄	semidirect
47. C₈×Dic₈₉	C₂×C₄	semidirect
51. #97×C₈₉	C₂×C₄	semidirect
52. C₄×C₈×C₈₉	C₂×C₄	semidirect
124. D₈₉	#1747	direct (N × H)
129. #1747	D₈₉	direct (N × H)
132. C₈₉	C₂×#1747	semidirect
173. #97	C₄×D₈₉	semidirect
177. C₄×C₈	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#405×D₈₉	2
#26020008	2
#26020007	2
#26020006	2
#338×D₈₉	2
H(3,ℤ/4)×D₈₉	2
C₈₉×#1747	2
C₂×#1747	89

Subgroups — #1747×D89 #36551929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1747×D₈₉ #36551929