Subgroups — #494×Dic₈₉ #36562866

Order 22784.

1086 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₈₉	C₂	1	4	yes	maximal
2	C₄⋊Q₈×Dic₈₉	C₂	1	2	yes	maximal
3	#26021114	C₂	1	4	yes	maximal
4	#26021055	C₂	1	2	yes	maximal
5	#26020412	C₂	1	1	yes	maximal, characteristic
6	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#494×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₉	C₂²	1	4	yes	—
9	Q₁₆×Dic₈₉	—	2	16	no	—
10	C₂×#15667946	C₂²	1	4	yes	—
11	C₂×#15667936	C₂²	1	8	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
14	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
15	#19151505	C₂²	1	4	yes	—
16	#19151497	—	2	4	no	—
17	#19151468	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₈₉	—	2	4	no	—
19	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₈₉	C₂²	1	4	yes	—
21	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
22	#494×C₈₉	C₄	1	8	yes	complemented
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×#10623494	C₂³	1	2	yes	—
25	Q₈×Dic₈₉	—	2	16	no	—
26	C₂×C₄⋊Dic₈₉	—	2	4	no	—
27	C₂×C₄⋊Dic₈₉	—	2	4	no	—
28	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
29	C₂×C₄×Dic₈₉	—	2	4	no	—
30	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
31	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
32	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
34	#15667946	—	2	8	no	—
35	#15667936	—	2	16	no	—
36	C₄×#10623494	—	2	2	no	—
37	C₈×Dic₈₉	—	2	4	no	—
38	C₈×Dic₈₉	D₄	1	8	yes	—
39	#15667919	—	2	8	no	—
40	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
41	C₂×Q₁₆×C₈₉	—	2	8	no	—
42	C₂×Q₁₆×C₈₉	—	2	8	no	—
43	C₂×Q₁₆×C₈₉	C₂×C₄	1	16	yes	—
44	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₈₉	—	2	4	no	—
46	C₄⋊Q₈×C₈₉	C₂×C₄	1	8	yes	—
47	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
50	C₂²×Dic₈₉	—	2	4	no	—
51	C₂²×Dic₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
53	C₂²×Dic₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×#10623494	—	2	2	no	—
55	C₂×#10623494	—	2	2	no	—
56	C₂×#10623494	C₂×D₄	1	4	yes	—
57	C₄⋊Dic₈₉	—	2	8	no	—
58	C₄⋊Dic₈₉	—	4	8	no	—
59	C₄⋊Dic₈₉	—	4	8	no	—
60	C₄×Dic₈₉	—	2	4	no	—
61	C₄×Dic₈₉	—	2	2	no	—
62	C₄×Dic₈₉	—	4	8	no	—
63	C₄×Dic₈₉	Q₁₆	1	8	yes	complemented
64	C₄×Dic₈₉	C₂×D₄	1	4	yes	—
65	C₂×Q₈×C₈₉	—	2	4	no	—
66	C₂×Q₈×C₈₉	—	2	8	no	—
67	C₂×Q₈×C₈₉	—	2	8	no	—
68	C₂×Q₈×C₈₉	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₈₉	—	2	32	no	—
73	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₈₉	#41	1	4	yes	—
76	C₄⋊C₄×C₈₉	—	2	16	no	—
77	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
78	C₄²×C₈₉	#41	1	2	yes	—
79	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
80	Q₈×C₈₉	—	2	16	no	—
81	Q₈×C₈₉	—	4	8	no	—
82	C₂×C₄×C₈₉	—	2	8	no	—
83	C₂×C₄×C₈₉	—	4	4	no	—
84	C₂×C₄×C₈₉	—	4	4	no	—
85	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₈₉	#135	1	4	yes	—
88	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
91	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
92	C₂×Dic₈₉	—	2	2	no	—
93	C₂×Dic₈₉	—	2	2	no	—
94	C₂×Dic₈₉	—	2	2	no	—
95	C₂×Dic₈₉	—	4	4	no	—
96	C₂×Dic₈₉	—	4	8	no	—
97	C₂×Dic₈₉	#127	1	2	yes	—
98	C₂×Dic₈₉	#127	1	2	yes	—
99	C₂×Dic₈₉	C₂×Q₁₆	1	4	yes	—
100	C₂×Dic₈₉	C₂×Q₁₆	1	4	yes	—
101	C₈×C₈₉	C₄×D₄	1	8	yes	—
102	#10623494	—	2	4	no	—
103	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
105	C₂²×C₈₉	#532	1	1	yes	characteristic
106	C₂²×C₈₉	C₂²×Q₁₆	1	2	yes	—
107	C₂²×C₈₉	C₂×#135	1	2	yes	—
108	C₄×C₈₉	—	4	8	no	—
109	C₄×C₈₉	C₄×Q₁₆	1	8	yes	complemented
110	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
111	C₄×C₈₉	#532	1	2	yes	—
112	Dic₈₉	—	2	2	no	—
113	Dic₈₉	—	2	4	no	—
114	Dic₈₉	—	8	4	no	—
115	Dic₈₉	#494	1	4	yes	direct product factor
116	C₄×#494	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
118	C₂×C₈₉	C₂×C₄×Q₁₆	1	2	yes	—
119	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
120	C₂×C₈₉	#3147	1	1	yes	characteristic
121	C₂×C₈₉	#3170	1	2	yes	—
122	C₂×#494	D₈₉	1	1	yes	characteristic
123	C₂×C₄×Q₁₆	—	89	4	no	—
124	C₄×C₄⋊Q₈	—	89	2	no	—
125	#1969	—	89	4	no	—
126	#1956	—	89	2	no	—
127	#1845	—	89	1	no	—
128	C₄²×C₈	—	89	1	no	—
129	C₈₉	C₄×#494	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	C₂×D₈₉	1	4	yes	—
131	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
132	C₂×C₄×Q₈	—	89	4	no	—
133	#494	Dic₈₉	1	8	yes	direct product factor
134	C₄×Q₁₆	—	178	16	no	—
135	C₂×#107	—	89	4	no	—
136	C₂×#103	—	89	8	no	—
137	C₂×C₄×C₈	—	89	2	no	—
138	C₂×C₄×C₈	—	89	2	no	—
139	C₂×C₄×C₈	—	89	1	no	—
140	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
141	#391	—	89	4	no	—
142	#389	—	178	4	no	—
143	#384	—	89	2	no	—
144	C₄×C₄⋊C₄	—	178	4	no	—
145	C₄³	—	89	1	no	—
146	C₂²×Q₈	C₂²×D₈₉	1	4	yes	—
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	2	8	no	—
149	C₂×Q₁₆	C₂×Dic₈₉	1	16	yes	—
150	C₂²×C₈	—	89	2	no	—
151	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×Dic₈₉	1	8	yes	—
154	C₄×Q₈	—	178	16	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	89	4	no	—
157	C₂×C₄⋊C₄	—	178	4	no	—
158	C₂×C₄⋊C₄	—	178	4	no	—
159	C₂×C₄²	—	89	2	no	—
160	C₂×C₄²	—	89	1	no	—
161	C₂×C₄²	—	89	2	no	—
162	C₂×C₄²	—	89	1	no	—
163	C₂×C₄²	—	178	4	no	—
164	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
165	#107	—	178	8	no	—
166	#103	—	178	16	no	—
167	C₄×C₈	—	89	8	no	—
168	C₄×C₈	—	178	4	no	—
169	C₄×C₈	—	178	2	no	—
170	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
171	#95	—	178	8	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	C₂²×Dic₈₉	1	8	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	89	1	no	—
178	C₂²×C₄	—	89	2	no	—
179	C₂²×C₄	—	89	1	no	—
180	C₂²×C₄	—	178	4	no	—
181	C₂²×C₄	D₄×D₈₉	1	2	yes	—
182	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
183	Q₁₆	—	2	32	no	—
184	C₂×C₈	—	89	4	no	—
185	C₂×C₈	—	178	2	no	—
186	C₂×C₈	—	178	2	no	—
187	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
188	C₂×C₈	D₄×D₈₉	1	4	yes	—
189	C₂×C₈	#13083573	1	4	yes	—
190	C₄⋊C₄	—	2	16	no	—
191	C₄⋊C₄	—	356	8	no	—
192	C₄⋊C₄	—	356	8	no	—
193	C₄⋊C₄	—	178	8	no	—
194	C₄²	—	89	8	no	—
195	C₄²	—	89	4	no	—
196	C₄²	—	356	8	no	—
197	C₄²	—	178	4	no	—
198	C₄²	—	178	2	no	—
199	C₄²	C₂²×Dic₈₉	1	2	yes	—
200	C₄²	#13083573	1	2	yes	—
201	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
202	Q₈	—	2	16	no	—
203	Q₈	—	4	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	89	2	no	—
208	C₂×C₄	—	89	4	no	—
209	C₂×C₄	—	89	2	no	—
210	C₂×C₄	—	89	4	no	—
211	C₂×C₄	—	356	4	no	—
212	C₂×C₄	—	356	8	no	—
213	C₂×C₄	—	178	2	no	—
214	C₂×C₄	—	178	2	no	—
215	C₂×C₄	—	178	2	no	—
216	C₂×C₄	D₄×Dic₈₉	1	4	yes	—
217	C₂×C₄	Q₁₆×D₈₉	1	4	yes	—
218	C₂×C₄	#15668035	1	4	yes	—
219	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
221	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
222	C₂×C₄	C₂×#13083573	1	2	yes	—
223	C₈	—	178	4	no	—
224	C₈	D₄×Dic₈₉	1	8	yes	—
225	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
226	C₂²	#127×D₈₉	1	1	yes	characteristic
227	C₂²	#19152146	1	1	yes	characteristic
228	C₂²	C₂×Q₁₆×D₈₉	1	2	yes	—
229	C₂²	C₂×#15668035	1	2	yes	—
230	C₄	—	4	8	no	—
231	C₄	—	89	4	no	—
232	C₄	—	178	2	no	—
233	C₄	—	178	4	no	—
234	C₄	—	712	4	no	—
235	C₄	Q₁₆×Dic₈₉	1	8	yes	complemented
236	C₄	C₂×D₄×Dic₈₉	1	2	yes	—
237	C₄	#19152146	1	2	yes	—
238	C₂	#127×Dic₈₉	1	1	yes	characteristic
239	C₂	C₂×Q₁₆×Dic₈₉	1	2	yes	—
240	C₂	#494×D₈₉	1	1	yes	characteristic
241	C₂	#26030681	1	1	yes	characteristic
242	C₂	#26030836	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#494×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#494	D₈₉
hypocenter	C₈₉	C₄×#494
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
22. #494×C₈₉	C₄	semidirect
63. C₄×Dic₈₉	Q₁₆	semidirect
109. C₄×C₈₉	C₄×Q₁₆	semidirect
115. Dic₈₉	#494	direct (N × H)
129. C₈₉	C₄×#494	semidirect
133. #494	Dic₈₉	direct (N × H)
235. C₄	Q₁₆×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×Dic₈₉	2
C₄⋊Q₈×Dic₈₉	2
#26021114	2
#26021055	2
#26020412	2
C₄×C₈×Dic₈₉	2
C₂×#494×C₈₉	2
C₄×#494	89

Subgroups — #494×Dic89 #36562866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #494×Dic₈₉ #36562866