Subgroups — C₈₉×#9656 #36387504

Order 22784.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1984	C₂	1	2	yes	maximal
3	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈×C₈₉	C₂	1	2	yes	maximal
5	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
8	C₄×Q₁₆×C₈₉	—	2	2	no	—
9	C₄×Q₁₆×C₈₉	—	2	2	no	—
10	C₂×#107×C₈₉	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
12	C₂×#103×C₈₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	4	no	—
17	#335×C₈₉	—	2	2	no	—
18	Q₈⋊C₈×C₈₉	—	2	4	no	—
19	C₈²×C₈₉	—	2	2	no	—
20	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
21	C₂×Q₁₆×C₈₉	—	2	2	no	—
22	C₂×Q₁₆×C₈₉	—	2	2	no	—
23	C₂×Q₁₆×C₈₉	C₈	1	4	yes	complemented
24	C₂²×C₈×C₈₉	—	2	1	no	—
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	D₄	1	2	yes	—
29	C₄×Q₈×C₈₉	—	2	2	no	—
30	C₄×Q₈×C₈₉	—	2	2	no	—
31	C₄×Q₈×C₈₉	—	2	2	no	—
32	C₄×Q₈×C₈₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₈₉	—	2	2	no	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	#107×C₈₉	—	2	2	no	—
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₄⋊C₈×C₈₉	—	2	2	no	—
40	#103×C₈₉	—	2	4	no	—
41	C₄×C₈×C₈₉	—	2	1	no	—
42	C₄×C₈×C₈₉	—	2	1	no	—
43	C₄×C₈×C₈₉	—	2	1	no	—
44	C₄×C₈×C₈₉	D₄	1	2	yes	—
45	C₄×C₈×C₈₉	D₄	1	2	yes	—
46	C₄×C₈×C₈₉	D₄	1	2	yes	—
47	C₂×Q₈×C₈₉	—	2	2	no	—
48	C₂×Q₈×C₈₉	—	2	2	no	—
49	C₂×Q₈×C₈₉	—	2	2	no	—
50	C₂×Q₈×C₈₉	—	2	2	no	—
51	C₂×Q₈×C₈₉	—	2	2	no	—
52	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	2	2	no	—
55	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
58	Q₁₆×C₈₉	—	2	8	no	—
59	C₂×C₈×C₈₉	—	2	4	no	—
60	C₂×C₈×C₈₉	—	2	2	no	—
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	—	2	1	no	—
63	C₂×C₈×C₈₉	—	2	1	no	—
64	C₂×C₈×C₈₉	—	2	2	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₂×C₈×C₈₉	—	4	4	no	—
67	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
71	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
72	C₄⋊C₄×C₈₉	—	2	2	no	—
73	C₄⋊C₄×C₈₉	—	2	2	no	—
74	C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₄⋊C₄×C₈₉	—	4	2	no	—
76	C₄²×C₈₉	—	4	2	no	—
77	C₄²×C₈₉	—	4	2	no	—
78	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₈₉	#41	1	1	yes	characteristic
82	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₈₉	—	2	4	no	—
84	Q₈×C₈₉	—	2	4	no	—
85	Q₈×C₈₉	—	4	2	no	—
86	Q₈×C₈₉	—	4	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	4	no	—
89	C₂×C₄×C₈₉	—	4	2	no	—
90	C₂×C₄×C₈₉	—	4	2	no	—
91	C₂×C₄×C₈₉	—	4	2	no	—
92	C₂×C₄×C₈₉	—	4	4	no	—
93	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₈₉	#135	1	2	yes	—
105	C₈×C₈₉	—	2	4	no	—
106	C₈×C₈₉	—	2	2	no	—
107	C₈×C₈₉	—	2	4	no	—
108	C₈×C₈₉	—	8	2	no	—
109	C₈×C₈₉	C₂²⋊C₈	1	4	yes	—
110	C₂²×C₈₉	#390	1	1	yes	characteristic
111	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
112	C₂²×C₈₉	#436	1	1	yes	characteristic
113	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
114	C₂²×C₈₉	#443	1	2	yes	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	—	4	4	no	—
118	C₄×C₈₉	—	8	2	no	—
119	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₈₉	#408	1	1	yes	characteristic
121	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
122	C₄×C₈₉	#436	1	1	yes	characteristic
123	C₄×C₈₉	#467	1	2	yes	—
124	C₄×C₈₉	#494	1	1	yes	characteristic
125	C₄×C₈₉	#495	1	1	yes	characteristic
126	#9656	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₈₉	C₈×Q₁₆	1	2	yes	—
128	C₂×C₈₉	#1591	1	2	yes	—
129	C₂×C₈₉	#1954	1	1	yes	characteristic
130	C₂×C₈₉	#1969	1	1	yes	characteristic
131	C₂×C₈₉	#1974	1	1	yes	characteristic
132	C₂×C₄×Q₁₆	C₂×C₈₉	1	1	yes	characteristic
133	#1984	C₂×C₈₉	1	2	yes	—
134	C₂×#335	C₂×C₈₉	1	1	yes	characteristic
135	C₂×Q₈⋊C₈	C₂×C₈₉	1	2	yes	—
136	C₂×C₈²	C₂×C₈₉	1	1	yes	characteristic
137	C₈₉	#9656	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×Q₁₆	C₄×C₈₉	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
140	C₄×Q₁₆	—	2	2	no	—
141	C₄×Q₁₆	—	2	2	no	—
142	C₂×#107	C₄×C₈₉	1	1	yes	characteristic
143	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
144	C₂×#103	C₄×C₈₉	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
148	#336	—	2	4	no	—
149	#335	—	2	2	no	—
150	Q₈⋊C₈	—	2	4	no	—
151	C₈²	—	2	2	no	—
152	C₂²×Q₈	C₂×C₄×C₈₉	1	2	yes	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	C₈×C₈₉	1	4	yes	complemented
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₈₉	1	2	yes	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
169	#107	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	#103	—	2	4	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	D₄×C₈₉	1	2	yes	—
177	C₄×C₈	D₄×C₈₉	1	2	yes	—
178	C₄×C₈	D₄×C₈₉	1	2	yes	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂×C₈×C₈₉	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
189	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
190	Q₁₆	—	2	8	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
200	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
201	C₂×C₈	C₂×C₈×C₈₉	1	2	yes	—
202	C₂×C₈	M₁₆×C₈₉	1	2	yes	—
203	C₂×C₈	Q₁₆×C₈₉	1	4	yes	complemented
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
213	C₄²	#41×C₈₉	1	1	yes	characteristic
214	C₂³	C₄×D₄×C₈₉	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
226	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
227	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
228	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
229	C₂×C₄	#124×C₈₉	1	1	yes	characteristic
230	C₂×C₄	#127×C₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
232	C₂×C₄	C₂×M₁₆×C₈₉	1	1	yes	characteristic
233	C₂×C₄	#131×C₈₉	1	1	yes	characteristic
234	C₂×C₄	#131×C₈₉	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
236	C₂×C₄	#135×C₈₉	1	2	yes	—
237	C₈	—	2	4	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	4	no	—
240	C₈	—	8	2	no	—
241	C₈	C₂²⋊C₈×C₈₉	1	4	yes	—
242	C₂²	#390×C₈₉	1	1	yes	characteristic
243	C₂²	C₈×D₄×C₈₉	1	1	yes	characteristic
244	C₂²	#436×C₈₉	1	1	yes	characteristic
245	C₂²	C₄×Q₁₆×C₈₉	1	2	yes	—
246	C₂²	#443×C₈₉	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	—	8	2	no	—
251	C₄	C₂×C₂²⋊C₈×C₈₉	1	1	yes	characteristic
252	C₄	#408×C₈₉	1	1	yes	characteristic
253	C₄	C₈×D₄×C₈₉	1	1	yes	characteristic
254	C₄	#436×C₈₉	1	1	yes	characteristic
255	C₄	#467×C₈₉	1	2	yes	—
256	C₄	#494×C₈₉	1	1	yes	characteristic
257	C₄	#495×C₈₉	1	1	yes	characteristic
258	C₂	C₈×Q₁₆×C₈₉	1	2	yes	—
259	C₂	C₈₉×#1591	1	2	yes	—
260	C₂	C₈₉×#1954	1	1	yes	characteristic
261	C₂	C₈₉×#1969	1	1	yes	characteristic
262	C₂	C₈₉×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
socle, prime radical	C₂³×C₈₉	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
23. C₂×Q₁₆×C₈₉	C₈	semidirect
71. C₂×C₈×C₈₉	Q₁₆	semidirect
126. #9656	C₈₉	direct (N × H)
137. C₈₉	#9656	direct (N × H)
155. C₂×Q₁₆	C₈×C₈₉	semidirect
203. C₂×C₈	Q₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₈₉	2
C₈₉×#1984	2
C₂×#335×C₈₉	2
C₂×Q₈⋊C₈×C₈₉	2
C₂×C₈²×C₈₉	2
#9656	89

Subgroups — C89×#9656 #36387504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9656 #36387504