Subgroups — C₈₉×#20824 #36398672

Order 22784.

930 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2285	C₂	1	2	yes	maximal
3	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1834	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1809	C₂	1	4	yes	maximal
7	C₈₉×#1776	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
9	C₂×#117×C₈₉	—	2	2	no	—
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₈₉	C₂²	1	4	yes	—
12	C₂×#106×C₈₉	C₂²	1	4	yes	—
13	C₄×M₁₆×C₈₉	C₄	1	8	yes	complemented
14	C₂×#97×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#384×C₈₉	C₂²	1	2	yes	—
18	#384×C₈₉	C₂²	1	2	yes	—
19	#382×C₈₉	—	2	2	no	—
20	#380×C₈₉	—	2	2	no	—
21	#380×C₈₉	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
23	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
24	#340×C₈₉	C₂²	1	8	yes	—
25	C₂³×C₄×C₈₉	D₄	1	2	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₈₉	—	2	4	no	—
28	C₂×M₁₆×C₈₉	—	2	4	no	—
29	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
32	#117×C₈₉	—	4	8	no	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
40	C₂×C₄²×C₈₉	—	2	4	no	—
41	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₈₉	D₄	1	2	yes	—
43	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	#107×C₈₉	—	2	8	no	—
47	#106×C₈₉	—	2	8	no	—
48	#97×C₈₉	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
51	#95×C₈₉	—	2	4	no	—
52	#95×C₈₉	—	2	4	no	—
53	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	—	2	4	no	—
55	C₂²×C₄×C₈₉	—	2	4	no	—
56	C₂²×C₄×C₈₉	—	2	2	no	—
57	C₂²×C₄×C₈₉	—	2	2	no	—
58	C₂²×C₄×C₈₉	—	2	4	no	—
59	C₂²×C₄×C₈₉	—	2	4	no	—
60	C₂²×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
61	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₈₉	—	2	16	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₈₉	#41	1	4	yes	—
73	C₂×C₈×C₈₉	#41	1	4	yes	—
74	C₄⋊C₄×C₈₉	—	2	8	no	—
75	C₄⋊C₄×C₈₉	—	2	8	no	—
76	C₄⋊C₄×C₈₉	—	4	4	no	—
77	C₄⋊C₄×C₈₉	—	4	4	no	—
78	C₄⋊C₄×C₈₉	—	4	4	no	—
79	C₄⋊C₄×C₈₉	—	4	4	no	—
80	C₂²⋊C₄×C₈₉	—	4	8	no	—
81	C₄²×C₈₉	—	4	8	no	—
82	C₄²×C₈₉	C₄⋊C₄	1	8	yes	—
83	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
84	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
85	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
86	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
87	C₂³×C₈₉	—	2	1	no	—
88	C₂³×C₈₉	—	2	1	no	—
89	C₂³×C₈₉	—	2	2	no	—
90	C₂³×C₈₉	—	2	2	no	—
91	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	2	4	no	—
99	C₂×C₄×C₈₉	—	2	16	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	—	4	4	no	—
102	C₂×C₄×C₈₉	—	4	4	no	—
103	C₂×C₄×C₈₉	—	4	4	no	—
104	C₂×C₄×C₈₉	—	4	4	no	—
105	C₂×C₄×C₈₉	—	4	4	no	—
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
110	C₂×C₄×C₈₉	#137	1	4	yes	—
111	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
115	C₈×C₈₉	—	2	4	no	—
116	C₈×C₈₉	#117	1	8	yes	complemented
117	C₂²×C₈₉	—	2	4	no	—
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	4	2	no	—
120	C₂²×C₈₉	—	4	1	no	—
121	C₂²×C₈₉	—	4	1	no	—
122	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₈₉	#532	1	1	yes	characteristic
124	C₂²×C₈₉	#532	1	1	yes	characteristic
125	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
126	C₂²×C₈₉	C₂×#137	1	2	yes	—
127	C₄×C₈₉	—	2	4	no	—
128	C₄×C₈₉	—	2	2	no	—
129	C₄×C₈₉	—	8	4	no	—
130	C₄×C₈₉	—	8	4	no	—
131	C₄×C₈₉	#428	1	8	yes	—
132	C₄×C₈₉	C₂×#117	1	2	yes	—
133	C₄×C₈₉	#533	1	2	yes	—
134	#20824	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₈₉	—	4	2	no	—
136	C₂×C₈₉	#2285	1	1	yes	characteristic
137	C₂×C₈₉	C₂×#428	1	2	yes	—
138	C₂×C₈₉	#3182	1	2	yes	—
139	C₂×C₈₉	#3184	1	1	yes	characteristic
140	C₂×C₈₉	#3185	1	1	yes	characteristic
141	C₂×C₄×M₁₆	C₂×C₈₉	1	1	yes	characteristic
142	#2285	C₂×C₈₉	1	2	yes	—
143	#1845	C₂×C₈₉	1	1	yes	characteristic, complemented
144	#1844	C₂×C₈₉	1	1	yes	characteristic, complemented
145	#1834	C₂×C₈₉	1	2	yes	complemented
146	#1809	C₂×C₈₉	1	4	yes	—
147	#1776	C₂×C₈₉	1	4	yes	complemented
148	C₈₉	#20824	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
150	C₂×#117	—	2	2	no	—
151	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
152	C₂×#107	C₂²×C₈₉	1	4	yes	—
153	C₂×#106	C₂²×C₈₉	1	4	yes	—
154	C₄×M₁₆	C₄×C₈₉	1	8	yes	complemented
155	C₂×#97	C₂²×C₈₉	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
158	#384	C₂²×C₈₉	1	2	yes	—
159	#384	C₂²×C₈₉	1	2	yes	—
160	#382	—	2	2	no	—
161	#380	—	2	2	no	—
162	#380	C₂²×C₈₉	1	4	yes	—
163	C₄×C₄⋊C₄	—	2	2	no	—
164	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
165	#340	C₂²×C₈₉	1	8	yes	—
166	C₂³×C₄	D₄×C₈₉	1	2	yes	—
167	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	C₂×C₄×C₈₉	1	8	yes	—
171	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
172	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
173	#117	—	4	8	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
179	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
180	C₂×C₂²⋊C₄	—	2	4	no	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	C₂×C₄×C₈₉	1	4	yes	—
183	C₂×C₄²	D₄×C₈₉	1	2	yes	—
184	C₂×C₄²	Q₈×C₈₉	1	4	yes	—
185	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
187	#107	—	2	8	no	—
188	#106	—	2	8	no	—
189	#97	C₂×C₄×C₈₉	1	8	yes	—
190	C₄×C₈	C₂×C₄×C₈₉	1	4	yes	—
191	C₄×C₈	C₂×C₄×C₈₉	1	4	yes	—
192	#95	—	2	4	no	—
193	#95	—	2	4	no	—
194	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₄⋊C₄×C₈₉	1	8	yes	—
202	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
206	C₂²×C₄	C₂×Q₈×C₈₉	1	2	yes	—
207	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
212	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
213	C₂×C₈	#41×C₈₉	1	4	yes	—
214	C₂×C₈	#41×C₈₉	1	4	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₄²	—	4	8	no	—
223	C₄²	C₄⋊C₄×C₈₉	1	8	yes	—
224	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
225	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
226	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
227	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
233	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
249	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
250	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
251	C₂×C₄	#137×C₈₉	1	4	yes	—
252	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
255	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	#117×C₈₉	1	8	yes	complemented
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	C₂²×C₄⋊C₄×C₈₉	1	1	yes	characteristic
264	C₂²	#532×C₈₉	1	1	yes	characteristic
265	C₂²	#532×C₈₉	1	1	yes	characteristic
266	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
267	C₂²	C₂×#137×C₈₉	1	2	yes	—
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	4	no	—
272	C₄	#428×C₈₉	1	8	yes	—
273	C₄	C₂×#117×C₈₉	1	2	yes	—
274	C₄	#533×C₈₉	1	2	yes	—
275	C₂	—	4	2	no	—
276	C₂	C₈₉×#2285	1	1	yes	characteristic
277	C₂	C₂×#428×C₈₉	1	2	yes	—
278	C₂	C₈₉×#3182	1	2	yes	—
279	C₂	C₈₉×#3184	1	1	yes	characteristic
280	C₂	C₈₉×#3185	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₈₉×#1845	C₂	semidirect
4. C₈₉×#1844	C₂	semidirect
5. C₈₉×#1834	C₂	semidirect
7. C₈₉×#1776	C₂	semidirect
13. C₄×M₁₆×C₈₉	C₄	semidirect
116. C₈×C₈₉	#117	semidirect
134. #20824	C₈₉	direct (N × H)
143. #1845	C₂×C₈₉	semidirect
144. #1844	C₂×C₈₉	semidirect
145. #1834	C₂×C₈₉	semidirect
147. #1776	C₂×C₈₉	semidirect
148. C₈₉	#20824	direct (N × H)
154. C₄×M₁₆	C₄×C₈₉	semidirect
257. C₈	#117×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×M₁₆×C₈₉	2
C₈₉×#2285	2
C₈₉×#1845	2
C₈₉×#1844	2
C₈₉×#1834	2
C₈₉×#1809	2
C₈₉×#1776	2
#20824	89

Subgroups — C89×#20824 #36398672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#20824 #36398672