Subgroups — C₈₉×#32849 #36410697

Order 22784.

604 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2100	C₂	1	2	yes	maximal
4	C₈₉×#2099	C₂	1	2	yes	maximal
5	C₈₉×#2088	C₂	1	4	yes	maximal
6	C₈₉×#2084	C₂	1	2	yes	maximal
7	C₈₉×#2084	C₂	1	2	yes	maximal
8	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₈₉	—	2	1	no	—
10	C₂×#125×C₈₉	—	2	1	no	—
11	C₂×#106×C₈₉	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
13	#389×C₈₉	—	2	1	no	—
14	#389×C₈₉	—	2	1	no	—
15	#389×C₈₉	—	2	1	no	—
16	#389×C₈₉	—	2	1	no	—
17	#389×C₈₉	C₂²	1	2	yes	—
18	#389×C₈₉	C₂²	1	2	yes	—
19	#389×C₈₉	C₂²	1	2	yes	—
20	#389×C₈₉	C₂²	1	2	yes	—
21	#384×C₈₉	C₂²	1	1	yes	characteristic
22	#384×C₈₉	C₂²	1	1	yes	characteristic
23	#384×C₈₉	C₂²	1	1	yes	characteristic
24	#384×C₈₉	C₂²	1	1	yes	characteristic
25	#382×C₈₉	—	2	1	no	—
26	#382×C₈₉	—	2	1	no	—
27	C₄³×C₈₉	C₂²	1	1	yes	characteristic
28	#340×C₈₉	C₂²	1	4	yes	—
29	#340×C₈₉	C₂²	1	4	yes	—
30	#340×C₈₉	C₂²	1	4	yes	—
31	#336×C₈₉	C₂²	1	2	yes	—
32	#336×C₈₉	C₂²	1	2	yes	—
33	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
34	#125×C₈₉	—	4	4	no	—
35	#125×C₈₉	—	4	4	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	#106×C₈₉	—	4	4	no	—
60	C₄⋊C₈×C₈₉	—	4	2	no	—
61	#95×C₈₉	—	2	2	no	—
62	#95×C₈₉	—	2	2	no	—
63	#95×C₈₉	—	2	2	no	—
64	#95×C₈₉	—	2	2	no	—
65	C₂²×C₄×C₈₉	—	2	2	no	—
66	C₂²×C₄×C₈₉	—	2	2	no	—
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	—	2	4	no	—
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₄⋊C₄×C₈₉	—	2	2	no	—
80	C₄⋊C₄×C₈₉	—	2	2	no	—
81	C₄⋊C₄×C₈₉	—	2	4	no	—
82	C₄⋊C₄×C₈₉	—	2	4	no	—
83	C₄⋊C₄×C₈₉	—	2	2	no	—
84	C₄⋊C₄×C₈₉	—	2	2	no	—
85	C₄⋊C₄×C₈₉	—	4	2	no	—
86	C₄⋊C₄×C₈₉	—	4	2	no	—
87	C₄⋊C₄×C₈₉	—	4	2	no	—
88	C₄⋊C₄×C₈₉	—	4	2	no	—
89	C₄⋊C₄×C₈₉	—	4	4	no	—
90	C₄⋊C₄×C₈₉	—	4	2	no	—
91	C₄⋊C₄×C₈₉	—	4	2	no	—
92	C₄⋊C₄×C₈₉	—	4	4	no	—
93	C₄⋊C₄×C₈₉	—	4	4	no	—
94	C₄⋊C₄×C₈₉	—	4	2	no	—
95	C₄⋊C₄×C₈₉	—	4	4	no	—
96	C₄⋊C₄×C₈₉	—	4	2	no	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₄²×C₈₉	—	2	2	no	—
99	C₄²×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	—	2	2	no	—
101	C₄²×C₈₉	—	2	2	no	—
102	C₄²×C₈₉	—	2	1	no	—
103	C₄²×C₈₉	—	4	1	no	—
104	C₄²×C₈₉	SD₁₆	1	4	yes	—
105	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₈₉	#137	1	2	yes	—
136	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
138	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
139	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₈×C₈₉	—	4	4	no	—
141	C₂²×C₈₉	#536	1	1	yes	characteristic
142	C₂²×C₈₉	#536	1	1	yes	characteristic
143	C₂²×C₈₉	#537	1	1	yes	characteristic
144	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
147	C₂²×C₈₉	#575	1	1	yes	characteristic
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	2	2	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	—	8	2	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	#563	1	2	yes	—
159	C₄×C₈₉	#564	1	2	yes	—
160	#32849	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₈₉	#2822	1	1	yes	characteristic
162	C₂×C₈₉	#3227	1	1	yes	characteristic
163	C₂×C₈₉	#3229	1	1	yes	characteristic
164	C₂×C₈₉	#3234	1	1	yes	characteristic
165	C₂×C₈₉	#3236	1	1	yes	characteristic
166	C₂×C₈₉	#3246	1	1	yes	characteristic
167	C₂×C₈₉	#3254	1	1	yes	characteristic
168	#2859	C₂×C₈₉	1	1	yes	characteristic
169	#2859	C₂×C₈₉	1	1	yes	characteristic
170	#2100	C₂×C₈₉	1	2	yes	—
171	#2099	C₂×C₈₉	1	2	yes	—
172	#2088	C₂×C₈₉	1	4	yes	—
173	#2084	C₂×C₈₉	1	2	yes	—
174	#2084	C₂×C₈₉	1	2	yes	—
175	#1831	C₂×C₈₉	1	1	yes	characteristic
176	C₈₉	#32849	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×#125	—	2	1	no	—
178	C₂×#125	—	2	1	no	—
179	C₂×#106	C₂²×C₈₉	1	4	yes	—
180	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
181	#389	—	2	1	no	—
182	#389	—	2	1	no	—
183	#389	—	2	1	no	—
184	#389	—	2	1	no	—
185	#389	C₂²×C₈₉	1	2	yes	—
186	#389	C₂²×C₈₉	1	2	yes	—
187	#389	C₂²×C₈₉	1	2	yes	—
188	#389	C₂²×C₈₉	1	2	yes	—
189	#384	C₂²×C₈₉	1	1	yes	characteristic
190	#384	C₂²×C₈₉	1	1	yes	characteristic
191	#384	C₂²×C₈₉	1	1	yes	characteristic
192	#384	C₂²×C₈₉	1	1	yes	characteristic
193	#382	—	2	1	no	—
194	#382	—	2	1	no	—
195	C₄³	C₂²×C₈₉	1	1	yes	characteristic
196	#340	C₂²×C₈₉	1	4	yes	—
197	#340	C₂²×C₈₉	1	4	yes	—
198	#340	C₂²×C₈₉	1	4	yes	—
199	#336	C₂²×C₈₉	1	2	yes	—
200	#336	C₂²×C₈₉	1	2	yes	—
201	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
202	#125	—	4	4	no	—
203	#125	—	4	4	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
220	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
221	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
222	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
223	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
227	#106	—	4	4	no	—
228	C₄⋊C₈	—	4	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
243	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	SD₁₆×C₈₉	1	4	yes	—
273	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
300	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
301	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
302	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
303	C₂×C₄	#137×C₈₉	1	2	yes	—
304	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
305	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
306	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
307	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₂²	#536×C₈₉	1	1	yes	characteristic
310	C₂²	#536×C₈₉	1	1	yes	characteristic
311	C₂²	#537×C₈₉	1	1	yes	characteristic
312	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
313	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
314	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
315	C₂²	#575×C₈₉	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#563×C₈₉	1	2	yes	—
327	C₄	#564×C₈₉	1	2	yes	—
328	C₂	C₈₉×#2822	1	1	yes	characteristic
329	C₂	C₈₉×#3227	1	1	yes	characteristic
330	C₂	C₈₉×#3229	1	1	yes	characteristic
331	C₂	C₈₉×#3234	1	1	yes	characteristic
332	C₂	C₈₉×#3236	1	1	yes	characteristic
333	C₂	C₈₉×#3246	1	1	yes	characteristic
334	C₂	C₈₉×#3254	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
160. #32849	C₈₉	direct (N × H)
176. C₈₉	#32849	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2859	2
C₈₉×#2100	2
C₈₉×#2099	2
C₈₉×#2088	2
C₈₉×#2084	2
C₈₉×#1831	2
#32849	89

Subgroups — C89×#32849 #36410697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#32849 #36410697