Subgroups — C₈₉×#9294 #36387142

Order 22784.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3208	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#1961	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1518	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	#533×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
10	#492×C₈₉	C₄	1	2	yes	complemented
11	#486×C₈₉	C₄	1	2	yes	complemented
12	#460×C₈₉	—	2	2	no	—
13	#450×C₈₉	—	2	2	no	—
14	#446×C₈₉	—	2	2	no	—
15	#446×C₈₉	—	2	2	no	—
16	#431×C₈₉	C₂²	1	1	yes	characteristic
17	#431×C₈₉	C₂²	1	1	yes	characteristic
18	#424×C₈₉	—	2	1	no	—
19	#423×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#102×C₈₉	—	2	1	no	—
21	C₂×#102×C₈₉	—	2	1	no	—
22	#405×C₈₉	—	2	1	no	—
23	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
24	#382×C₈₉	C₂²	1	1	yes	characteristic
25	#331×C₈₉	—	2	2	no	—
26	#328×C₈₉	—	2	2	no	—
27	C₂²×D₄×C₈₉	—	2	1	no	—
28	C₂²×D₄×C₈₉	—	2	1	no	—
29	C₂²×D₄×C₈₉	—	2	1	no	—
30	C₂×SD₁₆×C₈₉	—	2	4	no	—
31	C₂×M₁₆×C₈₉	—	2	1	no	—
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	#127×C₈₉	—	2	1	no	—
37	#127×C₈₉	—	2	1	no	—
38	#127×C₈₉	—	2	1	no	—
39	#127×C₈₉	C₂×C₄	1	2	yes	—
40	#125×C₈₉	—	2	1	no	—
41	C₂²⋊Q₈×C₈₉	—	2	1	no	—
42	C₂²⋊Q₈×C₈₉	D₄	1	2	yes	—
43	C₄×Q₈×C₈₉	—	2	1	no	—
44	C₄×Q₈×C₈₉	D₄	1	2	yes	—
45	#117×C₈₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	—	2	4	no	—
49	C₄⋊C₈×C₈₉	—	2	1	no	—
50	C₄⋊C₈×C₈₉	—	2	1	no	—
51	C₄⋊C₈×C₈₉	—	2	1	no	—
52	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
53	#102×C₈₉	—	2	4	no	—
54	#102×C₈₉	—	4	2	no	—
55	#102×C₈₉	—	4	2	no	—
56	C₂²⋊C₈×C₈₉	—	2	1	no	—
57	C₂²⋊C₈×C₈₉	—	2	1	no	—
58	#97×C₈₉	—	2	2	no	—
59	#97×C₈₉	—	2	1	no	—
60	#97×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	—	2	2	no	—
62	C₄×C₈×C₈₉	—	2	1	no	—
63	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
64	#95×C₈₉	—	2	1	no	—
65	#95×C₈₉	—	2	1	no	—
66	C₂⁴×C₈₉	—	4	1	no	—
67	C₂×Q₈×C₈₉	—	2	1	no	—
68	C₂×Q₈×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₂×D₄×C₈₉	—	2	2	no	—
70	C₂×D₄×C₈₉	—	2	2	no	—
71	C₂×D₄×C₈₉	—	2	2	no	—
72	C₂×D₄×C₈₉	—	4	1	no	—
73	C₂×D₄×C₈₉	—	4	1	no	—
74	C₂×D₄×C₈₉	—	4	1	no	—
75	C₂×D₄×C₈₉	—	4	1	no	—
76	C₂×D₄×C₈₉	—	4	1	no	—
77	C₂×D₄×C₈₉	—	4	1	no	—
78	C₂×D₄×C₈₉	—	4	1	no	—
79	C₂×D₄×C₈₉	—	8	2	no	—
80	C₂×D₄×C₈₉	—	8	2	no	—
81	C₂²×C₄×C₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
85	SD₁₆×C₈₉	—	8	4	no	—
86	M₁₆×C₈₉	—	4	1	no	—
87	M₁₆×C₈₉	—	4	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	1	no	—
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	4	1	no	—
96	C₂×C₈×C₈₉	SD₁₆	1	2	yes	complemented
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	2	no	—
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	—	4	1	no	—
103	C₄⋊C₄×C₈₉	—	4	1	no	—
104	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
105	C₂²⋊C₄×C₈₉	—	2	2	no	—
106	C₄²×C₈₉	—	2	2	no	—
107	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₈₉	#41	1	1	yes	characteristic
111	C₂³×C₈₉	—	4	2	no	—
112	C₂³×C₈₉	—	8	2	no	—
113	C₂³×C₈₉	—	8	2	no	—
114	C₂³×C₈₉	—	8	1	no	—
115	C₂³×C₈₉	—	8	1	no	—
116	C₂³×C₈₉	#121	1	1	yes	characteristic
117	Q₈×C₈₉	—	4	2	no	—
118	Q₈×C₈₉	—	8	1	no	—
119	D₄×C₈₉	—	4	4	no	—
120	D₄×C₈₉	—	4	2	no	—
121	D₄×C₈₉	—	4	2	no	—
122	D₄×C₈₉	—	8	2	no	—
123	D₄×C₈₉	—	8	2	no	—
124	D₄×C₈₉	—	8	2	no	—
125	D₄×C₈₉	—	8	1	no	—
126	D₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	8	1	no	—
132	C₂×C₄×C₈₉	—	8	1	no	—
133	C₂×C₄×C₈₉	—	8	1	no	—
134	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
142	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	—	4	2	no	—
146	C₈×C₈₉	—	4	1	no	—
147	C₈×C₈₉	—	4	2	no	—
148	C₈×C₈₉	—	8	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	8	2	no	—
152	C₂²×C₈₉	—	8	2	no	—
153	C₂²×C₈₉	—	8	2	no	—
154	C₂²×C₈₉	—	16	2	no	—
155	C₂²×C₈₉	—	16	2	no	—
156	C₂²×C₈₉	#386	1	1	yes	characteristic
157	C₂²×C₈₉	#460	1	1	yes	characteristic
158	C₂²×C₈₉	#464	1	1	yes	characteristic
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	4	2	no	—
164	C₄×C₈₉	—	4	2	no	—
165	C₄×C₈₉	—	16	1	no	—
166	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
167	C₄×C₈₉	#440	1	1	yes	characteristic
168	C₄×C₈₉	#448	1	1	yes	characteristic
169	C₄×C₈₉	#449	1	1	yes	characteristic
170	#9294	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂×C₈₉	—	2	1	no	—
172	C₂×C₈₉	—	2	1	no	—
173	C₂×C₈₉	—	16	2	no	—
174	C₂×C₈₉	#1880	1	1	yes	characteristic
175	C₂×C₈₉	#1903	1	1	yes	characteristic
176	C₂×C₈₉	#1910	1	1	yes	characteristic
177	#3208	C₂×C₈₉	1	1	yes	characteristic
178	#2961	C₂×C₈₉	1	1	yes	characteristic, complemented
179	#1961	C₂×C₈₉	1	1	yes	characteristic
180	#1545	C₂×C₈₉	1	1	yes	characteristic
181	#1518	C₂×C₈₉	1	1	yes	characteristic, complemented
182	#1286	C₂×C₈₉	1	1	yes	characteristic, complemented
183	#1275	C₂×C₈₉	1	1	yes	characteristic, complemented
184	C₈₉	#9294	1	1	yes	characteristic, direct product factor, Sylow
185	#533	C₂²×C₈₉	1	1	yes	characteristic
186	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
187	#492	C₄×C₈₉	1	2	yes	complemented
188	#486	C₄×C₈₉	1	2	yes	complemented
189	#460	—	2	2	no	—
190	#450	—	2	2	no	—
191	#446	—	2	2	no	—
192	#446	—	2	2	no	—
193	#431	C₂²×C₈₉	1	1	yes	characteristic
194	#431	C₂²×C₈₉	1	1	yes	characteristic
195	#424	—	2	1	no	—
196	#423	C₂²×C₈₉	1	1	yes	characteristic
197	C₂×#102	—	2	1	no	—
198	C₂×#102	—	2	1	no	—
199	#405	—	2	1	no	—
200	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
201	#382	C₂²×C₈₉	1	1	yes	characteristic
202	#331	—	2	2	no	—
203	#328	—	2	2	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
210	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
211	C₄⋊Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
212	C₄⋊Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
213	#127	—	2	1	no	—
214	#127	—	2	1	no	—
215	#127	—	2	1	no	—
216	#127	C₂×C₄×C₈₉	1	2	yes	—
217	#125	—	2	1	no	—
218	C₂²⋊Q₈	—	2	1	no	—
219	C₂²⋊Q₈	D₄×C₈₉	1	2	yes	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	D₄×C₈₉	1	2	yes	—
222	#117	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
230	#102	—	2	4	no	—
231	#102	—	4	2	no	—
232	#102	—	4	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	2	1	no	—
235	#97	—	2	2	no	—
236	#97	—	2	1	no	—
237	#97	—	2	1	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
241	#95	—	2	1	no	—
242	#95	—	2	1	no	—
243	C₂⁴	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	C₂²⋊C₄×C₈₉	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	8	2	no	—
257	C₂×D₄	—	8	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
261	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
262	SD₁₆	—	8	4	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	SD₁₆×C₈₉	1	2	yes	complemented
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
287	C₄²	#41×C₈₉	1	1	yes	characteristic
288	C₂³	—	4	2	no	—
289	C₂³	—	8	2	no	—
290	C₂³	—	8	2	no	—
291	C₂³	—	8	1	no	—
292	C₂³	—	8	1	no	—
293	C₂³	#121×C₈₉	1	1	yes	characteristic
294	Q₈	—	4	2	no	—
295	Q₈	—	8	1	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	1	no	—
303	D₄	—	8	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
312	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
313	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
314	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
315	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
316	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
317	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
318	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
319	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
320	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	4	1	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	16	2	no	—
332	C₂²	—	16	2	no	—
333	C₂²	#386×C₈₉	1	1	yes	characteristic
334	C₂²	#460×C₈₉	1	1	yes	characteristic
335	C₂²	#464×C₈₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	16	1	no	—
343	C₄	C₄×SD₁₆×C₈₉	1	1	yes	characteristic
344	C₄	#440×C₈₉	1	1	yes	characteristic
345	C₄	#448×C₈₉	1	1	yes	characteristic
346	C₄	#449×C₈₉	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	16	2	no	—
350	C₂	C₈₉×#1880	1	1	yes	characteristic
351	C₂	C₈₉×#1903	1	1	yes	characteristic
352	C₂	C₈₉×#1910	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#386
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2961	C₂	semidirect
5. C₈₉×#1518	C₂	semidirect
6. C₈₉×#1286	C₂	semidirect
7. C₈₉×#1275	C₂	semidirect
10. #492×C₈₉	C₄	semidirect
11. #486×C₈₉	C₄	semidirect
96. C₂×C₈×C₈₉	SD₁₆	semidirect
170. #9294	C₈₉	direct (N × H)
178. #2961	C₂×C₈₉	semidirect
181. #1518	C₂×C₈₉	semidirect
182. #1286	C₂×C₈₉	semidirect
183. #1275	C₂×C₈₉	semidirect
184. C₈₉	#9294	direct (N × H)
187. #492	C₄×C₈₉	semidirect
188. #486	C₄×C₈₉	semidirect
273. C₂×C₈	SD₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3208	2
C₈₉×#2961	2
C₈₉×#1961	2
C₈₉×#1545	2
C₈₉×#1518	2
C₈₉×#1286	2
C₈₉×#1275	2
#9294	89

Subgroups — C89×#9294 #36387142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9294 #36387142