Subgroups — C₈₉×#10281 #36388129

Order 22784.

690 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3206	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3205	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2858	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1962	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	#532×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#124×C₈₉	—	2	2	no	—
12	#493×C₈₉	—	2	1	no	—
13	#492×C₈₉	—	2	1	no	—
14	#487×C₈₉	—	2	1	no	—
15	#486×C₈₉	—	2	1	no	—
16	#461×C₈₉	—	2	2	no	—
17	#459×C₈₉	—	2	2	no	—
18	#448×C₈₉	—	2	2	no	—
19	#447×C₈₉	—	2	2	no	—
20	#431×C₈₉	C₂²	1	1	yes	characteristic
21	#431×C₈₉	C₂²	1	1	yes	characteristic
22	#423×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#104×C₈₉	—	2	2	no	—
24	#390×C₈₉	—	2	2	no	—
25	#390×C₈₉	—	2	2	no	—
26	C₄³×C₈₉	C₂²	1	1	yes	characteristic
27	#333×C₈₉	—	2	2	no	—
28	#331×C₈₉	—	2	2	no	—
29	C₂×#41×C₈₉	—	2	1	no	—
30	C₂²×Q₈×C₈₉	D₄	1	2	yes	—
31	C₂²×D₄×C₈₉	—	2	1	no	—
32	C₂²×D₄×C₈₉	—	2	1	no	—
33	C₂²×D₄×C₈₉	D₄	1	2	yes	—
34	C₂×SD₁₆×C₈₉	—	2	4	no	—
35	C₂×D₈×C₈₉	—	2	4	no	—
36	C₂×M₁₆×C₈₉	—	2	1	no	—
37	C₄⋊Q₈×C₈₉	—	2	1	no	—
38	C₄⋊Q₈×C₈₉	—	2	1	no	—
39	C₄⋊Q₈×C₈₉	—	2	1	no	—
40	C₄⋊Q₈×C₈₉	—	2	1	no	—
41	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
42	#127×C₈₉	—	2	1	no	—
43	#127×C₈₉	—	2	1	no	—
44	#127×C₈₉	—	2	1	no	—
45	#127×C₈₉	—	2	1	no	—
46	#127×C₈₉	D₄	1	1	yes	characteristic
47	#124×C₈₉	—	2	1	no	—
48	#124×C₈₉	—	4	8	no	—
49	#121×C₈₉	—	2	1	no	—
50	#121×C₈₉	—	2	1	no	—
51	C₄×D₄×C₈₉	—	2	1	no	—
52	C₄×D₄×C₈₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
56	C₂×C₄²×C₈₉	—	2	1	no	—
57	C₂×C₄²×C₈₉	—	2	1	no	—
58	C₂×C₄²×C₈₉	D₄	1	2	yes	—
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	—	2	1	no	—
62	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
63	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
64	#104×C₈₉	—	4	4	no	—
65	#103×C₈₉	—	2	4	no	—
66	#102×C₈₉	—	2	4	no	—
67	C₂²⋊C₈×C₈₉	—	2	1	no	—
68	C₂²⋊C₈×C₈₉	—	2	1	no	—
69	C₄×C₈×C₈₉	—	2	1	no	—
70	C₄×C₈×C₈₉	—	2	1	no	—
71	C₂⁴×C₈₉	—	2	2	no	—
72	#41×C₈₉	—	4	2	no	—
73	#41×C₈₉	—	4	2	no	—
74	C₂×Q₈×C₈₉	—	2	2	no	—
75	C₂×Q₈×C₈₉	—	2	1	no	—
76	C₂×Q₈×C₈₉	—	2	2	no	—
77	C₂×Q₈×C₈₉	—	4	4	no	—
78	C₂×D₄×C₈₉	—	2	2	no	—
79	C₂×D₄×C₈₉	—	2	1	no	—
80	C₂×D₄×C₈₉	—	2	1	no	—
81	C₂×D₄×C₈₉	—	2	1	no	—
82	C₂×D₄×C₈₉	—	2	2	no	—
83	C₂×D₄×C₈₉	—	4	4	no	—
84	C₂×D₄×C₈₉	—	4	4	no	—
85	C₂×D₄×C₈₉	—	4	1	no	—
86	C₂×D₄×C₈₉	—	4	1	no	—
87	C₂×D₄×C₈₉	—	4	1	no	—
88	C₂×D₄×C₈₉	—	4	1	no	—
89	C₂×D₄×C₈₉	—	4	1	no	—
90	C₂×D₄×C₈₉	—	4	2	no	—
91	C₂×D₄×C₈₉	—	4	2	no	—
92	C₂²×C₄×C₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	—	2	2	no	—
94	C₂²×C₄×C₈₉	—	4	1	no	—
95	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	SD₁₆×C₈₉	—	8	4	no	—
99	D₈×C₈₉	—	8	4	no	—
100	M₁₆×C₈₉	—	4	2	no	—
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	4	1	no	—
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	—	4	1	no	—
107	C₄⋊C₄×C₈₉	—	4	1	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	1	no	—
110	C₂²⋊C₄×C₈₉	—	4	16	no	—
111	C₂²⋊C₄×C₈₉	—	4	1	no	—
112	C₂²⋊C₄×C₈₉	—	4	1	no	—
113	C₄²×C₈₉	—	2	4	no	—
114	C₄²×C₈₉	—	2	2	no	—
115	C₄²×C₈₉	—	2	2	no	—
116	C₄²×C₈₉	—	4	2	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₉	#41	1	1	yes	characteristic
121	C₂³×C₈₉	—	4	4	no	—
122	C₂³×C₈₉	—	4	4	no	—
123	C₂³×C₈₉	—	4	1	no	—
124	C₂³×C₈₉	—	4	2	no	—
125	C₂³×C₈₉	—	4	4	no	—
126	C₂³×C₈₉	#120	1	1	yes	characteristic
127	Q₈×C₈₉	—	4	4	no	—
128	Q₈×C₈₉	—	4	2	no	—
129	Q₈×C₈₉	—	8	2	no	—
130	D₄×C₈₉	—	4	2	no	—
131	D₄×C₈₉	—	4	2	no	—
132	D₄×C₈₉	—	4	4	no	—
133	D₄×C₈₉	—	4	2	no	—
134	D₄×C₈₉	—	8	2	no	—
135	D₄×C₈₉	—	8	2	no	—
136	D₄×C₈₉	—	8	2	no	—
137	D₄×C₈₉	—	8	2	no	—
138	D₄×C₈₉	—	8	2	no	—
139	D₄×C₈₉	—	8	1	no	—
140	D₄×C₈₉	—	8	1	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	4	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	—	4	4	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	8	1	no	—
150	C₂×C₄×C₈₉	—	8	1	no	—
151	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
152	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
153	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
154	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
155	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
156	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
157	C₂×C₄×C₈₉	#135	1	2	yes	—
158	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
159	C₈×C₈₉	—	4	2	no	—
160	C₈×C₈₉	—	4	2	no	—
161	C₈×C₈₉	—	8	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	8	8	no	—
165	C₂²×C₈₉	—	8	1	no	—
166	C₂²×C₈₉	—	8	2	no	—
167	C₂²×C₈₉	—	8	2	no	—
168	C₂²×C₈₉	—	8	1	no	—
169	C₂²×C₈₉	—	8	1	no	—
170	C₂²×C₈₉	—	8	2	no	—
171	C₂²×C₈₉	#392	1	1	yes	characteristic
172	C₂²×C₈₉	#449	1	2	yes	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	4	4	no	—
176	C₄×C₈₉	—	8	2	no	—
177	C₄×C₈₉	—	8	1	no	—
178	C₄×C₈₉	#453	1	2	yes	—
179	C₄×C₈₉	#455	1	2	yes	—
180	C₄×C₈₉	#463	1	2	yes	—
181	C₄×C₈₉	#495	1	1	yes	characteristic
182	C₄×C₈₉	#496	1	1	yes	characteristic
183	#10281	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂×C₈₉	—	2	2	no	—
185	C₂×C₈₉	—	16	1	no	—
186	C₂×C₈₉	—	16	2	no	—
187	C₂×C₈₉	#2002	1	1	yes	characteristic
188	C₂×C₈₉	#2004	1	1	yes	characteristic
189	C₂×C₈₉	#2012	1	1	yes	characteristic
190	#3206	C₂×C₈₉	1	1	yes	characteristic, complemented
191	#3205	C₂×C₈₉	1	1	yes	characteristic
192	#2858	C₂×C₈₉	1	1	yes	characteristic, complemented
193	#1962	C₂×C₈₉	1	1	yes	characteristic, complemented
194	#1548	C₂×C₈₉	1	1	yes	characteristic, complemented
195	#1545	C₂×C₈₉	1	1	yes	characteristic, complemented
196	#1276	C₂×C₈₉	1	1	yes	characteristic, complemented
197	C₈₉	#10281	1	1	yes	characteristic, direct product factor, Sylow
198	#532	C₂²×C₈₉	1	1	yes	characteristic
199	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
200	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
201	C₂×#124	—	2	2	no	—
202	#493	—	2	1	no	—
203	#492	—	2	1	no	—
204	#487	—	2	1	no	—
205	#486	—	2	1	no	—
206	#461	—	2	2	no	—
207	#459	—	2	2	no	—
208	#448	—	2	2	no	—
209	#447	—	2	2	no	—
210	#431	C₂²×C₈₉	1	1	yes	characteristic
211	#431	C₂²×C₈₉	1	1	yes	characteristic
212	#423	C₂²×C₈₉	1	1	yes	characteristic
213	C₂×#104	—	2	2	no	—
214	#390	—	2	2	no	—
215	#390	—	2	2	no	—
216	C₄³	C₂²×C₈₉	1	1	yes	characteristic
217	#333	—	2	2	no	—
218	#331	—	2	2	no	—
219	C₂×#41	—	2	1	no	—
220	C₂²×Q₈	D₄×C₈₉	1	2	yes	—
221	C₂²×D₄	—	2	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	D₄×C₈₉	1	2	yes	—
224	C₂×SD₁₆	—	2	4	no	—
225	C₂×D₈	—	2	4	no	—
226	C₂×M₁₆	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
232	#127	—	2	1	no	—
233	#127	—	2	1	no	—
234	#127	—	2	1	no	—
235	#127	—	2	1	no	—
236	#127	D₄×C₈₉	1	1	yes	characteristic
237	#124	—	2	1	no	—
238	#124	—	4	8	no	—
239	#121	—	2	1	no	—
240	#121	—	2	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₂²⋊C₄	—	2	8	no	—
246	C₂×C₄²	—	2	1	no	—
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	D₄×C₈₉	1	2	yes	—
249	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
253	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
254	#104	—	4	4	no	—
255	#103	—	2	4	no	—
256	#102	—	2	4	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₂⁴	—	2	2	no	—
262	#41	—	4	2	no	—
263	#41	—	4	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	4	4	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	4	4	no	—
274	C₂×D₄	—	4	4	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
288	SD₁₆	—	8	4	no	—
289	D₈	—	8	4	no	—
290	M₁₆	—	4	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	4	16	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₄²	—	2	4	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	4	2	no	—
307	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
310	C₄²	#41×C₈₉	1	1	yes	characteristic
311	C₂³	—	4	4	no	—
312	C₂³	—	4	4	no	—
313	C₂³	—	4	1	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	4	no	—
316	C₂³	#120×C₈₉	1	1	yes	characteristic
317	Q₈	—	4	4	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	8	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	1	no	—
330	D₄	—	8	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
342	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
343	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
344	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
345	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
346	C₂×C₄	#127×C₈₉	1	1	yes	characteristic
347	C₂×C₄	#135×C₈₉	1	2	yes	—
348	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
349	C₈	—	4	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	8	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	8	8	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	—	8	2	no	—
361	C₂²	#392×C₈₉	1	1	yes	characteristic
362	C₂²	#449×C₈₉	1	2	yes	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	4	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	1	no	—
368	C₄	#453×C₈₉	1	2	yes	—
369	C₄	#455×C₈₉	1	2	yes	—
370	C₄	#463×C₈₉	1	2	yes	—
371	C₄	#495×C₈₉	1	1	yes	characteristic
372	C₄	#496×C₈₉	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	16	1	no	—
375	C₂	—	16	2	no	—
376	C₂	C₈₉×#2002	1	1	yes	characteristic
377	C₂	C₈₉×#2004	1	1	yes	characteristic
378	C₂	C₈₉×#2012	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₈₉×#3205	C₂
LCS	G ▷ C₂×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
1. C₈₉×#3206	C₂	semidirect
3. C₈₉×#2858	C₂	semidirect
4. C₈₉×#1962	C₂	semidirect
5. C₈₉×#1548	C₂	semidirect
6. C₈₉×#1545	C₂	semidirect
7. C₈₉×#1276	C₂	semidirect
183. #10281	C₈₉	direct (N × H)
190. #3206	C₂×C₈₉	semidirect
192. #2858	C₂×C₈₉	semidirect
193. #1962	C₂×C₈₉	semidirect
194. #1548	C₂×C₈₉	semidirect
195. #1545	C₂×C₈₉	semidirect
196. #1276	C₂×C₈₉	semidirect
197. C₈₉	#10281	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×#3206	2
C₈₉×#3205	2
C₈₉×#2858	2
C₈₉×#1962	2
C₈₉×#1548	2
C₈₉×#1545	2
C₈₉×#1276	2
#10281	89

Subgroups — C89×#10281 #36388129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#10281 #36388129