Subgroups — C₈₉×#13035 #36390883

Order 22784.

456 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2957	C₂	1	1	yes	maximal, characteristic
3	C₂×#362×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2143	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	#568×C₈₉	—	2	1	no	—
11	#510×C₈₉	—	2	2	no	—
12	#510×C₈₉	—	2	2	no	—
13	C₂×Q₃₂×C₈₉	C₄	1	2	yes	complemented
14	C₂×SD₃₂×C₈₉	C₄	1	2	yes	complemented
15	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
17	#444×C₈₉	—	2	2	no	—
18	#443×C₈₉	—	2	2	no	—
19	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
21	#421×C₈₉	—	2	1	no	—
22	#419×C₈₉	—	2	1	no	—
23	C₂×#101×C₈₉	—	2	1	no	—
24	#405×C₈₉	C₂²	1	1	yes	characteristic
25	#362×C₈₉	—	2	2	no	—
26	#359×C₈₉	—	2	2	no	—
27	C₂×#41×C₈₉	—	2	1	no	—
28	C₂²×Q₈×C₈₉	—	2	1	no	—
29	#135×C₈₉	—	2	1	no	—
30	#135×C₈₉	—	2	1	no	—
31	#135×C₈₉	D₄	1	2	yes	—
32	#135×C₈₉	D₄	1	2	yes	—
33	C₂×Q₁₆×C₈₉	—	2	1	no	—
34	C₂×Q₁₆×C₈₉	—	2	1	no	—
35	C₂×Q₁₆×C₈₉	—	2	1	no	—
36	C₂×Q₁₆×C₈₉	—	2	1	no	—
37	C₂×Q₁₆×C₈₉	—	2	1	no	—
38	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×SD₁₆×C₈₉	—	2	1	no	—
41	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	#131×C₈₉	—	2	4	no	—
43	#131×C₈₉	—	4	2	no	—
44	C₂×M₁₆×C₈₉	—	2	1	no	—
45	C₂×M₁₆×C₈₉	—	2	1	no	—
46	C₂×M₁₆×C₈₉	—	4	2	no	—
47	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	#117×C₈₉	D₄	1	1	yes	characteristic
50	Q₃₂×C₈₉	—	2	4	no	—
51	SD₃₂×C₈₉	—	2	4	no	—
52	M₃₂×C₈₉	—	2	1	no	—
53	M₃₂×C₈₉	—	2	1	no	—
54	M₃₂×C₈₉	—	2	1	no	—
55	M₃₂×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
58	#108×C₈₉	—	2	1	no	—
59	#108×C₈₉	—	2	1	no	—
60	#108×C₈₉	—	2	1	no	—
61	#108×C₈₉	D₄	1	2	yes	—
62	#104×C₈₉	—	2	4	no	—
63	#103×C₈₉	—	4	2	no	—
64	#101×C₈₉	—	4	2	no	—
65	#97×C₈₉	—	2	1	no	—
66	C₄×C₈×C₈₉	—	2	1	no	—
67	#41×C₈₉	—	2	2	no	—
68	#41×C₈₉	—	2	2	no	—
69	#41×C₈₉	—	4	1	no	—
70	#41×C₈₉	—	4	1	no	—
71	C₂×Q₈×C₈₉	—	2	1	no	—
72	C₂×Q₈×C₈₉	—	2	2	no	—
73	C₂×Q₈×C₈₉	—	4	1	no	—
74	C₂×Q₈×C₈₉	—	4	1	no	—
75	C₂×Q₈×C₈₉	—	4	1	no	—
76	C₂×Q₈×C₈₉	—	4	1	no	—
77	C₂×Q₈×C₈₉	—	4	1	no	—
78	C₂×D₄×C₈₉	—	2	1	no	—
79	C₂×D₄×C₈₉	—	4	1	no	—
80	C₂²×C₄×C₈₉	—	4	1	no	—
81	C₂²×C₄×C₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₈₉	—	2	2	no	—
84	Q₁₆×C₈₉	—	2	1	no	—
85	Q₁₆×C₈₉	—	2	2	no	—
86	Q₁₆×C₈₉	—	4	1	no	—
87	Q₁₆×C₈₉	—	4	1	no	—
88	Q₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
89	SD₁₆×C₈₉	—	2	2	no	—
90	SD₁₆×C₈₉	—	2	2	no	—
91	D₈×C₈₉	—	2	1	no	—
92	D₈×C₈₉	C₂²⋊C₄	1	2	yes	—
93	M₁₆×C₈₉	—	2	1	no	—
94	M₁₆×C₈₉	—	2	2	no	—
95	M₁₆×C₈₉	—	4	4	no	—
96	M₁₆×C₈₉	—	4	1	no	—
97	M₁₆×C₈₉	—	4	1	no	—
98	M₁₆×C₈₉	—	4	1	no	—
99	M₁₆×C₈₉	D₈	1	2	yes	—
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	4	4	no	—
102	C₂×C₈×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	4	1	no	—
104	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
110	C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₂²⋊C₄×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	—	2	1	no	—
113	C₁₆×C₈₉	—	2	2	no	—
114	C₁₆×C₈₉	—	4	1	no	—
115	C₂³×C₈₉	—	4	1	no	—
116	C₂³×C₈₉	#135	1	1	yes	characteristic
117	Q₈×C₈₉	—	2	2	no	—
118	Q₈×C₈₉	—	4	1	no	—
119	Q₈×C₈₉	—	4	2	no	—
120	Q₈×C₈₉	—	4	2	no	—
121	Q₈×C₈₉	—	8	1	no	—
122	Q₈×C₈₉	—	8	1	no	—
123	D₄×C₈₉	—	2	2	no	—
124	D₄×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	4	2	no	—
126	D₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	1	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	—	4	1	no	—
132	C₂×C₄×C₈₉	—	8	1	no	—
133	C₂×C₄×C₈₉	—	8	1	no	—
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
136	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₈×C₈₉	—	2	2	no	—
141	C₈×C₈₉	—	4	2	no	—
142	C₈×C₈₉	—	8	1	no	—
143	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
145	C₈×C₈₉	#121	1	1	yes	characteristic
146	C₈×C₈₉	#123	1	1	yes	characteristic
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	—	4	2	no	—
149	C₂²×C₈₉	—	16	1	no	—
150	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
151	C₂²×C₈₉	#449	1	1	yes	characteristic
152	C₂²×C₈₉	#463	1	1	yes	characteristic
153	C₄×C₈₉	—	4	2	no	—
154	C₄×C₈₉	—	8	1	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	#386	1	1	yes	characteristic
157	C₄×C₈₉	#442	1	1	yes	characteristic
158	C₄×C₈₉	#447	1	1	yes	characteristic
159	C₄×C₈₉	#459	1	1	yes	characteristic
160	#13035	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₈₉	—	2	1	no	—
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	—	8	2	no	—
165	C₂×C₈₉	#1879	1	1	yes	characteristic
166	C₂×#510	C₂×C₈₉	1	1	yes	characteristic
167	#2957	C₂×C₈₉	1	1	yes	characteristic
168	C₂×#362	C₂×C₈₉	1	1	yes	characteristic, complemented
169	#2143	C₂×C₈₉	1	1	yes	characteristic
170	#1979	C₂×C₈₉	1	1	yes	characteristic, complemented
171	#1385	C₂×C₈₉	1	1	yes	characteristic, complemented
172	#1378	C₂×C₈₉	1	1	yes	characteristic, complemented
173	C₈₉	#13035	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
175	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
176	#568	—	2	1	no	—
177	#510	—	2	2	no	—
178	#510	—	2	2	no	—
179	C₂×Q₃₂	C₄×C₈₉	1	2	yes	complemented
180	C₂×SD₃₂	C₄×C₈₉	1	2	yes	complemented
181	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
183	#444	—	2	2	no	—
184	#443	—	2	2	no	—
185	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
186	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
187	#421	—	2	1	no	—
188	#419	—	2	1	no	—
189	C₂×#101	—	2	1	no	—
190	#405	C₂²×C₈₉	1	1	yes	characteristic
191	#362	—	2	2	no	—
192	#359	—	2	2	no	—
193	C₂×#41	—	2	1	no	—
194	C₂²×Q₈	—	2	1	no	—
195	#135	—	2	1	no	—
196	#135	—	2	1	no	—
197	#135	D₄×C₈₉	1	2	yes	—
198	#135	D₄×C₈₉	1	2	yes	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₂×C₄×C₈₉	1	2	yes	—
205	C₂×Q₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
206	C₂×SD₁₆	—	2	1	no	—
207	C₂×D₈	C₂×C₄×C₈₉	1	1	yes	characteristic
208	#131	—	2	4	no	—
209	#131	—	4	2	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	4	2	no	—
213	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
215	#117	D₄×C₈₉	1	1	yes	characteristic
216	Q₃₂	—	2	4	no	—
217	SD₃₂	—	2	4	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	—
224	#108	—	2	1	no	—
225	#108	—	2	1	no	—
226	#108	—	2	1	no	—
227	#108	D₄×C₈₉	1	2	yes	—
228	#104	—	2	4	no	—
229	#103	—	4	2	no	—
230	#101	—	4	2	no	—
231	#97	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	4	1	no	—
253	Q₁₆	—	4	1	no	—
254	Q₁₆	C₂²⋊C₄×C₈₉	1	2	yes	—
255	SD₁₆	—	2	2	no	—
256	SD₁₆	—	2	2	no	—
257	D₈	—	2	1	no	—
258	D₈	C₂²⋊C₄×C₈₉	1	2	yes	—
259	M₁₆	—	2	1	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	4	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	D₈×C₈₉	1	2	yes	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
274	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
275	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₁₆	—	2	2	no	—
280	C₁₆	—	4	1	no	—
281	C₂³	—	4	1	no	—
282	C₂³	#135×C₈₉	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	8	1	no	—
288	Q₈	—	8	1	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
301	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
302	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
304	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
305	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	8	1	no	—
309	C₈	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
310	C₈	C₄×D₄×C₈₉	1	1	yes	characteristic
311	C₈	#121×C₈₉	1	1	yes	characteristic
312	C₈	#123×C₈₉	1	1	yes	characteristic
313	C₂²	—	4	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	16	1	no	—
316	C₂²	C₄×D₈×C₈₉	1	1	yes	characteristic
317	C₂²	#449×C₈₉	1	1	yes	characteristic
318	C₂²	#463×C₈₉	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	#386×C₈₉	1	1	yes	characteristic
323	C₄	#442×C₈₉	1	1	yes	characteristic
324	C₄	#447×C₈₉	1	1	yes	characteristic
325	C₄	#459×C₈₉	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	8	2	no	—
330	C₂	C₈₉×#1879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	#1879
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
prime radical	C₂×#135×C₈₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #405×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#362×C₈₉	C₂	semidirect
5. C₈₉×#1979	C₂	semidirect
6. C₈₉×#1385	C₂	semidirect
7. C₈₉×#1378	C₂	semidirect
13. C₂×Q₃₂×C₈₉	C₄	semidirect
14. C₂×SD₃₂×C₈₉	C₄	semidirect
160. #13035	C₈₉	direct (N × H)
168. C₂×#362	C₂×C₈₉	semidirect
170. #1979	C₂×C₈₉	semidirect
171. #1385	C₂×C₈₉	semidirect
172. #1378	C₂×C₈₉	semidirect
173. C₈₉	#13035	direct (N × H)
179. C₂×Q₃₂	C₄×C₈₉	semidirect
180. C₂×SD₃₂	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510×C₈₉	2
C₈₉×#2957	2
C₂×#362×C₈₉	2
C₈₉×#2143	2
C₈₉×#1979	2
C₈₉×#1385	2
C₈₉×#1378	2
#13035	89

Subgroups — C89×#13035 #36390883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#13035 #36390883