Subgroups — C₈₉×#21963 #36399811

Order 22784.

830 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1949	C₂	1	4	yes	maximal, complemented
5	C₈₉×#1948	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1947	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1809	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
9	C₂×#117×C₈₉	—	2	1	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	#423×C₈₉	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	#384×C₈₉	C₂²	1	2	yes	—
17	#383×C₈₉	—	2	1	no	—
18	#382×C₈₉	—	2	1	no	—
19	#382×C₈₉	C₂²	1	2	yes	—
20	#381×C₈₉	—	2	1	no	—
21	#380×C₈₉	—	2	1	no	—
22	#380×C₈₉	C₂²	1	2	yes	—
23	#380×C₈₉	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
25	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₈₉	—	2	1	no	—
29	#376×C₈₉	—	2	1	no	—
30	#340×C₈₉	C₂²	1	4	yes	—
31	#340×C₈₉	C₂²	1	4	yes	—
32	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₈₉	—	4	4	no	—
36	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
38	#117×C₈₉	—	4	4	no	—
39	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
46	C₂×C₄²×C₈₉	—	2	1	no	—
47	C₂×C₄²×C₈₉	—	2	1	no	—
48	C₂×C₄²×C₈₉	—	2	2	no	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	#107×C₈₉	—	4	4	no	—
56	#106×C₈₉	—	4	4	no	—
57	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
58	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
59	#95×C₈₉	—	2	2	no	—
60	#95×C₈₉	—	2	2	no	—
61	#95×C₈₉	—	2	2	no	—
62	#95×C₈₉	—	2	2	no	—
63	#95×C₈₉	—	2	2	no	—
64	#95×C₈₉	—	2	1	no	—
65	#95×C₈₉	—	2	1	no	—
66	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	—	2	2	no	—
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	4	no	—
74	C₂²×C₄×C₈₉	—	4	2	no	—
75	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₈₉	—	4	8	no	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄×C₈₉	—	2	2	no	—
90	C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₄⋊C₄×C₈₉	—	2	2	no	—
93	C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	—	2	4	no	—
95	C₄⋊C₄×C₈₉	—	4	4	no	—
96	C₄⋊C₄×C₈₉	—	4	4	no	—
97	C₂²⋊C₄×C₈₉	—	4	4	no	—
98	C₂²⋊C₄×C₈₉	—	4	4	no	—
99	C₄²×C₈₉	—	4	4	no	—
100	C₄²×C₈₉	—	4	2	no	—
101	C₄²×C₈₉	—	4	2	no	—
102	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
103	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
104	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
105	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
106	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
107	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
108	C₂³×C₈₉	—	2	2	no	—
109	C₂³×C₈₉	—	2	1	no	—
110	C₂³×C₈₉	—	2	1	no	—
111	C₂³×C₈₉	—	2	2	no	—
112	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	4	no	—
122	C₂×C₄×C₈₉	—	2	8	no	—
123	C₂×C₄×C₈₉	—	4	4	no	—
124	C₂×C₄×C₈₉	—	4	4	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	4	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	2	no	—
132	C₂×C₄×C₈₉	—	4	4	no	—
133	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₈₉	#137	1	2	yes	—
140	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
143	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
144	C₈×C₈₉	—	2	8	no	—
145	C₂²×C₈₉	—	2	2	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	4	2	no	—
149	C₂²×C₈₉	—	4	2	no	—
150	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₈₉	#537	1	2	yes	—
152	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
154	C₂²×C₈₉	#575	1	2	yes	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	8	4	no	—
160	C₄×C₈₉	—	8	2	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	#428	1	4	yes	—
163	C₄×C₈₉	#518	1	1	yes	characteristic
164	C₄×C₈₉	#519	1	1	yes	characteristic
165	C₄×C₈₉	#544	1	2	yes	—
166	#21963	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₈₉	—	4	2	no	—
168	C₂×C₈₉	#2316	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#428	1	1	yes	characteristic
170	C₂×C₈₉	#2911	1	1	yes	characteristic
171	C₂×C₈₉	#3251	1	2	yes	—
172	C₂×C₈₉	#3252	1	2	yes	—
173	C₂×#423	C₂×C₈₉	1	1	yes	characteristic
174	#2285	C₂×C₈₉	1	1	yes	characteristic
175	#2283	C₂×C₈₉	1	1	yes	characteristic
176	#1949	C₂×C₈₉	1	4	yes	complemented
177	#1948	C₂×C₈₉	1	2	yes	complemented
178	#1947	C₂×C₈₉	1	2	yes	complemented
179	#1809	C₂×C₈₉	1	4	yes	—
180	C₈₉	#21963	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
182	C₂×#117	—	2	1	no	—
183	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
184	C₂×#107	C₂²×C₈₉	1	4	yes	—
185	C₂×#106	C₂²×C₈₉	1	4	yes	—
186	#423	C₄×C₈₉	1	8	yes	complemented
187	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
188	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
189	#384	C₂²×C₈₉	1	2	yes	—
190	#383	—	2	1	no	—
191	#382	—	2	1	no	—
192	#382	C₂²×C₈₉	1	2	yes	—
193	#381	—	2	1	no	—
194	#380	—	2	1	no	—
195	#380	C₂²×C₈₉	1	2	yes	—
196	#380	C₂²×C₈₉	1	2	yes	—
197	C₄×C₄⋊C₄	—	2	1	no	—
198	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
199	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
200	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
201	C₄×C₂²⋊C₄	—	2	1	no	—
202	#376	—	2	1	no	—
203	#340	C₂²×C₈₉	1	4	yes	—
204	#340	C₂²×C₈₉	1	4	yes	—
205	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
206	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
207	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
208	C₂×M₁₆	—	4	4	no	—
209	C₂×M₁₆	C₂×C₄×C₈₉	1	8	yes	—
210	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
211	#117	—	4	4	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	C₂×C₄×C₈₉	1	4	yes	—
223	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
224	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
225	C₂×C₄²	Q₈×C₈₉	1	4	yes	—
226	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
228	#107	—	4	4	no	—
229	#106	—	4	4	no	—
230	C₄⋊C₈	C₂×C₄×C₈₉	1	8	yes	—
231	C₄⋊C₈	C₂×C₄×C₈₉	1	8	yes	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₄⋊C₄×C₈₉	1	4	yes	—
249	C₂²×C₄	C₄⋊C₄×C₈₉	1	4	yes	—
250	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×Q₈×C₈₉	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
258	M₁₆	—	4	8	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	C₂²×C₄×C₈₉	1	8	yes	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	—	4	4	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₄²	—	4	4	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₄⋊C₄×C₈₉	1	4	yes	—
276	C₄²	C₄⋊C₄×C₈₉	1	4	yes	—
277	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
278	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
279	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
280	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
281	C₂³	—	2	2	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
286	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
308	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
309	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
311	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
312	C₂×C₄	#137×C₈₉	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
315	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
316	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
317	C₈	—	2	8	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂²×C₄⋊C₄×C₈₉	1	1	yes	characteristic
324	C₂²	#537×C₈₉	1	2	yes	—
325	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
326	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
327	C₂²	#575×C₈₉	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	4	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	#428×C₈₉	1	4	yes	—
336	C₄	#518×C₈₉	1	1	yes	characteristic
337	C₄	#519×C₈₉	1	1	yes	characteristic
338	C₄	#544×C₈₉	1	2	yes	—
339	C₂	—	4	2	no	—
340	C₂	C₈₉×#2316	1	1	yes	characteristic
341	C₂	C₂×#428×C₈₉	1	1	yes	characteristic
342	C₂	C₈₉×#2911	1	1	yes	characteristic
343	C₂	C₈₉×#3251	1	2	yes	—
344	C₂	C₈₉×#3252	1	2	yes	—

	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
4. C₈₉×#1949	C₂	semidirect
5. C₈₉×#1948	C₂	semidirect
6. C₈₉×#1947	C₂	semidirect
13. #423×C₈₉	C₄	semidirect
166. #21963	C₈₉	direct (N × H)
176. #1949	C₂×C₈₉	semidirect
177. #1948	C₂×C₈₉	semidirect
178. #1947	C₂×C₈₉	semidirect
180. C₈₉	#21963	direct (N × H)
186. #423	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×#423×C₈₉	2
C₈₉×#2285	2
C₈₉×#2283	2
C₈₉×#1949	2
C₈₉×#1948	2
C₈₉×#1947	2
C₈₉×#1809	2
#21963	89

Subgroups — C89×#21963 #36399811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#21963 #36399811