Subgroups — C₈₉×#9769 #36387617

Order 22784.

450 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#1549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
10	#486×C₈₉	—	2	1	no	—
11	#486×C₈₉	—	2	1	no	—
12	#486×C₈₉	C₄	1	4	yes	—
13	#431×C₈₉	C₂²	1	1	yes	characteristic
14	#431×C₈₉	C₂²	1	1	yes	characteristic
15	#423×C₈₉	C₂²	1	1	yes	characteristic
16	#423×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#102×C₈₉	—	2	2	no	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#341×C₈₉	—	2	2	no	—
20	#333×C₈₉	—	2	2	no	—
21	#332×C₈₉	—	2	2	no	—
22	#331×C₈₉	—	2	2	no	—
23	#331×C₈₉	—	2	2	no	—
24	#322×C₈₉	—	2	1	no	—
25	C₈²×C₈₉	—	2	1	no	—
26	C₂²×Q₈×C₈₉	—	2	1	no	—
27	C₂²×D₄×C₈₉	—	2	1	no	—
28	C₂²×D₄×C₈₉	—	2	1	no	—
29	C₂²×D₄×C₈₉	—	2	1	no	—
30	C₂×M₁₆×C₈₉	—	2	1	no	—
31	C₂×M₁₆×C₈₉	—	2	1	no	—
32	C₂²×C₈×C₈₉	D₄	1	2	yes	—
33	C₄⋊Q₈×C₈₉	—	2	1	no	—
34	C₄⋊Q₈×C₈₉	—	2	1	no	—
35	C₄⋊Q₈×C₈₉	—	2	1	no	—
36	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
37	#127×C₈₉	—	2	1	no	—
38	#127×C₈₉	—	2	1	no	—
39	#127×C₈₉	—	2	1	no	—
40	#127×C₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₈₉	—	2	1	no	—
45	C₄⋊C₈×C₈₉	—	2	1	no	—
46	C₄⋊C₈×C₈₉	—	2	1	no	—
47	C₄⋊C₈×C₈₉	—	2	1	no	—
48	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
49	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
50	#102×C₈₉	—	2	4	no	—
51	#102×C₈₉	—	2	4	no	—
52	C₂²⋊C₈×C₈₉	—	2	1	no	—
53	C₂²⋊C₈×C₈₉	—	2	1	no	—
54	C₄×C₈×C₈₉	—	2	1	no	—
55	C₄×C₈×C₈₉	—	2	1	no	—
56	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
59	C₂⁴×C₈₉	—	4	1	no	—
60	C₂×Q₈×C₈₉	—	2	2	no	—
61	C₂×Q₈×C₈₉	—	4	1	no	—
62	C₂×Q₈×C₈₉	—	8	2	no	—
63	C₂×D₄×C₈₉	—	2	2	no	—
64	C₂×D₄×C₈₉	—	2	2	no	—
65	C₂×D₄×C₈₉	—	2	2	no	—
66	C₂×D₄×C₈₉	—	4	1	no	—
67	C₂×D₄×C₈₉	—	4	1	no	—
68	C₂×D₄×C₈₉	—	4	1	no	—
69	C₂×D₄×C₈₉	—	4	2	no	—
70	C₂×D₄×C₈₉	—	4	2	no	—
71	C₂×D₄×C₈₉	—	8	2	no	—
72	C₂×D₄×C₈₉	—	8	2	no	—
73	C₂²×C₄×C₈₉	—	4	1	no	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
77	M₁₆×C₈₉	—	4	2	no	—
78	M₁₆×C₈₉	—	4	2	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	—	4	1	no	—
84	C₂×C₈×C₈₉	—	4	1	no	—
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	4	1	no	—
90	C₄⋊C₄×C₈₉	—	4	1	no	—
91	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₈₉	—	4	2	no	—
96	C₂³×C₈₉	—	8	2	no	—
97	C₂³×C₈₉	—	8	2	no	—
98	C₂³×C₈₉	—	8	2	no	—
99	C₂³×C₈₉	#124	1	1	yes	characteristic
100	Q₈×C₈₉	—	8	2	no	—
101	Q₈×C₈₉	—	8	2	no	—
102	D₄×C₈₉	—	4	4	no	—
103	D₄×C₈₉	—	8	2	no	—
104	D₄×C₈₉	—	8	2	no	—
105	D₄×C₈₉	—	8	2	no	—
106	D₄×C₈₉	—	8	2	no	—
107	D₄×C₈₉	—	8	1	no	—
108	D₄×C₈₉	—	8	1	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	8	2	no	—
113	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₈₉	#135	1	2	yes	—
120	C₂×C₄×C₈₉	#135	1	2	yes	—
121	C₈×C₈₉	—	2	2	no	—
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	—	4	2	no	—
124	C₈×C₈₉	—	4	2	no	—
125	C₈×C₈₉	—	8	1	no	—
126	C₈×C₈₉	—	8	1	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	8	2	no	—
130	C₂²×C₈₉	—	8	2	no	—
131	C₂²×C₈₉	—	8	2	no	—
132	C₂²×C₈₉	—	16	4	no	—
133	C₂²×C₈₉	#390	1	1	yes	characteristic
134	C₂²×C₈₉	#488	1	2	yes	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	8	2	no	—
138	C₄×C₈₉	#416	1	2	yes	—
139	C₄×C₈₉	#471	1	2	yes	—
140	C₄×C₈₉	#472	1	2	yes	—
141	C₄×C₈₉	#495	1	1	yes	characteristic
142	C₄×C₈₉	#495	1	1	yes	characteristic
143	#9769	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₈₉	—	2	2	no	—
145	C₂×C₈₉	—	16	2	no	—
146	C₂×C₈₉	#1958	1	1	yes	characteristic
147	C₂×C₈₉	#1977	1	1	yes	characteristic
148	C₂×C₈₉	#1978	1	1	yes	characteristic
149	C₂×#486	C₂×C₈₉	1	1	yes	characteristic
150	#1837	C₂×C₈₉	1	1	yes	characteristic, complemented
151	#1549	C₂×C₈₉	1	1	yes	characteristic, complemented
152	#1548	C₂×C₈₉	1	1	yes	characteristic, complemented
153	#1545	C₂×C₈₉	1	1	yes	characteristic
154	#1545	C₂×C₈₉	1	1	yes	characteristic
155	#1450	C₂×C₈₉	1	1	yes	characteristic, complemented
156	C₈₉	#9769	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
158	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
159	#486	—	2	1	no	—
160	#486	—	2	1	no	—
161	#486	C₄×C₈₉	1	4	yes	—
162	#431	C₂²×C₈₉	1	1	yes	characteristic
163	#431	C₂²×C₈₉	1	1	yes	characteristic
164	#423	C₂²×C₈₉	1	1	yes	characteristic
165	#423	C₂²×C₈₉	1	1	yes	characteristic
166	C₂×#102	—	2	2	no	—
167	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
168	#341	—	2	2	no	—
169	#333	—	2	2	no	—
170	#332	—	2	2	no	—
171	#331	—	2	2	no	—
172	#331	—	2	2	no	—
173	#322	—	2	1	no	—
174	C₈²	—	2	1	no	—
175	C₂²×Q₈	—	2	1	no	—
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	2	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	D₄×C₈₉	1	2	yes	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	C₂×C₄×C₈₉	1	2	yes	—
186	#127	—	2	1	no	—
187	#127	—	2	1	no	—
188	#127	—	2	1	no	—
189	#127	C₂×C₄×C₈₉	1	2	yes	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
198	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
199	#102	—	2	4	no	—
200	#102	—	2	4	no	—
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
206	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
207	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
208	C₂⁴	—	4	1	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	8	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	8	2	no	—
221	C₂×D₄	—	8	2	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
224	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
225	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
235	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	—	8	2	no	—
248	C₂³	#124×C₈₉	1	1	yes	characteristic
249	Q₈	—	8	2	no	—
250	Q₈	—	8	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	D₄	—	8	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	8	1	no	—
257	D₄	—	8	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
265	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
266	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
267	C₂×C₄	#127×C₈₉	1	1	yes	characteristic
268	C₂×C₄	#135×C₈₉	1	2	yes	—
269	C₂×C₄	#135×C₈₉	1	2	yes	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	2	no	—
274	C₈	—	8	1	no	—
275	C₈	—	8	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	16	4	no	—
282	C₂²	#390×C₈₉	1	1	yes	characteristic
283	C₂²	#488×C₈₉	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	8	2	no	—
287	C₄	#416×C₈₉	1	2	yes	—
288	C₄	#471×C₈₉	1	2	yes	—
289	C₄	#472×C₈₉	1	2	yes	—
290	C₄	#495×C₈₉	1	1	yes	characteristic
291	C₄	#495×C₈₉	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	16	2	no	—
294	C₂	C₈₉×#1958	1	1	yes	characteristic
295	C₂	C₈₉×#1977	1	1	yes	characteristic
296	C₂	C₈₉×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#390
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×#1837	C₂	semidirect
3. C₈₉×#1549	C₂	semidirect
4. C₈₉×#1548	C₂	semidirect
7. C₈₉×#1450	C₂	semidirect
143. #9769	C₈₉	direct (N × H)
150. #1837	C₂×C₈₉	semidirect
151. #1549	C₂×C₈₉	semidirect
152. #1548	C₂×C₈₉	semidirect
155. #1450	C₂×C₈₉	semidirect
156. C₈₉	#9769	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#486×C₈₉	2
C₈₉×#1837	2
C₈₉×#1549	2
C₈₉×#1548	2
C₈₉×#1545	2
C₈₉×#1450	2
#9769	89

Subgroups — C89×#9769 #36387617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9769 #36387617