Subgroups — C₈₉×#9804 #36387652

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 296 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×#1966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#1965	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1451	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	2	no	—
11	#486×C₈₉	—	2	2	no	—
12	#431×C₈₉	C₂²	1	1	yes	characteristic
13	#423×C₈₉	C₂²	1	1	yes	characteristic
14	#423×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#102×C₈₉	C₄	1	4	yes	—
16	C₂×#101×C₈₉	—	2	2	no	—
17	C₂×#100×C₈₉	—	2	2	no	—
18	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#334×C₈₉	—	2	2	no	—
21	#332×C₈₉	—	2	2	no	—
22	#331×C₈₉	—	2	2	no	—
23	#322×C₈₉	—	2	2	no	—
24	C₂²×Q₈×C₈₉	—	2	1	no	—
25	C₂²×D₄×C₈₉	—	2	1	no	—
26	C₂²×D₄×C₈₉	—	2	1	no	—
27	C₂²×D₄×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₈₉	—	2	1	no	—
29	C₂×M₁₆×C₈₉	—	2	1	no	—
30	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₈₉	—	2	1	no	—
33	C₄⋊Q₈×C₈₉	—	2	1	no	—
34	C₄⋊Q₈×C₈₉	—	2	1	no	—
35	C₄⋊Q₈×C₈₉	—	2	1	no	—
36	#127×C₈₉	—	2	1	no	—
37	#127×C₈₉	—	2	1	no	—
38	#127×C₈₉	—	2	1	no	—
39	#127×C₈₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	—	2	1	no	—
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	D₄	1	2	yes	—
48	#102×C₈₉	—	2	8	no	—
49	#101×C₈₉	—	4	4	no	—
50	#100×C₈₉	—	4	4	no	—
51	C₂²⋊C₈×C₈₉	—	2	1	no	—
52	#97×C₈₉	—	2	1	no	—
53	C₄×C₈×C₈₉	—	2	1	no	—
54	C₄×C₈×C₈₉	—	2	1	no	—
55	C₄×C₈×C₈₉	D₄	1	2	yes	—
56	C₄×C₈×C₈₉	D₄	1	2	yes	—
57	C₂⁴×C₈₉	—	2	2	no	—
58	C₂×Q₈×C₈₉	—	4	1	no	—
59	C₂×Q₈×C₈₉	—	4	1	no	—
60	C₂×Q₈×C₈₉	—	4	2	no	—
61	C₂×Q₈×C₈₉	—	4	2	no	—
62	C₂×D₄×C₈₉	—	2	2	no	—
63	C₂×D₄×C₈₉	—	2	2	no	—
64	C₂×D₄×C₈₉	—	4	4	no	—
65	C₂×D₄×C₈₉	—	4	1	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	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
76	M₁₆×C₈₉	—	2	2	no	—
77	M₁₆×C₈₉	—	2	2	no	—
78	M₁₆×C₈₉	—	4	2	no	—
79	M₁₆×C₈₉	—	4	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	4	2	no	—
83	C₂×C₈×C₈₉	—	4	1	no	—
84	C₂×C₈×C₈₉	—	4	1	no	—
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₈₉	—	2	2	no	—
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	4	1	no	—
89	C₄⋊C₄×C₈₉	—	4	1	no	—
90	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	#41	1	1	yes	characteristic
94	C₂³×C₈₉	—	4	2	no	—
95	C₂³×C₈₉	—	4	4	no	—
96	C₂³×C₈₉	—	8	2	no	—
97	C₂³×C₈₉	—	8	2	no	—
98	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₈₉	—	8	2	no	—
100	Q₈×C₈₉	—	8	1	no	—
101	Q₈×C₈₉	—	8	1	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	—	4	4	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	#121	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
121	C₈×C₈₉	—	2	4	no	—
122	C₈×C₈₉	—	4	2	no	—
123	C₈×C₈₉	—	4	2	no	—
124	C₈×C₈₉	—	8	1	no	—
125	C₈×C₈₉	—	8	1	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	8	2	no	—
129	C₂²×C₈₉	—	8	2	no	—
130	C₂²×C₈₉	—	8	2	no	—
131	C₂²×C₈₉	—	16	4	no	—
132	C₂²×C₈₉	#390	1	1	yes	characteristic
133	C₂²×C₈₉	#442	1	2	yes	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	8	2	no	—
137	C₄×C₈₉	#413	1	2	yes	—
138	C₄×C₈₉	#469	1	2	yes	—
139	C₄×C₈₉	#471	1	2	yes	—
140	C₄×C₈₉	#496	1	1	yes	characteristic
141	C₄×C₈₉	#496	1	1	yes	characteristic
142	#9804	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₈₉	—	2	2	no	—
144	C₂×C₈₉	—	16	2	no	—
145	C₂×C₈₉	#1967	1	1	yes	characteristic
146	C₂×C₈₉	#1972	1	1	yes	characteristic
147	C₂×C₈₉	#1977	1	1	yes	characteristic
148	C₂×#486	C₂×C₈₉	1	1	yes	characteristic
149	#1966	C₂×C₈₉	1	1	yes	characteristic, complemented
150	#1965	C₂×C₈₉	1	1	yes	characteristic
151	#1566	C₂×C₈₉	1	1	yes	characteristic, complemented
152	#1546	C₂×C₈₉	1	1	yes	characteristic, complemented
153	#1545	C₂×C₈₉	1	1	yes	characteristic
154	#1451	C₂×C₈₉	1	1	yes	characteristic, complemented
155	C₈₉	#9804	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
157	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
158	#486	—	2	2	no	—
159	#486	—	2	2	no	—
160	#431	C₂²×C₈₉	1	1	yes	characteristic
161	#423	C₂²×C₈₉	1	1	yes	characteristic
162	#423	C₂²×C₈₉	1	1	yes	characteristic
163	C₂×#102	C₄×C₈₉	1	4	yes	—
164	C₂×#101	—	2	2	no	—
165	C₂×#100	—	2	2	no	—
166	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
168	#334	—	2	2	no	—
169	#332	—	2	2	no	—
170	#331	—	2	2	no	—
171	#322	—	2	2	no	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×D₄	—	2	1	no	—
174	C₂²×D₄	—	2	1	no	—
175	C₂²×D₄	C₂×C₄×C₈₉	1	2	yes	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
179	C₂²×C₈	C₂×C₄×C₈₉	1	2	yes	—
180	C₄⋊Q₈	—	2	1	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	#127	—	2	1	no	—
185	#127	—	2	1	no	—
186	#127	—	2	1	no	—
187	#127	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
190	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
196	#102	—	2	8	no	—
197	#101	—	4	4	no	—
198	#100	—	4	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	D₄×C₈₉	1	2	yes	—
204	C₄×C₈	D₄×C₈₉	1	2	yes	—
205	C₂⁴	—	2	2	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	8	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
223	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
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	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	C₂²⋊C₄×C₈₉	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
241	C₄²	#41×C₈₉	1	1	yes	characteristic
242	C₂³	—	4	2	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	C₄×D₄×C₈₉	1	1	yes	characteristic
247	Q₈	—	8	2	no	—
248	Q₈	—	8	1	no	—
249	Q₈	—	8	1	no	—
250	D₄	—	4	4	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	D₄	—	8	2	no	—
255	D₄	—	8	1	no	—
256	D₄	—	8	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
262	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
263	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
264	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
265	C₂×C₄	#124×C₈₉	1	1	yes	characteristic
266	C₂×C₄	#127×C₈₉	1	1	yes	characteristic
267	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
268	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
269	C₈	—	2	4	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	8	1	no	—
273	C₈	—	8	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	16	4	no	—
280	C₂²	#390×C₈₉	1	1	yes	characteristic
281	C₂²	#442×C₈₉	1	2	yes	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	8	2	no	—
285	C₄	#413×C₈₉	1	2	yes	—
286	C₄	#469×C₈₉	1	2	yes	—
287	C₄	#471×C₈₉	1	2	yes	—
288	C₄	#496×C₈₉	1	1	yes	characteristic
289	C₄	#496×C₈₉	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	16	2	no	—
292	C₂	C₈₉×#1967	1	1	yes	characteristic
293	C₂	C₈₉×#1972	1	1	yes	characteristic
294	C₂	C₈₉×#1977	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#390
commutator	C₂²×C₄	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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×#1966	C₂	semidirect
4. C₈₉×#1566	C₂	semidirect
5. C₈₉×#1546	C₂	semidirect
7. C₈₉×#1451	C₂	semidirect
142. #9804	C₈₉	direct (N × H)
149. #1966	C₂×C₈₉	semidirect
151. #1566	C₂×C₈₉	semidirect
152. #1546	C₂×C₈₉	semidirect
154. #1451	C₂×C₈₉	semidirect
155. C₈₉	#9804	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×#1966	2
C₈₉×#1965	2
C₈₉×#1566	2
C₈₉×#1546	2
C₈₉×#1545	2
C₈₉×#1451	2
#9804	89

Subgroups — C89×#9804 #36387652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9804 #36387652