Subgroups — C₈₉×#18698 #36396546

Order 22784.

834 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3254	C₂	1	2	yes	maximal
3	C₈₉×#3131	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1636	C₂	1	4	yes	maximal, complemented
5	C₈₉×#1626	C₂	1	4	yes	maximal, complemented
6	C₈₉×#1561	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1493	C₂	1	2	yes	maximal, complemented
8	#564×C₈₉	—	2	2	no	—
9	#557×C₈₉	—	2	2	no	—
10	#557×C₈₉	—	2	2	no	—
11	#557×C₈₉	C₂²	1	4	yes	—
12	#541×C₈₉	—	2	4	no	—
13	#541×C₈₉	—	2	2	no	—
14	#533×C₈₉	C₂²	1	2	yes	—
15	#533×C₈₉	C₂²	1	1	yes	characteristic
16	#532×C₈₉	C₂²	1	1	yes	characteristic
17	#488×C₈₉	—	2	2	no	—
18	#487×C₈₉	C₂²	1	2	yes	—
19	#486×C₈₉	C₂²	1	2	yes	—
20	#477×C₈₉	C₂²	1	4	yes	—
21	#474×C₈₉	—	2	2	no	—
22	#468×C₈₉	—	2	4	no	—
23	#461×C₈₉	—	2	2	no	—
24	#460×C₈₉	C₂²	1	4	yes	—
25	#423×C₈₉	C₂²	1	2	yes	—
26	#416×C₈₉	—	2	2	no	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	#332×C₈₉	C₂²	1	4	yes	—
29	Q₈⋊C₈×C₈₉	C₂²	1	8	yes	—
30	C₂×#41×C₈₉	—	2	1	no	—
31	C₂×SD₁₆×C₈₉	—	2	4	no	—
32	C₂×M₁₆×C₈₉	—	2	2	no	—
33	C₂²×C₈×C₈₉	—	2	1	no	—
34	C₄⋊Q₈×C₈₉	—	2	2	no	—
35	C₄⋊Q₈×C₈₉	—	2	4	no	—
36	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
37	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
38	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
40	#127×C₈₉	C₂³	1	1	yes	characteristic
41	#126×C₈₉	—	2	8	no	—
42	#125×C₈₉	—	2	2	no	—
43	#125×C₈₉	—	2	8	no	—
44	#125×C₈₉	—	2	8	no	—
45	#125×C₈₉	—	2	4	no	—
46	#125×C₈₉	—	2	8	no	—
47	#125×C₈₉	—	2	2	no	—
48	#125×C₈₉	D₄	1	2	yes	—
49	#124×C₈₉	—	2	1	no	—
50	C₂²⋊Q₈×C₈₉	—	2	2	no	—
51	C₂²⋊Q₈×C₈₉	—	2	2	no	—
52	C₂²⋊Q₈×C₈₉	D₄	1	4	yes	—
53	#121×C₈₉	—	2	2	no	—
54	C₄×Q₈×C₈₉	—	2	4	no	—
55	C₄×Q₈×C₈₉	—	2	4	no	—
56	C₄×Q₈×C₈₉	—	2	4	no	—
57	C₄×Q₈×C₈₉	—	2	2	no	—
58	C₄×Q₈×C₈₉	—	2	2	no	—
59	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
60	C₄×D₄×C₈₉	—	2	2	no	—
61	#117×C₈₉	—	2	2	no	—
62	#117×C₈₉	D₄	1	2	yes	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	#107×C₈₉	—	2	4	no	—
65	C₄⋊C₈×C₈₉	—	2	2	no	—
66	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
67	#103×C₈₉	—	2	4	no	—
68	#103×C₈₉	—	2	4	no	—
69	#102×C₈₉	—	2	4	no	—
70	#102×C₈₉	—	2	4	no	—
71	C₄×C₈×C₈₉	—	2	1	no	—
72	C₄×C₈×C₈₉	C₂³	1	2	yes	—
73	#41×C₈₉	—	4	4	no	—
74	C₂×Q₈×C₈₉	—	2	2	no	—
75	C₂×Q₈×C₈₉	—	2	2	no	—
76	C₂×Q₈×C₈₉	—	2	1	no	—
77	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₈₉	—	2	1	no	—
79	C₂×D₄×C₈₉	—	2	2	no	—
80	C₂×D₄×C₈₉	—	4	1	no	—
81	C₂²×C₄×C₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	SD₁₆×C₈₉	—	8	4	no	—
85	M₁₆×C₈₉	—	8	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₄⋊C₄×C₈₉	—	2	8	no	—
90	C₄⋊C₄×C₈₉	—	2	8	no	—
91	C₄⋊C₄×C₈₉	—	2	8	no	—
92	C₄⋊C₄×C₈₉	—	2	2	no	—
93	C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₄⋊C₄×C₈₉	—	2	4	no	—
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	4	no	—
99	C₄⋊C₄×C₈₉	—	2	2	no	—
100	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
102	C₂²⋊C₄×C₈₉	—	2	4	no	—
103	C₂²⋊C₄×C₈₉	—	2	2	no	—
104	C₂²⋊C₄×C₈₉	—	4	2	no	—
105	C₄²×C₈₉	—	2	8	no	—
106	C₄²×C₈₉	—	2	2	no	—
107	C₄²×C₈₉	—	2	4	no	—
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×D₄	1	2	yes	—
110	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₂³×C₈₉	—	4	1	no	—
112	C₂³×C₈₉	#120	1	1	yes	characteristic
113	Q₈×C₈₉	—	2	4	no	—
114	Q₈×C₈₉	—	2	4	no	—
115	Q₈×C₈₉	—	4	2	no	—
116	Q₈×C₈₉	—	4	2	no	—
117	Q₈×C₈₉	—	4	2	no	—
118	Q₈×C₈₉	—	4	4	no	—
119	D₄×C₈₉	—	4	2	no	—
120	D₄×C₈₉	—	4	2	no	—
121	D₄×C₈₉	—	4	2	no	—
122	D₄×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	4	no	—
127	C₂×C₄×C₈₉	—	2	4	no	—
128	C₂×C₄×C₈₉	—	2	4	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	#120	1	2	yes	—
132	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
135	C₈×C₈₉	—	4	2	no	—
136	C₈×C₈₉	—	4	4	no	—
137	C₂²×C₈₉	—	2	2	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	8	2	no	—
140	C₂²×C₈₉	—	8	1	no	—
141	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	4	4	no	—
145	C₄×C₈₉	—	4	4	no	—
146	C₄×C₈₉	—	4	4	no	—
147	C₄×C₈₉	—	8	1	no	—
148	C₄×C₈₉	#454	1	2	yes	—
149	C₄×C₈₉	#577	1	2	yes	—
150	C₄×C₈₉	#578	1	2	yes	—
151	#18698	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₈₉	—	4	1	no	—
153	C₂×C₈₉	—	16	1	no	—
154	C₂×C₈₉	#3013	1	2	yes	—
155	C₂×C₈₉	C₂×#454	1	1	yes	characteristic
156	#3556	C₂×C₈₉	1	1	yes	characteristic, complemented
157	#3254	C₂×C₈₉	1	2	yes	—
158	#3131	C₂×C₈₉	1	1	yes	characteristic
159	#1636	C₂×C₈₉	1	4	yes	complemented
160	#1626	C₂×C₈₉	1	4	yes	complemented
161	#1561	C₂×C₈₉	1	1	yes	characteristic, complemented
162	#1493	C₂×C₈₉	1	2	yes	complemented
163	C₈₉	#18698	1	1	yes	characteristic, direct product factor, Sylow
164	#564	—	2	2	no	—
165	#557	—	2	2	no	—
166	#557	—	2	2	no	—
167	#557	C₂²×C₈₉	1	4	yes	—
168	#541	—	2	4	no	—
169	#541	—	2	2	no	—
170	#533	C₂²×C₈₉	1	2	yes	—
171	#533	C₂²×C₈₉	1	1	yes	characteristic
172	#532	C₂²×C₈₉	1	1	yes	characteristic
173	#488	—	2	2	no	—
174	#487	C₂²×C₈₉	1	2	yes	—
175	#486	C₂²×C₈₉	1	2	yes	—
176	#477	C₂²×C₈₉	1	4	yes	—
177	#474	—	2	2	no	—
178	#468	—	2	4	no	—
179	#461	—	2	2	no	—
180	#460	C₂²×C₈₉	1	4	yes	—
181	#423	C₂²×C₈₉	1	2	yes	—
182	#416	—	2	2	no	—
183	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
184	#332	C₂²×C₈₉	1	4	yes	—
185	Q₈⋊C₈	C₂²×C₈₉	1	8	yes	—
186	C₂×#41	—	2	1	no	—
187	C₂×SD₁₆	—	2	4	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
193	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
194	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
195	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
196	#127	C₂³×C₈₉	1	1	yes	characteristic
197	#126	—	2	8	no	—
198	#125	—	2	2	no	—
199	#125	—	2	8	no	—
200	#125	—	2	8	no	—
201	#125	—	2	4	no	—
202	#125	—	2	8	no	—
203	#125	—	2	2	no	—
204	#125	D₄×C₈₉	1	2	yes	—
205	#124	—	2	1	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	D₄×C₈₉	1	4	yes	—
209	#121	—	2	2	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
216	C₄×D₄	—	2	2	no	—
217	#117	—	2	2	no	—
218	#117	D₄×C₈₉	1	2	yes	—
219	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
223	#103	—	2	4	no	—
224	#103	—	2	4	no	—
225	#102	—	2	4	no	—
226	#102	—	2	4	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	C₂³×C₈₉	1	2	yes	—
229	#41	—	4	4	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
240	SD₁₆	—	8	4	no	—
241	M₁₆	—	8	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
257	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
258	C₂²⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	—	2	8	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₈₉	1	2	yes	—
266	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
267	C₂³	—	4	1	no	—
268	C₂³	#120×C₈₉	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	2	4	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	4	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	#120×C₈₉	1	2	yes	—
288	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
291	C₈	—	4	2	no	—
292	C₈	—	4	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	1	no	—
297	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	1	no	—
304	C₄	#454×C₈₉	1	2	yes	—
305	C₄	#577×C₈₉	1	2	yes	—
306	C₄	#578×C₈₉	1	2	yes	—
307	C₂	—	4	1	no	—
308	C₂	—	16	1	no	—
309	C₂	C₈₉×#3013	1	2	yes	—
310	C₂	C₂×#454×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
prime radical	#532×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
1. C₈₉×#3556	C₂	semidirect
4. C₈₉×#1636	C₂	semidirect
5. C₈₉×#1626	C₂	semidirect
6. C₈₉×#1561	C₂	semidirect
7. C₈₉×#1493	C₂	semidirect
151. #18698	C₈₉	direct (N × H)
156. #3556	C₂×C₈₉	semidirect
159. #1636	C₂×C₈₉	semidirect
160. #1626	C₂×C₈₉	semidirect
161. #1561	C₂×C₈₉	semidirect
162. #1493	C₂×C₈₉	semidirect
163. C₈₉	#18698	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×#3556	2
C₈₉×#3254	2
C₈₉×#3131	2
C₈₉×#1636	2
C₈₉×#1626	2
C₈₉×#1561	2
C₈₉×#1493	2
#18698	89

Subgroups — C89×#18698 #36396546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#18698 #36396546