Subgroups — C₈₉×#23842 #36401690

Order 22784.

662 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2331	C₂	1	2	yes	maximal
2	C₈₉×#2085	C₂	1	2	yes	maximal
3	C₈₉×#2085	C₂	1	2	yes	maximal
4	C₈₉×#2058	C₂	1	2	yes	maximal
5	C₈₉×#2057	C₂	1	2	yes	maximal
6	C₈₉×#1776	C₂	1	2	yes	maximal
7	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₄×#106×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
11	C₂×#107×C₈₉	C₂²	1	2	yes	—
12	C₂×#107×C₈₉	C₂²	1	2	yes	—
13	C₂×#106×C₈₉	C₂²	1	2	yes	—
14	C₂×#106×C₈₉	C₂²	1	2	yes	—
15	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#391×C₈₉	—	2	2	no	—
18	#391×C₈₉	—	2	2	no	—
19	#389×C₈₉	C₂²	1	2	yes	—
20	#389×C₈₉	C₂²	1	2	yes	—
21	#387×C₈₉	—	2	2	no	—
22	#387×C₈₉	—	2	2	no	—
23	#382×C₈₉	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
26	#376×C₈₉	C₂²	1	1	yes	characteristic
27	#340×C₈₉	C₂²	1	1	yes	characteristic
28	#340×C₈₉	C₂²	1	2	yes	—
29	#340×C₈₉	C₂²	1	1	yes	characteristic
30	#340×C₈₉	C₂²	1	1	yes	characteristic
31	#340×C₈₉	C₂²	1	1	yes	characteristic
32	#340×C₈₉	C₂²	1	2	yes	—
33	#336×C₈₉	C₂²	1	2	yes	—
34	#336×C₈₉	C₂²	1	2	yes	—
35	C₂²×Q₈×C₈₉	—	2	2	no	—
36	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₈₉	—	2	4	no	—
40	C₄⋊Q₈×C₈₉	—	2	4	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₈₉	—	2	2	no	—
49	C₂×C₄²×C₈₉	—	2	2	no	—
50	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	#107×C₈₉	—	2	4	no	—
53	#107×C₈₉	—	2	4	no	—
54	#106×C₈₉	—	2	4	no	—
55	#106×C₈₉	—	2	4	no	—
56	#97×C₈₉	Q₈	1	4	yes	complemented
57	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
58	#95×C₈₉	—	2	2	no	—
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	D₄	1	1	yes	characteristic
65	#95×C₈₉	D₄	1	2	yes	—
66	#95×C₈₉	D₄	1	1	yes	characteristic
67	C₂×Q₈×C₈₉	—	2	8	no	—
68	C₂×Q₈×C₈₉	—	8	2	no	—
69	C₂×Q₈×C₈₉	—	8	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	2	no	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
86	C₄⋊C₄×C₈₉	—	2	8	no	—
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	2	2	no	—
90	C₄⋊C₄×C₈₉	—	2	2	no	—
91	C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₄⋊C₄×C₈₉	—	2	2	no	—
93	C₄⋊C₄×C₈₉	—	8	2	no	—
94	C₄⋊C₄×C₈₉	—	8	2	no	—
95	C₄⋊C₄×C₈₉	#41	1	4	yes	—
96	C₄⋊C₄×C₈₉	#41	1	4	yes	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₄²×C₈₉	—	2	2	no	—
99	C₄²×C₈₉	—	8	2	no	—
100	C₄²×C₈₉	—	8	2	no	—
101	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
102	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
103	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₈₉	—	8	8	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	4	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
131	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
132	C₈×C₈₉	—	2	2	no	—
133	C₈×C₈₉	—	2	4	no	—
134	C₈×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	#525	1	2	yes	—
136	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₈₉	#576	1	1	yes	characteristic
138	C₂²×C₈₉	#577	1	2	yes	—
139	C₂²×C₈₉	#578	1	1	yes	characteristic
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	4	2	no	—
143	C₄×C₈₉	—	4	4	no	—
144	C₄×C₈₉	—	4	4	no	—
145	C₄×C₈₉	—	4	2	no	—
146	C₄×C₈₉	—	8	2	no	—
147	C₄×C₈₉	—	8	2	no	—
148	C₄×C₈₉	#533	1	1	yes	characteristic
149	C₄×C₈₉	#533	1	2	yes	—
150	C₄×C₈₉	#533	1	1	yes	characteristic
151	#23842	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₈₉	#2399	1	1	yes	characteristic
153	C₂×C₈₉	#3118	1	2	yes	—
154	C₂×C₈₉	#3121	1	2	yes	—
155	C₂×C₈₉	#3163	1	1	yes	characteristic
156	C₂×C₈₉	#3164	1	1	yes	characteristic
157	#2331	C₂×C₈₉	1	2	yes	—
158	#2085	C₂×C₈₉	1	2	yes	—
159	#2085	C₂×C₈₉	1	2	yes	—
160	#2058	C₂×C₈₉	1	2	yes	—
161	#2057	C₂×C₈₉	1	2	yes	—
162	#1776	C₂×C₈₉	1	2	yes	—
163	C₄×#107	C₂×C₈₉	1	1	yes	characteristic
164	C₄×#106	C₂×C₈₉	1	1	yes	characteristic
165	#1746	C₂×C₈₉	1	1	yes	characteristic
166	C₈₉	#23842	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
168	C₂×#107	C₂²×C₈₉	1	2	yes	—
169	C₂×#107	C₂²×C₈₉	1	2	yes	—
170	C₂×#106	C₂²×C₈₉	1	2	yes	—
171	C₂×#106	C₂²×C₈₉	1	2	yes	—
172	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
174	#391	—	2	2	no	—
175	#391	—	2	2	no	—
176	#389	C₂²×C₈₉	1	2	yes	—
177	#389	C₂²×C₈₉	1	2	yes	—
178	#387	—	2	2	no	—
179	#387	—	2	2	no	—
180	#382	C₂²×C₈₉	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
182	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
183	#376	C₂²×C₈₉	1	1	yes	characteristic
184	#340	C₂²×C₈₉	1	1	yes	characteristic
185	#340	C₂²×C₈₉	1	2	yes	—
186	#340	C₂²×C₈₉	1	1	yes	characteristic
187	#340	C₂²×C₈₉	1	1	yes	characteristic
188	#340	C₂²×C₈₉	1	1	yes	characteristic
189	#340	C₂²×C₈₉	1	2	yes	—
190	#336	C₂²×C₈₉	1	2	yes	—
191	#336	C₂²×C₈₉	1	2	yes	—
192	C₂²×Q₈	—	2	2	no	—
193	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
208	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
209	#107	—	2	4	no	—
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	#106	—	2	4	no	—
213	#97	Q₈×C₈₉	1	4	yes	complemented
214	C₄×C₈	Q₈×C₈₉	1	4	yes	complemented
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	D₄×C₈₉	1	1	yes	characteristic
222	#95	D₄×C₈₉	1	2	yes	—
223	#95	D₄×C₈₉	1	1	yes	characteristic
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	—	8	2	no	—
226	C₂×Q₈	—	8	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
241	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
242	C₂×C₈	C₂×Q₈×C₈₉	1	4	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	—	8	2	no	—
252	C₄⋊C₄	#41×C₈₉	1	4	yes	—
253	C₄⋊C₄	#41×C₈₉	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	8	2	no	—
257	C₄²	—	8	2	no	—
258	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
259	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
260	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
261	Q₈	—	8	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
284	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
285	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
286	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
288	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₂²	#525×C₈₉	1	2	yes	—
293	C₂²	C₂×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
294	C₂²	#576×C₈₉	1	1	yes	characteristic
295	C₂²	#577×C₈₉	1	2	yes	—
296	C₂²	#578×C₈₉	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	#533×C₈₉	1	1	yes	characteristic
306	C₄	#533×C₈₉	1	2	yes	—
307	C₄	#533×C₈₉	1	1	yes	characteristic
308	C₂	C₈₉×#2399	1	1	yes	characteristic
309	C₂	C₈₉×#3118	1	2	yes	—
310	C₂	C₈₉×#3121	1	2	yes	—
311	C₂	C₈₉×#3163	1	1	yes	characteristic
312	C₂	C₈₉×#3164	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ #376×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
56. #97×C₈₉	Q₈	semidirect
57. C₄×C₈×C₈₉	Q₈	semidirect
151. #23842	C₈₉	direct (N × H)
166. C₈₉	#23842	direct (N × H)
213. #97	Q₈×C₈₉	semidirect
214. C₄×C₈	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2331	2
C₈₉×#2085	2
C₈₉×#2058	2
C₈₉×#2057	2
C₈₉×#1776	2
C₄×#107×C₈₉	2
C₄×#106×C₈₉	2
C₈₉×#1746	2
#23842	89

Subgroups — C89×#23842 #36401690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23842 #36401690