Subgroups — C₈₉×#23803 #36401651

Order 22784.

1142 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#440×C₈₉	C₂	1	4	yes	maximal, complemented
2	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2095	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2093	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2014	C₂	1	2	yes	maximal
7	C₈₉×#2013	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1745	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₈₉	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₈₉	—	2	2	no	—
14	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
15	#440×C₈₉	—	2	16	no	—
16	C₂×#107×C₈₉	C₂²	1	4	yes	—
17	C₂×#103×C₈₉	C₂²	1	4	yes	—
18	C₂×#102×C₈₉	C₂²	1	4	yes	—
19	C₂×#97×C₈₉	C₂²	1	2	yes	—
20	#391×C₈₉	C₂²	1	2	yes	—
21	#390×C₈₉	—	2	2	no	—
22	#390×C₈₉	C₂²	1	2	yes	—
23	#389×C₈₉	—	2	2	no	—
24	#384×C₈₉	C₂²	1	1	yes	characteristic
25	#384×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
27	C₄³×C₈₉	C₂²	1	1	yes	characteristic
28	#336×C₈₉	C₂²	1	2	yes	—
29	#336×C₈₉	C₂²	1	2	yes	—
30	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
31	C₂²×D₄×C₈₉	—	2	2	no	—
32	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
33	C₂³×C₄×C₈₉	—	2	2	no	—
34	C₂×SD₁₆×C₈₉	—	2	8	no	—
35	C₂×SD₁₆×C₈₉	—	2	8	no	—
36	C₂×SD₁₆×C₈₉	—	2	8	no	—
37	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₈₉	—	2	4	no	—
39	#127×C₈₉	—	2	4	no	—
40	C₄×Q₈×C₈₉	—	2	8	no	—
41	C₄×D₄×C₈₉	—	2	8	no	—
42	C₄×D₄×C₈₉	—	4	8	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
51	C₂×C₄²×C₈₉	—	2	2	no	—
52	C₂×C₄²×C₈₉	D₄	1	2	yes	—
53	C₂×C₄²×C₈₉	D₄	1	2	yes	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	#107×C₈₉	—	2	8	no	—
58	#103×C₈₉	—	2	8	no	—
59	#102×C₈₉	—	2	8	no	—
60	#97×C₈₉	D₄	1	8	yes	complemented
61	#95×C₈₉	—	2	4	no	—
62	C₂⁴×C₈₉	—	2	2	no	—
63	C₂×Q₈×C₈₉	—	2	2	no	—
64	C₂×Q₈×C₈₉	—	2	4	no	—
65	C₂×Q₈×C₈₉	—	2	4	no	—
66	C₂×Q₈×C₈₉	#41	1	4	yes	—
67	C₂×D₄×C₈₉	—	2	8	no	—
68	C₂×D₄×C₈₉	—	2	4	no	—
69	C₂×D₄×C₈₉	—	2	4	no	—
70	C₂×D₄×C₈₉	—	2	2	no	—
71	C₂×D₄×C₈₉	—	4	4	no	—
72	C₂×D₄×C₈₉	—	4	4	no	—
73	C₂×D₄×C₈₉	#41	1	4	yes	—
74	C₂²×C₄×C₈₉	—	2	2	no	—
75	C₂²×C₄×C₈₉	—	2	2	no	—
76	C₂²×C₄×C₈₉	—	2	2	no	—
77	C₂²×C₄×C₈₉	—	4	8	no	—
78	C₂²×C₄×C₈₉	—	4	4	no	—
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	2	yes	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₈₉	—	2	32	no	—
85	C₂×C₈×C₈₉	—	2	4	no	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	C₂×D₄	1	8	yes	—
88	C₄⋊C₄×C₈₉	—	2	8	no	—
89	C₄⋊C₄×C₈₉	—	2	4	no	—
90	C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₄⋊C₄×C₈₉	—	4	4	no	—
92	C₄⋊C₄×C₈₉	—	4	4	no	—
93	C₄⋊C₄×C₈₉	—	4	4	no	—
94	C₂²⋊C₄×C₈₉	—	4	4	no	—
95	C₂²⋊C₄×C₈₉	—	4	8	no	—
96	C₄²×C₈₉	—	2	4	no	—
97	C₄²×C₈₉	—	2	4	no	—
98	C₄²×C₈₉	—	2	2	no	—
99	C₄²×C₈₉	—	2	2	no	—
100	C₄²×C₈₉	—	4	4	no	—
101	C₄²×C₈₉	C₂×D₄	1	4	yes	—
102	C₂³×C₈₉	—	2	4	no	—
103	C₂³×C₈₉	—	4	2	no	—
104	C₂³×C₈₉	—	4	2	no	—
105	C₂³×C₈₉	—	4	4	no	—
106	C₂³×C₈₉	—	4	4	no	—
107	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₈₉	—	2	8	no	—
109	Q₈×C₈₉	—	4	4	no	—
110	D₄×C₈₉	—	2	8	no	—
111	D₄×C₈₉	—	4	16	no	—
112	D₄×C₈₉	—	4	4	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	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	4	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	4	no	—
125	C₂×C₄×C₈₉	—	4	4	no	—
126	C₂×C₄×C₈₉	—	8	8	no	—
127	C₂×C₄×C₈₉	#127	1	2	yes	—
128	C₂×C₄×C₈₉	#127	1	2	yes	—
129	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₈₉	#137	1	4	yes	—
131	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
134	C₈×C₈₉	—	2	8	no	—
135	C₂²×C₈₉	—	4	8	no	—
136	C₂²×C₈₉	—	4	4	no	—
137	C₂²×C₈₉	—	8	4	no	—
138	C₂²×C₈₉	—	8	2	no	—
139	C₂²×C₈₉	—	8	2	no	—
140	C₂²×C₈₉	#525	1	1	yes	characteristic
141	C₂²×C₈₉	#525	1	1	yes	characteristic
142	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
143	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
144	C₂²×C₈₉	C₂×#137	1	2	yes	—
145	C₄×C₈₉	—	2	4	no	—
146	C₄×C₈₉	—	2	4	no	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	4	4	no	—
150	C₄×C₈₉	—	8	2	no	—
151	C₄×C₈₉	—	8	2	no	—
152	C₄×C₈₉	#525	1	4	yes	—
153	#23803	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₈₉	—	8	4	no	—
155	C₂×C₈₉	#2397	1	1	yes	characteristic
156	C₂×C₈₉	#3110	1	2	yes	—
157	C₂×C₈₉	#3112	1	2	yes	—
158	C₂×C₈₉	#3151	1	1	yes	characteristic
159	C₂×C₈₉	#3152	1	1	yes	characteristic
160	C₂×#440	C₂×C₈₉	1	4	yes	complemented
161	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
162	C₄×#127	C₂×C₈₉	1	1	yes	characteristic
163	#2095	C₂×C₈₉	1	2	yes	complemented
164	#2093	C₂×C₈₉	1	2	yes	complemented
165	#2014	C₂×C₈₉	1	2	yes	—
166	#2013	C₂×C₈₉	1	2	yes	complemented
167	#1745	C₂×C₈₉	1	1	yes	characteristic, complemented
168	C₈₉	#23803	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×SD₁₆	C₂²×C₈₉	1	4	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
171	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
173	C₂×C₄×D₄	—	2	2	no	—
174	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
175	#440	—	2	16	no	—
176	C₂×#107	C₂²×C₈₉	1	4	yes	—
177	C₂×#103	C₂²×C₈₉	1	4	yes	—
178	C₂×#102	C₂²×C₈₉	1	4	yes	—
179	C₂×#97	C₂²×C₈₉	1	2	yes	—
180	#391	C₂²×C₈₉	1	2	yes	—
181	#390	—	2	2	no	—
182	#390	C₂²×C₈₉	1	2	yes	—
183	#389	—	2	2	no	—
184	#384	C₂²×C₈₉	1	1	yes	characteristic
185	#384	C₂²×C₈₉	1	1	yes	characteristic
186	C₄×C₄⋊C₄	—	2	2	no	—
187	C₄³	C₂²×C₈₉	1	1	yes	characteristic
188	#336	C₂²×C₈₉	1	2	yes	—
189	#336	C₂²×C₈₉	1	2	yes	—
190	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
191	C₂²×D₄	—	2	2	no	—
192	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
193	C₂³×C₄	—	2	2	no	—
194	C₂×SD₁₆	—	2	8	no	—
195	C₂×SD₁₆	—	2	8	no	—
196	C₂×SD₁₆	—	2	8	no	—
197	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
198	C₄⋊Q₈	—	2	4	no	—
199	#127	—	2	4	no	—
200	C₄×Q₈	—	2	8	no	—
201	C₄×D₄	—	2	8	no	—
202	C₄×D₄	—	4	8	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	4	no	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	D₄×C₈₉	1	2	yes	—
213	C₂×C₄²	D₄×C₈₉	1	2	yes	—
214	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
217	#107	—	2	8	no	—
218	#103	—	2	8	no	—
219	#102	—	2	8	no	—
220	#97	D₄×C₈₉	1	8	yes	complemented
221	#95	—	2	4	no	—
222	C₂⁴	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	#41×C₈₉	1	4	yes	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	#41×C₈₉	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	8	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
244	SD₁₆	—	2	32	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	C₂×D₄×C₈₉	1	8	yes	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	8	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	4	no	—
261	C₄²	C₂×D₄×C₈₉	1	4	yes	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
268	Q₈	—	2	8	no	—
269	Q₈	—	4	4	no	—
270	D₄	—	2	8	no	—
271	D₄	—	4	16	no	—
272	D₄	—	4	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	8	no	—
287	C₂×C₄	#127×C₈₉	1	2	yes	—
288	C₂×C₄	#127×C₈₉	1	2	yes	—
289	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
290	C₂×C₄	#137×C₈₉	1	4	yes	—
291	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
292	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
293	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	#525×C₈₉	1	1	yes	characteristic
301	C₂²	#525×C₈₉	1	1	yes	characteristic
302	C₂²	C₂×#127×C₈₉	1	1	yes	characteristic
303	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
304	C₂²	C₂×#137×C₈₉	1	2	yes	—
305	C₄	—	2	4	no	—
306	C₄	—	2	4	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	4	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	#525×C₈₉	1	4	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₈₉×#2397	1	1	yes	characteristic
315	C₂	C₈₉×#3110	1	2	yes	—
316	C₂	C₈₉×#3112	1	2	yes	—
317	C₂	C₈₉×#3151	1	1	yes	characteristic
318	C₂	C₈₉×#3152	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#440×C₈₉	C₂	semidirect
2. C₄×C₄⋊Q₈×C₈₉	C₂	semidirect
4. C₈₉×#2095	C₂	semidirect
5. C₈₉×#2093	C₂	semidirect
7. C₈₉×#2013	C₂	semidirect
8. C₈₉×#1745	C₂	semidirect
60. #97×C₈₉	D₄	semidirect
153. #23803	C₈₉	direct (N × H)
160. C₂×#440	C₂×C₈₉	semidirect
161. C₄×C₄⋊Q₈	C₂×C₈₉	semidirect
163. #2095	C₂×C₈₉	semidirect
164. #2093	C₂×C₈₉	semidirect
166. #2013	C₂×C₈₉	semidirect
167. #1745	C₂×C₈₉	semidirect
168. C₈₉	#23803	direct (N × H)
220. #97	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#440×C₈₉	2
C₄×C₄⋊Q₈×C₈₉	2
C₄×#127×C₈₉	2
C₈₉×#2095	2
C₈₉×#2093	2
C₈₉×#2014	2
C₈₉×#2013	2
C₈₉×#1745	2
#23803	89

Subgroups — C89×#23803 #36401651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23803 #36401651