Subgroups — C₈₉×#23552 #36401400

Order 22784.

1358 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₈₉	C₂	1	2	yes	maximal
3	C₈₉×#2296	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2058	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2057	C₂	1	4	yes	maximal, complemented
6	C₈₉×#1736	C₂	1	2	yes	maximal
7	C₈₉×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₈₉	—	2	1	no	—
9	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₈₉	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₈₉	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
13	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#428×C₈₉	—	2	8	no	—
19	C₂×#107×C₈₉	C₂²	1	4	yes	—
20	C₂×#106×C₈₉	C₂²	1	4	yes	—
21	#389×C₈₉	C₂²	1	2	yes	—
22	#389×C₈₉	C₂²	1	2	yes	—
23	#385×C₈₉	—	2	1	no	—
24	#385×C₈₉	—	2	1	no	—
25	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
26	#340×C₈₉	C₂²	1	8	yes	—
27	#336×C₈₉	C₂²	1	4	yes	—
28	C₂²×Q₈×C₈₉	—	2	8	no	—
29	C₂²×Q₈×C₈₉	—	2	4	no	—
30	C₂²×Q₈×C₈₉	—	4	8	no	—
31	C₂³×C₄×C₈₉	—	2	2	no	—
32	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₈₉	—	2	4	no	—
36	C₂×M₁₆×C₈₉	—	2	4	no	—
37	C₂×M₁₆×C₈₉	Q₈	1	8	yes	complemented
38	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₈₉	—	2	16	no	—
40	C₄⋊Q₈×C₈₉	—	2	16	no	—
41	#117×C₈₉	—	2	2	no	—
42	#117×C₈₉	—	2	4	no	—
43	#117×C₈₉	Q₈	1	4	yes	complemented
44	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₈₉	D₄	1	8	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
56	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄²×C₈₉	—	2	1	no	—
58	C₂×C₄²×C₈₉	—	2	4	no	—
59	C₂×C₄²×C₈₉	—	2	1	no	—
60	C₂×C₄²×C₈₉	D₄	1	2	yes	—
61	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
62	#107×C₈₉	—	2	8	no	—
63	#106×C₈₉	—	2	8	no	—
64	#95×C₈₉	—	2	2	no	—
65	#95×C₈₉	—	2	2	no	—
66	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₈₉	—	2	16	no	—
68	C₂×Q₈×C₈₉	—	4	32	no	—
69	C₂×Q₈×C₈₉	—	4	8	no	—
70	C₂×Q₈×C₈₉	—	4	8	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	4	no	—
74	C₂²×C₄×C₈₉	—	2	4	no	—
75	C₂²×C₄×C₈₉	—	2	8	no	—
76	C₂²×C₄×C₈₉	—	2	2	no	—
77	C₂²×C₄×C₈₉	—	2	4	no	—
78	C₂²×C₄×C₈₉	—	2	1	no	—
79	C₂²×C₄×C₈₉	—	2	1	no	—
80	C₂²×C₄×C₈₉	—	4	8	no	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
86	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₈₉	#41	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₈₉	—	2	16	no	—
90	C₂×C₈×C₈₉	—	2	4	no	—
91	C₂×C₈×C₈₉	—	2	4	no	—
92	C₂×C₈×C₈₉	C₂×Q₈	1	8	yes	complemented
93	C₄⋊C₄×C₈₉	—	2	16	no	—
94	C₄⋊C₄×C₈₉	—	2	8	no	—
95	C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	4	4	no	—
100	C₄⋊C₄×C₈₉	—	4	4	no	—
101	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
102	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	complemented
103	C₂²⋊C₄×C₈₉	—	2	4	no	—
104	C₂²⋊C₄×C₈₉	—	4	2	no	—
105	C₄²×C₈₉	—	2	4	no	—
106	C₄²×C₈₉	—	2	4	no	—
107	C₄²×C₈₉	—	2	4	no	—
108	C₄²×C₈₉	—	4	2	no	—
109	C₂³×C₈₉	—	2	2	no	—
110	C₂³×C₈₉	—	2	2	no	—
111	C₂³×C₈₉	#120	1	2	yes	—
112	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
113	C₂³×C₈₉	C₄⋊Q₈	1	2	yes	—
114	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₈₉	—	4	32	no	—
116	C₂×C₄×C₈₉	—	2	8	no	—
117	C₂×C₄×C₈₉	—	2	8	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	8	no	—
123	C₂×C₄×C₈₉	—	2	4	no	—
124	C₂×C₄×C₈₉	—	4	16	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	8	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	#120	1	2	yes	—
133	C₂×C₄×C₈₉	C₂²⋊Q₈	1	8	yes	—
134	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
136	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
138	C₂×C₄×C₈₉	#137	1	4	yes	—
139	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
141	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₈₉	—	2	8	no	—
144	C₂²×C₈₉	—	2	4	no	—
145	C₂²×C₈₉	—	2	4	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	#395	1	4	yes	—
149	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
150	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
153	C₂²×C₈₉	C₂×#137	1	2	yes	—
154	C₄×C₈₉	—	2	4	no	—
155	C₄×C₈₉	—	2	4	no	—
156	C₄×C₈₉	—	4	4	no	—
157	C₄×C₈₉	—	4	8	no	—
158	C₄×C₈₉	—	8	1	no	—
159	C₄×C₈₉	—	8	1	no	—
160	C₄×C₈₉	#395	1	4	yes	—
161	C₄×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
162	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
163	#23552	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₈₉	—	2	4	no	—
165	C₂×C₈₉	C₂×#395	1	1	yes	characteristic
166	C₂×C₈₉	#3003	1	1	yes	characteristic
167	C₂×C₈₉	#3004	1	1	yes	characteristic
168	C₂×C₈₉	#3079	1	2	yes	—
169	C₂×C₈₉	#3165	1	2	yes	—
170	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
171	C₂×#428	C₂×C₈₉	1	2	yes	—
172	#2296	C₂×C₈₉	1	1	yes	characteristic
173	#2058	C₂×C₈₉	1	4	yes	complemented
174	#2057	C₂×C₈₉	1	4	yes	complemented
175	#1736	C₂×C₈₉	1	2	yes	—
176	#1728	C₂×C₈₉	1	1	yes	characteristic
177	C₈₉	#23552	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³×Q₈	—	2	1	no	—
179	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
180	C₂×C₄⋊Q₈	—	2	8	no	—
181	C₂×C₄⋊Q₈	—	2	2	no	—
182	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	—
183	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
184	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	—	2	1	no	—
186	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
187	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
188	#428	—	2	8	no	—
189	C₂×#107	C₂²×C₈₉	1	4	yes	—
190	C₂×#106	C₂²×C₈₉	1	4	yes	—
191	#389	C₂²×C₈₉	1	2	yes	—
192	#389	C₂²×C₈₉	1	2	yes	—
193	#385	—	2	1	no	—
194	#385	—	2	1	no	—
195	C₄×C₄⋊C₄	—	2	2	no	—
196	#340	C₂²×C₈₉	1	8	yes	—
197	#336	C₂²×C₈₉	1	4	yes	—
198	C₂²×Q₈	—	2	8	no	—
199	C₂²×Q₈	—	2	4	no	—
200	C₂²×Q₈	—	4	8	no	—
201	C₂³×C₄	—	2	2	no	—
202	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
203	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
205	C₂×M₁₆	—	2	4	no	—
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	Q₈×C₈₉	1	8	yes	complemented
208	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
209	C₄⋊Q₈	—	2	16	no	—
210	C₄⋊Q₈	—	2	16	no	—
211	#117	—	2	2	no	—
212	#117	—	2	4	no	—
213	#117	Q₈×C₈₉	1	4	yes	complemented
214	C₂×C₄⋊C₄	—	2	8	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	4	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	D₄×C₈₉	1	8	yes	—
221	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
223	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
224	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
225	C₂×C₂²⋊C₄	—	2	1	no	—
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	—	2	4	no	—
229	C₂×C₄²	—	2	1	no	—
230	C₂×C₄²	D₄×C₈₉	1	2	yes	—
231	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
232	#107	—	2	8	no	—
233	#106	—	2	8	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
237	C₂×Q₈	—	2	16	no	—
238	C₂×Q₈	—	4	32	no	—
239	C₂×Q₈	—	4	8	no	—
240	C₂×Q₈	—	4	8	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	4	8	no	—
251	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
255	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
256	C₂²×C₄	C₂×Q₈×C₈₉	1	2	yes	—
257	C₂²×C₄	#41×C₈₉	1	2	yes	—
258	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
259	M₁₆	—	2	16	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	C₂×Q₈×C₈₉	1	8	yes	complemented
263	C₄⋊C₄	—	2	16	no	—
264	C₄⋊C₄	—	2	8	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
272	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	complemented
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	4	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	#120×C₈₉	1	2	yes	—
282	C₂³	C₂²⋊Q₈×C₈₉	1	2	yes	—
283	C₂³	C₄⋊Q₈×C₈₉	1	2	yes	—
284	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
285	Q₈	—	4	32	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	#120×C₈₉	1	2	yes	—
303	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
304	C₂×C₄	C₂²⋊Q₈×C₈₉	1	8	yes	—
305	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
306	C₂×C₄	C₄⋊Q₈×C₈₉	1	8	yes	—
307	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
308	C₂×C₄	#137×C₈₉	1	4	yes	—
309	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
310	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
311	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
312	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
313	C₈	—	2	8	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	#395×C₈₉	1	4	yes	—
319	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
320	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
321	C₂²	C₂×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
322	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
323	C₂²	C₂×#137×C₈₉	1	2	yes	—
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	8	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	#395×C₈₉	1	4	yes	—
331	C₄	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
332	C₄	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
333	C₂	—	2	4	no	—
334	C₂	C₂×#395×C₈₉	1	1	yes	characteristic
335	C₂	C₈₉×#3003	1	1	yes	characteristic
336	C₂	C₈₉×#3004	1	1	yes	characteristic
337	C₂	C₈₉×#3079	1	2	yes	—
338	C₂	C₈₉×#3165	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈₉×#2058	C₂	semidirect
5. C₈₉×#2057	C₂	semidirect
37. C₂×M₁₆×C₈₉	Q₈	semidirect
43. #117×C₈₉	Q₈	semidirect
92. C₂×C₈×C₈₉	C₂×Q₈	semidirect
102. C₄⋊C₄×C₈₉	C₂×Q₈	semidirect
163. #23552	C₈₉	direct (N × H)
173. #2058	C₂×C₈₉	semidirect
174. #2057	C₂×C₈₉	semidirect
177. C₈₉	#23552	direct (N × H)
207. C₂×M₁₆	Q₈×C₈₉	semidirect
213. #117	Q₈×C₈₉	semidirect
262. C₂×C₈	C₂×Q₈×C₈₉	semidirect
272. C₄⋊C₄	C₂×Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₈₉	2
C₂×#428×C₈₉	2
C₈₉×#2296	2
C₈₉×#2058	2
C₈₉×#2057	2
C₈₉×#1736	2
C₈₉×#1728	2
#23552	89

Subgroups — C89×#23552 #36401400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23552 #36401400