Subgroups — C₈₉×#23551 #36401399

Order 22784.

1154 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₈₉	C₂	1	2	yes	maximal
3	C₈₉×#2296	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2059	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2058	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2057	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1735	C₂	1	2	yes	maximal
8	C₂×#336×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
10	#533×C₈₉	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
12	C₂×#125×C₈₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
14	C₂×C₄×Q₈×C₈₉	—	2	2	no	—
15	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#117×C₈₉	C₂²	1	2	yes	—
17	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	#427×C₈₉	—	2	8	no	—
20	C₂×#107×C₈₉	C₂²	1	4	yes	—
21	C₂×#106×C₈₉	C₂²	1	4	yes	—
22	#389×C₈₉	C₂²	1	2	yes	—
23	#389×C₈₉	C₂²	1	2	yes	—
24	#385×C₈₉	—	2	2	no	—
25	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
26	#340×C₈₉	C₂²	1	4	yes	—
27	#340×C₈₉	C₂²	1	4	yes	—
28	#336×C₈₉	C₂²	1	1	yes	characteristic
29	#336×C₈₉	C₂²	1	1	yes	characteristic
30	#336×C₈₉	C₂²	1	2	yes	—
31	C₂²×Q₈×C₈₉	—	2	2	no	—
32	C₂³×C₄×C₈₉	D₄	1	2	yes	—
33	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	—	2	4	no	—
35	C₂²×C₈×C₈₉	—	2	4	no	—
36	C₂²×C₈×C₈₉	Q₈	1	8	yes	complemented
37	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₈₉	—	2	8	no	—
40	#125×C₈₉	—	2	8	no	—
41	C₂²⋊Q₈×C₈₉	—	2	16	no	—
42	C₄×Q₈×C₈₉	—	2	16	no	—
43	#117×C₈₉	—	2	4	no	—
44	#117×C₈₉	—	2	2	no	—
45	#117×C₈₉	—	2	4	no	—
46	#117×C₈₉	D₄	1	8	yes	—
47	#117×C₈₉	Q₈	1	4	yes	complemented
48	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
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	2	no	—
57	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₄²×C₈₉	—	2	4	no	—
59	C₂×C₄²×C₈₉	—	2	1	no	—
60	C₂×C₄²×C₈₉	—	2	2	no	—
61	C₂×C₄²×C₈₉	—	2	1	no	—
62	C₂×C₄²×C₈₉	D₄	1	2	yes	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	#107×C₈₉	—	2	8	no	—
66	#106×C₈₉	—	2	8	no	—
67	#95×C₈₉	—	2	2	no	—
68	#95×C₈₉	—	2	2	no	—
69	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×Q₈×C₈₉	—	2	8	no	—
71	C₂×Q₈×C₈₉	—	4	4	no	—
72	C₂×Q₈×C₈₉	—	4	4	no	—
73	C₂²×C₄×C₈₉	—	2	4	no	—
74	C₂²×C₄×C₈₉	—	2	2	no	—
75	C₂²×C₄×C₈₉	—	2	4	no	—
76	C₂²×C₄×C₈₉	—	2	4	no	—
77	C₂²×C₄×C₈₉	—	2	2	no	—
78	C₂²×C₄×C₈₉	—	4	2	no	—
79	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₈₉	#41	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₈₉	—	2	16	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
93	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₈₉	—	2	16	no	—
95	C₄⋊C₄×C₈₉	—	2	8	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	—	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	—
103	C₂²⋊C₄×C₈₉	—	2	8	no	—
104	C₂²⋊C₄×C₈₉	—	2	4	no	—
105	C₂²⋊C₄×C₈₉	—	4	2	no	—
106	C₄²×C₈₉	—	2	4	no	—
107	C₄²×C₈₉	—	2	8	no	—
108	C₄²×C₈₉	—	2	4	no	—
109	C₄²×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	4	2	no	—
112	C₂³×C₈₉	—	2	2	no	—
113	C₂³×C₈₉	—	2	2	no	—
114	C₂³×C₈₉	#120	1	2	yes	—
115	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
116	C₂³×C₈₉	C₄⋊Q₈	1	2	yes	—
117	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₈₉	—	4	16	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	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	4	8	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	4	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	4	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	#120	1	2	yes	—
135	C₂×C₄×C₈₉	C₂²⋊Q₈	1	8	yes	—
136	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
138	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
139	C₂×C₄×C₈₉	#135	1	4	yes	—
140	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
142	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₈₉	—	2	4	no	—
145	C₈×C₈₉	—	2	4	no	—
146	C₂²×C₈₉	—	2	4	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	2	4	no	—
150	C₂²×C₈₉	#395	1	4	yes	—
151	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#135	1	2	yes	—
155	C₂²×C₈₉	#575	1	2	yes	—
156	C₄×C₈₉	—	2	4	no	—
157	C₄×C₈₉	—	4	4	no	—
158	C₄×C₈₉	—	4	2	no	—
159	C₄×C₈₉	—	4	8	no	—
160	C₄×C₈₉	—	8	2	no	—
161	C₄×C₈₉	#395	1	4	yes	—
162	C₄×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
163	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
164	#23551	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₈₉	—	2	4	no	—
166	C₂×C₈₉	C₂×#395	1	1	yes	characteristic
167	C₂×C₈₉	#3002	1	2	yes	—
168	C₂×C₈₉	#3076	1	2	yes	—
169	C₂×C₈₉	#3160	1	1	yes	characteristic
170	C₂×C₈₉	#3162	1	1	yes	characteristic
171	C₂×#533	C₂×C₈₉	1	1	yes	characteristic
172	C₂×#427	C₂×C₈₉	1	2	yes	—
173	#2296	C₂×C₈₉	1	1	yes	characteristic
174	#2059	C₂×C₈₉	1	4	yes	complemented
175	#2058	C₂×C₈₉	1	2	yes	complemented
176	#2057	C₂×C₈₉	1	2	yes	complemented
177	#1735	C₂×C₈₉	1	2	yes	—
178	C₂×#336	C₂×C₈₉	1	1	yes	characteristic
179	C₈₉	#23551	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×C₈	C₂²×C₈₉	1	2	yes	—
181	#533	—	2	8	no	—
182	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
183	C₂×#125	C₂²×C₈₉	1	2	yes	—
184	C₂×C₂²⋊Q₈	—	2	2	no	—
185	C₂×C₄×Q₈	—	2	2	no	—
186	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
187	C₂×#117	C₂²×C₈₉	1	2	yes	—
188	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
189	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
190	#427	—	2	8	no	—
191	C₂×#107	C₂²×C₈₉	1	4	yes	—
192	C₂×#106	C₂²×C₈₉	1	4	yes	—
193	#389	C₂²×C₈₉	1	2	yes	—
194	#389	C₂²×C₈₉	1	2	yes	—
195	#385	—	2	2	no	—
196	C₄×C₄⋊C₄	—	2	2	no	—
197	#340	C₂²×C₈₉	1	4	yes	—
198	#340	C₂²×C₈₉	1	4	yes	—
199	#336	C₂²×C₈₉	1	1	yes	characteristic
200	#336	C₂²×C₈₉	1	1	yes	characteristic
201	#336	C₂²×C₈₉	1	2	yes	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂³×C₄	D₄×C₈₉	1	2	yes	—
204	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	Q₈×C₈₉	1	8	yes	complemented
208	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
209	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
210	C₄⋊Q₈	—	2	8	no	—
211	#125	—	2	8	no	—
212	C₂²⋊Q₈	—	2	16	no	—
213	C₄×Q₈	—	2	16	no	—
214	#117	—	2	4	no	—
215	#117	—	2	2	no	—
216	#117	—	2	4	no	—
217	#117	D₄×C₈₉	1	8	yes	—
218	#117	Q₈×C₈₉	1	4	yes	complemented
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
225	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	1	no	—
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	—	2	4	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	D₄×C₈₉	1	2	yes	—
234	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
236	#107	—	2	8	no	—
237	#106	—	2	8	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
254	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
255	C₂²×C₄	C₂×Q₈×C₈₉	1	2	yes	—
256	C₂²×C₄	#41×C₈₉	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
258	C₂×C₈	—	2	16	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂×Q₈×C₈₉	1	4	yes	—
264	C₂×C₈	C₂×Q₈×C₈₉	1	4	yes	—
265	C₄⋊C₄	—	2	16	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
273	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	—
274	C₂²⋊C₄	—	2	8	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	8	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	#120×C₈₉	1	2	yes	—
286	C₂³	C₂²⋊Q₈×C₈₉	1	2	yes	—
287	C₂³	C₄⋊Q₈×C₈₉	1	2	yes	—
288	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
289	Q₈	—	4	16	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	#120×C₈₉	1	2	yes	—
306	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
307	C₂×C₄	C₂²⋊Q₈×C₈₉	1	8	yes	—
308	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
309	C₂×C₄	C₄⋊Q₈×C₈₉	1	8	yes	—
310	C₂×C₄	#135×C₈₉	1	4	yes	—
311	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
312	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
313	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
315	C₈	—	2	4	no	—
316	C₈	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	#395×C₈₉	1	4	yes	—
322	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
323	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
324	C₂²	C₂×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
325	C₂²	C₂×#135×C₈₉	1	2	yes	—
326	C₂²	#575×C₈₉	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	4	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	8	no	—
331	C₄	—	8	2	no	—
332	C₄	#395×C₈₉	1	4	yes	—
333	C₄	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
334	C₄	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	C₂×#395×C₈₉	1	1	yes	characteristic
337	C₂	C₈₉×#3002	1	2	yes	—
338	C₂	C₈₉×#3076	1	2	yes	—
339	C₂	C₈₉×#3160	1	1	yes	characteristic
340	C₂	C₈₉×#3162	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₈₉×#2059	C₂	semidirect
5. C₈₉×#2058	C₂	semidirect
6. C₈₉×#2057	C₂	semidirect
36. C₂²×C₈×C₈₉	Q₈	semidirect
47. #117×C₈₉	Q₈	semidirect
164. #23551	C₈₉	direct (N × H)
174. #2059	C₂×C₈₉	semidirect
175. #2058	C₂×C₈₉	semidirect
176. #2057	C₂×C₈₉	semidirect
179. C₈₉	#23551	direct (N × H)
207. C₂²×C₈	Q₈×C₈₉	semidirect
218. #117	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₈₉	2
C₂×#427×C₈₉	2
C₈₉×#2296	2
C₈₉×#2059	2
C₈₉×#2058	2
C₈₉×#2057	2
C₈₉×#1735	2
C₂×#336×C₈₉	2
#23551	89

Subgroups — C89×#23551 #36401399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23551 #36401399