Subgroups — C₈₉×#29219 #36407067

Order 22784.

778 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2868	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2099	C₂	1	4	yes	maximal
4	C₈₉×#1990	C₂	1	2	yes	maximal
5	C₈₉×#1988	C₂	1	4	yes	maximal
6	C₈₉×#1834	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
15	#391×C₈₉	—	2	1	no	—
16	#389×C₈₉	—	2	1	no	—
17	#389×C₈₉	C₂²	1	2	yes	—
18	#384×C₈₉	—	2	1	no	—
19	#384×C₈₉	—	2	1	no	—
20	#384×C₈₉	C₂²	1	1	yes	characteristic
21	#384×C₈₉	C₂²	1	2	yes	—
22	#384×C₈₉	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
24	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
25	C₄³×C₈₉	C₂²	1	1	yes	characteristic
26	#340×C₈₉	C₂²	1	8	yes	—
27	#340×C₈₉	C₂²	1	8	yes	—
28	#336×C₈₉	C₂²	1	2	yes	—
29	C₂²×Q₈×C₈₉	—	2	1	no	—
30	C₂²×Q₈×C₈₉	—	2	2	no	—
31	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₈₉	—	2	2	no	—
33	C₄⋊Q₈×C₈₉	—	2	8	no	—
34	C₄⋊Q₈×C₈₉	—	4	4	no	—
35	C₄⋊Q₈×C₈₉	—	4	4	no	—
36	C₄⋊Q₈×C₈₉	Q₈	1	4	yes	complemented
37	C₄×Q₈×C₈₉	—	4	4	no	—
38	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	2	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	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	1	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	4	yes	—
49	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₈₉	—	2	2	no	—
51	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₈₉	—	2	4	no	—
59	C₄×C₈×C₈₉	—	2	4	no	—
60	#95×C₈₉	—	2	2	no	—
61	#95×C₈₉	—	2	2	no	—
62	C₂×Q₈×C₈₉	—	2	4	no	—
63	C₂×Q₈×C₈₉	—	4	4	no	—
64	C₂×Q₈×C₈₉	—	4	2	no	—
65	C₂×Q₈×C₈₉	—	4	4	no	—
66	C₂×Q₈×C₈₉	—	4	4	no	—
67	C₂×Q₈×C₈₉	—	4	2	no	—
68	C₂²×C₄×C₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	—	2	2	no	—
70	C₂²×C₄×C₈₉	—	2	4	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	—	2	4	no	—
81	C₄⋊C₄×C₈₉	—	2	4	no	—
82	C₄⋊C₄×C₈₉	—	2	2	no	—
83	C₄⋊C₄×C₈₉	—	2	8	no	—
84	C₄⋊C₄×C₈₉	—	2	4	no	—
85	C₄⋊C₄×C₈₉	—	4	4	no	—
86	C₄⋊C₄×C₈₉	—	4	4	no	—
87	C₄⋊C₄×C₈₉	—	4	2	no	—
88	C₄⋊C₄×C₈₉	—	4	4	no	—
89	C₄⋊C₄×C₈₉	—	4	4	no	—
90	C₄⋊C₄×C₈₉	—	4	4	no	—
91	C₄⋊C₄×C₈₉	—	4	4	no	—
92	C₄⋊C₄×C₈₉	—	4	2	no	—
93	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
94	C₄²×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	—	2	2	no	—
97	C₄²×C₈₉	—	4	4	no	—
98	C₄²×C₈₉	D₈	1	4	yes	—
99	C₄²×C₈₉	SD₁₆	1	4	yes	—
100	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
101	C₄²×C₈₉	#41	1	2	yes	—
102	C₄²×C₈₉	#41	1	4	yes	—
103	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₈₉	—	4	8	no	—
105	Q₈×C₈₉	—	8	8	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	—	4	4	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	4	4	no	—
119	C₂×C₄×C₈₉	—	4	4	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₈₉	#124	1	2	yes	—
124	C₂×C₄×C₈₉	#124	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₈₉	#137	1	2	yes	—
132	C₂×C₄×C₈₉	#137	1	2	yes	—
133	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
136	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
137	C₈×C₈₉	—	4	4	no	—
138	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
140	C₂²×C₈₉	#536	1	1	yes	characteristic
141	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
142	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
143	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
144	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	4	4	no	—
149	C₄×C₈₉	—	8	4	no	—
150	C₄×C₈₉	—	8	2	no	—
151	C₄×C₈₉	#474	1	4	yes	—
152	C₄×C₈₉	#475	1	4	yes	—
153	C₄×C₈₉	#486	1	4	yes	—
154	C₄×C₈₉	#556	1	2	yes	—
155	C₄×C₈₉	#557	1	2	yes	—
156	#29219	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₈₉	#2568	1	1	yes	characteristic
158	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
159	C₂×C₈₉	C₂×#475	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
161	C₂×C₈₉	#3139	1	1	yes	characteristic
162	C₂×C₈₉	#3226	1	1	yes	characteristic
163	C₂×C₈₉	#3229	1	1	yes	characteristic
164	#2868	C₂×C₈₉	1	1	yes	characteristic
165	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
166	#2099	C₂×C₈₉	1	4	yes	—
167	#1990	C₂×C₈₉	1	2	yes	—
168	#1988	C₂×C₈₉	1	4	yes	—
169	#1834	C₂×C₈₉	1	2	yes	—
170	C₄×C₄⋊C₈	C₂×C₈₉	1	1	yes	characteristic
171	C₈₉	#29219	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×C₄⋊Q₈	—	2	1	no	—
173	C₂×C₄⋊Q₈	—	2	1	no	—
174	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
175	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
176	C₂×C₄×Q₈	—	2	1	no	—
177	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
179	#391	—	2	1	no	—
180	#389	—	2	1	no	—
181	#389	C₂²×C₈₉	1	2	yes	—
182	#384	—	2	1	no	—
183	#384	—	2	1	no	—
184	#384	C₂²×C₈₉	1	1	yes	characteristic
185	#384	C₂²×C₈₉	1	2	yes	—
186	#384	C₂²×C₈₉	1	2	yes	—
187	C₄×C₄⋊C₄	—	2	1	no	—
188	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
189	C₄³	C₂²×C₈₉	1	1	yes	characteristic
190	#340	C₂²×C₈₉	1	8	yes	—
191	#340	C₂²×C₈₉	1	8	yes	—
192	#336	C₂²×C₈₉	1	2	yes	—
193	C₂²×Q₈	—	2	1	no	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	2	8	no	—
198	C₄⋊Q₈	—	4	4	no	—
199	C₄⋊Q₈	—	4	4	no	—
200	C₄⋊Q₈	Q₈×C₈₉	1	4	yes	complemented
201	C₄×Q₈	—	4	4	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	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	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
212	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
216	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
222	C₄⋊C₈	—	2	4	no	—
223	C₄×C₈	—	2	4	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	4	no	—
262	C₄²	D₈×C₈₉	1	4	yes	—
263	C₄²	SD₁₆×C₈₉	1	4	yes	—
264	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
265	C₄²	#41×C₈₉	1	2	yes	—
266	C₄²	#41×C₈₉	1	4	yes	—
267	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
268	Q₈	—	4	8	no	—
269	Q₈	—	8	8	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
286	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
287	C₂×C₄	#124×C₈₉	1	2	yes	—
288	C₂×C₄	#124×C₈₉	1	2	yes	—
289	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
290	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
291	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
292	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
293	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
294	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
295	C₂×C₄	#137×C₈₉	1	2	yes	—
296	C₂×C₄	#137×C₈₉	1	2	yes	—
297	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
299	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
300	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
301	C₈	—	4	4	no	—
302	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
303	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
304	C₂²	#536×C₈₉	1	1	yes	characteristic
305	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
306	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
307	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
308	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	4	no	—
314	C₄	—	8	2	no	—
315	C₄	#474×C₈₉	1	4	yes	—
316	C₄	#475×C₈₉	1	4	yes	—
317	C₄	#486×C₈₉	1	4	yes	—
318	C₄	#556×C₈₉	1	2	yes	—
319	C₄	#557×C₈₉	1	2	yes	—
320	C₂	C₈₉×#2568	1	1	yes	characteristic
321	C₂	C₂×#474×C₈₉	1	1	yes	characteristic
322	C₂	C₂×#475×C₈₉	1	1	yes	characteristic
323	C₂	C₂×#486×C₈₉	1	1	yes	characteristic
324	C₂	C₈₉×#3139	1	1	yes	characteristic
325	C₂	C₈₉×#3226	1	1	yes	characteristic
326	C₂	C₈₉×#3229	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 ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₄⋊Q₈×C₈₉	Q₈	semidirect
156. #29219	C₈₉	direct (N × H)
171. C₈₉	#29219	direct (N × H)
200. C₄⋊Q₈	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×#2868	2
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2099	2
C₈₉×#1990	2
C₈₉×#1988	2
C₈₉×#1834	2
C₄×C₄⋊C₈×C₈₉	2
#29219	89

Subgroups — C89×#29219 #36407067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29219 #36407067