Subgroups — C₈₉×#19257 #36397105

Order 22784.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3167	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3110	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3109	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1683	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1681	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1671	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1665	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1552	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1551	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₈₉	—	2	1	no	—
13	#533×C₈₉	C₂²	1	1	yes	characteristic
14	#532×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	#498×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	#492×C₈₉	C₂²	1	1	yes	characteristic, complemented
18	#487×C₈₉	C₂²	1	1	yes	characteristic, complemented
19	#486×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	#484×C₈₉	—	2	1	no	—
21	#481×C₈₉	—	2	1	no	—
22	#476×C₈₉	C₂²	1	2	yes	—
23	#474×C₈₉	C₂²	1	2	yes	—
24	#465×C₈₉	—	2	1	no	—
25	#465×C₈₉	—	2	1	no	—
26	#461×C₈₉	C₂²	1	2	yes	—
27	#459×C₈₉	C₂²	1	2	yes	—
28	#450×C₈₉	—	2	1	no	—
29	#447×C₈₉	—	2	1	no	—
30	#442×C₈₉	—	2	2	no	—
31	#440×C₈₉	—	2	2	no	—
32	C₄×SD₁₆×C₈₉	—	2	1	no	—
33	C₄×SD₁₆×C₈₉	—	2	1	no	—
34	C₈×D₄×C₈₉	—	2	1	no	—
35	C₈×D₄×C₈₉	—	2	1	no	—
36	C₈×D₄×C₈₉	C₂²	1	2	yes	—
37	C₈×D₄×C₈₉	C₂²	1	2	yes	—
38	#432×C₈₉	C₂²	1	1	yes	characteristic
39	#432×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
43	#335×C₈₉	C₂²	1	2	yes	complemented
44	#334×C₈₉	C₂²	1	2	yes	complemented
45	#333×C₈₉	C₂²	1	2	yes	complemented
46	#332×C₈₉	C₂²	1	2	yes	complemented
47	#332×C₈₉	C₂²	1	2	yes	complemented
48	#331×C₈₉	C₂²	1	2	yes	complemented
49	C₂×#41×C₈₉	—	2	1	no	—
50	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₈₉	—	2	1	no	—
52	C₂×SD₁₆×C₈₉	—	2	2	no	—
53	C₂×SD₁₆×C₈₉	—	2	1	no	—
54	C₂×SD₁₆×C₈₉	—	2	1	no	—
55	C₂×D₈×C₈₉	—	2	2	no	—
56	C₂²×C₈×C₈₉	—	2	1	no	—
57	C₂²×C₈×C₈₉	—	2	2	no	—
58	C₂²×C₈×C₈₉	—	2	2	no	—
59	C₂²×C₈×C₈₉	—	4	4	no	—
60	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	#127×C₈₉	C₂³	1	1	yes	characteristic
66	#125×C₈₉	—	2	1	no	—
67	#124×C₈₉	—	2	1	no	—
68	C₂²⋊Q₈×C₈₉	—	2	1	no	—
69	C₂²⋊Q₈×C₈₉	—	2	1	no	—
70	#121×C₈₉	—	2	1	no	—
71	#121×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₄×Q₈×C₈₉	—	2	1	no	—
74	C₄×D₄×C₈₉	—	2	1	no	—
75	C₄×D₄×C₈₉	—	2	1	no	—
76	C₄×D₄×C₈₉	—	2	1	no	—
77	C₄×D₄×C₈₉	—	2	1	no	—
78	C₄×D₄×C₈₉	D₄	1	2	yes	—
79	C₄×D₄×C₈₉	C₂³	1	2	yes	—
80	#117×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
83	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
84	#107×C₈₉	—	2	2	no	—
85	#106×C₈₉	—	2	1	no	—
86	#106×C₈₉	—	2	1	no	—
87	#106×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	—	2	1	no	—
89	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
90	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
91	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
92	#103×C₈₉	—	2	1	no	—
93	#103×C₈₉	—	2	1	no	—
94	#103×C₈₉	—	2	2	no	—
95	#102×C₈₉	—	2	2	no	—
96	#102×C₈₉	—	2	2	no	—
97	#102×C₈₉	—	2	1	no	—
98	#102×C₈₉	—	2	1	no	—
99	#102×C₈₉	—	2	2	no	—
100	C₂²⋊C₈×C₈₉	—	2	1	no	—
101	C₂²⋊C₈×C₈₉	—	2	1	no	—
102	C₂²⋊C₈×C₈₉	—	2	2	no	—
103	C₂²⋊C₈×C₈₉	—	2	2	no	—
104	#97×C₈₉	—	2	1	no	—
105	#97×C₈₉	—	2	1	no	—
106	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
107	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
108	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
110	C₂⁴×C₈₉	—	2	1	no	—
111	#41×C₈₉	—	8	2	no	—
112	C₂×Q₈×C₈₉	—	2	1	no	—
113	C₂×Q₈×C₈₉	—	2	1	no	—
114	C₂×Q₈×C₈₉	—	2	1	no	—
115	C₂×D₄×C₈₉	—	2	4	no	—
116	C₂×D₄×C₈₉	—	2	1	no	—
117	C₂×D₄×C₈₉	—	2	1	no	—
118	C₂×D₄×C₈₉	—	2	1	no	—
119	C₂×D₄×C₈₉	—	2	1	no	—
120	C₂×D₄×C₈₉	—	4	1	no	—
121	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₈₉	—	2	1	no	—
123	C₂²×C₄×C₈₉	—	2	2	no	—
124	C₂²×C₄×C₈₉	—	2	2	no	—
125	C₂²×C₄×C₈₉	—	4	2	no	—
126	C₂²×C₄×C₈₉	—	4	2	no	—
127	C₂²×C₄×C₈₉	—	4	1	no	—
128	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
131	SD₁₆×C₈₉	—	4	2	no	—
132	SD₁₆×C₈₉	—	4	4	no	—
133	SD₁₆×C₈₉	—	4	2	no	—
134	D₈×C₈₉	—	4	4	no	—
135	C₂×C₈×C₈₉	—	2	2	no	—
136	C₂×C₈×C₈₉	—	2	2	no	—
137	C₂×C₈×C₈₉	—	2	2	no	—
138	C₂×C₈×C₈₉	—	2	2	no	—
139	C₂×C₈×C₈₉	—	4	1	no	—
140	C₂×C₈×C₈₉	—	4	4	no	—
141	C₂×C₈×C₈₉	—	4	2	no	—
142	C₂×C₈×C₈₉	—	4	4	no	—
143	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₄⋊C₄×C₈₉	—	2	1	no	—
148	C₄⋊C₄×C₈₉	—	2	1	no	—
149	C₄⋊C₄×C₈₉	—	2	1	no	—
150	C₄⋊C₄×C₈₉	—	2	1	no	—
151	C₄⋊C₄×C₈₉	—	2	1	no	—
152	C₄⋊C₄×C₈₉	—	2	1	no	—
153	C₄⋊C₄×C₈₉	—	2	1	no	—
154	C₄⋊C₄×C₈₉	—	4	1	no	—
155	C₄⋊C₄×C₈₉	—	4	1	no	—
156	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₈₉	—	2	2	no	—
158	C₂²⋊C₄×C₈₉	—	2	2	no	—
159	C₂²⋊C₄×C₈₉	—	4	1	no	—
160	C₂²⋊C₄×C₈₉	—	4	1	no	—
161	C₂²⋊C₄×C₈₉	—	4	1	no	—
162	C₄²×C₈₉	—	4	1	no	—
163	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₈₉	#41	1	1	yes	characteristic
166	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
167	C₂³×C₈₉	—	2	4	no	—
168	C₂³×C₈₉	—	2	2	no	—
169	C₂³×C₈₉	—	4	2	no	—
170	C₂³×C₈₉	—	4	2	no	—
171	C₂³×C₈₉	—	4	1	no	—
172	C₂³×C₈₉	#121	1	1	yes	characteristic
173	Q₈×C₈₉	—	4	2	no	—
174	Q₈×C₈₉	—	4	1	no	—
175	Q₈×C₈₉	—	4	1	no	—
176	Q₈×C₈₉	—	8	1	no	—
177	D₄×C₈₉	—	2	4	no	—
178	D₄×C₈₉	—	2	2	no	—
179	D₄×C₈₉	—	4	2	no	—
180	D₄×C₈₉	—	4	1	no	—
181	D₄×C₈₉	—	4	1	no	—
182	D₄×C₈₉	—	8	1	no	—
183	D₄×C₈₉	—	8	2	no	—
184	D₄×C₈₉	#135	1	4	yes	—
185	C₂×C₄×C₈₉	—	2	2	no	—
186	C₂×C₄×C₈₉	—	2	2	no	—
187	C₂×C₄×C₈₉	—	2	1	no	—
188	C₂×C₄×C₈₉	—	2	1	no	—
189	C₂×C₄×C₈₉	—	2	2	no	—
190	C₂×C₄×C₈₉	—	4	4	no	—
191	C₂×C₄×C₈₉	—	4	2	no	—
192	C₂×C₄×C₈₉	—	4	2	no	—
193	C₂×C₄×C₈₉	—	4	2	no	—
194	C₂×C₄×C₈₉	—	4	1	no	—
195	C₂×C₄×C₈₉	—	4	1	no	—
196	C₂×C₄×C₈₉	—	4	1	no	—
197	C₂×C₄×C₈₉	—	8	2	no	—
198	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
199	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
200	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
201	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
204	C₈×C₈₉	—	2	1	no	—
205	C₈×C₈₉	—	2	1	no	—
206	C₈×C₈₉	—	2	1	no	—
207	C₈×C₈₉	—	4	2	no	—
208	C₈×C₈₉	—	4	2	no	—
209	C₈×C₈₉	—	4	2	no	—
210	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
211	C₂²×C₈₉	—	2	1	no	—
212	C₂²×C₈₉	—	2	1	no	—
213	C₂²×C₈₉	—	2	4	no	—
214	C₂²×C₈₉	—	4	8	no	—
215	C₂²×C₈₉	—	4	2	no	—
216	C₂²×C₈₉	—	4	2	no	—
217	C₂²×C₈₉	—	8	1	no	—
218	C₂²×C₈₉	—	8	1	no	—
219	C₂²×C₈₉	—	8	1	no	—
220	C₂²×C₈₉	#471	1	2	yes	—
221	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
222	C₄×C₈₉	—	2	1	no	—
223	C₄×C₈₉	—	2	1	no	—
224	C₄×C₈₉	—	2	1	no	—
225	C₄×C₈₉	—	2	1	no	—
226	C₄×C₈₉	—	4	2	no	—
227	C₄×C₈₉	—	8	1	no	—
228	C₄×C₈₉	—	8	1	no	—
229	C₄×C₈₉	—	8	1	no	—
230	C₄×C₈₉	C₂×#135	1	2	yes	—
231	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₄×C₈₉	#575	1	1	yes	characteristic
233	#19257	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂×C₈₉	—	2	2	no	—
235	C₂×C₈₉	—	4	4	no	—
236	C₂×C₈₉	—	16	1	no	—
237	C₂×C₈₉	#3036	1	1	yes	characteristic
238	C₂×C₈₉	#3050	1	1	yes	characteristic
239	C₂×C₈₉	C₂×#471	1	1	yes	characteristic
240	#3167	C₂×C₈₉	1	1	yes	characteristic, complemented
241	#3110	C₂×C₈₉	1	1	yes	characteristic, complemented
242	#3109	C₂×C₈₉	1	1	yes	characteristic
243	C₂×C₈×D₄	C₂×C₈₉	1	1	yes	characteristic, complemented
244	#1683	C₂×C₈₉	1	2	yes	complemented
245	#1681	C₂×C₈₉	1	2	yes	complemented
246	#1671	C₂×C₈₉	1	2	yes	complemented
247	#1665	C₂×C₈₉	1	2	yes	complemented
248	#1568	C₂×C₈₉	1	1	yes	characteristic, complemented
249	#1552	C₂×C₈₉	1	1	yes	characteristic, complemented
250	#1551	C₂×C₈₉	1	1	yes	characteristic, complemented
251	C₈₉	#19257	1	1	yes	characteristic, direct product factor, Sylow
252	C₂³×C₈	—	2	1	no	—
253	#533	C₂²×C₈₉	1	1	yes	characteristic
254	#532	C₂²×C₈₉	1	1	yes	characteristic
255	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
256	#498	C₂²×C₈₉	1	1	yes	characteristic, complemented
257	#492	C₂²×C₈₉	1	1	yes	characteristic, complemented
258	#487	C₂²×C₈₉	1	1	yes	characteristic, complemented
259	#486	C₂²×C₈₉	1	1	yes	characteristic, complemented
260	#484	—	2	1	no	—
261	#481	—	2	1	no	—
262	#476	C₂²×C₈₉	1	2	yes	—
263	#474	C₂²×C₈₉	1	2	yes	—
264	#465	—	2	1	no	—
265	#465	—	2	1	no	—
266	#461	C₂²×C₈₉	1	2	yes	—
267	#459	C₂²×C₈₉	1	2	yes	—
268	#450	—	2	1	no	—
269	#447	—	2	1	no	—
270	#442	—	2	2	no	—
271	#440	—	2	2	no	—
272	C₄×SD₁₆	—	2	1	no	—
273	C₄×SD₁₆	—	2	1	no	—
274	C₈×D₄	—	2	1	no	—
275	C₈×D₄	—	2	1	no	—
276	C₈×D₄	C₂²×C₈₉	1	2	yes	—
277	C₈×D₄	C₂²×C₈₉	1	2	yes	—
278	#432	C₂²×C₈₉	1	1	yes	characteristic
279	#432	C₂²×C₈₉	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
281	C₂×C₂²⋊C₈	—	2	1	no	—
282	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
283	#335	C₂²×C₈₉	1	2	yes	complemented
284	#334	C₂²×C₈₉	1	2	yes	complemented
285	#333	C₂²×C₈₉	1	2	yes	complemented
286	#332	C₂²×C₈₉	1	2	yes	complemented
287	#332	C₂²×C₈₉	1	2	yes	complemented
288	#331	C₂²×C₈₉	1	2	yes	complemented
289	C₂×#41	—	2	1	no	—
290	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
291	C₂³×C₄	—	2	1	no	—
292	C₂×SD₁₆	—	2	2	no	—
293	C₂×SD₁₆	—	2	1	no	—
294	C₂×SD₁₆	—	2	1	no	—
295	C₂×D₈	—	2	2	no	—
296	C₂²×C₈	—	2	1	no	—
297	C₂²×C₈	—	2	2	no	—
298	C₂²×C₈	—	2	2	no	—
299	C₂²×C₈	—	4	4	no	—
300	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
301	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
302	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
303	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
304	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
305	#127	C₂³×C₈₉	1	1	yes	characteristic
306	#125	—	2	1	no	—
307	#124	—	2	1	no	—
308	C₂²⋊Q₈	—	2	1	no	—
309	C₂²⋊Q₈	—	2	1	no	—
310	#121	—	2	1	no	—
311	#121	—	2	1	no	—
312	C₄×Q₈	—	2	1	no	—
313	C₄×Q₈	—	2	1	no	—
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	D₄×C₈₉	1	2	yes	—
319	C₄×D₄	C₂³×C₈₉	1	2	yes	—
320	#117	—	2	1	no	—
321	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
322	C₂×C₂²⋊C₄	—	2	1	no	—
323	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
324	#107	—	2	2	no	—
325	#106	—	2	1	no	—
326	#106	—	2	1	no	—
327	#106	—	2	2	no	—
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
330	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
331	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
332	#103	—	2	1	no	—
333	#103	—	2	1	no	—
334	#103	—	2	2	no	—
335	#102	—	2	2	no	—
336	#102	—	2	2	no	—
337	#102	—	2	1	no	—
338	#102	—	2	1	no	—
339	#102	—	2	2	no	—
340	C₂²⋊C₈	—	2	1	no	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	C₂²⋊C₈	—	2	2	no	—
344	#97	—	2	1	no	—
345	#97	—	2	1	no	—
346	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
347	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
348	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
349	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
350	C₂⁴	—	2	1	no	—
351	#41	—	8	2	no	—
352	C₂×Q₈	—	2	1	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×D₄	—	2	4	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	1	no	—
368	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
370	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
371	SD₁₆	—	4	2	no	—
372	SD₁₆	—	4	4	no	—
373	SD₁₆	—	4	2	no	—
374	D₈	—	4	4	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	4	1	no	—
380	C₂×C₈	—	4	4	no	—
381	C₂×C₈	—	4	2	no	—
382	C₂×C₈	—	4	4	no	—
383	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
384	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
385	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
386	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	4	1	no	—
395	C₄⋊C₄	—	4	1	no	—
396	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
397	C₂²⋊C₄	—	2	2	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	4	1	no	—
400	C₂²⋊C₄	—	4	1	no	—
401	C₂²⋊C₄	—	4	1	no	—
402	C₄²	—	4	1	no	—
403	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
404	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
405	C₄²	#41×C₈₉	1	1	yes	characteristic
406	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
407	C₂³	—	2	4	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	1	no	—
412	C₂³	#121×C₈₉	1	1	yes	characteristic
413	Q₈	—	4	2	no	—
414	Q₈	—	4	1	no	—
415	Q₈	—	4	1	no	—
416	Q₈	—	8	1	no	—
417	D₄	—	2	4	no	—
418	D₄	—	2	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	1	no	—
421	D₄	—	4	1	no	—
422	D₄	—	8	1	no	—
423	D₄	—	8	2	no	—
424	D₄	#135×C₈₉	1	4	yes	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	8	2	no	—
438	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
439	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
440	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
441	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
444	C₈	—	2	1	no	—
445	C₈	—	2	1	no	—
446	C₈	—	2	1	no	—
447	C₈	—	4	2	no	—
448	C₈	—	4	2	no	—
449	C₈	—	4	2	no	—
450	C₈	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
451	C₂²	—	2	1	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	4	no	—
454	C₂²	—	4	8	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	8	1	no	—
458	C₂²	—	8	1	no	—
459	C₂²	—	8	1	no	—
460	C₂²	#471×C₈₉	1	2	yes	—
461	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
462	C₄	—	2	1	no	—
463	C₄	—	2	1	no	—
464	C₄	—	2	1	no	—
465	C₄	—	2	1	no	—
466	C₄	—	4	2	no	—
467	C₄	—	8	1	no	—
468	C₄	—	8	1	no	—
469	C₄	—	8	1	no	—
470	C₄	C₂×#135×C₈₉	1	2	yes	—
471	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
472	C₄	#575×C₈₉	1	1	yes	characteristic
473	C₂	—	2	2	no	—
474	C₂	—	4	4	no	—
475	C₂	—	16	1	no	—
476	C₂	C₈₉×#3036	1	1	yes	characteristic
477	C₂	C₈₉×#3050	1	1	yes	characteristic
478	C₂	C₂×#471×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
prime radical	C₈₉×#3109	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3167	C₂	semidirect
2. C₈₉×#3110	C₂	semidirect
4. C₂×C₈×D₄×C₈₉	C₂	semidirect
5. C₈₉×#1683	C₂	semidirect
6. C₈₉×#1681	C₂	semidirect
7. C₈₉×#1671	C₂	semidirect
8. C₈₉×#1665	C₂	semidirect
9. C₈₉×#1568	C₂	semidirect
10. C₈₉×#1552	C₂	semidirect
11. C₈₉×#1551	C₂	semidirect
16. #498×C₈₉	C₂²	semidirect
17. #492×C₈₉	C₂²	semidirect
18. #487×C₈₉	C₂²	semidirect
19. #486×C₈₉	C₂²	semidirect
43. #335×C₈₉	C₂²	semidirect
44. #334×C₈₉	C₂²	semidirect
45. #333×C₈₉	C₂²	semidirect
46. #332×C₈₉	C₂²	semidirect
47. #332×C₈₉	C₂²	semidirect
48. #331×C₈₉	C₂²	semidirect
233. #19257	C₈₉	direct (N × H)
240. #3167	C₂×C₈₉	semidirect
241. #3110	C₂×C₈₉	semidirect
243. C₂×C₈×D₄	C₂×C₈₉	semidirect
244. #1683	C₂×C₈₉	semidirect
245. #1681	C₂×C₈₉	semidirect
246. #1671	C₂×C₈₉	semidirect
247. #1665	C₂×C₈₉	semidirect
248. #1568	C₂×C₈₉	semidirect
249. #1552	C₂×C₈₉	semidirect
250. #1551	C₂×C₈₉	semidirect
251. C₈₉	#19257	direct (N × H)
256. #498	C₂²×C₈₉	semidirect
257. #492	C₂²×C₈₉	semidirect
258. #487	C₂²×C₈₉	semidirect
259. #486	C₂²×C₈₉	semidirect
283. #335	C₂²×C₈₉	semidirect
284. #334	C₂²×C₈₉	semidirect
285. #333	C₂²×C₈₉	semidirect
286. #332	C₂²×C₈₉	semidirect
287. #332	C₂²×C₈₉	semidirect
288. #331	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3167	2
C₈₉×#3110	2
C₈₉×#3109	2
C₂×C₈×D₄×C₈₉	2
C₈₉×#1683	2
C₈₉×#1681	2
C₈₉×#1671	2
C₈₉×#1665	2
C₈₉×#1568	2
C₈₉×#1552	2
C₈₉×#1551	2
#19257	89

Subgroups — C89×#19257 #36397105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#19257 #36397105