Subgroups — C₈₉×#19174 #36397022

Order 22784.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3138	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3104	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#460×C₈₉	C₂	1	1	yes	maximal, characteristic
4	Q₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1702	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1698	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1682	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1678	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1550	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₈₉	—	2	1	no	—
13	#533×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	#490×C₈₉	—	2	2	no	—
17	#486×C₈₉	C₂²	1	2	yes	—
18	#486×C₈₉	C₂²	1	2	yes	—
19	#484×C₈₉	—	2	1	no	—
20	#477×C₈₉	C₂²	1	1	yes	characteristic, complemented
21	#477×C₈₉	C₂²	1	1	yes	characteristic, complemented
22	#462×C₈₉	C₂²	1	1	yes	characteristic, complemented
23	#462×C₈₉	C₂²	1	1	yes	characteristic, complemented
24	#460×C₈₉	—	2	1	no	—
25	#460×C₈₉	—	2	1	no	—
26	#460×C₈₉	C₂²	1	2	yes	—
27	#460×C₈₉	C₂²	1	2	yes	—
28	#451×C₈₉	—	2	1	no	—
29	#446×C₈₉	—	2	2	no	—
30	#446×C₈₉	C₂²	1	2	yes	complemented
31	C₈×Q₈×C₈₉	C₂²	1	2	yes	complemented
32	C₄×Q₁₆×C₈₉	—	2	1	no	—
33	C₄×Q₁₆×C₈₉	—	2	1	no	—
34	#431×C₈₉	C₂²	1	1	yes	characteristic
35	#423×C₈₉	—	2	1	no	—
36	#423×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
38	#418×C₈₉	—	2	1	no	—
39	#418×C₈₉	—	2	1	no	—
40	C₂×#102×C₈₉	—	2	1	no	—
41	C₄×M₁₆×C₈₉	—	2	1	no	—
42	C₄×M₁₆×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	#331×C₈₉	C₂²	1	2	yes	—
48	#331×C₈₉	C₂²	1	2	yes	—
49	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₈₉	—	2	1	no	—
51	C₂²×D₄×C₈₉	—	2	1	no	—
52	C₂²×D₄×C₈₉	—	2	1	no	—
53	C₂×Q₁₆×C₈₉	—	2	1	no	—
54	C₂×Q₁₆×C₈₉	—	2	1	no	—
55	C₂×SD₁₆×C₈₉	—	2	2	no	—
56	C₂×SD₁₆×C₈₉	—	2	2	no	—
57	C₂×SD₁₆×C₈₉	—	4	4	no	—
58	C₂×M₁₆×C₈₉	—	2	1	no	—
59	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	—	2	1	no	—
62	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
64	#127×C₈₉	—	2	1	no	—
65	#127×C₈₉	—	2	1	no	—
66	#127×C₈₉	—	2	1	no	—
67	#127×C₈₉	C₂³	1	2	yes	—
68	#125×C₈₉	—	2	1	no	—
69	C₂²⋊Q₈×C₈₉	—	2	1	no	—
70	C₂²⋊Q₈×C₈₉	—	2	1	no	—
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₄×Q₈×C₈₉	—	2	1	no	—
74	C₄×Q₈×C₈₉	—	2	1	no	—
75	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
79	#117×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₈₉	—	2	1	no	—
83	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
84	#107×C₈₉	—	2	1	no	—
85	#107×C₈₉	—	2	1	no	—
86	C₄⋊C₈×C₈₉	—	2	2	no	—
87	C₄⋊C₈×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	—	2	1	no	—
89	C₄⋊C₈×C₈₉	—	2	1	no	—
90	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
91	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
92	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
94	#103×C₈₉	—	2	1	no	—
95	#103×C₈₉	—	2	1	no	—
96	#103×C₈₉	—	2	1	no	—
97	#103×C₈₉	—	2	1	no	—
98	#102×C₈₉	—	2	2	no	—
99	#102×C₈₉	—	2	2	no	—
100	#102×C₈₉	—	2	2	no	—
101	#102×C₈₉	—	2	2	no	—
102	#102×C₈₉	—	2	2	no	—
103	#102×C₈₉	—	2	2	no	—
104	C₂²⋊C₈×C₈₉	—	2	1	no	—
105	#97×C₈₉	—	2	2	no	—
106	#97×C₈₉	D₄	1	2	yes	—
107	C₄×C₈×C₈₉	—	2	2	no	—
108	C₄×C₈×C₈₉	—	2	1	no	—
109	C₄×C₈×C₈₉	C₂³	1	2	yes	—
110	C₂⁴×C₈₉	—	4	1	no	—
111	C₂×Q₈×C₈₉	—	2	1	no	—
112	C₂×Q₈×C₈₉	—	2	2	no	—
113	C₂×Q₈×C₈₉	—	2	1	no	—
114	C₂×Q₈×C₈₉	—	2	1	no	—
115	C₂×Q₈×C₈₉	—	4	1	no	—
116	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₈₉	—	2	2	no	—
119	C₂×D₄×C₈₉	—	2	2	no	—
120	C₂×D₄×C₈₉	—	2	2	no	—
121	C₂×D₄×C₈₉	—	4	2	no	—
122	C₂×D₄×C₈₉	—	4	2	no	—
123	C₂×D₄×C₈₉	—	4	1	no	—
124	C₂×D₄×C₈₉	—	4	1	no	—
125	C₂×D₄×C₈₉	—	4	1	no	—
126	C₂×D₄×C₈₉	—	4	2	no	—
127	C₂×D₄×C₈₉	—	4	2	no	—
128	C₂×D₄×C₈₉	—	8	2	no	—
129	C₂²×C₄×C₈₉	—	2	1	no	—
130	C₂²×C₄×C₈₉	—	2	1	no	—
131	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
134	Q₁₆×C₈₉	—	4	2	no	—
135	Q₁₆×C₈₉	—	4	2	no	—
136	SD₁₆×C₈₉	—	4	4	no	—
137	SD₁₆×C₈₉	—	4	4	no	—
138	M₁₆×C₈₉	—	2	2	no	—
139	M₁₆×C₈₉	—	2	2	no	—
140	M₁₆×C₈₉	—	4	2	no	—
141	C₂×C₈×C₈₉	—	2	2	no	—
142	C₂×C₈×C₈₉	—	2	1	no	—
143	C₂×C₈×C₈₉	—	2	1	no	—
144	C₂×C₈×C₈₉	—	2	2	no	—
145	C₂×C₈×C₈₉	—	4	2	no	—
146	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
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	—	2	1	no	—
155	C₄⋊C₄×C₈₉	—	2	1	no	—
156	C₄⋊C₄×C₈₉	—	2	1	no	—
157	C₄⋊C₄×C₈₉	—	4	1	no	—
158	C₄⋊C₄×C₈₉	—	4	1	no	—
159	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
160	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
161	C₂²⋊C₄×C₈₉	—	4	1	no	—
162	C₄²×C₈₉	—	2	1	no	—
163	C₄²×C₈₉	—	2	1	no	—
164	C₄²×C₈₉	—	2	1	no	—
165	C₄²×C₈₉	—	2	1	no	—
166	C₄²×C₈₉	—	4	1	no	—
167	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₈₉	#41	1	1	yes	characteristic
170	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₂³×C₈₉	—	4	2	no	—
172	C₂³×C₈₉	—	8	2	no	—
173	C₂³×C₈₉	—	8	2	no	—
174	C₂³×C₈₉	—	8	2	no	—
175	C₂³×C₈₉	#121	1	1	yes	characteristic
176	Q₈×C₈₉	—	2	1	no	—
177	Q₈×C₈₉	—	2	1	no	—
178	Q₈×C₈₉	—	2	1	no	—
179	Q₈×C₈₉	—	4	2	no	—
180	Q₈×C₈₉	—	4	1	no	—
181	Q₈×C₈₉	—	4	1	no	—
182	Q₈×C₈₉	—	8	1	no	—
183	Q₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
184	D₄×C₈₉	—	4	2	no	—
185	D₄×C₈₉	—	4	2	no	—
186	D₄×C₈₉	—	4	2	no	—
187	D₄×C₈₉	—	4	2	no	—
188	D₄×C₈₉	—	8	2	no	—
189	D₄×C₈₉	—	8	2	no	—
190	D₄×C₈₉	—	8	2	no	—
191	D₄×C₈₉	—	8	2	no	—
192	C₂×C₄×C₈₉	—	2	1	no	—
193	C₂×C₄×C₈₉	—	2	1	no	—
194	C₂×C₄×C₈₉	—	2	1	no	—
195	C₂×C₄×C₈₉	—	2	1	no	—
196	C₂×C₄×C₈₉	—	2	1	no	—
197	C₂×C₄×C₈₉	—	2	1	no	—
198	C₂×C₄×C₈₉	—	4	1	no	—
199	C₂×C₄×C₈₉	—	4	2	no	—
200	C₂×C₄×C₈₉	—	4	1	no	—
201	C₂×C₄×C₈₉	—	4	1	no	—
202	C₂×C₄×C₈₉	—	8	1	no	—
203	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
204	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
205	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
206	C₂×C₄×C₈₉	#135	1	2	yes	—
207	C₂×C₄×C₈₉	#137	1	2	yes	—
208	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
211	C₈×C₈₉	—	2	2	no	—
212	C₈×C₈₉	—	2	2	no	—
213	C₈×C₈₉	—	4	2	no	—
214	C₈×C₈₉	—	4	2	no	—
215	C₈×C₈₉	—	4	1	no	—
216	C₈×C₈₉	—	4	1	no	—
217	C₂²×C₈₉	—	2	1	no	—
218	C₂²×C₈₉	—	2	1	no	—
219	C₂²×C₈₉	—	8	2	no	—
220	C₂²×C₈₉	—	8	2	no	—
221	C₂²×C₈₉	—	8	2	no	—
222	C₂²×C₈₉	—	16	4	no	—
223	C₂²×C₈₉	#464	1	2	yes	—
224	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
225	C₄×C₈₉	—	2	1	no	—
226	C₄×C₈₉	—	2	2	no	—
227	C₄×C₈₉	—	2	1	no	—
228	C₄×C₈₉	—	2	1	no	—
229	C₄×C₈₉	—	4	1	no	—
230	C₄×C₈₉	—	4	1	no	—
231	C₄×C₈₉	—	4	1	no	—
232	C₄×C₈₉	—	8	1	no	—
233	C₄×C₈₉	—	8	1	no	—
234	C₄×C₈₉	#464	1	2	yes	—
235	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
236	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
237	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
238	C₄×C₈₉	#575	1	1	yes	characteristic
239	#19174	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₈₉	—	2	2	no	—
241	C₂×C₈₉	—	16	2	no	—
242	C₂×C₈₉	C₂×#464	1	1	yes	characteristic
243	C₂×C₈₉	#3055	1	1	yes	characteristic
244	C₂×C₈₉	#3068	1	1	yes	characteristic
245	#3138	C₂×C₈₉	1	1	yes	characteristic
246	#3104	C₂×C₈₉	1	1	yes	characteristic, complemented
247	C₂×#460	C₂×C₈₉	1	1	yes	characteristic
248	Q₈×M₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
249	#1702	C₂×C₈₉	1	2	yes	complemented
250	#1698	C₂×C₈₉	1	2	yes	complemented
251	#1682	C₂×C₈₉	1	2	yes	complemented
252	#1678	C₂×C₈₉	1	2	yes	complemented
253	#1565	C₂×C₈₉	1	1	yes	characteristic, complemented
254	#1550	C₂×C₈₉	1	1	yes	characteristic, complemented
255	#1545	C₂×C₈₉	1	1	yes	characteristic
256	C₈₉	#19174	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²×SD₁₆	—	2	1	no	—
258	#533	C₂²×C₈₉	1	1	yes	characteristic
259	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
260	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
261	#490	—	2	2	no	—
262	#486	C₂²×C₈₉	1	2	yes	—
263	#486	C₂²×C₈₉	1	2	yes	—
264	#484	—	2	1	no	—
265	#477	C₂²×C₈₉	1	1	yes	characteristic, complemented
266	#477	C₂²×C₈₉	1	1	yes	characteristic, complemented
267	#462	C₂²×C₈₉	1	1	yes	characteristic, complemented
268	#462	C₂²×C₈₉	1	1	yes	characteristic, complemented
269	#460	—	2	1	no	—
270	#460	—	2	1	no	—
271	#460	C₂²×C₈₉	1	2	yes	—
272	#460	C₂²×C₈₉	1	2	yes	—
273	#451	—	2	1	no	—
274	#446	—	2	2	no	—
275	#446	C₂²×C₈₉	1	2	yes	complemented
276	C₈×Q₈	C₂²×C₈₉	1	2	yes	complemented
277	C₄×Q₁₆	—	2	1	no	—
278	C₄×Q₁₆	—	2	1	no	—
279	#431	C₂²×C₈₉	1	1	yes	characteristic
280	#423	—	2	1	no	—
281	#423	C₂²×C₈₉	1	1	yes	characteristic
282	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
283	#418	—	2	1	no	—
284	#418	—	2	1	no	—
285	C₂×#102	—	2	1	no	—
286	C₄×M₁₆	—	2	1	no	—
287	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
288	#335	C₂²×C₈₉	1	2	yes	complemented
289	#334	C₂²×C₈₉	1	2	yes	complemented
290	#333	C₂²×C₈₉	1	2	yes	complemented
291	#332	C₂²×C₈₉	1	2	yes	complemented
292	#331	C₂²×C₈₉	1	2	yes	—
293	#331	C₂²×C₈₉	1	2	yes	—
294	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	—	2	1	no	—
298	C₂×Q₁₆	—	2	1	no	—
299	C₂×Q₁₆	—	2	1	no	—
300	C₂×SD₁₆	—	2	2	no	—
301	C₂×SD₁₆	—	2	2	no	—
302	C₂×SD₁₆	—	4	4	no	—
303	C₂×M₁₆	—	2	1	no	—
304	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
305	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
306	C₂²×C₈	—	2	1	no	—
307	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
308	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
309	#127	—	2	1	no	—
310	#127	—	2	1	no	—
311	#127	—	2	1	no	—
312	#127	C₂³×C₈₉	1	2	yes	—
313	#125	—	2	1	no	—
314	C₂²⋊Q₈	—	2	1	no	—
315	C₂²⋊Q₈	—	2	1	no	—
316	C₄×Q₈	—	2	1	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	—	2	1	no	—
319	C₄×Q₈	—	2	1	no	—
320	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
321	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
322	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
323	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
324	#117	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
329	#107	—	2	1	no	—
330	#107	—	2	1	no	—
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	2	2	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	2	1	no	—
335	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
336	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
337	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
338	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
339	#103	—	2	1	no	—
340	#103	—	2	1	no	—
341	#103	—	2	1	no	—
342	#103	—	2	1	no	—
343	#102	—	2	2	no	—
344	#102	—	2	2	no	—
345	#102	—	2	2	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	2	2	no	—
349	C₂²⋊C₈	—	2	1	no	—
350	#97	—	2	2	no	—
351	#97	D₄×C₈₉	1	2	yes	—
352	C₄×C₈	—	2	2	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	C₂³×C₈₉	1	2	yes	—
355	C₂⁴	—	4	1	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	4	1	no	—
361	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
362	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	4	2	no	—
367	C₂×D₄	—	4	2	no	—
368	C₂×D₄	—	4	1	no	—
369	C₂×D₄	—	4	1	no	—
370	C₂×D₄	—	4	1	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	—	8	2	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
378	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
379	Q₁₆	—	4	2	no	—
380	Q₁₆	—	4	2	no	—
381	SD₁₆	—	4	4	no	—
382	SD₁₆	—	4	4	no	—
383	M₁₆	—	2	2	no	—
384	M₁₆	—	2	2	no	—
385	M₁₆	—	4	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	4	2	no	—
391	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
392	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	4	1	no	—
403	C₄⋊C₄	—	4	1	no	—
404	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
405	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
406	C₂²⋊C₄	—	4	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	4	1	no	—
412	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
413	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
414	C₄²	#41×C₈₉	1	1	yes	characteristic
415	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
416	C₂³	—	4	2	no	—
417	C₂³	—	8	2	no	—
418	C₂³	—	8	2	no	—
419	C₂³	—	8	2	no	—
420	C₂³	#121×C₈₉	1	1	yes	characteristic
421	Q₈	—	2	1	no	—
422	Q₈	—	2	1	no	—
423	Q₈	—	2	1	no	—
424	Q₈	—	4	2	no	—
425	Q₈	—	4	1	no	—
426	Q₈	—	4	1	no	—
427	Q₈	—	8	1	no	—
428	Q₈	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
429	D₄	—	4	2	no	—
430	D₄	—	4	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	4	2	no	—
433	D₄	—	8	2	no	—
434	D₄	—	8	2	no	—
435	D₄	—	8	2	no	—
436	D₄	—	8	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	8	1	no	—
448	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
449	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
450	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
451	C₂×C₄	#135×C₈₉	1	2	yes	—
452	C₂×C₄	#137×C₈₉	1	2	yes	—
453	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
454	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
455	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
456	C₈	—	2	2	no	—
457	C₈	—	2	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₈	—	4	1	no	—
461	C₈	—	4	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	2	1	no	—
464	C₂²	—	8	2	no	—
465	C₂²	—	8	2	no	—
466	C₂²	—	8	2	no	—
467	C₂²	—	16	4	no	—
468	C₂²	#464×C₈₉	1	2	yes	—
469	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	1	no	—
477	C₄	—	8	1	no	—
478	C₄	—	8	1	no	—
479	C₄	#464×C₈₉	1	2	yes	—
480	C₄	C₂×#135×C₈₉	1	1	yes	characteristic
481	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
482	C₄	C₂×#137×C₈₉	1	1	yes	characteristic
483	C₄	#575×C₈₉	1	1	yes	characteristic
484	C₂	—	2	2	no	—
485	C₂	—	16	2	no	—
486	C₂	C₂×#464×C₈₉	1	1	yes	characteristic
487	C₂	C₈₉×#3055	1	1	yes	characteristic
488	C₂	C₈₉×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
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
2. C₈₉×#3104	C₂	semidirect
4. Q₈×M₁₆×C₈₉	C₂	semidirect
5. C₈₉×#1702	C₂	semidirect
6. C₈₉×#1698	C₂	semidirect
7. C₈₉×#1682	C₂	semidirect
8. C₈₉×#1678	C₂	semidirect
9. C₈₉×#1565	C₂	semidirect
10. C₈₉×#1550	C₂	semidirect
20. #477×C₈₉	C₂²	semidirect
21. #477×C₈₉	C₂²	semidirect
22. #462×C₈₉	C₂²	semidirect
23. #462×C₈₉	C₂²	semidirect
30. #446×C₈₉	C₂²	semidirect
31. C₈×Q₈×C₈₉	C₂²	semidirect
43. #335×C₈₉	C₂²	semidirect
44. #334×C₈₉	C₂²	semidirect
45. #333×C₈₉	C₂²	semidirect
46. #332×C₈₉	C₂²	semidirect
239. #19174	C₈₉	direct (N × H)
246. #3104	C₂×C₈₉	semidirect
248. Q₈×M₁₆	C₂×C₈₉	semidirect
249. #1702	C₂×C₈₉	semidirect
250. #1698	C₂×C₈₉	semidirect
251. #1682	C₂×C₈₉	semidirect
252. #1678	C₂×C₈₉	semidirect
253. #1565	C₂×C₈₉	semidirect
254. #1550	C₂×C₈₉	semidirect
256. C₈₉	#19174	direct (N × H)
265. #477	C₂²×C₈₉	semidirect
266. #477	C₂²×C₈₉	semidirect
267. #462	C₂²×C₈₉	semidirect
268. #462	C₂²×C₈₉	semidirect
275. #446	C₂²×C₈₉	semidirect
276. C₈×Q₈	C₂²×C₈₉	semidirect
288. #335	C₂²×C₈₉	semidirect
289. #334	C₂²×C₈₉	semidirect
290. #333	C₂²×C₈₉	semidirect
291. #332	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×#3138	2
C₈₉×#3104	2
C₂×#460×C₈₉	2
Q₈×M₁₆×C₈₉	2
C₈₉×#1702	2
C₈₉×#1698	2
C₈₉×#1682	2
C₈₉×#1678	2
C₈₉×#1565	2
C₈₉×#1550	2
C₈₉×#1545	2
#19174	89

Subgroups — C89×#19174 #36397022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#19174 #36397022