Subgroups — C₈₉×#25444 #36403292

Order 22784.

934 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3155	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3062	C₂	1	2	yes	maximal, complemented
4	C₈₉×#3060	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#444×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2037	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2006	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1972	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1907	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
13	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
14	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#131×C₈₉	C₂²	1	2	yes	—
18	#537×C₈₉	C₂²	1	2	yes	—
19	#535×C₈₉	C₂²	1	2	yes	—
20	#496×C₈₉	—	2	2	no	—
21	#495×C₈₉	—	2	1	no	—
22	#493×C₈₉	—	2	1	no	—
23	#472×C₈₉	—	2	4	no	—
24	#470×C₈₉	—	2	1	no	—
25	#469×C₈₉	—	2	1	no	—
26	#469×C₈₉	—	2	2	no	—
27	#468×C₈₉	—	2	2	no	—
28	#466×C₈₉	—	2	2	no	—
29	#465×C₈₉	—	2	2	no	—
30	#465×C₈₉	—	2	2	no	—
31	#444×C₈₉	—	2	4	no	—
32	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
33	#428×C₈₉	C₂²	1	2	yes	—
34	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#104×C₈₉	C₂²	1	2	yes	—
36	C₂×#104×C₈₉	C₂²	1	1	yes	characteristic
37	#417×C₈₉	C₂²	1	2	yes	—
38	#416×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#102×C₈₉	C₂²	1	2	yes	—
40	C₂×#100×C₈₉	C₂²	1	2	yes	—
41	#405×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
43	#390×C₈₉	C₂²	1	2	yes	—
44	#343×C₈₉	C₂²	1	2	yes	—
45	#341×C₈₉	C₂²	1	1	yes	characteristic
46	#339×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×#41×C₈₉	C₂³	1	2	yes	—
49	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
50	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
51	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
52	AGL(1,ℤ/8)×C₈₉	—	4	2	no	—
53	AGL(1,ℤ/8)×C₈₉	—	4	2	no	—
54	#135×C₈₉	—	2	2	no	—
55	#135×C₈₉	—	2	2	no	—
56	C₂×Q₁₆×C₈₉	D₄	1	1	yes	characteristic
57	C₂×SD₁₆×C₈₉	—	2	2	no	—
58	C₂×SD₁₆×C₈₉	—	2	2	no	—
59	C₂×SD₁₆×C₈₉	D₄	1	2	yes	—
60	C₂×D₈×C₈₉	—	2	2	no	—
61	C₂×D₈×C₈₉	—	2	2	no	—
62	C₂×D₈×C₈₉	—	2	2	no	—
63	C₂×D₈×C₈₉	—	2	1	no	—
64	C₂×D₈×C₈₉	—	2	1	no	—
65	C₂×D₈×C₈₉	—	2	2	no	—
66	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
67	#131×C₈₉	—	2	4	no	—
68	#131×C₈₉	—	2	2	no	—
69	#131×C₈₉	—	2	2	no	—
70	#131×C₈₉	—	4	2	no	—
71	C₂×M₁₆×C₈₉	—	2	2	no	—
72	C₂×M₁₆×C₈₉	—	2	1	no	—
73	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	—	2	1	no	—
77	C₂²×C₈×C₈₉	—	2	2	no	—
78	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	#127×C₈₉	—	2	2	no	—
81	#124×C₈₉	—	2	2	no	—
82	C₂²⋊Q₈×C₈₉	—	2	2	no	—
83	C₂²⋊Q₈×C₈₉	—	2	2	no	—
84	#121×C₈₉	—	2	2	no	—
85	#121×C₈₉	—	2	2	no	—
86	#121×C₈₉	—	2	2	no	—
87	#121×C₈₉	—	2	2	no	—
88	#121×C₈₉	—	4	2	no	—
89	#120×C₈₉	—	4	2	no	—
90	#117×C₈₉	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₈₉	—	4	2	no	—
93	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
94	#108×C₈₉	D₄	1	4	yes	complemented
95	#107×C₈₉	—	2	2	no	—
96	#106×C₈₉	—	2	1	no	—
97	#106×C₈₉	—	2	1	no	—
98	#106×C₈₉	—	2	2	no	—
99	#104×C₈₉	—	2	4	no	—
100	#104×C₈₉	—	2	2	no	—
101	#103×C₈₉	—	2	2	no	—
102	#103×C₈₉	D₄	1	2	yes	—
103	#102×C₈₉	—	2	2	no	—
104	#102×C₈₉	—	2	2	no	—
105	#102×C₈₉	—	2	2	no	—
106	#102×C₈₉	D₄	1	2	yes	—
107	#100×C₈₉	—	2	4	no	—
108	#97×C₈₉	—	2	2	no	—
109	#97×C₈₉	—	2	1	no	—
110	C₄×C₈×C₈₉	—	2	1	no	—
111	C₂⁴×C₈₉	—	2	2	no	—
112	#41×C₈₉	—	2	2	no	—
113	#41×C₈₉	—	2	1	no	—
114	#41×C₈₉	—	2	1	no	—
115	#41×C₈₉	—	2	1	no	—
116	#41×C₈₉	—	2	2	no	—
117	#41×C₈₉	—	2	2	no	—
118	#41×C₈₉	—	2	2	no	—
119	#41×C₈₉	—	2	1	no	—
120	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×D₄×C₈₉	—	2	2	no	—
123	C₂×D₄×C₈₉	—	2	1	no	—
124	C₂×D₄×C₈₉	—	2	2	no	—
125	C₂×D₄×C₈₉	—	2	2	no	—
126	C₂×D₄×C₈₉	—	2	2	no	—
127	C₂×D₄×C₈₉	—	2	2	no	—
128	C₂×D₄×C₈₉	—	2	2	no	—
129	C₂×D₄×C₈₉	—	2	2	no	—
130	C₂×D₄×C₈₉	—	4	2	no	—
131	C₂×D₄×C₈₉	—	4	2	no	—
132	C₂×D₄×C₈₉	—	4	2	no	—
133	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₈₉	—	2	2	no	—
136	C₂²×C₄×C₈₉	—	2	1	no	—
137	C₂²×C₄×C₈₉	—	2	1	no	—
138	C₂²×C₄×C₈₉	—	2	2	no	—
139	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₈₉	—	2	2	no	—
141	SD₁₆×C₈₉	—	2	4	no	—
142	SD₁₆×C₈₉	—	4	4	no	—
143	SD₁₆×C₈₉	—	4	4	no	—
144	D₈×C₈₉	—	2	2	no	—
145	D₈×C₈₉	—	4	4	no	—
146	D₈×C₈₉	—	4	4	no	—
147	D₈×C₈₉	—	4	2	no	—
148	D₈×C₈₉	—	4	2	no	—
149	M₁₆×C₈₉	—	2	2	no	—
150	M₁₆×C₈₉	—	2	1	no	—
151	M₁₆×C₈₉	—	2	1	no	—
152	M₁₆×C₈₉	—	2	1	no	—
153	M₁₆×C₈₉	—	4	4	no	—
154	M₁₆×C₈₉	—	4	2	no	—
155	M₁₆×C₈₉	C₂×D₄	1	4	yes	—
156	M₁₆×C₈₉	#41	1	2	yes	—
157	C₂×C₈×C₈₉	—	2	1	no	—
158	C₂×C₈×C₈₉	—	2	1	no	—
159	C₂×C₈×C₈₉	—	2	2	no	—
160	C₂×C₈×C₈₉	—	2	1	no	—
161	C₂×C₈×C₈₉	—	2	1	no	—
162	C₂×C₈×C₈₉	—	4	4	no	—
163	C₂×C₈×C₈₉	—	4	2	no	—
164	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₈₉	#41	1	2	yes	—
169	C₄⋊C₄×C₈₉	—	2	2	no	—
170	C₄⋊C₄×C₈₉	—	2	2	no	—
171	C₄⋊C₄×C₈₉	—	4	2	no	—
172	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
173	C₂²⋊C₄×C₈₉	—	2	1	no	—
174	C₂²⋊C₄×C₈₉	—	4	2	no	—
175	C₂²⋊C₄×C₈₉	—	4	2	no	—
176	C₂²⋊C₄×C₈₉	—	4	2	no	—
177	C₂²⋊C₄×C₈₉	—	4	2	no	—
178	C₂²⋊C₄×C₈₉	—	8	4	no	—
179	C₄²×C₈₉	—	2	1	no	—
180	C₄²×C₈₉	—	2	1	no	—
181	C₄²×C₈₉	—	2	1	no	—
182	C₂³×C₈₉	—	2	1	no	—
183	C₂³×C₈₉	—	2	2	no	—
184	C₂³×C₈₉	—	4	2	no	—
185	C₂³×C₈₉	—	4	2	no	—
186	C₂³×C₈₉	—	4	2	no	—
187	C₂³×C₈₉	—	8	2	no	—
188	C₂³×C₈₉	—	8	2	no	—
189	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
190	Q₈×C₈₉	—	2	2	no	—
191	Q₈×C₈₉	—	2	1	no	—
192	Q₈×C₈₉	—	2	1	no	—
193	Q₈×C₈₉	—	2	2	no	—
194	D₄×C₈₉	—	2	2	no	—
195	D₄×C₈₉	—	2	1	no	—
196	D₄×C₈₉	—	2	1	no	—
197	D₄×C₈₉	—	2	2	no	—
198	D₄×C₈₉	—	4	2	no	—
199	D₄×C₈₉	—	4	1	no	—
200	D₄×C₈₉	—	4	2	no	—
201	D₄×C₈₉	—	4	1	no	—
202	D₄×C₈₉	—	4	2	no	—
203	D₄×C₈₉	—	4	2	no	—
204	D₄×C₈₉	—	4	2	no	—
205	D₄×C₈₉	—	4	2	no	—
206	D₄×C₈₉	—	16	2	no	—
207	D₄×C₈₉	—	16	2	no	—
208	D₄×C₈₉	—	16	2	no	—
209	C₂×C₄×C₈₉	—	2	2	no	—
210	C₂×C₄×C₈₉	—	2	2	no	—
211	C₂×C₄×C₈₉	—	2	1	no	—
212	C₂×C₄×C₈₉	—	4	1	no	—
213	C₂×C₄×C₈₉	—	4	2	no	—
214	C₂×C₄×C₈₉	—	4	2	no	—
215	C₂×C₄×C₈₉	—	4	1	no	—
216	C₂×C₄×C₈₉	—	4	2	no	—
217	C₂×C₄×C₈₉	—	4	2	no	—
218	C₂×C₄×C₈₉	—	4	1	no	—
219	C₂×C₄×C₈₉	—	8	2	no	—
220	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
226	C₈×C₈₉	—	2	2	no	—
227	C₈×C₈₉	—	2	4	no	—
228	C₈×C₈₉	—	2	2	no	—
229	C₈×C₈₉	—	4	1	no	—
230	C₈×C₈₉	#121	1	2	yes	—
231	C₈×C₈₉	#121	1	2	yes	—
232	C₂²×C₈₉	—	2	2	no	—
233	C₂²×C₈₉	—	4	1	no	—
234	C₂²×C₈₉	—	4	2	no	—
235	C₂²×C₈₉	—	4	2	no	—
236	C₂²×C₈₉	—	4	2	no	—
237	C₂²×C₈₉	—	4	2	no	—
238	C₂²×C₈₉	—	8	4	no	—
239	C₂²×C₈₉	—	8	2	no	—
240	C₂²×C₈₉	—	8	2	no	—
241	C₂²×C₈₉	—	8	2	no	—
242	C₂²×C₈₉	—	16	2	no	—
243	C₂²×C₈₉	#540	1	1	yes	characteristic
244	C₂²×C₈₉	D₄²	1	1	yes	characteristic
245	C₂²×C₈₉	#550	1	1	yes	characteristic
246	C₄×C₈₉	—	4	2	no	—
247	C₄×C₈₉	—	4	1	no	—
248	C₄×C₈₉	—	4	2	no	—
249	C₄×C₈₉	—	8	1	no	—
250	C₄×C₈₉	—	8	2	no	—
251	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
252	C₄×C₈₉	#532	1	1	yes	characteristic
253	C₄×C₈₉	D₄²	1	1	yes	characteristic
254	C₄×C₈₉	#546	1	1	yes	characteristic
255	#25444	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂×C₈₉	—	2	1	no	—
257	C₂×C₈₉	—	2	1	no	—
258	C₂×C₈₉	—	8	2	no	—
259	C₂×C₈₉	—	8	1	no	—
260	C₂×C₈₉	—	16	2	no	—
261	C₂×C₈₉	#2428	1	1	yes	characteristic
262	C₂×C₈₉	#3288	1	2	yes	—
263	#3155	C₂×C₈₉	1	1	yes	characteristic, complemented
264	C₂×#472	C₂×C₈₉	1	1	yes	characteristic, complemented
265	#3062	C₂×C₈₉	1	2	yes	complemented
266	#3060	C₂×C₈₉	1	1	yes	characteristic, complemented
267	C₂×#444	C₂×C₈₉	1	1	yes	characteristic, complemented
268	#2037	C₂×C₈₉	1	2	yes	complemented
269	#2006	C₂×C₈₉	1	2	yes	complemented
270	#1972	C₂×C₈₉	1	1	yes	characteristic, complemented
271	#1951	C₂×C₈₉	1	1	yes	characteristic, complemented
272	#1907	C₂×C₈₉	1	2	yes	complemented
273	#1765	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₈₉	#25444	1	1	yes	characteristic, direct product factor, Sylow
275	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
276	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
277	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
278	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
279	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
280	C₂×#131	C₂²×C₈₉	1	2	yes	—
281	#537	C₂²×C₈₉	1	2	yes	—
282	#535	C₂²×C₈₉	1	2	yes	—
283	#496	—	2	2	no	—
284	#495	—	2	1	no	—
285	#493	—	2	1	no	—
286	#472	—	2	4	no	—
287	#470	—	2	1	no	—
288	#469	—	2	1	no	—
289	#469	—	2	2	no	—
290	#468	—	2	2	no	—
291	#466	—	2	2	no	—
292	#465	—	2	2	no	—
293	#465	—	2	2	no	—
294	#444	—	2	4	no	—
295	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
296	#428	C₂²×C₈₉	1	2	yes	—
297	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
298	C₂×#104	C₂²×C₈₉	1	2	yes	—
299	C₂×#104	C₂²×C₈₉	1	1	yes	characteristic
300	#417	C₂²×C₈₉	1	2	yes	—
301	#416	C₂²×C₈₉	1	1	yes	characteristic
302	C₂×#102	C₂²×C₈₉	1	2	yes	—
303	C₂×#100	C₂²×C₈₉	1	2	yes	—
304	#405	C₂²×C₈₉	1	1	yes	characteristic
305	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
306	#390	C₂²×C₈₉	1	2	yes	—
307	#343	C₂²×C₈₉	1	2	yes	—
308	#341	C₂²×C₈₉	1	1	yes	characteristic
309	#339	C₂²×C₈₉	1	1	yes	characteristic
310	C₂×#41	C₂³×C₈₉	1	2	yes	—
311	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
312	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
313	AGL(1,ℤ/8)	—	2	4	no	—
314	AGL(1,ℤ/8)	—	2	4	no	—
315	AGL(1,ℤ/8)	—	4	2	no	—
316	AGL(1,ℤ/8)	—	4	2	no	—
317	#135	—	2	2	no	—
318	#135	—	2	2	no	—
319	C₂×Q₁₆	D₄×C₈₉	1	1	yes	characteristic
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×SD₁₆	D₄×C₈₉	1	2	yes	—
323	C₂×D₈	—	2	2	no	—
324	C₂×D₈	—	2	2	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂×D₈	—	2	1	no	—
327	C₂×D₈	—	2	1	no	—
328	C₂×D₈	—	2	2	no	—
329	C₂×D₈	D₄×C₈₉	1	1	yes	characteristic
330	#131	—	2	4	no	—
331	#131	—	2	2	no	—
332	#131	—	2	2	no	—
333	#131	—	4	2	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	—	2	1	no	—
336	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
337	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
338	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
339	C₂²×C₈	—	2	1	no	—
340	C₂²×C₈	—	2	2	no	—
341	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
342	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
343	#127	—	2	2	no	—
344	#124	—	2	2	no	—
345	C₂²⋊Q₈	—	2	2	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	#121	—	2	2	no	—
348	#121	—	2	2	no	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	4	2	no	—
352	#120	—	4	2	no	—
353	#117	C₂³×C₈₉	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
355	C₂×C₂²⋊C₄	—	4	2	no	—
356	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
357	#108	D₄×C₈₉	1	4	yes	complemented
358	#107	—	2	2	no	—
359	#106	—	2	1	no	—
360	#106	—	2	1	no	—
361	#106	—	2	2	no	—
362	#104	—	2	4	no	—
363	#104	—	2	2	no	—
364	#103	—	2	2	no	—
365	#103	D₄×C₈₉	1	2	yes	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	D₄×C₈₉	1	2	yes	—
370	#100	—	2	4	no	—
371	#97	—	2	2	no	—
372	#97	—	2	1	no	—
373	C₄×C₈	—	2	1	no	—
374	C₂⁴	—	2	2	no	—
375	#41	—	2	2	no	—
376	#41	—	2	1	no	—
377	#41	—	2	1	no	—
378	#41	—	2	1	no	—
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	#41	—	2	2	no	—
382	#41	—	2	1	no	—
383	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
384	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂×D₄	—	4	2	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
397	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
403	Q₁₆	—	2	2	no	—
404	SD₁₆	—	2	4	no	—
405	SD₁₆	—	4	4	no	—
406	SD₁₆	—	4	4	no	—
407	D₈	—	2	2	no	—
408	D₈	—	4	4	no	—
409	D₈	—	4	4	no	—
410	D₈	—	4	2	no	—
411	D₈	—	4	2	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	1	no	—
414	M₁₆	—	2	1	no	—
415	M₁₆	—	2	1	no	—
416	M₁₆	—	4	4	no	—
417	M₁₆	—	4	2	no	—
418	M₁₆	C₂×D₄×C₈₉	1	4	yes	—
419	M₁₆	#41×C₈₉	1	2	yes	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	4	4	no	—
426	C₂×C₈	—	4	2	no	—
427	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
428	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
430	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
431	C₂×C₈	#41×C₈₉	1	2	yes	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	4	2	no	—
435	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	4	2	no	—
438	C₂²⋊C₄	—	4	2	no	—
439	C₂²⋊C₄	—	4	2	no	—
440	C₂²⋊C₄	—	4	2	no	—
441	C₂²⋊C₄	—	8	4	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	8	2	no	—
451	C₂³	—	8	2	no	—
452	C₂³	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	1	no	—
455	Q₈	—	2	1	no	—
456	Q₈	—	2	2	no	—
457	D₄	—	2	2	no	—
458	D₄	—	2	1	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	1	no	—
463	D₄	—	4	2	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	2	no	—
466	D₄	—	4	2	no	—
467	D₄	—	4	2	no	—
468	D₄	—	4	2	no	—
469	D₄	—	16	2	no	—
470	D₄	—	16	2	no	—
471	D₄	—	16	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	8	2	no	—
483	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
488	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	4	no	—
491	C₈	—	2	2	no	—
492	C₈	—	4	1	no	—
493	C₈	#121×C₈₉	1	2	yes	—
494	C₈	#121×C₈₉	1	2	yes	—
495	C₂²	—	2	2	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	8	4	no	—
502	C₂²	—	8	2	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	16	2	no	—
506	C₂²	#540×C₈₉	1	1	yes	characteristic
507	C₂²	D₄²×C₈₉	1	1	yes	characteristic
508	C₂²	#550×C₈₉	1	1	yes	characteristic
509	C₄	—	4	2	no	—
510	C₄	—	4	1	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	1	no	—
513	C₄	—	8	2	no	—
514	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
515	C₄	#532×C₈₉	1	1	yes	characteristic
516	C₄	D₄²×C₈₉	1	1	yes	characteristic
517	C₄	#546×C₈₉	1	1	yes	characteristic
518	C₂	—	2	1	no	—
519	C₂	—	2	1	no	—
520	C₂	—	8	2	no	—
521	C₂	—	8	1	no	—
522	C₂	—	16	2	no	—
523	C₂	C₈₉×#2428	1	1	yes	characteristic
524	C₂	C₈₉×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3155	C₂	semidirect
2. C₂×#472×C₈₉	C₂	semidirect
3. C₈₉×#3062	C₂	semidirect
4. C₈₉×#3060	C₂	semidirect
5. C₂×#444×C₈₉	C₂	semidirect
6. C₈₉×#2037	C₂	semidirect
7. C₈₉×#2006	C₂	semidirect
8. C₈₉×#1972	C₂	semidirect
9. C₈₉×#1951	C₂	semidirect
10. C₈₉×#1907	C₂	semidirect
11. C₈₉×#1765	C₂	semidirect
94. #108×C₈₉	D₄	semidirect
255. #25444	C₈₉	direct (N × H)
263. #3155	C₂×C₈₉	semidirect
264. C₂×#472	C₂×C₈₉	semidirect
265. #3062	C₂×C₈₉	semidirect
266. #3060	C₂×C₈₉	semidirect
267. C₂×#444	C₂×C₈₉	semidirect
268. #2037	C₂×C₈₉	semidirect
269. #2006	C₂×C₈₉	semidirect
270. #1972	C₂×C₈₉	semidirect
271. #1951	C₂×C₈₉	semidirect
272. #1907	C₂×C₈₉	semidirect
273. #1765	C₂×C₈₉	semidirect
274. C₈₉	#25444	direct (N × H)
357. #108	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3155	2
C₂×#472×C₈₉	2
C₈₉×#3062	2
C₈₉×#3060	2
C₂×#444×C₈₉	2
C₈₉×#2037	2
C₈₉×#2006	2
C₈₉×#1972	2
C₈₉×#1951	2
C₈₉×#1907	2
C₈₉×#1765	2
#25444	89

Subgroups — C89×#25444 #36403292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25444 #36403292