Subgroups — C₈₉×#25441 #36403289

Order 22784.

938 conjugacy classes of subgroups.

The subgroup lattice has 548 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×#3061	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#443×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×#1970	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1907	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1764	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	#469×C₈₉	—	2	2	no	—
25	#468×C₈₉	—	2	2	no	—
26	#468×C₈₉	—	2	2	no	—
27	#467×C₈₉	—	2	1	no	—
28	#466×C₈₉	—	2	1	no	—
29	#466×C₈₉	—	2	2	no	—
30	#465×C₈₉	—	2	2	no	—
31	#443×C₈₉	—	2	4	no	—
32	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
33	#428×C₈₉	C₂²	1	2	yes	—
34	C₂×#107×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×C₄×C₈×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	1	no	—
72	C₂×M₁₆×C₈₉	—	2	2	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	2	no	—
77	C₂²×C₈×C₈₉	—	2	1	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	1	no	—
96	#107×C₈₉	—	2	1	no	—
97	#107×C₈₉	—	2	2	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	1	yes	characteristic
103	#103×C₈₉	D₄	1	1	yes	characteristic
104	#102×C₈₉	—	2	2	no	—
105	#102×C₈₉	—	2	2	no	—
106	#102×C₈₉	—	2	2	no	—
107	#102×C₈₉	D₄	1	1	yes	characteristic
108	#102×C₈₉	D₄	1	1	yes	characteristic
109	#100×C₈₉	—	2	4	no	—
110	#97×C₈₉	—	2	1	no	—
111	C₄×C₈×C₈₉	—	2	2	no	—
112	C₄×C₈×C₈₉	—	2	1	no	—
113	C₂⁴×C₈₉	—	2	2	no	—
114	#41×C₈₉	—	2	2	no	—
115	#41×C₈₉	—	2	1	no	—
116	#41×C₈₉	—	2	1	no	—
117	#41×C₈₉	—	2	1	no	—
118	#41×C₈₉	—	2	2	no	—
119	#41×C₈₉	—	2	2	no	—
120	#41×C₈₉	—	2	2	no	—
121	#41×C₈₉	—	2	1	no	—
122	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
123	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×D₄×C₈₉	—	2	2	no	—
125	C₂×D₄×C₈₉	—	2	1	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	—	2	2	no	—
131	C₂×D₄×C₈₉	—	2	2	no	—
132	C₂×D₄×C₈₉	—	4	2	no	—
133	C₂×D₄×C₈₉	—	4	2	no	—
134	C₂×D₄×C₈₉	—	4	2	no	—
135	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₈₉	—	2	1	no	—
138	C₂²×C₄×C₈₉	—	2	2	no	—
139	C₂²×C₄×C₈₉	—	2	1	no	—
140	C₂²×C₄×C₈₉	—	2	2	no	—
141	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
142	Q₁₆×C₈₉	—	2	2	no	—
143	SD₁₆×C₈₉	—	2	4	no	—
144	SD₁₆×C₈₉	—	4	4	no	—
145	SD₁₆×C₈₉	—	4	4	no	—
146	D₈×C₈₉	—	2	2	no	—
147	D₈×C₈₉	—	4	4	no	—
148	D₈×C₈₉	—	4	4	no	—
149	D₈×C₈₉	—	4	2	no	—
150	D₈×C₈₉	—	4	2	no	—
151	M₁₆×C₈₉	—	2	2	no	—
152	M₁₆×C₈₉	—	2	1	no	—
153	M₁₆×C₈₉	—	2	1	no	—
154	M₁₆×C₈₉	—	2	1	no	—
155	M₁₆×C₈₉	—	4	4	no	—
156	M₁₆×C₈₉	—	4	2	no	—
157	M₁₆×C₈₉	C₂×D₄	1	4	yes	—
158	M₁₆×C₈₉	#41	1	2	yes	—
159	C₂×C₈×C₈₉	—	2	1	no	—
160	C₂×C₈×C₈₉	—	2	1	no	—
161	C₂×C₈×C₈₉	—	2	2	no	—
162	C₂×C₈×C₈₉	—	2	1	no	—
163	C₂×C₈×C₈₉	—	2	1	no	—
164	C₂×C₈×C₈₉	—	4	4	no	—
165	C₂×C₈×C₈₉	—	4	2	no	—
166	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
168	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
169	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
171	C₂×C₈×C₈₉	#41	1	2	yes	—
172	C₄⋊C₄×C₈₉	—	2	2	no	—
173	C₄⋊C₄×C₈₉	—	2	2	no	—
174	C₄⋊C₄×C₈₉	—	4	2	no	—
175	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
176	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂²⋊C₄×C₈₉	—	2	1	no	—
178	C₂²⋊C₄×C₈₉	—	4	2	no	—
179	C₂²⋊C₄×C₈₉	—	4	2	no	—
180	C₂²⋊C₄×C₈₉	—	4	2	no	—
181	C₂²⋊C₄×C₈₉	—	4	2	no	—
182	C₂²⋊C₄×C₈₉	—	8	4	no	—
183	C₄²×C₈₉	—	2	1	no	—
184	C₄²×C₈₉	—	2	1	no	—
185	C₄²×C₈₉	—	2	1	no	—
186	C₂³×C₈₉	—	2	1	no	—
187	C₂³×C₈₉	—	2	2	no	—
188	C₂³×C₈₉	—	4	2	no	—
189	C₂³×C₈₉	—	4	2	no	—
190	C₂³×C₈₉	—	4	2	no	—
191	C₂³×C₈₉	—	8	2	no	—
192	C₂³×C₈₉	—	8	2	no	—
193	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
194	Q₈×C₈₉	—	2	2	no	—
195	Q₈×C₈₉	—	2	1	no	—
196	Q₈×C₈₉	—	2	1	no	—
197	Q₈×C₈₉	—	2	2	no	—
198	D₄×C₈₉	—	2	2	no	—
199	D₄×C₈₉	—	2	1	no	—
200	D₄×C₈₉	—	2	1	no	—
201	D₄×C₈₉	—	2	2	no	—
202	D₄×C₈₉	—	4	2	no	—
203	D₄×C₈₉	—	4	1	no	—
204	D₄×C₈₉	—	4	2	no	—
205	D₄×C₈₉	—	4	1	no	—
206	D₄×C₈₉	—	4	2	no	—
207	D₄×C₈₉	—	4	2	no	—
208	D₄×C₈₉	—	4	2	no	—
209	D₄×C₈₉	—	4	2	no	—
210	D₄×C₈₉	—	16	2	no	—
211	D₄×C₈₉	—	16	2	no	—
212	D₄×C₈₉	—	16	2	no	—
213	C₂×C₄×C₈₉	—	2	2	no	—
214	C₂×C₄×C₈₉	—	2	1	no	—
215	C₂×C₄×C₈₉	—	2	1	no	—
216	C₂×C₄×C₈₉	—	2	2	no	—
217	C₂×C₄×C₈₉	—	2	1	no	—
218	C₂×C₄×C₈₉	—	4	1	no	—
219	C₂×C₄×C₈₉	—	4	1	no	—
220	C₂×C₄×C₈₉	—	4	1	no	—
221	C₂×C₄×C₈₉	—	4	2	no	—
222	C₂×C₄×C₈₉	—	4	2	no	—
223	C₂×C₄×C₈₉	—	4	1	no	—
224	C₂×C₄×C₈₉	—	4	2	no	—
225	C₂×C₄×C₈₉	—	8	2	no	—
226	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
232	C₈×C₈₉	—	2	1	no	—
233	C₈×C₈₉	—	2	1	no	—
234	C₈×C₈₉	—	2	4	no	—
235	C₈×C₈₉	—	2	2	no	—
236	C₈×C₈₉	—	4	1	no	—
237	C₈×C₈₉	#121	1	2	yes	—
238	C₈×C₈₉	#121	1	2	yes	—
239	C₂²×C₈₉	—	2	1	no	—
240	C₂²×C₈₉	—	2	1	no	—
241	C₂²×C₈₉	—	4	1	no	—
242	C₂²×C₈₉	—	4	2	no	—
243	C₂²×C₈₉	—	4	2	no	—
244	C₂²×C₈₉	—	4	1	no	—
245	C₂²×C₈₉	—	4	2	no	—
246	C₂²×C₈₉	—	4	1	no	—
247	C₂²×C₈₉	—	8	4	no	—
248	C₂²×C₈₉	—	8	2	no	—
249	C₂²×C₈₉	—	8	2	no	—
250	C₂²×C₈₉	—	8	2	no	—
251	C₂²×C₈₉	—	16	2	no	—
252	C₂²×C₈₉	#540	1	1	yes	characteristic
253	C₂²×C₈₉	D₄²	1	1	yes	characteristic
254	C₂²×C₈₉	#550	1	1	yes	characteristic
255	C₄×C₈₉	—	4	2	no	—
256	C₄×C₈₉	—	4	2	no	—
257	C₄×C₈₉	—	4	1	no	—
258	C₄×C₈₉	—	4	1	no	—
259	C₄×C₈₉	—	4	1	no	—
260	C₄×C₈₉	—	8	2	no	—
261	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
262	C₄×C₈₉	#532	1	1	yes	characteristic
263	C₄×C₈₉	D₄²	1	1	yes	characteristic
264	C₄×C₈₉	#546	1	1	yes	characteristic
265	#25441	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₈₉	—	2	1	no	—
267	C₂×C₈₉	—	2	1	no	—
268	C₂×C₈₉	—	8	2	no	—
269	C₂×C₈₉	—	8	1	no	—
270	C₂×C₈₉	—	16	2	no	—
271	C₂×C₈₉	#2428	1	1	yes	characteristic
272	C₂×C₈₉	#3290	1	1	yes	characteristic
273	C₂×C₈₉	#3292	1	1	yes	characteristic
274	#3155	C₂×C₈₉	1	1	yes	characteristic, complemented
275	C₂×#472	C₂×C₈₉	1	1	yes	characteristic, complemented
276	#3062	C₂×C₈₉	1	2	yes	complemented
277	#3061	C₂×C₈₉	1	1	yes	characteristic, complemented
278	C₂×#443	C₂×C₈₉	1	1	yes	characteristic, complemented
279	#2037	C₂×C₈₉	1	2	yes	complemented
280	#2006	C₂×C₈₉	1	2	yes	complemented
281	#1970	C₂×C₈₉	1	1	yes	characteristic, complemented
282	#1952	C₂×C₈₉	1	1	yes	characteristic, complemented
283	#1907	C₂×C₈₉	1	2	yes	complemented
284	#1764	C₂×C₈₉	1	1	yes	characteristic, complemented
285	C₈₉	#25441	1	1	yes	characteristic, direct product factor, Sylow
286	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
287	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
288	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
289	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
290	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
291	C₂×#131	C₂²×C₈₉	1	2	yes	—
292	#537	C₂²×C₈₉	1	2	yes	—
293	#535	C₂²×C₈₉	1	2	yes	—
294	#496	—	2	2	no	—
295	#495	—	2	1	no	—
296	#493	—	2	1	no	—
297	#472	—	2	4	no	—
298	#469	—	2	2	no	—
299	#468	—	2	2	no	—
300	#468	—	2	2	no	—
301	#467	—	2	1	no	—
302	#466	—	2	1	no	—
303	#466	—	2	2	no	—
304	#465	—	2	2	no	—
305	#443	—	2	4	no	—
306	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
307	#428	C₂²×C₈₉	1	2	yes	—
308	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
309	C₂×#104	C₂²×C₈₉	1	2	yes	—
310	C₂×#104	C₂²×C₈₉	1	1	yes	characteristic
311	#417	C₂²×C₈₉	1	2	yes	—
312	#416	C₂²×C₈₉	1	1	yes	characteristic
313	C₂×#102	C₂²×C₈₉	1	2	yes	—
314	C₂×#100	C₂²×C₈₉	1	2	yes	—
315	#405	C₂²×C₈₉	1	1	yes	characteristic
316	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
317	#390	C₂²×C₈₉	1	2	yes	—
318	#343	C₂²×C₈₉	1	2	yes	—
319	#341	C₂²×C₈₉	1	1	yes	characteristic
320	#339	C₂²×C₈₉	1	1	yes	characteristic
321	C₂×#41	C₂³×C₈₉	1	2	yes	—
322	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
323	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
324	AGL(1,ℤ/8)	—	2	4	no	—
325	AGL(1,ℤ/8)	—	2	4	no	—
326	AGL(1,ℤ/8)	—	4	2	no	—
327	AGL(1,ℤ/8)	—	4	2	no	—
328	#135	—	2	2	no	—
329	#135	—	2	2	no	—
330	C₂×Q₁₆	D₄×C₈₉	1	1	yes	characteristic
331	C₂×SD₁₆	—	2	2	no	—
332	C₂×SD₁₆	—	2	2	no	—
333	C₂×SD₁₆	D₄×C₈₉	1	2	yes	—
334	C₂×D₈	—	2	2	no	—
335	C₂×D₈	—	2	2	no	—
336	C₂×D₈	—	2	2	no	—
337	C₂×D₈	—	2	1	no	—
338	C₂×D₈	—	2	1	no	—
339	C₂×D₈	—	2	2	no	—
340	C₂×D₈	D₄×C₈₉	1	1	yes	characteristic
341	#131	—	2	4	no	—
342	#131	—	2	2	no	—
343	#131	—	2	2	no	—
344	#131	—	4	2	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	2	2	no	—
347	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
348	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
349	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
350	C₂²×C₈	—	2	2	no	—
351	C₂²×C₈	—	2	1	no	—
352	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
354	#127	—	2	2	no	—
355	#124	—	2	2	no	—
356	C₂²⋊Q₈	—	2	2	no	—
357	C₂²⋊Q₈	—	2	2	no	—
358	#121	—	2	2	no	—
359	#121	—	2	2	no	—
360	#121	—	2	2	no	—
361	#121	—	2	2	no	—
362	#121	—	4	2	no	—
363	#120	—	4	2	no	—
364	#117	C₂³×C₈₉	1	1	yes	characteristic
365	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
366	C₂×C₂²⋊C₄	—	4	2	no	—
367	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
368	#108	D₄×C₈₉	1	4	yes	complemented
369	#107	—	2	1	no	—
370	#107	—	2	1	no	—
371	#107	—	2	2	no	—
372	#106	—	2	2	no	—
373	#104	—	2	4	no	—
374	#104	—	2	2	no	—
375	#103	—	2	2	no	—
376	#103	D₄×C₈₉	1	1	yes	characteristic
377	#103	D₄×C₈₉	1	1	yes	characteristic
378	#102	—	2	2	no	—
379	#102	—	2	2	no	—
380	#102	—	2	2	no	—
381	#102	D₄×C₈₉	1	1	yes	characteristic
382	#102	D₄×C₈₉	1	1	yes	characteristic
383	#100	—	2	4	no	—
384	#97	—	2	1	no	—
385	C₄×C₈	—	2	2	no	—
386	C₄×C₈	—	2	1	no	—
387	C₂⁴	—	2	2	no	—
388	#41	—	2	2	no	—
389	#41	—	2	1	no	—
390	#41	—	2	1	no	—
391	#41	—	2	1	no	—
392	#41	—	2	2	no	—
393	#41	—	2	2	no	—
394	#41	—	2	2	no	—
395	#41	—	2	1	no	—
396	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
397	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	—	4	2	no	—
408	C₂×D₄	—	4	2	no	—
409	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
410	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
416	Q₁₆	—	2	2	no	—
417	SD₁₆	—	2	4	no	—
418	SD₁₆	—	4	4	no	—
419	SD₁₆	—	4	4	no	—
420	D₈	—	2	2	no	—
421	D₈	—	4	4	no	—
422	D₈	—	4	4	no	—
423	D₈	—	4	2	no	—
424	D₈	—	4	2	no	—
425	M₁₆	—	2	2	no	—
426	M₁₆	—	2	1	no	—
427	M₁₆	—	2	1	no	—
428	M₁₆	—	2	1	no	—
429	M₁₆	—	4	4	no	—
430	M₁₆	—	4	2	no	—
431	M₁₆	C₂×D₄×C₈₉	1	4	yes	—
432	M₁₆	#41×C₈₉	1	2	yes	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	4	4	no	—
439	C₂×C₈	—	4	2	no	—
440	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
441	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
442	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
443	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
444	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
445	C₂×C₈	#41×C₈₉	1	2	yes	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	4	2	no	—
449	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
450	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
451	C₂²⋊C₄	—	2	1	no	—
452	C₂²⋊C₄	—	4	2	no	—
453	C₂²⋊C₄	—	4	2	no	—
454	C₂²⋊C₄	—	4	2	no	—
455	C₂²⋊C₄	—	4	2	no	—
456	C₂²⋊C₄	—	8	4	no	—
457	C₄²	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₄²	—	2	1	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	2	no	—
465	C₂³	—	8	2	no	—
466	C₂³	—	8	2	no	—
467	C₂³	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
468	Q₈	—	2	2	no	—
469	Q₈	—	2	1	no	—
470	Q₈	—	2	1	no	—
471	Q₈	—	2	2	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	1	no	—
474	D₄	—	2	1	no	—
475	D₄	—	2	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	1	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	1	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	2	no	—
483	D₄	—	4	2	no	—
484	D₄	—	16	2	no	—
485	D₄	—	16	2	no	—
486	D₄	—	16	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	8	2	no	—
500	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
503	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
504	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
505	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
506	C₈	—	2	1	no	—
507	C₈	—	2	1	no	—
508	C₈	—	2	4	no	—
509	C₈	—	2	2	no	—
510	C₈	—	4	1	no	—
511	C₈	#121×C₈₉	1	2	yes	—
512	C₈	#121×C₈₉	1	2	yes	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	1	no	—
521	C₂²	—	8	4	no	—
522	C₂²	—	8	2	no	—
523	C₂²	—	8	2	no	—
524	C₂²	—	8	2	no	—
525	C₂²	—	16	2	no	—
526	C₂²	#540×C₈₉	1	1	yes	characteristic
527	C₂²	D₄²×C₈₉	1	1	yes	characteristic
528	C₂²	#550×C₈₉	1	1	yes	characteristic
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	1	no	—
533	C₄	—	4	1	no	—
534	C₄	—	8	2	no	—
535	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
536	C₄	#532×C₈₉	1	1	yes	characteristic
537	C₄	D₄²×C₈₉	1	1	yes	characteristic
538	C₄	#546×C₈₉	1	1	yes	characteristic
539	C₂	—	2	1	no	—
540	C₂	—	2	1	no	—
541	C₂	—	8	2	no	—
542	C₂	—	8	1	no	—
543	C₂	—	16	2	no	—
544	C₂	C₈₉×#2428	1	1	yes	characteristic
545	C₂	C₈₉×#3290	1	1	yes	characteristic
546	C₂	C₈₉×#3292	1	1	yes	characteristic

	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×#3061	C₂	semidirect
5. C₂×#443×C₈₉	C₂	semidirect
6. C₈₉×#2037	C₂	semidirect
7. C₈₉×#2006	C₂	semidirect
8. C₈₉×#1970	C₂	semidirect
9. C₈₉×#1952	C₂	semidirect
10. C₈₉×#1907	C₂	semidirect
11. C₈₉×#1764	C₂	semidirect
94. #108×C₈₉	D₄	semidirect
265. #25441	C₈₉	direct (N × H)
274. #3155	C₂×C₈₉	semidirect
275. C₂×#472	C₂×C₈₉	semidirect
276. #3062	C₂×C₈₉	semidirect
277. #3061	C₂×C₈₉	semidirect
278. C₂×#443	C₂×C₈₉	semidirect
279. #2037	C₂×C₈₉	semidirect
280. #2006	C₂×C₈₉	semidirect
281. #1970	C₂×C₈₉	semidirect
282. #1952	C₂×C₈₉	semidirect
283. #1907	C₂×C₈₉	semidirect
284. #1764	C₂×C₈₉	semidirect
285. C₈₉	#25441	direct (N × H)
368. #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×#3061	2
C₂×#443×C₈₉	2
C₈₉×#2037	2
C₈₉×#2006	2
C₈₉×#1970	2
C₈₉×#1952	2
C₈₉×#1907	2
C₈₉×#1764	2
#25441	89

Subgroups — C89×#25441 #36403289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25441 #36403289