Subgroups — C₈₉×#19170 #36397018

Order 22784.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3151	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3103	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#460×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1688	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1686	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1668	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1666	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₈₉	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₈₉	—	2	1	no	—
14	C₂²×SD₁₆×C₈₉	—	2	1	no	—
15	#532×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	#496×C₈₉	—	2	2	no	—
19	#493×C₈₉	C₂²	1	2	yes	—
20	#492×C₈₉	C₂²	1	2	yes	—
21	#481×C₈₉	—	2	1	no	—
22	#475×C₈₉	C₂²	1	1	yes	characteristic, complemented
23	#475×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	#460×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#460×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	#448×C₈₉	—	2	1	no	—
31	#446×C₈₉	—	2	2	no	—
32	#446×C₈₉	C₂²	1	2	yes	complemented
33	C₈×Q₈×C₈₉	C₂²	1	2	yes	complemented
34	C₄×SD₁₆×C₈₉	—	2	1	no	—
35	C₄×SD₁₆×C₈₉	—	2	1	no	—
36	#431×C₈₉	C₂²	1	1	yes	characteristic
37	#423×C₈₉	—	2	1	no	—
38	#423×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
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	#332×C₈₉	C₂²	1	2	yes	complemented
46	#331×C₈₉	C₂²	1	2	yes	—
47	#331×C₈₉	C₂²	1	2	yes	—
48	#331×C₈₉	C₂²	1	2	yes	complemented
49	C₂×#41×C₈₉	—	2	1	no	—
50	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₈₉	—	2	1	no	—
52	C₂²×D₄×C₈₉	—	2	1	no	—
53	C₂²×D₄×C₈₉	—	2	1	no	—
54	AGL(1,ℤ/8)×C₈₉	—	4	4	no	—
55	AGL(1,ℤ/8)×C₈₉	—	4	4	no	—
56	C₂×SD₁₆×C₈₉	—	2	1	no	—
57	C₂×SD₁₆×C₈₉	—	2	1	no	—
58	C₂×SD₁₆×C₈₉	—	2	2	no	—
59	C₂×SD₁₆×C₈₉	—	2	2	no	—
60	C₂×SD₁₆×C₈₉	—	2	2	no	—
61	C₂×SD₁₆×C₈₉	—	2	2	no	—
62	C₂×SD₁₆×C₈₉	—	4	4	no	—
63	C₂×D₈×C₈₉	—	2	2	no	—
64	C₂×D₈×C₈₉	—	2	2	no	—
65	C₂×M₁₆×C₈₉	—	2	1	no	—
66	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
67	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	—	2	1	no	—
69	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
70	#127×C₈₉	—	2	1	no	—
71	#127×C₈₉	—	2	1	no	—
72	#127×C₈₉	—	2	1	no	—
73	#127×C₈₉	C₂³	1	2	yes	—
74	#127×C₈₉	C₂³	1	1	yes	characteristic
75	#124×C₈₉	—	2	1	no	—
76	#121×C₈₉	—	2	1	no	—
77	#121×C₈₉	—	2	1	no	—
78	C₄×Q₈×C₈₉	—	2	1	no	—
79	C₄×Q₈×C₈₉	—	2	1	no	—
80	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
83	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₈₉	—	2	1	no	—
85	C₄×D₄×C₈₉	—	2	1	no	—
86	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₈₉	—	2	1	no	—
89	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
90	#106×C₈₉	—	2	1	no	—
91	#106×C₈₉	—	2	1	no	—
92	C₄⋊C₈×C₈₉	—	2	2	no	—
93	C₄⋊C₈×C₈₉	—	2	2	no	—
94	C₄⋊C₈×C₈₉	—	2	1	no	—
95	C₄⋊C₈×C₈₉	—	2	1	no	—
96	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
97	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
98	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
100	#103×C₈₉	—	2	1	no	—
101	#103×C₈₉	—	2	1	no	—
102	#102×C₈₉	—	2	2	no	—
103	#102×C₈₉	—	2	2	no	—
104	#102×C₈₉	—	2	1	no	—
105	#102×C₈₉	—	2	1	no	—
106	C₂²⋊C₈×C₈₉	—	2	1	no	—
107	#97×C₈₉	—	2	2	no	—
108	#97×C₈₉	D₄	1	2	yes	—
109	C₄×C₈×C₈₉	—	2	2	no	—
110	C₄×C₈×C₈₉	—	2	1	no	—
111	C₄×C₈×C₈₉	C₂³	1	2	yes	—
112	C₂⁴×C₈₉	—	4	1	no	—
113	#41×C₈₉	—	4	2	no	—
114	#41×C₈₉	—	4	2	no	—
115	C₂×Q₈×C₈₉	—	2	1	no	—
116	C₂×Q₈×C₈₉	—	2	2	no	—
117	C₂×Q₈×C₈₉	—	2	1	no	—
118	C₂×Q₈×C₈₉	—	4	1	no	—
119	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×D₄×C₈₉	—	2	1	no	—
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	1	no	—
126	C₂×D₄×C₈₉	—	2	2	no	—
127	C₂×D₄×C₈₉	—	4	2	no	—
128	C₂×D₄×C₈₉	—	4	2	no	—
129	C₂×D₄×C₈₉	—	4	1	no	—
130	C₂×D₄×C₈₉	—	4	1	no	—
131	C₂×D₄×C₈₉	—	4	1	no	—
132	C₂×D₄×C₈₉	—	4	1	no	—
133	C₂×D₄×C₈₉	—	4	2	no	—
134	C₂×D₄×C₈₉	—	4	2	no	—
135	C₂×D₄×C₈₉	—	8	2	no	—
136	C₂²×C₄×C₈₉	—	2	1	no	—
137	C₂²×C₄×C₈₉	—	2	1	no	—
138	C₂²×C₄×C₈₉	—	4	1	no	—
139	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
142	SD₁₆×C₈₉	—	4	4	no	—
143	SD₁₆×C₈₉	—	4	2	no	—
144	SD₁₆×C₈₉	—	4	4	no	—
145	SD₁₆×C₈₉	—	4	2	no	—
146	SD₁₆×C₈₉	—	4	4	no	—
147	SD₁₆×C₈₉	—	4	4	no	—
148	D₈×C₈₉	—	4	4	no	—
149	D₈×C₈₉	—	4	4	no	—
150	M₁₆×C₈₉	—	2	2	no	—
151	M₁₆×C₈₉	—	4	2	no	—
152	M₁₆×C₈₉	SD₁₆	1	4	yes	complemented
153	C₂×C₈×C₈₉	—	2	2	no	—
154	C₂×C₈×C₈₉	—	2	1	no	—
155	C₂×C₈×C₈₉	—	2	1	no	—
156	C₂×C₈×C₈₉	—	2	2	no	—
157	C₂×C₈×C₈₉	—	4	2	no	—
158	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₈₉	—	2	1	no	—
161	C₄⋊C₄×C₈₉	—	2	1	no	—
162	C₄⋊C₄×C₈₉	—	2	1	no	—
163	C₄⋊C₄×C₈₉	—	2	1	no	—
164	C₄⋊C₄×C₈₉	—	2	1	no	—
165	C₄⋊C₄×C₈₉	—	2	1	no	—
166	C₄⋊C₄×C₈₉	—	2	1	no	—
167	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
168	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
169	C₂²⋊C₄×C₈₉	—	4	1	no	—
170	C₂²⋊C₄×C₈₉	—	4	1	no	—
171	C₄²×C₈₉	—	2	1	no	—
172	C₄²×C₈₉	—	2	1	no	—
173	C₄²×C₈₉	—	2	1	no	—
174	C₄²×C₈₉	—	2	1	no	—
175	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₈₉	#41	1	1	yes	characteristic
178	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
179	C₂³×C₈₉	—	4	1	no	—
180	C₂³×C₈₉	—	4	2	no	—
181	C₂³×C₈₉	—	8	2	no	—
182	C₂³×C₈₉	—	8	2	no	—
183	C₂³×C₈₉	—	8	2	no	—
184	C₂³×C₈₉	#121	1	1	yes	characteristic
185	Q₈×C₈₉	—	2	1	no	—
186	Q₈×C₈₉	—	2	1	no	—
187	Q₈×C₈₉	—	2	1	no	—
188	Q₈×C₈₉	—	4	2	no	—
189	Q₈×C₈₉	—	4	1	no	—
190	Q₈×C₈₉	—	4	1	no	—
191	Q₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	complemented
192	D₄×C₈₉	—	4	1	no	—
193	D₄×C₈₉	—	4	2	no	—
194	D₄×C₈₉	—	4	1	no	—
195	D₄×C₈₉	—	4	2	no	—
196	D₄×C₈₉	—	4	2	no	—
197	D₄×C₈₉	—	4	1	no	—
198	D₄×C₈₉	—	4	1	no	—
199	D₄×C₈₉	—	4	2	no	—
200	D₄×C₈₉	—	8	1	no	—
201	D₄×C₈₉	—	8	2	no	—
202	D₄×C₈₉	—	8	2	no	—
203	D₄×C₈₉	—	8	2	no	—
204	D₄×C₈₉	—	8	2	no	—
205	D₄×C₈₉	—	8	2	no	—
206	C₂×C₄×C₈₉	—	2	1	no	—
207	C₂×C₄×C₈₉	—	2	1	no	—
208	C₂×C₄×C₈₉	—	2	1	no	—
209	C₂×C₄×C₈₉	—	2	1	no	—
210	C₂×C₄×C₈₉	—	2	1	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	1	no	—
215	C₂×C₄×C₈₉	—	8	1	no	—
216	C₂×C₄×C₈₉	—	8	2	no	—
217	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
218	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
219	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
220	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
221	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
222	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
225	C₈×C₈₉	—	2	2	no	—
226	C₈×C₈₉	—	4	2	no	—
227	C₈×C₈₉	—	4	2	no	—
228	C₈×C₈₉	—	4	1	no	—
229	C₈×C₈₉	—	4	1	no	—
230	C₈×C₈₉	C₂×SD₁₆	1	4	yes	complemented
231	C₂²×C₈₉	—	2	1	no	—
232	C₂²×C₈₉	—	2	1	no	—
233	C₂²×C₈₉	—	8	1	no	—
234	C₂²×C₈₉	—	8	2	no	—
235	C₂²×C₈₉	—	8	2	no	—
236	C₂²×C₈₉	—	8	1	no	—
237	C₂²×C₈₉	—	8	2	no	—
238	C₂²×C₈₉	—	8	1	no	—
239	C₂²×C₈₉	—	16	4	no	—
240	C₂²×C₈₉	#460	1	2	yes	—
241	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
242	C₄×C₈₉	—	2	1	no	—
243	C₄×C₈₉	—	2	2	no	—
244	C₄×C₈₉	—	2	1	no	—
245	C₄×C₈₉	—	2	1	no	—
246	C₄×C₈₉	—	4	1	no	—
247	C₄×C₈₉	—	4	1	no	—
248	C₄×C₈₉	—	4	1	no	—
249	C₄×C₈₉	—	8	1	no	—
250	C₄×C₈₉	#460	1	2	yes	—
251	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
252	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₄×C₈₉	#575	1	1	yes	characteristic
255	#19170	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂×C₈₉	—	2	2	no	—
257	C₂×C₈₉	—	16	1	no	—
258	C₂×C₈₉	—	16	2	no	—
259	C₂×C₈₉	C₂×#460	1	1	yes	characteristic
260	C₂×C₈₉	#3055	1	1	yes	characteristic
261	C₂×C₈₉	#3068	1	1	yes	characteristic
262	#3151	C₂×C₈₉	1	1	yes	characteristic
263	#3103	C₂×C₈₉	1	1	yes	characteristic, complemented
264	C₂×#460	C₂×C₈₉	1	1	yes	characteristic, complemented
265	Q₈×M₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
266	#1688	C₂×C₈₉	1	2	yes	complemented
267	#1686	C₂×C₈₉	1	2	yes	complemented
268	#1668	C₂×C₈₉	1	2	yes	complemented
269	#1666	C₂×C₈₉	1	2	yes	complemented
270	#1565	C₂×C₈₉	1	1	yes	characteristic, complemented
271	#1547	C₂×C₈₉	1	1	yes	characteristic, complemented
272	#1545	C₂×C₈₉	1	1	yes	characteristic, complemented
273	C₈₉	#19170	1	1	yes	characteristic, direct product factor, Sylow
274	C₂×AGL(1,ℤ/8)	—	2	1	no	—
275	C₂×AGL(1,ℤ/8)	—	2	1	no	—
276	C₂²×SD₁₆	—	2	1	no	—
277	#532	C₂²×C₈₉	1	1	yes	characteristic
278	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
279	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
280	#496	—	2	2	no	—
281	#493	C₂²×C₈₉	1	2	yes	—
282	#492	C₂²×C₈₉	1	2	yes	—
283	#481	—	2	1	no	—
284	#475	C₂²×C₈₉	1	1	yes	characteristic, complemented
285	#475	C₂²×C₈₉	1	1	yes	characteristic, complemented
286	#460	—	2	1	no	—
287	#460	—	2	1	no	—
288	#460	C₂²×C₈₉	1	2	yes	—
289	#460	C₂²×C₈₉	1	2	yes	—
290	#460	C₂²×C₈₉	1	1	yes	characteristic, complemented
291	#460	C₂²×C₈₉	1	1	yes	characteristic, complemented
292	#448	—	2	1	no	—
293	#446	—	2	2	no	—
294	#446	C₂²×C₈₉	1	2	yes	complemented
295	C₈×Q₈	C₂²×C₈₉	1	2	yes	complemented
296	C₄×SD₁₆	—	2	1	no	—
297	C₄×SD₁₆	—	2	1	no	—
298	#431	C₂²×C₈₉	1	1	yes	characteristic
299	#423	—	2	1	no	—
300	#423	C₂²×C₈₉	1	1	yes	characteristic
301	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
302	C₂×#102	—	2	1	no	—
303	C₄×M₁₆	—	2	1	no	—
304	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
305	#335	C₂²×C₈₉	1	2	yes	complemented
306	#334	C₂²×C₈₉	1	2	yes	complemented
307	#332	C₂²×C₈₉	1	2	yes	complemented
308	#331	C₂²×C₈₉	1	2	yes	—
309	#331	C₂²×C₈₉	1	2	yes	—
310	#331	C₂²×C₈₉	1	2	yes	complemented
311	C₂×#41	—	2	1	no	—
312	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	—	2	1	no	—
316	AGL(1,ℤ/8)	—	4	4	no	—
317	AGL(1,ℤ/8)	—	4	4	no	—
318	C₂×SD₁₆	—	2	2	no	—
319	C₂×SD₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	1	no	—
321	C₂×SD₁₆	—	2	1	no	—
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×SD₁₆	—	2	2	no	—
324	C₂×SD₁₆	—	4	4	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂×D₈	—	2	2	no	—
327	C₂×M₁₆	—	2	1	no	—
328	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
329	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
330	C₂²×C₈	—	2	1	no	—
331	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
332	#127	—	2	1	no	—
333	#127	—	2	1	no	—
334	#127	—	2	1	no	—
335	#127	C₂³×C₈₉	1	2	yes	—
336	#127	C₂³×C₈₉	1	1	yes	characteristic
337	#124	—	2	1	no	—
338	#121	—	2	1	no	—
339	#121	—	2	1	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
343	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
344	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
345	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
352	#106	—	2	1	no	—
353	#106	—	2	1	no	—
354	C₄⋊C₈	—	2	2	no	—
355	C₄⋊C₈	—	2	2	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
359	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
360	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
361	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
362	#103	—	2	1	no	—
363	#103	—	2	1	no	—
364	#102	—	2	2	no	—
365	#102	—	2	2	no	—
366	#102	—	2	1	no	—
367	#102	—	2	1	no	—
368	C₂²⋊C₈	—	2	1	no	—
369	#97	—	2	2	no	—
370	#97	D₄×C₈₉	1	2	yes	—
371	C₄×C₈	—	2	2	no	—
372	C₄×C₈	—	2	1	no	—
373	C₄×C₈	C₂³×C₈₉	1	2	yes	—
374	C₂⁴	—	4	1	no	—
375	#41	—	4	2	no	—
376	#41	—	4	2	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	4	1	no	—
381	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
382	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	—	4	1	no	—
392	C₂×D₄	—	4	1	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂×D₄	—	4	1	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	—	4	2	no	—
397	C₂×D₄	—	8	2	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
403	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
404	SD₁₆	—	4	4	no	—
405	SD₁₆	—	4	4	no	—
406	SD₁₆	—	4	2	no	—
407	SD₁₆	—	4	4	no	—
408	SD₁₆	—	4	2	no	—
409	SD₁₆	—	4	4	no	—
410	D₈	—	4	4	no	—
411	D₈	—	4	4	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	4	2	no	—
414	M₁₆	SD₁₆×C₈₉	1	4	yes	complemented
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	4	2	no	—
420	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
421	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
430	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
431	C₂²⋊C₄	—	4	1	no	—
432	C₂²⋊C₄	—	4	1	no	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	1	no	—
437	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
439	C₄²	#41×C₈₉	1	1	yes	characteristic
440	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
441	C₂³	—	4	1	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	8	2	no	—
444	C₂³	—	8	2	no	—
445	C₂³	—	8	2	no	—
446	C₂³	#121×C₈₉	1	1	yes	characteristic
447	Q₈	—	2	1	no	—
448	Q₈	—	2	1	no	—
449	Q₈	—	2	1	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	1	no	—
452	Q₈	—	4	1	no	—
453	Q₈	AGL(1,ℤ/8)×C₈₉	1	2	yes	complemented
454	D₄	—	4	1	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	1	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	2	no	—
462	D₄	—	8	1	no	—
463	D₄	—	8	2	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	2	no	—
466	D₄	—	8	2	no	—
467	D₄	—	8	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	8	1	no	—
478	C₂×C₄	—	8	2	no	—
479	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
480	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
481	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
482	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
483	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
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	C₈	—	2	2	no	—
488	C₈	—	4	2	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	1	no	—
491	C₈	—	4	1	no	—
492	C₈	C₂×SD₁₆×C₈₉	1	4	yes	complemented
493	C₂²	—	2	1	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	1	no	—
497	C₂²	—	8	1	no	—
498	C₂²	—	8	2	no	—
499	C₂²	—	8	2	no	—
500	C₂²	—	8	1	no	—
501	C₂²	—	16	4	no	—
502	C₂²	#460×C₈₉	1	2	yes	—
503	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
504	C₄	—	2	1	no	—
505	C₄	—	2	1	no	—
506	C₄	—	2	1	no	—
507	C₄	—	2	2	no	—
508	C₄	—	4	1	no	—
509	C₄	—	4	1	no	—
510	C₄	—	4	1	no	—
511	C₄	—	8	1	no	—
512	C₄	#460×C₈₉	1	2	yes	—
513	C₄	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
514	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
515	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
516	C₄	#575×C₈₉	1	1	yes	characteristic
517	C₂	—	2	2	no	—
518	C₂	—	16	1	no	—
519	C₂	—	16	2	no	—
520	C₂	C₂×#460×C₈₉	1	1	yes	characteristic
521	C₂	C₈₉×#3055	1	1	yes	characteristic
522	C₂	C₈₉×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
prime radical	C₈₉×#3151	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
2. C₈₉×#3103	C₂	semidirect
3. C₂×#460×C₈₉	C₂	semidirect
4. Q₈×M₁₆×C₈₉	C₂	semidirect
5. C₈₉×#1688	C₂	semidirect
6. C₈₉×#1686	C₂	semidirect
7. C₈₉×#1668	C₂	semidirect
8. C₈₉×#1666	C₂	semidirect
9. C₈₉×#1565	C₂	semidirect
10. C₈₉×#1547	C₂	semidirect
11. C₈₉×#1545	C₂	semidirect
22. #475×C₈₉	C₂²	semidirect
23. #475×C₈₉	C₂²	semidirect
28. #460×C₈₉	C₂²	semidirect
29. #460×C₈₉	C₂²	semidirect
32. #446×C₈₉	C₂²	semidirect
33. C₈×Q₈×C₈₉	C₂²	semidirect
43. #335×C₈₉	C₂²	semidirect
44. #334×C₈₉	C₂²	semidirect
45. #332×C₈₉	C₂²	semidirect
48. #331×C₈₉	C₂²	semidirect
152. M₁₆×C₈₉	SD₁₆	semidirect
191. Q₈×C₈₉	AGL(1,ℤ/8)	semidirect
230. C₈×C₈₉	C₂×SD₁₆	semidirect
255. #19170	C₈₉	direct (N × H)
263. #3103	C₂×C₈₉	semidirect
264. C₂×#460	C₂×C₈₉	semidirect
265. Q₈×M₁₆	C₂×C₈₉	semidirect
266. #1688	C₂×C₈₉	semidirect
267. #1686	C₂×C₈₉	semidirect
268. #1668	C₂×C₈₉	semidirect
269. #1666	C₂×C₈₉	semidirect
270. #1565	C₂×C₈₉	semidirect
271. #1547	C₂×C₈₉	semidirect
272. #1545	C₂×C₈₉	semidirect
273. C₈₉	#19170	direct (N × H)
284. #475	C₂²×C₈₉	semidirect
285. #475	C₂²×C₈₉	semidirect
290. #460	C₂²×C₈₉	semidirect
291. #460	C₂²×C₈₉	semidirect
294. #446	C₂²×C₈₉	semidirect
295. C₈×Q₈	C₂²×C₈₉	semidirect
305. #335	C₂²×C₈₉	semidirect
306. #334	C₂²×C₈₉	semidirect
307. #332	C₂²×C₈₉	semidirect
310. #331	C₂²×C₈₉	semidirect
414. M₁₆	SD₁₆×C₈₉	semidirect
453. Q₈	AGL(1,ℤ/8)×C₈₉	semidirect
492. C₈	C₂×SD₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3151	2
C₈₉×#3103	2
C₂×#460×C₈₉	2
Q₈×M₁₆×C₈₉	2
C₈₉×#1688	2
C₈₉×#1686	2
C₈₉×#1668	2
C₈₉×#1666	2
C₈₉×#1565	2
C₈₉×#1547	2
C₈₉×#1545	2
#19170	89

Subgroups — C89×#19170 #36397018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#19170 #36397018