Subgroups — C₈₉×#60489 #36438337

Order 22784.

2002 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3428	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3084	C₂	1	4	yes	maximal, complemented
4	C₈₉×#3083	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3082	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3081	C₂	1	4	yes	maximal, complemented
7	C₈₉×#3076	C₂	1	8	yes	maximal, complemented
8	C₈₉×#2911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#427×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2896	C₂	1	1	yes	maximal, characteristic
12	C₂×#416×C₈₉	C₂	1	1	yes	maximal, characteristic
13	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#544×C₈₉	—	2	2	no	—
17	#544×C₈₉	C₂²	1	4	yes	—
18	#544×C₈₉	C₂²	1	4	yes	complemented
19	#543×C₈₉	—	2	2	no	—
20	#533×C₈₉	—	2	2	no	—
21	#533×C₈₉	—	2	2	no	—
22	#533×C₈₉	C₂²	1	4	yes	complemented
23	#520×C₈₉	—	2	2	no	—
24	#520×C₈₉	C₂²	1	4	yes	—
25	#519×C₈₉	C₂²	1	2	yes	—
26	#518×C₈₉	—	2	2	no	—
27	#518×C₈₉	C₂²	1	2	yes	—
28	C₄×#41×C₈₉	—	2	2	no	—
29	C₄×#41×C₈₉	C₂²	1	4	yes	—
30	C₂×#117×C₈₉	—	2	2	no	—
31	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
35	#479×C₈₉	C₂²	1	8	yes	complemented
36	#479×C₈₉	C₂²	1	8	yes	complemented
37	#478×C₈₉	C₂²	1	8	yes	complemented
38	#478×C₈₉	C₂²	1	8	yes	complemented
39	#477×C₈₉	C₂²	1	8	yes	complemented
40	#476×C₈₉	C₂²	1	8	yes	complemented
41	#475×C₈₉	C₂²	1	8	yes	complemented
42	#474×C₈₉	C₂²	1	8	yes	complemented
43	#428×C₈₉	C₂²	1	4	yes	—
44	#428×C₈₉	C₂²	1	4	yes	complemented
45	#427×C₈₉	C₂²	1	4	yes	—
46	#427×C₈₉	C₂²	1	4	yes	complemented
47	#427×C₈₉	C₂²	1	4	yes	complemented
48	C₂×#107×C₈₉	C₂²	1	2	yes	complemented
49	C₂×#106×C₈₉	C₂²	1	2	yes	complemented
50	#424×C₈₉	C₂²	1	4	yes	—
51	#424×C₈₉	C₂²	1	2	yes	complemented
52	#424×C₈₉	C₂²	1	2	yes	complemented
53	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	complemented
54	#419×C₈₉	C₂²	1	2	yes	—
55	#418×C₈₉	C₂²	1	2	yes	—
56	#417×C₈₉	C₂²	1	4	yes	—
57	#416×C₈₉	C₂²	1	4	yes	—
58	#416×C₈₉	C₂²	1	4	yes	—
59	#416×C₈₉	C₂²	1	2	yes	—
60	#416×C₈₉	C₂²	1	2	yes	—
61	C₂×#103×C₈₉	C₂²	1	2	yes	—
62	C₂×#102×C₈₉	C₂²	1	2	yes	—
63	C₂×#41×C₈₉	—	2	1	no	—
64	C₂×#41×C₈₉	—	2	1	no	—
65	C₂×#41×C₈₉	—	2	4	no	—
66	C₂×#41×C₈₉	Q₈	1	8	yes	complemented
67	C₂×#41×C₈₉	C₂³	1	2	yes	—
68	C₂×#41×C₈₉	C₂³	1	2	yes	—
69	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₈₉	—	2	1	no	—
71	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₈₉	—	2	1	no	—
73	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₈₉	—	2	4	no	—
75	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
76	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
77	C₂²×C₈×C₈₉	—	2	4	no	—
78	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
80	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
81	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₈₉	—	2	4	no	—
84	C₄⋊Q₈×C₈₉	—	2	4	no	—
85	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
86	#125×C₈₉	—	2	4	no	—
87	#125×C₈₉	—	2	4	no	—
88	#125×C₈₉	C₂³	1	8	yes	—
89	C₂²⋊Q₈×C₈₉	—	2	8	no	—
90	C₂²⋊Q₈×C₈₉	—	2	8	no	—
91	C₂²⋊Q₈×C₈₉	—	2	8	no	—
92	C₄×Q₈×C₈₉	—	2	4	no	—
93	C₄×Q₈×C₈₉	—	2	8	no	—
94	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
95	C₄×D₄×C₈₉	—	2	8	no	—
96	C₄×D₄×C₈₉	—	2	4	no	—
97	C₄×D₄×C₈₉	—	2	8	no	—
98	C₄×D₄×C₈₉	C₂³	1	8	yes	—
99	#117×C₈₉	—	2	1	no	—
100	#117×C₈₉	—	2	1	no	—
101	#117×C₈₉	—	2	1	no	—
102	#117×C₈₉	—	2	1	no	—
103	#117×C₈₉	—	2	1	no	—
104	#117×C₈₉	—	2	1	no	—
105	#117×C₈₉	—	2	4	no	—
106	#117×C₈₉	—	2	4	no	—
107	#117×C₈₉	—	2	4	no	—
108	#117×C₈₉	D₄	1	2	yes	—
109	#117×C₈₉	D₄	1	2	yes	—
110	#117×C₈₉	C₂³	1	2	yes	—
111	#117×C₈₉	C₂³	1	2	yes	—
112	#117×C₈₉	C₂³	1	2	yes	—
113	#117×C₈₉	C₂³	1	2	yes	—
114	#117×C₈₉	C₂³	1	2	yes	—
115	#117×C₈₉	C₂³	1	2	yes	—
116	#117×C₈₉	C₂³	1	2	yes	—
117	#117×C₈₉	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
119	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
120	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
127	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
128	C₂×C₄²×C₈₉	—	2	1	no	—
129	C₂×C₄²×C₈₉	—	2	1	no	—
130	C₂×C₄²×C₈₉	—	2	1	no	—
131	C₂×C₄²×C₈₉	—	2	4	no	—
132	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
133	#107×C₈₉	C₂³	1	4	yes	—
134	#107×C₈₉	C₂³	1	4	yes	—
135	#107×C₈₉	C₂³	1	8	yes	—
136	#106×C₈₉	C₂³	1	4	yes	—
137	#106×C₈₉	C₂³	1	4	yes	—
138	#106×C₈₉	C₂³	1	8	yes	—
139	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
140	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
141	#103×C₈₉	C₂³	1	4	yes	—
142	#103×C₈₉	C₂³	1	4	yes	—
143	#103×C₈₉	C₂³	1	8	yes	—
144	#102×C₈₉	C₂³	1	4	yes	—
145	#102×C₈₉	C₂³	1	4	yes	—
146	#102×C₈₉	C₂³	1	8	yes	—
147	C₂⁴×C₈₉	—	2	1	no	—
148	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
149	#41×C₈₉	—	2	8	no	—
150	#41×C₈₉	—	2	8	no	—
151	#41×C₈₉	—	2	4	no	—
152	#41×C₈₉	—	2	4	no	—
153	#41×C₈₉	C₂×Q₈	1	8	yes	—
154	#41×C₈₉	C₂×Q₈	1	8	yes	—
155	C₂×Q₈×C₈₉	—	2	1	no	—
156	C₂×Q₈×C₈₉	—	2	8	no	—
157	C₂×Q₈×C₈₉	—	2	2	no	—
158	C₂×Q₈×C₈₉	C₂×Q₈	1	4	yes	—
159	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
160	C₂×D₄×C₈₉	—	2	8	no	—
161	C₂×D₄×C₈₉	—	2	2	no	—
162	C₂×D₄×C₈₉	—	2	2	no	—
163	C₂×D₄×C₈₉	—	2	1	no	—
164	C₂×D₄×C₈₉	—	2	2	no	—
165	C₂×D₄×C₈₉	C₂×Q₈	1	4	yes	—
166	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₈₉	—	2	1	no	—
168	C₂²×C₄×C₈₉	—	2	8	no	—
169	C₂²×C₄×C₈₉	—	2	2	no	—
170	C₂²×C₄×C₈₉	—	2	2	no	—
171	C₂²×C₄×C₈₉	—	2	2	no	—
172	C₂²×C₄×C₈₉	—	2	2	no	—
173	C₂²×C₄×C₈₉	—	2	2	no	—
174	C₂²×C₄×C₈₉	—	2	8	no	—
175	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
176	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
180	C₂²×C₄×C₈₉	#41	1	4	yes	—
181	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
184	M₁₆×C₈₉	—	2	4	no	—
185	M₁₆×C₈₉	—	2	4	no	—
186	C₂×C₈×C₈₉	—	2	4	no	—
187	C₂×C₈×C₈₉	—	2	4	no	—
188	C₂×C₈×C₈₉	—	2	2	no	—
189	C₂×C₈×C₈₉	—	2	2	no	—
190	C₂×C₈×C₈₉	—	2	4	no	—
191	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
192	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
193	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₈₉	—	2	2	no	—
195	C₄⋊C₄×C₈₉	—	2	8	no	—
196	C₄⋊C₄×C₈₉	—	2	2	no	—
197	C₄⋊C₄×C₈₉	—	2	2	no	—
198	C₄⋊C₄×C₈₉	—	2	8	no	—
199	C₄⋊C₄×C₈₉	—	2	8	no	—
200	C₄⋊C₄×C₈₉	—	2	8	no	—
201	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
202	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
203	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
204	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
205	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
206	C₂²⋊C₄×C₈₉	—	2	2	no	—
207	C₂²⋊C₄×C₈₉	—	2	2	no	—
208	C₂²⋊C₄×C₈₉	—	2	2	no	—
209	C₂²⋊C₄×C₈₉	—	2	2	no	—
210	C₂²⋊C₄×C₈₉	—	2	2	no	—
211	C₂²⋊C₄×C₈₉	—	2	2	no	—
212	C₂²⋊C₄×C₈₉	—	2	8	no	—
213	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
214	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
215	C₄²×C₈₉	—	2	8	no	—
216	C₄²×C₈₉	—	2	2	no	—
217	C₄²×C₈₉	—	2	2	no	—
218	C₄²×C₈₉	—	2	2	no	—
219	C₄²×C₈₉	—	2	2	no	—
220	C₄²×C₈₉	—	2	2	no	—
221	C₄²×C₈₉	—	2	2	no	—
222	C₄²×C₈₉	C₂×D₄	1	4	yes	—
223	C₄²×C₈₉	C₂⁴	1	4	yes	—
224	C₂³×C₈₉	—	2	2	no	—
225	C₂³×C₈₉	—	2	4	no	—
226	C₂³×C₈₉	—	4	2	no	—
227	C₂³×C₈₉	—	8	2	no	—
228	C₂³×C₈₉	C₂²⋊Q₈	1	4	yes	—
229	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
230	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
231	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₈₉	—	2	4	no	—
233	Q₈×C₈₉	—	2	2	no	—
234	Q₈×C₈₉	—	8	8	no	—
235	Q₈×C₈₉	C₂²×Q₈	1	4	yes	—
236	D₄×C₈₉	—	2	8	no	—
237	D₄×C₈₉	—	2	8	no	—
238	D₄×C₈₉	—	2	4	no	—
239	D₄×C₈₉	—	2	2	no	—
240	D₄×C₈₉	C₂²×Q₈	1	4	yes	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	2	2	no	—
243	C₂×C₄×C₈₉	—	2	8	no	—
244	C₂×C₄×C₈₉	—	2	8	no	—
245	C₂×C₄×C₈₉	—	2	4	no	—
246	C₂×C₄×C₈₉	—	2	4	no	—
247	C₂×C₄×C₈₉	—	2	4	no	—
248	C₂×C₄×C₈₉	—	4	4	no	—
249	C₂×C₄×C₈₉	—	8	8	no	—
250	C₂×C₄×C₈₉	—	8	2	no	—
251	C₂×C₄×C₈₉	—	8	2	no	—
252	C₂×C₄×C₈₉	—	8	2	no	—
253	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
254	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
255	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
256	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
257	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
261	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
262	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
263	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
264	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
265	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
266	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
267	C₈×C₈₉	—	2	2	no	—
268	C₈×C₈₉	—	2	2	no	—
269	C₈×C₈₉	—	2	4	no	—
270	C₂²×C₈₉	—	2	2	no	—
271	C₂²×C₈₉	—	2	2	no	—
272	C₂²×C₈₉	—	2	2	no	—
273	C₂²×C₈₉	—	2	4	no	—
274	C₂²×C₈₉	—	4	4	no	—
275	C₂²×C₈₉	—	4	4	no	—
276	C₂²×C₈₉	—	8	4	no	—
277	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
278	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
279	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
280	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
281	C₄×C₈₉	—	2	8	no	—
282	C₄×C₈₉	—	2	4	no	—
283	C₄×C₈₉	—	4	4	no	—
284	C₄×C₈₉	—	4	4	no	—
285	C₄×C₈₉	—	4	4	no	—
286	C₄×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
287	C₄×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
288	C₄×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
289	C₄×C₈₉	C₂³×Q₈	1	1	yes	characteristic
290	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
291	#60489	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₈₉	—	2	2	no	—
293	C₂×C₈₉	—	2	2	no	—
294	C₂×C₈₉	—	2	2	no	—
295	C₂×C₈₉	—	4	4	no	—
296	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
297	C₂×C₈₉	#3584	1	2	yes	—
298	#3458	C₂×C₈₉	1	1	yes	characteristic, complemented
299	#3428	C₂×C₈₉	1	1	yes	characteristic
300	#3084	C₂×C₈₉	1	4	yes	complemented
301	#3083	C₂×C₈₉	1	4	yes	complemented
302	#3082	C₂×C₈₉	1	4	yes	complemented
303	#3081	C₂×C₈₉	1	4	yes	complemented
304	#3076	C₂×C₈₉	1	8	yes	complemented
305	#2911	C₂×C₈₉	1	1	yes	characteristic, complemented
306	C₂×#427	C₂×C₈₉	1	1	yes	characteristic, complemented
307	C₂×#424	C₂×C₈₉	1	1	yes	characteristic, complemented
308	#2896	C₂×C₈₉	1	1	yes	characteristic
309	C₂×#416	C₂×C₈₉	1	1	yes	characteristic
310	C₈₉	#60489	1	1	yes	characteristic, direct product factor, Sylow
311	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
312	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
313	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
314	#544	—	2	2	no	—
315	#544	C₂²×C₈₉	1	4	yes	—
316	#544	C₂²×C₈₉	1	4	yes	complemented
317	#543	—	2	2	no	—
318	#533	—	2	2	no	—
319	#533	—	2	2	no	—
320	#533	C₂²×C₈₉	1	4	yes	complemented
321	#520	—	2	2	no	—
322	#520	C₂²×C₈₉	1	4	yes	—
323	#519	C₂²×C₈₉	1	2	yes	—
324	#518	—	2	2	no	—
325	#518	C₂²×C₈₉	1	2	yes	—
326	C₄×#41	—	2	2	no	—
327	C₄×#41	C₂²×C₈₉	1	4	yes	—
328	C₂×#117	—	2	2	no	—
329	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
330	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
331	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
332	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
333	#479	C₂²×C₈₉	1	8	yes	complemented
334	#479	C₂²×C₈₉	1	8	yes	complemented
335	#478	C₂²×C₈₉	1	8	yes	complemented
336	#478	C₂²×C₈₉	1	8	yes	complemented
337	#477	C₂²×C₈₉	1	8	yes	complemented
338	#476	C₂²×C₈₉	1	8	yes	complemented
339	#475	C₂²×C₈₉	1	8	yes	complemented
340	#474	C₂²×C₈₉	1	8	yes	complemented
341	#428	C₂²×C₈₉	1	4	yes	—
342	#428	C₂²×C₈₉	1	4	yes	complemented
343	#427	C₂²×C₈₉	1	4	yes	—
344	#427	C₂²×C₈₉	1	4	yes	complemented
345	#427	C₂²×C₈₉	1	4	yes	complemented
346	C₂×#107	C₂²×C₈₉	1	2	yes	complemented
347	C₂×#106	C₂²×C₈₉	1	2	yes	complemented
348	#424	C₂²×C₈₉	1	4	yes	—
349	#424	C₂²×C₈₉	1	2	yes	complemented
350	#424	C₂²×C₈₉	1	2	yes	complemented
351	C₂×C₄⋊C₈	C₂²×C₈₉	1	4	yes	complemented
352	#419	C₂²×C₈₉	1	2	yes	—
353	#418	C₂²×C₈₉	1	2	yes	—
354	#417	C₂²×C₈₉	1	4	yes	—
355	#416	C₂²×C₈₉	1	4	yes	—
356	#416	C₂²×C₈₉	1	4	yes	—
357	#416	C₂²×C₈₉	1	2	yes	—
358	#416	C₂²×C₈₉	1	2	yes	—
359	C₂×#103	C₂²×C₈₉	1	2	yes	—
360	C₂×#102	C₂²×C₈₉	1	2	yes	—
361	C₂×#41	—	2	1	no	—
362	C₂×#41	—	2	1	no	—
363	C₂×#41	—	2	4	no	—
364	C₂×#41	Q₈×C₈₉	1	8	yes	complemented
365	C₂×#41	C₂³×C₈₉	1	2	yes	—
366	C₂×#41	C₂³×C₈₉	1	2	yes	—
367	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
368	C₂²×D₄	—	2	1	no	—
369	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
370	C₂³×C₄	—	2	1	no	—
371	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
372	C₂×M₁₆	—	2	4	no	—
373	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
374	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
375	C₂²×C₈	—	2	4	no	—
376	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
377	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
378	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
380	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
381	C₄⋊Q₈	—	2	4	no	—
382	C₄⋊Q₈	—	2	4	no	—
383	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
384	#125	—	2	4	no	—
385	#125	—	2	4	no	—
386	#125	C₂³×C₈₉	1	8	yes	—
387	C₂²⋊Q₈	—	2	8	no	—
388	C₂²⋊Q₈	—	2	8	no	—
389	C₂²⋊Q₈	—	2	8	no	—
390	C₄×Q₈	—	2	4	no	—
391	C₄×Q₈	—	2	8	no	—
392	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
393	C₄×D₄	—	2	8	no	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	8	no	—
396	C₄×D₄	C₂³×C₈₉	1	8	yes	—
397	#117	—	2	1	no	—
398	#117	—	2	1	no	—
399	#117	—	2	1	no	—
400	#117	—	2	1	no	—
401	#117	—	2	1	no	—
402	#117	—	2	1	no	—
403	#117	—	2	4	no	—
404	#117	—	2	4	no	—
405	#117	—	2	4	no	—
406	#117	D₄×C₈₉	1	2	yes	—
407	#117	D₄×C₈₉	1	2	yes	—
408	#117	C₂³×C₈₉	1	2	yes	—
409	#117	C₂³×C₈₉	1	2	yes	—
410	#117	C₂³×C₈₉	1	2	yes	—
411	#117	C₂³×C₈₉	1	2	yes	—
412	#117	C₂³×C₈₉	1	2	yes	—
413	#117	C₂³×C₈₉	1	2	yes	—
414	#117	C₂³×C₈₉	1	2	yes	—
415	#117	C₂³×C₈₉	1	2	yes	—
416	C₂×C₄⋊C₄	—	2	4	no	—
417	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	4	no	—
425	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	—	2	1	no	—
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	4	no	—
430	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
431	#107	C₂³×C₈₉	1	4	yes	—
432	#107	C₂³×C₈₉	1	4	yes	—
433	#107	C₂³×C₈₉	1	8	yes	—
434	#106	C₂³×C₈₉	1	4	yes	—
435	#106	C₂³×C₈₉	1	4	yes	—
436	#106	C₂³×C₈₉	1	8	yes	—
437	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
438	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
439	#103	C₂³×C₈₉	1	4	yes	—
440	#103	C₂³×C₈₉	1	4	yes	—
441	#103	C₂³×C₈₉	1	8	yes	—
442	#102	C₂³×C₈₉	1	4	yes	—
443	#102	C₂³×C₈₉	1	4	yes	—
444	#102	C₂³×C₈₉	1	8	yes	—
445	C₂⁴	—	2	1	no	—
446	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
447	#41	—	2	8	no	—
448	#41	—	2	8	no	—
449	#41	—	2	4	no	—
450	#41	—	2	4	no	—
451	#41	C₂×Q₈×C₈₉	1	8	yes	—
452	#41	C₂×Q₈×C₈₉	1	8	yes	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	8	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	C₂×Q₈×C₈₉	1	4	yes	—
457	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
458	C₂×D₄	—	2	8	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	C₂×Q₈×C₈₉	1	4	yes	—
464	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
474	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
477	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
478	C₂²×C₄	#41×C₈₉	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
482	M₁₆	—	2	4	no	—
483	M₁₆	—	2	4	no	—
484	C₂×C₈	—	2	4	no	—
485	C₂×C₈	—	2	4	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	4	no	—
489	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
490	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
491	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	8	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	8	no	—
497	C₄⋊C₄	—	2	8	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
500	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
501	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
502	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
503	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	8	no	—
511	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
512	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	8	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	C₂×D₄×C₈₉	1	4	yes	—
521	C₄²	C₂⁴×C₈₉	1	4	yes	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	8	2	no	—
526	C₂³	C₂²⋊Q₈×C₈₉	1	4	yes	—
527	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
528	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
529	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
530	Q₈	—	2	4	no	—
531	Q₈	—	2	2	no	—
532	Q₈	—	8	8	no	—
533	Q₈	C₂²×Q₈×C₈₉	1	4	yes	—
534	D₄	—	2	8	no	—
535	D₄	—	2	8	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	2	no	—
538	D₄	C₂²×Q₈×C₈₉	1	4	yes	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	8	8	no	—
548	C₂×C₄	—	8	2	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	—	8	2	no	—
551	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
552	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
553	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
554	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
555	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
556	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
557	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
558	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
559	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
560	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
561	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
562	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
563	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
564	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
565	C₈	—	2	2	no	—
566	C₈	—	2	2	no	—
567	C₈	—	2	4	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	8	4	no	—
575	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
576	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
577	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
578	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
579	C₄	—	2	8	no	—
580	C₄	—	2	4	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	4	no	—
584	C₄	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
585	C₄	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
586	C₄	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
587	C₄	C₂³×Q₈×C₈₉	1	1	yes	characteristic
588	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
589	C₂	—	2	2	no	—
590	C₂	—	2	2	no	—
591	C₂	—	2	2	no	—
592	C₂	—	4	4	no	—
593	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
594	C₂	C₈₉×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂³×D₄
prime radical	C₂²×#41×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3458	C₂	semidirect
3. C₈₉×#3084	C₂	semidirect
4. C₈₉×#3083	C₂	semidirect
5. C₈₉×#3082	C₂	semidirect
6. C₈₉×#3081	C₂	semidirect
7. C₈₉×#3076	C₂	semidirect
8. C₈₉×#2911	C₂	semidirect
9. C₂×#427×C₈₉	C₂	semidirect
10. C₂×#424×C₈₉	C₂	semidirect
18. #544×C₈₉	C₂²	semidirect
22. #533×C₈₉	C₂²	semidirect
35. #479×C₈₉	C₂²	semidirect
36. #479×C₈₉	C₂²	semidirect
37. #478×C₈₉	C₂²	semidirect
38. #478×C₈₉	C₂²	semidirect
39. #477×C₈₉	C₂²	semidirect
40. #476×C₈₉	C₂²	semidirect
41. #475×C₈₉	C₂²	semidirect
42. #474×C₈₉	C₂²	semidirect
44. #428×C₈₉	C₂²	semidirect
46. #427×C₈₉	C₂²	semidirect
47. #427×C₈₉	C₂²	semidirect
48. C₂×#107×C₈₉	C₂²	semidirect
49. C₂×#106×C₈₉	C₂²	semidirect
51. #424×C₈₉	C₂²	semidirect
52. #424×C₈₉	C₂²	semidirect
53. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
66. C₂×#41×C₈₉	Q₈	semidirect
291. #60489	C₈₉	direct (N × H)
298. #3458	C₂×C₈₉	semidirect
300. #3084	C₂×C₈₉	semidirect
301. #3083	C₂×C₈₉	semidirect
302. #3082	C₂×C₈₉	semidirect
303. #3081	C₂×C₈₉	semidirect
304. #3076	C₂×C₈₉	semidirect
305. #2911	C₂×C₈₉	semidirect
306. C₂×#427	C₂×C₈₉	semidirect
307. C₂×#424	C₂×C₈₉	semidirect
310. C₈₉	#60489	direct (N × H)
316. #544	C₂²×C₈₉	semidirect
320. #533	C₂²×C₈₉	semidirect
333. #479	C₂²×C₈₉	semidirect
334. #479	C₂²×C₈₉	semidirect
335. #478	C₂²×C₈₉	semidirect
336. #478	C₂²×C₈₉	semidirect
337. #477	C₂²×C₈₉	semidirect
338. #476	C₂²×C₈₉	semidirect
339. #475	C₂²×C₈₉	semidirect
340. #474	C₂²×C₈₉	semidirect
342. #428	C₂²×C₈₉	semidirect
344. #427	C₂²×C₈₉	semidirect
345. #427	C₂²×C₈₉	semidirect
346. C₂×#107	C₂²×C₈₉	semidirect
347. C₂×#106	C₂²×C₈₉	semidirect
349. #424	C₂²×C₈₉	semidirect
350. #424	C₂²×C₈₉	semidirect
351. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
364. C₂×#41	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3458	2
C₈₉×#3428	2
C₈₉×#3084	2
C₈₉×#3083	2
C₈₉×#3082	2
C₈₉×#3081	2
C₈₉×#3076	2
C₈₉×#2911	2
C₂×#427×C₈₉	2
C₂×#424×C₈₉	2
C₈₉×#2896	2
C₂×#416×C₈₉	2
#60489	89

Subgroups — C89×#60489 #36438337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60489 #36438337