Subgroups — C₈₉×#25514 #36403362

Order 22784.

1092 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#542×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#468×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#467×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2625	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2057	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2042	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2024	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2015	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1894	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1893	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1878	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1877	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#1863	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1862	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈₉×#1729	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#542×C₈₉	—	2	4	no	—
21	#542×C₈₉	—	2	4	no	—
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#126×C₈₉	—	2	1	no	—
24	C₂×#124×C₈₉	—	2	1	no	—
25	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#123×C₈₉	—	2	1	no	—
27	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
28	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
34	#468×C₈₉	—	2	4	no	—
35	#467×C₈₉	—	2	4	no	—
36	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
46	#400×C₈₉	—	2	1	no	—
47	#398×C₈₉	—	2	1	no	—
48	#397×C₈₉	—	2	1	no	—
49	#397×C₈₉	—	2	1	no	—
50	#391×C₈₉	C₂²	1	1	yes	characteristic
51	#391×C₈₉	C₂²	1	1	yes	characteristic
52	#389×C₈₉	C₂²	1	1	yes	characteristic
53	#389×C₈₉	C₂²	1	1	yes	characteristic
54	#380×C₈₉	C₂²	1	1	yes	characteristic
55	#340×C₈₉	C₂²	1	1	yes	characteristic
56	#340×C₈₉	C₂²	1	1	yes	characteristic
57	#340×C₈₉	C₂²	1	1	yes	characteristic
58	#340×C₈₉	C₂²	1	1	yes	characteristic
59	#336×C₈₉	C₂²	1	1	yes	characteristic
60	#336×C₈₉	C₂²	1	1	yes	characteristic
61	#336×C₈₉	C₂²	1	1	yes	characteristic
62	#336×C₈₉	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₈₉	—	2	1	no	—
64	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₈₉	—	2	1	no	—
67	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₈₉	—	2	1	no	—
69	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₈₉	—	2	2	no	—
71	C₂×Q₁₆×C₈₉	—	2	2	no	—
72	C₂×Q₁₆×C₈₉	—	4	1	no	—
73	C₂×SD₁₆×C₈₉	—	2	2	no	—
74	C₂×SD₁₆×C₈₉	—	2	2	no	—
75	C₂×SD₁₆×C₈₉	—	4	1	no	—
76	C₂×M₁₆×C₈₉	—	2	2	no	—
77	C₂×M₁₆×C₈₉	—	2	2	no	—
78	C₂×M₁₆×C₈₉	—	4	1	no	—
79	C₂²×C₈×C₈₉	—	2	2	no	—
80	C₂²×C₈×C₈₉	—	2	2	no	—
81	C₂²×C₈×C₈₉	—	4	1	no	—
82	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈×C₈₉	—	2	2	no	—
87	C₄⋊Q₈×C₈₉	—	2	2	no	—
88	#126×C₈₉	—	2	8	no	—
89	#124×C₈₉	—	2	8	no	—
90	#124×C₈₉	—	2	2	no	—
91	#124×C₈₉	—	2	2	no	—
92	#123×C₈₉	—	2	8	no	—
93	C₂²⋊Q₈×C₈₉	—	2	8	no	—
94	C₂²⋊Q₈×C₈₉	—	2	2	no	—
95	C₂²⋊Q₈×C₈₉	—	2	2	no	—
96	C₂²⋊Q₈×C₈₉	—	2	2	no	—
97	C₂²⋊Q₈×C₈₉	D₄	1	4	yes	—
98	#121×C₈₉	—	2	2	no	—
99	#121×C₈₉	D₄	1	4	yes	—
100	C₄×Q₈×C₈₉	—	2	2	no	—
101	C₄×Q₈×C₈₉	—	2	2	no	—
102	C₄×D₄×C₈₉	—	2	2	no	—
103	C₄×D₄×C₈₉	—	2	2	no	—
104	#117×C₈₉	—	2	2	no	—
105	#117×C₈₉	—	2	2	no	—
106	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
127	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₈₉	—	2	1	no	—
129	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
131	#107×C₈₉	—	2	2	no	—
132	#107×C₈₉	—	2	2	no	—
133	#103×C₈₉	—	2	2	no	—
134	#103×C₈₉	—	2	2	no	—
135	#103×C₈₉	—	2	2	no	—
136	#103×C₈₉	—	2	2	no	—
137	#103×C₈₉	—	4	1	no	—
138	#102×C₈₉	—	2	2	no	—
139	#102×C₈₉	—	2	2	no	—
140	#102×C₈₉	—	4	1	no	—
141	#95×C₈₉	—	2	1	no	—
142	#95×C₈₉	—	2	1	no	—
143	#95×C₈₉	—	2	1	no	—
144	#95×C₈₉	—	2	1	no	—
145	#95×C₈₉	—	2	1	no	—
146	#95×C₈₉	—	2	1	no	—
147	#95×C₈₉	—	2	1	no	—
148	C₂⁴×C₈₉	—	2	1	no	—
149	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₈₉	—	2	4	no	—
151	C₂×Q₈×C₈₉	—	2	1	no	—
152	C₂×Q₈×C₈₉	—	2	1	no	—
153	C₂×Q₈×C₈₉	—	2	1	no	—
154	C₂×Q₈×C₈₉	—	2	1	no	—
155	C₂×Q₈×C₈₉	—	2	1	no	—
156	C₂×Q₈×C₈₉	—	2	1	no	—
157	C₂×Q₈×C₈₉	—	2	1	no	—
158	C₂×Q₈×C₈₉	—	2	1	no	—
159	C₂×Q₈×C₈₉	—	2	1	no	—
160	C₂×Q₈×C₈₉	—	2	1	no	—
161	C₂×Q₈×C₈₉	—	2	1	no	—
162	C₂×Q₈×C₈₉	—	8	1	no	—
163	C₂×Q₈×C₈₉	—	8	1	no	—
164	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
165	C₂×D₄×C₈₉	—	2	4	no	—
166	C₂×D₄×C₈₉	—	2	1	no	—
167	C₂×D₄×C₈₉	—	2	1	no	—
168	C₂×D₄×C₈₉	—	2	1	no	—
169	C₂×D₄×C₈₉	—	2	1	no	—
170	C₂×D₄×C₈₉	—	2	1	no	—
171	C₂×D₄×C₈₉	—	8	1	no	—
172	C₂×D₄×C₈₉	—	8	1	no	—
173	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₈₉	—	2	1	no	—
175	C₂²×C₄×C₈₉	—	2	1	no	—
176	C₂²×C₄×C₈₉	—	2	1	no	—
177	C₂²×C₄×C₈₉	—	2	1	no	—
178	C₂²×C₄×C₈₉	—	2	1	no	—
179	C₂²×C₄×C₈₉	—	2	1	no	—
180	C₂²×C₄×C₈₉	—	2	1	no	—
181	C₂²×C₄×C₈₉	—	2	4	no	—
182	C₂²×C₄×C₈₉	—	2	1	no	—
183	C₂²×C₄×C₈₉	—	2	1	no	—
184	C₂²×C₄×C₈₉	—	2	1	no	—
185	C₂²×C₄×C₈₉	—	2	1	no	—
186	C₂²×C₄×C₈₉	—	8	1	no	—
187	C₂²×C₄×C₈₉	—	8	1	no	—
188	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
196	Q₁₆×C₈₉	—	4	4	no	—
197	SD₁₆×C₈₉	—	4	4	no	—
198	M₁₆×C₈₉	—	4	4	no	—
199	C₂×C₈×C₈₉	—	2	1	no	—
200	C₂×C₈×C₈₉	—	2	1	no	—
201	C₂×C₈×C₈₉	—	2	1	no	—
202	C₂×C₈×C₈₉	—	2	1	no	—
203	C₂×C₈×C₈₉	—	2	1	no	—
204	C₂×C₈×C₈₉	—	2	1	no	—
205	C₂×C₈×C₈₉	—	2	1	no	—
206	C₂×C₈×C₈₉	—	2	1	no	—
207	C₂×C₈×C₈₉	—	2	1	no	—
208	C₂×C₈×C₈₉	—	2	1	no	—
209	C₂×C₈×C₈₉	—	4	4	no	—
210	C₂×C₈×C₈₉	#41	1	2	yes	—
211	C₂×C₈×C₈₉	#41	1	2	yes	—
212	C₄⋊C₄×C₈₉	—	2	4	no	—
213	C₄⋊C₄×C₈₉	—	2	4	no	—
214	C₄⋊C₄×C₈₉	—	2	4	no	—
215	C₄⋊C₄×C₈₉	—	2	2	no	—
216	C₄⋊C₄×C₈₉	—	2	2	no	—
217	C₄⋊C₄×C₈₉	—	2	1	no	—
218	C₄⋊C₄×C₈₉	—	2	1	no	—
219	C₄⋊C₄×C₈₉	—	2	1	no	—
220	C₄⋊C₄×C₈₉	—	2	1	no	—
221	C₄⋊C₄×C₈₉	—	2	1	no	—
222	C₄⋊C₄×C₈₉	—	2	1	no	—
223	C₄⋊C₄×C₈₉	—	2	1	no	—
224	C₄⋊C₄×C₈₉	—	2	1	no	—
225	C₄⋊C₄×C₈₉	—	4	2	no	—
226	C₄⋊C₄×C₈₉	—	8	1	no	—
227	C₄⋊C₄×C₈₉	—	8	1	no	—
228	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
229	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
230	C₂²⋊C₄×C₈₉	—	2	4	no	—
231	C₂²⋊C₄×C₈₉	—	2	4	no	—
232	C₂²⋊C₄×C₈₉	—	2	4	no	—
233	C₂²⋊C₄×C₈₉	—	2	4	no	—
234	C₂²⋊C₄×C₈₉	—	2	2	no	—
235	C₂²⋊C₄×C₈₉	—	2	2	no	—
236	C₂²⋊C₄×C₈₉	—	2	4	no	—
237	C₂²⋊C₄×C₈₉	—	4	2	no	—
238	C₂²⋊C₄×C₈₉	—	8	1	no	—
239	C₄²×C₈₉	—	2	4	no	—
240	C₄²×C₈₉	—	2	1	no	—
241	C₄²×C₈₉	—	2	1	no	—
242	C₄²×C₈₉	—	2	1	no	—
243	C₄²×C₈₉	—	2	1	no	—
244	C₂³×C₈₉	—	2	1	no	—
245	C₂³×C₈₉	—	2	1	no	—
246	C₂³×C₈₉	—	2	1	no	—
247	C₂³×C₈₉	—	2	1	no	—
248	C₂³×C₈₉	—	2	1	no	—
249	C₂³×C₈₉	—	2	1	no	—
250	C₂³×C₈₉	—	2	2	no	—
251	C₂³×C₈₉	—	4	1	no	—
252	C₂³×C₈₉	—	4	1	no	—
253	C₂³×C₈₉	—	4	1	no	—
254	C₂³×C₈₉	—	4	1	no	—
255	C₂³×C₈₉	—	4	1	no	—
256	C₂³×C₈₉	—	4	1	no	—
257	C₂³×C₈₉	#120	1	2	yes	—
258	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₈₉	—	2	2	no	—
260	Q₈×C₈₉	—	2	2	no	—
261	Q₈×C₈₉	—	4	1	no	—
262	Q₈×C₈₉	—	4	1	no	—
263	Q₈×C₈₉	—	4	1	no	—
264	Q₈×C₈₉	—	4	1	no	—
265	Q₈×C₈₉	—	4	1	no	—
266	Q₈×C₈₉	—	4	1	no	—
267	Q₈×C₈₉	—	8	4	no	—
268	D₄×C₈₉	—	2	2	no	—
269	D₄×C₈₉	—	2	2	no	—
270	D₄×C₈₉	—	4	1	no	—
271	D₄×C₈₉	—	4	1	no	—
272	D₄×C₈₉	—	8	4	no	—
273	C₂×C₄×C₈₉	—	2	1	no	—
274	C₂×C₄×C₈₉	—	2	1	no	—
275	C₂×C₄×C₈₉	—	2	2	no	—
276	C₂×C₄×C₈₉	—	2	2	no	—
277	C₂×C₄×C₈₉	—	2	2	no	—
278	C₂×C₄×C₈₉	—	2	2	no	—
279	C₂×C₄×C₈₉	—	2	1	no	—
280	C₂×C₄×C₈₉	—	2	1	no	—
281	C₂×C₄×C₈₉	—	2	2	no	—
282	C₂×C₄×C₈₉	—	2	1	no	—
283	C₂×C₄×C₈₉	—	2	1	no	—
284	C₂×C₄×C₈₉	—	2	2	no	—
285	C₂×C₄×C₈₉	—	2	2	no	—
286	C₂×C₄×C₈₉	—	2	1	no	—
287	C₂×C₄×C₈₉	—	2	1	no	—
288	C₂×C₄×C₈₉	—	2	1	no	—
289	C₂×C₄×C₈₉	—	2	1	no	—
290	C₂×C₄×C₈₉	—	2	1	no	—
291	C₂×C₄×C₈₉	—	2	1	no	—
292	C₂×C₄×C₈₉	—	2	1	no	—
293	C₂×C₄×C₈₉	—	2	1	no	—
294	C₂×C₄×C₈₉	—	4	1	no	—
295	C₂×C₄×C₈₉	—	4	1	no	—
296	C₂×C₄×C₈₉	—	4	1	no	—
297	C₂×C₄×C₈₉	—	4	1	no	—
298	C₂×C₄×C₈₉	—	4	1	no	—
299	C₂×C₄×C₈₉	—	4	1	no	—
300	C₂×C₄×C₈₉	—	4	1	no	—
301	C₂×C₄×C₈₉	—	4	1	no	—
302	C₂×C₄×C₈₉	—	4	1	no	—
303	C₂×C₄×C₈₉	—	4	1	no	—
304	C₂×C₄×C₈₉	—	4	1	no	—
305	C₂×C₄×C₈₉	—	4	1	no	—
306	C₂×C₄×C₈₉	—	4	1	no	—
307	C₂×C₄×C₈₉	—	4	1	no	—
308	C₂×C₄×C₈₉	—	4	1	no	—
309	C₂×C₄×C₈₉	—	4	1	no	—
310	C₂×C₄×C₈₉	—	4	1	no	—
311	C₂×C₄×C₈₉	—	4	1	no	—
312	C₂×C₄×C₈₉	—	8	4	no	—
313	C₂×C₄×C₈₉	#120	1	2	yes	—
314	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
317	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
318	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
319	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
320	C₈×C₈₉	—	2	2	no	—
321	C₈×C₈₉	—	2	2	no	—
322	C₈×C₈₉	—	4	1	no	—
323	C₈×C₈₉	—	4	1	no	—
324	C₂²×C₈₉	—	2	2	no	—
325	C₂²×C₈₉	—	2	2	no	—
326	C₂²×C₈₉	—	2	2	no	—
327	C₂²×C₈₉	—	4	1	no	—
328	C₂²×C₈₉	—	4	1	no	—
329	C₂²×C₈₉	—	4	1	no	—
330	C₂²×C₈₉	—	4	1	no	—
331	C₂²×C₈₉	—	4	2	no	—
332	C₂²×C₈₉	—	4	2	no	—
333	C₂²×C₈₉	—	4	2	no	—
334	C₂²×C₈₉	—	8	1	no	—
335	C₂²×C₈₉	—	8	1	no	—
336	C₂²×C₈₉	—	8	1	no	—
337	C₂²×C₈₉	—	8	1	no	—
338	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
339	C₂²×C₈₉	#532	1	1	yes	characteristic
340	C₂²×C₈₉	#537	1	1	yes	characteristic
341	C₂²×C₈₉	#576	1	1	yes	characteristic
342	C₂²×C₈₉	#577	1	1	yes	characteristic
343	C₂²×C₈₉	#578	1	1	yes	characteristic
344	C₂²×C₈₉	#578	1	1	yes	characteristic
345	C₄×C₈₉	—	2	2	no	—
346	C₄×C₈₉	—	4	2	no	—
347	C₄×C₈₉	—	4	1	no	—
348	C₄×C₈₉	—	4	1	no	—
349	C₄×C₈₉	—	4	1	no	—
350	C₄×C₈₉	—	4	1	no	—
351	C₄×C₈₉	—	4	2	no	—
352	C₄×C₈₉	—	4	2	no	—
353	C₄×C₈₉	—	8	1	no	—
354	C₄×C₈₉	—	8	1	no	—
355	C₄×C₈₉	—	8	1	no	—
356	C₄×C₈₉	—	8	1	no	—
357	C₄×C₈₉	#546	1	1	yes	characteristic
358	C₄×C₈₉	#546	1	1	yes	characteristic
359	C₄×C₈₉	#547	1	1	yes	characteristic
360	C₄×C₈₉	#547	1	1	yes	characteristic
361	#25514	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
362	C₂×C₈₉	—	4	2	no	—
363	C₂×C₈₉	—	8	2	no	—
364	C₂×C₈₉	#2430	1	1	yes	characteristic
365	C₂×C₈₉	#3011	1	1	yes	characteristic
366	C₂×C₈₉	#3012	1	1	yes	characteristic
367	C₂×C₈₉	#3175	1	1	yes	characteristic
368	C₂×C₈₉	#3178	1	1	yes	characteristic
369	C₂×C₈₉	#3258	1	1	yes	characteristic
370	C₂×C₈₉	#3262	1	1	yes	characteristic
371	C₂×#542	C₂×C₈₉	1	1	yes	characteristic
372	C₂×#468	C₂×C₈₉	1	1	yes	characteristic
373	C₂×#467	C₂×C₈₉	1	1	yes	characteristic, complemented
374	#2625	C₂×C₈₉	1	1	yes	characteristic, complemented
375	#2057	C₂×C₈₉	1	1	yes	characteristic, complemented
376	#2042	C₂×C₈₉	1	1	yes	characteristic, complemented
377	#2024	C₂×C₈₉	1	1	yes	characteristic, complemented
378	#2015	C₂×C₈₉	1	1	yes	characteristic, complemented
379	#1894	C₂×C₈₉	1	1	yes	characteristic, complemented
380	#1893	C₂×C₈₉	1	1	yes	characteristic
381	#1878	C₂×C₈₉	1	1	yes	characteristic, complemented
382	#1877	C₂×C₈₉	1	1	yes	characteristic, complemented
383	#1863	C₂×C₈₉	1	1	yes	characteristic, complemented
384	#1862	C₂×C₈₉	1	1	yes	characteristic, complemented
385	#1729	C₂×C₈₉	1	1	yes	characteristic, complemented
386	C₈₉	#25514	1	1	yes	characteristic, direct product factor, Sylow
387	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
388	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic
389	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
390	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
391	#542	—	2	4	no	—
392	#542	—	2	4	no	—
393	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
394	C₂×#126	—	2	1	no	—
395	C₂×#124	—	2	1	no	—
396	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
397	C₂×#123	—	2	1	no	—
398	C₂×C₂²⋊Q₈	—	2	1	no	—
399	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
400	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
401	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
402	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
403	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
404	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
405	#468	—	2	4	no	—
406	#467	—	2	4	no	—
407	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
408	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
409	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
410	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
411	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
412	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
413	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
414	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
415	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
416	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
417	#400	—	2	1	no	—
418	#398	—	2	1	no	—
419	#397	—	2	1	no	—
420	#397	—	2	1	no	—
421	#391	C₂²×C₈₉	1	1	yes	characteristic
422	#391	C₂²×C₈₉	1	1	yes	characteristic
423	#389	C₂²×C₈₉	1	1	yes	characteristic
424	#389	C₂²×C₈₉	1	1	yes	characteristic
425	#380	C₂²×C₈₉	1	1	yes	characteristic
426	#340	C₂²×C₈₉	1	1	yes	characteristic
427	#340	C₂²×C₈₉	1	1	yes	characteristic
428	#340	C₂²×C₈₉	1	1	yes	characteristic
429	#340	C₂²×C₈₉	1	1	yes	characteristic
430	#336	C₂²×C₈₉	1	1	yes	characteristic
431	#336	C₂²×C₈₉	1	1	yes	characteristic
432	#336	C₂²×C₈₉	1	1	yes	characteristic
433	#336	C₂²×C₈₉	1	1	yes	characteristic
434	C₂²×Q₈	—	2	1	no	—
435	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
436	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
437	C₂²×D₄	—	2	1	no	—
438	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
439	C₂³×C₄	—	2	1	no	—
440	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
441	C₂×Q₁₆	—	2	2	no	—
442	C₂×Q₁₆	—	2	2	no	—
443	C₂×Q₁₆	—	4	1	no	—
444	C₂×SD₁₆	—	2	2	no	—
445	C₂×SD₁₆	—	2	2	no	—
446	C₂×SD₁₆	—	4	1	no	—
447	C₂×M₁₆	—	2	2	no	—
448	C₂×M₁₆	—	2	2	no	—
449	C₂×M₁₆	—	4	1	no	—
450	C₂²×C₈	—	2	2	no	—
451	C₂²×C₈	—	2	2	no	—
452	C₂²×C₈	—	4	1	no	—
453	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
454	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
455	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
456	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
457	C₄⋊Q₈	—	2	2	no	—
458	C₄⋊Q₈	—	2	2	no	—
459	#126	—	2	8	no	—
460	#124	—	2	8	no	—
461	#124	—	2	2	no	—
462	#124	—	2	2	no	—
463	#123	—	2	8	no	—
464	C₂²⋊Q₈	—	2	8	no	—
465	C₂²⋊Q₈	—	2	2	no	—
466	C₂²⋊Q₈	—	2	2	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	D₄×C₈₉	1	4	yes	—
469	#121	—	2	2	no	—
470	#121	D₄×C₈₉	1	4	yes	—
471	C₄×Q₈	—	2	2	no	—
472	C₄×Q₈	—	2	2	no	—
473	C₄×D₄	—	2	2	no	—
474	C₄×D₄	—	2	2	no	—
475	#117	—	2	2	no	—
476	#117	—	2	2	no	—
477	C₂×C₄⋊C₄	—	2	1	no	—
478	C₂×C₄⋊C₄	—	2	1	no	—
479	C₂×C₄⋊C₄	—	2	1	no	—
480	C₂×C₄⋊C₄	—	2	1	no	—
481	C₂×C₄⋊C₄	—	2	1	no	—
482	C₂×C₄⋊C₄	—	2	1	no	—
483	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
484	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
485	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
486	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
487	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
488	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
489	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
490	C₂×C₂²⋊C₄	—	2	1	no	—
491	C₂×C₂²⋊C₄	—	2	1	no	—
492	C₂×C₂²⋊C₄	—	2	1	no	—
493	C₂×C₂²⋊C₄	—	2	1	no	—
494	C₂×C₂²⋊C₄	—	2	1	no	—
495	C₂×C₂²⋊C₄	—	2	1	no	—
496	C₂×C₂²⋊C₄	—	2	1	no	—
497	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
498	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
499	C₂×C₄²	—	2	1	no	—
500	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
501	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
502	#107	—	2	2	no	—
503	#107	—	2	2	no	—
504	#103	—	2	2	no	—
505	#103	—	2	2	no	—
506	#103	—	2	2	no	—
507	#103	—	2	2	no	—
508	#103	—	4	1	no	—
509	#102	—	2	2	no	—
510	#102	—	2	2	no	—
511	#102	—	4	1	no	—
512	#95	—	2	1	no	—
513	#95	—	2	1	no	—
514	#95	—	2	1	no	—
515	#95	—	2	1	no	—
516	#95	—	2	1	no	—
517	#95	—	2	1	no	—
518	#95	—	2	1	no	—
519	C₂⁴	—	2	1	no	—
520	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
521	C₂×Q₈	—	2	4	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	—	2	1	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	8	1	no	—
534	C₂×Q₈	—	8	1	no	—
535	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
536	C₂×D₄	—	2	4	no	—
537	C₂×D₄	—	2	1	no	—
538	C₂×D₄	—	2	1	no	—
539	C₂×D₄	—	2	1	no	—
540	C₂×D₄	—	2	1	no	—
541	C₂×D₄	—	2	1	no	—
542	C₂×D₄	—	8	1	no	—
543	C₂×D₄	—	8	1	no	—
544	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	1	no	—
549	C₂²×C₄	—	2	1	no	—
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	—	8	1	no	—
558	C₂²×C₄	—	8	1	no	—
559	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
560	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
561	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
562	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
563	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
564	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
565	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
566	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
567	Q₁₆	—	4	4	no	—
568	SD₁₆	—	4	4	no	—
569	M₁₆	—	4	4	no	—
570	C₂×C₈	—	2	1	no	—
571	C₂×C₈	—	2	1	no	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	2	1	no	—
580	C₂×C₈	—	4	4	no	—
581	C₂×C₈	#41×C₈₉	1	2	yes	—
582	C₂×C₈	#41×C₈₉	1	2	yes	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	4	2	no	—
597	C₄⋊C₄	—	8	1	no	—
598	C₄⋊C₄	—	8	1	no	—
599	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
600	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	4	no	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	—	2	4	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	4	no	—
608	C₂²⋊C₄	—	4	2	no	—
609	C₂²⋊C₄	—	8	1	no	—
610	C₄²	—	2	4	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	2	1	no	—
613	C₄²	—	2	1	no	—
614	C₄²	—	2	1	no	—
615	C₂³	—	2	1	no	—
616	C₂³	—	2	1	no	—
617	C₂³	—	2	1	no	—
618	C₂³	—	2	1	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	2	no	—
622	C₂³	—	4	1	no	—
623	C₂³	—	4	1	no	—
624	C₂³	—	4	1	no	—
625	C₂³	—	4	1	no	—
626	C₂³	—	4	1	no	—
627	C₂³	—	4	1	no	—
628	C₂³	#120×C₈₉	1	2	yes	—
629	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
630	Q₈	—	2	2	no	—
631	Q₈	—	2	2	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	8	4	no	—
639	D₄	—	2	2	no	—
640	D₄	—	2	2	no	—
641	D₄	—	4	1	no	—
642	D₄	—	4	1	no	—
643	D₄	—	8	4	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	1	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	—	8	4	no	—
684	C₂×C₄	#120×C₈₉	1	2	yes	—
685	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
686	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
687	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
688	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
689	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
690	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
691	C₈	—	2	2	no	—
692	C₈	—	2	2	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₂²	—	2	2	no	—
696	C₂²	—	2	2	no	—
697	C₂²	—	2	2	no	—
698	C₂²	—	4	1	no	—
699	C₂²	—	4	1	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	—	4	2	no	—
705	C₂²	—	8	1	no	—
706	C₂²	—	8	1	no	—
707	C₂²	—	8	1	no	—
708	C₂²	—	8	1	no	—
709	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
710	C₂²	#532×C₈₉	1	1	yes	characteristic
711	C₂²	#537×C₈₉	1	1	yes	characteristic
712	C₂²	#576×C₈₉	1	1	yes	characteristic
713	C₂²	#577×C₈₉	1	1	yes	characteristic
714	C₂²	#578×C₈₉	1	1	yes	characteristic
715	C₂²	#578×C₈₉	1	1	yes	characteristic
716	C₄	—	2	2	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	1	no	—
720	C₄	—	4	1	no	—
721	C₄	—	4	1	no	—
722	C₄	—	4	2	no	—
723	C₄	—	4	2	no	—
724	C₄	—	8	1	no	—
725	C₄	—	8	1	no	—
726	C₄	—	8	1	no	—
727	C₄	—	8	1	no	—
728	C₄	#546×C₈₉	1	1	yes	characteristic
729	C₄	#546×C₈₉	1	1	yes	characteristic
730	C₄	#547×C₈₉	1	1	yes	characteristic
731	C₄	#547×C₈₉	1	1	yes	characteristic
732	C₂	—	4	2	no	—
733	C₂	—	8	2	no	—
734	C₂	C₈₉×#2430	1	1	yes	characteristic
735	C₂	C₈₉×#3011	1	1	yes	characteristic
736	C₂	C₈₉×#3012	1	1	yes	characteristic
737	C₂	C₈₉×#3175	1	1	yes	characteristic
738	C₂	C₈₉×#3178	1	1	yes	characteristic
739	C₂	C₈₉×#3258	1	1	yes	characteristic
740	C₂	C₈₉×#3262	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#121×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#467×C₈₉	C₂	semidirect
4. C₈₉×#2625	C₂	semidirect
5. C₈₉×#2057	C₂	semidirect
6. C₈₉×#2042	C₂	semidirect
7. C₈₉×#2024	C₂	semidirect
8. C₈₉×#2015	C₂	semidirect
9. C₈₉×#1894	C₂	semidirect
11. C₈₉×#1878	C₂	semidirect
12. C₈₉×#1877	C₂	semidirect
13. C₈₉×#1863	C₂	semidirect
14. C₈₉×#1862	C₂	semidirect
15. C₈₉×#1729	C₂	semidirect
361. #25514	C₈₉	direct (N × H)
373. C₂×#467	C₂×C₈₉	semidirect
374. #2625	C₂×C₈₉	semidirect
375. #2057	C₂×C₈₉	semidirect
376. #2042	C₂×C₈₉	semidirect
377. #2024	C₂×C₈₉	semidirect
378. #2015	C₂×C₈₉	semidirect
379. #1894	C₂×C₈₉	semidirect
381. #1878	C₂×C₈₉	semidirect
382. #1877	C₂×C₈₉	semidirect
383. #1863	C₂×C₈₉	semidirect
384. #1862	C₂×C₈₉	semidirect
385. #1729	C₂×C₈₉	semidirect
386. C₈₉	#25514	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#542×C₈₉	2
C₂×#468×C₈₉	2
C₂×#467×C₈₉	2
C₈₉×#2625	2
C₈₉×#2057	2
C₈₉×#2042	2
C₈₉×#2024	2
C₈₉×#2015	2
C₈₉×#1894	2
C₈₉×#1893	2
C₈₉×#1878	2
C₈₉×#1877	2
C₈₉×#1863	2
C₈₉×#1862	2
C₈₉×#1729	2
#25514	89

Subgroups — C89×#25514 #36403362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25514 #36403362