Subgroups — C₈₉×#26846 #36404694

Order 22784.

800 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#489×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#441×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2451	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2448	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2065	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2056	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2015	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1968	C₂	1	1	yes	maximal, characteristic
10	C₈₉×#1936	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1935	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1933	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#1880	C₂	1	1	yes	maximal, characteristic
14	C₈₉×#1865	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈₉×#1770	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	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	#489×C₈₉	—	2	4	no	—
23	#441×C₈₉	—	2	4	no	—
24	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	#399×C₈₉	—	2	1	no	—
39	#398×C₈₉	—	2	1	no	—
40	#397×C₈₉	—	2	1	no	—
41	#395×C₈₉	—	2	1	no	—
42	#391×C₈₉	—	2	1	no	—
43	#391×C₈₉	C₂²	1	1	yes	characteristic
44	#391×C₈₉	C₂²	1	1	yes	characteristic
45	#391×C₈₉	C₂²	1	1	yes	characteristic
46	#390×C₈₉	C₂²	1	1	yes	characteristic
47	#390×C₈₉	C₂²	1	1	yes	characteristic
48	#389×C₈₉	C₂²	1	1	yes	characteristic
49	#389×C₈₉	C₂²	1	1	yes	characteristic
50	#389×C₈₉	C₂²	1	1	yes	characteristic
51	#388×C₈₉	—	2	1	no	—
52	#388×C₈₉	—	2	1	no	—
53	#387×C₈₉	—	2	1	no	—
54	C₄×C₄⋊C₄×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	C₂²×Q₈×C₈₉	—	2	1	no	—
62	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₈₉	—	2	2	no	—
66	C₂×Q₁₆×C₈₉	—	2	2	no	—
67	C₂×Q₁₆×C₈₉	—	2	2	no	—
68	C₂×SD₁₆×C₈₉	—	2	2	no	—
69	C₂×SD₁₆×C₈₉	—	2	2	no	—
70	C₂×SD₁₆×C₈₉	—	4	1	no	—
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₈₉	—	2	2	no	—
76	C₄⋊Q₈×C₈₉	—	4	1	no	—
77	#124×C₈₉	—	2	2	no	—
78	#124×C₈₉	—	4	1	no	—
79	C₄×Q₈×C₈₉	—	2	2	no	—
80	C₄×Q₈×C₈₉	—	2	2	no	—
81	C₄×Q₈×C₈₉	—	4	1	no	—
82	C₄×Q₈×C₈₉	—	4	1	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₂×C₄²×C₈₉	—	2	1	no	—
103	C₂×C₄²×C₈₉	—	2	1	no	—
104	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	#106×C₈₉	—	2	2	no	—
108	#106×C₈₉	—	4	1	no	—
109	C₄⋊C₈×C₈₉	—	2	2	no	—
110	#103×C₈₉	—	2	2	no	—
111	#103×C₈₉	—	2	2	no	—
112	#103×C₈₉	—	2	2	no	—
113	#103×C₈₉	—	2	2	no	—
114	#103×C₈₉	D₄	1	4	yes	—
115	#102×C₈₉	—	2	2	no	—
116	#102×C₈₉	—	2	2	no	—
117	#102×C₈₉	—	2	2	no	—
118	#97×C₈₉	—	2	2	no	—
119	#97×C₈₉	—	2	2	no	—
120	C₄×C₈×C₈₉	—	2	2	no	—
121	#95×C₈₉	—	2	1	no	—
122	#95×C₈₉	—	2	1	no	—
123	#95×C₈₉	—	2	1	no	—
124	#95×C₈₉	—	2	1	no	—
125	#95×C₈₉	—	2	1	no	—
126	#95×C₈₉	—	2	1	no	—
127	#95×C₈₉	—	2	1	no	—
128	#95×C₈₉	—	2	1	no	—
129	#95×C₈₉	—	2	1	no	—
130	#95×C₈₉	D₄	1	1	yes	characteristic
131	#95×C₈₉	D₄	1	1	yes	characteristic
132	C₂⁴×C₈₉	—	2	1	no	—
133	C₂×Q₈×C₈₉	—	2	1	no	—
134	C₂×Q₈×C₈₉	—	2	1	no	—
135	C₂×Q₈×C₈₉	—	2	1	no	—
136	C₂×Q₈×C₈₉	—	2	1	no	—
137	C₂×Q₈×C₈₉	—	2	1	no	—
138	C₂×Q₈×C₈₉	—	2	1	no	—
139	C₂×Q₈×C₈₉	—	2	1	no	—
140	C₂×Q₈×C₈₉	—	2	1	no	—
141	C₂×Q₈×C₈₉	—	2	1	no	—
142	C₂×Q₈×C₈₉	—	2	1	no	—
143	C₂×Q₈×C₈₉	—	4	2	no	—
144	C₂×Q₈×C₈₉	—	4	2	no	—
145	C₂×Q₈×C₈₉	—	8	1	no	—
146	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
147	C₂×Q₈×C₈₉	#41	1	2	yes	—
148	C₂×D₄×C₈₉	—	2	1	no	—
149	C₂×D₄×C₈₉	—	2	1	no	—
150	C₂×D₄×C₈₉	—	2	1	no	—
151	C₂×D₄×C₈₉	—	2	1	no	—
152	C₂×D₄×C₈₉	—	2	1	no	—
153	C₂×D₄×C₈₉	—	2	1	no	—
154	C₂²×C₄×C₈₉	—	2	1	no	—
155	C₂²×C₄×C₈₉	—	2	1	no	—
156	C₂²×C₄×C₈₉	—	2	1	no	—
157	C₂²×C₄×C₈₉	—	2	1	no	—
158	C₂²×C₄×C₈₉	—	2	1	no	—
159	C₂²×C₄×C₈₉	—	2	1	no	—
160	C₂²×C₄×C₈₉	—	2	1	no	—
161	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₈₉	—	2	8	no	—
169	SD₁₆×C₈₉	—	4	4	no	—
170	C₂×C₈×C₈₉	—	2	1	no	—
171	C₂×C₈×C₈₉	—	2	1	no	—
172	C₂×C₈×C₈₉	—	2	1	no	—
173	C₂×C₈×C₈₉	—	2	1	no	—
174	C₂×C₈×C₈₉	—	2	1	no	—
175	C₂×C₈×C₈₉	—	2	1	no	—
176	C₂×C₈×C₈₉	—	2	1	no	—
177	C₂×C₈×C₈₉	—	2	1	no	—
178	C₂×C₈×C₈₉	—	2	1	no	—
179	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
180	C₂×C₈×C₈₉	#41	1	2	yes	—
181	C₂×C₈×C₈₉	#41	1	2	yes	—
182	C₄⋊C₄×C₈₉	—	2	2	no	—
183	C₄⋊C₄×C₈₉	—	2	1	no	—
184	C₄⋊C₄×C₈₉	—	2	1	no	—
185	C₄⋊C₄×C₈₉	—	2	1	no	—
186	C₄⋊C₄×C₈₉	—	2	1	no	—
187	C₄⋊C₄×C₈₉	—	2	1	no	—
188	C₄⋊C₄×C₈₉	—	2	1	no	—
189	C₄⋊C₄×C₈₉	—	2	1	no	—
190	C₄⋊C₄×C₈₉	—	2	1	no	—
191	C₄⋊C₄×C₈₉	—	2	1	no	—
192	C₄⋊C₄×C₈₉	—	4	1	no	—
193	C₄⋊C₄×C₈₉	—	4	2	no	—
194	C₄⋊C₄×C₈₉	—	4	2	no	—
195	C₄⋊C₄×C₈₉	—	4	2	no	—
196	C₄⋊C₄×C₈₉	—	8	1	no	—
197	C₄⋊C₄×C₈₉	—	8	1	no	—
198	C₄⋊C₄×C₈₉	—	8	1	no	—
199	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
200	C₂²⋊C₄×C₈₉	—	4	2	no	—
201	C₂²⋊C₄×C₈₉	—	4	2	no	—
202	C₂²⋊C₄×C₈₉	—	4	2	no	—
203	C₂²⋊C₄×C₈₉	—	4	2	no	—
204	C₂²⋊C₄×C₈₉	—	8	1	no	—
205	C₂²⋊C₄×C₈₉	—	8	1	no	—
206	C₄²×C₈₉	—	2	1	no	—
207	C₄²×C₈₉	—	2	1	no	—
208	C₄²×C₈₉	—	2	1	no	—
209	C₄²×C₈₉	—	2	1	no	—
210	C₄²×C₈₉	—	2	1	no	—
211	C₄²×C₈₉	—	2	1	no	—
212	C₄²×C₈₉	—	4	2	no	—
213	C₄²×C₈₉	—	4	2	no	—
214	C₂³×C₈₉	—	4	1	no	—
215	C₂³×C₈₉	—	4	1	no	—
216	C₂³×C₈₉	—	4	1	no	—
217	C₂³×C₈₉	—	4	1	no	—
218	C₂³×C₈₉	—	4	1	no	—
219	C₂³×C₈₉	—	4	1	no	—
220	C₂³×C₈₉	—	4	1	no	—
221	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
222	Q₈×C₈₉	—	2	2	no	—
223	Q₈×C₈₉	—	2	2	no	—
224	Q₈×C₈₉	—	2	2	no	—
225	Q₈×C₈₉	—	2	2	no	—
226	Q₈×C₈₉	—	4	1	no	—
227	Q₈×C₈₉	—	4	1	no	—
228	Q₈×C₈₉	—	4	1	no	—
229	Q₈×C₈₉	—	4	1	no	—
230	Q₈×C₈₉	—	8	4	no	—
231	D₄×C₈₉	—	4	1	no	—
232	D₄×C₈₉	—	4	1	no	—
233	D₄×C₈₉	—	4	1	no	—
234	D₄×C₈₉	—	4	1	no	—
235	C₂×C₄×C₈₉	—	2	1	no	—
236	C₂×C₄×C₈₉	—	2	1	no	—
237	C₂×C₄×C₈₉	—	2	1	no	—
238	C₂×C₄×C₈₉	—	2	1	no	—
239	C₂×C₄×C₈₉	—	2	1	no	—
240	C₂×C₄×C₈₉	—	2	1	no	—
241	C₂×C₄×C₈₉	—	2	1	no	—
242	C₂×C₄×C₈₉	—	2	2	no	—
243	C₂×C₄×C₈₉	—	2	2	no	—
244	C₂×C₄×C₈₉	—	2	2	no	—
245	C₂×C₄×C₈₉	—	2	1	no	—
246	C₂×C₄×C₈₉	—	2	1	no	—
247	C₂×C₄×C₈₉	—	2	1	no	—
248	C₂×C₄×C₈₉	—	2	1	no	—
249	C₂×C₄×C₈₉	—	2	1	no	—
250	C₂×C₄×C₈₉	—	2	1	no	—
251	C₂×C₄×C₈₉	—	2	1	no	—
252	C₂×C₄×C₈₉	—	2	1	no	—
253	C₂×C₄×C₈₉	—	2	1	no	—
254	C₂×C₄×C₈₉	—	2	1	no	—
255	C₂×C₄×C₈₉	—	4	1	no	—
256	C₂×C₄×C₈₉	—	4	1	no	—
257	C₂×C₄×C₈₉	—	4	1	no	—
258	C₂×C₄×C₈₉	—	4	1	no	—
259	C₂×C₄×C₈₉	—	4	1	no	—
260	C₂×C₄×C₈₉	—	4	1	no	—
261	C₂×C₄×C₈₉	—	4	1	no	—
262	C₂×C₄×C₈₉	—	4	1	no	—
263	C₂×C₄×C₈₉	—	4	1	no	—
264	C₂×C₄×C₈₉	—	4	1	no	—
265	C₂×C₄×C₈₉	—	4	1	no	—
266	C₂×C₄×C₈₉	—	4	1	no	—
267	C₂×C₄×C₈₉	—	4	1	no	—
268	C₂×C₄×C₈₉	—	4	1	no	—
269	C₂×C₄×C₈₉	—	4	1	no	—
270	C₂×C₄×C₈₉	—	4	1	no	—
271	C₂×C₄×C₈₉	—	4	1	no	—
272	C₂×C₄×C₈₉	—	4	1	no	—
273	C₂×C₄×C₈₉	#137	1	2	yes	—
274	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
276	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
279	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
280	C₈×C₈₉	—	2	2	no	—
281	C₈×C₈₉	—	2	2	no	—
282	C₈×C₈₉	—	2	2	no	—
283	C₈×C₈₉	—	4	1	no	—
284	C₂²×C₈₉	—	8	1	no	—
285	C₂²×C₈₉	—	8	1	no	—
286	C₂²×C₈₉	—	8	1	no	—
287	C₂²×C₈₉	—	8	1	no	—
288	C₂²×C₈₉	—	8	1	no	—
289	C₂²×C₈₉	—	8	1	no	—
290	C₂²×C₈₉	—	8	1	no	—
291	C₂²×C₈₉	#532	1	1	yes	characteristic
292	C₂²×C₈₉	D₄²	1	1	yes	characteristic
293	C₂²×C₈₉	#548	1	1	yes	characteristic
294	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
295	C₂²×C₈₉	#575	1	1	yes	characteristic
296	C₂²×C₈₉	#577	1	1	yes	characteristic
297	C₂²×C₈₉	#577	1	1	yes	characteristic
298	C₄×C₈₉	—	2	2	no	—
299	C₄×C₈₉	—	4	1	no	—
300	C₄×C₈₉	—	4	2	no	—
301	C₄×C₈₉	—	4	2	no	—
302	C₄×C₈₉	—	4	1	no	—
303	C₄×C₈₉	—	4	1	no	—
304	C₄×C₈₉	—	4	2	no	—
305	C₄×C₈₉	—	4	1	no	—
306	C₄×C₈₉	—	4	1	no	—
307	C₄×C₈₉	—	8	1	no	—
308	C₄×C₈₉	—	8	1	no	—
309	C₄×C₈₉	—	8	1	no	—
310	C₄×C₈₉	—	8	1	no	—
311	C₄×C₈₉	#525	1	1	yes	characteristic
312	C₄×C₈₉	#542	1	1	yes	characteristic
313	C₄×C₈₉	#547	1	1	yes	characteristic
314	C₄×C₈₉	#551	1	1	yes	characteristic
315	#26846	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂×C₈₉	—	16	1	no	—
317	C₂×C₈₉	#2461	1	1	yes	characteristic
318	C₂×C₈₉	#3173	1	1	yes	characteristic
319	C₂×C₈₉	#3181	1	1	yes	characteristic
320	C₂×C₈₉	#3275	1	1	yes	characteristic
321	C₂×C₈₉	#3278	1	1	yes	characteristic
322	C₂×C₈₉	#3369	1	1	yes	characteristic
323	C₂×C₈₉	#3371	1	1	yes	characteristic
324	C₂×#489	C₂×C₈₉	1	1	yes	characteristic
325	C₂×#441	C₂×C₈₉	1	1	yes	characteristic, complemented
326	#2451	C₂×C₈₉	1	1	yes	characteristic, complemented
327	#2448	C₂×C₈₉	1	1	yes	characteristic
328	#2065	C₂×C₈₉	1	1	yes	characteristic
329	#2058	C₂×C₈₉	1	1	yes	characteristic, complemented
330	#2056	C₂×C₈₉	1	1	yes	characteristic, complemented
331	#2015	C₂×C₈₉	1	1	yes	characteristic
332	#1968	C₂×C₈₉	1	1	yes	characteristic
333	#1936	C₂×C₈₉	1	1	yes	characteristic
334	#1935	C₂×C₈₉	1	1	yes	characteristic, complemented
335	#1933	C₂×C₈₉	1	1	yes	characteristic, complemented
336	#1880	C₂×C₈₉	1	1	yes	characteristic
337	#1865	C₂×C₈₉	1	1	yes	characteristic, complemented
338	#1770	C₂×C₈₉	1	1	yes	characteristic, complemented
339	C₈₉	#26846	1	1	yes	characteristic, direct product factor, Sylow
340	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
341	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic
342	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
343	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
345	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
346	#489	—	2	4	no	—
347	#441	—	2	4	no	—
348	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
349	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
350	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
351	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
352	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
353	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
354	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
355	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
356	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
357	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
358	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
359	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
360	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
361	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
362	#399	—	2	1	no	—
363	#398	—	2	1	no	—
364	#397	—	2	1	no	—
365	#395	—	2	1	no	—
366	#391	—	2	1	no	—
367	#391	C₂²×C₈₉	1	1	yes	characteristic
368	#391	C₂²×C₈₉	1	1	yes	characteristic
369	#391	C₂²×C₈₉	1	1	yes	characteristic
370	#390	C₂²×C₈₉	1	1	yes	characteristic
371	#390	C₂²×C₈₉	1	1	yes	characteristic
372	#389	C₂²×C₈₉	1	1	yes	characteristic
373	#389	C₂²×C₈₉	1	1	yes	characteristic
374	#389	C₂²×C₈₉	1	1	yes	characteristic
375	#388	—	2	1	no	—
376	#388	—	2	1	no	—
377	#387	—	2	1	no	—
378	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
379	#340	C₂²×C₈₉	1	1	yes	characteristic
380	#340	C₂²×C₈₉	1	1	yes	characteristic
381	#340	C₂²×C₈₉	1	1	yes	characteristic
382	#340	C₂²×C₈₉	1	1	yes	characteristic
383	#336	C₂²×C₈₉	1	1	yes	characteristic
384	#336	C₂²×C₈₉	1	1	yes	characteristic
385	C₂²×Q₈	—	2	1	no	—
386	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
387	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
389	C₂×Q₁₆	—	2	2	no	—
390	C₂×Q₁₆	—	2	2	no	—
391	C₂×Q₁₆	—	2	2	no	—
392	C₂×SD₁₆	—	2	2	no	—
393	C₂×SD₁₆	—	2	2	no	—
394	C₂×SD₁₆	—	4	1	no	—
395	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
399	C₄⋊Q₈	—	2	2	no	—
400	C₄⋊Q₈	—	4	1	no	—
401	#124	—	2	2	no	—
402	#124	—	4	1	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	—	2	2	no	—
405	C₄×Q₈	—	4	1	no	—
406	C₄×Q₈	—	4	1	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
414	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
420	C₂×C₂²⋊C₄	—	2	1	no	—
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	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	—	2	1	no	—
428	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
431	#106	—	2	2	no	—
432	#106	—	4	1	no	—
433	C₄⋊C₈	—	2	2	no	—
434	#103	—	2	2	no	—
435	#103	—	2	2	no	—
436	#103	—	2	2	no	—
437	#103	—	2	2	no	—
438	#103	D₄×C₈₉	1	4	yes	—
439	#102	—	2	2	no	—
440	#102	—	2	2	no	—
441	#102	—	2	2	no	—
442	#97	—	2	2	no	—
443	#97	—	2	2	no	—
444	C₄×C₈	—	2	2	no	—
445	#95	—	2	1	no	—
446	#95	—	2	1	no	—
447	#95	—	2	1	no	—
448	#95	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	D₄×C₈₉	1	1	yes	characteristic
455	#95	D₄×C₈₉	1	1	yes	characteristic
456	C₂⁴	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	4	2	no	—
468	C₂×Q₈	—	4	2	no	—
469	C₂×Q₈	—	8	1	no	—
470	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
471	C₂×Q₈	#41×C₈₉	1	2	yes	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
490	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
492	Q₁₆	—	2	8	no	—
493	SD₁₆	—	4	4	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
504	C₂×C₈	#41×C₈₉	1	2	yes	—
505	C₂×C₈	#41×C₈₉	1	2	yes	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	4	1	no	—
517	C₄⋊C₄	—	4	2	no	—
518	C₄⋊C₄	—	4	2	no	—
519	C₄⋊C₄	—	4	2	no	—
520	C₄⋊C₄	—	8	1	no	—
521	C₄⋊C₄	—	8	1	no	—
522	C₄⋊C₄	—	8	1	no	—
523	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
524	C₂²⋊C₄	—	4	2	no	—
525	C₂²⋊C₄	—	4	2	no	—
526	C₂²⋊C₄	—	4	2	no	—
527	C₂²⋊C₄	—	4	2	no	—
528	C₂²⋊C₄	—	8	1	no	—
529	C₂²⋊C₄	—	8	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	2	1	no	—
535	C₄²	—	2	1	no	—
536	C₄²	—	4	2	no	—
537	C₄²	—	4	2	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	—	4	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	1	no	—
545	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
546	Q₈	—	2	2	no	—
547	Q₈	—	2	2	no	—
548	Q₈	—	2	2	no	—
549	Q₈	—	2	2	no	—
550	Q₈	—	4	1	no	—
551	Q₈	—	4	1	no	—
552	Q₈	—	4	1	no	—
553	Q₈	—	4	1	no	—
554	Q₈	—	8	4	no	—
555	D₄	—	4	1	no	—
556	D₄	—	4	1	no	—
557	D₄	—	4	1	no	—
558	D₄	—	4	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	#137×C₈₉	1	2	yes	—
598	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
599	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
600	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
601	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
602	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
603	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
604	C₈	—	2	2	no	—
605	C₈	—	2	2	no	—
606	C₈	—	2	2	no	—
607	C₈	—	4	1	no	—
608	C₂²	—	8	1	no	—
609	C₂²	—	8	1	no	—
610	C₂²	—	8	1	no	—
611	C₂²	—	8	1	no	—
612	C₂²	—	8	1	no	—
613	C₂²	—	8	1	no	—
614	C₂²	—	8	1	no	—
615	C₂²	#532×C₈₉	1	1	yes	characteristic
616	C₂²	D₄²×C₈₉	1	1	yes	characteristic
617	C₂²	#548×C₈₉	1	1	yes	characteristic
618	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
619	C₂²	#575×C₈₉	1	1	yes	characteristic
620	C₂²	#577×C₈₉	1	1	yes	characteristic
621	C₂²	#577×C₈₉	1	1	yes	characteristic
622	C₄	—	2	2	no	—
623	C₄	—	4	1	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	1	no	—
631	C₄	—	8	1	no	—
632	C₄	—	8	1	no	—
633	C₄	—	8	1	no	—
634	C₄	—	8	1	no	—
635	C₄	#525×C₈₉	1	1	yes	characteristic
636	C₄	#542×C₈₉	1	1	yes	characteristic
637	C₄	#547×C₈₉	1	1	yes	characteristic
638	C₄	#551×C₈₉	1	1	yes	characteristic
639	C₂	—	16	1	no	—
640	C₂	C₈₉×#2461	1	1	yes	characteristic
641	C₂	C₈₉×#3173	1	1	yes	characteristic
642	C₂	C₈₉×#3181	1	1	yes	characteristic
643	C₂	C₈₉×#3275	1	1	yes	characteristic
644	C₂	C₈₉×#3278	1	1	yes	characteristic
645	C₂	C₈₉×#3369	1	1	yes	characteristic
646	C₂	C₈₉×#3371	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#441×C₈₉	C₂	semidirect
3. C₈₉×#2451	C₂	semidirect
6. C₈₉×#2058	C₂	semidirect
7. C₈₉×#2056	C₂	semidirect
11. C₈₉×#1935	C₂	semidirect
12. C₈₉×#1933	C₂	semidirect
14. C₈₉×#1865	C₂	semidirect
15. C₈₉×#1770	C₂	semidirect
315. #26846	C₈₉	direct (N × H)
325. C₂×#441	C₂×C₈₉	semidirect
326. #2451	C₂×C₈₉	semidirect
329. #2058	C₂×C₈₉	semidirect
330. #2056	C₂×C₈₉	semidirect
334. #1935	C₂×C₈₉	semidirect
335. #1933	C₂×C₈₉	semidirect
337. #1865	C₂×C₈₉	semidirect
338. #1770	C₂×C₈₉	semidirect
339. C₈₉	#26846	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×#489×C₈₉	2
C₂×#441×C₈₉	2
C₈₉×#2451	2
C₈₉×#2448	2
C₈₉×#2065	2
C₈₉×#2058	2
C₈₉×#2056	2
C₈₉×#2015	2
C₈₉×#1968	2
C₈₉×#1936	2
C₈₉×#1935	2
C₈₉×#1933	2
C₈₉×#1880	2
C₈₉×#1865	2
C₈₉×#1770	2
#26846	89

Subgroups — C89×#26846 #36404694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#26846 #36404694