Subgroups — C₈₉×#29414 #36407262

Order 22784.

868 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#121×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2036	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2034	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2000	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1927	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1925	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1920	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1918	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1880	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×#1817	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₈₉	—	2	1	no	—
21	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#103×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×#102×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#391×C₈₉	C₂²	1	1	yes	characteristic
35	#390×C₈₉	C₂²	1	1	yes	characteristic
36	#389×C₈₉	C₂²	1	1	yes	characteristic
37	#389×C₈₉	C₂²	1	1	yes	characteristic
38	#388×C₈₉	—	2	1	no	—
39	#387×C₈₉	—	2	1	no	—
40	#386×C₈₉	—	2	1	no	—
41	#385×C₈₉	—	2	1	no	—
42	#380×C₈₉	C₂²	1	1	yes	characteristic
43	#380×C₈₉	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
47	C₄×C₂²⋊C₄×C₈₉	—	2	1	no	—
48	C₄×C₂²⋊C₄×C₈₉	—	2	1	no	—
49	#340×C₈₉	C₂²	1	1	yes	characteristic
50	#340×C₈₉	C₂²	1	1	yes	characteristic
51	#340×C₈₉	C₂²	1	1	yes	characteristic
52	#340×C₈₉	C₂²	1	1	yes	characteristic
53	#340×C₈₉	C₂²	1	1	yes	characteristic
54	#340×C₈₉	C₂²	1	1	yes	characteristic
55	#336×C₈₉	C₂²	1	1	yes	characteristic
56	#336×C₈₉	C₂²	1	1	yes	characteristic
57	#336×C₈₉	C₂²	1	1	yes	characteristic
58	#336×C₈₉	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₈₉	—	2	1	no	—
61	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₈₉	—	2	1	no	—
63	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×SD₁₆×C₈₉	—	2	2	no	—
67	C₂×SD₁₆×C₈₉	—	2	2	no	—
68	C₂×SD₁₆×C₈₉	—	4	1	no	—
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂²⋊Q₈×C₈₉	—	2	2	no	—
74	C₂²⋊Q₈×C₈₉	—	2	2	no	—
75	#121×C₈₉	—	2	2	no	—
76	#121×C₈₉	—	2	2	no	—
77	C₄×Q₈×C₈₉	—	4	1	no	—
78	C₄×Q₈×C₈₉	—	4	1	no	—
79	C₄×D₄×C₈₉	—	4	1	no	—
80	C₄×D₄×C₈₉	—	4	1	no	—
81	C₄×D₄×C₈₉	—	4	4	no	—
82	C₂×C₄⋊C₄×C₈₉	—	2	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	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
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×C₄²×C₈₉	—	2	1	no	—
100	C₂×C₄²×C₈₉	—	2	1	no	—
101	C₂×C₄²×C₈₉	—	2	1	no	—
102	C₂×C₄²×C₈₉	—	4	1	no	—
103	C₂×C₄²×C₈₉	—	4	1	no	—
104	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	#107×C₈₉	—	2	2	no	—
109	#107×C₈₉	—	4	1	no	—
110	#106×C₈₉	—	2	2	no	—
111	#103×C₈₉	—	2	2	no	—
112	#103×C₈₉	—	2	2	no	—
113	#103×C₈₉	—	4	1	no	—
114	#102×C₈₉	—	2	2	no	—
115	#102×C₈₉	—	2	2	no	—
116	#102×C₈₉	—	4	1	no	—
117	C₂²⋊C₈×C₈₉	—	4	1	no	—
118	C₂²⋊C₈×C₈₉	—	4	1	no	—
119	#95×C₈₉	—	2	1	no	—
120	#95×C₈₉	—	2	1	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	C₂⁴×C₈₉	—	2	1	no	—
128	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Q₈×C₈₉	—	2	1	no	—
130	C₂×Q₈×C₈₉	—	2	1	no	—
131	C₂×Q₈×C₈₉	—	2	1	no	—
132	C₂×Q₈×C₈₉	—	2	1	no	—
133	C₂×Q₈×C₈₉	—	2	1	no	—
134	C₂×Q₈×C₈₉	#41	1	2	yes	—
135	C₂×D₄×C₈₉	—	2	4	no	—
136	C₂×D₄×C₈₉	—	2	1	no	—
137	C₂×D₄×C₈₉	—	2	1	no	—
138	C₂×D₄×C₈₉	—	2	1	no	—
139	C₂×D₄×C₈₉	—	2	1	no	—
140	C₂×D₄×C₈₉	—	2	1	no	—
141	C₂×D₄×C₈₉	—	4	2	no	—
142	C₂×D₄×C₈₉	—	4	2	no	—
143	C₂×D₄×C₈₉	#41	1	2	yes	—
144	C₂²×C₄×C₈₉	—	2	2	no	—
145	C₂²×C₄×C₈₉	—	2	1	no	—
146	C₂²×C₄×C₈₉	—	2	1	no	—
147	C₂²×C₄×C₈₉	—	2	1	no	—
148	C₂²×C₄×C₈₉	—	2	2	no	—
149	C₂²×C₄×C₈₉	—	2	2	no	—
150	C₂²×C₄×C₈₉	—	2	1	no	—
151	C₂²×C₄×C₈₉	—	2	1	no	—
152	C₂²×C₄×C₈₉	—	2	1	no	—
153	C₂²×C₄×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	—	4	1	no	—
161	C₂²×C₄×C₈₉	—	4	1	no	—
162	C₂²×C₄×C₈₉	—	4	1	no	—
163	C₂²×C₄×C₈₉	—	4	2	no	—
164	C₂²×C₄×C₈₉	—	4	2	no	—
165	C₂²×C₄×C₈₉	—	4	2	no	—
166	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
172	SD₁₆×C₈₉	—	4	4	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	—	2	1	no	—
180	C₂×C₈×C₈₉	—	2	1	no	—
181	C₂×C₈×C₈₉	—	2	1	no	—
182	C₂×C₈×C₈₉	—	2	1	no	—
183	C₂×C₈×C₈₉	—	2	1	no	—
184	C₂×C₈×C₈₉	—	2	1	no	—
185	C₄⋊C₄×C₈₉	—	2	4	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	—	2	1	no	—
193	C₄⋊C₄×C₈₉	—	2	1	no	—
194	C₄⋊C₄×C₈₉	—	2	1	no	—
195	C₄⋊C₄×C₈₉	—	2	1	no	—
196	C₄⋊C₄×C₈₉	—	4	2	no	—
197	C₄⋊C₄×C₈₉	—	4	2	no	—
198	C₄⋊C₄×C₈₉	—	4	2	no	—
199	C₄⋊C₄×C₈₉	#41	1	2	yes	—
200	C₄⋊C₄×C₈₉	#41	1	2	yes	—
201	C₂²⋊C₄×C₈₉	—	2	4	no	—
202	C₂²⋊C₄×C₈₉	—	2	4	no	—
203	C₂²⋊C₄×C₈₉	—	4	2	no	—
204	C₂²⋊C₄×C₈₉	—	4	2	no	—
205	C₂²⋊C₄×C₈₉	—	4	2	no	—
206	C₂²⋊C₄×C₈₉	—	4	2	no	—
207	C₂²⋊C₄×C₈₉	—	4	2	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	1	no	—
213	C₄²×C₈₉	—	4	1	no	—
214	C₄²×C₈₉	—	4	2	no	—
215	C₄²×C₈₉	—	4	2	no	—
216	C₄²×C₈₉	—	4	2	no	—
217	C₂³×C₈₉	—	2	1	no	—
218	C₂³×C₈₉	—	2	1	no	—
219	C₂³×C₈₉	—	2	1	no	—
220	C₂³×C₈₉	—	2	1	no	—
221	C₂³×C₈₉	—	2	1	no	—
222	C₂³×C₈₉	—	2	1	no	—
223	C₂³×C₈₉	—	2	2	no	—
224	C₂³×C₈₉	—	4	1	no	—
225	C₂³×C₈₉	—	4	1	no	—
226	C₂³×C₈₉	—	4	1	no	—
227	C₂³×C₈₉	—	4	1	no	—
228	C₂³×C₈₉	—	4	1	no	—
229	C₂³×C₈₉	—	4	1	no	—
230	C₂³×C₈₉	#137	1	2	yes	—
231	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₈₉	—	2	2	no	—
233	Q₈×C₈₉	—	2	2	no	—
234	Q₈×C₈₉	—	4	1	no	—
235	Q₈×C₈₉	—	4	1	no	—
236	D₄×C₈₉	—	2	2	no	—
237	D₄×C₈₉	—	2	2	no	—
238	D₄×C₈₉	—	4	8	no	—
239	D₄×C₈₉	—	4	1	no	—
240	D₄×C₈₉	—	4	1	no	—
241	C₂×C₄×C₈₉	—	2	1	no	—
242	C₂×C₄×C₈₉	—	2	1	no	—
243	C₂×C₄×C₈₉	—	2	1	no	—
244	C₂×C₄×C₈₉	—	2	1	no	—
245	C₂×C₄×C₈₉	—	2	1	no	—
246	C₂×C₄×C₈₉	—	2	2	no	—
247	C₂×C₄×C₈₉	—	2	2	no	—
248	C₂×C₄×C₈₉	—	2	2	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	—	2	1	no	—
256	C₂×C₄×C₈₉	—	2	1	no	—
257	C₂×C₄×C₈₉	—	4	4	no	—
258	C₂×C₄×C₈₉	—	4	4	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	—	4	1	no	—
274	C₂×C₄×C₈₉	—	4	1	no	—
275	C₂×C₄×C₈₉	—	4	1	no	—
276	C₂×C₄×C₈₉	—	4	1	no	—
277	C₂×C₄×C₈₉	—	4	1	no	—
278	C₂×C₄×C₈₉	—	4	1	no	—
279	C₂×C₄×C₈₉	—	4	1	no	—
280	C₂×C₄×C₈₉	—	4	1	no	—
281	C₂×C₄×C₈₉	—	8	4	no	—
282	C₂×C₄×C₈₉	#135	1	2	yes	—
283	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
284	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
289	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
290	C₈×C₈₉	—	4	1	no	—
291	C₈×C₈₉	—	4	1	no	—
292	C₈×C₈₉	—	4	1	no	—
293	C₈×C₈₉	—	4	1	no	—
294	C₂²×C₈₉	—	2	2	no	—
295	C₂²×C₈₉	—	2	2	no	—
296	C₂²×C₈₉	—	2	2	no	—
297	C₂²×C₈₉	—	4	1	no	—
298	C₂²×C₈₉	—	4	1	no	—
299	C₂²×C₈₉	—	4	1	no	—
300	C₂²×C₈₉	—	4	1	no	—
301	C₂²×C₈₉	—	4	2	no	—
302	C₂²×C₈₉	—	4	2	no	—
303	C₂²×C₈₉	—	4	2	no	—
304	C₂²×C₈₉	—	8	1	no	—
305	C₂²×C₈₉	—	8	1	no	—
306	C₂²×C₈₉	—	8	1	no	—
307	C₂²×C₈₉	—	8	1	no	—
308	C₂²×C₈₉	#525	1	1	yes	characteristic
309	C₂²×C₈₉	#525	1	1	yes	characteristic
310	C₂²×C₈₉	#534	1	1	yes	characteristic
311	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
312	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
314	C₂²×C₈₉	#575	1	1	yes	characteristic
315	C₄×C₈₉	—	2	2	no	—
316	C₄×C₈₉	—	2	2	no	—
317	C₄×C₈₉	—	4	2	no	—
318	C₄×C₈₉	—	4	2	no	—
319	C₄×C₈₉	—	4	1	no	—
320	C₄×C₈₉	—	4	2	no	—
321	C₄×C₈₉	—	4	1	no	—
322	C₄×C₈₉	—	4	1	no	—
323	C₄×C₈₉	—	8	1	no	—
324	C₄×C₈₉	—	8	1	no	—
325	C₄×C₈₉	—	8	1	no	—
326	C₄×C₈₉	—	8	1	no	—
327	C₄×C₈₉	#551	1	1	yes	characteristic
328	C₄×C₈₉	#553	1	1	yes	characteristic
329	C₄×C₈₉	#553	1	1	yes	characteristic
330	C₄×C₈₉	#559	1	1	yes	characteristic
331	#29414	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
332	C₂×C₈₉	—	4	2	no	—
333	C₂×C₈₉	—	8	2	no	—
334	C₂×C₈₉	#2600	1	1	yes	characteristic
335	C₂×C₈₉	#3102	1	1	yes	characteristic
336	C₂×C₈₉	#3106	1	1	yes	characteristic
337	C₂×C₈₉	#3110	1	1	yes	characteristic
338	C₂×C₈₉	#3117	1	1	yes	characteristic
339	C₂×C₈₉	#3190	1	1	yes	characteristic
340	C₂×C₈₉	#3194	1	1	yes	characteristic
341	C₄×C₂²⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
342	C₄×#121	C₂×C₈₉	1	1	yes	characteristic
343	#2076	C₂×C₈₉	1	1	yes	characteristic, complemented
344	#2036	C₂×C₈₉	1	1	yes	characteristic, complemented
345	#2034	C₂×C₈₉	1	1	yes	characteristic
346	#2000	C₂×C₈₉	1	1	yes	characteristic
347	#1927	C₂×C₈₉	1	1	yes	characteristic, complemented
348	#1925	C₂×C₈₉	1	1	yes	characteristic, complemented
349	#1920	C₂×C₈₉	1	1	yes	characteristic, complemented
350	#1918	C₂×C₈₉	1	1	yes	characteristic, complemented
351	#1880	C₂×C₈₉	1	1	yes	characteristic, complemented
352	#1877	C₂×C₈₉	1	1	yes	characteristic, complemented
353	#1863	C₂×C₈₉	1	1	yes	characteristic, complemented
354	#1862	C₂×C₈₉	1	1	yes	characteristic, complemented
355	#1817	C₂×C₈₉	1	1	yes	characteristic, complemented
356	C₈₉	#29414	1	1	yes	characteristic, direct product factor, Sylow
357	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic
358	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
359	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
360	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
361	C₂×C₄×D₄	—	2	1	no	—
362	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
363	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
364	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
365	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
366	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
367	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
368	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
369	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
370	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
371	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
372	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
373	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
374	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
375	#391	C₂²×C₈₉	1	1	yes	characteristic
376	#390	C₂²×C₈₉	1	1	yes	characteristic
377	#389	C₂²×C₈₉	1	1	yes	characteristic
378	#389	C₂²×C₈₉	1	1	yes	characteristic
379	#388	—	2	1	no	—
380	#387	—	2	1	no	—
381	#386	—	2	1	no	—
382	#385	—	2	1	no	—
383	#380	C₂²×C₈₉	1	1	yes	characteristic
384	#380	C₂²×C₈₉	1	1	yes	characteristic
385	C₄×C₄⋊C₄	—	2	1	no	—
386	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
387	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
388	C₄×C₂²⋊C₄	—	2	1	no	—
389	C₄×C₂²⋊C₄	—	2	1	no	—
390	#340	C₂²×C₈₉	1	1	yes	characteristic
391	#340	C₂²×C₈₉	1	1	yes	characteristic
392	#340	C₂²×C₈₉	1	1	yes	characteristic
393	#340	C₂²×C₈₉	1	1	yes	characteristic
394	#340	C₂²×C₈₉	1	1	yes	characteristic
395	#340	C₂²×C₈₉	1	1	yes	characteristic
396	#336	C₂²×C₈₉	1	1	yes	characteristic
397	#336	C₂²×C₈₉	1	1	yes	characteristic
398	#336	C₂²×C₈₉	1	1	yes	characteristic
399	#336	C₂²×C₈₉	1	1	yes	characteristic
400	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
401	C₂²×D₄	—	2	1	no	—
402	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
403	C₂³×C₄	—	2	1	no	—
404	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
405	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
406	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
407	C₂×SD₁₆	—	2	2	no	—
408	C₂×SD₁₆	—	2	2	no	—
409	C₂×SD₁₆	—	4	1	no	—
410	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
411	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
412	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
413	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
414	C₂²⋊Q₈	—	2	2	no	—
415	C₂²⋊Q₈	—	2	2	no	—
416	#121	—	2	2	no	—
417	#121	—	2	2	no	—
418	C₄×Q₈	—	4	1	no	—
419	C₄×Q₈	—	4	1	no	—
420	C₄×D₄	—	4	1	no	—
421	C₄×D₄	—	4	1	no	—
422	C₄×D₄	—	4	4	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	—	2	1	no	—
443	C₂×C₄²	—	4	1	no	—
444	C₂×C₄²	—	4	1	no	—
445	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
446	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
449	#107	—	2	2	no	—
450	#107	—	4	1	no	—
451	#106	—	2	2	no	—
452	#103	—	2	2	no	—
453	#103	—	2	2	no	—
454	#103	—	4	1	no	—
455	#102	—	2	2	no	—
456	#102	—	2	2	no	—
457	#102	—	4	1	no	—
458	C₂²⋊C₈	—	4	1	no	—
459	C₂²⋊C₈	—	4	1	no	—
460	#95	—	2	1	no	—
461	#95	—	2	1	no	—
462	#95	—	2	1	no	—
463	#95	—	2	1	no	—
464	#95	—	2	1	no	—
465	#95	—	2	1	no	—
466	#95	—	2	1	no	—
467	#95	—	2	1	no	—
468	C₂⁴	—	2	1	no	—
469	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	#41×C₈₉	1	2	yes	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	4	2	no	—
483	C₂×D₄	—	4	2	no	—
484	C₂×D₄	#41×C₈₉	1	2	yes	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	4	1	no	—
502	C₂²×C₄	—	4	1	no	—
503	C₂²×C₄	—	4	1	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
510	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
511	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
512	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
513	SD₁₆	—	4	4	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	1	no	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	4	2	no	—
538	C₄⋊C₄	—	4	2	no	—
539	C₄⋊C₄	—	4	2	no	—
540	C₄⋊C₄	#41×C₈₉	1	2	yes	—
541	C₄⋊C₄	#41×C₈₉	1	2	yes	—
542	C₂²⋊C₄	—	2	4	no	—
543	C₂²⋊C₄	—	2	4	no	—
544	C₂²⋊C₄	—	4	2	no	—
545	C₂²⋊C₄	—	4	2	no	—
546	C₂²⋊C₄	—	4	2	no	—
547	C₂²⋊C₄	—	4	2	no	—
548	C₂²⋊C₄	—	4	2	no	—
549	C₄²	—	2	1	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	4	1	no	—
554	C₄²	—	4	1	no	—
555	C₄²	—	4	2	no	—
556	C₄²	—	4	2	no	—
557	C₄²	—	4	2	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	2	1	no	—
560	C₂³	—	2	1	no	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	#137×C₈₉	1	2	yes	—
572	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	4	1	no	—
576	Q₈	—	4	1	no	—
577	D₄	—	2	2	no	—
578	D₄	—	2	2	no	—
579	D₄	—	4	8	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	4	4	no	—
599	C₂×C₄	—	4	4	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	8	4	no	—
623	C₂×C₄	#135×C₈₉	1	2	yes	—
624	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
625	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
626	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
627	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
628	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
629	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
630	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
631	C₈	—	4	1	no	—
632	C₈	—	4	1	no	—
633	C₈	—	4	1	no	—
634	C₈	—	4	1	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	4	1	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	8	1	no	—
646	C₂²	—	8	1	no	—
647	C₂²	—	8	1	no	—
648	C₂²	—	8	1	no	—
649	C₂²	#525×C₈₉	1	1	yes	characteristic
650	C₂²	#525×C₈₉	1	1	yes	characteristic
651	C₂²	#534×C₈₉	1	1	yes	characteristic
652	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
653	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
654	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
655	C₂²	#575×C₈₉	1	1	yes	characteristic
656	C₄	—	2	2	no	—
657	C₄	—	2	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	2	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	1	no	—
664	C₄	—	8	1	no	—
665	C₄	—	8	1	no	—
666	C₄	—	8	1	no	—
667	C₄	—	8	1	no	—
668	C₄	#551×C₈₉	1	1	yes	characteristic
669	C₄	#553×C₈₉	1	1	yes	characteristic
670	C₄	#553×C₈₉	1	1	yes	characteristic
671	C₄	#559×C₈₉	1	1	yes	characteristic
672	C₂	—	4	2	no	—
673	C₂	—	8	2	no	—
674	C₂	C₈₉×#2600	1	1	yes	characteristic
675	C₂	C₈₉×#3102	1	1	yes	characteristic
676	C₂	C₈₉×#3106	1	1	yes	characteristic
677	C₂	C₈₉×#3110	1	1	yes	characteristic
678	C₂	C₈₉×#3117	1	1	yes	characteristic
679	C₂	C₈₉×#3190	1	1	yes	characteristic
680	C₂	C₈₉×#3194	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 ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₂²⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2076	C₂	semidirect
4. C₈₉×#2036	C₂	semidirect
7. C₈₉×#1927	C₂	semidirect
8. C₈₉×#1925	C₂	semidirect
9. C₈₉×#1920	C₂	semidirect
10. C₈₉×#1918	C₂	semidirect
11. C₈₉×#1880	C₂	semidirect
12. C₈₉×#1877	C₂	semidirect
13. C₈₉×#1863	C₂	semidirect
14. C₈₉×#1862	C₂	semidirect
15. C₈₉×#1817	C₂	semidirect
331. #29414	C₈₉	direct (N × H)
341. C₄×C₂²⋊Q₈	C₂×C₈₉	semidirect
343. #2076	C₂×C₈₉	semidirect
344. #2036	C₂×C₈₉	semidirect
347. #1927	C₂×C₈₉	semidirect
348. #1925	C₂×C₈₉	semidirect
349. #1920	C₂×C₈₉	semidirect
350. #1918	C₂×C₈₉	semidirect
351. #1880	C₂×C₈₉	semidirect
352. #1877	C₂×C₈₉	semidirect
353. #1863	C₂×C₈₉	semidirect
354. #1862	C₂×C₈₉	semidirect
355. #1817	C₂×C₈₉	semidirect
356. C₈₉	#29414	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊Q₈×C₈₉	2
C₄×#121×C₈₉	2
C₈₉×#2076	2
C₈₉×#2036	2
C₈₉×#2034	2
C₈₉×#2000	2
C₈₉×#1927	2
C₈₉×#1925	2
C₈₉×#1920	2
C₈₉×#1918	2
C₈₉×#1880	2
C₈₉×#1877	2
C₈₉×#1863	2
C₈₉×#1862	2
C₈₉×#1817	2
#29414	89

Subgroups — C89×#29414 #36407262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29414 #36407262