Subgroups — C₈₉×#26102 #36403950

Order 22784.

708 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2552	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2357	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2094	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2093	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2067	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2065	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1947	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1940	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1919	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#1847	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1802	C₂	1	1	yes	maximal, characteristic
15	C₈₉×#1762	C₂	1	1	yes	maximal, characteristic
16	C₂×#125×C₈₉	—	2	1	no	—
17	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#102×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×#97×C₈₉	C₂²	1	1	yes	characteristic
32	#398×C₈₉	—	2	1	no	—
33	#398×C₈₉	—	2	1	no	—
34	#390×C₈₉	C₂²	1	1	yes	characteristic
35	#390×C₈₉	C₂²	1	1	yes	characteristic
36	#389×C₈₉	—	2	1	no	—
37	#389×C₈₉	C₂²	1	1	yes	characteristic
38	#389×C₈₉	C₂²	1	1	yes	characteristic
39	#388×C₈₉	—	2	1	no	—
40	#388×C₈₉	—	2	1	no	—
41	#384×C₈₉	C₂²	1	1	yes	characteristic
42	#384×C₈₉	C₂²	1	1	yes	characteristic
43	#384×C₈₉	C₂²	1	1	yes	characteristic
44	#382×C₈₉	C₂²	1	1	yes	characteristic
45	#381×C₈₉	—	2	1	no	—
46	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
49	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₈₉	—	2	1	no	—
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	#340×C₈₉	C₂²	1	1	yes	characteristic
56	#340×C₈₉	C₂²	1	1	yes	characteristic
57	#336×C₈₉	C₂²	1	1	yes	characteristic
58	#336×C₈₉	C₂²	1	1	yes	characteristic
59	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₈₉	—	2	1	no	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	#125×C₈₉	—	4	4	no	—
66	C₄×D₄×C₈₉	—	2	2	no	—
67	C₄×D₄×C₈₉	Q₈	1	4	yes	—
68	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
69	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
71	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
74	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
90	C₂×C₄²×C₈₉	—	2	1	no	—
91	C₂×C₄²×C₈₉	—	2	1	no	—
92	C₂×C₄²×C₈₉	—	2	1	no	—
93	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
96	#107×C₈₉	—	2	2	no	—
97	#107×C₈₉	—	4	1	no	—
98	#107×C₈₉	—	4	1	no	—
99	#106×C₈₉	—	2	2	no	—
100	#106×C₈₉	—	4	1	no	—
101	#106×C₈₉	—	4	1	no	—
102	C₄⋊C₈×C₈₉	—	2	2	no	—
103	C₄⋊C₈×C₈₉	—	2	2	no	—
104	C₄⋊C₈×C₈₉	—	2	2	no	—
105	#102×C₈₉	—	2	2	no	—
106	#102×C₈₉	—	2	2	no	—
107	#102×C₈₉	—	4	1	no	—
108	#102×C₈₉	—	4	1	no	—
109	#97×C₈₉	—	2	2	no	—
110	#95×C₈₉	—	2	1	no	—
111	#95×C₈₉	—	2	1	no	—
112	#95×C₈₉	—	2	1	no	—
113	#95×C₈₉	—	2	1	no	—
114	#95×C₈₉	—	2	1	no	—
115	#95×C₈₉	—	2	1	no	—
116	#95×C₈₉	—	2	1	no	—
117	#95×C₈₉	D₄	1	1	yes	characteristic
118	#95×C₈₉	D₄	1	1	yes	characteristic
119	C₂⁴×C₈₉	—	2	1	no	—
120	C₂×D₄×C₈₉	—	2	1	no	—
121	C₂×D₄×C₈₉	—	2	1	no	—
122	C₂×D₄×C₈₉	—	2	1	no	—
123	C₂×D₄×C₈₉	—	2	1	no	—
124	C₂×D₄×C₈₉	—	2	1	no	—
125	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
126	C₂²×C₄×C₈₉	—	2	1	no	—
127	C₂²×C₄×C₈₉	—	2	4	no	—
128	C₂²×C₄×C₈₉	—	2	1	no	—
129	C₂²×C₄×C₈₉	—	2	1	no	—
130	C₂²×C₄×C₈₉	—	2	1	no	—
131	C₂²×C₄×C₈₉	—	2	1	no	—
132	C₂²×C₄×C₈₉	—	2	1	no	—
133	C₂²×C₄×C₈₉	—	2	1	no	—
134	C₂²×C₄×C₈₉	—	8	1	no	—
135	C₂²×C₄×C₈₉	—	8	1	no	—
136	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
143	C₂×C₈×C₈₉	—	2	1	no	—
144	C₂×C₈×C₈₉	—	2	1	no	—
145	C₂×C₈×C₈₉	—	2	1	no	—
146	C₂×C₈×C₈₉	—	2	1	no	—
147	C₂×C₈×C₈₉	—	2	1	no	—
148	C₂×C₈×C₈₉	—	2	1	no	—
149	C₂×C₈×C₈₉	—	2	1	no	—
150	C₂×C₈×C₈₉	—	2	1	no	—
151	C₂×C₈×C₈₉	—	2	1	no	—
152	C₂×C₈×C₈₉	—	2	1	no	—
153	C₂×C₈×C₈₉	#41	1	2	yes	—
154	C₂×C₈×C₈₉	#41	1	2	yes	—
155	C₄⋊C₄×C₈₉	—	2	2	no	—
156	C₄⋊C₄×C₈₉	—	2	2	no	—
157	C₄⋊C₄×C₈₉	—	2	1	no	—
158	C₄⋊C₄×C₈₉	—	2	1	no	—
159	C₄⋊C₄×C₈₉	—	2	1	no	—
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	—	2	1	no	—
168	C₄⋊C₄×C₈₉	—	2	1	no	—
169	C₄⋊C₄×C₈₉	—	2	1	no	—
170	C₄⋊C₄×C₈₉	—	4	2	no	—
171	C₄⋊C₄×C₈₉	—	4	2	no	—
172	C₄⋊C₄×C₈₉	—	4	2	no	—
173	C₄⋊C₄×C₈₉	—	4	2	no	—
174	C₄⋊C₄×C₈₉	—	4	2	no	—
175	C₄⋊C₄×C₈₉	—	8	1	no	—
176	C₄⋊C₄×C₈₉	—	8	1	no	—
177	C₄⋊C₄×C₈₉	—	8	1	no	—
178	C₄⋊C₄×C₈₉	C₂×Q₈	1	2	yes	—
179	C₂²⋊C₄×C₈₉	—	2	4	no	—
180	C₂²⋊C₄×C₈₉	—	4	2	no	—
181	C₂²⋊C₄×C₈₉	—	4	2	no	—
182	C₂²⋊C₄×C₈₉	—	8	1	no	—
183	C₂²⋊C₄×C₈₉	—	8	1	no	—
184	C₄²×C₈₉	—	2	1	no	—
185	C₄²×C₈₉	—	2	1	no	—
186	C₄²×C₈₉	—	2	1	no	—
187	C₄²×C₈₉	—	2	1	no	—
188	C₄²×C₈₉	—	4	2	no	—
189	C₄²×C₈₉	—	8	1	no	—
190	C₄²×C₈₉	—	8	1	no	—
191	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
192	C₄²×C₈₉	#41	1	2	yes	—
193	C₂³×C₈₉	—	2	2	no	—
194	C₂³×C₈₉	—	4	1	no	—
195	C₂³×C₈₉	—	4	1	no	—
196	C₂³×C₈₉	—	4	1	no	—
197	C₂³×C₈₉	—	4	1	no	—
198	C₂³×C₈₉	—	4	1	no	—
199	C₂³×C₈₉	—	4	1	no	—
200	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
201	D₄×C₈₉	—	2	2	no	—
202	D₄×C₈₉	—	2	2	no	—
203	D₄×C₈₉	—	4	1	no	—
204	D₄×C₈₉	—	4	1	no	—
205	C₂×C₄×C₈₉	—	2	1	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	—	2	1	no	—
213	C₂×C₄×C₈₉	—	2	2	no	—
214	C₂×C₄×C₈₉	—	2	1	no	—
215	C₂×C₄×C₈₉	—	2	1	no	—
216	C₂×C₄×C₈₉	—	2	1	no	—
217	C₂×C₄×C₈₉	—	2	1	no	—
218	C₂×C₄×C₈₉	—	2	1	no	—
219	C₂×C₄×C₈₉	—	2	1	no	—
220	C₂×C₄×C₈₉	—	4	1	no	—
221	C₂×C₄×C₈₉	—	4	1	no	—
222	C₂×C₄×C₈₉	—	4	1	no	—
223	C₂×C₄×C₈₉	—	4	1	no	—
224	C₂×C₄×C₈₉	—	4	1	no	—
225	C₂×C₄×C₈₉	—	4	1	no	—
226	C₂×C₄×C₈₉	—	4	1	no	—
227	C₂×C₄×C₈₉	—	4	1	no	—
228	C₂×C₄×C₈₉	—	4	1	no	—
229	C₂×C₄×C₈₉	—	4	1	no	—
230	C₂×C₄×C₈₉	—	4	1	no	—
231	C₂×C₄×C₈₉	—	4	1	no	—
232	C₂×C₄×C₈₉	—	4	1	no	—
233	C₂×C₄×C₈₉	—	4	1	no	—
234	C₂×C₄×C₈₉	—	4	1	no	—
235	C₂×C₄×C₈₉	—	4	1	no	—
236	C₂×C₄×C₈₉	—	4	1	no	—
237	C₂×C₄×C₈₉	—	4	1	no	—
238	C₂×C₄×C₈₉	—	4	1	no	—
239	C₂×C₄×C₈₉	—	4	1	no	—
240	C₂×C₄×C₈₉	—	8	4	no	—
241	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
242	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
243	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
244	C₂×C₄×C₈₉	#137	1	2	yes	—
245	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
246	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
247	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
251	C₈×C₈₉	—	2	2	no	—
252	C₈×C₈₉	—	2	2	no	—
253	C₈×C₈₉	—	4	1	no	—
254	C₈×C₈₉	—	4	1	no	—
255	C₂²×C₈₉	—	4	2	no	—
256	C₂²×C₈₉	—	4	2	no	—
257	C₂²×C₈₉	—	4	2	no	—
258	C₂²×C₈₉	—	8	1	no	—
259	C₂²×C₈₉	—	8	1	no	—
260	C₂²×C₈₉	—	8	1	no	—
261	C₂²×C₈₉	—	8	1	no	—
262	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂²×C₈₉	#547	1	1	yes	characteristic
264	C₂²×C₈₉	#550	1	1	yes	characteristic
265	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
267	C₂²×C₈₉	#576	1	1	yes	characteristic
268	C₂²×C₈₉	#577	1	1	yes	characteristic
269	C₄×C₈₉	—	2	2	no	—
270	C₄×C₈₉	—	2	2	no	—
271	C₄×C₈₉	—	2	2	no	—
272	C₄×C₈₉	—	2	2	no	—
273	C₄×C₈₉	—	4	2	no	—
274	C₄×C₈₉	—	4	1	no	—
275	C₄×C₈₉	—	4	1	no	—
276	C₄×C₈₉	—	8	1	no	—
277	C₄×C₈₉	—	8	1	no	—
278	C₄×C₈₉	—	8	1	no	—
279	C₄×C₈₉	—	8	1	no	—
280	C₄×C₈₉	—	8	1	no	—
281	C₄×C₈₉	—	8	1	no	—
282	C₄×C₈₉	#533	1	1	yes	characteristic
283	C₄×C₈₉	#544	1	1	yes	characteristic
284	C₄×C₈₉	#552	1	1	yes	characteristic
285	C₄×C₈₉	#553	1	1	yes	characteristic
286	#26102	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₈₉	—	8	2	no	—
288	C₂×C₈₉	#2441	1	1	yes	characteristic
289	C₂×C₈₉	#3079	1	1	yes	characteristic
290	C₂×C₈₉	#3081	1	1	yes	characteristic
291	C₂×C₈₉	#3345	1	1	yes	characteristic
292	C₂×C₈₉	#3346	1	1	yes	characteristic
293	C₂×C₈₉	#3399	1	1	yes	characteristic
294	C₂×C₈₉	#3400	1	1	yes	characteristic
295	#2552	C₂×C₈₉	1	1	yes	characteristic, complemented
296	#2357	C₂×C₈₉	1	1	yes	characteristic
297	#2094	C₂×C₈₉	1	1	yes	characteristic
298	#2093	C₂×C₈₉	1	1	yes	characteristic
299	#2067	C₂×C₈₉	1	1	yes	characteristic
300	#2065	C₂×C₈₉	1	1	yes	characteristic
301	#1949	C₂×C₈₉	1	1	yes	characteristic, complemented
302	#1948	C₂×C₈₉	1	1	yes	characteristic, complemented
303	#1947	C₂×C₈₉	1	1	yes	characteristic, complemented
304	#1940	C₂×C₈₉	1	1	yes	characteristic, complemented
305	#1921	C₂×C₈₉	1	1	yes	characteristic, complemented
306	#1919	C₂×C₈₉	1	1	yes	characteristic, complemented
307	#1847	C₂×C₈₉	1	1	yes	characteristic, complemented
308	#1802	C₂×C₈₉	1	1	yes	characteristic
309	#1762	C₂×C₈₉	1	1	yes	characteristic
310	C₈₉	#26102	1	1	yes	characteristic, direct product factor, Sylow
311	C₂×#125	—	2	1	no	—
312	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
313	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
314	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
315	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
316	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
317	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
318	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
319	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
320	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
321	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
322	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
323	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
324	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
325	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
326	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
327	#398	—	2	1	no	—
328	#398	—	2	1	no	—
329	#390	C₂²×C₈₉	1	1	yes	characteristic
330	#390	C₂²×C₈₉	1	1	yes	characteristic
331	#389	—	2	1	no	—
332	#389	C₂²×C₈₉	1	1	yes	characteristic
333	#389	C₂²×C₈₉	1	1	yes	characteristic
334	#388	—	2	1	no	—
335	#388	—	2	1	no	—
336	#384	C₂²×C₈₉	1	1	yes	characteristic
337	#384	C₂²×C₈₉	1	1	yes	characteristic
338	#384	C₂²×C₈₉	1	1	yes	characteristic
339	#382	C₂²×C₈₉	1	1	yes	characteristic
340	#381	—	2	1	no	—
341	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
342	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
343	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
344	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
345	C₄×C₂²⋊C₄	—	2	1	no	—
346	#340	C₂²×C₈₉	1	1	yes	characteristic
347	#340	C₂²×C₈₉	1	1	yes	characteristic
348	#340	C₂²×C₈₉	1	1	yes	characteristic
349	#340	C₂²×C₈₉	1	1	yes	characteristic
350	#340	C₂²×C₈₉	1	1	yes	characteristic
351	#340	C₂²×C₈₉	1	1	yes	characteristic
352	#336	C₂²×C₈₉	1	1	yes	characteristic
353	#336	C₂²×C₈₉	1	1	yes	characteristic
354	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
355	C₂³×C₄	—	2	1	no	—
356	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
357	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
358	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
359	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
360	#125	—	4	4	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	Q₈×C₈₉	1	4	yes	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
372	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	—	2	1	no	—
388	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
391	#107	—	2	2	no	—
392	#107	—	4	1	no	—
393	#107	—	4	1	no	—
394	#106	—	2	2	no	—
395	#106	—	4	1	no	—
396	#106	—	4	1	no	—
397	C₄⋊C₈	—	2	2	no	—
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	—	2	2	no	—
400	#102	—	2	2	no	—
401	#102	—	2	2	no	—
402	#102	—	4	1	no	—
403	#102	—	4	1	no	—
404	#97	—	2	2	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	#95	—	2	1	no	—
410	#95	—	2	1	no	—
411	#95	—	2	1	no	—
412	#95	D₄×C₈₉	1	1	yes	characteristic
413	#95	D₄×C₈₉	1	1	yes	characteristic
414	C₂⁴	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	8	1	no	—
430	C₂²×C₄	—	8	1	no	—
431	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	#41×C₈₉	1	2	yes	—
449	C₂×C₈	#41×C₈₉	1	2	yes	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	4	2	no	—
466	C₄⋊C₄	—	4	2	no	—
467	C₄⋊C₄	—	4	2	no	—
468	C₄⋊C₄	—	4	2	no	—
469	C₄⋊C₄	—	4	2	no	—
470	C₄⋊C₄	—	8	1	no	—
471	C₄⋊C₄	—	8	1	no	—
472	C₄⋊C₄	—	8	1	no	—
473	C₄⋊C₄	C₂×Q₈×C₈₉	1	2	yes	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	4	2	no	—
476	C₂²⋊C₄	—	4	2	no	—
477	C₂²⋊C₄	—	8	1	no	—
478	C₂²⋊C₄	—	8	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	4	2	no	—
484	C₄²	—	8	1	no	—
485	C₄²	—	8	1	no	—
486	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
487	C₄²	#41×C₈₉	1	2	yes	—
488	C₂³	—	2	2	no	—
489	C₂³	—	4	1	no	—
490	C₂³	—	4	1	no	—
491	C₂³	—	4	1	no	—
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	1	no	—
495	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
496	D₄	—	2	2	no	—
497	D₄	—	2	2	no	—
498	D₄	—	4	1	no	—
499	D₄	—	4	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	8	4	no	—
536	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
537	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
538	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
539	C₂×C₄	#137×C₈₉	1	2	yes	—
540	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
541	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
542	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
543	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
544	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
545	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
546	C₈	—	2	2	no	—
547	C₈	—	2	2	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	8	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
558	C₂²	#547×C₈₉	1	1	yes	characteristic
559	C₂²	#550×C₈₉	1	1	yes	characteristic
560	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
561	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
562	C₂²	#576×C₈₉	1	1	yes	characteristic
563	C₂²	#577×C₈₉	1	1	yes	characteristic
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	4	2	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	8	1	no	—
572	C₄	—	8	1	no	—
573	C₄	—	8	1	no	—
574	C₄	—	8	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	#533×C₈₉	1	1	yes	characteristic
578	C₄	#544×C₈₉	1	1	yes	characteristic
579	C₄	#552×C₈₉	1	1	yes	characteristic
580	C₄	#553×C₈₉	1	1	yes	characteristic
581	C₂	—	8	2	no	—
582	C₂	C₈₉×#2441	1	1	yes	characteristic
583	C₂	C₈₉×#3079	1	1	yes	characteristic
584	C₂	C₈₉×#3081	1	1	yes	characteristic
585	C₂	C₈₉×#3345	1	1	yes	characteristic
586	C₂	C₈₉×#3346	1	1	yes	characteristic
587	C₂	C₈₉×#3399	1	1	yes	characteristic
588	C₂	C₈₉×#3400	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
1. C₈₉×#2552	C₂	semidirect
7. C₈₉×#1949	C₂	semidirect
8. C₈₉×#1948	C₂	semidirect
9. C₈₉×#1947	C₂	semidirect
10. C₈₉×#1940	C₂	semidirect
11. C₈₉×#1921	C₂	semidirect
12. C₈₉×#1919	C₂	semidirect
13. C₈₉×#1847	C₂	semidirect
286. #26102	C₈₉	direct (N × H)
295. #2552	C₂×C₈₉	semidirect
301. #1949	C₂×C₈₉	semidirect
302. #1948	C₂×C₈₉	semidirect
303. #1947	C₂×C₈₉	semidirect
304. #1940	C₂×C₈₉	semidirect
305. #1921	C₂×C₈₉	semidirect
306. #1919	C₂×C₈₉	semidirect
307. #1847	C₂×C₈₉	semidirect
310. C₈₉	#26102	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2552	2
C₈₉×#2357	2
C₈₉×#2094	2
C₈₉×#2093	2
C₈₉×#2067	2
C₈₉×#2065	2
C₈₉×#1949	2
C₈₉×#1948	2
C₈₉×#1947	2
C₈₉×#1940	2
C₈₉×#1921	2
C₈₉×#1919	2
C₈₉×#1847	2
C₈₉×#1802	2
C₈₉×#1762	2
#26102	89

Subgroups — C89×#26102 #36403950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#26102 #36403950