Subgroups — C₈₉×#35287 #36413135

Order 22784.

1886 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2860	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2854	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2647	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2641	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2629	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2619	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2593	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2554	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2468	C₂	1	4	yes	maximal, complemented
10	C₈₉×#2457	C₂	1	4	yes	maximal, complemented
11	C₈₉×#2324	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2319	C₂	1	2	yes	maximal, complemented
13	C₄×#127×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#125×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#126×C₈₉	C₂²	1	4	yes	—
18	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#123×C₈₉	C₂²	1	4	yes	—
20	C₂×#121×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
23	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
26	#400×C₈₉	C₂²	1	4	yes	—
27	#398×C₈₉	C₂²	1	4	yes	—
28	#398×C₈₉	C₂²	1	4	yes	—
29	#396×C₈₉	C₂²	1	4	yes	—
30	#394×C₈₉	C₂²	1	4	yes	—
31	#394×C₈₉	C₂²	1	4	yes	—
32	#394×C₈₉	C₂²	1	4	yes	—
33	#392×C₈₉	C₂²	1	4	yes	—
34	#390×C₈₉	C₂²	1	4	yes	—
35	#390×C₈₉	C₂²	1	4	yes	—
36	#390×C₈₉	C₂²	1	2	yes	—
37	#390×C₈₉	C₂²	1	2	yes	—
38	#390×C₈₉	C₂²	1	2	yes	—
39	#390×C₈₉	C₂²	1	2	yes	—
40	#389×C₈₉	C₂²	1	2	yes	—
41	#389×C₈₉	C₂²	1	2	yes	—
42	#389×C₈₉	C₂²	1	2	yes	—
43	#389×C₈₉	C₂²	1	2	yes	—
44	#388×C₈₉	C₂²	1	4	yes	—
45	#388×C₈₉	C₂²	1	4	yes	—
46	#388×C₈₉	C₂²	1	4	yes	—
47	#388×C₈₉	C₂²	1	4	yes	—
48	#388×C₈₉	C₂²	1	4	yes	—
49	#388×C₈₉	C₂²	1	4	yes	—
50	#388×C₈₉	C₂²	1	4	yes	—
51	#388×C₈₉	C₂²	1	4	yes	—
52	#387×C₈₉	C₂²	1	4	yes	—
53	#386×C₈₉	C₂²	1	4	yes	—
54	#386×C₈₉	C₂²	1	4	yes	—
55	#385×C₈₉	C₂²	1	4	yes	—
56	#384×C₈₉	C₂²	1	1	yes	characteristic
57	#382×C₈₉	C₂²	1	1	yes	characteristic
58	#381×C₈₉	C₂²	1	4	yes	—
59	#380×C₈₉	C₂²	1	4	yes	—
60	#380×C₈₉	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
67	#376×C₈₉	C₂²	1	1	yes	characteristic
68	#376×C₈₉	C₂²	1	1	yes	characteristic
69	C₄³×C₈₉	C₂²	1	1	yes	characteristic
70	C₄³×C₈₉	C₂²	1	1	yes	characteristic
71	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
72	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
73	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
74	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
75	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
77	#127×C₈₉	—	2	4	no	—
78	#126×C₈₉	—	4	8	no	—
79	#125×C₈₉	—	2	4	no	—
80	#123×C₈₉	—	4	8	no	—
81	#121×C₈₉	—	2	8	no	—
82	C₄×D₄×C₈₉	—	2	8	no	—
83	C₄×D₄×C₈₉	—	4	8	no	—
84	#117×C₈₉	—	2	4	no	—
85	#117×C₈₉	—	2	4	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	2	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
112	C₂×C₄²×C₈₉	—	2	4	no	—
113	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
117	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
118	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
120	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
121	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
122	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
123	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
125	#95×C₈₉	C₂³	1	2	yes	—
126	#95×C₈₉	C₂³	1	2	yes	—
127	#95×C₈₉	C₂³	1	2	yes	—
128	#95×C₈₉	C₂³	1	2	yes	—
129	#95×C₈₉	C₂³	1	4	yes	—
130	#95×C₈₉	C₂³	1	4	yes	—
131	#95×C₈₉	C₂³	1	4	yes	—
132	#95×C₈₉	C₂³	1	4	yes	—
133	#95×C₈₉	C₂³	1	4	yes	—
134	#95×C₈₉	C₂³	1	4	yes	—
135	#95×C₈₉	C₂³	1	4	yes	—
136	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
137	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₈₉	—	2	8	no	—
139	C₂×D₄×C₈₉	—	2	4	no	—
140	C₂×D₄×C₈₉	—	2	4	no	—
141	C₂×D₄×C₈₉	—	2	8	no	—
142	C₂×D₄×C₈₉	—	2	4	no	—
143	C₂×D₄×C₈₉	—	4	4	no	—
144	C₂×D₄×C₈₉	—	4	4	no	—
145	C₂×D₄×C₈₉	—	4	4	no	—
146	C₂×D₄×C₈₉	—	4	4	no	—
147	C₂²×C₄×C₈₉	—	2	2	no	—
148	C₂²×C₄×C₈₉	—	2	2	no	—
149	C₂²×C₄×C₈₉	—	2	4	no	—
150	C₂²×C₄×C₈₉	—	2	4	no	—
151	C₂²×C₄×C₈₉	—	2	8	no	—
152	C₂²×C₄×C₈₉	—	4	2	no	—
153	C₂²×C₄×C₈₉	—	4	4	no	—
154	C₂²×C₄×C₈₉	—	4	4	no	—
155	C₂²×C₄×C₈₉	#41	1	4	yes	—
156	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₈₉	—	2	2	no	—
170	C₄⋊C₄×C₈₉	—	2	4	no	—
171	C₄⋊C₄×C₈₉	—	2	2	no	—
172	C₄⋊C₄×C₈₉	—	2	4	no	—
173	C₄⋊C₄×C₈₉	—	2	4	no	—
174	C₄⋊C₄×C₈₉	—	2	8	no	—
175	C₄⋊C₄×C₈₉	—	2	4	no	—
176	C₄⋊C₄×C₈₉	—	2	8	no	—
177	C₄⋊C₄×C₈₉	—	4	4	no	—
178	C₄⋊C₄×C₈₉	—	4	4	no	—
179	C₄⋊C₄×C₈₉	—	4	4	no	—
180	C₄⋊C₄×C₈₉	—	4	4	no	—
181	C₂²⋊C₄×C₈₉	—	2	2	no	—
182	C₂²⋊C₄×C₈₉	—	2	4	no	—
183	C₂²⋊C₄×C₈₉	—	2	2	no	—
184	C₂²⋊C₄×C₈₉	—	2	8	no	—
185	C₂²⋊C₄×C₈₉	—	2	8	no	—
186	C₂²⋊C₄×C₈₉	—	2	8	no	—
187	C₂²⋊C₄×C₈₉	—	2	8	no	—
188	C₂²⋊C₄×C₈₉	—	4	4	no	—
189	C₂²⋊C₄×C₈₉	—	4	4	no	—
190	C₂²⋊C₄×C₈₉	—	4	4	no	—
191	C₂²⋊C₄×C₈₉	—	4	4	no	—
192	C₂²⋊C₄×C₈₉	—	4	4	no	—
193	C₂²⋊C₄×C₈₉	—	4	4	no	—
194	C₂²⋊C₄×C₈₉	—	4	4	no	—
195	C₄²×C₈₉	—	2	2	no	—
196	C₄²×C₈₉	—	2	4	no	—
197	C₄²×C₈₉	—	2	2	no	—
198	C₄²×C₈₉	—	2	2	no	—
199	C₄²×C₈₉	—	2	2	no	—
200	C₄²×C₈₉	—	2	4	no	—
201	C₄²×C₈₉	—	2	4	no	—
202	C₄²×C₈₉	—	4	2	no	—
203	C₄²×C₈₉	—	4	4	no	—
204	C₄²×C₈₉	—	4	2	no	—
205	C₄²×C₈₉	#41	1	4	yes	—
206	C₄²×C₈₉	#41	1	4	yes	—
207	C₂³×C₈₉	—	2	1	no	—
208	C₂³×C₈₉	—	2	1	no	—
209	C₂³×C₈₉	—	2	2	no	—
210	C₂³×C₈₉	—	2	2	no	—
211	C₂³×C₈₉	—	2	4	no	—
212	C₂³×C₈₉	—	2	4	no	—
213	C₂³×C₈₉	—	2	4	no	—
214	C₂³×C₈₉	—	2	4	no	—
215	C₂³×C₈₉	—	2	4	no	—
216	C₂³×C₈₉	—	2	4	no	—
217	C₂³×C₈₉	—	2	4	no	—
218	C₂³×C₈₉	C₂×#41	1	2	yes	—
219	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
220	D₄×C₈₉	—	4	16	no	—
221	D₄×C₈₉	—	4	8	no	—
222	D₄×C₈₉	—	8	8	no	—
223	C₂×C₄×C₈₉	—	2	2	no	—
224	C₂×C₄×C₈₉	—	2	1	no	—
225	C₂×C₄×C₈₉	—	2	1	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	2	no	—
228	C₂×C₄×C₈₉	—	2	2	no	—
229	C₂×C₄×C₈₉	—	2	2	no	—
230	C₂×C₄×C₈₉	—	2	2	no	—
231	C₂×C₄×C₈₉	—	2	8	no	—
232	C₂×C₄×C₈₉	—	2	2	no	—
233	C₂×C₄×C₈₉	—	2	2	no	—
234	C₂×C₄×C₈₉	—	2	1	no	—
235	C₂×C₄×C₈₉	—	2	1	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	2	4	no	—
238	C₂×C₄×C₈₉	—	2	4	no	—
239	C₂×C₄×C₈₉	—	2	2	no	—
240	C₂×C₄×C₈₉	—	2	2	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	2	4	no	—
243	C₂×C₄×C₈₉	—	2	4	no	—
244	C₂×C₄×C₈₉	—	2	2	no	—
245	C₂×C₄×C₈₉	—	2	4	no	—
246	C₂×C₄×C₈₉	—	2	2	no	—
247	C₂×C₄×C₈₉	—	2	4	no	—
248	C₂×C₄×C₈₉	—	2	2	no	—
249	C₂×C₄×C₈₉	—	2	4	no	—
250	C₂×C₄×C₈₉	—	2	2	no	—
251	C₂×C₄×C₈₉	—	4	8	no	—
252	C₂×C₄×C₈₉	—	8	8	no	—
253	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
254	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
255	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
256	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
257	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
258	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
259	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
260	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
261	C₂²×C₈₉	—	2	2	no	—
262	C₂²×C₈₉	—	2	4	no	—
263	C₂²×C₈₉	—	4	2	no	—
264	C₂²×C₈₉	—	4	1	no	—
265	C₂²×C₈₉	—	4	1	no	—
266	C₂²×C₈₉	—	4	4	no	—
267	C₂²×C₈₉	—	4	4	no	—
268	C₂²×C₈₉	—	4	4	no	—
269	C₂²×C₈₉	—	4	4	no	—
270	C₂²×C₈₉	—	4	4	no	—
271	C₂²×C₈₉	—	4	4	no	—
272	C₂²×C₈₉	—	4	4	no	—
273	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
274	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
275	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
276	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
277	C₂²×C₈₉	#585	1	2	yes	—
278	C₄×C₈₉	—	2	2	no	—
279	C₄×C₈₉	—	2	4	no	—
280	C₄×C₈₉	—	2	4	no	—
281	C₄×C₈₉	—	2	2	no	—
282	C₄×C₈₉	—	2	2	no	—
283	C₄×C₈₉	—	4	2	no	—
284	C₄×C₈₉	—	4	4	no	—
285	C₄×C₈₉	—	4	2	no	—
286	C₄×C₈₉	—	4	4	no	—
287	C₄×C₈₉	—	4	1	no	—
288	C₄×C₈₉	—	4	4	no	—
289	C₄×C₈₉	—	4	1	no	—
290	C₄×C₈₉	#529	1	4	yes	—
291	#35287	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₈₉	—	4	2	no	—
293	C₂×C₈₉	—	8	4	no	—
294	C₂×C₈₉	C₂×#529	1	1	yes	characteristic
295	C₂×C₈₉	#3460	1	2	yes	—
296	C₂×C₈₉	#3494	1	2	yes	—
297	C₂×C₈₉	#3508	1	1	yes	characteristic
298	C₂×C₈₉	#3512	1	1	yes	characteristic
299	#2860	C₂×C₈₉	1	2	yes	complemented
300	#2854	C₂×C₈₉	1	2	yes	complemented
301	#2647	C₂×C₈₉	1	1	yes	characteristic, complemented
302	#2641	C₂×C₈₉	1	1	yes	characteristic, complemented
303	#2629	C₂×C₈₉	1	1	yes	characteristic, complemented
304	#2619	C₂×C₈₉	1	1	yes	characteristic
305	#2593	C₂×C₈₉	1	4	yes	complemented
306	#2554	C₂×C₈₉	1	4	yes	complemented
307	#2468	C₂×C₈₉	1	4	yes	complemented
308	#2457	C₂×C₈₉	1	4	yes	complemented
309	#2324	C₂×C₈₉	1	2	yes	complemented
310	#2319	C₂×C₈₉	1	2	yes	complemented
311	C₄×#127	C₂×C₈₉	1	1	yes	characteristic, complemented
312	C₄×#125	C₂×C₈₉	1	1	yes	characteristic, complemented
313	C₄×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
314	C₈₉	#35287	1	1	yes	characteristic, direct product factor, Sylow
315	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
316	C₂×#126	C₂²×C₈₉	1	4	yes	—
317	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
318	C₂×#123	C₂²×C₈₉	1	4	yes	—
319	C₂×#121	C₂²×C₈₉	1	2	yes	—
320	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
321	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
322	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
323	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
324	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
325	#400	C₂²×C₈₉	1	4	yes	—
326	#398	C₂²×C₈₉	1	4	yes	—
327	#398	C₂²×C₈₉	1	4	yes	—
328	#396	C₂²×C₈₉	1	4	yes	—
329	#394	C₂²×C₈₉	1	4	yes	—
330	#394	C₂²×C₈₉	1	4	yes	—
331	#394	C₂²×C₈₉	1	4	yes	—
332	#392	C₂²×C₈₉	1	4	yes	—
333	#390	C₂²×C₈₉	1	4	yes	—
334	#390	C₂²×C₈₉	1	2	yes	—
335	#390	C₂²×C₈₉	1	2	yes	—
336	#390	C₂²×C₈₉	1	2	yes	—
337	#390	C₂²×C₈₉	1	2	yes	—
338	#390	C₂²×C₈₉	1	4	yes	—
339	#389	C₂²×C₈₉	1	2	yes	—
340	#389	C₂²×C₈₉	1	2	yes	—
341	#389	C₂²×C₈₉	1	2	yes	—
342	#389	C₂²×C₈₉	1	2	yes	—
343	#388	C₂²×C₈₉	1	4	yes	—
344	#388	C₂²×C₈₉	1	4	yes	—
345	#388	C₂²×C₈₉	1	4	yes	—
346	#388	C₂²×C₈₉	1	4	yes	—
347	#388	C₂²×C₈₉	1	4	yes	—
348	#388	C₂²×C₈₉	1	4	yes	—
349	#388	C₂²×C₈₉	1	4	yes	—
350	#388	C₂²×C₈₉	1	4	yes	—
351	#387	C₂²×C₈₉	1	4	yes	—
352	#386	C₂²×C₈₉	1	4	yes	—
353	#386	C₂²×C₈₉	1	4	yes	—
354	#385	C₂²×C₈₉	1	4	yes	—
355	#384	C₂²×C₈₉	1	1	yes	characteristic
356	#382	C₂²×C₈₉	1	1	yes	characteristic
357	#381	C₂²×C₈₉	1	4	yes	—
358	#380	C₂²×C₈₉	1	4	yes	—
359	#380	C₂²×C₈₉	1	2	yes	—
360	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
361	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
362	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
363	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
365	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
366	#376	C₂²×C₈₉	1	1	yes	characteristic
367	#376	C₂²×C₈₉	1	1	yes	characteristic
368	C₄³	C₂²×C₈₉	1	1	yes	characteristic
369	C₄³	C₂²×C₈₉	1	1	yes	characteristic
370	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
371	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
372	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
373	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
374	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
376	#127	—	2	4	no	—
377	#126	—	4	8	no	—
378	#125	—	2	4	no	—
379	#123	—	4	8	no	—
380	#121	—	2	8	no	—
381	C₄×D₄	—	2	8	no	—
382	C₄×D₄	—	4	8	no	—
383	#117	—	2	4	no	—
384	#117	—	2	4	no	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
394	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
396	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
411	C₂×C₄²	—	2	4	no	—
412	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
415	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
416	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
419	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
420	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
421	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
422	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
424	#95	C₂³×C₈₉	1	2	yes	—
425	#95	C₂³×C₈₉	1	2	yes	—
426	#95	C₂³×C₈₉	1	2	yes	—
427	#95	C₂³×C₈₉	1	2	yes	—
428	#95	C₂³×C₈₉	1	4	yes	—
429	#95	C₂³×C₈₉	1	4	yes	—
430	#95	C₂³×C₈₉	1	4	yes	—
431	#95	C₂³×C₈₉	1	4	yes	—
432	#95	C₂³×C₈₉	1	4	yes	—
433	#95	C₂³×C₈₉	1	4	yes	—
434	#95	C₂³×C₈₉	1	4	yes	—
435	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
436	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
437	C₂×D₄	—	2	8	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	8	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	4	4	no	—
443	C₂×D₄	—	4	4	no	—
444	C₂×D₄	—	4	4	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	8	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	—	4	4	no	—
454	C₂²×C₄	#41×C₈₉	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
461	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
464	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
465	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
467	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	4	4	no	—
477	C₄⋊C₄	—	4	4	no	—
478	C₄⋊C₄	—	4	4	no	—
479	C₄⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	—	2	2	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	4	4	no	—
488	C₂²⋊C₄	—	4	4	no	—
489	C₂²⋊C₄	—	4	4	no	—
490	C₂²⋊C₄	—	4	4	no	—
491	C₂²⋊C₄	—	4	4	no	—
492	C₂²⋊C₄	—	4	4	no	—
493	C₂²⋊C₄	—	4	4	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	2	4	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	4	2	no	—
502	C₄²	—	4	4	no	—
503	C₄²	—	4	2	no	—
504	C₄²	#41×C₈₉	1	4	yes	—
505	C₄²	#41×C₈₉	1	4	yes	—
506	C₂³	—	2	1	no	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	4	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	4	no	—
513	C₂³	—	2	4	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	4	no	—
517	C₂³	C₂×#41×C₈₉	1	2	yes	—
518	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
519	D₄	—	4	16	no	—
520	D₄	—	4	8	no	—
521	D₄	—	8	8	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	8	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	4	8	no	—
551	C₂×C₄	—	8	8	no	—
552	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
553	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
554	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
555	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
556	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
557	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
558	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
559	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
573	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
574	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
575	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
576	C₂²	#585×C₈₉	1	2	yes	—
577	C₄	—	2	2	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	4	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	2	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	4	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	#529×C₈₉	1	4	yes	—
590	C₂	—	4	2	no	—
591	C₂	—	8	4	no	—
592	C₂	C₂×#529×C₈₉	1	1	yes	characteristic
593	C₂	C₈₉×#3460	1	2	yes	—
594	C₂	C₈₉×#3494	1	2	yes	—
595	C₂	C₈₉×#3508	1	1	yes	characteristic
596	C₂	C₈₉×#3512	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₈₉×#2619	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2860	C₂	semidirect
2. C₈₉×#2854	C₂	semidirect
3. C₈₉×#2647	C₂	semidirect
4. C₈₉×#2641	C₂	semidirect
5. C₈₉×#2629	C₂	semidirect
7. C₈₉×#2593	C₂	semidirect
8. C₈₉×#2554	C₂	semidirect
9. C₈₉×#2468	C₂	semidirect
10. C₈₉×#2457	C₂	semidirect
11. C₈₉×#2324	C₂	semidirect
12. C₈₉×#2319	C₂	semidirect
13. C₄×#127×C₈₉	C₂	semidirect
14. C₄×#125×C₈₉	C₂	semidirect
15. C₄×#117×C₈₉	C₂	semidirect
291. #35287	C₈₉	direct (N × H)
299. #2860	C₂×C₈₉	semidirect
300. #2854	C₂×C₈₉	semidirect
301. #2647	C₂×C₈₉	semidirect
302. #2641	C₂×C₈₉	semidirect
303. #2629	C₂×C₈₉	semidirect
305. #2593	C₂×C₈₉	semidirect
306. #2554	C₂×C₈₉	semidirect
307. #2468	C₂×C₈₉	semidirect
308. #2457	C₂×C₈₉	semidirect
309. #2324	C₂×C₈₉	semidirect
310. #2319	C₂×C₈₉	semidirect
311. C₄×#127	C₂×C₈₉	semidirect
312. C₄×#125	C₂×C₈₉	semidirect
313. C₄×#117	C₂×C₈₉	semidirect
314. C₈₉	#35287	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×#2860	2
C₈₉×#2854	2
C₈₉×#2647	2
C₈₉×#2641	2
C₈₉×#2629	2
C₈₉×#2619	2
C₈₉×#2593	2
C₈₉×#2554	2
C₈₉×#2468	2
C₈₉×#2457	2
C₈₉×#2324	2
C₈₉×#2319	2
C₄×#127×C₈₉	2
C₄×#125×C₈₉	2
C₄×#117×C₈₉	2
#35287	89

Subgroups — C89×#35287 #36413135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35287 #36413135