Subgroups — C₈₉×#24939 #36402787

Order 22784.

834 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2487	C₂	1	1	yes	maximal, characteristic
2	D₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2083	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2076	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2076	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1921	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1921	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1869	C₂	1	1	yes	maximal, characteristic
10	C₈₉×#1852	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1852	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1824	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#1824	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1808	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#340×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₈₉	—	2	1	no	—
18	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#107×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×#107×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#400×C₈₉	—	2	1	no	—
30	#399×C₈₉	—	2	1	no	—
31	#389×C₈₉	—	2	1	no	—
32	#389×C₈₉	C₂²	1	1	yes	characteristic
33	#389×C₈₉	C₂²	1	1	yes	characteristic
34	#386×C₈₉	—	2	1	no	—
35	#385×C₈₉	—	2	1	no	—
36	#385×C₈₉	—	2	1	no	—
37	#385×C₈₉	—	2	1	no	—
38	#384×C₈₉	C₂²	1	1	yes	characteristic
39	#380×C₈₉	C₂²	1	1	yes	characteristic
40	#380×C₈₉	C₂²	1	1	yes	characteristic
41	#380×C₈₉	C₂²	1	1	yes	characteristic
42	#380×C₈₉	C₂²	1	1	yes	characteristic
43	#380×C₈₉	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
45	#340×C₈₉	C₂²	1	1	yes	characteristic
46	#340×C₈₉	C₂²	1	1	yes	characteristic
47	#340×C₈₉	C₂²	1	1	yes	characteristic
48	#340×C₈₉	C₂²	1	1	yes	characteristic
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	#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	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₈₉	—	2	1	no	—
61	C₂³×C₄×C₈₉	—	2	1	no	—
62	C₂³×C₄×C₈₉	—	2	1	no	—
63	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	—	2	2	no	—
66	C₂²×C₈×C₈₉	—	2	2	no	—
67	C₂²×C₈×C₈₉	—	4	1	no	—
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
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	C₄×D₄×C₈₉	—	2	2	no	—
73	C₄×D₄×C₈₉	—	2	2	no	—
74	C₄×D₄×C₈₉	—	4	4	no	—
75	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
78	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	—	2	1	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	—	4	1	no	—
87	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
88	C₂×C₄⋊C₄×C₈₉	D₄	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×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×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₈₉	—	2	1	no	—
105	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	#107×C₈₉	—	2	2	no	—
107	#107×C₈₉	—	2	2	no	—
108	#107×C₈₉	—	4	1	no	—
109	#107×C₈₉	—	4	1	no	—
110	C₂²⋊C₈×C₈₉	—	2	2	no	—
111	C₂²⋊C₈×C₈₉	—	2	2	no	—
112	C₂²⋊C₈×C₈₉	—	4	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	—	2	1	no	—
118	#95×C₈₉	—	2	1	no	—
119	#95×C₈₉	—	2	1	no	—
120	#95×C₈₉	—	2	1	no	—
121	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×D₄×C₈₉	—	2	2	no	—
124	C₂×D₄×C₈₉	—	2	2	no	—
125	C₂×D₄×C₈₉	Q₁₆	1	4	yes	—
126	C₂²×C₄×C₈₉	—	2	1	no	—
127	C₂²×C₄×C₈₉	—	2	1	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	—	2	1	no	—
135	C₂²×C₄×C₈₉	—	2	1	no	—
136	C₂²×C₄×C₈₉	—	2	1	no	—
137	C₂²×C₄×C₈₉	—	2	1	no	—
138	C₂²×C₄×C₈₉	—	2	1	no	—
139	C₂²×C₄×C₈₉	—	2	1	no	—
140	C₂²×C₄×C₈₉	—	2	1	no	—
141	C₂²×C₄×C₈₉	—	2	1	no	—
142	C₂²×C₄×C₈₉	—	2	1	no	—
143	C₂²×C₄×C₈₉	—	2	1	no	—
144	C₂²×C₄×C₈₉	—	4	2	no	—
145	C₂²×C₄×C₈₉	—	4	2	no	—
146	C₂²×C₄×C₈₉	—	4	2	no	—
147	C₂²×C₄×C₈₉	—	4	2	no	—
148	C₂²×C₄×C₈₉	—	4	2	no	—
149	C₂²×C₄×C₈₉	—	8	1	no	—
150	C₂²×C₄×C₈₉	—	8	1	no	—
151	C₂²×C₄×C₈₉	—	8	1	no	—
152	C₂²×C₄×C₈₉	—	8	1	no	—
153	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₈₉	#41	1	2	yes	—
158	C₂²×C₄×C₈₉	#41	1	2	yes	—
159	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
160	C₂×C₈×C₈₉	—	2	1	no	—
161	C₂×C₈×C₈₉	—	2	1	no	—
162	C₂×C₈×C₈₉	—	2	1	no	—
163	C₂×C₈×C₈₉	—	2	1	no	—
164	C₂×C₈×C₈₉	—	2	1	no	—
165	C₂×C₈×C₈₉	—	2	1	no	—
166	C₂×C₈×C₈₉	—	2	1	no	—
167	C₂×C₈×C₈₉	—	2	1	no	—
168	C₂×C₈×C₈₉	—	2	1	no	—
169	C₂×C₈×C₈₉	—	2	1	no	—
170	C₂×C₈×C₈₉	—	2	1	no	—
171	C₂×C₈×C₈₉	—	4	4	no	—
172	C₂×C₈×C₈₉	#41	1	2	yes	—
173	C₄⋊C₄×C₈₉	—	2	2	no	—
174	C₄⋊C₄×C₈₉	—	2	1	no	—
175	C₄⋊C₄×C₈₉	—	2	1	no	—
176	C₄⋊C₄×C₈₉	—	2	1	no	—
177	C₄⋊C₄×C₈₉	—	2	1	no	—
178	C₄⋊C₄×C₈₉	—	2	1	no	—
179	C₄⋊C₄×C₈₉	—	2	1	no	—
180	C₄⋊C₄×C₈₉	—	2	1	no	—
181	C₄⋊C₄×C₈₉	—	2	1	no	—
182	C₄⋊C₄×C₈₉	—	2	1	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	—	4	2	no	—
190	C₄⋊C₄×C₈₉	—	4	2	no	—
191	C₄⋊C₄×C₈₉	—	4	2	no	—
192	C₄⋊C₄×C₈₉	—	4	2	no	—
193	C₄⋊C₄×C₈₉	—	8	1	no	—
194	C₄⋊C₄×C₈₉	—	8	1	no	—
195	C₄⋊C₄×C₈₉	—	8	1	no	—
196	C₄⋊C₄×C₈₉	#41	1	2	yes	—
197	C₂²⋊C₄×C₈₉	—	2	2	no	—
198	C₂²⋊C₄×C₈₉	—	2	2	no	—
199	C₂²⋊C₄×C₈₉	—	4	2	no	—
200	C₂²⋊C₄×C₈₉	—	4	2	no	—
201	C₂²⋊C₄×C₈₉	—	4	2	no	—
202	C₂²⋊C₄×C₈₉	—	8	1	no	—
203	C₂²⋊C₄×C₈₉	—	8	1	no	—
204	C₄²×C₈₉	—	2	1	no	—
205	C₄²×C₈₉	—	2	1	no	—
206	C₄²×C₈₉	—	4	2	no	—
207	C₂³×C₈₉	—	2	1	no	—
208	C₂³×C₈₉	—	2	1	no	—
209	C₂³×C₈₉	—	2	1	no	—
210	C₂³×C₈₉	—	2	1	no	—
211	C₂³×C₈₉	—	2	1	no	—
212	C₂³×C₈₉	—	2	1	no	—
213	C₂³×C₈₉	—	2	1	no	—
214	C₂³×C₈₉	—	2	1	no	—
215	C₂³×C₈₉	—	2	1	no	—
216	C₂³×C₈₉	—	2	1	no	—
217	C₂³×C₈₉	#123	1	2	yes	—
218	C₂³×C₈₉	C₂×Q₁₆	1	2	yes	—
219	C₂³×C₈₉	C₂×Q₁₆	1	2	yes	—
220	C₂³×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
221	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
222	D₄×C₈₉	—	2	8	no	—
223	C₂×C₄×C₈₉	—	2	1	no	—
224	C₂×C₄×C₈₉	—	2	1	no	—
225	C₂×C₄×C₈₉	—	2	1	no	—
226	C₂×C₄×C₈₉	—	2	1	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	1	no	—
232	C₂×C₄×C₈₉	—	2	1	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	1	no	—
237	C₂×C₄×C₈₉	—	2	1	no	—
238	C₂×C₄×C₈₉	—	4	1	no	—
239	C₂×C₄×C₈₉	—	4	1	no	—
240	C₂×C₄×C₈₉	—	4	1	no	—
241	C₂×C₄×C₈₉	—	4	1	no	—
242	C₂×C₄×C₈₉	—	4	1	no	—
243	C₂×C₄×C₈₉	—	4	1	no	—
244	C₂×C₄×C₈₉	—	4	1	no	—
245	C₂×C₄×C₈₉	—	4	1	no	—
246	C₂×C₄×C₈₉	—	4	1	no	—
247	C₂×C₄×C₈₉	—	4	1	no	—
248	C₂×C₄×C₈₉	—	4	1	no	—
249	C₂×C₄×C₈₉	—	4	1	no	—
250	C₂×C₄×C₈₉	—	4	1	no	—
251	C₂×C₄×C₈₉	—	4	1	no	—
252	C₂×C₄×C₈₉	—	4	1	no	—
253	C₂×C₄×C₈₉	—	4	1	no	—
254	C₂×C₄×C₈₉	—	4	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	—	8	4	no	—
266	C₂×C₄×C₈₉	—	8	4	no	—
267	C₂×C₄×C₈₉	—	8	4	no	—
268	C₂×C₄×C₈₉	#123	1	2	yes	—
269	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
270	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
275	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
276	C₈×C₈₉	—	2	2	no	—
277	C₈×C₈₉	—	4	1	no	—
278	C₈×C₈₉	—	4	1	no	—
279	C₈×C₈₉	—	4	1	no	—
280	C₂²×C₈₉	—	2	2	no	—
281	C₂²×C₈₉	—	2	2	no	—
282	C₂²×C₈₉	—	2	2	no	—
283	C₂²×C₈₉	—	2	2	no	—
284	C₂²×C₈₉	—	2	2	no	—
285	C₂²×C₈₉	—	2	2	no	—
286	C₂²×C₈₉	—	2	2	no	—
287	C₂²×C₈₉	—	2	2	no	—
288	C₂²×C₈₉	—	2	2	no	—
289	C₂²×C₈₉	—	4	1	no	—
290	C₂²×C₈₉	—	4	1	no	—
291	C₂²×C₈₉	—	4	1	no	—
292	C₂²×C₈₉	—	4	1	no	—
293	C₂²×C₈₉	#484	1	4	yes	—
294	C₂²×C₈₉	C₂×#123	1	1	yes	characteristic
295	C₂²×C₈₉	#546	1	1	yes	characteristic
296	C₂²×C₈₉	#547	1	1	yes	characteristic
297	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
298	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂²×C₈₉	#576	1	1	yes	characteristic
300	C₂²×C₈₉	#577	1	1	yes	characteristic
301	C₄×C₈₉	—	2	2	no	—
302	C₄×C₈₉	—	2	2	no	—
303	C₄×C₈₉	—	4	1	no	—
304	C₄×C₈₉	—	4	1	no	—
305	C₄×C₈₉	—	4	2	no	—
306	C₄×C₈₉	—	8	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	—	8	1	no	—
312	C₄×C₈₉	—	8	1	no	—
313	C₄×C₈₉	#529	1	1	yes	characteristic
314	C₄×C₈₉	#539	1	1	yes	characteristic
315	C₄×C₈₉	#551	1	1	yes	characteristic
316	C₄×C₈₉	#552	1	1	yes	characteristic
317	#24939	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
318	C₂×C₈₉	—	2	4	no	—
319	C₂×C₈₉	—	4	2	no	—
320	C₂×C₈₉	#2419	1	1	yes	characteristic
321	C₂×C₈₉	C₂×#484	1	1	yes	characteristic
322	C₂×C₈₉	#3096	1	1	yes	characteristic
323	C₂×C₈₉	#3299	1	1	yes	characteristic
324	C₂×C₈₉	#3318	1	1	yes	characteristic
325	C₂×C₈₉	#3338	1	1	yes	characteristic
326	C₂×C₈₉	#3345	1	1	yes	characteristic
327	#2487	C₂×C₈₉	1	1	yes	characteristic
328	D₄×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic
329	#2100	C₂×C₈₉	1	1	yes	characteristic, complemented
330	#2083	C₂×C₈₉	1	1	yes	characteristic, complemented
331	#2076	C₂×C₈₉	1	1	yes	characteristic, complemented
332	#2076	C₂×C₈₉	1	1	yes	characteristic, complemented
333	#1921	C₂×C₈₉	1	1	yes	characteristic, complemented
334	#1921	C₂×C₈₉	1	1	yes	characteristic, complemented
335	#1869	C₂×C₈₉	1	1	yes	characteristic
336	#1852	C₂×C₈₉	1	1	yes	characteristic, complemented
337	#1852	C₂×C₈₉	1	1	yes	characteristic, complemented
338	#1824	C₂×C₈₉	1	1	yes	characteristic, complemented
339	#1824	C₂×C₈₉	1	1	yes	characteristic, complemented
340	#1808	C₂×C₈₉	1	1	yes	characteristic, complemented
341	C₂×#340	C₂×C₈₉	1	1	yes	characteristic, complemented
342	C₈₉	#24939	1	1	yes	characteristic, direct product factor, Sylow
343	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
344	C₂×C₄×D₄	—	2	1	no	—
345	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
346	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
347	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
348	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
349	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
350	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
351	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
352	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
353	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
354	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
355	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
356	#400	—	2	1	no	—
357	#399	—	2	1	no	—
358	#389	—	2	1	no	—
359	#389	C₂²×C₈₉	1	1	yes	characteristic
360	#389	C₂²×C₈₉	1	1	yes	characteristic
361	#386	—	2	1	no	—
362	#385	—	2	1	no	—
363	#385	—	2	1	no	—
364	#385	—	2	1	no	—
365	#384	C₂²×C₈₉	1	1	yes	characteristic
366	#380	C₂²×C₈₉	1	1	yes	characteristic
367	#380	C₂²×C₈₉	1	1	yes	characteristic
368	#380	C₂²×C₈₉	1	1	yes	characteristic
369	#380	C₂²×C₈₉	1	1	yes	characteristic
370	#380	C₂²×C₈₉	1	1	yes	characteristic
371	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
372	#340	C₂²×C₈₉	1	1	yes	characteristic
373	#340	C₂²×C₈₉	1	1	yes	characteristic
374	#340	C₂²×C₈₉	1	1	yes	characteristic
375	#340	C₂²×C₈₉	1	1	yes	characteristic
376	#340	C₂²×C₈₉	1	1	yes	characteristic
377	#340	C₂²×C₈₉	1	1	yes	characteristic
378	#340	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	#340	C₂²×C₈₉	1	1	yes	characteristic
384	#336	C₂²×C₈₉	1	1	yes	characteristic
385	#336	C₂²×C₈₉	1	1	yes	characteristic
386	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
387	C₂³×C₄	—	2	1	no	—
388	C₂³×C₄	—	2	1	no	—
389	C₂³×C₄	—	2	1	no	—
390	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
392	C₂²×C₈	—	2	2	no	—
393	C₂²×C₈	—	2	2	no	—
394	C₂²×C₈	—	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×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	4	4	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	—	2	2	no	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	—	2	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	—	4	1	no	—
414	C₂×C₄⋊C₄	—	4	1	no	—
415	C₂×C₄⋊C₄	D₄×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×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
433	#107	—	2	2	no	—
434	#107	—	2	2	no	—
435	#107	—	4	1	no	—
436	#107	—	4	1	no	—
437	C₂²⋊C₈	—	2	2	no	—
438	C₂²⋊C₈	—	2	2	no	—
439	C₂²⋊C₈	—	4	1	no	—
440	#95	—	2	1	no	—
441	#95	—	2	1	no	—
442	#95	—	2	1	no	—
443	#95	—	2	1	no	—
444	#95	—	2	1	no	—
445	#95	—	2	1	no	—
446	#95	—	2	1	no	—
447	#95	—	2	1	no	—
448	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
449	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	Q₁₆×C₈₉	1	4	yes	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	2	no	—
476	C₂²×C₄	—	8	1	no	—
477	C₂²×C₄	—	8	1	no	—
478	C₂²×C₄	—	8	1	no	—
479	C₂²×C₄	—	8	1	no	—
480	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
484	C₂²×C₄	#41×C₈₉	1	2	yes	—
485	C₂²×C₄	#41×C₈₉	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	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	—	4	4	no	—
499	C₂×C₈	#41×C₈₉	1	2	yes	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	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	2	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	#41×C₈₉	1	2	yes	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	4	2	no	—
527	C₂²⋊C₄	—	4	2	no	—
528	C₂²⋊C₄	—	4	2	no	—
529	C₂²⋊C₄	—	8	1	no	—
530	C₂²⋊C₄	—	8	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	4	2	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	2	1	no	—
536	C₂³	—	2	1	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	1	no	—
540	C₂³	—	2	1	no	—
541	C₂³	—	2	1	no	—
542	C₂³	—	2	1	no	—
543	C₂³	—	2	1	no	—
544	C₂³	#123×C₈₉	1	2	yes	—
545	C₂³	C₂×Q₁₆×C₈₉	1	2	yes	—
546	C₂³	C₂×Q₁₆×C₈₉	1	2	yes	—
547	C₂³	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
548	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
549	D₄	—	2	8	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	2	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	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	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	—	8	4	no	—
593	C₂×C₄	—	8	4	no	—
594	C₂×C₄	—	8	4	no	—
595	C₂×C₄	#123×C₈₉	1	2	yes	—
596	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
597	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
598	C₂×C₄	C₂×#41×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	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
602	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
603	C₈	—	2	2	no	—
604	C₈	—	4	1	no	—
605	C₈	—	4	1	no	—
606	C₈	—	4	1	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	2	no	—
614	C₂²	—	2	2	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	4	1	no	—
617	C₂²	—	4	1	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	1	no	—
620	C₂²	#484×C₈₉	1	4	yes	—
621	C₂²	C₂×#123×C₈₉	1	1	yes	characteristic
622	C₂²	#546×C₈₉	1	1	yes	characteristic
623	C₂²	#547×C₈₉	1	1	yes	characteristic
624	C₂²	C₂²×Q₁₆×C₈₉	1	1	yes	characteristic
625	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
626	C₂²	#576×C₈₉	1	1	yes	characteristic
627	C₂²	#577×C₈₉	1	1	yes	characteristic
628	C₄	—	2	2	no	—
629	C₄	—	2	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	1	no	—
632	C₄	—	4	1	no	—
633	C₄	—	8	1	no	—
634	C₄	—	8	1	no	—
635	C₄	—	8	1	no	—
636	C₄	—	8	1	no	—
637	C₄	—	8	1	no	—
638	C₄	—	8	1	no	—
639	C₄	—	8	1	no	—
640	C₄	#529×C₈₉	1	1	yes	characteristic
641	C₄	#539×C₈₉	1	1	yes	characteristic
642	C₄	#551×C₈₉	1	1	yes	characteristic
643	C₄	#552×C₈₉	1	1	yes	characteristic
644	C₂	—	2	4	no	—
645	C₂	—	4	2	no	—
646	C₂	C₈₉×#2419	1	1	yes	characteristic
647	C₂	C₂×#484×C₈₉	1	1	yes	characteristic
648	C₂	C₈₉×#3096	1	1	yes	characteristic
649	C₂	C₈₉×#3299	1	1	yes	characteristic
650	C₂	C₈₉×#3318	1	1	yes	characteristic
651	C₂	C₈₉×#3338	1	1	yes	characteristic
652	C₂	C₈₉×#3345	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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄×D₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#2100	C₂	semidirect
4. C₈₉×#2083	C₂	semidirect
5. C₈₉×#2076	C₂	semidirect
6. C₈₉×#2076	C₂	semidirect
7. C₈₉×#1921	C₂	semidirect
8. C₈₉×#1921	C₂	semidirect
10. C₈₉×#1852	C₂	semidirect
11. C₈₉×#1852	C₂	semidirect
12. C₈₉×#1824	C₂	semidirect
13. C₈₉×#1824	C₂	semidirect
14. C₈₉×#1808	C₂	semidirect
15. C₂×#340×C₈₉	C₂	semidirect
317. #24939	C₈₉	direct (N × H)
329. #2100	C₂×C₈₉	semidirect
330. #2083	C₂×C₈₉	semidirect
331. #2076	C₂×C₈₉	semidirect
332. #2076	C₂×C₈₉	semidirect
333. #1921	C₂×C₈₉	semidirect
334. #1921	C₂×C₈₉	semidirect
336. #1852	C₂×C₈₉	semidirect
337. #1852	C₂×C₈₉	semidirect
338. #1824	C₂×C₈₉	semidirect
339. #1824	C₂×C₈₉	semidirect
340. #1808	C₂×C₈₉	semidirect
341. C₂×#340	C₂×C₈₉	semidirect
342. C₈₉	#24939	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×#2487	2
D₄×C₄⋊C₄×C₈₉	2
C₈₉×#2100	2
C₈₉×#2083	2
C₈₉×#2076	2
C₈₉×#1921	2
C₈₉×#1869	2
C₈₉×#1852	2
C₈₉×#1824	2
C₈₉×#1808	2
C₂×#340×C₈₉	2
#24939	89

Subgroups — C89×#24939 #36402787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#24939 #36402787