Subgroups — C₈₉×#22891 #36400739

Order 22784.

1062 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#453×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2946	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2897	C₂	1	2	yes	maximal, complemented
4	C₈₉×#1961	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1931	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1907	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1860	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1765	C₂	1	2	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
12	C₂×#131×C₈₉	C₂²	1	2	yes	—
13	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
14	#520×C₈₉	C₂²	1	2	yes	—
15	#518×C₈₉	C₂²	1	1	yes	characteristic, complemented
16	#453×C₈₉	C₄	1	8	yes	complemented
17	#442×C₈₉	—	2	2	no	—
18	#440×C₈₉	—	2	2	no	—
19	C₄×SD₁₆×C₈₉	—	2	2	no	—
20	C₄×D₈×C₈₉	—	2	2	no	—
21	#428×C₈₉	C₂²	1	2	yes	complemented
22	C₂×#104×C₈₉	C₂²	1	2	yes	—
23	#418×C₈₉	—	2	2	no	—
24	#418×C₈₉	C₂²	1	2	yes	—
25	#417×C₈₉	—	2	2	no	—
26	#417×C₈₉	C₂²	1	2	yes	—
27	#416×C₈₉	—	2	2	no	—
28	#416×C₈₉	C₂²	1	2	yes	—
29	C₂×#102×C₈₉	—	2	2	no	—
30	C₂×#102×C₈₉	C₂²	1	2	yes	—
31	C₂×#102×C₈₉	C₂²	1	2	yes	complemented
32	C₂×#100×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#100×C₈₉	C₂²	1	1	yes	characteristic, complemented
34	#405×C₈₉	C₂²	1	2	yes	complemented
35	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic, complemented
36	#382×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	#343×C₈₉	C₂²	1	2	yes	complemented
38	#341×C₈₉	C₂²	1	2	yes	complemented
39	#339×C₈₉	C₂²	1	2	yes	complemented
40	H(5,𝔽₂)×C₈₉	—	2	4	no	—
41	H(5,𝔽₂)×C₈₉	—	4	1	no	—
42	H(5,𝔽₂)×C₈₉	C₂×C₄	1	4	yes	complemented
43	C₂×#41×C₈₉	—	2	2	no	—
44	C₂×#41×C₈₉	C₂³	1	2	yes	—
45	C₂²×D₄×C₈₉	—	2	2	no	—
46	C₂²×D₄×C₈₉	—	4	1	no	—
47	C₂²×D₄×C₈₉	—	4	1	no	—
48	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
50	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
51	AGL(1,ℤ/8)×C₈₉	C₂×C₄	1	8	yes	complemented
52	C₂×SD₁₆×C₈₉	—	2	2	no	—
53	C₂×D₈×C₈₉	—	2	2	no	—
54	#131×C₈₉	—	2	4	no	—
55	#131×C₈₉	—	4	2	no	—
56	C₂×M₁₆×C₈₉	—	2	2	no	—
57	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
58	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
59	C₂²×C₈×C₈₉	—	2	2	no	—
60	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
61	#127×C₈₉	—	4	1	no	—
62	#127×C₈₉	C₂×C₄	1	4	yes	complemented
63	C₄×Q₈×C₈₉	—	2	2	no	—
64	C₄×D₄×C₈₉	—	2	2	no	—
65	C₄×D₄×C₈₉	—	2	2	no	—
66	C₄×D₄×C₈₉	—	2	2	no	—
67	C₄×D₄×C₈₉	—	4	2	no	—
68	#117×C₈₉	—	2	2	no	—
69	#117×C₈₉	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
71	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
73	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	#107×C₈₉	—	2	2	no	—
75	#106×C₈₉	—	2	2	no	—
76	#104×C₈₉	C₂×C₄	1	8	yes	complemented
77	#103×C₈₉	—	2	2	no	—
78	#103×C₈₉	D₄	1	2	yes	complemented
79	#103×C₈₉	D₄	1	2	yes	complemented
80	#102×C₈₉	—	2	2	no	—
81	#102×C₈₉	—	2	2	no	—
82	#102×C₈₉	—	2	2	no	—
83	#102×C₈₉	—	2	2	no	—
84	#102×C₈₉	—	2	2	no	—
85	#102×C₈₉	—	2	2	no	—
86	#102×C₈₉	—	2	2	no	—
87	#102×C₈₉	D₄	1	2	yes	complemented
88	#102×C₈₉	D₄	1	2	yes	complemented
89	#100×C₈₉	—	4	1	no	—
90	#100×C₈₉	C₂×C₄	1	4	yes	complemented
91	#97×C₈₉	—	2	2	no	—
92	#97×C₈₉	—	4	1	no	—
93	C₄×C₈×C₈₉	—	2	2	no	—
94	#95×C₈₉	—	4	1	no	—
95	C₂⁴×C₈₉	—	2	1	no	—
96	C₂⁴×C₈₉	—	2	1	no	—
97	C₂⁴×C₈₉	—	4	1	no	—
98	#41×C₈₉	—	2	2	no	—
99	#41×C₈₉	—	2	2	no	—
100	#41×C₈₉	—	2	8	no	—
101	#41×C₈₉	—	4	4	no	—
102	#41×C₈₉	C₂²×C₄	1	4	yes	—
103	#41×C₈₉	#41	1	4	yes	—
104	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	2	8	no	—
107	C₂×D₄×C₈₉	—	2	2	no	—
108	C₂×D₄×C₈₉	—	2	2	no	—
109	C₂×D₄×C₈₉	—	2	2	no	—
110	C₂×D₄×C₈₉	—	2	2	no	—
111	C₂×D₄×C₈₉	—	2	1	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂×D₄×C₈₉	—	4	4	no	—
114	C₂×D₄×C₈₉	—	4	2	no	—
115	C₂×D₄×C₈₉	—	4	4	no	—
116	C₂×D₄×C₈₉	—	4	4	no	—
117	C₂×D₄×C₈₉	—	4	4	no	—
118	C₂×D₄×C₈₉	—	8	4	no	—
119	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
121	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₈₉	—	2	2	no	—
123	C₂²×C₄×C₈₉	—	2	1	no	—
124	C₂²×C₄×C₈₉	—	2	2	no	—
125	C₂²×C₄×C₈₉	—	4	1	no	—
126	C₂²×C₄×C₈₉	—	4	2	no	—
127	C₂²×C₄×C₈₉	—	4	2	no	—
128	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
129	SD₁₆×C₈₉	—	2	8	no	—
130	D₈×C₈₉	—	2	8	no	—
131	M₁₆×C₈₉	—	2	2	no	—
132	M₁₆×C₈₉	—	2	2	no	—
133	M₁₆×C₈₉	—	2	2	no	—
134	M₁₆×C₈₉	—	4	4	no	—
135	M₁₆×C₈₉	C₂²×C₄	1	4	yes	—
136	C₂×C₈×C₈₉	—	2	2	no	—
137	C₂×C₈×C₈₉	—	2	2	no	—
138	C₂×C₈×C₈₉	—	2	2	no	—
139	C₂×C₈×C₈₉	—	2	2	no	—
140	C₂×C₈×C₈₉	—	4	4	no	—
141	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₈₉	—	2	2	no	—
144	C₄⋊C₄×C₈₉	—	2	2	no	—
145	C₄⋊C₄×C₈₉	—	4	2	no	—
146	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
148	C₂²⋊C₄×C₈₉	—	2	2	no	—
149	C₂²⋊C₄×C₈₉	—	4	1	no	—
150	C₂²⋊C₄×C₈₉	—	4	1	no	—
151	C₂²⋊C₄×C₈₉	—	4	2	no	—
152	C₄²×C₈₉	—	2	2	no	—
153	C₄²×C₈₉	—	2	1	no	—
154	C₄²×C₈₉	—	4	2	no	—
155	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
156	C₂³×C₈₉	—	2	2	no	—
157	C₂³×C₈₉	—	2	2	no	—
158	C₂³×C₈₉	—	2	2	no	—
159	C₂³×C₈₉	—	4	4	no	—
160	C₂³×C₈₉	—	4	1	no	—
161	C₂³×C₈₉	—	4	1	no	—
162	C₂³×C₈₉	—	4	1	no	—
163	C₂³×C₈₉	—	4	1	no	—
164	C₂³×C₈₉	—	8	2	no	—
165	C₂³×C₈₉	—	8	2	no	—
166	C₂³×C₈₉	—	16	2	no	—
167	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
168	Q₈×C₈₉	—	2	2	no	—
169	Q₈×C₈₉	C₄×D₄	1	4	yes	—
170	D₄×C₈₉	—	2	4	no	—
171	D₄×C₈₉	—	2	4	no	—
172	D₄×C₈₉	—	2	4	no	—
173	D₄×C₈₉	—	2	4	no	—
174	D₄×C₈₉	—	2	2	no	—
175	D₄×C₈₉	—	4	8	no	—
176	D₄×C₈₉	—	4	2	no	—
177	D₄×C₈₉	—	4	2	no	—
178	D₄×C₈₉	—	8	8	no	—
179	D₄×C₈₉	C₄×D₄	1	4	yes	—
180	C₂×C₄×C₈₉	—	2	2	no	—
181	C₂×C₄×C₈₉	—	2	4	no	—
182	C₂×C₄×C₈₉	—	2	4	no	—
183	C₂×C₄×C₈₉	—	2	2	no	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	2	no	—
186	C₂×C₄×C₈₉	—	4	4	no	—
187	C₂×C₄×C₈₉	—	4	2	no	—
188	C₂×C₄×C₈₉	—	4	2	no	—
189	C₂×C₄×C₈₉	—	8	2	no	—
190	C₂×C₄×C₈₉	—	16	2	no	—
191	C₂×C₄×C₈₉	—	16	2	no	—
192	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
195	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
196	C₈×C₈₉	—	2	4	no	—
197	C₈×C₈₉	—	2	2	no	—
198	C₈×C₈₉	—	2	2	no	—
199	C₈×C₈₉	—	4	2	no	—
200	C₂²×C₈₉	—	2	2	no	—
201	C₂²×C₈₉	—	2	4	no	—
202	C₂²×C₈₉	—	4	2	no	—
203	C₂²×C₈₉	—	4	2	no	—
204	C₂²×C₈₉	—	4	2	no	—
205	C₂²×C₈₉	—	4	2	no	—
206	C₂²×C₈₉	—	8	2	no	—
207	C₂²×C₈₉	—	8	2	no	—
208	C₂²×C₈₉	—	8	2	no	—
209	C₂²×C₈₉	—	8	2	no	—
210	C₂²×C₈₉	—	16	1	no	—
211	C₂²×C₈₉	—	16	1	no	—
212	C₂²×C₈₉	—	16	4	no	—
213	C₂²×C₈₉	#518	1	1	yes	characteristic
214	C₂²×C₈₉	D₄²	1	1	yes	characteristic
215	C₂²×C₈₉	#551	1	1	yes	characteristic
216	C₄×C₈₉	—	2	4	no	—
217	C₄×C₈₉	—	4	2	no	—
218	C₄×C₈₉	—	4	2	no	—
219	C₄×C₈₉	—	4	2	no	—
220	C₄×C₈₉	—	8	2	no	—
221	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
222	C₄×C₈₉	#546	1	2	yes	—
223	#22891	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₈₉	—	2	1	no	—
225	C₂×C₈₉	—	2	1	no	—
226	C₂×C₈₉	—	4	4	no	—
227	C₂×C₈₉	—	8	2	no	—
228	C₂×C₈₉	—	8	2	no	—
229	C₂×C₈₉	#2358	1	1	yes	characteristic
230	C₂×C₈₉	#3288	1	2	yes	—
231	C₂×#453	C₂×C₈₉	1	1	yes	characteristic
232	#2946	C₂×C₈₉	1	2	yes	complemented
233	#2897	C₂×C₈₉	1	2	yes	complemented
234	#1961	C₂×C₈₉	1	1	yes	characteristic, complemented
235	#1931	C₂×C₈₉	1	2	yes	complemented
236	#1907	C₂×C₈₉	1	2	yes	complemented
237	#1887	C₂×C₈₉	1	1	yes	characteristic, complemented
238	#1860	C₂×C₈₉	1	2	yes	complemented
239	#1765	C₂×C₈₉	1	2	yes	complemented
240	C₈₉	#22891	1	1	yes	characteristic, direct product factor, Sylow
241	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	1	yes	characteristic
242	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
243	C₂×#131	C₂²×C₈₉	1	2	yes	—
244	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
245	#520	C₂²×C₈₉	1	2	yes	—
246	#518	C₂²×C₈₉	1	1	yes	characteristic, complemented
247	#453	C₄×C₈₉	1	8	yes	complemented
248	#442	—	2	2	no	—
249	#440	—	2	2	no	—
250	C₄×SD₁₆	—	2	2	no	—
251	C₄×D₈	—	2	2	no	—
252	#428	C₂²×C₈₉	1	2	yes	complemented
253	C₂×#104	C₂²×C₈₉	1	2	yes	—
254	#418	—	2	2	no	—
255	#418	C₂²×C₈₉	1	2	yes	—
256	#417	—	2	2	no	—
257	#417	C₂²×C₈₉	1	2	yes	—
258	#416	—	2	2	no	—
259	#416	C₂²×C₈₉	1	2	yes	—
260	C₂×#102	—	2	2	no	—
261	C₂×#102	C₂²×C₈₉	1	2	yes	—
262	C₂×#102	C₂²×C₈₉	1	2	yes	complemented
263	C₂×#100	C₂²×C₈₉	1	1	yes	characteristic
264	C₂×#100	C₂²×C₈₉	1	1	yes	characteristic, complemented
265	#405	C₂²×C₈₉	1	2	yes	complemented
266	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic, complemented
267	#382	C₂²×C₈₉	1	1	yes	characteristic, complemented
268	#343	C₂²×C₈₉	1	2	yes	complemented
269	#341	C₂²×C₈₉	1	2	yes	complemented
270	#339	C₂²×C₈₉	1	2	yes	complemented
271	H(5,𝔽₂)	—	2	4	no	—
272	H(5,𝔽₂)	—	4	1	no	—
273	H(5,𝔽₂)	C₂×C₄×C₈₉	1	4	yes	complemented
274	C₂×#41	—	2	2	no	—
275	C₂×#41	C₂³×C₈₉	1	2	yes	—
276	C₂²×D₄	—	2	2	no	—
277	C₂²×D₄	—	4	1	no	—
278	C₂²×D₄	—	4	1	no	—
279	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
280	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
281	AGL(1,ℤ/8)	—	2	4	no	—
282	AGL(1,ℤ/8)	C₂×C₄×C₈₉	1	8	yes	complemented
283	C₂×SD₁₆	—	2	2	no	—
284	C₂×D₈	—	2	2	no	—
285	#131	—	2	4	no	—
286	#131	—	4	2	no	—
287	C₂×M₁₆	—	2	2	no	—
288	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
289	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
290	C₂²×C₈	—	2	2	no	—
291	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
292	#127	—	4	1	no	—
293	#127	C₂×C₄×C₈₉	1	4	yes	complemented
294	C₄×Q₈	—	2	2	no	—
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	—	2	2	no	—
297	C₄×D₄	—	2	2	no	—
298	C₄×D₄	—	4	2	no	—
299	#117	—	2	2	no	—
300	#117	C₂³×C₈₉	1	2	yes	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
303	C₂×C₂²⋊C₄	—	4	1	no	—
304	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
305	#107	—	2	2	no	—
306	#106	—	2	2	no	—
307	#104	C₂×C₄×C₈₉	1	8	yes	complemented
308	#103	—	2	2	no	—
309	#103	D₄×C₈₉	1	2	yes	complemented
310	#103	D₄×C₈₉	1	2	yes	complemented
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	#102	—	2	2	no	—
316	#102	—	2	2	no	—
317	#102	—	2	2	no	—
318	#102	D₄×C₈₉	1	2	yes	complemented
319	#102	D₄×C₈₉	1	2	yes	complemented
320	#100	—	4	1	no	—
321	#100	C₂×C₄×C₈₉	1	4	yes	complemented
322	#97	—	2	2	no	—
323	#97	—	4	1	no	—
324	C₄×C₈	—	2	2	no	—
325	#95	—	4	1	no	—
326	C₂⁴	—	2	1	no	—
327	C₂⁴	—	2	1	no	—
328	C₂⁴	—	4	1	no	—
329	#41	—	2	2	no	—
330	#41	—	2	2	no	—
331	#41	—	2	8	no	—
332	#41	—	4	4	no	—
333	#41	C₂²×C₄×C₈₉	1	4	yes	—
334	#41	#41×C₈₉	1	4	yes	—
335	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	4	4	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂×D₄	—	4	4	no	—
348	C₂×D₄	—	4	4	no	—
349	C₂×D₄	—	8	4	no	—
350	C₂×D₄	C₂²×C₄×C₈₉	1	2	yes	—
351	C₂×D₄	C₂²×C₄×C₈₉	1	2	yes	—
352	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	1	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	—	4	2	no	—
359	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
360	SD₁₆	—	2	8	no	—
361	D₈	—	2	8	no	—
362	M₁₆	—	2	2	no	—
363	M₁₆	—	2	2	no	—
364	M₁₆	—	2	2	no	—
365	M₁₆	—	4	4	no	—
366	M₁₆	C₂²×C₄×C₈₉	1	4	yes	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	4	4	no	—
372	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
378	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
379	C₂²⋊C₄	—	2	2	no	—
380	C₂²⋊C₄	—	4	1	no	—
381	C₂²⋊C₄	—	4	1	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	4	2	no	—
386	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	4	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	8	2	no	—
396	C₂³	—	8	2	no	—
397	C₂³	—	16	2	no	—
398	C₂³	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
399	Q₈	—	2	2	no	—
400	Q₈	C₄×D₄×C₈₉	1	4	yes	—
401	D₄	—	2	4	no	—
402	D₄	—	2	4	no	—
403	D₄	—	2	4	no	—
404	D₄	—	2	4	no	—
405	D₄	—	2	2	no	—
406	D₄	—	4	8	no	—
407	D₄	—	4	2	no	—
408	D₄	—	4	2	no	—
409	D₄	—	8	8	no	—
410	D₄	C₄×D₄×C₈₉	1	4	yes	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	4	4	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	8	2	no	—
421	C₂×C₄	—	16	2	no	—
422	C₂×C₄	—	16	2	no	—
423	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
424	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
425	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
426	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	—	2	2	no	—
429	C₈	—	2	2	no	—
430	C₈	—	4	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	8	2	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	16	1	no	—
442	C₂²	—	16	1	no	—
443	C₂²	—	16	4	no	—
444	C₂²	#518×C₈₉	1	1	yes	characteristic
445	C₂²	D₄²×C₈₉	1	1	yes	characteristic
446	C₂²	#551×C₈₉	1	1	yes	characteristic
447	C₄	—	2	4	no	—
448	C₄	—	4	2	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	8	2	no	—
452	C₄	C₂×C₄×D₄×C₈₉	1	2	yes	—
453	C₄	#546×C₈₉	1	2	yes	—
454	C₂	—	2	1	no	—
455	C₂	—	2	1	no	—
456	C₂	—	4	4	no	—
457	C₂	—	8	2	no	—
458	C₂	—	8	2	no	—
459	C₂	C₈₉×#2358	1	1	yes	characteristic
460	C₂	C₈₉×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	D₄²
commutator	C₂×C₄	C₂³×C₄×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#453×C₈₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2946	C₂	semidirect
3. C₈₉×#2897	C₂	semidirect
4. C₈₉×#1961	C₂	semidirect
5. C₈₉×#1931	C₂	semidirect
6. C₈₉×#1907	C₂	semidirect
7. C₈₉×#1887	C₂	semidirect
8. C₈₉×#1860	C₂	semidirect
9. C₈₉×#1765	C₂	semidirect
15. #518×C₈₉	C₂²	semidirect
16. #453×C₈₉	C₄	semidirect
21. #428×C₈₉	C₂²	semidirect
31. C₂×#102×C₈₉	C₂²	semidirect
33. C₂×#100×C₈₉	C₂²	semidirect
34. #405×C₈₉	C₂²	semidirect
35. C₂×#97×C₈₉	C₂²	semidirect
36. #382×C₈₉	C₂²	semidirect
37. #343×C₈₉	C₂²	semidirect
38. #341×C₈₉	C₂²	semidirect
39. #339×C₈₉	C₂²	semidirect
42. H(5,𝔽₂)×C₈₉	C₂×C₄	semidirect
51. AGL(1,ℤ/8)×C₈₉	C₂×C₄	semidirect
62. #127×C₈₉	C₂×C₄	semidirect
76. #104×C₈₉	C₂×C₄	semidirect
78. #103×C₈₉	D₄	semidirect
79. #103×C₈₉	D₄	semidirect
87. #102×C₈₉	D₄	semidirect
88. #102×C₈₉	D₄	semidirect
90. #100×C₈₉	C₂×C₄	semidirect
223. #22891	C₈₉	direct (N × H)
232. #2946	C₂×C₈₉	semidirect
233. #2897	C₂×C₈₉	semidirect
234. #1961	C₂×C₈₉	semidirect
235. #1931	C₂×C₈₉	semidirect
236. #1907	C₂×C₈₉	semidirect
237. #1887	C₂×C₈₉	semidirect
238. #1860	C₂×C₈₉	semidirect
239. #1765	C₂×C₈₉	semidirect
240. C₈₉	#22891	direct (N × H)
246. #518	C₂²×C₈₉	semidirect
247. #453	C₄×C₈₉	semidirect
252. #428	C₂²×C₈₉	semidirect
262. C₂×#102	C₂²×C₈₉	semidirect
264. C₂×#100	C₂²×C₈₉	semidirect
265. #405	C₂²×C₈₉	semidirect
266. C₂×#97	C₂²×C₈₉	semidirect
267. #382	C₂²×C₈₉	semidirect
268. #343	C₂²×C₈₉	semidirect
269. #341	C₂²×C₈₉	semidirect
270. #339	C₂²×C₈₉	semidirect
273. H(5,𝔽₂)	C₂×C₄×C₈₉	semidirect
282. AGL(1,ℤ/8)	C₂×C₄×C₈₉	semidirect
293. #127	C₂×C₄×C₈₉	semidirect
307. #104	C₂×C₄×C₈₉	semidirect
309. #103	D₄×C₈₉	semidirect
310. #103	D₄×C₈₉	semidirect
318. #102	D₄×C₈₉	semidirect
319. #102	D₄×C₈₉	semidirect
321. #100	C₂×C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#453×C₈₉	2
C₈₉×#2946	2
C₈₉×#2897	2
C₈₉×#1961	2
C₈₉×#1931	2
C₈₉×#1907	2
C₈₉×#1887	2
C₈₉×#1860	2
C₈₉×#1765	2
#22891	89

Subgroups — C89×#22891 #36400739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#22891 #36400739