Subgroups — C₈₉×#25432 #36403280

Order 22784.

912 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#473×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3058	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#3043	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2046	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2004	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1888	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₈₉	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
17	#535×C₈₉	C₂²	1	2	yes	—
18	C₂×#124×C₈₉	C₂²	1	2	yes	—
19	#519×C₈₉	C₂²	1	2	yes	—
20	#495×C₈₉	—	2	4	no	—
21	#473×C₈₉	—	2	4	no	—
22	#469×C₈₉	—	2	2	no	—
23	#466×C₈₉	—	2	2	no	—
24	#464×C₈₉	—	2	4	no	—
25	#460×C₈₉	—	2	4	no	—
26	#441×C₈₉	—	2	2	no	—
27	C₄×Q₁₆×C₈₉	—	2	2	no	—
28	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
29	#427×C₈₉	C₂²	1	1	yes	characteristic
30	#424×C₈₉	C₂²	1	2	yes	—
31	C₂×#104×C₈₉	C₂²	1	1	yes	characteristic
32	#419×C₈₉	C₂²	1	2	yes	—
33	#418×C₈₉	C₂²	1	2	yes	—
34	#418×C₈₉	C₂²	1	2	yes	—
35	#416×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#101×C₈₉	C₂²	1	2	yes	—
37	C₂×#100×C₈₉	C₂²	1	2	yes	—
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
40	#341×C₈₉	C₂²	1	1	yes	characteristic
41	#339×C₈₉	C₂²	1	1	yes	characteristic
42	#339×C₈₉	C₂²	1	2	yes	—
43	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
45	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
46	#137×C₈₉	—	2	4	no	—
47	#137×C₈₉	—	4	2	no	—
48	C₂×Q₁₆×C₈₉	—	2	2	no	—
49	C₂×Q₁₆×C₈₉	—	2	1	no	—
50	C₂×Q₁₆×C₈₉	—	2	1	no	—
51	C₂×Q₁₆×C₈₉	—	2	2	no	—
52	C₂×Q₁₆×C₈₉	D₄	1	4	yes	complemented
53	C₂×SD₁₆×C₈₉	—	2	4	no	—
54	C₂×SD₁₆×C₈₉	—	2	2	no	—
55	C₂×SD₁₆×C₈₉	—	2	4	no	—
56	C₂×SD₁₆×C₈₉	—	4	2	no	—
57	C₂×D₈×C₈₉	—	2	1	no	—
58	C₂×D₈×C₈₉	—	2	1	no	—
59	C₂×D₈×C₈₉	—	2	2	no	—
60	C₂×D₈×C₈₉	—	2	2	no	—
61	#131×C₈₉	—	2	2	no	—
62	#131×C₈₉	—	2	2	no	—
63	C₂×M₁₆×C₈₉	—	2	1	no	—
64	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
66	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	—	2	1	no	—
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	#127×C₈₉	—	2	2	no	—
71	#124×C₈₉	—	2	2	no	—
72	#124×C₈₉	—	2	4	no	—
73	#124×C₈₉	—	2	4	no	—
74	#121×C₈₉	—	2	4	no	—
75	#120×C₈₉	—	4	2	no	—
76	C₄×Q₈×C₈₉	—	2	4	no	—
77	C₄×Q₈×C₈₉	—	2	4	no	—
78	#117×C₈₉	—	4	2	no	—
79	#117×C₈₉	C₂³	1	1	yes	characteristic
80	#117×C₈₉	C₂³	1	2	yes	—
81	C₂×C₂²⋊C₄×C₈₉	—	4	2	no	—
82	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	#108×C₈₉	—	2	2	no	—
84	#107×C₈₉	D₄	1	2	yes	—
85	#106×C₈₉	D₄	1	2	yes	—
86	C₄⋊C₈×C₈₉	—	2	4	no	—
87	#104×C₈₉	—	2	2	no	—
88	#103×C₈₉	—	2	4	no	—
89	#103×C₈₉	D₄	1	2	yes	—
90	#102×C₈₉	—	2	4	no	—
91	#102×C₈₉	—	2	4	no	—
92	#102×C₈₉	D₄	1	2	yes	—
93	#101×C₈₉	—	2	4	no	—
94	#100×C₈₉	—	4	2	no	—
95	#97×C₈₉	—	2	1	no	—
96	C₄×C₈×C₈₉	—	2	2	no	—
97	C₄×C₈×C₈₉	—	2	1	no	—
98	C₂⁴×C₈₉	—	2	2	no	—
99	#41×C₈₉	—	2	1	no	—
100	#41×C₈₉	—	2	1	no	—
101	#41×C₈₉	—	2	1	no	—
102	#41×C₈₉	—	2	1	no	—
103	C₂×Q₈×C₈₉	—	2	2	no	—
104	C₂×Q₈×C₈₉	—	2	2	no	—
105	C₂×Q₈×C₈₉	—	2	2	no	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	2	2	no	—
108	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
109	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×D₄×C₈₉	—	2	1	no	—
111	C₂×D₄×C₈₉	—	2	2	no	—
112	C₂×D₄×C₈₉	—	2	2	no	—
113	C₂×D₄×C₈₉	—	2	2	no	—
114	C₂×D₄×C₈₉	—	2	2	no	—
115	C₂×D₄×C₈₉	—	2	2	no	—
116	C₂×D₄×C₈₉	—	2	2	no	—
117	C₂×D₄×C₈₉	—	4	4	no	—
118	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	—	2	1	no	—
120	C₂²×C₄×C₈₉	—	2	2	no	—
121	C₂²×C₄×C₈₉	—	2	1	no	—
122	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
123	Q₁₆×C₈₉	—	2	4	no	—
124	Q₁₆×C₈₉	—	2	4	no	—
125	Q₁₆×C₈₉	—	4	4	no	—
126	SD₁₆×C₈₉	—	4	8	no	—
127	SD₁₆×C₈₉	—	4	4	no	—
128	D₈×C₈₉	—	4	2	no	—
129	D₈×C₈₉	—	4	2	no	—
130	M₁₆×C₈₉	—	2	1	no	—
131	M₁₆×C₈₉	—	2	2	no	—
132	M₁₆×C₈₉	—	2	1	no	—
133	M₁₆×C₈₉	—	2	1	no	—
134	M₁₆×C₈₉	—	2	1	no	—
135	M₁₆×C₈₉	—	2	2	no	—
136	M₁₆×C₈₉	—	4	2	no	—
137	C₂×C₈×C₈₉	—	2	1	no	—
138	C₂×C₈×C₈₉	—	2	1	no	—
139	C₂×C₈×C₈₉	—	2	1	no	—
140	C₂×C₈×C₈₉	—	2	1	no	—
141	C₂×C₈×C₈₉	—	2	2	no	—
142	C₂×C₈×C₈₉	—	4	2	no	—
143	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₈₉	#41	1	2	yes	—
148	C₄⋊C₄×C₈₉	—	4	4	no	—
149	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
151	C₂²⋊C₄×C₈₉	—	2	1	no	—
152	C₂²⋊C₄×C₈₉	—	2	2	no	—
153	C₂²⋊C₄×C₈₉	—	4	8	no	—
154	C₂²⋊C₄×C₈₉	—	4	4	no	—
155	C₄²×C₈₉	—	2	1	no	—
156	C₄²×C₈₉	—	2	1	no	—
157	C₄²×C₈₉	—	2	2	no	—
158	C₄²×C₈₉	—	4	4	no	—
159	C₄²×C₈₉	#41	1	2	yes	—
160	C₂³×C₈₉	—	2	1	no	—
161	C₂³×C₈₉	—	4	2	no	—
162	C₂³×C₈₉	—	4	2	no	—
163	C₂³×C₈₉	—	4	2	no	—
164	C₂³×C₈₉	—	8	4	no	—
165	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈×C₈₉	—	2	4	no	—
167	Q₈×C₈₉	—	2	1	no	—
168	Q₈×C₈₉	—	2	4	no	—
169	Q₈×C₈₉	—	2	1	no	—
170	Q₈×C₈₉	—	4	2	no	—
171	Q₈×C₈₉	—	4	2	no	—
172	D₄×C₈₉	—	2	1	no	—
173	D₄×C₈₉	—	2	1	no	—
174	D₄×C₈₉	—	4	1	no	—
175	D₄×C₈₉	—	4	1	no	—
176	D₄×C₈₉	—	4	2	no	—
177	D₄×C₈₉	—	4	2	no	—
178	D₄×C₈₉	—	4	2	no	—
179	D₄×C₈₉	—	4	2	no	—
180	D₄×C₈₉	—	16	4	no	—
181	C₂×C₄×C₈₉	—	2	2	no	—
182	C₂×C₄×C₈₉	—	2	4	no	—
183	C₂×C₄×C₈₉	—	2	1	no	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	4	no	—
186	C₂×C₄×C₈₉	—	4	2	no	—
187	C₂×C₄×C₈₉	—	4	2	no	—
188	C₂×C₄×C₈₉	—	4	2	no	—
189	C₂×C₄×C₈₉	—	4	1	no	—
190	C₂×C₄×C₈₉	—	4	1	no	—
191	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
197	C₈×C₈₉	—	2	2	no	—
198	C₈×C₈₉	—	4	1	no	—
199	C₈×C₈₉	—	4	2	no	—
200	C₈×C₈₉	—	4	1	no	—
201	C₈×C₈₉	#121	1	2	yes	—
202	C₈×C₈₉	#121	1	2	yes	—
203	C₂²×C₈₉	—	2	2	no	—
204	C₂²×C₈₉	—	4	1	no	—
205	C₂²×C₈₉	—	4	2	no	—
206	C₂²×C₈₉	—	8	4	no	—
207	C₂²×C₈₉	—	8	2	no	—
208	C₂²×C₈₉	—	8	4	no	—
209	C₂²×C₈₉	—	16	2	no	—
210	C₂²×C₈₉	#540	1	1	yes	characteristic
211	C₂²×C₈₉	D₄²	1	1	yes	characteristic
212	C₂²×C₈₉	#546	1	1	yes	characteristic
213	C₄×C₈₉	—	4	2	no	—
214	C₄×C₈₉	—	4	1	no	—
215	C₄×C₈₉	—	4	4	no	—
216	C₄×C₈₉	—	4	2	no	—
217	C₄×C₈₉	—	4	4	no	—
218	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
219	C₄×C₈₉	#532	1	1	yes	characteristic
220	C₄×C₈₉	D₄²	1	1	yes	characteristic
221	C₄×C₈₉	#550	1	1	yes	characteristic
222	#25432	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂×C₈₉	—	2	1	no	—
224	C₂×C₈₉	—	2	1	no	—
225	C₂×C₈₉	—	8	1	no	—
226	C₂×C₈₉	—	16	2	no	—
227	C₂×C₈₉	#2428	1	1	yes	characteristic
228	C₂×C₈₉	#3290	1	2	yes	—
229	C₂×#495	C₂×C₈₉	1	1	yes	characteristic, complemented
230	C₂×#473	C₂×C₈₉	1	1	yes	characteristic, complemented
231	#3058	C₂×C₈₉	1	1	yes	characteristic
232	#3043	C₂×C₈₉	1	2	yes	complemented
233	#2950	C₂×C₈₉	1	1	yes	characteristic, complemented
234	#2046	C₂×C₈₉	1	2	yes	complemented
235	#2004	C₂×C₈₉	1	2	yes	complemented
236	#1978	C₂×C₈₉	1	1	yes	characteristic, complemented
237	#1945	C₂×C₈₉	1	1	yes	characteristic, complemented
238	#1888	C₂×C₈₉	1	2	yes	complemented
239	#1764	C₂×C₈₉	1	1	yes	characteristic, complemented
240	C₈₉	#25432	1	1	yes	characteristic, direct product factor, Sylow
241	C₂×#137	C₂²×C₈₉	1	2	yes	—
242	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
243	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
244	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
245	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
246	#535	C₂²×C₈₉	1	2	yes	—
247	C₂×#124	C₂²×C₈₉	1	2	yes	—
248	#519	C₂²×C₈₉	1	2	yes	—
249	#495	—	2	4	no	—
250	#473	—	2	4	no	—
251	#469	—	2	2	no	—
252	#466	—	2	2	no	—
253	#464	—	2	4	no	—
254	#460	—	2	4	no	—
255	#441	—	2	2	no	—
256	C₄×Q₁₆	—	2	2	no	—
257	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
258	#427	C₂²×C₈₉	1	1	yes	characteristic
259	#424	C₂²×C₈₉	1	2	yes	—
260	C₂×#104	C₂²×C₈₉	1	1	yes	characteristic
261	#419	C₂²×C₈₉	1	2	yes	—
262	#418	C₂²×C₈₉	1	2	yes	—
263	#418	C₂²×C₈₉	1	2	yes	—
264	#416	C₂²×C₈₉	1	1	yes	characteristic
265	C₂×#101	C₂²×C₈₉	1	2	yes	—
266	C₂×#100	C₂²×C₈₉	1	2	yes	—
267	#405	C₂²×C₈₉	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
269	#341	C₂²×C₈₉	1	1	yes	characteristic
270	#339	C₂²×C₈₉	1	1	yes	characteristic
271	#339	C₂²×C₈₉	1	2	yes	—
272	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
273	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
274	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
275	#137	—	2	4	no	—
276	#137	—	4	2	no	—
277	C₂×Q₁₆	—	2	2	no	—
278	C₂×Q₁₆	—	2	1	no	—
279	C₂×Q₁₆	—	2	1	no	—
280	C₂×Q₁₆	—	2	2	no	—
281	C₂×Q₁₆	D₄×C₈₉	1	4	yes	complemented
282	C₂×SD₁₆	—	2	4	no	—
283	C₂×SD₁₆	—	2	2	no	—
284	C₂×SD₁₆	—	2	4	no	—
285	C₂×SD₁₆	—	4	2	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂×D₈	—	2	1	no	—
288	C₂×D₈	—	2	2	no	—
289	C₂×D₈	—	2	2	no	—
290	#131	—	2	2	no	—
291	#131	—	2	2	no	—
292	C₂×M₁₆	—	2	1	no	—
293	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
295	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
296	C₂²×C₈	—	2	1	no	—
297	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
299	#127	—	2	2	no	—
300	#124	—	2	4	no	—
301	#124	—	2	2	no	—
302	#124	—	2	4	no	—
303	#121	—	2	4	no	—
304	#120	—	4	2	no	—
305	C₄×Q₈	—	2	4	no	—
306	C₄×Q₈	—	2	4	no	—
307	#117	—	4	2	no	—
308	#117	C₂³×C₈₉	1	1	yes	characteristic
309	#117	C₂³×C₈₉	1	2	yes	—
310	C₂×C₂²⋊C₄	—	4	2	no	—
311	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
312	#108	—	2	2	no	—
313	#107	D₄×C₈₉	1	2	yes	—
314	#106	D₄×C₈₉	1	2	yes	—
315	C₄⋊C₈	—	2	4	no	—
316	#104	—	2	2	no	—
317	#103	—	2	4	no	—
318	#103	D₄×C₈₉	1	2	yes	—
319	#102	—	2	4	no	—
320	#102	—	2	4	no	—
321	#102	D₄×C₈₉	1	2	yes	—
322	#101	—	2	4	no	—
323	#100	—	4	2	no	—
324	#97	—	2	1	no	—
325	C₄×C₈	—	2	2	no	—
326	C₄×C₈	—	2	1	no	—
327	C₂⁴	—	2	2	no	—
328	#41	—	2	1	no	—
329	#41	—	2	1	no	—
330	#41	—	2	1	no	—
331	#41	—	2	1	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
338	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
352	Q₁₆	—	2	4	no	—
353	Q₁₆	—	2	4	no	—
354	Q₁₆	—	4	4	no	—
355	SD₁₆	—	4	8	no	—
356	SD₁₆	—	4	4	no	—
357	D₈	—	4	2	no	—
358	D₈	—	4	2	no	—
359	M₁₆	—	2	1	no	—
360	M₁₆	—	2	2	no	—
361	M₁₆	—	2	1	no	—
362	M₁₆	—	2	1	no	—
363	M₁₆	—	2	1	no	—
364	M₁₆	—	2	2	no	—
365	M₁₆	—	4	2	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	4	2	no	—
372	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
374	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
376	C₂×C₈	#41×C₈₉	1	2	yes	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
379	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	2	no	—
382	C₂²⋊C₄	—	4	8	no	—
383	C₂²⋊C₄	—	4	4	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	4	4	no	—
388	C₄²	#41×C₈₉	1	2	yes	—
389	C₂³	—	2	1	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	8	4	no	—
394	C₂³	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
395	Q₈	—	2	4	no	—
396	Q₈	—	2	1	no	—
397	Q₈	—	2	4	no	—
398	Q₈	—	2	1	no	—
399	Q₈	—	4	2	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	1	no	—
402	D₄	—	2	1	no	—
403	D₄	—	4	1	no	—
404	D₄	—	4	1	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	D₄	—	4	2	no	—
409	D₄	—	16	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
425	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
426	C₈	—	2	2	no	—
427	C₈	—	4	1	no	—
428	C₈	—	4	2	no	—
429	C₈	—	4	1	no	—
430	C₈	#121×C₈₉	1	2	yes	—
431	C₈	#121×C₈₉	1	2	yes	—
432	C₂²	—	2	2	no	—
433	C₂²	—	4	1	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	4	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	4	no	—
438	C₂²	—	16	2	no	—
439	C₂²	#540×C₈₉	1	1	yes	characteristic
440	C₂²	D₄²×C₈₉	1	1	yes	characteristic
441	C₂²	#546×C₈₉	1	1	yes	characteristic
442	C₄	—	4	2	no	—
443	C₄	—	4	1	no	—
444	C₄	—	4	4	no	—
445	C₄	—	4	4	no	—
446	C₄	—	4	2	no	—
447	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
448	C₄	#532×C₈₉	1	1	yes	characteristic
449	C₄	D₄²×C₈₉	1	1	yes	characteristic
450	C₄	#550×C₈₉	1	1	yes	characteristic
451	C₂	—	2	1	no	—
452	C₂	—	2	1	no	—
453	C₂	—	8	1	no	—
454	C₂	—	16	2	no	—
455	C₂	C₈₉×#2428	1	1	yes	characteristic
456	C₂	C₈₉×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₈₉×#3058	C₂
LCS	G ▷ C₂²×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
1. C₂×#495×C₈₉	C₂	semidirect
2. C₂×#473×C₈₉	C₂	semidirect
4. C₈₉×#3043	C₂	semidirect
5. C₈₉×#2950	C₂	semidirect
6. C₈₉×#2046	C₂	semidirect
7. C₈₉×#2004	C₂	semidirect
8. C₈₉×#1978	C₂	semidirect
9. C₈₉×#1945	C₂	semidirect
10. C₈₉×#1888	C₂	semidirect
11. C₈₉×#1764	C₂	semidirect
52. C₂×Q₁₆×C₈₉	D₄	semidirect
222. #25432	C₈₉	direct (N × H)
229. C₂×#495	C₂×C₈₉	semidirect
230. C₂×#473	C₂×C₈₉	semidirect
232. #3043	C₂×C₈₉	semidirect
233. #2950	C₂×C₈₉	semidirect
234. #2046	C₂×C₈₉	semidirect
235. #2004	C₂×C₈₉	semidirect
236. #1978	C₂×C₈₉	semidirect
237. #1945	C₂×C₈₉	semidirect
238. #1888	C₂×C₈₉	semidirect
239. #1764	C₂×C₈₉	semidirect
240. C₈₉	#25432	direct (N × H)
281. C₂×Q₁₆	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#495×C₈₉	2
C₂×#473×C₈₉	2
C₈₉×#3058	2
C₈₉×#3043	2
C₈₉×#2950	2
C₈₉×#2046	2
C₈₉×#2004	2
C₈₉×#1978	2
C₈₉×#1945	2
C₈₉×#1888	2
C₈₉×#1764	2
#25432	89

Subgroups — C89×#25432 #36403280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25432 #36403280