Subgroups — C₈₉×#25053 #36402901

Order 22784.

736 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3098	C₂	1	2	yes	maximal, complemented
2	C₈₉×#3061	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2951	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2078	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2039	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1974	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1952	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1859	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1816	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
14	#537×C₈₉	C₂²	1	2	yes	—
15	#520×C₈₉	C₂²	1	2	yes	—
16	#485×C₈₉	—	2	2	no	—
17	#484×C₈₉	—	2	2	no	—
18	#482×C₈₉	—	2	2	no	—
19	#480×C₈₉	—	2	2	no	—
20	#468×C₈₉	—	2	2	no	—
21	#467×C₈₉	—	2	1	no	—
22	#466×C₈₉	—	2	1	no	—
23	#440×C₈₉	—	2	2	no	—
24	C₄×Q₁₆×C₈₉	—	2	1	no	—
25	C₄×D₈×C₈₉	—	2	1	no	—
26	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
27	#428×C₈₉	C₂²	1	2	yes	—
28	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#107×C₈₉	C₂²	1	2	yes	—
30	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#104×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#104×C₈₉	C₂²	1	2	yes	—
33	#417×C₈₉	C₂²	1	2	yes	—
34	#417×C₈₉	C₂²	1	2	yes	—
35	#417×C₈₉	C₂²	1	2	yes	—
36	#416×C₈₉	C₂²	1	1	yes	characteristic
37	#408×C₈₉	C₂²	1	2	yes	—
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
40	#384×C₈₉	C₂²	1	2	yes	—
41	#343×C₈₉	C₂²	1	2	yes	—
42	#343×C₈₉	C₂²	1	2	yes	—
43	#341×C₈₉	C₂²	1	1	yes	characteristic
44	#339×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#41×C₈₉	C₂³	1	2	yes	—
46	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
47	#135×C₈₉	—	2	2	no	—
48	#135×C₈₉	—	2	2	no	—
49	C₂×Q₁₆×C₈₉	D₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₈₉	D₄	1	2	yes	—
51	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
52	#131×C₈₉	—	2	2	no	—
53	#131×C₈₉	—	2	2	no	—
54	C₂×M₁₆×C₈₉	—	2	1	no	—
55	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
57	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	—	2	1	no	—
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²⋊Q₈×C₈₉	—	2	2	no	—
62	C₂²⋊Q₈×C₈₉	—	2	2	no	—
63	#121×C₈₉	—	2	2	no	—
64	#121×C₈₉	—	2	2	no	—
65	#121×C₈₉	—	4	2	no	—
66	C₄×Q₈×C₈₉	—	2	2	no	—
67	C₄×D₄×C₈₉	—	2	2	no	—
68	C₄×D₄×C₈₉	—	4	2	no	—
69	#117×C₈₉	—	2	2	no	—
70	#117×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
72	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	#108×C₈₉	—	2	2	no	—
77	#107×C₈₉	—	2	1	no	—
78	#107×C₈₉	—	2	1	no	—
79	#107×C₈₉	—	2	1	no	—
80	#107×C₈₉	—	2	1	no	—
81	#107×C₈₉	—	2	2	no	—
82	#107×C₈₉	—	2	2	no	—
83	#107×C₈₉	—	2	2	no	—
84	#106×C₈₉	—	2	2	no	—
85	#104×C₈₉	—	2	2	no	—
86	#104×C₈₉	—	2	4	no	—
87	#103×C₈₉	—	2	2	no	—
88	#103×C₈₉	—	2	2	no	—
89	#103×C₈₉	—	2	2	no	—
90	#103×C₈₉	D₄	1	1	yes	characteristic
91	#103×C₈₉	D₄	1	1	yes	characteristic
92	#102×C₈₉	—	2	2	no	—
93	#102×C₈₉	—	2	2	no	—
94	#102×C₈₉	—	2	2	no	—
95	#102×C₈₉	D₄	1	1	yes	characteristic
96	#102×C₈₉	D₄	1	1	yes	characteristic
97	C₂²⋊C₈×C₈₉	—	2	2	no	—
98	C₂²⋊C₈×C₈₉	—	2	2	no	—
99	#97×C₈₉	—	2	1	no	—
100	C₄×C₈×C₈₉	—	2	1	no	—
101	C₄×C₈×C₈₉	—	2	2	no	—
102	#41×C₈₉	—	2	2	no	—
103	#41×C₈₉	—	2	1	no	—
104	#41×C₈₉	—	2	1	no	—
105	#41×C₈₉	—	2	1	no	—
106	#41×C₈₉	—	2	2	no	—
107	#41×C₈₉	—	2	2	no	—
108	#41×C₈₉	—	2	2	no	—
109	#41×C₈₉	—	2	1	no	—
110	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×D₄×C₈₉	—	2	2	no	—
113	C₂×D₄×C₈₉	—	2	1	no	—
114	C₂×D₄×C₈₉	—	4	2	no	—
115	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₈₉	—	2	1	no	—
118	C₂²×C₄×C₈₉	—	2	2	no	—
119	C₂²×C₄×C₈₉	—	2	1	no	—
120	C₂²×C₄×C₈₉	—	2	2	no	—
121	C₂²×C₄×C₈₉	—	2	2	no	—
122	C₂²×C₄×C₈₉	—	4	2	no	—
123	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	Q₁₆×C₈₉	—	2	2	no	—
125	SD₁₆×C₈₉	—	2	4	no	—
126	D₈×C₈₉	—	2	2	no	—
127	M₁₆×C₈₉	—	2	2	no	—
128	M₁₆×C₈₉	—	2	1	no	—
129	M₁₆×C₈₉	—	2	2	no	—
130	M₁₆×C₈₉	—	2	1	no	—
131	M₁₆×C₈₉	—	2	1	no	—
132	M₁₆×C₈₉	—	2	1	no	—
133	M₁₆×C₈₉	—	4	2	no	—
134	C₂×C₈×C₈₉	—	2	1	no	—
135	C₂×C₈×C₈₉	—	2	1	no	—
136	C₂×C₈×C₈₉	—	2	1	no	—
137	C₂×C₈×C₈₉	—	2	2	no	—
138	C₂×C₈×C₈₉	—	2	1	no	—
139	C₂×C₈×C₈₉	—	4	2	no	—
140	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₈₉	#41	1	2	yes	—
145	C₂×C₈×C₈₉	#41	1	2	yes	—
146	C₄⋊C₄×C₈₉	—	2	2	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₄⋊C₄×C₈₉	—	2	2	no	—
149	C₄⋊C₄×C₈₉	—	2	2	no	—
150	C₄⋊C₄×C₈₉	—	4	2	no	—
151	C₄⋊C₄×C₈₉	—	4	2	no	—
152	C₄⋊C₄×C₈₉	—	8	4	no	—
153	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
154	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂²⋊C₄×C₈₉	—	2	1	no	—
156	C₂²⋊C₄×C₈₉	—	4	2	no	—
157	C₂²⋊C₄×C₈₉	—	4	2	no	—
158	C₂²⋊C₄×C₈₉	—	4	2	no	—
159	C₄²×C₈₉	—	2	1	no	—
160	C₄²×C₈₉	—	2	1	no	—
161	C₄²×C₈₉	—	4	2	no	—
162	C₄²×C₈₉	#41	1	2	yes	—
163	C₂³×C₈₉	—	2	2	no	—
164	C₂³×C₈₉	—	2	1	no	—
165	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈×C₈₉	—	2	2	no	—
167	Q₈×C₈₉	—	2	1	no	—
168	Q₈×C₈₉	—	2	1	no	—
169	Q₈×C₈₉	—	2	2	no	—
170	D₄×C₈₉	—	2	2	no	—
171	D₄×C₈₉	—	2	1	no	—
172	D₄×C₈₉	—	2	1	no	—
173	D₄×C₈₉	—	2	2	no	—
174	D₄×C₈₉	—	4	2	no	—
175	D₄×C₈₉	—	4	1	no	—
176	D₄×C₈₉	—	4	2	no	—
177	D₄×C₈₉	—	4	1	no	—
178	D₄×C₈₉	—	16	2	no	—
179	C₂×C₄×C₈₉	—	2	2	no	—
180	C₂×C₄×C₈₉	—	2	2	no	—
181	C₂×C₄×C₈₉	—	2	2	no	—
182	C₂×C₄×C₈₉	—	2	1	no	—
183	C₂×C₄×C₈₉	—	2	1	no	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	1	no	—
186	C₂×C₄×C₈₉	—	4	1	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	2	no	—
191	C₂×C₄×C₈₉	—	4	2	no	—
192	C₂×C₄×C₈₉	—	8	2	no	—
193	C₂×C₄×C₈₉	—	8	2	no	—
194	C₂×C₄×C₈₉	—	16	2	no	—
195	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈×C₈₉	—	2	1	no	—
202	C₈×C₈₉	—	2	1	no	—
203	C₈×C₈₉	—	4	1	no	—
204	C₈×C₈₉	—	4	2	no	—
205	C₈×C₈₉	—	4	1	no	—
206	C₈×C₈₉	#123	1	2	yes	—
207	C₈×C₈₉	#123	1	2	yes	—
208	C₂²×C₈₉	—	2	1	no	—
209	C₂²×C₈₉	—	2	1	no	—
210	C₂²×C₈₉	—	4	1	no	—
211	C₂²×C₈₉	—	4	2	no	—
212	C₂²×C₈₉	—	4	2	no	—
213	C₂²×C₈₉	—	4	1	no	—
214	C₂²×C₈₉	—	4	2	no	—
215	C₂²×C₈₉	—	4	1	no	—
216	C₂²×C₈₉	#542	1	1	yes	characteristic
217	C₂²×C₈₉	D₄²	1	1	yes	characteristic
218	C₂²×C₈₉	#553	1	1	yes	characteristic
219	C₄×C₈₉	—	2	4	no	—
220	C₄×C₈₉	—	4	2	no	—
221	C₄×C₈₉	—	4	1	no	—
222	C₄×C₈₉	—	4	1	no	—
223	C₄×C₈₉	—	4	1	no	—
224	C₄×C₈₉	—	8	2	no	—
225	C₄×C₈₉	—	8	2	no	—
226	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
227	C₄×C₈₉	#532	1	1	yes	characteristic
228	C₄×C₈₉	#547	1	1	yes	characteristic
229	C₄×C₈₉	#547	1	1	yes	characteristic
230	#25053	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂×C₈₉	—	2	1	no	—
232	C₂×C₈₉	—	2	1	no	—
233	C₂×C₈₉	—	8	2	no	—
234	C₂×C₈₉	—	8	1	no	—
235	C₂×C₈₉	#2422	1	1	yes	characteristic
236	C₂×C₈₉	#3292	1	1	yes	characteristic
237	C₂×C₈₉	#3293	1	1	yes	characteristic
238	#3098	C₂×C₈₉	1	2	yes	complemented
239	#3061	C₂×C₈₉	1	1	yes	characteristic
240	#2951	C₂×C₈₉	1	1	yes	characteristic, complemented
241	#2078	C₂×C₈₉	1	2	yes	complemented
242	#2039	C₂×C₈₉	1	2	yes	complemented
243	#1974	C₂×C₈₉	1	1	yes	characteristic, complemented
244	#1952	C₂×C₈₉	1	1	yes	characteristic, complemented
245	#1859	C₂×C₈₉	1	2	yes	complemented
246	#1845	C₂×C₈₉	1	1	yes	characteristic, complemented
247	#1816	C₂×C₈₉	1	1	yes	characteristic, complemented
248	#1764	C₂×C₈₉	1	1	yes	characteristic, complemented
249	C₈₉	#25053	1	1	yes	characteristic, direct product factor, Sylow
250	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
251	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
252	#537	C₂²×C₈₉	1	2	yes	—
253	#520	C₂²×C₈₉	1	2	yes	—
254	#485	—	2	2	no	—
255	#484	—	2	2	no	—
256	#482	—	2	2	no	—
257	#480	—	2	2	no	—
258	#468	—	2	2	no	—
259	#467	—	2	1	no	—
260	#466	—	2	1	no	—
261	#440	—	2	2	no	—
262	C₄×Q₁₆	—	2	1	no	—
263	C₄×D₈	—	2	1	no	—
264	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
265	#428	C₂²×C₈₉	1	2	yes	—
266	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
267	C₂×#107	C₂²×C₈₉	1	2	yes	—
268	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
269	C₂×#104	C₂²×C₈₉	1	2	yes	—
270	C₂×#104	C₂²×C₈₉	1	1	yes	characteristic
271	#417	C₂²×C₈₉	1	2	yes	—
272	#417	C₂²×C₈₉	1	2	yes	—
273	#417	C₂²×C₈₉	1	2	yes	—
274	#416	C₂²×C₈₉	1	1	yes	characteristic
275	#408	C₂²×C₈₉	1	2	yes	—
276	#405	C₂²×C₈₉	1	1	yes	characteristic
277	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
278	#384	C₂²×C₈₉	1	2	yes	—
279	#343	C₂²×C₈₉	1	2	yes	—
280	#343	C₂²×C₈₉	1	2	yes	—
281	#341	C₂²×C₈₉	1	1	yes	characteristic
282	#339	C₂²×C₈₉	1	1	yes	characteristic
283	C₂×#41	C₂³×C₈₉	1	2	yes	—
284	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
285	#135	—	2	2	no	—
286	#135	—	2	2	no	—
287	C₂×Q₁₆	D₄×C₈₉	1	1	yes	characteristic
288	C₂×SD₁₆	D₄×C₈₉	1	2	yes	—
289	C₂×D₈	D₄×C₈₉	1	1	yes	characteristic
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	1	yes	characteristic
295	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
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	C₂²⋊Q₈	—	2	2	no	—
300	C₂²⋊Q₈	—	2	2	no	—
301	#121	—	2	2	no	—
302	#121	—	2	2	no	—
303	#121	—	4	2	no	—
304	C₄×Q₈	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	4	2	no	—
307	#117	—	2	2	no	—
308	#117	C₂³×C₈₉	1	1	yes	characteristic
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	4	2	no	—
311	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
313	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
314	#108	—	2	2	no	—
315	#107	—	2	1	no	—
316	#107	—	2	1	no	—
317	#107	—	2	1	no	—
318	#107	—	2	1	no	—
319	#107	—	2	2	no	—
320	#107	—	2	2	no	—
321	#107	—	2	2	no	—
322	#106	—	2	2	no	—
323	#104	—	2	4	no	—
324	#104	—	2	2	no	—
325	#103	—	2	2	no	—
326	#103	—	2	2	no	—
327	#103	—	2	2	no	—
328	#103	D₄×C₈₉	1	1	yes	characteristic
329	#103	D₄×C₈₉	1	1	yes	characteristic
330	#102	—	2	2	no	—
331	#102	—	2	2	no	—
332	#102	—	2	2	no	—
333	#102	D₄×C₈₉	1	1	yes	characteristic
334	#102	D₄×C₈₉	1	1	yes	characteristic
335	C₂²⋊C₈	—	2	2	no	—
336	C₂²⋊C₈	—	2	2	no	—
337	#97	—	2	1	no	—
338	C₄×C₈	—	2	1	no	—
339	C₄×C₈	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	2	1	no	—
342	#41	—	2	1	no	—
343	#41	—	2	1	no	—
344	#41	—	2	2	no	—
345	#41	—	2	2	no	—
346	#41	—	2	2	no	—
347	#41	—	2	1	no	—
348	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
349	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
354	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	4	2	no	—
361	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
362	Q₁₆	—	2	2	no	—
363	SD₁₆	—	2	4	no	—
364	D₈	—	2	2	no	—
365	M₁₆	—	2	2	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	—	2	1	no	—
369	M₁₆	—	2	1	no	—
370	M₁₆	—	2	1	no	—
371	M₁₆	—	4	2	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	4	2	no	—
378	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
379	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
380	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
381	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
382	C₂×C₈	#41×C₈₉	1	2	yes	—
383	C₂×C₈	#41×C₈₉	1	2	yes	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	8	4	no	—
391	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
392	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
393	C₂²⋊C₄	—	2	1	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	2	no	—
396	C₂²⋊C₄	—	4	2	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	4	2	no	—
400	C₄²	#41×C₈₉	1	2	yes	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	1	no	—
403	C₂³	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
404	Q₈	—	2	2	no	—
405	Q₈	—	2	1	no	—
406	Q₈	—	2	1	no	—
407	Q₈	—	2	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	2	1	no	—
410	D₄	—	2	1	no	—
411	D₄	—	2	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	1	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	1	no	—
416	D₄	—	16	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	8	2	no	—
431	C₂×C₄	—	8	2	no	—
432	C₂×C₄	—	16	2	no	—
433	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
434	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
438	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
439	C₈	—	2	1	no	—
440	C₈	—	2	1	no	—
441	C₈	—	4	1	no	—
442	C₈	—	4	2	no	—
443	C₈	—	4	1	no	—
444	C₈	#123×C₈₉	1	2	yes	—
445	C₈	#123×C₈₉	1	2	yes	—
446	C₂²	—	2	1	no	—
447	C₂²	—	2	1	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	1	no	—
454	C₂²	#542×C₈₉	1	1	yes	characteristic
455	C₂²	D₄²×C₈₉	1	1	yes	characteristic
456	C₂²	#553×C₈₉	1	1	yes	characteristic
457	C₄	—	2	4	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	1	no	—
460	C₄	—	4	1	no	—
461	C₄	—	4	1	no	—
462	C₄	—	8	2	no	—
463	C₄	—	8	2	no	—
464	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
465	C₄	#532×C₈₉	1	1	yes	characteristic
466	C₄	#547×C₈₉	1	1	yes	characteristic
467	C₄	#547×C₈₉	1	1	yes	characteristic
468	C₂	—	2	1	no	—
469	C₂	—	2	1	no	—
470	C₂	—	8	2	no	—
471	C₂	—	8	1	no	—
472	C₂	C₈₉×#2422	1	1	yes	characteristic
473	C₂	C₈₉×#3292	1	1	yes	characteristic
474	C₂	C₈₉×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₈₉×#3061	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. C₈₉×#3098	C₂	semidirect
3. C₈₉×#2951	C₂	semidirect
4. C₈₉×#2078	C₂	semidirect
5. C₈₉×#2039	C₂	semidirect
6. C₈₉×#1974	C₂	semidirect
7. C₈₉×#1952	C₂	semidirect
8. C₈₉×#1859	C₂	semidirect
9. C₈₉×#1845	C₂	semidirect
10. C₈₉×#1816	C₂	semidirect
11. C₈₉×#1764	C₂	semidirect
230. #25053	C₈₉	direct (N × H)
238. #3098	C₂×C₈₉	semidirect
240. #2951	C₂×C₈₉	semidirect
241. #2078	C₂×C₈₉	semidirect
242. #2039	C₂×C₈₉	semidirect
243. #1974	C₂×C₈₉	semidirect
244. #1952	C₂×C₈₉	semidirect
245. #1859	C₂×C₈₉	semidirect
246. #1845	C₂×C₈₉	semidirect
247. #1816	C₂×C₈₉	semidirect
248. #1764	C₂×C₈₉	semidirect
249. C₈₉	#25053	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×#3098	2
C₈₉×#3061	2
C₈₉×#2951	2
C₈₉×#2078	2
C₈₉×#2039	2
C₈₉×#1974	2
C₈₉×#1952	2
C₈₉×#1859	2
C₈₉×#1845	2
C₈₉×#1816	2
C₈₉×#1764	2
#25053	89

Subgroups — C89×#25053 #36402901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25053 #36402901