Subgroups — C₈₉×#25361 #36403209

Order 22784.

732 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3141	C₂	1	1	yes	maximal, characteristic
2	C₂×#475×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3037	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2935	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2079	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2052	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1908	C₂	1	2	yes	maximal, complemented
10	C₈₉×#1815	C₂	1	2	yes	maximal, complemented
11	C₄×#101×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₈₉	C₂²	1	2	yes	—
13	C₂×#131×C₈₉	C₂²	1	2	yes	—
14	#532×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	#489×C₈₉	—	2	2	no	—
18	#487×C₈₉	—	2	1	no	—
19	#486×C₈₉	—	2	1	no	—
20	#475×C₈₉	—	2	1	no	—
21	#475×C₈₉	—	2	2	no	—
22	#475×C₈₉	—	2	1	no	—
23	#464×C₈₉	—	2	2	no	—
24	#462×C₈₉	—	2	1	no	—
25	#460×C₈₉	—	2	1	no	—
26	#436×C₈₉	—	2	1	no	—
27	#435×C₈₉	—	2	2	no	—
28	C₈×D₄×C₈₉	—	2	1	no	—
29	C₂×#108×C₈₉	C₂²	1	2	yes	—
30	C₂×#106×C₈₉	C₂²	1	2	yes	—
31	#423×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
33	#417×C₈₉	C₂²	1	2	yes	—
34	#417×C₈₉	C₂²	1	2	yes	—
35	#417×C₈₉	C₂²	1	2	yes	—
36	C₂×#103×C₈₉	C₂²	1	2	yes	—
37	C₂×#101×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×#101×C₈₉	C₂²	1	1	yes	characteristic
39	#408×C₈₉	C₂²	1	2	yes	—
40	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#343×C₈₉	C₂²	1	2	yes	—
43	#343×C₈₉	C₂²	1	1	yes	characteristic
44	#343×C₈₉	C₂²	1	2	yes	—
45	#343×C₈₉	C₂²	1	1	yes	characteristic
46	#341×C₈₉	C₂²	1	2	yes	—
47	C₂×#41×C₈₉	C₂³	1	2	yes	—
48	C₂²×Q₈×C₈₉	—	2	1	no	—
49	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
50	#137×C₈₉	—	2	4	no	—
51	#137×C₈₉	—	4	2	no	—
52	C₂×Q₁₆×C₈₉	—	2	2	no	—
53	C₂×SD₁₆×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	1	yes	characteristic
58	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	—	2	2	no	—
63	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₈₉	—	2	1	no	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	C₄⋊Q₈×C₈₉	—	2	1	no	—
67	C₄⋊Q₈×C₈₉	—	2	1	no	—
68	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
69	#127×C₈₉	D₄	1	1	yes	characteristic
70	#124×C₈₉	D₄	1	2	yes	—
71	#121×C₈₉	—	2	1	no	—
72	#121×C₈₉	—	2	1	no	—
73	C₄×Q₈×C₈₉	—	2	1	no	—
74	C₄×Q₈×C₈₉	—	2	1	no	—
75	C₄×Q₈×C₈₉	—	2	1	no	—
76	C₄×Q₈×C₈₉	—	2	1	no	—
77	C₄×D₄×C₈₉	—	2	1	no	—
78	C₄×D₄×C₈₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₈₉	—	2	1	no	—
84	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
85	#108×C₈₉	—	4	2	no	—
86	#106×C₈₉	—	2	2	no	—
87	#106×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	—	2	1	no	—
89	C₄⋊C₈×C₈₉	—	2	1	no	—
90	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
91	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
92	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
93	#103×C₈₉	—	2	2	no	—
94	#103×C₈₉	—	2	2	no	—
95	#103×C₈₉	—	2	2	no	—
96	#103×C₈₉	—	2	2	no	—
97	#103×C₈₉	—	2	2	no	—
98	#102×C₈₉	—	2	2	no	—
99	#102×C₈₉	—	2	2	no	—
100	#102×C₈₉	—	2	2	no	—
101	#101×C₈₉	—	2	2	no	—
102	#101×C₈₉	—	2	2	no	—
103	C₂²⋊C₈×C₈₉	—	2	2	no	—
104	C₂²⋊C₈×C₈₉	—	2	2	no	—
105	#97×C₈₉	—	2	1	no	—
106	#97×C₈₉	—	2	1	no	—
107	C₄×C₈×C₈₉	—	2	1	no	—
108	C₄×C₈×C₈₉	—	2	1	no	—
109	#41×C₈₉	—	2	2	no	—
110	#41×C₈₉	—	2	2	no	—
111	#41×C₈₉	—	2	2	no	—
112	#41×C₈₉	—	2	2	no	—
113	C₂×Q₈×C₈₉	—	2	1	no	—
114	C₂×Q₈×C₈₉	—	2	1	no	—
115	C₂×Q₈×C₈₉	—	2	1	no	—
116	C₂×Q₈×C₈₉	—	2	1	no	—
117	C₂×Q₈×C₈₉	—	2	1	no	—
118	C₂×Q₈×C₈₉	—	4	1	no	—
119	C₂×Q₈×C₈₉	—	4	1	no	—
120	C₂×Q₈×C₈₉	—	8	2	no	—
121	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
122	C₂×Q₈×C₈₉	#41	1	2	yes	—
123	C₂×D₄×C₈₉	—	2	2	no	—
124	C₂×D₄×C₈₉	—	2	1	no	—
125	C₂×D₄×C₈₉	—	2	1	no	—
126	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₈₉	—	2	1	no	—
128	C₂²×C₄×C₈₉	—	2	2	no	—
129	C₂²×C₄×C₈₉	—	2	1	no	—
130	C₂²×C₄×C₈₉	—	2	2	no	—
131	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
133	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	Q₁₆×C₈₉	—	4	4	no	—
135	SD₁₆×C₈₉	—	4	4	no	—
136	M₁₆×C₈₉	—	2	1	no	—
137	M₁₆×C₈₉	—	2	1	no	—
138	M₁₆×C₈₉	—	2	1	no	—
139	M₁₆×C₈₉	—	2	1	no	—
140	M₁₆×C₈₉	—	2	2	no	—
141	M₁₆×C₈₉	—	2	1	no	—
142	M₁₆×C₈₉	—	2	2	no	—
143	M₁₆×C₈₉	—	2	1	no	—
144	M₁₆×C₈₉	—	2	1	no	—
145	M₁₆×C₈₉	—	4	4	no	—
146	M₁₆×C₈₉	#41	1	2	yes	—
147	C₂×C₈×C₈₉	—	2	1	no	—
148	C₂×C₈×C₈₉	—	2	2	no	—
149	C₂×C₈×C₈₉	—	2	2	no	—
150	C₂×C₈×C₈₉	—	2	1	no	—
151	C₂×C₈×C₈₉	—	2	1	no	—
152	C₂×C₈×C₈₉	—	2	1	no	—
153	C₂×C₈×C₈₉	—	2	2	no	—
154	C₂×C₈×C₈₉	—	4	4	no	—
155	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₈₉	—	2	1	no	—
158	C₄⋊C₄×C₈₉	—	2	1	no	—
159	C₄⋊C₄×C₈₉	—	2	1	no	—
160	C₄⋊C₄×C₈₉	—	2	2	no	—
161	C₄⋊C₄×C₈₉	—	2	2	no	—
162	C₄⋊C₄×C₈₉	—	2	1	no	—
163	C₄⋊C₄×C₈₉	—	4	1	no	—
164	C₄⋊C₄×C₈₉	—	4	1	no	—
165	C₄⋊C₄×C₈₉	—	4	1	no	—
166	C₄⋊C₄×C₈₉	—	4	1	no	—
167	C₂²⋊C₄×C₈₉	—	2	2	no	—
168	C₂²⋊C₄×C₈₉	—	2	2	no	—
169	C₄²×C₈₉	—	4	1	no	—
170	C₄²×C₈₉	—	4	1	no	—
171	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
175	C₂³×C₈₉	—	2	2	no	—
176	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
177	Q₈×C₈₉	—	2	2	no	—
178	Q₈×C₈₉	—	2	2	no	—
179	Q₈×C₈₉	—	2	2	no	—
180	Q₈×C₈₉	—	2	2	no	—
181	Q₈×C₈₉	—	4	1	no	—
182	Q₈×C₈₉	—	4	1	no	—
183	Q₈×C₈₉	—	8	2	no	—
184	Q₈×C₈₉	—	8	2	no	—
185	D₄×C₈₉	—	2	2	no	—
186	D₄×C₈₉	—	2	2	no	—
187	D₄×C₈₉	—	4	2	no	—
188	D₄×C₈₉	—	4	2	no	—
189	D₄×C₈₉	—	4	2	no	—
190	D₄×C₈₉	—	8	1	no	—
191	C₂×C₄×C₈₉	—	2	2	no	—
192	C₂×C₄×C₈₉	—	2	2	no	—
193	C₂×C₄×C₈₉	—	2	2	no	—
194	C₂×C₄×C₈₉	—	4	1	no	—
195	C₂×C₄×C₈₉	—	4	1	no	—
196	C₂×C₄×C₈₉	—	4	1	no	—
197	C₂×C₄×C₈₉	—	4	2	no	—
198	C₂×C₄×C₈₉	—	4	2	no	—
199	C₂×C₄×C₈₉	—	4	1	no	—
200	C₂×C₄×C₈₉	—	4	2	no	—
201	C₂×C₄×C₈₉	—	8	1	no	—
202	C₂×C₄×C₈₉	—	8	2	no	—
203	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
204	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
205	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
206	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
207	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₈₉	—	2	2	no	—
214	C₈×C₈₉	—	2	2	no	—
215	C₈×C₈₉	—	2	2	no	—
216	C₈×C₈₉	—	4	1	no	—
217	C₈×C₈₉	—	4	2	no	—
218	C₈×C₈₉	—	4	1	no	—
219	C₈×C₈₉	—	4	1	no	—
220	C₂²×C₈₉	—	2	1	no	—
221	C₂²×C₈₉	—	2	1	no	—
222	C₂²×C₈₉	—	4	2	no	—
223	C₂²×C₈₉	—	4	2	no	—
224	C₂²×C₈₉	—	4	2	no	—
225	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
226	C₂²×C₈₉	#529	1	1	yes	characteristic
227	C₂²×C₈₉	#542	1	1	yes	characteristic
228	C₄×C₈₉	—	2	1	no	—
229	C₄×C₈₉	—	2	1	no	—
230	C₄×C₈₉	—	2	1	no	—
231	C₄×C₈₉	—	4	2	no	—
232	C₄×C₈₉	—	4	2	no	—
233	C₄×C₈₉	—	8	2	no	—
234	C₄×C₈₉	—	8	1	no	—
235	C₄×C₈₉	#471	1	2	yes	—
236	C₄×C₈₉	#546	1	1	yes	characteristic
237	C₄×C₈₉	#546	1	1	yes	characteristic
238	C₄×C₈₉	#547	1	1	yes	characteristic
239	C₄×C₈₉	#548	1	1	yes	characteristic
240	#25361	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₈₉	—	2	1	no	—
242	C₂×C₈₉	—	2	1	no	—
243	C₂×C₈₉	—	8	2	no	—
244	C₂×C₈₉	#2427	1	1	yes	characteristic
245	C₂×C₈₉	C₂×#471	1	1	yes	characteristic
246	C₂×C₈₉	#3069	1	1	yes	characteristic
247	#3141	C₂×C₈₉	1	1	yes	characteristic
248	C₂×#475	C₂×C₈₉	1	1	yes	characteristic, complemented
249	#3037	C₂×C₈₉	1	1	yes	characteristic
250	#2935	C₂×C₈₉	1	1	yes	characteristic
251	#2079	C₂×C₈₉	1	2	yes	complemented
252	#2052	C₂×C₈₉	1	2	yes	complemented
253	#1997	C₂×C₈₉	1	1	yes	characteristic, complemented
254	#1950	C₂×C₈₉	1	1	yes	characteristic, complemented
255	#1908	C₂×C₈₉	1	2	yes	complemented
256	#1815	C₂×C₈₉	1	2	yes	complemented
257	C₄×#101	C₂×C₈₉	1	1	yes	characteristic, complemented
258	C₈₉	#25361	1	1	yes	characteristic, direct product factor, Sylow
259	C₂×#137	C₂²×C₈₉	1	2	yes	—
260	C₂×#131	C₂²×C₈₉	1	2	yes	—
261	#532	C₂²×C₈₉	1	1	yes	characteristic
262	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
263	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
264	#489	—	2	2	no	—
265	#487	—	2	1	no	—
266	#486	—	2	1	no	—
267	#475	—	2	1	no	—
268	#475	—	2	2	no	—
269	#475	—	2	1	no	—
270	#464	—	2	2	no	—
271	#462	—	2	1	no	—
272	#460	—	2	1	no	—
273	#436	—	2	1	no	—
274	#435	—	2	2	no	—
275	C₈×D₄	—	2	1	no	—
276	C₂×#108	C₂²×C₈₉	1	2	yes	—
277	C₂×#106	C₂²×C₈₉	1	2	yes	—
278	#423	C₂²×C₈₉	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
280	#417	C₂²×C₈₉	1	2	yes	—
281	#417	C₂²×C₈₉	1	2	yes	—
282	#417	C₂²×C₈₉	1	2	yes	—
283	C₂×#103	C₂²×C₈₉	1	2	yes	—
284	C₂×#101	C₂²×C₈₉	1	1	yes	characteristic
285	C₂×#101	C₂²×C₈₉	1	1	yes	characteristic
286	#408	C₂²×C₈₉	1	2	yes	—
287	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
288	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
289	#343	C₂²×C₈₉	1	2	yes	—
290	#343	C₂²×C₈₉	1	1	yes	characteristic
291	#343	C₂²×C₈₉	1	2	yes	—
292	#343	C₂²×C₈₉	1	1	yes	characteristic
293	#341	C₂²×C₈₉	1	2	yes	—
294	C₂×#41	C₂³×C₈₉	1	2	yes	—
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
297	#137	—	2	4	no	—
298	#137	—	4	2	no	—
299	C₂×Q₁₆	—	2	2	no	—
300	C₂×SD₁₆	—	2	2	no	—
301	#131	—	2	4	no	—
302	#131	—	4	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
305	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
306	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
307	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
308	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
309	C₂²×C₈	—	2	2	no	—
310	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	1	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
316	#127	D₄×C₈₉	1	1	yes	characteristic
317	#124	D₄×C₈₉	1	2	yes	—
318	#121	—	2	1	no	—
319	#121	—	2	1	no	—
320	C₄×Q₈	—	2	1	no	—
321	C₄×Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	1	no	—
323	C₄×Q₈	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
330	C₂×C₄²	—	2	1	no	—
331	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
332	#108	—	4	2	no	—
333	#106	—	2	2	no	—
334	#106	—	2	2	no	—
335	C₄⋊C₈	—	2	1	no	—
336	C₄⋊C₈	—	2	1	no	—
337	C₄⋊C₈	D₄×C₈₉	1	1	yes	characteristic
338	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
339	C₄⋊C₈	D₄×C₈₉	1	1	yes	characteristic
340	#103	—	2	2	no	—
341	#103	—	2	2	no	—
342	#103	—	2	2	no	—
343	#103	—	2	2	no	—
344	#103	—	2	2	no	—
345	#102	—	2	2	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#101	—	2	2	no	—
349	#101	—	2	2	no	—
350	C₂²⋊C₈	—	2	2	no	—
351	C₂²⋊C₈	—	2	2	no	—
352	#97	—	2	1	no	—
353	#97	—	2	1	no	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	—	2	1	no	—
356	#41	—	2	2	no	—
357	#41	—	2	2	no	—
358	#41	—	2	2	no	—
359	#41	—	2	2	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	4	1	no	—
366	C₂×Q₈	—	4	1	no	—
367	C₂×Q₈	—	8	2	no	—
368	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
369	C₂×Q₈	#41×C₈₉	1	2	yes	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
379	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
381	Q₁₆	—	4	4	no	—
382	SD₁₆	—	4	4	no	—
383	M₁₆	—	2	1	no	—
384	M₁₆	—	2	1	no	—
385	M₁₆	—	2	1	no	—
386	M₁₆	—	2	1	no	—
387	M₁₆	—	2	2	no	—
388	M₁₆	—	2	1	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	—	2	1	no	—
391	M₁₆	—	2	1	no	—
392	M₁₆	—	4	4	no	—
393	M₁₆	#41×C₈₉	1	2	yes	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	4	4	no	—
402	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
403	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	4	1	no	—
411	C₄⋊C₄	—	4	1	no	—
412	C₄⋊C₄	—	4	1	no	—
413	C₄⋊C₄	—	4	1	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₄²	—	4	1	no	—
417	C₄²	—	4	1	no	—
418	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
419	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
420	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
422	C₂³	—	2	2	no	—
423	C₂³	C₂×#41×C₈₉	1	1	yes	characteristic
424	Q₈	—	2	2	no	—
425	Q₈	—	2	2	no	—
426	Q₈	—	2	2	no	—
427	Q₈	—	2	2	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	1	no	—
430	Q₈	—	8	2	no	—
431	Q₈	—	8	2	no	—
432	D₄	—	2	2	no	—
433	D₄	—	2	2	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	8	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	8	1	no	—
449	C₂×C₄	—	8	2	no	—
450	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
451	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
452	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
453	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
454	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
455	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
456	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
457	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
458	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
459	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
460	C₈	—	2	2	no	—
461	C₈	—	2	2	no	—
462	C₈	—	2	2	no	—
463	C₈	—	4	1	no	—
464	C₈	—	4	2	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
473	C₂²	#529×C₈₉	1	1	yes	characteristic
474	C₂²	#542×C₈₉	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	8	2	no	—
481	C₄	—	8	1	no	—
482	C₄	#471×C₈₉	1	2	yes	—
483	C₄	#546×C₈₉	1	1	yes	characteristic
484	C₄	#546×C₈₉	1	1	yes	characteristic
485	C₄	#547×C₈₉	1	1	yes	characteristic
486	C₄	#548×C₈₉	1	1	yes	characteristic
487	C₂	—	2	1	no	—
488	C₂	—	2	1	no	—
489	C₂	—	8	2	no	—
490	C₂	C₈₉×#2427	1	1	yes	characteristic
491	C₂	C₂×#471×C₈₉	1	1	yes	characteristic
492	C₂	C₈₉×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	#532×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#475×C₈₉	C₂	semidirect
5. C₈₉×#2079	C₂	semidirect
6. C₈₉×#2052	C₂	semidirect
7. C₈₉×#1997	C₂	semidirect
8. C₈₉×#1950	C₂	semidirect
9. C₈₉×#1908	C₂	semidirect
10. C₈₉×#1815	C₂	semidirect
11. C₄×#101×C₈₉	C₂	semidirect
240. #25361	C₈₉	direct (N × H)
248. C₂×#475	C₂×C₈₉	semidirect
251. #2079	C₂×C₈₉	semidirect
252. #2052	C₂×C₈₉	semidirect
253. #1997	C₂×C₈₉	semidirect
254. #1950	C₂×C₈₉	semidirect
255. #1908	C₂×C₈₉	semidirect
256. #1815	C₂×C₈₉	semidirect
257. C₄×#101	C₂×C₈₉	semidirect
258. C₈₉	#25361	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×#3141	2
C₂×#475×C₈₉	2
C₈₉×#3037	2
C₈₉×#2935	2
C₈₉×#2079	2
C₈₉×#2052	2
C₈₉×#1997	2
C₈₉×#1950	2
C₈₉×#1908	2
C₈₉×#1815	2
C₄×#101×C₈₉	2
#25361	89

Subgroups — C89×#25361 #36403209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25361 #36403209