Subgroups — C₈₉×#25603 #36403451

Order 22784.

1172 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#497×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#489×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2850	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2001	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2000	C₂	1	2	yes	maximal
7	C₈₉×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1880	C₂	1	2	yes	maximal, complemented
10	C₈₉×#1878	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1749	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
14	#533×C₈₉	—	2	4	no	—
15	#533×C₈₉	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#125×C₈₉	—	2	1	no	—
19	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
21	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
26	C₂×#117×C₈₉	—	2	1	no	—
27	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
28	#497×C₈₉	—	2	4	no	—
29	#489×C₈₉	—	2	4	no	—
30	C₂×#103×C₈₉	C₂²	1	2	yes	—
31	C₂×#103×C₈₉	C₂²	1	2	yes	—
32	C₂×#103×C₈₉	C₂²	1	2	yes	—
33	C₂×#102×C₈₉	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
35	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
37	#391×C₈₉	C₂²	1	1	yes	characteristic
38	#390×C₈₉	C₂²	1	1	yes	characteristic
39	#388×C₈₉	—	2	1	no	—
40	#388×C₈₉	—	2	1	no	—
41	#386×C₈₉	—	2	1	no	—
42	#386×C₈₉	—	2	1	no	—
43	#382×C₈₉	C₂²	1	1	yes	characteristic
44	#340×C₈₉	C₂²	1	2	yes	—
45	#340×C₈₉	C₂²	1	2	yes	—
46	#336×C₈₉	C₂²	1	2	yes	—
47	C₂²×Q₈×C₈₉	—	2	1	no	—
48	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
50	C₂²×D₄×C₈₉	—	2	1	no	—
51	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×Q₁₆×C₈₉	—	2	4	no	—
56	C₂×Q₁₆×C₈₉	—	2	4	no	—
57	C₂×Q₁₆×C₈₉	—	4	2	no	—
58	C₂×SD₁₆×C₈₉	—	2	4	no	—
59	C₂×SD₁₆×C₈₉	—	2	4	no	—
60	C₂×SD₁₆×C₈₉	—	4	2	no	—
61	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
62	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₈₉	—	2	2	no	—
64	C₄⋊Q₈×C₈₉	—	2	2	no	—
65	C₄⋊Q₈×C₈₉	D₄	1	4	yes	complemented
66	C₄⋊Q₈×C₈₉	D₄	1	4	yes	complemented
67	#125×C₈₉	—	2	8	no	—
68	#124×C₈₉	—	2	2	no	—
69	#124×C₈₉	—	2	2	no	—
70	C₂²⋊Q₈×C₈₉	—	2	8	no	—
71	C₂²⋊Q₈×C₈₉	—	2	4	no	—
72	C₂²⋊Q₈×C₈₉	—	2	4	no	—
73	C₂²⋊Q₈×C₈₉	—	4	2	no	—
74	#121×C₈₉	—	4	2	no	—
75	C₄×Q₈×C₈₉	—	2	8	no	—
76	C₄×Q₈×C₈₉	—	2	4	no	—
77	C₄×Q₈×C₈₉	—	2	4	no	—
78	#117×C₈₉	—	2	8	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	2	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
88	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
95	C₂×C₄²×C₈₉	—	2	2	no	—
96	C₂×C₄²×C₈₉	—	2	2	no	—
97	C₂×C₄²×C₈₉	—	2	2	no	—
98	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
102	#103×C₈₉	—	2	4	no	—
103	#103×C₈₉	—	4	2	no	—
104	#103×C₈₉	—	4	2	no	—
105	#102×C₈₉	—	2	4	no	—
106	C₂²⋊C₈×C₈₉	—	2	4	no	—
107	#97×C₈₉	—	2	2	no	—
108	#97×C₈₉	—	2	2	no	—
109	#95×C₈₉	—	2	1	no	—
110	#95×C₈₉	—	2	1	no	—
111	#95×C₈₉	—	2	1	no	—
112	#95×C₈₉	—	2	1	no	—
113	C₂⁴×C₈₉	—	2	1	no	—
114	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Q₈×C₈₉	—	2	4	no	—
116	C₂×Q₈×C₈₉	—	2	2	no	—
117	C₂×Q₈×C₈₉	—	2	1	no	—
118	C₂×Q₈×C₈₉	—	2	1	no	—
119	C₂×Q₈×C₈₉	—	2	2	no	—
120	C₂×Q₈×C₈₉	—	2	2	no	—
121	C₂×Q₈×C₈₉	—	2	2	no	—
122	C₂×Q₈×C₈₉	—	2	2	no	—
123	C₂×Q₈×C₈₉	—	2	2	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	4	2	no	—
126	C₂×Q₈×C₈₉	—	4	2	no	—
127	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
128	C₂×D₄×C₈₉	—	2	1	no	—
129	C₂×D₄×C₈₉	—	2	1	no	—
130	C₂×D₄×C₈₉	—	2	2	no	—
131	C₂×D₄×C₈₉	—	2	2	no	—
132	C₂×D₄×C₈₉	—	4	2	no	—
133	C₂×D₄×C₈₉	—	4	2	no	—
134	C₂×D₄×C₈₉	—	4	2	no	—
135	C₂²×C₄×C₈₉	—	2	2	no	—
136	C₂²×C₄×C₈₉	—	2	2	no	—
137	C₂²×C₄×C₈₉	—	2	2	no	—
138	C₂²×C₄×C₈₉	—	2	2	no	—
139	C₂²×C₄×C₈₉	—	2	2	no	—
140	C₂²×C₄×C₈₉	—	2	2	no	—
141	C₂²×C₄×C₈₉	—	2	1	no	—
142	C₂²×C₄×C₈₉	—	2	1	no	—
143	C₂²×C₄×C₈₉	—	2	2	no	—
144	C₂²×C₄×C₈₉	—	2	2	no	—
145	C₂²×C₄×C₈₉	—	2	1	no	—
146	C₂²×C₄×C₈₉	—	2	1	no	—
147	C₂²×C₄×C₈₉	—	2	1	no	—
148	C₂²×C₄×C₈₉	—	4	1	no	—
149	C₂²×C₄×C₈₉	—	4	1	no	—
150	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
156	Q₁₆×C₈₉	—	4	8	no	—
157	SD₁₆×C₈₉	—	4	8	no	—
158	C₂×C₈×C₈₉	—	2	2	no	—
159	C₂×C₈×C₈₉	—	2	2	no	—
160	C₂×C₈×C₈₉	—	2	2	no	—
161	C₂×C₈×C₈₉	—	2	2	no	—
162	C₂×C₈×C₈₉	—	2	2	no	—
163	C₂×C₈×C₈₉	#41	1	4	yes	—
164	C₄⋊C₄×C₈₉	—	2	8	no	—
165	C₄⋊C₄×C₈₉	—	2	8	no	—
166	C₄⋊C₄×C₈₉	—	2	4	no	—
167	C₄⋊C₄×C₈₉	—	2	4	no	—
168	C₄⋊C₄×C₈₉	—	2	4	no	—
169	C₄⋊C₄×C₈₉	—	2	2	no	—
170	C₄⋊C₄×C₈₉	—	2	2	no	—
171	C₄⋊C₄×C₈₉	—	2	2	no	—
172	C₄⋊C₄×C₈₉	—	4	2	no	—
173	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
174	C₂²⋊C₄×C₈₉	—	2	8	no	—
175	C₂²⋊C₄×C₈₉	—	4	2	no	—
176	C₂²⋊C₄×C₈₉	—	4	2	no	—
177	C₂²⋊C₄×C₈₉	—	4	2	no	—
178	C₂²⋊C₄×C₈₉	—	4	2	no	—
179	C₂²⋊C₄×C₈₉	—	4	2	no	—
180	C₂²⋊C₄×C₈₉	—	4	2	no	—
181	C₄²×C₈₉	—	2	8	no	—
182	C₄²×C₈₉	—	2	2	no	—
183	C₄²×C₈₉	—	2	2	no	—
184	C₄²×C₈₉	—	2	1	no	—
185	C₄²×C₈₉	—	2	1	no	—
186	C₄²×C₈₉	C₂×D₄	1	2	yes	—
187	C₄²×C₈₉	C₂×D₄	1	2	yes	—
188	C₂³×C₈₉	—	2	1	no	—
189	C₂³×C₈₉	—	2	1	no	—
190	C₂³×C₈₉	—	2	1	no	—
191	C₂³×C₈₉	—	2	1	no	—
192	C₂³×C₈₉	—	2	1	no	—
193	C₂³×C₈₉	—	2	1	no	—
194	C₂³×C₈₉	—	4	1	no	—
195	C₂³×C₈₉	—	4	1	no	—
196	C₂³×C₈₉	—	4	1	no	—
197	C₂³×C₈₉	—	4	1	no	—
198	C₂³×C₈₉	—	4	1	no	—
199	C₂³×C₈₉	—	4	1	no	—
200	C₂³×C₈₉	—	4	1	no	—
201	C₂³×C₈₉	#137	1	2	yes	—
202	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₈₉	—	2	4	no	—
204	Q₈×C₈₉	—	2	4	no	—
205	Q₈×C₈₉	—	4	8	no	—
206	Q₈×C₈₉	—	4	2	no	—
207	Q₈×C₈₉	—	4	2	no	—
208	Q₈×C₈₉	—	4	2	no	—
209	Q₈×C₈₉	—	4	2	no	—
210	D₄×C₈₉	—	4	2	no	—
211	D₄×C₈₉	—	4	2	no	—
212	D₄×C₈₉	—	8	4	no	—
213	C₂×C₄×C₈₉	—	2	8	no	—
214	C₂×C₄×C₈₉	—	2	1	no	—
215	C₂×C₄×C₈₉	—	2	1	no	—
216	C₂×C₄×C₈₉	—	2	1	no	—
217	C₂×C₄×C₈₉	—	2	1	no	—
218	C₂×C₄×C₈₉	—	2	1	no	—
219	C₂×C₄×C₈₉	—	2	1	no	—
220	C₂×C₄×C₈₉	—	2	1	no	—
221	C₂×C₄×C₈₉	—	2	1	no	—
222	C₂×C₄×C₈₉	—	2	4	no	—
223	C₂×C₄×C₈₉	—	2	4	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	1	no	—
228	C₂×C₄×C₈₉	—	4	4	no	—
229	C₂×C₄×C₈₉	—	4	2	no	—
230	C₂×C₄×C₈₉	—	4	2	no	—
231	C₂×C₄×C₈₉	—	4	2	no	—
232	C₂×C₄×C₈₉	—	4	2	no	—
233	C₂×C₄×C₈₉	—	4	2	no	—
234	C₂×C₄×C₈₉	—	4	2	no	—
235	C₂×C₄×C₈₉	—	4	1	no	—
236	C₂×C₄×C₈₉	—	4	1	no	—
237	C₂×C₄×C₈₉	—	4	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	#120	1	2	yes	—
245	C₂×C₄×C₈₉	#120	1	2	yes	—
246	C₂×C₄×C₈₉	#137	1	2	yes	—
247	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
250	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
251	C₈×C₈₉	—	2	4	no	—
252	C₈×C₈₉	—	4	2	no	—
253	C₂²×C₈₉	—	2	2	no	—
254	C₂²×C₈₉	—	2	2	no	—
255	C₂²×C₈₉	—	2	2	no	—
256	C₂²×C₈₉	—	4	1	no	—
257	C₂²×C₈₉	—	4	1	no	—
258	C₂²×C₈₉	—	4	1	no	—
259	C₂²×C₈₉	—	4	1	no	—
260	C₂²×C₈₉	—	8	1	no	—
261	C₂²×C₈₉	—	8	1	no	—
262	C₂²×C₈₉	—	8	1	no	—
263	C₂²×C₈₉	—	8	1	no	—
264	C₂²×C₈₉	—	8	1	no	—
265	C₂²×C₈₉	—	8	1	no	—
266	C₂²×C₈₉	—	8	1	no	—
267	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
268	C₂²×C₈₉	#532	1	1	yes	characteristic
269	C₂²×C₈₉	#535	1	1	yes	characteristic
270	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
271	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
272	C₂²×C₈₉	#575	1	1	yes	characteristic
273	C₂²×C₈₉	#575	1	1	yes	characteristic
274	C₄×C₈₉	—	2	2	no	—
275	C₄×C₈₉	—	2	2	no	—
276	C₄×C₈₉	—	2	2	no	—
277	C₄×C₈₉	—	4	1	no	—
278	C₄×C₈₉	—	4	4	no	—
279	C₄×C₈₉	—	4	1	no	—
280	C₄×C₈₉	—	4	4	no	—
281	C₄×C₈₉	—	8	1	no	—
282	C₄×C₈₉	—	8	1	no	—
283	C₄×C₈₉	—	8	1	no	—
284	C₄×C₈₉	#546	1	2	yes	—
285	C₄×C₈₉	#550	1	2	yes	—
286	#25603	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₈₉	—	4	2	no	—
288	C₂×C₈₉	—	16	1	no	—
289	C₂×C₈₉	#2431	1	1	yes	characteristic
290	C₂×C₈₉	#3004	1	1	yes	characteristic
291	C₂×C₈₉	#3005	1	1	yes	characteristic
292	C₂×C₈₉	#3172	1	1	yes	characteristic
293	C₂×C₈₉	#3181	1	1	yes	characteristic
294	C₂×C₈₉	#3215	1	1	yes	characteristic
295	C₂×C₈₉	#3216	1	1	yes	characteristic
296	C₂×#533	C₂×C₈₉	1	1	yes	characteristic, complemented
297	C₂×#497	C₂×C₈₉	1	1	yes	characteristic, complemented
298	C₂×#489	C₂×C₈₉	1	1	yes	characteristic, complemented
299	#2850	C₂×C₈₉	1	1	yes	characteristic
300	#2001	C₂×C₈₉	1	2	yes	complemented
301	#2000	C₂×C₈₉	1	2	yes	—
302	#1993	C₂×C₈₉	1	1	yes	characteristic, complemented
303	#1960	C₂×C₈₉	1	1	yes	characteristic, complemented
304	#1880	C₂×C₈₉	1	2	yes	complemented
305	#1878	C₂×C₈₉	1	2	yes	complemented
306	#1749	C₂×C₈₉	1	1	yes	characteristic, complemented
307	C₈₉	#25603	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²×Q₁₆	C₂²×C₈₉	1	2	yes	—
309	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
310	#533	—	2	4	no	—
311	#533	—	2	4	no	—
312	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
313	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
314	C₂×#125	—	2	1	no	—
315	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
316	C₂×C₂²⋊Q₈	—	2	1	no	—
317	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
318	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
319	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
320	C₂×C₄×Q₈	—	2	1	no	—
321	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
322	C₂×#117	—	2	1	no	—
323	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
324	#497	—	2	4	no	—
325	#489	—	2	4	no	—
326	C₂×#103	C₂²×C₈₉	1	2	yes	—
327	C₂×#103	C₂²×C₈₉	1	2	yes	—
328	C₂×#103	C₂²×C₈₉	1	2	yes	—
329	C₂×#102	C₂²×C₈₉	1	2	yes	—
330	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
331	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
332	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
333	#391	C₂²×C₈₉	1	1	yes	characteristic
334	#390	C₂²×C₈₉	1	1	yes	characteristic
335	#388	—	2	1	no	—
336	#388	—	2	1	no	—
337	#386	—	2	1	no	—
338	#386	—	2	1	no	—
339	#382	C₂²×C₈₉	1	1	yes	characteristic
340	#340	C₂²×C₈₉	1	2	yes	—
341	#340	C₂²×C₈₉	1	2	yes	—
342	#336	C₂²×C₈₉	1	2	yes	—
343	C₂²×Q₈	—	2	1	no	—
344	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
345	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
346	C₂²×D₄	—	2	1	no	—
347	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
348	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
349	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
351	C₂×Q₁₆	—	2	4	no	—
352	C₂×Q₁₆	—	2	4	no	—
353	C₂×Q₁₆	—	4	2	no	—
354	C₂×SD₁₆	—	2	4	no	—
355	C₂×SD₁₆	—	2	4	no	—
356	C₂×SD₁₆	—	4	2	no	—
357	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
358	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
359	C₄⋊Q₈	—	2	2	no	—
360	C₄⋊Q₈	—	2	2	no	—
361	C₄⋊Q₈	D₄×C₈₉	1	4	yes	complemented
362	C₄⋊Q₈	D₄×C₈₉	1	4	yes	complemented
363	#125	—	2	8	no	—
364	#124	—	2	2	no	—
365	#124	—	2	2	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	—	2	4	no	—
368	C₂²⋊Q₈	—	2	4	no	—
369	C₂²⋊Q₈	—	4	2	no	—
370	#121	—	4	2	no	—
371	C₄×Q₈	—	2	8	no	—
372	C₄×Q₈	—	2	4	no	—
373	C₄×Q₈	—	2	4	no	—
374	#117	—	2	8	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	—	2	2	no	—
393	C₂×C₄²	—	2	2	no	—
394	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
395	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
398	#103	—	2	4	no	—
399	#103	—	4	2	no	—
400	#103	—	4	2	no	—
401	#102	—	2	4	no	—
402	C₂²⋊C₈	—	2	4	no	—
403	#97	—	2	2	no	—
404	#97	—	2	2	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	C₂⁴	—	2	1	no	—
410	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
411	C₂×Q₈	—	2	4	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	1	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	4	2	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	4	2	no	—
429	C₂×D₄	—	4	2	no	—
430	C₂×D₄	—	4	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
452	Q₁₆	—	4	8	no	—
453	SD₁₆	—	4	8	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	#41×C₈₉	1	4	yes	—
460	C₄⋊C₄	—	2	8	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	4	2	no	—
469	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
470	C₂²⋊C₄	—	2	8	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	—	4	2	no	—
477	C₄²	—	2	8	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	C₂×D₄×C₈₉	1	2	yes	—
483	C₄²	C₂×D₄×C₈₉	1	2	yes	—
484	C₂³	—	2	1	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	—	2	1	no	—
489	C₂³	—	2	1	no	—
490	C₂³	—	4	1	no	—
491	C₂³	—	4	1	no	—
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	1	no	—
495	C₂³	—	4	1	no	—
496	C₂³	—	4	1	no	—
497	C₂³	#137×C₈₉	1	2	yes	—
498	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
499	Q₈	—	2	4	no	—
500	Q₈	—	2	4	no	—
501	Q₈	—	4	8	no	—
502	Q₈	—	4	2	no	—
503	Q₈	—	4	2	no	—
504	Q₈	—	4	2	no	—
505	Q₈	—	4	2	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	D₄	—	8	4	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	#120×C₈₉	1	2	yes	—
541	C₂×C₄	#120×C₈₉	1	2	yes	—
542	C₂×C₄	#137×C₈₉	1	2	yes	—
543	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
544	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
545	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
546	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
547	C₈	—	2	4	no	—
548	C₈	—	4	2	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	—	8	1	no	—
559	C₂²	—	8	1	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
564	C₂²	#532×C₈₉	1	1	yes	characteristic
565	C₂²	#535×C₈₉	1	1	yes	characteristic
566	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
567	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
568	C₂²	#575×C₈₉	1	1	yes	characteristic
569	C₂²	#575×C₈₉	1	1	yes	characteristic
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	1	no	—
576	C₄	—	4	4	no	—
577	C₄	—	8	1	no	—
578	C₄	—	8	1	no	—
579	C₄	—	8	1	no	—
580	C₄	#546×C₈₉	1	2	yes	—
581	C₄	#550×C₈₉	1	2	yes	—
582	C₂	—	4	2	no	—
583	C₂	—	16	1	no	—
584	C₂	C₈₉×#2431	1	1	yes	characteristic
585	C₂	C₈₉×#3004	1	1	yes	characteristic
586	C₂	C₈₉×#3005	1	1	yes	characteristic
587	C₂	C₈₉×#3172	1	1	yes	characteristic
588	C₂	C₈₉×#3181	1	1	yes	characteristic
589	C₂	C₈₉×#3215	1	1	yes	characteristic
590	C₂	C₈₉×#3216	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#121×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#533×C₈₉	C₂	semidirect
2. C₂×#497×C₈₉	C₂	semidirect
3. C₂×#489×C₈₉	C₂	semidirect
5. C₈₉×#2001	C₂	semidirect
7. C₈₉×#1993	C₂	semidirect
8. C₈₉×#1960	C₂	semidirect
9. C₈₉×#1880	C₂	semidirect
10. C₈₉×#1878	C₂	semidirect
11. C₈₉×#1749	C₂	semidirect
65. C₄⋊Q₈×C₈₉	D₄	semidirect
66. C₄⋊Q₈×C₈₉	D₄	semidirect
286. #25603	C₈₉	direct (N × H)
296. C₂×#533	C₂×C₈₉	semidirect
297. C₂×#497	C₂×C₈₉	semidirect
298. C₂×#489	C₂×C₈₉	semidirect
300. #2001	C₂×C₈₉	semidirect
302. #1993	C₂×C₈₉	semidirect
303. #1960	C₂×C₈₉	semidirect
304. #1880	C₂×C₈₉	semidirect
305. #1878	C₂×C₈₉	semidirect
306. #1749	C₂×C₈₉	semidirect
307. C₈₉	#25603	direct (N × H)
361. C₄⋊Q₈	D₄×C₈₉	semidirect
362. C₄⋊Q₈	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×#533×C₈₉	2
C₂×#497×C₈₉	2
C₂×#489×C₈₉	2
C₈₉×#2850	2
C₈₉×#2001	2
C₈₉×#2000	2
C₈₉×#1993	2
C₈₉×#1960	2
C₈₉×#1880	2
C₈₉×#1878	2
C₈₉×#1749	2
#25603	89

Subgroups — C89×#25603 #36403451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25603 #36403451