Subgroups — C₈₉×#60524 #36438372

Order 22784.

2116 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3489	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3099	C₂	1	3	yes	maximal, complemented
4	C₈₉×#3098	C₂	1	3	yes	maximal, complemented
5	C₈₉×#3097	C₂	1	3	yes	maximal, complemented
6	C₈₉×#3097	C₂	1	3	yes	maximal, complemented
7	C₈₉×#3095	C₂	1	3	yes	maximal, complemented
8	C₈₉×#3093	C₂	1	3	yes	maximal, complemented
9	C₈₉×#3091	C₂	1	3	yes	maximal, complemented
10	C₂×#485×C₈₉	C₂	1	3	yes	maximal, complemented
11	C₈₉×#2913	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#2912	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#2898	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#2898	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈₉×#2880	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×2₋¹⁺⁴×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
20	D₄×Q₈×C₈₉	—	2	1	no	—
21	#548×C₈₉	—	2	1	no	—
22	#547×C₈₉	—	2	3	no	—
23	#546×C₈₉	—	2	3	no	—
24	#537×C₈₉	C₂²	1	3	yes	—
25	#536×C₈₉	C₂²	1	3	yes	—
26	#525×C₈₉	C₂²	1	3	yes	—
27	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	3	yes	—
28	#520×C₈₉	C₂²	1	3	yes	—
29	#520×C₈₉	C₂²	1	3	yes	—
30	#520×C₈₉	C₂²	1	1	yes	characteristic
31	#520×C₈₉	C₂²	1	1	yes	characteristic
32	#520×C₈₉	C₂²	1	1	yes	characteristic
33	#519×C₈₉	—	2	1	no	—
34	#518×C₈₉	—	2	3	no	—
35	C₄×#41×C₈₉	—	2	3	no	—
36	C₄×#41×C₈₉	—	2	1	no	—
37	C₂×#117×C₈₉	C₂²	1	3	yes	—
38	C₂×#117×C₈₉	C₂²	1	3	yes	—
39	#485×C₈₉	C₂²	1	3	yes	—
40	#485×C₈₉	C₂²	1	6	yes	—
41	#485×C₈₉	C₂²	1	3	yes	—
42	#485×C₈₉	C₂²	1	6	yes	complemented
43	#485×C₈₉	C₂²	1	6	yes	complemented
44	#485×C₈₉	C₂²	1	6	yes	complemented
45	#484×C₈₉	C₂²	1	3	yes	complemented
46	#484×C₈₉	C₂²	1	6	yes	complemented
47	#483×C₈₉	C₂²	1	3	yes	complemented
48	#483×C₈₉	C₂²	1	6	yes	complemented
49	#482×C₈₉	C₂²	1	1	yes	characteristic
50	#482×C₈₉	C₂²	1	3	yes	—
51	#482×C₈₉	C₂²	1	6	yes	complemented
52	#481×C₈₉	C₂²	1	3	yes	complemented
53	#480×C₈₉	C₂²	1	3	yes	complemented
54	#428×C₈₉	C₂²	1	3	yes	complemented
55	#428×C₈₉	C₂²	1	3	yes	complemented
56	#428×C₈₉	C₂²	1	3	yes	complemented
57	#428×C₈₉	C₂²	1	3	yes	complemented
58	#427×C₈₉	C₂²	1	3	yes	complemented
59	#427×C₈₉	C₂²	1	3	yes	complemented
60	C₂×#107×C₈₉	C₂²	1	3	yes	complemented
61	C₂×#106×C₈₉	C₂²	1	3	yes	complemented
62	#419×C₈₉	C₂²	1	3	yes	complemented
63	#419×C₈₉	C₂²	1	3	yes	complemented
64	#417×C₈₉	C₂²	1	3	yes	complemented
65	#417×C₈₉	C₂²	1	3	yes	complemented
66	#416×C₈₉	C₂²	1	3	yes	complemented
67	#416×C₈₉	C₂²	1	3	yes	complemented
68	C₂×#103×C₈₉	C₂²	1	3	yes	complemented
69	C₂×#103×C₈₉	C₂²	1	3	yes	complemented
70	#408×C₈₉	C₂²	1	3	yes	—
71	#408×C₈₉	C₂²	1	3	yes	—
72	#407×C₈₉	C₂²	1	3	yes	—
73	C₂×C₂²⋊C₈×C₈₉	C₂²	1	3	yes	—
74	2₋¹⁺⁴×C₈₉	—	2	4	no	—
75	2₋¹⁺⁴×C₈₉	—	2	4	no	—
76	C₂×#41×C₈₉	—	2	4	no	—
77	C₂×#41×C₈₉	—	2	4	no	—
78	C₂×#41×C₈₉	—	2	3	no	—
79	C₂×#41×C₈₉	D₄	1	3	yes	—
80	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
81	C₂×#41×C₈₉	C₂³	1	3	yes	—
82	C₂×#41×C₈₉	C₂³	1	3	yes	—
83	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₈₉	—	2	1	no	—
85	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
86	C₂²×Q₈×C₈₉	C₂³	1	3	yes	—
87	C₂²×D₄×C₈₉	D₄	1	3	yes	—
88	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
89	#131×C₈₉	—	2	4	no	—
90	#131×C₈₉	—	2	4	no	—
91	C₂×M₁₆×C₈₉	C₂³	1	3	yes	—
92	C₂×M₁₆×C₈₉	C₂³	1	3	yes	—
93	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
94	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
95	C₄⋊Q₈×C₈₉	—	2	3	no	—
96	#124×C₈₉	—	2	3	no	—
97	#123×C₈₉	—	2	6	no	—
98	#123×C₈₉	—	2	3	no	—
99	C₂²⋊Q₈×C₈₉	—	2	3	no	—
100	C₂²⋊Q₈×C₈₉	—	2	3	no	—
101	C₂²⋊Q₈×C₈₉	—	2	3	no	—
102	C₂²⋊Q₈×C₈₉	C₂³	1	6	yes	—
103	C₂²⋊Q₈×C₈₉	C₂³	1	3	yes	—
104	C₂²⋊Q₈×C₈₉	C₂³	1	3	yes	—
105	#121×C₈₉	—	2	6	no	—
106	#121×C₈₉	—	2	3	no	—
107	#121×C₈₉	C₂³	1	3	yes	—
108	#121×C₈₉	C₂³	1	1	yes	characteristic
109	#120×C₈₉	—	2	3	no	—
110	C₄×Q₈×C₈₉	—	2	3	no	—
111	C₄×Q₈×C₈₉	—	2	3	no	—
112	C₄×Q₈×C₈₉	—	2	1	no	—
113	C₄×Q₈×C₈₉	—	2	1	no	—
114	C₄×Q₈×C₈₉	—	2	1	no	—
115	C₄×D₄×C₈₉	—	2	6	no	—
116	C₄×D₄×C₈₉	—	2	6	no	—
117	C₄×D₄×C₈₉	—	2	3	no	—
118	C₄×D₄×C₈₉	—	2	3	no	—
119	C₄×D₄×C₈₉	—	2	3	no	—
120	C₄×D₄×C₈₉	—	2	3	no	—
121	C₄×D₄×C₈₉	—	2	3	no	—
122	C₄×D₄×C₈₉	—	2	3	no	—
123	C₄×D₄×C₈₉	—	2	3	no	—
124	#117×C₈₉	—	2	3	no	—
125	#117×C₈₉	—	2	3	no	—
126	#117×C₈₉	—	2	3	no	—
127	#117×C₈₉	—	2	3	no	—
128	#117×C₈₉	C₂³	1	3	yes	—
129	#117×C₈₉	C₂³	1	3	yes	—
130	#117×C₈₉	C₂³	1	3	yes	—
131	#117×C₈₉	C₂³	1	1	yes	characteristic
132	#117×C₈₉	C₂³	1	3	yes	—
133	#117×C₈₉	C₂³	1	6	yes	—
134	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
135	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
136	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
137	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
138	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
139	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
140	C₂×C₂²⋊C₄×C₈₉	—	2	3	no	—
141	C₂×C₂²⋊C₄×C₈₉	—	2	3	no	—
142	C₂×C₂²⋊C₄×C₈₉	—	2	3	no	—
143	C₂×C₄²×C₈₉	—	2	3	no	—
144	C₂×C₄²×C₈₉	—	2	3	no	—
145	#107×C₈₉	C₂³	1	3	yes	—
146	#107×C₈₉	C₂³	1	6	yes	—
147	#107×C₈₉	C₂³	1	1	yes	characteristic
148	#107×C₈₉	C₂³	1	3	yes	—
149	#107×C₈₉	C₂³	1	3	yes	—
150	#106×C₈₉	C₂³	1	3	yes	—
151	#106×C₈₉	C₂³	1	6	yes	—
152	#106×C₈₉	C₂³	1	1	yes	characteristic
153	#106×C₈₉	C₂³	1	3	yes	—
154	#106×C₈₉	C₂³	1	3	yes	—
155	#103×C₈₉	C₂³	1	3	yes	—
156	#103×C₈₉	C₂³	1	3	yes	—
157	#103×C₈₉	C₂³	1	6	yes	—
158	#103×C₈₉	C₂³	1	6	yes	—
159	#103×C₈₉	C₂³	1	3	yes	—
160	#103×C₈₉	C₂³	1	3	yes	—
161	#102×C₈₉	C₂³	1	1	yes	characteristic
162	#102×C₈₉	C₂³	1	1	yes	characteristic
163	#102×C₈₉	C₂³	1	3	yes	—
164	#102×C₈₉	C₂³	1	3	yes	—
165	C₂²⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
166	C₂²⋊C₈×C₈₉	C₂³	1	3	yes	—
167	C₂²⋊C₈×C₈₉	C₂³	1	3	yes	—
168	C₂²⋊C₈×C₈₉	C₂³	1	3	yes	—
169	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
170	C₂⁴×C₈₉	C₂×D₄	1	3	yes	—
171	#41×C₈₉	—	2	4	no	—
172	#41×C₈₉	—	2	12	no	—
173	#41×C₈₉	—	2	6	no	—
174	#41×C₈₉	—	2	6	no	—
175	#41×C₈₉	—	2	6	no	—
176	#41×C₈₉	—	2	6	no	—
177	#41×C₈₉	—	4	12	no	—
178	#41×C₈₉	#41	1	4	yes	—
179	#41×C₈₉	#41	1	4	yes	—
180	C₂×Q₈×C₈₉	—	2	4	no	—
181	C₂×Q₈×C₈₉	—	2	3	no	—
182	C₂×Q₈×C₈₉	—	2	6	no	—
183	C₂×Q₈×C₈₉	—	2	6	no	—
184	C₂×Q₈×C₈₉	—	4	4	no	—
185	C₂×Q₈×C₈₉	C₂×D₄	1	3	yes	—
186	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
187	C₂×Q₈×C₈₉	C₂⁴	1	3	yes	—
188	C₂×D₄×C₈₉	—	2	12	no	—
189	C₂×D₄×C₈₉	—	2	1	no	—
190	C₂×D₄×C₈₉	—	2	3	no	—
191	C₂×D₄×C₈₉	—	2	3	no	—
192	C₂×D₄×C₈₉	C₂×D₄	1	6	yes	—
193	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
194	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
195	C₂×D₄×C₈₉	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₈₉	—	2	6	no	—
197	C₂²×C₄×C₈₉	—	2	6	no	—
198	C₂²×C₄×C₈₉	—	2	3	no	—
199	C₂²×C₄×C₈₉	—	2	3	no	—
200	C₂²×C₄×C₈₉	—	2	3	no	—
201	C₂²×C₄×C₈₉	—	2	3	no	—
202	C₂²×C₄×C₈₉	—	2	3	no	—
203	C₂²×C₄×C₈₉	—	2	3	no	—
204	C₂²×C₄×C₈₉	—	2	3	no	—
205	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
206	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
207	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
208	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
209	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
211	M₁₆×C₈₉	—	2	6	no	—
212	M₁₆×C₈₉	—	2	6	no	—
213	C₂×C₈×C₈₉	—	2	6	no	—
214	C₂×C₈×C₈₉	—	2	6	no	—
215	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
216	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
217	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
218	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
219	C₄⋊C₄×C₈₉	—	2	3	no	—
220	C₄⋊C₄×C₈₉	—	2	6	no	—
221	C₄⋊C₄×C₈₉	—	2	3	no	—
222	C₄⋊C₄×C₈₉	—	2	3	no	—
223	C₄⋊C₄×C₈₉	—	2	3	no	—
224	C₄⋊C₄×C₈₉	—	2	3	no	—
225	C₄⋊C₄×C₈₉	—	2	3	no	—
226	C₄⋊C₄×C₈₉	—	2	3	no	—
227	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
228	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
229	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
230	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
231	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
232	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
233	C₂²⋊C₄×C₈₉	—	2	3	no	—
234	C₂²⋊C₄×C₈₉	—	2	1	no	—
235	C₂²⋊C₄×C₈₉	—	2	3	no	—
236	C₂²⋊C₄×C₈₉	—	2	6	no	—
237	C₂²⋊C₄×C₈₉	—	2	3	no	—
238	C₂²⋊C₄×C₈₉	—	2	3	no	—
239	C₂²⋊C₄×C₈₉	—	2	6	no	—
240	C₂²⋊C₄×C₈₉	—	2	3	no	—
241	C₂²⋊C₄×C₈₉	—	2	3	no	—
242	C₂²⋊C₄×C₈₉	—	2	1	no	—
243	C₂²⋊C₄×C₈₉	—	2	3	no	—
244	C₄²×C₈₉	—	2	3	no	—
245	C₄²×C₈₉	—	2	1	no	—
246	C₄²×C₈₉	—	2	6	no	—
247	C₄²×C₈₉	—	2	3	no	—
248	C₄²×C₈₉	—	2	3	no	—
249	C₄²×C₈₉	—	2	3	no	—
250	C₂³×C₈₉	—	2	6	no	—
251	C₂³×C₈₉	—	2	1	no	—
252	C₂³×C₈₉	—	4	3	no	—
253	C₂³×C₈₉	—	4	3	no	—
254	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
255	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
256	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
257	Q₈×C₈₉	—	2	6	no	—
258	Q₈×C₈₉	—	2	3	no	—
259	Q₈×C₈₉	—	2	3	no	—
260	Q₈×C₈₉	—	4	6	no	—
261	Q₈×C₈₉	#123	1	4	yes	—
262	D₄×C₈₉	—	2	12	no	—
263	D₄×C₈₉	—	2	6	no	—
264	D₄×C₈₉	—	2	1	no	—
265	D₄×C₈₉	—	2	1	no	—
266	D₄×C₈₉	—	4	2	no	—
267	D₄×C₈₉	—	4	6	no	—
268	D₄×C₈₉	—	8	3	no	—
269	D₄×C₈₉	#123	1	12	yes	—
270	C₂×C₄×C₈₉	—	2	6	no	—
271	C₂×C₄×C₈₉	—	2	6	no	—
272	C₂×C₄×C₈₉	—	2	6	no	—
273	C₂×C₄×C₈₉	—	2	1	no	—
274	C₂×C₄×C₈₉	—	2	1	no	—
275	C₂×C₄×C₈₉	—	2	3	no	—
276	C₂×C₄×C₈₉	—	2	3	no	—
277	C₂×C₄×C₈₉	—	2	3	no	—
278	C₂×C₄×C₈₉	—	2	1	no	—
279	C₂×C₄×C₈₉	—	2	3	no	—
280	C₂×C₄×C₈₉	—	2	1	no	—
281	C₂×C₄×C₈₉	—	2	1	no	—
282	C₂×C₄×C₈₉	—	2	3	no	—
283	C₂×C₄×C₈₉	—	2	3	no	—
284	C₂×C₄×C₈₉	—	4	6	no	—
285	C₂×C₄×C₈₉	—	4	6	no	—
286	C₂×C₄×C₈₉	—	8	3	no	—
287	C₂×C₄×C₈₉	—	8	3	no	—
288	C₂×C₄×C₈₉	—	8	3	no	—
289	C₂×C₄×C₈₉	—	8	3	no	—
290	C₂×C₄×C₈₉	—	8	3	no	—
291	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
293	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
294	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
295	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
296	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
297	C₈×C₈₉	—	2	1	no	—
298	C₈×C₈₉	—	2	1	no	—
299	C₈×C₈₉	—	2	3	no	—
300	C₈×C₈₉	—	2	3	no	—
301	C₂²×C₈₉	—	2	3	no	—
302	C₂²×C₈₉	—	2	3	no	—
303	C₂²×C₈₉	—	2	3	no	—
304	C₂²×C₈₉	—	2	1	no	—
305	C₂²×C₈₉	—	2	3	no	—
306	C₂²×C₈₉	—	2	1	no	—
307	C₂²×C₈₉	—	2	1	no	—
308	C₂²×C₈₉	—	2	3	no	—
309	C₂²×C₈₉	—	4	1	no	—
310	C₂²×C₈₉	—	4	1	no	—
311	C₂²×C₈₉	—	4	1	no	—
312	C₂²×C₈₉	—	8	6	no	—
313	C₂²×C₈₉	C₂×#123	1	6	yes	—
314	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
315	C₄×C₈₉	—	2	1	no	—
316	C₄×C₈₉	—	2	3	no	—
317	C₄×C₈₉	—	4	3	no	—
318	C₄×C₈₉	—	4	1	no	—
319	C₄×C₈₉	—	4	3	no	—
320	C₄×C₈₉	—	4	3	no	—
321	C₄×C₈₉	—	4	1	no	—
322	C₄×C₈₉	—	4	1	no	—
323	C₄×C₈₉	—	4	3	no	—
324	C₄×C₈₉	C₂×#123	1	6	yes	—
325	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
326	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
327	#60524	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
328	C₂×C₈₉	—	2	6	no	—
329	C₂×C₈₉	—	4	3	no	—
330	C₂×C₈₉	—	4	1	no	—
331	C₂×C₈₉	—	8	1	no	—
332	C₂×C₈₉	C₂²×#123	1	1	yes	characteristic
333	C₂×C₈₉	#3584	1	1	yes	characteristic
334	C₂×C₈₉	#3586	1	1	yes	characteristic
335	#3489	C₂×C₈₉	1	1	yes	characteristic
336	#3428	C₂×C₈₉	1	1	yes	characteristic, complemented
337	#3099	C₂×C₈₉	1	3	yes	complemented
338	#3098	C₂×C₈₉	1	3	yes	complemented
339	#3097	C₂×C₈₉	1	3	yes	complemented
340	#3097	C₂×C₈₉	1	3	yes	complemented
341	#3095	C₂×C₈₉	1	3	yes	complemented
342	#3093	C₂×C₈₉	1	3	yes	complemented
343	#3091	C₂×C₈₉	1	3	yes	complemented
344	C₂×#485	C₂×C₈₉	1	3	yes	complemented
345	#2913	C₂×C₈₉	1	1	yes	characteristic, complemented
346	#2912	C₂×C₈₉	1	1	yes	characteristic, complemented
347	#2898	C₂×C₈₉	1	1	yes	characteristic, complemented
348	#2898	C₂×C₈₉	1	1	yes	characteristic, complemented
349	#2880	C₂×C₈₉	1	1	yes	characteristic, complemented
350	C₈₉	#60524	1	1	yes	characteristic, direct product factor, Sylow
351	C₂×2₋¹⁺⁴	C₂²×C₈₉	1	1	yes	characteristic
352	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
353	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
354	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
355	D₄×Q₈	—	2	1	no	—
356	#548	—	2	1	no	—
357	#547	—	2	3	no	—
358	#546	—	2	3	no	—
359	#537	C₂²×C₈₉	1	3	yes	—
360	#536	C₂²×C₈₉	1	3	yes	—
361	#525	C₂²×C₈₉	1	3	yes	—
362	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	3	yes	—
363	#520	C₂²×C₈₉	1	3	yes	—
364	#520	C₂²×C₈₉	1	3	yes	—
365	#520	C₂²×C₈₉	1	1	yes	characteristic
366	#520	C₂²×C₈₉	1	1	yes	characteristic
367	#520	C₂²×C₈₉	1	1	yes	characteristic
368	#519	—	2	1	no	—
369	#518	—	2	3	no	—
370	C₄×#41	—	2	3	no	—
371	C₄×#41	—	2	1	no	—
372	C₂×#117	C₂²×C₈₉	1	3	yes	—
373	C₂×#117	C₂²×C₈₉	1	3	yes	—
374	#485	C₂²×C₈₉	1	3	yes	—
375	#485	C₂²×C₈₉	1	6	yes	—
376	#485	C₂²×C₈₉	1	3	yes	—
377	#485	C₂²×C₈₉	1	6	yes	complemented
378	#485	C₂²×C₈₉	1	6	yes	complemented
379	#485	C₂²×C₈₉	1	6	yes	complemented
380	#484	C₂²×C₈₉	1	3	yes	complemented
381	#484	C₂²×C₈₉	1	6	yes	complemented
382	#483	C₂²×C₈₉	1	3	yes	complemented
383	#483	C₂²×C₈₉	1	6	yes	complemented
384	#482	C₂²×C₈₉	1	1	yes	characteristic
385	#482	C₂²×C₈₉	1	3	yes	—
386	#482	C₂²×C₈₉	1	6	yes	complemented
387	#481	C₂²×C₈₉	1	3	yes	complemented
388	#480	C₂²×C₈₉	1	3	yes	complemented
389	#428	C₂²×C₈₉	1	3	yes	complemented
390	#428	C₂²×C₈₉	1	3	yes	complemented
391	#428	C₂²×C₈₉	1	3	yes	complemented
392	#428	C₂²×C₈₉	1	3	yes	complemented
393	#427	C₂²×C₈₉	1	3	yes	complemented
394	#427	C₂²×C₈₉	1	3	yes	complemented
395	C₂×#107	C₂²×C₈₉	1	3	yes	complemented
396	C₂×#106	C₂²×C₈₉	1	3	yes	complemented
397	#419	C₂²×C₈₉	1	3	yes	complemented
398	#419	C₂²×C₈₉	1	3	yes	complemented
399	#417	C₂²×C₈₉	1	3	yes	complemented
400	#417	C₂²×C₈₉	1	3	yes	complemented
401	#416	C₂²×C₈₉	1	3	yes	complemented
402	#416	C₂²×C₈₉	1	3	yes	complemented
403	C₂×#103	C₂²×C₈₉	1	3	yes	complemented
404	C₂×#103	C₂²×C₈₉	1	3	yes	complemented
405	#408	C₂²×C₈₉	1	3	yes	—
406	#408	C₂²×C₈₉	1	3	yes	—
407	#407	C₂²×C₈₉	1	3	yes	—
408	C₂×C₂²⋊C₈	C₂²×C₈₉	1	3	yes	—
409	2₋¹⁺⁴	—	2	4	no	—
410	2₋¹⁺⁴	—	2	4	no	—
411	C₂×#41	—	2	4	no	—
412	C₂×#41	—	2	4	no	—
413	C₂×#41	—	2	3	no	—
414	C₂×#41	D₄×C₈₉	1	3	yes	—
415	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
416	C₂×#41	C₂³×C₈₉	1	3	yes	—
417	C₂×#41	C₂³×C₈₉	1	3	yes	—
418	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
419	C₂²×Q₈	—	2	1	no	—
420	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
421	C₂²×Q₈	C₂³×C₈₉	1	3	yes	—
422	C₂²×D₄	D₄×C₈₉	1	3	yes	—
423	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
424	#131	—	2	4	no	—
425	#131	—	2	4	no	—
426	C₂×M₁₆	C₂³×C₈₉	1	3	yes	—
427	C₂×M₁₆	C₂³×C₈₉	1	3	yes	—
428	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
429	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
430	C₄⋊Q₈	—	2	3	no	—
431	#124	—	2	3	no	—
432	#123	—	2	6	no	—
433	#123	—	2	3	no	—
434	C₂²⋊Q₈	—	2	3	no	—
435	C₂²⋊Q₈	—	2	3	no	—
436	C₂²⋊Q₈	—	2	3	no	—
437	C₂²⋊Q₈	C₂³×C₈₉	1	3	yes	—
438	C₂²⋊Q₈	C₂³×C₈₉	1	6	yes	—
439	C₂²⋊Q₈	C₂³×C₈₉	1	3	yes	—
440	#121	—	2	6	no	—
441	#121	—	2	3	no	—
442	#121	C₂³×C₈₉	1	1	yes	characteristic
443	#121	C₂³×C₈₉	1	3	yes	—
444	#120	—	2	3	no	—
445	C₄×Q₈	—	2	3	no	—
446	C₄×Q₈	—	2	3	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×D₄	—	2	6	no	—
451	C₄×D₄	—	2	6	no	—
452	C₄×D₄	—	2	3	no	—
453	C₄×D₄	—	2	3	no	—
454	C₄×D₄	—	2	3	no	—
455	C₄×D₄	—	2	3	no	—
456	C₄×D₄	—	2	3	no	—
457	C₄×D₄	—	2	3	no	—
458	C₄×D₄	—	2	3	no	—
459	#117	—	2	3	no	—
460	#117	—	2	3	no	—
461	#117	—	2	3	no	—
462	#117	—	2	3	no	—
463	#117	C₂³×C₈₉	1	3	yes	—
464	#117	C₂³×C₈₉	1	1	yes	characteristic
465	#117	C₂³×C₈₉	1	3	yes	—
466	#117	C₂³×C₈₉	1	3	yes	—
467	#117	C₂³×C₈₉	1	6	yes	—
468	#117	C₂³×C₈₉	1	3	yes	—
469	C₂×C₄⋊C₄	—	2	3	no	—
470	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
471	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
472	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
473	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
474	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
475	C₂×C₂²⋊C₄	—	2	3	no	—
476	C₂×C₂²⋊C₄	—	2	3	no	—
477	C₂×C₂²⋊C₄	—	2	3	no	—
478	C₂×C₄²	—	2	3	no	—
479	C₂×C₄²	—	2	3	no	—
480	#107	C₂³×C₈₉	1	3	yes	—
481	#107	C₂³×C₈₉	1	6	yes	—
482	#107	C₂³×C₈₉	1	1	yes	characteristic
483	#107	C₂³×C₈₉	1	3	yes	—
484	#107	C₂³×C₈₉	1	3	yes	—
485	#106	C₂³×C₈₉	1	3	yes	—
486	#106	C₂³×C₈₉	1	6	yes	—
487	#106	C₂³×C₈₉	1	1	yes	characteristic
488	#106	C₂³×C₈₉	1	3	yes	—
489	#106	C₂³×C₈₉	1	3	yes	—
490	#103	C₂³×C₈₉	1	3	yes	—
491	#103	C₂³×C₈₉	1	3	yes	—
492	#103	C₂³×C₈₉	1	6	yes	—
493	#103	C₂³×C₈₉	1	6	yes	—
494	#103	C₂³×C₈₉	1	3	yes	—
495	#103	C₂³×C₈₉	1	3	yes	—
496	#102	C₂³×C₈₉	1	1	yes	characteristic
497	#102	C₂³×C₈₉	1	1	yes	characteristic
498	#102	C₂³×C₈₉	1	3	yes	—
499	#102	C₂³×C₈₉	1	3	yes	—
500	C₂²⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
501	C₂²⋊C₈	C₂³×C₈₉	1	3	yes	—
502	C₂²⋊C₈	C₂³×C₈₉	1	3	yes	—
503	C₂²⋊C₈	C₂³×C₈₉	1	3	yes	—
504	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
505	C₂⁴	C₂×D₄×C₈₉	1	3	yes	—
506	#41	—	2	4	no	—
507	#41	—	2	12	no	—
508	#41	—	2	6	no	—
509	#41	—	2	6	no	—
510	#41	—	2	6	no	—
511	#41	—	2	6	no	—
512	#41	—	4	12	no	—
513	#41	#41×C₈₉	1	4	yes	—
514	#41	#41×C₈₉	1	4	yes	—
515	C₂×Q₈	—	2	4	no	—
516	C₂×Q₈	—	2	3	no	—
517	C₂×Q₈	—	2	6	no	—
518	C₂×Q₈	—	2	6	no	—
519	C₂×Q₈	—	4	4	no	—
520	C₂×Q₈	C₂×D₄×C₈₉	1	3	yes	—
521	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
522	C₂×Q₈	C₂⁴×C₈₉	1	3	yes	—
523	C₂×D₄	—	2	12	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	3	no	—
526	C₂×D₄	—	2	3	no	—
527	C₂×D₄	C₂×D₄×C₈₉	1	6	yes	—
528	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
529	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
530	C₂×D₄	C₂⁴×C₈₉	1	1	yes	characteristic
531	C₂²×C₄	—	2	6	no	—
532	C₂²×C₄	—	2	6	no	—
533	C₂²×C₄	—	2	3	no	—
534	C₂²×C₄	—	2	3	no	—
535	C₂²×C₄	—	2	3	no	—
536	C₂²×C₄	—	2	3	no	—
537	C₂²×C₄	—	2	3	no	—
538	C₂²×C₄	—	2	3	no	—
539	C₂²×C₄	—	2	3	no	—
540	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
541	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
542	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
543	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
544	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
545	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
546	M₁₆	—	2	6	no	—
547	M₁₆	—	2	6	no	—
548	C₂×C₈	—	2	6	no	—
549	C₂×C₈	—	2	6	no	—
550	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
551	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
552	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
553	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
554	C₄⋊C₄	—	2	3	no	—
555	C₄⋊C₄	—	2	6	no	—
556	C₄⋊C₄	—	2	3	no	—
557	C₄⋊C₄	—	2	3	no	—
558	C₄⋊C₄	—	2	3	no	—
559	C₄⋊C₄	—	2	3	no	—
560	C₄⋊C₄	—	2	3	no	—
561	C₄⋊C₄	—	2	3	no	—
562	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
563	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
564	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
565	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
566	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
567	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
568	C₂²⋊C₄	—	2	3	no	—
569	C₂²⋊C₄	—	2	1	no	—
570	C₂²⋊C₄	—	2	3	no	—
571	C₂²⋊C₄	—	2	6	no	—
572	C₂²⋊C₄	—	2	3	no	—
573	C₂²⋊C₄	—	2	3	no	—
574	C₂²⋊C₄	—	2	6	no	—
575	C₂²⋊C₄	—	2	3	no	—
576	C₂²⋊C₄	—	2	3	no	—
577	C₂²⋊C₄	—	2	1	no	—
578	C₂²⋊C₄	—	2	3	no	—
579	C₄²	—	2	3	no	—
580	C₄²	—	2	1	no	—
581	C₄²	—	2	6	no	—
582	C₄²	—	2	3	no	—
583	C₄²	—	2	3	no	—
584	C₄²	—	2	3	no	—
585	C₂³	—	2	6	no	—
586	C₂³	—	2	1	no	—
587	C₂³	—	4	3	no	—
588	C₂³	—	4	3	no	—
589	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
590	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
591	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
592	Q₈	—	2	6	no	—
593	Q₈	—	2	3	no	—
594	Q₈	—	2	3	no	—
595	Q₈	—	4	6	no	—
596	Q₈	#123×C₈₉	1	4	yes	—
597	D₄	—	2	12	no	—
598	D₄	—	2	6	no	—
599	D₄	—	2	1	no	—
600	D₄	—	2	1	no	—
601	D₄	—	4	2	no	—
602	D₄	—	4	6	no	—
603	D₄	—	8	3	no	—
604	D₄	#123×C₈₉	1	12	yes	—
605	C₂×C₄	—	2	6	no	—
606	C₂×C₄	—	2	6	no	—
607	C₂×C₄	—	2	6	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	3	no	—
611	C₂×C₄	—	2	3	no	—
612	C₂×C₄	—	2	3	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	3	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	3	no	—
618	C₂×C₄	—	2	3	no	—
619	C₂×C₄	—	4	6	no	—
620	C₂×C₄	—	4	6	no	—
621	C₂×C₄	—	8	3	no	—
622	C₂×C₄	—	8	3	no	—
623	C₂×C₄	—	8	3	no	—
624	C₂×C₄	—	8	3	no	—
625	C₂×C₄	—	8	3	no	—
626	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
627	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
628	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
629	C₂×C₄	C₂×#41×C₈₉	1	6	yes	—
630	C₂×C₄	C₂×#41×C₈₉	1	6	yes	—
631	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
632	C₈	—	2	1	no	—
633	C₈	—	2	1	no	—
634	C₈	—	2	3	no	—
635	C₈	—	2	3	no	—
636	C₂²	—	2	3	no	—
637	C₂²	—	2	3	no	—
638	C₂²	—	2	3	no	—
639	C₂²	—	2	1	no	—
640	C₂²	—	2	3	no	—
641	C₂²	—	2	1	no	—
642	C₂²	—	2	1	no	—
643	C₂²	—	2	3	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	8	6	no	—
648	C₂²	C₂×#123×C₈₉	1	6	yes	—
649	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
650	C₄	—	2	1	no	—
651	C₄	—	2	3	no	—
652	C₄	—	4	3	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	3	no	—
655	C₄	—	4	3	no	—
656	C₄	—	4	1	no	—
657	C₄	—	4	1	no	—
658	C₄	—	4	3	no	—
659	C₄	C₂×#123×C₈₉	1	6	yes	—
660	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
661	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
662	C₂	—	2	6	no	—
663	C₂	—	4	3	no	—
664	C₂	—	4	1	no	—
665	C₂	—	8	1	no	—
666	C₂	C₂²×#123×C₈₉	1	1	yes	characteristic
667	C₂	C₈₉×#3584	1	1	yes	characteristic
668	C₂	C₈₉×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₈₉×#3489	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3428	C₂	semidirect
3. C₈₉×#3099	C₂	semidirect
4. C₈₉×#3098	C₂	semidirect
5. C₈₉×#3097	C₂	semidirect
6. C₈₉×#3097	C₂	semidirect
7. C₈₉×#3095	C₂	semidirect
8. C₈₉×#3093	C₂	semidirect
9. C₈₉×#3091	C₂	semidirect
10. C₂×#485×C₈₉	C₂	semidirect
11. C₈₉×#2913	C₂	semidirect
12. C₈₉×#2912	C₂	semidirect
13. C₈₉×#2898	C₂	semidirect
14. C₈₉×#2898	C₂	semidirect
15. C₈₉×#2880	C₂	semidirect
42. #485×C₈₉	C₂²	semidirect
43. #485×C₈₉	C₂²	semidirect
44. #485×C₈₉	C₂²	semidirect
45. #484×C₈₉	C₂²	semidirect
46. #484×C₈₉	C₂²	semidirect
47. #483×C₈₉	C₂²	semidirect
48. #483×C₈₉	C₂²	semidirect
51. #482×C₈₉	C₂²	semidirect
52. #481×C₈₉	C₂²	semidirect
53. #480×C₈₉	C₂²	semidirect
54. #428×C₈₉	C₂²	semidirect
55. #428×C₈₉	C₂²	semidirect
56. #428×C₈₉	C₂²	semidirect
57. #428×C₈₉	C₂²	semidirect
58. #427×C₈₉	C₂²	semidirect
59. #427×C₈₉	C₂²	semidirect
60. C₂×#107×C₈₉	C₂²	semidirect
61. C₂×#106×C₈₉	C₂²	semidirect
62. #419×C₈₉	C₂²	semidirect
63. #419×C₈₉	C₂²	semidirect
64. #417×C₈₉	C₂²	semidirect
65. #417×C₈₉	C₂²	semidirect
66. #416×C₈₉	C₂²	semidirect
67. #416×C₈₉	C₂²	semidirect
68. C₂×#103×C₈₉	C₂²	semidirect
69. C₂×#103×C₈₉	C₂²	semidirect
327. #60524	C₈₉	direct (N × H)
336. #3428	C₂×C₈₉	semidirect
337. #3099	C₂×C₈₉	semidirect
338. #3098	C₂×C₈₉	semidirect
339. #3097	C₂×C₈₉	semidirect
340. #3097	C₂×C₈₉	semidirect
341. #3095	C₂×C₈₉	semidirect
342. #3093	C₂×C₈₉	semidirect
343. #3091	C₂×C₈₉	semidirect
344. C₂×#485	C₂×C₈₉	semidirect
345. #2913	C₂×C₈₉	semidirect
346. #2912	C₂×C₈₉	semidirect
347. #2898	C₂×C₈₉	semidirect
348. #2898	C₂×C₈₉	semidirect
349. #2880	C₂×C₈₉	semidirect
350. C₈₉	#60524	direct (N × H)
377. #485	C₂²×C₈₉	semidirect
378. #485	C₂²×C₈₉	semidirect
379. #485	C₂²×C₈₉	semidirect
380. #484	C₂²×C₈₉	semidirect
381. #484	C₂²×C₈₉	semidirect
382. #483	C₂²×C₈₉	semidirect
383. #483	C₂²×C₈₉	semidirect
386. #482	C₂²×C₈₉	semidirect
387. #481	C₂²×C₈₉	semidirect
388. #480	C₂²×C₈₉	semidirect
389. #428	C₂²×C₈₉	semidirect
390. #428	C₂²×C₈₉	semidirect
391. #428	C₂²×C₈₉	semidirect
392. #428	C₂²×C₈₉	semidirect
393. #427	C₂²×C₈₉	semidirect
394. #427	C₂²×C₈₉	semidirect
395. C₂×#107	C₂²×C₈₉	semidirect
396. C₂×#106	C₂²×C₈₉	semidirect
397. #419	C₂²×C₈₉	semidirect
398. #419	C₂²×C₈₉	semidirect
399. #417	C₂²×C₈₉	semidirect
400. #417	C₂²×C₈₉	semidirect
401. #416	C₂²×C₈₉	semidirect
402. #416	C₂²×C₈₉	semidirect
403. C₂×#103	C₂²×C₈₉	semidirect
404. C₂×#103	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3489	2
C₈₉×#3428	2
C₈₉×#3099	2
C₈₉×#3098	2
C₈₉×#3097	2
C₈₉×#3095	2
C₈₉×#3093	2
C₈₉×#3091	2
C₂×#485×C₈₉	2
C₈₉×#2913	2
C₈₉×#2912	2
C₈₉×#2898	2
C₈₉×#2880	2
#60524	89

Subgroups — C89×#60524 #36438372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60524 #36438372