Subgroups — C₈₉×#60498 #36438346

Order 22784.

2006 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×2₋¹⁺⁴×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3083	C₂	1	6	yes	maximal, complemented
4	C₈₉×#3082	C₂	1	6	yes	maximal, complemented
5	C₈₉×#3079	C₂	1	3	yes	maximal, complemented
6	C₈₉×#3078	C₂	1	3	yes	maximal, complemented
7	C₈₉×#3076	C₂	1	3	yes	maximal, complemented
8	C₂×#477×C₈₉	C₂	1	3	yes	maximal, complemented
9	C₈₉×#2913	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2898	C₂	1	2	yes	maximal
12	C₂×2₋¹⁺⁴×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#131×C₈₉	C₂²	1	2	yes	—
14	Q₈²×C₈₉	—	2	1	no	—
15	#557×C₈₉	—	2	1	no	—
16	#554×C₈₉	—	2	3	no	—
17	D₄×Q₈×C₈₉	—	2	3	no	—
18	#544×C₈₉	C₂²	1	6	yes	—
19	#533×C₈₉	C₂²	1	3	yes	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	3	yes	—
21	#520×C₈₉	—	2	3	no	—
22	#520×C₈₉	C₂²	1	3	yes	—
23	#520×C₈₉	C₂²	1	2	yes	—
24	#520×C₈₉	C₂²	1	1	yes	characteristic
25	#519×C₈₉	—	2	1	no	—
26	#519×C₈₉	C₂²	1	3	yes	—
27	C₄×#41×C₈₉	—	2	3	no	—
28	C₄×#41×C₈₉	C₂²	1	3	yes	—
29	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
31	C₂×C₄×Q₈×C₈₉	C₂²	1	3	yes	—
32	#479×C₈₉	C₂²	1	6	yes	—
33	#479×C₈₉	C₂²	1	12	yes	—
34	#478×C₈₉	C₂²	1	6	yes	—
35	#477×C₈₉	C₂²	1	3	yes	—
36	#477×C₈₉	C₂²	1	6	yes	—
37	#477×C₈₉	C₂²	1	12	yes	—
38	#476×C₈₉	C₂²	1	3	yes	—
39	#475×C₈₉	C₂²	1	3	yes	—
40	#475×C₈₉	C₂²	1	6	yes	—
41	#474×C₈₉	C₂²	1	1	yes	characteristic
42	#474×C₈₉	C₂²	1	6	yes	—
43	#428×C₈₉	C₂²	1	6	yes	—
44	#428×C₈₉	C₂²	1	6	yes	—
45	#427×C₈₉	C₂²	1	6	yes	—
46	C₂×#107×C₈₉	C₂²	1	6	yes	—
47	#424×C₈₉	C₂²	1	6	yes	—
48	#423×C₈₉	C₂²	1	3	yes	—
49	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
50	#419×C₈₉	C₂²	1	6	yes	—
51	#417×C₈₉	C₂²	1	6	yes	—
52	#416×C₈₉	C₂²	1	6	yes	—
53	C₂×#103×C₈₉	C₂²	1	6	yes	—
54	2₋¹⁺⁴×C₈₉	—	2	4	no	—
55	2₋¹⁺⁴×C₈₉	Q₈	1	8	yes	complemented
56	C₂×#41×C₈₉	—	2	3	no	—
57	C₂×#41×C₈₉	C₂³	1	3	yes	—
58	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₈₉	—	2	1	no	—
60	C₂²×Q₈×C₈₉	—	2	3	no	—
61	C₂²×Q₈×C₈₉	C₂³	1	3	yes	—
62	#131×C₈₉	—	2	8	no	—
63	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
64	C₂²×C₈×C₈₉	C₂³	1	6	yes	—
65	C₄⋊Q₈×C₈₉	—	2	6	no	—
66	C₄⋊Q₈×C₈₉	—	2	3	no	—
67	C₄⋊Q₈×C₈₉	—	2	3	no	—
68	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₈₉	C₂³	1	6	yes	—
70	C₄⋊Q₈×C₈₉	C₂³	1	3	yes	—
71	#125×C₈₉	—	2	6	no	—
72	#125×C₈₉	C₂³	1	6	yes	—
73	C₂²⋊Q₈×C₈₉	—	2	3	no	—
74	C₂²⋊Q₈×C₈₉	—	2	3	no	—
75	C₂²⋊Q₈×C₈₉	—	2	12	no	—
76	C₄×Q₈×C₈₉	—	2	1	no	—
77	C₄×Q₈×C₈₉	—	2	2	no	—
78	C₄×Q₈×C₈₉	—	2	3	no	—
79	C₄×Q₈×C₈₉	—	2	3	no	—
80	C₄×Q₈×C₈₉	—	2	3	no	—
81	C₄×Q₈×C₈₉	—	2	3	no	—
82	C₄×Q₈×C₈₉	—	2	3	no	—
83	C₄×Q₈×C₈₉	—	2	3	no	—
84	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
86	C₄×Q₈×C₈₉	C₂³	1	3	yes	—
87	C₄×Q₈×C₈₉	C₂³	1	3	yes	—
88	C₄×Q₈×C₈₉	C₂³	1	6	yes	—
89	C₄×D₄×C₈₉	—	2	3	no	—
90	C₄×D₄×C₈₉	—	2	6	no	—
91	C₄×D₄×C₈₉	—	2	3	no	—
92	C₄×D₄×C₈₉	—	2	3	no	—
93	C₄×D₄×C₈₉	—	2	6	no	—
94	C₄×D₄×C₈₉	D₄	1	3	yes	—
95	C₄×D₄×C₈₉	D₄	1	3	yes	—
96	C₄×D₄×C₈₉	C₂³	1	3	yes	—
97	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	#117×C₈₉	—	2	3	no	—
99	#117×C₈₉	—	2	6	no	—
100	#117×C₈₉	—	2	3	no	—
101	#117×C₈₉	C₂³	1	6	yes	—
102	#117×C₈₉	C₂³	1	3	yes	—
103	#117×C₈₉	C₂³	1	3	yes	—
104	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
105	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
106	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
107	C₂×C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
109	C₂×C₄²×C₈₉	—	2	3	no	—
110	C₂×C₄²×C₈₉	—	2	3	no	—
111	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
112	#107×C₈₉	C₂³	1	6	yes	—
113	#107×C₈₉	C₂³	1	12	yes	—
114	#107×C₈₉	C₂³	1	2	yes	—
115	#106×C₈₉	C₂³	1	6	yes	—
116	#106×C₈₉	C₂³	1	6	yes	—
117	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
119	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
120	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
121	#103×C₈₉	C₂³	1	6	yes	—
122	#103×C₈₉	C₂³	1	12	yes	—
123	#103×C₈₉	C₂³	1	6	yes	—
124	#102×C₈₉	C₂³	1	2	yes	—
125	#102×C₈₉	C₂³	1	6	yes	—
126	#41×C₈₉	—	2	24	no	—
127	#41×C₈₉	—	2	6	no	—
128	#41×C₈₉	C₂×Q₈	1	4	yes	—
129	#41×C₈₉	C₂×Q₈	1	12	yes	complemented
130	#41×C₈₉	#41	1	4	yes	—
131	C₂×Q₈×C₈₉	—	2	8	no	—
132	C₂×Q₈×C₈₉	—	2	3	no	—
133	C₂×Q₈×C₈₉	—	2	1	no	—
134	C₂×Q₈×C₈₉	—	2	3	no	—
135	C₂×Q₈×C₈₉	—	2	6	no	—
136	C₂×Q₈×C₈₉	—	2	6	no	—
137	C₂×Q₈×C₈₉	—	8	6	no	—
138	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₈₉	C₂×Q₈	1	12	yes	complemented
140	C₂×Q₈×C₈₉	C₂⁴	1	3	yes	—
141	C₂×D₄×C₈₉	—	2	3	no	—
142	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
143	C₂×D₄×C₈₉	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₈₉	—	2	3	no	—
145	C₂²×C₄×C₈₉	—	2	3	no	—
146	C₂²×C₄×C₈₉	—	2	3	no	—
147	C₂²×C₄×C₈₉	—	2	6	no	—
148	C₂²×C₄×C₈₉	—	2	1	no	—
149	C₂²×C₄×C₈₉	—	2	3	no	—
150	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
151	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
152	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
153	M₁₆×C₈₉	—	2	12	no	—
154	C₂×C₈×C₈₉	—	2	12	no	—
155	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
156	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
157	C₄⋊C₄×C₈₉	—	2	6	no	—
158	C₄⋊C₄×C₈₉	—	2	3	no	—
159	C₄⋊C₄×C₈₉	—	2	3	no	—
160	C₄⋊C₄×C₈₉	—	2	3	no	—
161	C₄⋊C₄×C₈₉	—	2	6	no	—
162	C₄⋊C₄×C₈₉	—	2	6	no	—
163	C₄⋊C₄×C₈₉	—	2	3	no	—
164	C₄⋊C₄×C₈₉	—	2	1	no	—
165	C₄⋊C₄×C₈₉	—	2	6	no	—
166	C₄⋊C₄×C₈₉	—	2	6	no	—
167	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
170	C₄⋊C₄×C₈₉	C₂⁴	1	6	yes	—
171	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
173	C₂²⋊C₄×C₈₉	—	2	3	no	—
174	C₂²⋊C₄×C₈₉	—	2	6	no	—
175	C₂²⋊C₄×C₈₉	—	2	1	no	—
176	C₂²⋊C₄×C₈₉	—	2	3	no	—
177	C₂²⋊C₄×C₈₉	C₂×D₄	1	6	yes	—
178	C₄²×C₈₉	—	2	3	no	—
179	C₄²×C₈₉	—	2	3	no	—
180	C₄²×C₈₉	—	2	6	no	—
181	C₄²×C₈₉	—	2	3	no	—
182	C₄²×C₈₉	—	2	6	no	—
183	C₄²×C₈₉	C₂×D₄	1	3	yes	—
184	C₄²×C₈₉	C₂×D₄	1	3	yes	—
185	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
186	C₄²×C₈₉	C₂⁴	1	3	yes	—
187	C₂³×C₈₉	—	2	1	no	—
188	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
189	Q₈×C₈₉	—	2	12	no	—
190	Q₈×C₈₉	—	2	3	no	—
191	Q₈×C₈₉	—	4	6	no	—
192	Q₈×C₈₉	—	4	2	no	—
193	Q₈×C₈₉	—	8	6	no	—
194	Q₈×C₈₉	C₂²⋊Q₈	1	4	yes	—
195	Q₈×C₈₉	C₂²×Q₈	1	6	yes	—
196	D₄×C₈₉	—	2	12	no	—
197	D₄×C₈₉	—	2	1	no	—
198	D₄×C₈₉	C₂²⋊Q₈	1	12	yes	—
199	D₄×C₈₉	C₂²×Q₈	1	2	yes	—
200	C₂×C₄×C₈₉	—	2	12	no	—
201	C₂×C₄×C₈₉	—	2	3	no	—
202	C₂×C₄×C₈₉	—	2	12	no	—
203	C₂×C₄×C₈₉	—	2	6	no	—
204	C₂×C₄×C₈₉	—	2	1	no	—
205	C₂×C₄×C₈₉	—	2	3	no	—
206	C₂×C₄×C₈₉	—	2	2	no	—
207	C₂×C₄×C₈₉	—	2	6	no	—
208	C₂×C₄×C₈₉	—	4	3	no	—
209	C₂×C₄×C₈₉	—	4	2	no	—
210	C₂×C₄×C₈₉	—	8	3	no	—
211	C₂×C₄×C₈₉	—	8	3	no	—
212	C₂×C₄×C₈₉	—	8	6	no	—
213	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
214	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
217	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₈₉	C₂²×Q₈	1	6	yes	—
219	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
220	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
221	C₈×C₈₉	—	2	2	no	—
222	C₈×C₈₉	—	2	6	no	—
223	C₂²×C₈₉	—	2	3	no	—
224	C₂²×C₈₉	—	2	3	no	—
225	C₂²×C₈₉	—	2	2	no	—
226	C₂²×C₈₉	—	4	1	no	—
227	C₂²×C₈₉	—	4	1	no	—
228	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	6	yes	—
229	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
230	C₄×C₈₉	—	2	1	no	—
231	C₄×C₈₉	—	2	3	no	—
232	C₄×C₈₉	—	2	6	no	—
233	C₄×C₈₉	—	2	3	no	—
234	C₄×C₈₉	—	4	3	no	—
235	C₄×C₈₉	—	4	6	no	—
236	C₄×C₈₉	—	4	1	no	—
237	C₄×C₈₉	—	4	2	no	—
238	C₄×C₈₉	C₂×C₂²⋊Q₈	1	6	yes	—
239	C₄×C₈₉	#578	1	2	yes	—
240	C₄×C₈₉	C₂³×Q₈	1	1	yes	characteristic
241	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
242	#60498	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₈₉	—	2	6	no	—
244	C₂×C₈₉	—	4	2	no	—
245	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₈₉	C₂×#578	1	1	yes	characteristic
247	C₂×C₈₉	#3585	1	1	yes	characteristic
248	#3503	C₂×C₈₉	1	1	yes	characteristic, complemented
249	C₄×2₋¹⁺⁴	C₂×C₈₉	1	1	yes	characteristic
250	#3083	C₂×C₈₉	1	6	yes	complemented
251	#3082	C₂×C₈₉	1	6	yes	complemented
252	#3079	C₂×C₈₉	1	3	yes	complemented
253	#3078	C₂×C₈₉	1	3	yes	complemented
254	#3076	C₂×C₈₉	1	3	yes	complemented
255	C₂×#477	C₂×C₈₉	1	3	yes	complemented
256	#2913	C₂×C₈₉	1	2	yes	complemented
257	#2906	C₂×C₈₉	1	1	yes	characteristic, complemented
258	#2898	C₂×C₈₉	1	2	yes	—
259	C₈₉	#60498	1	1	yes	characteristic, direct product factor, Sylow
260	C₂×2₋¹⁺⁴	C₂²×C₈₉	1	1	yes	characteristic
261	C₂×#131	C₂²×C₈₉	1	2	yes	—
262	Q₈²	—	2	1	no	—
263	#557	—	2	1	no	—
264	#554	—	2	3	no	—
265	D₄×Q₈	—	2	3	no	—
266	#544	C₂²×C₈₉	1	6	yes	—
267	#533	C₂²×C₈₉	1	3	yes	—
268	C₂×C₄⋊Q₈	C₂²×C₈₉	1	3	yes	—
269	#520	—	2	3	no	—
270	#520	C₂²×C₈₉	1	3	yes	—
271	#520	C₂²×C₈₉	1	2	yes	—
272	#520	C₂²×C₈₉	1	1	yes	characteristic
273	#519	—	2	1	no	—
274	#519	C₂²×C₈₉	1	3	yes	—
275	C₄×#41	—	2	3	no	—
276	C₄×#41	C₂²×C₈₉	1	3	yes	—
277	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
278	C₂×C₄×Q₈	—	2	1	no	—
279	C₂×C₄×Q₈	C₂²×C₈₉	1	3	yes	—
280	#479	C₂²×C₈₉	1	6	yes	—
281	#479	C₂²×C₈₉	1	12	yes	—
282	#478	C₂²×C₈₉	1	6	yes	—
283	#477	C₂²×C₈₉	1	3	yes	—
284	#477	C₂²×C₈₉	1	6	yes	—
285	#477	C₂²×C₈₉	1	12	yes	—
286	#476	C₂²×C₈₉	1	3	yes	—
287	#475	C₂²×C₈₉	1	3	yes	—
288	#475	C₂²×C₈₉	1	6	yes	—
289	#474	C₂²×C₈₉	1	1	yes	characteristic
290	#474	C₂²×C₈₉	1	6	yes	—
291	#428	C₂²×C₈₉	1	6	yes	—
292	#428	C₂²×C₈₉	1	6	yes	—
293	#427	C₂²×C₈₉	1	6	yes	—
294	C₂×#107	C₂²×C₈₉	1	6	yes	—
295	#424	C₂²×C₈₉	1	6	yes	—
296	#423	C₂²×C₈₉	1	3	yes	—
297	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
298	#419	C₂²×C₈₉	1	6	yes	—
299	#417	C₂²×C₈₉	1	6	yes	—
300	#416	C₂²×C₈₉	1	6	yes	—
301	C₂×#103	C₂²×C₈₉	1	6	yes	—
302	2₋¹⁺⁴	—	2	4	no	—
303	2₋¹⁺⁴	Q₈×C₈₉	1	8	yes	complemented
304	C₂×#41	—	2	3	no	—
305	C₂×#41	C₂³×C₈₉	1	3	yes	—
306	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
307	C₂²×Q₈	—	2	1	no	—
308	C₂²×Q₈	—	2	3	no	—
309	C₂²×Q₈	C₂³×C₈₉	1	3	yes	—
310	#131	—	2	8	no	—
311	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
312	C₂²×C₈	C₂³×C₈₉	1	6	yes	—
313	C₄⋊Q₈	—	2	6	no	—
314	C₄⋊Q₈	—	2	3	no	—
315	C₄⋊Q₈	—	2	3	no	—
316	C₄⋊Q₈	C₂³×C₈₉	1	3	yes	—
317	C₄⋊Q₈	C₂³×C₈₉	1	6	yes	—
318	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
319	#125	—	2	6	no	—
320	#125	C₂³×C₈₉	1	6	yes	—
321	C₂²⋊Q₈	—	2	3	no	—
322	C₂²⋊Q₈	—	2	3	no	—
323	C₂²⋊Q₈	—	2	12	no	—
324	C₄×Q₈	—	2	1	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	3	no	—
327	C₄×Q₈	—	2	3	no	—
328	C₄×Q₈	—	2	3	no	—
329	C₄×Q₈	—	2	3	no	—
330	C₄×Q₈	—	2	3	no	—
331	C₄×Q₈	—	2	3	no	—
332	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
333	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
334	C₄×Q₈	C₂³×C₈₉	1	3	yes	—
335	C₄×Q₈	C₂³×C₈₉	1	6	yes	—
336	C₄×Q₈	C₂³×C₈₉	1	3	yes	—
337	C₄×D₄	—	2	3	no	—
338	C₄×D₄	—	2	6	no	—
339	C₄×D₄	—	2	3	no	—
340	C₄×D₄	—	2	3	no	—
341	C₄×D₄	—	2	6	no	—
342	C₄×D₄	D₄×C₈₉	1	3	yes	—
343	C₄×D₄	D₄×C₈₉	1	3	yes	—
344	C₄×D₄	C₂³×C₈₉	1	1	yes	characteristic
345	C₄×D₄	C₂³×C₈₉	1	3	yes	—
346	#117	—	2	3	no	—
347	#117	—	2	6	no	—
348	#117	—	2	3	no	—
349	#117	C₂³×C₈₉	1	3	yes	—
350	#117	C₂³×C₈₉	1	6	yes	—
351	#117	C₂³×C₈₉	1	3	yes	—
352	C₂×C₄⋊C₄	—	2	3	no	—
353	C₂×C₄⋊C₄	—	2	3	no	—
354	C₂×C₄⋊C₄	—	2	3	no	—
355	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
356	C₂×C₄⋊C₄	C₂³×C₈₉	1	6	yes	—
357	C₂×C₄²	—	2	3	no	—
358	C₂×C₄²	—	2	3	no	—
359	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
360	#107	C₂³×C₈₉	1	6	yes	—
361	#107	C₂³×C₈₉	1	12	yes	—
362	#107	C₂³×C₈₉	1	2	yes	—
363	#106	C₂³×C₈₉	1	6	yes	—
364	#106	C₂³×C₈₉	1	6	yes	—
365	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
366	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
367	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
368	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
369	#103	C₂³×C₈₉	1	6	yes	—
370	#103	C₂³×C₈₉	1	12	yes	—
371	#103	C₂³×C₈₉	1	6	yes	—
372	#102	C₂³×C₈₉	1	2	yes	—
373	#102	C₂³×C₈₉	1	6	yes	—
374	#41	—	2	24	no	—
375	#41	—	2	6	no	—
376	#41	C₂×Q₈×C₈₉	1	4	yes	—
377	#41	C₂×Q₈×C₈₉	1	12	yes	complemented
378	#41	#41×C₈₉	1	4	yes	—
379	C₂×Q₈	—	2	8	no	—
380	C₂×Q₈	—	2	3	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	3	no	—
383	C₂×Q₈	—	2	6	no	—
384	C₂×Q₈	—	2	6	no	—
385	C₂×Q₈	—	8	6	no	—
386	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
387	C₂×Q₈	C₂×Q₈×C₈₉	1	12	yes	complemented
388	C₂×Q₈	C₂⁴×C₈₉	1	3	yes	—
389	C₂×D₄	—	2	3	no	—
390	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
391	C₂×D₄	C₂⁴×C₈₉	1	1	yes	characteristic
392	C₂²×C₄	—	2	3	no	—
393	C₂²×C₄	—	2	3	no	—
394	C₂²×C₄	—	2	3	no	—
395	C₂²×C₄	—	2	6	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	3	no	—
398	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
399	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
400	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
401	M₁₆	—	2	12	no	—
402	C₂×C₈	—	2	12	no	—
403	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
404	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
405	C₄⋊C₄	—	2	6	no	—
406	C₄⋊C₄	—	2	3	no	—
407	C₄⋊C₄	—	2	3	no	—
408	C₄⋊C₄	—	2	3	no	—
409	C₄⋊C₄	—	2	6	no	—
410	C₄⋊C₄	—	2	6	no	—
411	C₄⋊C₄	—	2	3	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	6	no	—
414	C₄⋊C₄	—	2	6	no	—
415	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
416	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
417	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
418	C₄⋊C₄	C₂⁴×C₈₉	1	6	yes	—
419	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
420	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
421	C₂²⋊C₄	—	2	3	no	—
422	C₂²⋊C₄	—	2	6	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₂²⋊C₄	—	2	3	no	—
425	C₂²⋊C₄	C₂×D₄×C₈₉	1	6	yes	—
426	C₄²	—	2	3	no	—
427	C₄²	—	2	3	no	—
428	C₄²	—	2	6	no	—
429	C₄²	—	2	3	no	—
430	C₄²	—	2	6	no	—
431	C₄²	C₂×D₄×C₈₉	1	3	yes	—
432	C₄²	C₂×D₄×C₈₉	1	3	yes	—
433	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
434	C₄²	C₂⁴×C₈₉	1	3	yes	—
435	C₂³	—	2	1	no	—
436	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
437	Q₈	—	2	12	no	—
438	Q₈	—	2	3	no	—
439	Q₈	—	4	6	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	8	6	no	—
442	Q₈	C₂²⋊Q₈×C₈₉	1	4	yes	—
443	Q₈	C₂²×Q₈×C₈₉	1	6	yes	—
444	D₄	—	2	12	no	—
445	D₄	—	2	1	no	—
446	D₄	C₂²⋊Q₈×C₈₉	1	12	yes	—
447	D₄	C₂²×Q₈×C₈₉	1	2	yes	—
448	C₂×C₄	—	2	12	no	—
449	C₂×C₄	—	2	3	no	—
450	C₂×C₄	—	2	12	no	—
451	C₂×C₄	—	2	6	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	3	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	6	no	—
456	C₂×C₄	—	4	3	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	8	3	no	—
459	C₂×C₄	—	8	3	no	—
460	C₂×C₄	—	8	6	no	—
461	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
462	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
463	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
465	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
466	C₂×C₄	C₂²×Q₈×C₈₉	1	6	yes	—
467	C₂×C₄	C₂×#41×C₈₉	1	6	yes	—
468	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	2	6	no	—
471	C₂²	—	2	3	no	—
472	C₂²	—	2	3	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	6	yes	—
477	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
478	C₄	—	2	3	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	3	no	—
481	C₄	—	2	6	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	3	no	—
485	C₄	—	4	6	no	—
486	C₄	C₂×C₂²⋊Q₈×C₈₉	1	6	yes	—
487	C₄	#578×C₈₉	1	2	yes	—
488	C₄	C₂³×Q₈×C₈₉	1	1	yes	characteristic
489	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
490	C₂	—	2	6	no	—
491	C₂	—	4	2	no	—
492	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
493	C₂	C₂×#578×C₈₉	1	1	yes	characteristic
494	C₂	C₈₉×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂×2₋¹⁺⁴×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3503	C₂	semidirect
3. C₈₉×#3083	C₂	semidirect
4. C₈₉×#3082	C₂	semidirect
5. C₈₉×#3079	C₂	semidirect
6. C₈₉×#3078	C₂	semidirect
7. C₈₉×#3076	C₂	semidirect
8. C₂×#477×C₈₉	C₂	semidirect
9. C₈₉×#2913	C₂	semidirect
10. C₈₉×#2906	C₂	semidirect
55. 2₋¹⁺⁴×C₈₉	Q₈	semidirect
129. #41×C₈₉	C₂×Q₈	semidirect
139. C₂×Q₈×C₈₉	C₂×Q₈	semidirect
242. #60498	C₈₉	direct (N × H)
248. #3503	C₂×C₈₉	semidirect
250. #3083	C₂×C₈₉	semidirect
251. #3082	C₂×C₈₉	semidirect
252. #3079	C₂×C₈₉	semidirect
253. #3078	C₂×C₈₉	semidirect
254. #3076	C₂×C₈₉	semidirect
255. C₂×#477	C₂×C₈₉	semidirect
256. #2913	C₂×C₈₉	semidirect
257. #2906	C₂×C₈₉	semidirect
259. C₈₉	#60498	direct (N × H)
303. 2₋¹⁺⁴	Q₈×C₈₉	semidirect
377. #41	C₂×Q₈×C₈₉	semidirect
387. C₂×Q₈	C₂×Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3503	2
C₄×2₋¹⁺⁴×C₈₉	2
C₈₉×#3083	2
C₈₉×#3082	2
C₈₉×#3079	2
C₈₉×#3078	2
C₈₉×#3076	2
C₂×#477×C₈₉	2
C₈₉×#2913	2
C₈₉×#2906	2
C₈₉×#2898	2
#60498	89

Subgroups — C89×#60498 #36438346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60498 #36438346