Subgroups — C₈₉×#23647 #36401495

Order 22784.

914 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2896	C₂	1	1	yes	maximal, characteristic
3	C₂×#417×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2879	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2079	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2078	C₂	1	4	yes	maximal, complemented
7	C₈₉×#1742	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#343×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1738	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
17	#428×C₈₉	—	2	2	no	—
18	#428×C₈₉	—	2	2	no	—
19	C₂×#107×C₈₉	C₂²	1	2	yes	complemented
20	C₂×#106×C₈₉	C₂²	1	2	yes	complemented
21	#419×C₈₉	—	2	1	no	—
22	#418×C₈₉	—	2	1	no	—
23	#417×C₈₉	—	2	2	no	—
24	#417×C₈₉	C₂²	1	2	yes	—
25	#417×C₈₉	C₂²	1	4	yes	—
26	#417×C₈₉	C₂²	1	2	yes	—
27	#416×C₈₉	—	2	2	no	—
28	C₂×#103×C₈₉	—	2	1	no	—
29	C₂×#102×C₈₉	—	2	1	no	—
30	#408×C₈₉	—	2	2	no	—
31	#408×C₈₉	C₂²	1	4	yes	—
32	#407×C₈₉	—	2	2	no	—
33	#343×C₈₉	C₂²	1	2	yes	complemented
34	#343×C₈₉	C₂²	1	4	yes	complemented
35	#343×C₈₉	C₂²	1	2	yes	complemented
36	#341×C₈₉	C₂²	1	4	yes	complemented
37	C₂×#41×C₈₉	—	2	1	no	—
38	C₂×#41×C₈₉	—	2	1	no	—
39	C₂×#41×C₈₉	—	2	2	no	—
40	C₂×#41×C₈₉	—	2	2	no	—
41	C₂×#41×C₈₉	—	2	2	no	—
42	C₂×#41×C₈₉	—	2	2	no	—
43	C₂×#41×C₈₉	D₄	1	2	yes	—
44	C₂×#41×C₈₉	C₂³	1	2	yes	—
45	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₈₉	—	2	1	no	—
47	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₈₉	—	2	1	no	—
49	C₂³×C₄×C₈₉	—	2	1	no	—
50	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₈₉	—	2	2	no	—
52	C₂×M₁₆×C₈₉	—	2	2	no	—
53	C₂×M₁₆×C₈₉	—	2	1	no	—
54	C₂×M₁₆×C₈₉	—	2	1	no	—
55	C₂×M₁₆×C₈₉	—	2	1	no	—
56	C₂×M₁₆×C₈₉	—	2	2	no	—
57	C₂×M₁₆×C₈₉	—	2	2	no	—
58	C₂×M₁₆×C₈₉	—	2	2	no	—
59	C₂×M₁₆×C₈₉	—	2	2	no	—
60	C₂×M₁₆×C₈₉	—	2	2	no	—
61	C₂×M₁₆×C₈₉	—	2	2	no	—
62	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
63	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
64	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
65	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
66	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
67	C₂²×C₈×C₈₉	—	2	1	no	—
68	C₂²×C₈×C₈₉	—	2	1	no	—
69	C₂²×C₈×C₈₉	—	2	1	no	—
70	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
71	#117×C₈₉	—	2	1	no	—
72	#117×C₈₉	—	2	1	no	—
73	#117×C₈₉	—	2	1	no	—
74	#117×C₈₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
84	C₂×C₄²×C₈₉	—	2	1	no	—
85	#107×C₈₉	—	2	4	no	—
86	#106×C₈₉	—	2	4	no	—
87	#103×C₈₉	—	2	4	no	—
88	#103×C₈₉	—	2	2	no	—
89	#103×C₈₉	—	2	2	no	—
90	#102×C₈₉	—	2	4	no	—
91	#102×C₈₉	—	2	2	no	—
92	#102×C₈₉	—	2	2	no	—
93	C₂²⋊C₈×C₈₉	—	2	8	no	—
94	C₂⁴×C₈₉	—	2	1	no	—
95	C₂⁴×C₈₉	#41	1	1	yes	characteristic
96	#41×C₈₉	—	2	4	no	—
97	#41×C₈₉	—	2	4	no	—
98	#41×C₈₉	—	2	2	no	—
99	#41×C₈₉	—	2	2	no	—
100	#41×C₈₉	—	2	2	no	—
101	#41×C₈₉	—	2	2	no	—
102	#41×C₈₉	—	4	2	no	—
103	#41×C₈₉	—	4	2	no	—
104	#41×C₈₉	—	4	2	no	—
105	#41×C₈₉	—	4	2	no	—
106	C₂×Q₈×C₈₉	—	2	1	no	—
107	C₂×Q₈×C₈₉	—	2	2	no	—
108	C₂×Q₈×C₈₉	—	2	2	no	—
109	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
110	C₂×D₄×C₈₉	—	2	2	no	—
111	C₂×D₄×C₈₉	—	2	2	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂×D₄×C₈₉	—	2	2	no	—
114	C₂×D₄×C₈₉	—	2	2	no	—
115	C₂×D₄×C₈₉	—	4	2	no	—
116	C₂×D₄×C₈₉	—	4	2	no	—
117	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₈₉	—	2	1	no	—
119	C₂²×C₄×C₈₉	—	2	2	no	—
120	C₂²×C₄×C₈₉	—	2	2	no	—
121	C₂²×C₄×C₈₉	—	2	1	no	—
122	C₂²×C₄×C₈₉	—	2	1	no	—
123	C₂²×C₄×C₈₉	—	2	2	no	—
124	C₂²×C₄×C₈₉	—	4	2	no	—
125	C₂²×C₄×C₈₉	—	4	2	no	—
126	C₂²×C₄×C₈₉	—	4	1	no	—
127	C₂²×C₄×C₈₉	—	4	1	no	—
128	C₂²×C₄×C₈₉	—	8	2	no	—
129	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
134	C₂²×C₄×C₈₉	#41	1	2	yes	—
135	C₂²×C₄×C₈₉	#41	1	2	yes	—
136	C₂²×C₄×C₈₉	#41	1	2	yes	—
137	C₂²×C₄×C₈₉	#41	1	2	yes	—
138	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
139	M₁₆×C₈₉	—	2	4	no	—
140	M₁₆×C₈₉	—	2	4	no	—
141	M₁₆×C₈₉	—	2	2	no	—
142	M₁₆×C₈₉	—	2	2	no	—
143	M₁₆×C₈₉	—	2	2	no	—
144	M₁₆×C₈₉	—	2	2	no	—
145	M₁₆×C₈₉	—	2	2	no	—
146	M₁₆×C₈₉	—	2	2	no	—
147	M₁₆×C₈₉	—	4	2	no	—
148	M₁₆×C₈₉	—	4	2	no	—
149	M₁₆×C₈₉	—	4	2	no	—
150	C₂×C₈×C₈₉	—	2	2	no	—
151	C₂×C₈×C₈₉	—	2	2	no	—
152	C₂×C₈×C₈₉	—	2	2	no	—
153	C₂×C₈×C₈₉	—	2	2	no	—
154	C₂×C₈×C₈₉	—	2	2	no	—
155	C₂×C₈×C₈₉	—	4	2	no	—
156	C₂×C₈×C₈₉	—	4	2	no	—
157	C₂×C₈×C₈₉	—	4	2	no	—
158	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
159	C₄⋊C₄×C₈₉	—	2	2	no	—
160	C₄⋊C₄×C₈₉	—	2	1	no	—
161	C₄⋊C₄×C₈₉	—	2	1	no	—
162	C₄⋊C₄×C₈₉	—	2	1	no	—
163	C₄⋊C₄×C₈₉	—	2	2	no	—
164	C₄⋊C₄×C₈₉	—	2	1	no	—
165	C₄⋊C₄×C₈₉	—	4	1	no	—
166	C₄⋊C₄×C₈₉	—	4	1	no	—
167	C₂²⋊C₄×C₈₉	—	4	1	no	—
168	C₂²⋊C₄×C₈₉	—	4	1	no	—
169	C₄²×C₈₉	—	4	1	no	—
170	C₄²×C₈₉	—	4	1	no	—
171	C₂³×C₈₉	—	2	2	no	—
172	C₂³×C₈₉	—	4	1	no	—
173	C₂³×C₈₉	—	4	1	no	—
174	C₂³×C₈₉	—	4	2	no	—
175	C₂³×C₈₉	—	8	1	no	—
176	C₂³×C₈₉	—	8	1	no	—
177	C₂³×C₈₉	#121	1	1	yes	characteristic
178	C₂³×C₈₉	#121	1	1	yes	characteristic
179	C₂³×C₈₉	#126	1	2	yes	—
180	C₂³×C₈₉	#126	1	2	yes	—
181	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
182	Q₈×C₈₉	—	2	2	no	—
183	Q₈×C₈₉	—	2	2	no	—
184	Q₈×C₈₉	—	4	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	4	no	—
190	D₄×C₈₉	—	4	2	no	—
191	C₂×C₄×C₈₉	—	2	2	no	—
192	C₂×C₄×C₈₉	—	4	2	no	—
193	C₂×C₄×C₈₉	—	4	2	no	—
194	C₂×C₄×C₈₉	—	4	2	no	—
195	C₂×C₄×C₈₉	—	4	4	no	—
196	C₂×C₄×C₈₉	—	4	1	no	—
197	C₂×C₄×C₈₉	—	4	1	no	—
198	C₂×C₄×C₈₉	—	4	2	no	—
199	C₂×C₄×C₈₉	—	8	2	no	—
200	C₂×C₄×C₈₉	—	8	2	no	—
201	C₂×C₄×C₈₉	—	8	1	no	—
202	C₂×C₄×C₈₉	—	8	1	no	—
203	C₂×C₄×C₈₉	—	8	2	no	—
204	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
205	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
206	C₂×C₄×C₈₉	#123	1	2	yes	—
207	C₂×C₄×C₈₉	#123	1	2	yes	—
208	C₂×C₄×C₈₉	#123	1	2	yes	—
209	C₂×C₄×C₈₉	#123	1	2	yes	—
210	C₂×C₄×C₈₉	#123	1	2	yes	—
211	C₂×C₄×C₈₉	#123	1	2	yes	—
212	C₂×C₄×C₈₉	#124	1	2	yes	—
213	C₂×C₄×C₈₉	#124	1	2	yes	—
214	C₂×C₄×C₈₉	#126	1	2	yes	—
215	C₂×C₄×C₈₉	#126	1	2	yes	—
216	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
222	C₈×C₈₉	—	2	4	no	—
223	C₈×C₈₉	—	4	2	no	—
224	C₈×C₈₉	—	4	2	no	—
225	C₈×C₈₉	—	4	2	no	—
226	C₂²×C₈₉	—	2	1	no	—
227	C₂²×C₈₉	—	2	1	no	—
228	C₂²×C₈₉	—	2	1	no	—
229	C₂²×C₈₉	—	2	1	no	—
230	C₂²×C₈₉	—	2	1	no	—
231	C₂²×C₈₉	—	2	1	no	—
232	C₂²×C₈₉	—	4	2	no	—
233	C₂²×C₈₉	—	4	2	no	—
234	C₂²×C₈₉	—	4	2	no	—
235	C₂²×C₈₉	—	4	2	no	—
236	C₂²×C₈₉	—	4	2	no	—
237	C₂²×C₈₉	—	8	4	no	—
238	C₂²×C₈₉	#397	1	2	yes	—
239	C₂²×C₈₉	#397	1	2	yes	—
240	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
241	C₂²×C₈₉	C₂×#126	1	1	yes	characteristic
242	C₂²×C₈₉	C₂×#126	1	1	yes	characteristic
243	C₄×C₈₉	—	4	2	no	—
244	C₄×C₈₉	—	8	1	no	—
245	C₄×C₈₉	—	8	2	no	—
246	C₄×C₈₉	—	8	1	no	—
247	C₄×C₈₉	#397	1	2	yes	—
248	C₄×C₈₉	#397	1	2	yes	—
249	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
250	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
251	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
252	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
253	#23647	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂×C₈₉	—	2	2	no	—
255	C₂×C₈₉	—	2	1	no	—
256	C₂×C₈₉	—	2	1	no	—
257	C₂×C₈₉	—	2	2	no	—
258	C₂×C₈₉	—	8	2	no	—
259	C₂×C₈₉	C₂×#397	1	1	yes	characteristic
260	C₂×C₈₉	#3068	1	1	yes	characteristic
261	C₂×C₈₉	#3069	1	1	yes	characteristic
262	C₂×#428	C₂×C₈₉	1	1	yes	characteristic, complemented
263	#2896	C₂×C₈₉	1	1	yes	characteristic
264	C₂×#417	C₂×C₈₉	1	1	yes	characteristic
265	#2879	C₂×C₈₉	1	1	yes	characteristic
266	#2079	C₂×C₈₉	1	4	yes	complemented
267	#2078	C₂×C₈₉	1	4	yes	complemented
268	#1742	C₂×C₈₉	1	1	yes	characteristic, complemented
269	C₂×#343	C₂×C₈₉	1	1	yes	characteristic, complemented
270	#1738	C₂×C₈₉	1	1	yes	characteristic, complemented
271	C₈₉	#23647	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
273	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
274	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
275	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
276	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
277	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
278	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
279	#428	—	2	2	no	—
280	#428	—	2	2	no	—
281	C₂×#107	C₂²×C₈₉	1	2	yes	complemented
282	C₂×#106	C₂²×C₈₉	1	2	yes	complemented
283	#419	—	2	1	no	—
284	#418	—	2	1	no	—
285	#417	—	2	2	no	—
286	#417	C₂²×C₈₉	1	2	yes	—
287	#417	C₂²×C₈₉	1	4	yes	—
288	#417	C₂²×C₈₉	1	2	yes	—
289	#416	—	2	2	no	—
290	C₂×#103	—	2	1	no	—
291	C₂×#102	—	2	1	no	—
292	#408	—	2	2	no	—
293	#408	C₂²×C₈₉	1	4	yes	—
294	#407	—	2	2	no	—
295	#343	C₂²×C₈₉	1	2	yes	complemented
296	#343	C₂²×C₈₉	1	4	yes	complemented
297	#343	C₂²×C₈₉	1	2	yes	complemented
298	#341	C₂²×C₈₉	1	4	yes	complemented
299	C₂×#41	—	2	1	no	—
300	C₂×#41	—	2	1	no	—
301	C₂×#41	—	2	2	no	—
302	C₂×#41	—	2	2	no	—
303	C₂×#41	—	2	2	no	—
304	C₂×#41	—	2	2	no	—
305	C₂×#41	D₄×C₈₉	1	2	yes	—
306	C₂×#41	C₂³×C₈₉	1	2	yes	—
307	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
308	C₂²×D₄	—	2	1	no	—
309	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	1	no	—
312	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
313	C₂×M₁₆	—	2	2	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	—	2	1	no	—
316	C₂×M₁₆	—	2	1	no	—
317	C₂×M₁₆	—	2	1	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	—	2	2	no	—
320	C₂×M₁₆	—	2	2	no	—
321	C₂×M₁₆	—	2	2	no	—
322	C₂×M₁₆	—	2	2	no	—
323	C₂×M₁₆	—	2	2	no	—
324	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
325	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
326	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
327	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
328	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
329	C₂²×C₈	—	2	1	no	—
330	C₂²×C₈	—	2	1	no	—
331	C₂²×C₈	—	2	1	no	—
332	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
333	#117	—	2	1	no	—
334	#117	—	2	1	no	—
335	#117	—	2	1	no	—
336	#117	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₄²	—	2	1	no	—
347	#107	—	2	4	no	—
348	#106	—	2	4	no	—
349	#103	—	2	4	no	—
350	#103	—	2	2	no	—
351	#103	—	2	2	no	—
352	#102	—	2	4	no	—
353	#102	—	2	2	no	—
354	#102	—	2	2	no	—
355	C₂²⋊C₈	—	2	8	no	—
356	C₂⁴	—	2	1	no	—
357	C₂⁴	#41×C₈₉	1	1	yes	characteristic
358	#41	—	2	4	no	—
359	#41	—	2	4	no	—
360	#41	—	2	2	no	—
361	#41	—	2	2	no	—
362	#41	—	2	2	no	—
363	#41	—	2	2	no	—
364	#41	—	4	2	no	—
365	#41	—	4	2	no	—
366	#41	—	4	2	no	—
367	#41	—	4	2	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	4	2	no	—
388	C₂²×C₄	—	4	1	no	—
389	C₂²×C₄	—	4	1	no	—
390	C₂²×C₄	—	8	2	no	—
391	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
395	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
396	C₂²×C₄	#41×C₈₉	1	2	yes	—
397	C₂²×C₄	#41×C₈₉	1	2	yes	—
398	C₂²×C₄	#41×C₈₉	1	2	yes	—
399	C₂²×C₄	#41×C₈₉	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
401	M₁₆	—	2	4	no	—
402	M₁₆	—	2	4	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	2	2	no	—
409	M₁₆	—	4	2	no	—
410	M₁₆	—	4	2	no	—
411	M₁₆	—	4	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	4	2	no	—
418	C₂×C₈	—	4	2	no	—
419	C₂×C₈	—	4	2	no	—
420	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	4	1	no	—
428	C₄⋊C₄	—	4	1	no	—
429	C₂²⋊C₄	—	4	1	no	—
430	C₂²⋊C₄	—	4	1	no	—
431	C₄²	—	4	1	no	—
432	C₄²	—	4	1	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	8	1	no	—
438	C₂³	—	8	1	no	—
439	C₂³	#121×C₈₉	1	1	yes	characteristic
440	C₂³	#121×C₈₉	1	1	yes	characteristic
441	C₂³	#126×C₈₉	1	2	yes	—
442	C₂³	#126×C₈₉	1	2	yes	—
443	C₂³	C₂×#41×C₈₉	1	1	yes	characteristic
444	Q₈	—	2	2	no	—
445	Q₈	—	2	2	no	—
446	Q₈	—	4	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	2	no	—
449	D₄	—	4	4	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	8	2	no	—
462	C₂×C₄	—	8	2	no	—
463	C₂×C₄	—	8	1	no	—
464	C₂×C₄	—	8	1	no	—
465	C₂×C₄	—	8	2	no	—
466	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
467	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
468	C₂×C₄	#123×C₈₉	1	2	yes	—
469	C₂×C₄	#123×C₈₉	1	2	yes	—
470	C₂×C₄	#123×C₈₉	1	2	yes	—
471	C₂×C₄	#123×C₈₉	1	2	yes	—
472	C₂×C₄	#123×C₈₉	1	2	yes	—
473	C₂×C₄	#123×C₈₉	1	2	yes	—
474	C₂×C₄	#124×C₈₉	1	2	yes	—
475	C₂×C₄	#124×C₈₉	1	2	yes	—
476	C₂×C₄	#126×C₈₉	1	2	yes	—
477	C₂×C₄	#126×C₈₉	1	2	yes	—
478	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
484	C₈	—	2	4	no	—
485	C₈	—	4	2	no	—
486	C₈	—	4	2	no	—
487	C₈	—	4	2	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	8	4	no	—
500	C₂²	#397×C₈₉	1	2	yes	—
501	C₂²	#397×C₈₉	1	2	yes	—
502	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
503	C₂²	C₂×#126×C₈₉	1	1	yes	characteristic
504	C₂²	C₂×#126×C₈₉	1	1	yes	characteristic
505	C₄	—	4	2	no	—
506	C₄	—	8	2	no	—
507	C₄	—	8	1	no	—
508	C₄	—	8	1	no	—
509	C₄	#397×C₈₉	1	2	yes	—
510	C₄	#397×C₈₉	1	2	yes	—
511	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
512	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
513	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
514	C₄	C₂×#124×C₈₉	1	1	yes	characteristic
515	C₂	—	2	2	no	—
516	C₂	—	2	2	no	—
517	C₂	—	2	1	no	—
518	C₂	—	2	1	no	—
519	C₂	—	8	2	no	—
520	C₂	C₂×#397×C₈₉	1	1	yes	characteristic
521	C₂	C₈₉×#3068	1	1	yes	characteristic
522	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	C₂²×#41×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#428×C₈₉	C₂	semidirect
5. C₈₉×#2079	C₂	semidirect
6. C₈₉×#2078	C₂	semidirect
7. C₈₉×#1742	C₂	semidirect
8. C₂×#343×C₈₉	C₂	semidirect
9. C₈₉×#1738	C₂	semidirect
19. C₂×#107×C₈₉	C₂²	semidirect
20. C₂×#106×C₈₉	C₂²	semidirect
33. #343×C₈₉	C₂²	semidirect
34. #343×C₈₉	C₂²	semidirect
35. #343×C₈₉	C₂²	semidirect
36. #341×C₈₉	C₂²	semidirect
253. #23647	C₈₉	direct (N × H)
262. C₂×#428	C₂×C₈₉	semidirect
266. #2079	C₂×C₈₉	semidirect
267. #2078	C₂×C₈₉	semidirect
268. #1742	C₂×C₈₉	semidirect
269. C₂×#343	C₂×C₈₉	semidirect
270. #1738	C₂×C₈₉	semidirect
271. C₈₉	#23647	direct (N × H)
281. C₂×#107	C₂²×C₈₉	semidirect
282. C₂×#106	C₂²×C₈₉	semidirect
295. #343	C₂²×C₈₉	semidirect
296. #343	C₂²×C₈₉	semidirect
297. #343	C₂²×C₈₉	semidirect
298. #341	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#428×C₈₉	2
C₈₉×#2896	2
C₂×#417×C₈₉	2
C₈₉×#2879	2
C₈₉×#2079	2
C₈₉×#2078	2
C₈₉×#1742	2
C₂×#343×C₈₉	2
C₈₉×#1738	2
#23647	89

Subgroups — C89×#23647 #36401495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23647 #36401495