Subgroups — C₈₉×#10792 #36388640

Order 22784.

1204 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3428	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂×#344×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1740	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×H(3,ℤ/4)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×H(3,ℤ/4)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	#520×C₈₉	—	2	2	no	—
12	#520×C₈₉	—	2	2	no	—
13	#519×C₈₉	—	2	1	no	—
14	#518×C₈₉	—	2	1	no	—
15	#518×C₈₉	—	2	2	no	—
16	C₄×#41×C₈₉	—	2	2	no	—
17	C₄×#41×C₈₉	C₄	1	4	yes	complemented
18	C₂×#117×C₈₉	—	2	2	no	—
19	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
23	#421×C₈₉	—	2	2	no	—
24	#421×C₈₉	—	2	2	no	—
25	C₂×#104×C₈₉	C₄	1	4	yes	complemented
26	#419×C₈₉	—	2	1	no	—
27	#418×C₈₉	—	2	1	no	—
28	#417×C₈₉	—	2	2	no	—
29	#416×C₈₉	C₄	1	4	yes	complemented
30	#344×C₈₉	—	2	2	no	—
31	#342×C₈₉	—	2	2	no	—
32	H(3,ℤ/4)×C₈₉	—	2	2	no	—
33	H(3,ℤ/4)×C₈₉	—	2	2	no	—
34	C₂×#41×C₈₉	—	2	1	no	—
35	C₂×#41×C₈₉	—	2	1	no	—
36	C₂×#41×C₈₉	—	2	1	no	—
37	C₂×#41×C₈₉	—	2	1	no	—
38	C₂×#41×C₈₉	—	2	1	no	—
39	C₂×#41×C₈₉	—	2	1	no	—
40	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
41	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
43	C₂×#41×C₈₉	D₄	1	2	yes	—
44	C₂×#41×C₈₉	D₄	1	2	yes	—
45	C₂×#41×C₈₉	Q₈	1	2	yes	—
46	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₈₉	—	2	1	no	—
48	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₈₉	—	2	1	no	—
50	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₈₉	—	2	1	no	—
52	C₂×M₁₆×C₈₉	—	2	1	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	1	no	—
57	C₂×M₁₆×C₈₉	—	2	1	no	—
58	C₂×M₁₆×C₈₉	—	2	1	no	—
59	C₂×M₁₆×C₈₉	—	2	1	no	—
60	C₂×M₁₆×C₈₉	—	2	1	no	—
61	C₂×M₁₆×C₈₉	—	2	1	no	—
62	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
63	C₂²×C₈×C₈₉	—	2	1	no	—
64	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	complemented
65	C₄×Q₈×C₈₉	—	2	4	no	—
66	C₄×Q₈×C₈₉	—	2	4	no	—
67	C₄×D₄×C₈₉	—	2	8	no	—
68	C₄×D₄×C₈₉	—	2	4	no	—
69	C₄×D₄×C₈₉	—	2	4	no	—
70	C₄×D₄×C₈₉	—	2	8	no	—
71	#117×C₈₉	—	2	4	no	—
72	#117×C₈₉	—	2	4	no	—
73	#117×C₈₉	—	2	1	no	—
74	#117×C₈₉	—	2	1	no	—
75	#117×C₈₉	—	2	1	no	—
76	#117×C₈₉	—	2	1	no	—
77	#117×C₈₉	—	2	1	no	—
78	#117×C₈₉	—	2	1	no	—
79	#117×C₈₉	—	2	1	no	—
80	#117×C₈₉	—	2	1	no	—
81	#117×C₈₉	—	2	1	no	—
82	#117×C₈₉	—	2	4	no	—
83	#117×C₈₉	—	2	1	no	—
84	#117×C₈₉	—	2	1	no	—
85	#117×C₈₉	C₂×C₄	1	2	yes	—
86	#117×C₈₉	C₂×C₄	1	2	yes	complemented
87	#117×C₈₉	D₄	1	2	yes	—
88	#117×C₈₉	D₄	1	2	yes	—
89	#117×C₈₉	D₄	1	2	yes	—
90	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
92	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
100	C₂×C₄²×C₈₉	—	2	1	no	—
101	C₂×C₄²×C₈₉	—	2	4	no	—
102	C₂×C₄²×C₈₉	—	2	1	no	—
103	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
104	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	complemented
105	#104×C₈₉	—	2	8	no	—
106	#103×C₈₉	—	2	4	no	—
107	#102×C₈₉	—	2	4	no	—
108	C₂⁴×C₈₉	—	2	1	no	—
109	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
110	#41×C₈₉	—	2	2	no	—
111	#41×C₈₉	—	2	2	no	—
112	#41×C₈₉	—	2	2	no	—
113	#41×C₈₉	—	2	2	no	—
114	#41×C₈₉	—	2	2	no	—
115	#41×C₈₉	—	2	2	no	—
116	#41×C₈₉	—	2	2	no	—
117	#41×C₈₉	—	2	2	no	—
118	#41×C₈₉	—	2	2	no	—
119	#41×C₈₉	—	2	2	no	—
120	#41×C₈₉	—	2	2	no	—
121	#41×C₈₉	—	2	2	no	—
122	#41×C₈₉	C₄²	1	4	yes	—
123	#41×C₈₉	C₂²⋊C₄	1	4	yes	—
124	#41×C₈₉	C₄⋊C₄	1	4	yes	—
125	#41×C₈₉	C₄⋊C₄	1	4	yes	—
126	C₂×Q₈×C₈₉	—	2	1	no	—
127	C₂×Q₈×C₈₉	—	2	1	no	—
128	C₂×Q₈×C₈₉	—	2	1	no	—
129	C₂×Q₈×C₈₉	C₂²⋊C₄	1	2	yes	—
130	C₂×Q₈×C₈₉	C₂²⋊C₄	1	2	yes	—
131	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
132	C₂×D₄×C₈₉	—	2	2	no	—
133	C₂×D₄×C₈₉	—	2	2	no	—
134	C₂×D₄×C₈₉	—	2	2	no	—
135	C₂×D₄×C₈₉	—	2	2	no	—
136	C₂×D₄×C₈₉	—	2	2	no	—
137	C₂×D₄×C₈₉	—	2	2	no	—
138	C₂×D₄×C₈₉	—	2	1	no	—
139	C₂×D₄×C₈₉	—	2	1	no	—
140	C₂×D₄×C₈₉	—	2	1	no	—
141	C₂×D₄×C₈₉	C₂²⋊C₄	1	2	yes	—
142	C₂×D₄×C₈₉	C₂²⋊C₄	1	2	yes	—
143	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
144	C₂²×C₄×C₈₉	—	2	1	no	—
145	C₂²×C₄×C₈₉	—	2	2	no	—
146	C₂²×C₄×C₈₉	—	2	2	no	—
147	C₂²×C₄×C₈₉	—	2	2	no	—
148	C₂²×C₄×C₈₉	—	2	2	no	—
149	C₂²×C₄×C₈₉	—	2	2	no	—
150	C₂²×C₄×C₈₉	—	2	2	no	—
151	C₂²×C₄×C₈₉	—	2	8	no	—
152	C₂²×C₄×C₈₉	—	2	2	no	—
153	C₂²×C₄×C₈₉	—	2	2	no	—
154	C₂²×C₄×C₈₉	—	2	2	no	—
155	C₂²×C₄×C₈₉	—	4	1	no	—
156	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
166	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
167	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
168	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
169	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
170	M₁₆×C₈₉	—	2	2	no	—
171	M₁₆×C₈₉	—	2	2	no	—
172	M₁₆×C₈₉	—	4	1	no	—
173	M₁₆×C₈₉	—	4	1	no	—
174	M₁₆×C₈₉	—	4	1	no	—
175	M₁₆×C₈₉	—	4	1	no	—
176	M₁₆×C₈₉	—	4	1	no	—
177	M₁₆×C₈₉	—	4	1	no	—
178	C₂×C₈×C₈₉	—	2	2	no	—
179	C₂×C₈×C₈₉	—	2	2	no	—
180	C₂×C₈×C₈₉	—	2	2	no	—
181	C₂×C₈×C₈₉	—	2	2	no	—
182	C₂×C₈×C₈₉	—	4	1	no	—
183	C₂×C₈×C₈₉	—	4	1	no	—
184	C₄⋊C₄×C₈₉	—	2	8	no	—
185	C₄⋊C₄×C₈₉	—	2	2	no	—
186	C₄⋊C₄×C₈₉	—	2	2	no	—
187	C₄⋊C₄×C₈₉	—	2	2	no	—
188	C₄⋊C₄×C₈₉	—	2	2	no	—
189	C₄⋊C₄×C₈₉	—	2	2	no	—
190	C₄⋊C₄×C₈₉	—	4	1	no	—
191	C₄⋊C₄×C₈₉	—	4	1	no	—
192	C₄⋊C₄×C₈₉	—	4	1	no	—
193	C₂²⋊C₄×C₈₉	—	2	8	no	—
194	C₂²⋊C₄×C₈₉	—	2	2	no	—
195	C₂²⋊C₄×C₈₉	—	2	2	no	—
196	C₂²⋊C₄×C₈₉	—	2	2	no	—
197	C₂²⋊C₄×C₈₉	—	2	2	no	—
198	C₂²⋊C₄×C₈₉	—	4	1	no	—
199	C₂²⋊C₄×C₈₉	—	4	1	no	—
200	C₂²⋊C₄×C₈₉	—	4	1	no	—
201	C₂²⋊C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
202	C₄²×C₈₉	—	2	8	no	—
203	C₄²×C₈₉	—	2	2	no	—
204	C₄²×C₈₉	—	2	2	no	—
205	C₄²×C₈₉	—	2	2	no	—
206	C₄²×C₈₉	—	2	2	no	—
207	C₄²×C₈₉	—	2	2	no	—
208	C₄²×C₈₉	—	2	2	no	—
209	C₄²×C₈₉	—	4	1	no	—
210	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
211	C₂³×C₈₉	—	2	2	no	—
212	C₂³×C₈₉	—	2	2	no	—
213	C₂³×C₈₉	—	2	2	no	—
214	C₂³×C₈₉	—	8	1	no	—
215	C₂³×C₈₉	—	8	2	no	—
216	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
218	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
219	C₂³×C₈₉	#120	1	1	yes	characteristic
220	C₂³×C₈₉	#120	1	1	yes	characteristic
221	C₂³×C₈₉	#121	1	1	yes	characteristic
222	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
223	Q₈×C₈₉	—	2	2	no	—
224	Q₈×C₈₉	—	2	2	no	—
225	Q₈×C₈₉	—	2	2	no	—
226	Q₈×C₈₉	#95	1	4	yes	—
227	D₄×C₈₉	—	2	4	no	—
228	D₄×C₈₉	—	2	4	no	—
229	D₄×C₈₉	—	2	4	no	—
230	D₄×C₈₉	—	2	4	no	—
231	D₄×C₈₉	—	2	2	no	—
232	D₄×C₈₉	—	2	2	no	—
233	D₄×C₈₉	—	2	2	no	—
234	D₄×C₈₉	#95	1	4	yes	—
235	C₂×C₄×C₈₉	—	2	2	no	—
236	C₂×C₄×C₈₉	—	2	2	no	—
237	C₂×C₄×C₈₉	—	2	2	no	—
238	C₂×C₄×C₈₉	—	2	4	no	—
239	C₂×C₄×C₈₉	—	2	4	no	—
240	C₂×C₄×C₈₉	—	2	4	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	2	4	no	—
243	C₂×C₄×C₈₉	—	2	4	no	—
244	C₂×C₄×C₈₉	—	4	2	no	—
245	C₂×C₄×C₈₉	—	4	2	no	—
246	C₂×C₄×C₈₉	—	8	2	no	—
247	C₂×C₄×C₈₉	—	8	2	no	—
248	C₂×C₄×C₈₉	—	8	8	no	—
249	C₂×C₄×C₈₉	—	8	2	no	—
250	C₂×C₄×C₈₉	—	8	2	no	—
251	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
252	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
260	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
261	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
262	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
263	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
264	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
265	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
266	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
267	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
268	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
269	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
270	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
271	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
272	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
273	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
274	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
275	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
276	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
277	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
278	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
279	C₈×C₈₉	—	4	2	no	—
280	C₈×C₈₉	—	4	2	no	—
281	C₂²×C₈₉	—	2	4	no	—
282	C₂²×C₈₉	—	4	1	no	—
283	C₂²×C₈₉	—	4	1	no	—
284	C₂²×C₈₉	—	4	1	no	—
285	C₂²×C₈₉	—	4	1	no	—
286	C₂²×C₈₉	—	4	1	no	—
287	C₂²×C₈₉	—	4	1	no	—
288	C₂²×C₈₉	—	4	1	no	—
289	C₂²×C₈₉	—	4	4	no	—
290	C₂²×C₈₉	—	8	2	no	—
291	C₂²×C₈₉	—	8	2	no	—
292	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
293	C₂²×C₈₉	#379	1	1	yes	characteristic
294	C₂²×C₈₉	#380	1	1	yes	characteristic
295	C₂²×C₈₉	#385	1	1	yes	characteristic
296	C₂²×C₈₉	#385	1	1	yes	characteristic
297	C₂²×C₈₉	#386	1	1	yes	characteristic
298	C₂²×C₈₉	#386	1	1	yes	characteristic
299	C₄×C₈₉	—	2	4	no	—
300	C₄×C₈₉	—	4	4	no	—
301	C₄×C₈₉	—	4	4	no	—
302	C₄×C₈₉	—	8	2	no	—
303	C₄×C₈₉	—	8	2	no	—
304	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
305	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
306	C₄×C₈₉	#380	1	1	yes	characteristic
307	C₄×C₈₉	#381	1	1	yes	characteristic
308	C₄×C₈₉	#385	1	1	yes	characteristic
309	C₄×C₈₉	#385	1	1	yes	characteristic
310	C₄×C₈₉	#386	1	1	yes	characteristic
311	C₄×C₈₉	#386	1	1	yes	characteristic
312	#10792	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
313	C₂×C₈₉	—	2	1	no	—
314	C₂×C₈₉	—	2	1	no	—
315	C₂×C₈₉	—	2	1	no	—
316	C₂×C₈₉	—	2	1	no	—
317	C₂×C₈₉	—	2	1	no	—
318	C₂×C₈₉	—	2	1	no	—
319	C₂×C₈₉	—	2	1	no	—
320	C₂×C₈₉	—	4	4	no	—
321	C₂×C₈₉	C₂²⋊#95	1	1	yes	characteristic
322	#3428	C₂×C₈₉	1	1	yes	characteristic
323	C₂×#421	C₂×C₈₉	1	1	yes	characteristic
324	#2896	C₂×C₈₉	1	1	yes	characteristic
325	C₂×#344	C₂×C₈₉	1	1	yes	characteristic, complemented
326	#1740	C₂×C₈₉	1	1	yes	characteristic, complemented
327	C₂×H(3,ℤ/4)	C₂×C₈₉	1	1	yes	characteristic, complemented
328	C₂×H(3,ℤ/4)	C₂×C₈₉	1	1	yes	characteristic, complemented
329	C₈₉	#10792	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
331	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
332	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
333	#520	—	2	2	no	—
334	#520	—	2	2	no	—
335	#519	—	2	1	no	—
336	#518	—	2	1	no	—
337	#518	—	2	2	no	—
338	C₄×#41	—	2	2	no	—
339	C₄×#41	C₄×C₈₉	1	4	yes	complemented
340	C₂×#117	—	2	2	no	—
341	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
342	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
343	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
344	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
345	#421	—	2	2	no	—
346	#421	—	2	2	no	—
347	C₂×#104	C₄×C₈₉	1	4	yes	complemented
348	#419	—	2	1	no	—
349	#418	—	2	1	no	—
350	#417	—	2	2	no	—
351	#416	C₄×C₈₉	1	4	yes	complemented
352	#344	—	2	2	no	—
353	#342	—	2	2	no	—
354	H(3,ℤ/4)	—	2	2	no	—
355	H(3,ℤ/4)	—	2	2	no	—
356	C₂×#41	—	2	1	no	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	1	no	—
359	C₂×#41	—	2	1	no	—
360	C₂×#41	—	2	1	no	—
361	C₂×#41	—	2	1	no	—
362	C₂×#41	C₂×C₄×C₈₉	1	2	yes	—
363	C₂×#41	C₂×C₄×C₈₉	1	2	yes	—
364	C₂×#41	C₂×C₄×C₈₉	1	2	yes	—
365	C₂×#41	D₄×C₈₉	1	2	yes	—
366	C₂×#41	D₄×C₈₉	1	2	yes	—
367	C₂×#41	Q₈×C₈₉	1	2	yes	—
368	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
369	C₂²×D₄	—	2	1	no	—
370	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
371	C₂³×C₄	—	2	1	no	—
372	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
373	C₂×M₁₆	—	2	1	no	—
374	C₂×M₁₆	—	2	1	no	—
375	C₂×M₁₆	—	2	1	no	—
376	C₂×M₁₆	—	2	1	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	1	no	—
380	C₂×M₁₆	—	2	1	no	—
381	C₂×M₁₆	—	2	1	no	—
382	C₂×M₁₆	—	2	1	no	—
383	C₂×M₁₆	—	2	1	no	—
384	C₂×M₁₆	C₂×C₄×C₈₉	1	2	yes	complemented
385	C₂²×C₈	—	2	1	no	—
386	C₂²×C₈	C₂×C₄×C₈₉	1	2	yes	complemented
387	C₄×Q₈	—	2	4	no	—
388	C₄×Q₈	—	2	4	no	—
389	C₄×D₄	—	2	8	no	—
390	C₄×D₄	—	2	4	no	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	—	2	8	no	—
393	#117	—	2	4	no	—
394	#117	—	2	4	no	—
395	#117	—	2	1	no	—
396	#117	—	2	1	no	—
397	#117	—	2	1	no	—
398	#117	—	2	1	no	—
399	#117	—	2	1	no	—
400	#117	—	2	1	no	—
401	#117	—	2	1	no	—
402	#117	—	2	1	no	—
403	#117	—	2	1	no	—
404	#117	—	2	4	no	—
405	#117	—	2	1	no	—
406	#117	—	2	1	no	—
407	#117	C₂×C₄×C₈₉	1	2	yes	—
408	#117	C₂×C₄×C₈₉	1	2	yes	complemented
409	#117	D₄×C₈₉	1	2	yes	—
410	#117	D₄×C₈₉	1	2	yes	—
411	#117	D₄×C₈₉	1	2	yes	—
412	C₂×C₄⋊C₄	—	2	4	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	—	2	4	no	—
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	1	no	—
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
422	C₂×C₄²	—	2	1	no	—
423	C₂×C₄²	—	2	4	no	—
424	C₂×C₄²	—	2	1	no	—
425	C₂×C₄²	C₂×C₄×C₈₉	1	2	yes	—
426	C₂×C₄²	C₂×C₄×C₈₉	1	2	yes	complemented
427	#104	—	2	8	no	—
428	#103	—	2	4	no	—
429	#102	—	2	4	no	—
430	C₂⁴	—	2	1	no	—
431	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
432	#41	—	2	2	no	—
433	#41	—	2	2	no	—
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	#41	—	2	2	no	—
437	#41	—	2	2	no	—
438	#41	—	2	2	no	—
439	#41	—	2	2	no	—
440	#41	—	2	2	no	—
441	#41	—	2	2	no	—
442	#41	—	2	2	no	—
443	#41	—	2	2	no	—
444	#41	C₄²×C₈₉	1	4	yes	—
445	#41	C₂²⋊C₄×C₈₉	1	4	yes	—
446	#41	C₄⋊C₄×C₈₉	1	4	yes	—
447	#41	C₄⋊C₄×C₈₉	1	4	yes	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	C₂²⋊C₄×C₈₉	1	2	yes	—
452	C₂×Q₈	C₂²⋊C₄×C₈₉	1	2	yes	—
453	C₂×Q₈	C₄⋊C₄×C₈₉	1	2	yes	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	C₂²⋊C₄×C₈₉	1	2	yes	—
464	C₂×D₄	C₂²⋊C₄×C₈₉	1	2	yes	—
465	C₂×D₄	C₄⋊C₄×C₈₉	1	2	yes	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	8	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	4	1	no	—
478	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
487	C₂²×C₄	C₂×Q₈×C₈₉	1	1	yes	characteristic
488	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
489	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
490	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
491	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
492	M₁₆	—	2	2	no	—
493	M₁₆	—	2	2	no	—
494	M₁₆	—	4	1	no	—
495	M₁₆	—	4	1	no	—
496	M₁₆	—	4	1	no	—
497	M₁₆	—	4	1	no	—
498	M₁₆	—	4	1	no	—
499	M₁₆	—	4	1	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	4	1	no	—
505	C₂×C₈	—	4	1	no	—
506	C₄⋊C₄	—	2	8	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	4	1	no	—
513	C₄⋊C₄	—	4	1	no	—
514	C₄⋊C₄	—	4	1	no	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	4	1	no	—
521	C₂²⋊C₄	—	4	1	no	—
522	C₂²⋊C₄	—	4	1	no	—
523	C₂²⋊C₄	C₂²⋊C₄×C₈₉	1	4	yes	—
524	C₄²	—	2	8	no	—
525	C₄²	—	2	2	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	2	no	—
531	C₄²	—	4	1	no	—
532	C₄²	C₂²⋊C₄×C₈₉	1	4	yes	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	8	1	no	—
537	C₂³	—	8	2	no	—
538	C₂³	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
539	C₂³	C₄×D₄×C₈₉	1	1	yes	characteristic
540	C₂³	C₄×D₄×C₈₉	1	1	yes	characteristic
541	C₂³	#120×C₈₉	1	1	yes	characteristic
542	C₂³	#120×C₈₉	1	1	yes	characteristic
543	C₂³	#121×C₈₉	1	1	yes	characteristic
544	C₂³	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
545	Q₈	—	2	2	no	—
546	Q₈	—	2	2	no	—
547	Q₈	—	2	2	no	—
548	Q₈	#95×C₈₉	1	4	yes	—
549	D₄	—	2	4	no	—
550	D₄	—	2	4	no	—
551	D₄	—	2	4	no	—
552	D₄	—	2	4	no	—
553	D₄	—	2	2	no	—
554	D₄	—	2	2	no	—
555	D₄	—	2	2	no	—
556	D₄	#95×C₈₉	1	4	yes	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	8	2	no	—
569	C₂×C₄	—	8	2	no	—
570	C₂×C₄	—	8	8	no	—
571	C₂×C₄	—	8	2	no	—
572	C₂×C₄	—	8	2	no	—
573	C₂×C₄	C₂×C₄²×C₈₉	1	1	yes	characteristic
574	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
575	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
577	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
578	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
579	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
580	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
581	C₂×C₄	#117×C₈₉	1	1	yes	characteristic
582	C₂×C₄	#117×C₈₉	1	1	yes	characteristic
583	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
584	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
585	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
586	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
587	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
588	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
589	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
590	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
591	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
592	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
593	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
594	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
595	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
596	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
597	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
598	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
599	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
600	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
601	C₈	—	4	2	no	—
602	C₈	—	4	2	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	1	no	—
609	C₂²	—	4	1	no	—
610	C₂²	—	4	1	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	8	2	no	—
613	C₂²	—	8	2	no	—
614	C₂²	C₄×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
615	C₂²	#379×C₈₉	1	1	yes	characteristic
616	C₂²	#380×C₈₉	1	1	yes	characteristic
617	C₂²	#385×C₈₉	1	1	yes	characteristic
618	C₂²	#385×C₈₉	1	1	yes	characteristic
619	C₂²	#386×C₈₉	1	1	yes	characteristic
620	C₂²	#386×C₈₉	1	1	yes	characteristic
621	C₄	—	2	4	no	—
622	C₄	—	4	4	no	—
623	C₄	—	4	4	no	—
624	C₄	—	8	2	no	—
625	C₄	—	8	2	no	—
626	C₄	C₂×#95×C₈₉	1	1	yes	characteristic
627	C₄	C₄×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
628	C₄	#380×C₈₉	1	1	yes	characteristic
629	C₄	#381×C₈₉	1	1	yes	characteristic
630	C₄	#385×C₈₉	1	1	yes	characteristic
631	C₄	#385×C₈₉	1	1	yes	characteristic
632	C₄	#386×C₈₉	1	1	yes	characteristic
633	C₄	#386×C₈₉	1	1	yes	characteristic
634	C₂	—	2	1	no	—
635	C₂	—	2	1	no	—
636	C₂	—	2	1	no	—
637	C₂	—	2	1	no	—
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	2	1	no	—
641	C₂	—	4	4	no	—
642	C₂	C₈₉×C₂²⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₈₉
Frattini, cosocle	C₂³×C₄	C₂³×C₈₉
socle	C₂×C₈₉	C₂²⋊#95
prime radical	C₂²×#41×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#344×C₈₉	C₂	semidirect
5. C₈₉×#1740	C₂	semidirect
6. C₂×H(3,ℤ/4)×C₈₉	C₂	semidirect
7. C₂×H(3,ℤ/4)×C₈₉	C₂	semidirect
17. C₄×#41×C₈₉	C₄	semidirect
25. C₂×#104×C₈₉	C₄	semidirect
29. #416×C₈₉	C₄	semidirect
62. C₂×M₁₆×C₈₉	C₂×C₄	semidirect
64. C₂²×C₈×C₈₉	C₂×C₄	semidirect
86. #117×C₈₉	C₂×C₄	semidirect
104. C₂×C₄²×C₈₉	C₂×C₄	semidirect
312. #10792	C₈₉	direct (N × H)
325. C₂×#344	C₂×C₈₉	semidirect
326. #1740	C₂×C₈₉	semidirect
327. C₂×H(3,ℤ/4)	C₂×C₈₉	semidirect
328. C₂×H(3,ℤ/4)	C₂×C₈₉	semidirect
329. C₈₉	#10792	direct (N × H)
339. C₄×#41	C₄×C₈₉	semidirect
347. C₂×#104	C₄×C₈₉	semidirect
351. #416	C₄×C₈₉	semidirect
384. C₂×M₁₆	C₂×C₄×C₈₉	semidirect
386. C₂²×C₈	C₂×C₄×C₈₉	semidirect
408. #117	C₂×C₄×C₈₉	semidirect
426. C₂×C₄²	C₂×C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3428	2
C₂×#421×C₈₉	2
C₈₉×#2896	2
C₂×#344×C₈₉	2
C₈₉×#1740	2
C₂×H(3,ℤ/4)×C₈₉	2
#10792	89

Subgroups — C89×#10792 #36388640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#10792 #36388640