Subgroups — C₈₉×#25318 #36403166

Order 22784.

1034 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#459×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2542	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2071	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2054	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1879	C₂	1	2	yes	maximal, complemented
10	C₈₉×#1802	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
17	C₂×C₄×D₄×C₈₉	—	2	1	no	—
18	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
20	#486×C₈₉	—	2	4	no	—
21	#459×C₈₉	—	2	4	no	—
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
24	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#102×C₈₉	C₂²	1	2	yes	—
26	C₂×#102×C₈₉	C₂²	1	2	yes	—
27	C₂×#102×C₈₉	C₂²	1	2	yes	—
28	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	#399×C₈₉	—	2	2	no	—
31	#390×C₈₉	—	2	1	no	—
32	#390×C₈₉	C₂²	1	2	yes	—
33	#390×C₈₉	C₂²	1	2	yes	—
34	#386×C₈₉	—	2	2	no	—
35	#384×C₈₉	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
38	#340×C₈₉	C₂²	1	2	yes	—
39	#340×C₈₉	C₂²	1	1	yes	characteristic
40	#340×C₈₉	C₂²	1	2	yes	—
41	#340×C₈₉	C₂²	1	1	yes	characteristic
42	#336×C₈₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₈₉	—	2	1	no	—
44	C₂²×D₄×C₈₉	—	2	1	no	—
45	C₂²×D₄×C₈₉	—	2	1	no	—
46	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
47	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₈₉	—	2	1	no	—
49	C₂³×C₄×C₈₉	—	2	2	no	—
50	C₂×D₈×C₈₉	—	2	4	no	—
51	C₂×D₈×C₈₉	—	2	4	no	—
52	C₂×D₈×C₈₉	—	4	2	no	—
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₈₉	—	2	2	no	—
57	C₄⋊Q₈×C₈₉	—	2	2	no	—
58	#127×C₈₉	—	2	2	no	—
59	#127×C₈₉	D₄	1	4	yes	—
60	C₂²⋊Q₈×C₈₉	—	4	4	no	—
61	C₂²⋊Q₈×C₈₉	—	4	4	no	—
62	C₄×D₄×C₈₉	—	2	4	no	—
63	C₄×D₄×C₈₉	—	2	2	no	—
64	C₄×D₄×C₈₉	—	2	2	no	—
65	C₄×D₄×C₈₉	—	4	2	no	—
66	C₄×D₄×C₈₉	—	4	4	no	—
67	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
69	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
82	C₂×C₄²×C₈₉	—	2	1	no	—
83	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
84	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₈₉	—	2	2	no	—
86	C₄⋊C₈×C₈₉	—	4	2	no	—
87	#102×C₈₉	—	2	4	no	—
88	#102×C₈₉	—	2	4	no	—
89	#102×C₈₉	—	2	4	no	—
90	#102×C₈₉	—	2	2	no	—
91	#102×C₈₉	—	2	2	no	—
92	C₄×C₈×C₈₉	—	2	2	no	—
93	#95×C₈₉	—	2	1	no	—
94	#95×C₈₉	D₄	1	2	yes	—
95	C₂⁴×C₈₉	—	2	1	no	—
96	C₂⁴×C₈₉	—	2	2	no	—
97	C₂×Q₈×C₈₉	—	4	2	no	—
98	C₂×Q₈×C₈₉	—	4	2	no	—
99	C₂×Q₈×C₈₉	—	4	2	no	—
100	C₂×D₄×C₈₉	—	2	4	no	—
101	C₂×D₄×C₈₉	—	2	4	no	—
102	C₂×D₄×C₈₉	—	2	1	no	—
103	C₂×D₄×C₈₉	—	2	1	no	—
104	C₂×D₄×C₈₉	—	2	1	no	—
105	C₂×D₄×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂×D₄×C₈₉	—	2	2	no	—
108	C₂×D₄×C₈₉	—	2	2	no	—
109	C₂×D₄×C₈₉	—	2	1	no	—
110	C₂×D₄×C₈₉	—	2	2	no	—
111	C₂×D₄×C₈₉	—	2	1	no	—
112	C₂×D₄×C₈₉	—	4	2	no	—
113	C₂×D₄×C₈₉	—	4	2	no	—
114	C₂×D₄×C₈₉	—	8	1	no	—
115	C₂×D₄×C₈₉	—	8	1	no	—
116	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
117	C₂×D₄×C₈₉	#41	1	2	yes	—
118	C₂²×C₄×C₈₉	—	2	1	no	—
119	C₂²×C₄×C₈₉	—	2	1	no	—
120	C₂²×C₄×C₈₉	—	2	2	no	—
121	C₂²×C₄×C₈₉	—	2	1	no	—
122	C₂²×C₄×C₈₉	—	4	4	no	—
123	C₂²×C₄×C₈₉	—	4	2	no	—
124	C₂²×C₄×C₈₉	—	4	4	no	—
125	C₂²×C₄×C₈₉	—	4	2	no	—
126	C₂²×C₄×C₈₉	—	4	2	no	—
127	C₂²×C₄×C₈₉	—	8	2	no	—
128	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
133	D₈×C₈₉	—	4	8	no	—
134	C₂×C₈×C₈₉	—	2	1	no	—
135	C₂×C₈×C₈₉	—	2	1	no	—
136	C₂×C₈×C₈₉	—	2	2	no	—
137	C₂×C₈×C₈₉	—	2	1	no	—
138	C₂×C₈×C₈₉	—	2	1	no	—
139	C₂×C₈×C₈₉	—	2	2	no	—
140	C₂×C₈×C₈₉	—	2	1	no	—
141	C₂×C₈×C₈₉	—	2	2	no	—
142	C₂×C₈×C₈₉	—	2	1	no	—
143	C₄⋊C₄×C₈₉	—	2	2	no	—
144	C₄⋊C₄×C₈₉	—	2	1	no	—
145	C₄⋊C₄×C₈₉	—	2	1	no	—
146	C₄⋊C₄×C₈₉	—	2	2	no	—
147	C₄⋊C₄×C₈₉	—	2	1	no	—
148	C₄⋊C₄×C₈₉	—	2	1	no	—
149	C₄⋊C₄×C₈₉	—	2	2	no	—
150	C₄⋊C₄×C₈₉	—	2	1	no	—
151	C₄⋊C₄×C₈₉	—	4	4	no	—
152	C₄⋊C₄×C₈₉	—	4	2	no	—
153	C₄⋊C₄×C₈₉	—	4	4	no	—
154	C₄⋊C₄×C₈₉	D₈	1	4	yes	complemented
155	C₄⋊C₄×C₈₉	#41	1	2	yes	—
156	C₂²⋊C₄×C₈₉	—	4	4	no	—
157	C₂²⋊C₄×C₈₉	—	4	4	no	—
158	C₂²⋊C₄×C₈₉	—	4	2	no	—
159	C₂²⋊C₄×C₈₉	—	4	4	no	—
160	C₂²⋊C₄×C₈₉	—	4	4	no	—
161	C₂²⋊C₄×C₈₉	—	8	2	no	—
162	C₄²×C₈₉	—	2	2	no	—
163	C₄²×C₈₉	—	2	2	no	—
164	C₄²×C₈₉	—	4	2	no	—
165	C₄²×C₈₉	C₂×D₄	1	2	yes	—
166	C₄²×C₈₉	#41	1	2	yes	—
167	C₂³×C₈₉	—	2	4	no	—
168	C₂³×C₈₉	—	2	2	no	—
169	C₂³×C₈₉	—	4	1	no	—
170	C₂³×C₈₉	—	4	2	no	—
171	C₂³×C₈₉	—	4	1	no	—
172	C₂³×C₈₉	—	4	2	no	—
173	C₂³×C₈₉	—	4	1	no	—
174	C₂³×C₈₉	—	4	2	no	—
175	C₂³×C₈₉	—	4	1	no	—
176	C₂³×C₈₉	—	4	2	no	—
177	C₂³×C₈₉	—	4	2	no	—
178	C₂³×C₈₉	—	4	2	no	—
179	C₂³×C₈₉	—	4	1	no	—
180	C₂³×C₈₉	—	4	1	no	—
181	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₈₉	—	8	4	no	—
183	D₄×C₈₉	—	2	2	no	—
184	D₄×C₈₉	—	2	2	no	—
185	D₄×C₈₉	—	2	4	no	—
186	D₄×C₈₉	—	2	4	no	—
187	D₄×C₈₉	—	4	2	no	—
188	D₄×C₈₉	—	4	1	no	—
189	D₄×C₈₉	—	4	2	no	—
190	D₄×C₈₉	—	4	1	no	—
191	D₄×C₈₉	—	4	8	no	—
192	D₄×C₈₉	—	8	4	no	—
193	C₂×C₄×C₈₉	—	2	2	no	—
194	C₂×C₄×C₈₉	—	2	2	no	—
195	C₂×C₄×C₈₉	—	2	1	no	—
196	C₂×C₄×C₈₉	—	2	1	no	—
197	C₂×C₄×C₈₉	—	2	2	no	—
198	C₂×C₄×C₈₉	—	2	2	no	—
199	C₂×C₄×C₈₉	—	2	2	no	—
200	C₂×C₄×C₈₉	—	2	2	no	—
201	C₂×C₄×C₈₉	—	4	1	no	—
202	C₂×C₄×C₈₉	—	4	1	no	—
203	C₂×C₄×C₈₉	—	4	1	no	—
204	C₂×C₄×C₈₉	—	4	2	no	—
205	C₂×C₄×C₈₉	—	4	1	no	—
206	C₂×C₄×C₈₉	—	4	1	no	—
207	C₂×C₄×C₈₉	—	4	2	no	—
208	C₂×C₄×C₈₉	—	4	1	no	—
209	C₂×C₄×C₈₉	—	4	2	no	—
210	C₂×C₄×C₈₉	—	4	1	no	—
211	C₂×C₄×C₈₉	—	4	1	no	—
212	C₂×C₄×C₈₉	—	8	8	no	—
213	C₂×C₄×C₈₉	—	8	4	no	—
214	C₂×C₄×C₈₉	#121	1	2	yes	—
215	C₂×C₄×C₈₉	#121	1	2	yes	—
216	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
217	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
218	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
219	C₂×C₄×C₈₉	C₂²×D₄	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	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₈₉	—	4	1	no	—
226	C₈×C₈₉	—	4	2	no	—
227	C₈×C₈₉	—	4	1	no	—
228	C₂²×C₈₉	—	4	2	no	—
229	C₂²×C₈₉	—	4	2	no	—
230	C₂²×C₈₉	—	4	2	no	—
231	C₂²×C₈₉	—	4	4	no	—
232	C₂²×C₈₉	—	4	4	no	—
233	C₂²×C₈₉	—	4	4	no	—
234	C₂²×C₈₉	—	8	2	no	—
235	C₂²×C₈₉	—	8	1	no	—
236	C₂²×C₈₉	—	8	2	no	—
237	C₂²×C₈₉	—	8	2	no	—
238	C₂²×C₈₉	—	8	1	no	—
239	C₂²×C₈₉	—	8	2	no	—
240	C₂²×C₈₉	—	8	1	no	—
241	C₂²×C₈₉	—	8	1	no	—
242	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
243	C₂²×C₈₉	#546	1	1	yes	characteristic
244	C₂²×C₈₉	#548	1	1	yes	characteristic
245	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
246	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂²×C₈₉	#577	1	1	yes	characteristic
248	C₂²×C₈₉	#577	1	1	yes	characteristic
249	C₄×C₈₉	—	2	4	no	—
250	C₄×C₈₉	—	2	2	no	—
251	C₄×C₈₉	—	4	2	no	—
252	C₄×C₈₉	—	4	2	no	—
253	C₄×C₈₉	—	8	1	no	—
254	C₄×C₈₉	—	8	1	no	—
255	C₄×C₈₉	—	8	2	no	—
256	C₄×C₈₉	#459	1	4	yes	—
257	C₄×C₈₉	#529	1	1	yes	characteristic
258	C₄×C₈₉	#542	1	1	yes	characteristic
259	C₄×C₈₉	#546	1	1	yes	characteristic
260	C₄×C₈₉	#547	1	1	yes	characteristic
261	#25318	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂×C₈₉	—	8	2	no	—
263	C₂×C₈₉	—	8	4	no	—
264	C₂×C₈₉	#2427	1	1	yes	characteristic
265	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
266	C₂×C₈₉	#3042	1	1	yes	characteristic
267	C₂×C₈₉	#3294	1	1	yes	characteristic
268	C₂×C₈₉	#3332	1	1	yes	characteristic
269	C₂×C₈₉	#3358	1	1	yes	characteristic
270	C₂×C₈₉	#3373	1	1	yes	characteristic
271	C₂×#486	C₂×C₈₉	1	1	yes	characteristic, complemented
272	C₂×#459	C₂×C₈₉	1	1	yes	characteristic
273	#2542	C₂×C₈₉	1	1	yes	characteristic, complemented
274	#2366	C₂×C₈₉	1	1	yes	characteristic, complemented
275	#2071	C₂×C₈₉	1	2	yes	complemented
276	#2054	C₂×C₈₉	1	2	yes	complemented
277	#1988	C₂×C₈₉	1	1	yes	characteristic, complemented
278	#1939	C₂×C₈₉	1	1	yes	characteristic, complemented
279	#1879	C₂×C₈₉	1	2	yes	complemented
280	#1802	C₂×C₈₉	1	2	yes	complemented
281	C₄×#102	C₂×C₈₉	1	1	yes	characteristic, complemented
282	C₈₉	#25318	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
284	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
285	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
286	C₂×C₂²⋊Q₈	—	2	1	no	—
287	C₂×C₂²⋊Q₈	—	2	1	no	—
288	C₂×C₄×D₄	—	2	1	no	—
289	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
291	#486	—	2	4	no	—
292	#459	—	2	4	no	—
293	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
295	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
296	C₂×#102	C₂²×C₈₉	1	2	yes	—
297	C₂×#102	C₂²×C₈₉	1	2	yes	—
298	C₂×#102	C₂²×C₈₉	1	2	yes	—
299	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
300	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
301	#399	—	2	2	no	—
302	#390	—	2	1	no	—
303	#390	C₂²×C₈₉	1	2	yes	—
304	#390	C₂²×C₈₉	1	2	yes	—
305	#386	—	2	2	no	—
306	#384	C₂²×C₈₉	1	2	yes	—
307	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
308	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
309	#340	C₂²×C₈₉	1	2	yes	—
310	#340	C₂²×C₈₉	1	1	yes	characteristic
311	#340	C₂²×C₈₉	1	2	yes	—
312	#340	C₂²×C₈₉	1	1	yes	characteristic
313	#336	C₂²×C₈₉	1	2	yes	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
318	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂³×C₄	—	2	2	no	—
321	C₂×D₈	—	2	4	no	—
322	C₂×D₈	—	2	4	no	—
323	C₂×D₈	—	4	2	no	—
324	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
325	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	#127	—	2	2	no	—
330	#127	D₄×C₈₉	1	4	yes	—
331	C₂²⋊Q₈	—	4	4	no	—
332	C₂²⋊Q₈	—	4	4	no	—
333	C₄×D₄	—	2	4	no	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	—	4	2	no	—
337	C₄×D₄	—	4	4	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
342	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
355	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
356	C₄⋊C₈	—	2	2	no	—
357	C₄⋊C₈	—	4	2	no	—
358	#102	—	2	4	no	—
359	#102	—	2	4	no	—
360	#102	—	2	4	no	—
361	#102	—	2	2	no	—
362	#102	—	2	2	no	—
363	C₄×C₈	—	2	2	no	—
364	#95	—	2	1	no	—
365	#95	D₄×C₈₉	1	2	yes	—
366	C₂⁴	—	2	1	no	—
367	C₂⁴	—	2	2	no	—
368	C₂×Q₈	—	4	2	no	—
369	C₂×Q₈	—	4	2	no	—
370	C₂×Q₈	—	4	2	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	8	1	no	—
386	C₂×D₄	—	8	1	no	—
387	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
388	C₂×D₄	#41×C₈₉	1	2	yes	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	4	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	8	2	no	—
399	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
402	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
404	D₈	—	4	8	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	4	4	no	—
423	C₄⋊C₄	—	4	2	no	—
424	C₄⋊C₄	—	4	4	no	—
425	C₄⋊C₄	D₈×C₈₉	1	4	yes	complemented
426	C₄⋊C₄	#41×C₈₉	1	2	yes	—
427	C₂²⋊C₄	—	4	4	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	4	2	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	—	8	2	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	4	2	no	—
436	C₄²	C₂×D₄×C₈₉	1	2	yes	—
437	C₄²	#41×C₈₉	1	2	yes	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	4	1	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	1	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	1	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	1	no	—
451	C₂³	—	4	1	no	—
452	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
453	Q₈	—	8	4	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	4	no	—
457	D₄	—	2	4	no	—
458	D₄	—	4	8	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	1	no	—
463	D₄	—	8	4	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	8	8	no	—
484	C₂×C₄	—	8	4	no	—
485	C₂×C₄	#121×C₈₉	1	2	yes	—
486	C₂×C₄	#121×C₈₉	1	2	yes	—
487	C₂×C₄	C₂×D₈×C₈₉	1	4	yes	—
488	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
489	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
490	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
491	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
494	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
495	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
496	C₈	—	4	1	no	—
497	C₈	—	4	2	no	—
498	C₈	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	1	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	—	8	1	no	—
510	C₂²	—	8	2	no	—
511	C₂²	—	8	1	no	—
512	C₂²	—	8	1	no	—
513	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
514	C₂²	#546×C₈₉	1	1	yes	characteristic
515	C₂²	#548×C₈₉	1	1	yes	characteristic
516	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
517	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
518	C₂²	#577×C₈₉	1	1	yes	characteristic
519	C₂²	#577×C₈₉	1	1	yes	characteristic
520	C₄	—	2	4	no	—
521	C₄	—	2	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	8	1	no	—
525	C₄	—	8	1	no	—
526	C₄	—	8	2	no	—
527	C₄	#459×C₈₉	1	4	yes	—
528	C₄	#529×C₈₉	1	1	yes	characteristic
529	C₄	#542×C₈₉	1	1	yes	characteristic
530	C₄	#546×C₈₉	1	1	yes	characteristic
531	C₄	#547×C₈₉	1	1	yes	characteristic
532	C₂	—	8	2	no	—
533	C₂	—	8	4	no	—
534	C₂	C₈₉×#2427	1	1	yes	characteristic
535	C₂	C₂×#459×C₈₉	1	1	yes	characteristic
536	C₂	C₈₉×#3042	1	1	yes	characteristic
537	C₂	C₈₉×#3294	1	1	yes	characteristic
538	C₂	C₈₉×#3332	1	1	yes	characteristic
539	C₂	C₈₉×#3358	1	1	yes	characteristic
540	C₂	C₈₉×#3373	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#459×C₈₉	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄×C₄⋊C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#486×C₈₉	C₂	semidirect
3. C₈₉×#2542	C₂	semidirect
4. C₈₉×#2366	C₂	semidirect
5. C₈₉×#2071	C₂	semidirect
6. C₈₉×#2054	C₂	semidirect
7. C₈₉×#1988	C₂	semidirect
8. C₈₉×#1939	C₂	semidirect
9. C₈₉×#1879	C₂	semidirect
10. C₈₉×#1802	C₂	semidirect
11. C₄×#102×C₈₉	C₂	semidirect
154. C₄⋊C₄×C₈₉	D₈	semidirect
261. #25318	C₈₉	direct (N × H)
271. C₂×#486	C₂×C₈₉	semidirect
273. #2542	C₂×C₈₉	semidirect
274. #2366	C₂×C₈₉	semidirect
275. #2071	C₂×C₈₉	semidirect
276. #2054	C₂×C₈₉	semidirect
277. #1988	C₂×C₈₉	semidirect
278. #1939	C₂×C₈₉	semidirect
279. #1879	C₂×C₈₉	semidirect
280. #1802	C₂×C₈₉	semidirect
281. C₄×#102	C₂×C₈₉	semidirect
282. C₈₉	#25318	direct (N × H)
425. C₄⋊C₄	D₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₈₉	2
C₂×#459×C₈₉	2
C₈₉×#2542	2
C₈₉×#2366	2
C₈₉×#2071	2
C₈₉×#2054	2
C₈₉×#1988	2
C₈₉×#1939	2
C₈₉×#1879	2
C₈₉×#1802	2
C₄×#102×C₈₉	2
#25318	89

Subgroups — C89×#25318 #36403166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25318 #36403166