Subgroups — C₈₉×#35239 #36413087

Order 22784.

1766 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2857	C₂	1	2	yes	maximal
2	C₈₉×#2851	C₂	1	2	yes	maximal
3	C₈₉×#2666	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2571	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2570	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2552	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2552	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2551	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2550	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2442	C₂	1	2	yes	maximal
11	C₈₉×#2440	C₂	1	2	yes	maximal
12	C₈₉×#2438	C₂	1	4	yes	maximal
13	C₈₉×#2331	C₂	1	2	yes	maximal, complemented
14	C₈₉×#2330	C₂	1	2	yes	maximal, complemented
15	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic
16	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic
17	C₂×C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	C₂×#125×C₈₉	C₂²	1	2	yes	—
21	C₂×#125×C₈₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
23	C₂×#117×C₈₉	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
27	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
28	#400×C₈₉	C₂²	1	4	yes	—
29	#400×C₈₉	C₂²	1	4	yes	—
30	#398×C₈₉	C₂²	1	4	yes	—
31	#398×C₈₉	C₂²	1	4	yes	—
32	#398×C₈₉	C₂²	1	4	yes	—
33	#398×C₈₉	C₂²	1	4	yes	—
34	#395×C₈₉	C₂²	1	4	yes	—
35	#395×C₈₉	C₂²	1	4	yes	—
36	#391×C₈₉	C₂²	1	4	yes	—
37	#391×C₈₉	C₂²	1	4	yes	—
38	#389×C₈₉	C₂²	1	2	yes	—
39	#389×C₈₉	C₂²	1	2	yes	—
40	#389×C₈₉	C₂²	1	4	yes	—
41	#389×C₈₉	C₂²	1	2	yes	—
42	#389×C₈₉	C₂²	1	2	yes	—
43	#389×C₈₉	C₂²	1	2	yes	—
44	#389×C₈₉	C₂²	1	2	yes	—
45	#389×C₈₉	C₂²	1	2	yes	—
46	#389×C₈₉	C₂²	1	2	yes	—
47	#389×C₈₉	C₂²	1	4	yes	—
48	#387×C₈₉	C₂²	1	4	yes	—
49	#387×C₈₉	C₂²	1	4	yes	—
50	#387×C₈₉	C₂²	1	4	yes	—
51	#387×C₈₉	C₂²	1	4	yes	—
52	#385×C₈₉	C₂²	1	4	yes	—
53	#385×C₈₉	C₂²	1	4	yes	—
54	#384×C₈₉	C₂²	1	1	yes	characteristic
55	#384×C₈₉	C₂²	1	2	yes	—
56	#384×C₈₉	C₂²	1	1	yes	characteristic
57	#384×C₈₉	C₂²	1	2	yes	—
58	#383×C₈₉	C₂²	1	2	yes	—
59	#382×C₈₉	C₂²	1	1	yes	characteristic
60	#382×C₈₉	C₂²	1	1	yes	characteristic
61	#382×C₈₉	C₂²	1	2	yes	—
62	#382×C₈₉	C₂²	1	2	yes	—
63	#382×C₈₉	C₂²	1	2	yes	—
64	#381×C₈₉	C₂²	1	2	yes	—
65	#380×C₈₉	C₂²	1	1	yes	characteristic
66	#380×C₈₉	C₂²	1	2	yes	—
67	#380×C₈₉	C₂²	1	1	yes	characteristic
68	#380×C₈₉	C₂²	1	4	yes	—
69	#380×C₈₉	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
73	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
74	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
75	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
79	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
81	#376×C₈₉	C₂²	1	2	yes	—
82	C₂×#95×C₈₉	C₂²	1	2	yes	—
83	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
84	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
85	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
86	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
89	C₄⋊Q₈×C₈₉	—	2	8	no	—
90	C₄⋊Q₈×C₈₉	—	4	4	no	—
91	#125×C₈₉	—	2	8	no	—
92	#125×C₈₉	—	4	4	no	—
93	C₂²⋊Q₈×C₈₉	—	4	8	no	—
94	#117×C₈₉	—	4	4	no	—
95	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
119	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
120	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
122	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
125	C₂×C₄²×C₈₉	—	4	4	no	—
126	C₂×C₄²×C₈₉	Q₈	1	8	yes	complemented
127	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
128	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
129	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
130	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
131	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
132	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
133	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
134	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
138	#95×C₈₉	C₂³	1	2	yes	—
139	#95×C₈₉	C₂³	1	2	yes	—
140	#95×C₈₉	C₂³	1	2	yes	—
141	#95×C₈₉	C₂³	1	2	yes	—
142	#95×C₈₉	C₂³	1	2	yes	—
143	#95×C₈₉	C₂³	1	2	yes	—
144	#95×C₈₉	C₂³	1	2	yes	—
145	#95×C₈₉	C₂³	1	2	yes	—
146	#95×C₈₉	C₂³	1	2	yes	—
147	#95×C₈₉	C₂³	1	2	yes	—
148	#95×C₈₉	C₂³	1	4	yes	—
149	#95×C₈₉	C₂³	1	4	yes	—
150	#95×C₈₉	C₂³	1	2	yes	—
151	#95×C₈₉	C₂³	1	2	yes	—
152	#95×C₈₉	C₂³	1	4	yes	—
153	#95×C₈₉	C₂³	1	4	yes	—
154	#95×C₈₉	C₂³	1	4	yes	—
155	#95×C₈₉	C₂³	1	4	yes	—
156	#95×C₈₉	C₂³	1	4	yes	—
157	#95×C₈₉	C₂³	1	4	yes	—
158	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
159	C₂×Q₈×C₈₉	—	2	8	no	—
160	C₂×Q₈×C₈₉	—	4	4	no	—
161	C₂×Q₈×C₈₉	—	4	4	no	—
162	C₂²×C₄×C₈₉	—	2	2	no	—
163	C₂²×C₄×C₈₉	—	2	2	no	—
164	C₂²×C₄×C₈₉	—	2	2	no	—
165	C₂²×C₄×C₈₉	—	2	2	no	—
166	C₂²×C₄×C₈₉	—	2	2	no	—
167	C₂²×C₄×C₈₉	—	2	2	no	—
168	C₂²×C₄×C₈₉	—	2	4	no	—
169	C₂²×C₄×C₈₉	—	2	4	no	—
170	C₂²×C₄×C₈₉	—	2	4	no	—
171	C₂²×C₄×C₈₉	—	2	4	no	—
172	C₂²×C₄×C₈₉	—	4	2	no	—
173	C₂²×C₄×C₈₉	—	4	2	no	—
174	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
175	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
176	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
177	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
193	C₄⋊C₄×C₈₉	—	2	4	no	—
194	C₄⋊C₄×C₈₉	—	2	4	no	—
195	C₄⋊C₄×C₈₉	—	2	2	no	—
196	C₄⋊C₄×C₈₉	—	2	2	no	—
197	C₄⋊C₄×C₈₉	—	2	8	no	—
198	C₄⋊C₄×C₈₉	—	2	8	no	—
199	C₄⋊C₄×C₈₉	—	2	4	no	—
200	C₄⋊C₄×C₈₉	—	2	8	no	—
201	C₄⋊C₄×C₈₉	—	2	4	no	—
202	C₄⋊C₄×C₈₉	—	2	2	no	—
203	C₄⋊C₄×C₈₉	—	2	4	no	—
204	C₄⋊C₄×C₈₉	—	2	2	no	—
205	C₄⋊C₄×C₈₉	—	2	2	no	—
206	C₄⋊C₄×C₈₉	—	2	4	no	—
207	C₄⋊C₄×C₈₉	—	2	2	no	—
208	C₄⋊C₄×C₈₉	—	4	2	no	—
209	C₄⋊C₄×C₈₉	—	4	2	no	—
210	C₄⋊C₄×C₈₉	—	4	4	no	—
211	C₄⋊C₄×C₈₉	—	4	4	no	—
212	C₄⋊C₄×C₈₉	—	4	4	no	—
213	C₄⋊C₄×C₈₉	—	4	4	no	—
214	C₄⋊C₄×C₈₉	—	4	4	no	—
215	C₄⋊C₄×C₈₉	—	4	4	no	—
216	C₄⋊C₄×C₈₉	#41	1	4	yes	—
217	C₄⋊C₄×C₈₉	#41	1	4	yes	—
218	C₂²⋊C₄×C₈₉	—	2	8	no	—
219	C₂²⋊C₄×C₈₉	—	2	4	no	—
220	C₂²⋊C₄×C₈₉	—	2	4	no	—
221	C₄²×C₈₉	—	2	4	no	—
222	C₄²×C₈₉	—	2	4	no	—
223	C₄²×C₈₉	—	2	4	no	—
224	C₄²×C₈₉	—	2	4	no	—
225	C₄²×C₈₉	—	4	2	no	—
226	C₄²×C₈₉	—	4	2	no	—
227	C₄²×C₈₉	—	4	4	no	—
228	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
229	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
230	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
231	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
232	C₂³×C₈₉	—	2	1	no	—
233	C₂³×C₈₉	—	2	1	no	—
234	C₂³×C₈₉	—	2	1	no	—
235	C₂³×C₈₉	—	2	1	no	—
236	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
237	C₂³×C₈₉	H(5,𝔽₂)	1	2	yes	—
238	C₂³×C₈₉	2₋¹⁺⁴	1	2	yes	—
239	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
240	Q₈×C₈₉	—	8	8	no	—
241	C₂×C₄×C₈₉	—	2	8	no	—
242	C₂×C₄×C₈₉	—	2	1	no	—
243	C₂×C₄×C₈₉	—	2	1	no	—
244	C₂×C₄×C₈₉	—	2	2	no	—
245	C₂×C₄×C₈₉	—	2	2	no	—
246	C₂×C₄×C₈₉	—	2	1	no	—
247	C₂×C₄×C₈₉	—	2	1	no	—
248	C₂×C₄×C₈₉	—	2	1	no	—
249	C₂×C₄×C₈₉	—	2	2	no	—
250	C₂×C₄×C₈₉	—	2	1	no	—
251	C₂×C₄×C₈₉	—	2	1	no	—
252	C₂×C₄×C₈₉	—	2	1	no	—
253	C₂×C₄×C₈₉	—	2	2	no	—
254	C₂×C₄×C₈₉	—	2	8	no	—
255	C₂×C₄×C₈₉	—	2	8	no	—
256	C₂×C₄×C₈₉	—	2	2	no	—
257	C₂×C₄×C₈₉	—	2	2	no	—
258	C₂×C₄×C₈₉	—	2	2	no	—
259	C₂×C₄×C₈₉	—	2	2	no	—
260	C₂×C₄×C₈₉	—	2	4	no	—
261	C₂×C₄×C₈₉	—	2	2	no	—
262	C₂×C₄×C₈₉	—	2	2	no	—
263	C₂×C₄×C₈₉	—	2	4	no	—
264	C₂×C₄×C₈₉	—	2	2	no	—
265	C₂×C₄×C₈₉	—	2	2	no	—
266	C₂×C₄×C₈₉	—	2	4	no	—
267	C₂×C₄×C₈₉	—	2	4	no	—
268	C₂×C₄×C₈₉	—	2	2	no	—
269	C₂×C₄×C₈₉	—	2	2	no	—
270	C₂×C₄×C₈₉	—	2	2	no	—
271	C₂×C₄×C₈₉	—	2	4	no	—
272	C₂×C₄×C₈₉	—	2	2	no	—
273	C₂×C₄×C₈₉	—	2	2	no	—
274	C₂×C₄×C₈₉	—	2	2	no	—
275	C₂×C₄×C₈₉	—	2	2	no	—
276	C₂×C₄×C₈₉	—	2	2	no	—
277	C₂×C₄×C₈₉	—	2	4	no	—
278	C₂×C₄×C₈₉	—	2	2	no	—
279	C₂×C₄×C₈₉	—	2	2	no	—
280	C₂×C₄×C₈₉	—	4	8	no	—
281	C₂×C₄×C₈₉	—	4	8	no	—
282	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
283	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
284	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
285	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
286	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
287	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
288	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
289	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
290	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
291	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
292	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
293	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
294	C₂²×C₈₉	—	2	2	no	—
295	C₂²×C₈₉	—	2	2	no	—
296	C₂²×C₈₉	—	2	2	no	—
297	C₂²×C₈₉	—	2	2	no	—
298	C₂²×C₈₉	—	2	2	no	—
299	C₂²×C₈₉	—	2	2	no	—
300	C₂²×C₈₉	#544	1	4	yes	—
301	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
302	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
303	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
304	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
306	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
308	C₄×C₈₉	—	2	2	no	—
309	C₄×C₈₉	—	2	2	no	—
310	C₄×C₈₉	—	2	2	no	—
311	C₄×C₈₉	—	2	2	no	—
312	C₄×C₈₉	—	2	4	no	—
313	C₄×C₈₉	—	2	4	no	—
314	C₄×C₈₉	—	4	2	no	—
315	C₄×C₈₉	—	4	4	no	—
316	C₄×C₈₉	—	4	2	no	—
317	C₄×C₈₉	—	4	2	no	—
318	C₄×C₈₉	—	4	2	no	—
319	C₄×C₈₉	—	4	2	no	—
320	C₄×C₈₉	—	4	4	no	—
321	C₄×C₈₉	—	4	2	no	—
322	C₄×C₈₉	#533	1	4	yes	—
323	C₄×C₈₉	#544	1	4	yes	—
324	#35239	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
325	C₂×C₈₉	—	2	4	no	—
326	C₂×C₈₉	C₂×#533	1	1	yes	characteristic
327	C₂×C₈₉	C₂×#544	1	1	yes	characteristic
328	C₂×C₈₉	C₂×#544	1	1	yes	characteristic
329	C₂×C₈₉	#3519	1	1	yes	characteristic
330	C₂×C₈₉	#3520	1	1	yes	characteristic
331	C₂×C₈₉	#3522	1	1	yes	characteristic
332	C₂×C₈₉	#3523	1	1	yes	characteristic
333	#2857	C₂×C₈₉	1	2	yes	—
334	#2851	C₂×C₈₉	1	2	yes	—
335	#2666	C₂×C₈₉	1	4	yes	complemented
336	#2571	C₂×C₈₉	1	1	yes	characteristic, complemented
337	#2570	C₂×C₈₉	1	2	yes	complemented
338	#2552	C₂×C₈₉	1	1	yes	characteristic, complemented
339	#2552	C₂×C₈₉	1	1	yes	characteristic, complemented
340	#2551	C₂×C₈₉	1	1	yes	characteristic, complemented
341	#2550	C₂×C₈₉	1	2	yes	complemented
342	#2442	C₂×C₈₉	1	2	yes	—
343	#2440	C₂×C₈₉	1	2	yes	—
344	#2438	C₂×C₈₉	1	4	yes	—
345	#2331	C₂×C₈₉	1	2	yes	complemented
346	#2330	C₂×C₈₉	1	2	yes	complemented
347	#2285	C₂×C₈₉	1	1	yes	characteristic
348	#2283	C₂×C₈₉	1	1	yes	characteristic
349	C₂×C₄×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic
350	C₈₉	#35239	1	1	yes	characteristic, direct product factor, Sylow
351	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
352	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
353	C₂×#125	C₂²×C₈₉	1	2	yes	—
354	C₂×#125	C₂²×C₈₉	1	2	yes	—
355	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
356	C₂×#117	C₂²×C₈₉	1	2	yes	—
357	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
358	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
359	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
360	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
361	#400	C₂²×C₈₉	1	4	yes	—
362	#400	C₂²×C₈₉	1	4	yes	—
363	#398	C₂²×C₈₉	1	4	yes	—
364	#398	C₂²×C₈₉	1	4	yes	—
365	#398	C₂²×C₈₉	1	4	yes	—
366	#398	C₂²×C₈₉	1	4	yes	—
367	#395	C₂²×C₈₉	1	4	yes	—
368	#395	C₂²×C₈₉	1	4	yes	—
369	#391	C₂²×C₈₉	1	4	yes	—
370	#391	C₂²×C₈₉	1	4	yes	—
371	#389	C₂²×C₈₉	1	2	yes	—
372	#389	C₂²×C₈₉	1	2	yes	—
373	#389	C₂²×C₈₉	1	2	yes	—
374	#389	C₂²×C₈₉	1	2	yes	—
375	#389	C₂²×C₈₉	1	4	yes	—
376	#389	C₂²×C₈₉	1	4	yes	—
377	#389	C₂²×C₈₉	1	2	yes	—
378	#389	C₂²×C₈₉	1	2	yes	—
379	#389	C₂²×C₈₉	1	2	yes	—
380	#389	C₂²×C₈₉	1	2	yes	—
381	#387	C₂²×C₈₉	1	4	yes	—
382	#387	C₂²×C₈₉	1	4	yes	—
383	#387	C₂²×C₈₉	1	4	yes	—
384	#387	C₂²×C₈₉	1	4	yes	—
385	#385	C₂²×C₈₉	1	4	yes	—
386	#385	C₂²×C₈₉	1	4	yes	—
387	#384	C₂²×C₈₉	1	2	yes	—
388	#384	C₂²×C₈₉	1	1	yes	characteristic
389	#384	C₂²×C₈₉	1	2	yes	—
390	#384	C₂²×C₈₉	1	1	yes	characteristic
391	#383	C₂²×C₈₉	1	2	yes	—
392	#382	C₂²×C₈₉	1	1	yes	characteristic
393	#382	C₂²×C₈₉	1	2	yes	—
394	#382	C₂²×C₈₉	1	2	yes	—
395	#382	C₂²×C₈₉	1	2	yes	—
396	#382	C₂²×C₈₉	1	1	yes	characteristic
397	#381	C₂²×C₈₉	1	2	yes	—
398	#380	C₂²×C₈₉	1	1	yes	characteristic
399	#380	C₂²×C₈₉	1	2	yes	—
400	#380	C₂²×C₈₉	1	1	yes	characteristic
401	#380	C₂²×C₈₉	1	4	yes	—
402	#380	C₂²×C₈₉	1	2	yes	—
403	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
404	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
405	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
406	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
407	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
408	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
409	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
410	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
411	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
412	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
413	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
414	#376	C₂²×C₈₉	1	2	yes	—
415	C₂×#95	C₂²×C₈₉	1	2	yes	—
416	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
417	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
418	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
419	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
420	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
421	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
422	C₄⋊Q₈	—	2	8	no	—
423	C₄⋊Q₈	—	4	4	no	—
424	#125	—	2	8	no	—
425	#125	—	4	4	no	—
426	C₂²⋊Q₈	—	4	8	no	—
427	#117	—	4	4	no	—
428	C₂×C₄⋊C₄	—	2	4	no	—
429	C₂×C₄⋊C₄	—	2	4	no	—
430	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
438	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
439	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
441	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
442	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
446	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
448	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
449	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
450	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
451	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
452	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
453	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
454	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
455	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
457	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
458	C₂×C₄²	—	4	4	no	—
459	C₂×C₄²	Q₈×C₈₉	1	8	yes	complemented
460	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
461	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
462	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
463	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
464	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
465	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
466	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
467	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
468	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
469	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
470	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
471	#95	C₂³×C₈₉	1	2	yes	—
472	#95	C₂³×C₈₉	1	2	yes	—
473	#95	C₂³×C₈₉	1	2	yes	—
474	#95	C₂³×C₈₉	1	2	yes	—
475	#95	C₂³×C₈₉	1	2	yes	—
476	#95	C₂³×C₈₉	1	2	yes	—
477	#95	C₂³×C₈₉	1	2	yes	—
478	#95	C₂³×C₈₉	1	2	yes	—
479	#95	C₂³×C₈₉	1	2	yes	—
480	#95	C₂³×C₈₉	1	2	yes	—
481	#95	C₂³×C₈₉	1	4	yes	—
482	#95	C₂³×C₈₉	1	4	yes	—
483	#95	C₂³×C₈₉	1	2	yes	—
484	#95	C₂³×C₈₉	1	2	yes	—
485	#95	C₂³×C₈₉	1	4	yes	—
486	#95	C₂³×C₈₉	1	4	yes	—
487	#95	C₂³×C₈₉	1	4	yes	—
488	#95	C₂³×C₈₉	1	4	yes	—
489	#95	C₂³×C₈₉	1	4	yes	—
490	#95	C₂³×C₈₉	1	4	yes	—
491	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
492	C₂×Q₈	—	2	8	no	—
493	C₂×Q₈	—	4	4	no	—
494	C₂×Q₈	—	4	4	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
508	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
509	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
510	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
511	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
514	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
515	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
517	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
518	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
520	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
521	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
522	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
525	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	8	no	—
531	C₄⋊C₄	—	2	8	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	8	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	2	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	4	2	no	—
542	C₄⋊C₄	—	4	2	no	—
543	C₄⋊C₄	—	4	4	no	—
544	C₄⋊C₄	—	4	4	no	—
545	C₄⋊C₄	—	4	4	no	—
546	C₄⋊C₄	—	4	4	no	—
547	C₄⋊C₄	—	4	4	no	—
548	C₄⋊C₄	—	4	4	no	—
549	C₄⋊C₄	#41×C₈₉	1	4	yes	—
550	C₄⋊C₄	#41×C₈₉	1	4	yes	—
551	C₂²⋊C₄	—	2	8	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₄²	—	2	4	no	—
555	C₄²	—	2	4	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	4	no	—
558	C₄²	—	4	2	no	—
559	C₄²	—	4	2	no	—
560	C₄²	—	4	4	no	—
561	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
562	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
563	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
564	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
570	C₂³	H(5,𝔽₂)×C₈₉	1	2	yes	—
571	C₂³	2₋¹⁺⁴×C₈₉	1	2	yes	—
572	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
573	Q₈	—	8	8	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	8	no	—
588	C₂×C₄	—	2	8	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	4	8	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
616	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
617	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
618	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
619	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
620	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
621	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
622	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
623	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
624	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
625	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
626	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	2	no	—
633	C₂²	#544×C₈₉	1	4	yes	—
634	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
635	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
636	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
637	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
638	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
639	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
640	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
641	C₄	—	2	2	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	2	2	no	—
645	C₄	—	2	4	no	—
646	C₄	—	2	4	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	4	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	4	no	—
654	C₄	—	4	2	no	—
655	C₄	#533×C₈₉	1	4	yes	—
656	C₄	#544×C₈₉	1	4	yes	—
657	C₂	—	2	4	no	—
658	C₂	C₂×#533×C₈₉	1	1	yes	characteristic
659	C₂	C₂×#544×C₈₉	1	1	yes	characteristic
660	C₂	C₂×#544×C₈₉	1	1	yes	characteristic
661	C₂	C₈₉×#3519	1	1	yes	characteristic
662	C₂	C₈₉×#3520	1	1	yes	characteristic
663	C₂	C₈₉×#3522	1	1	yes	characteristic
664	C₂	C₈₉×#3523	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁴×C₈₉	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#2666	C₂	semidirect
4. C₈₉×#2571	C₂	semidirect
5. C₈₉×#2570	C₂	semidirect
6. C₈₉×#2552	C₂	semidirect
7. C₈₉×#2552	C₂	semidirect
8. C₈₉×#2551	C₂	semidirect
9. C₈₉×#2550	C₂	semidirect
13. C₈₉×#2331	C₂	semidirect
14. C₈₉×#2330	C₂	semidirect
126. C₂×C₄²×C₈₉	Q₈	semidirect
324. #35239	C₈₉	direct (N × H)
335. #2666	C₂×C₈₉	semidirect
336. #2571	C₂×C₈₉	semidirect
337. #2570	C₂×C₈₉	semidirect
338. #2552	C₂×C₈₉	semidirect
339. #2552	C₂×C₈₉	semidirect
340. #2551	C₂×C₈₉	semidirect
341. #2550	C₂×C₈₉	semidirect
345. #2331	C₂×C₈₉	semidirect
346. #2330	C₂×C₈₉	semidirect
350. C₈₉	#35239	direct (N × H)
459. C₂×C₄²	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×#2857	2
C₈₉×#2851	2
C₈₉×#2666	2
C₈₉×#2571	2
C₈₉×#2570	2
C₈₉×#2552	2
C₈₉×#2551	2
C₈₉×#2550	2
C₈₉×#2442	2
C₈₉×#2440	2
C₈₉×#2438	2
C₈₉×#2331	2
C₈₉×#2330	2
C₈₉×#2285	2
C₈₉×#2283	2
C₂×C₄×C₄⋊C₄×C₈₉	2
#35239	89

Subgroups — C89×#35239 #36413087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35239 #36413087