Subgroups — C₈₉×#35233 #36413081

Order 22784.

2902 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2849	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2636	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2598	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2566	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2544	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2534	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2503	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#2501	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2435	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#2433	C₂	1	1	yes	maximal, characteristic
13	C₈₉×#2328	C₂	1	2	yes	maximal, complemented
14	C₈₉×#2326	C₂	1	2	yes	maximal, complemented
15	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
18	C₂×#127×C₈₉	C₂²	1	2	yes	—
19	#529×C₈₉	—	2	8	no	—
20	C₂×#126×C₈₉	C₂²	1	2	yes	complemented
21	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic, complemented
22	C₂×#124×C₈₉	C₂²	1	2	yes	—
23	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic, complemented
24	C₂×#123×C₈₉	C₂²	1	2	yes	—
25	C₂×#123×C₈₉	C₂²	1	4	yes	—
26	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#120×C₈₉	C₂²	1	4	yes	—
29	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₄×D₄×C₈₉	—	2	8	no	—
31	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	C₂×#117×C₈₉	C₂²	1	2	yes	complemented
38	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
40	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
41	#400×C₈₉	C₂²	1	4	yes	complemented
42	#399×C₈₉	C₂²	1	4	yes	complemented
43	#396×C₈₉	C₂²	1	8	yes	complemented
44	#394×C₈₉	C₂²	1	4	yes	—
45	#394×C₈₉	C₂²	1	8	yes	complemented
46	#392×C₈₉	C₂²	1	4	yes	—
47	#391×C₈₉	C₂²	1	2	yes	complemented
48	#390×C₈₉	C₂²	1	2	yes	—
49	#390×C₈₉	C₂²	1	2	yes	—
50	#390×C₈₉	C₂²	1	2	yes	complemented
51	#388×C₈₉	C₂²	1	8	yes	complemented
52	#386×C₈₉	C₂²	1	8	yes	complemented
53	#386×C₈₉	C₂²	1	8	yes	complemented
54	#386×C₈₉	C₂²	1	8	yes	complemented
55	#385×C₈₉	C₂²	1	8	yes	complemented
56	#385×C₈₉	C₂²	1	8	yes	complemented
57	#384×C₈₉	C₂²	1	2	yes	complemented
58	#383×C₈₉	C₂²	1	2	yes	complemented
59	#382×C₈₉	C₂²	1	2	yes	complemented
60	#382×C₈₉	C₂²	1	2	yes	complemented
61	#381×C₈₉	C₂²	1	2	yes	—
62	#381×C₈₉	C₂²	1	4	yes	—
63	#380×C₈₉	C₂²	1	2	yes	—
64	#379×C₈₉	C₂²	1	4	yes	—
65	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
66	C₄×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
67	C₄×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
68	#376×C₈₉	C₂²	1	2	yes	complemented
69	C₂⁵×C₈₉	C₂³	1	2	yes	—
70	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₈₉	—	2	8	no	—
72	C₂²×D₄×C₈₉	—	2	4	no	—
73	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
75	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
76	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
77	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
78	C₂³×C₄×C₈₉	—	2	8	no	—
79	C₂³×C₄×C₈₉	—	4	8	no	—
80	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
81	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
84	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
85	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
86	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
87	#127×C₈₉	—	2	8	no	—
88	#126×C₈₉	—	2	8	no	—
89	#125×C₈₉	—	2	4	no	—
90	#124×C₈₉	—	2	8	no	—
91	#124×C₈₉	—	2	4	no	—
92	#123×C₈₉	—	2	8	no	—
93	#123×C₈₉	—	4	8	no	—
94	C₂²⋊Q₈×C₈₉	—	2	4	no	—
95	#121×C₈₉	—	2	4	no	—
96	#120×C₈₉	—	4	8	no	—
97	C₄×Q₈×C₈₉	—	2	4	no	—
98	C₄×D₄×C₈₉	—	2	8	no	—
99	C₄×D₄×C₈₉	—	2	8	no	—
100	C₄×D₄×C₈₉	—	2	8	no	—
101	C₄×D₄×C₈₉	—	2	8	no	—
102	C₄×D₄×C₈₉	—	2	8	no	—
103	C₄×D₄×C₈₉	—	2	4	no	—
104	#117×C₈₉	—	2	8	no	—
105	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
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	1	yes	characteristic
109	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
116	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
117	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
124	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
127	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
128	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
129	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
130	C₂×C₄²×C₈₉	D₄	1	8	yes	complemented
131	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
132	C₂×C₄²×C₈₉	C₂³	1	2	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	4	yes	—
139	#95×C₈₉	C₂³	1	4	yes	—
140	#95×C₈₉	C₂³	1	2	yes	—
141	#95×C₈₉	C₂³	1	2	yes	—
142	#95×C₈₉	C₂³	1	4	yes	—
143	#95×C₈₉	C₂³	1	8	yes	—
144	#95×C₈₉	C₂³	1	8	yes	—
145	C₂⁴×C₈₉	—	2	4	no	—
146	C₂⁴×C₈₉	—	2	8	no	—
147	C₂⁴×C₈₉	—	2	8	no	—
148	C₂⁴×C₈₉	—	4	4	no	—
149	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
150	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
151	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
152	C₂×Q₈×C₈₉	—	2	2	no	—
153	C₂×Q₈×C₈₉	—	2	4	no	—
154	C₂×D₄×C₈₉	—	2	32	no	—
155	C₂×D₄×C₈₉	—	2	8	no	—
156	C₂×D₄×C₈₉	—	2	4	no	—
157	C₂×D₄×C₈₉	—	2	8	no	—
158	C₂×D₄×C₈₉	—	2	16	no	—
159	C₂×D₄×C₈₉	—	2	4	no	—
160	C₂×D₄×C₈₉	—	2	4	no	—
161	C₂×D₄×C₈₉	—	2	2	no	—
162	C₂×D₄×C₈₉	—	2	4	no	—
163	C₂×D₄×C₈₉	—	2	4	no	—
164	C₂×D₄×C₈₉	—	4	8	no	—
165	C₂×D₄×C₈₉	—	4	8	no	—
166	C₂×D₄×C₈₉	#41	1	8	yes	—
167	C₂²×C₄×C₈₉	—	2	4	no	—
168	C₂²×C₄×C₈₉	—	2	4	no	—
169	C₂²×C₄×C₈₉	—	2	4	no	—
170	C₂²×C₄×C₈₉	—	2	8	no	—
171	C₂²×C₄×C₈₉	—	2	4	no	—
172	C₂²×C₄×C₈₉	—	2	8	no	—
173	C₂²×C₄×C₈₉	—	2	8	no	—
174	C₂²×C₄×C₈₉	—	4	8	no	—
175	C₂²×C₄×C₈₉	—	4	8	no	—
176	C₂²×C₄×C₈₉	—	4	2	no	—
177	C₂²×C₄×C₈₉	—	4	4	no	—
178	C₂²×C₄×C₈₉	—	4	32	no	—
179	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
194	C₄⋊C₄×C₈₉	—	2	4	no	—
195	C₄⋊C₄×C₈₉	—	2	4	no	—
196	C₄⋊C₄×C₈₉	—	2	4	no	—
197	C₄⋊C₄×C₈₉	—	2	4	no	—
198	C₄⋊C₄×C₈₉	—	2	2	no	—
199	C₄⋊C₄×C₈₉	—	2	4	no	—
200	C₄⋊C₄×C₈₉	—	2	2	no	—
201	C₄⋊C₄×C₈₉	—	2	4	no	—
202	C₄⋊C₄×C₈₉	—	4	4	no	—
203	C₄⋊C₄×C₈₉	—	4	8	no	—
204	C₂²⋊C₄×C₈₉	—	2	4	no	—
205	C₂²⋊C₄×C₈₉	—	2	16	no	—
206	C₂²⋊C₄×C₈₉	—	2	4	no	—
207	C₂²⋊C₄×C₈₉	—	2	4	no	—
208	C₂²⋊C₄×C₈₉	—	2	8	no	—
209	C₂²⋊C₄×C₈₉	—	2	8	no	—
210	C₂²⋊C₄×C₈₉	—	2	4	no	—
211	C₂²⋊C₄×C₈₉	—	2	4	no	—
212	C₂²⋊C₄×C₈₉	—	2	4	no	—
213	C₂²⋊C₄×C₈₉	—	2	4	no	—
214	C₂²⋊C₄×C₈₉	—	4	8	no	—
215	C₂²⋊C₄×C₈₉	—	4	8	no	—
216	C₂²⋊C₄×C₈₉	—	4	8	no	—
217	C₄²×C₈₉	—	2	4	no	—
218	C₄²×C₈₉	—	2	4	no	—
219	C₄²×C₈₉	—	2	4	no	—
220	C₄²×C₈₉	C₂×D₄	1	4	yes	—
221	C₄²×C₈₉	C₂×D₄	1	4	yes	—
222	C₄²×C₈₉	C₂×D₄	1	4	yes	—
223	C₄²×C₈₉	C₂×D₄	1	4	yes	—
224	C₂³×C₈₉	—	2	2	no	—
225	C₂³×C₈₉	—	2	2	no	—
226	C₂³×C₈₉	—	2	16	no	—
227	C₂³×C₈₉	—	2	4	no	—
228	C₂³×C₈₉	—	2	4	no	—
229	C₂³×C₈₉	—	2	8	no	—
230	C₂³×C₈₉	—	2	8	no	—
231	C₂³×C₈₉	—	2	2	no	—
232	C₂³×C₈₉	—	2	4	no	—
233	C₂³×C₈₉	—	2	4	no	—
234	C₂³×C₈₉	—	2	4	no	—
235	C₂³×C₈₉	—	4	32	no	—
236	C₂³×C₈₉	—	4	8	no	—
237	C₂³×C₈₉	—	4	8	no	—
238	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
239	C₂³×C₈₉	C₂×#41	1	8	yes	—
240	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
241	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
242	Q₈×C₈₉	—	4	4	no	—
243	D₄×C₈₉	—	2	16	no	—
244	D₄×C₈₉	—	2	16	no	—
245	D₄×C₈₉	—	4	8	no	—
246	D₄×C₈₉	—	4	4	no	—
247	D₄×C₈₉	—	8	16	no	—
248	C₂×C₄×C₈₉	—	2	8	no	—
249	C₂×C₄×C₈₉	—	2	2	no	—
250	C₂×C₄×C₈₉	—	2	2	no	—
251	C₂×C₄×C₈₉	—	2	2	no	—
252	C₂×C₄×C₈₉	—	2	2	no	—
253	C₂×C₄×C₈₉	—	2	2	no	—
254	C₂×C₄×C₈₉	—	2	8	no	—
255	C₂×C₄×C₈₉	—	2	2	no	—
256	C₂×C₄×C₈₉	—	2	2	no	—
257	C₂×C₄×C₈₉	—	2	2	no	—
258	C₂×C₄×C₈₉	—	2	4	no	—
259	C₂×C₄×C₈₉	—	2	4	no	—
260	C₂×C₄×C₈₉	—	2	4	no	—
261	C₂×C₄×C₈₉	—	2	4	no	—
262	C₂×C₄×C₈₉	—	2	4	no	—
263	C₂×C₄×C₈₉	—	2	4	no	—
264	C₂×C₄×C₈₉	—	2	4	no	—
265	C₂×C₄×C₈₉	—	2	8	no	—
266	C₂×C₄×C₈₉	—	2	4	no	—
267	C₂×C₄×C₈₉	—	2	4	no	—
268	C₂×C₄×C₈₉	—	2	4	no	—
269	C₂×C₄×C₈₉	—	4	4	no	—
270	C₂×C₄×C₈₉	—	4	8	no	—
271	C₂×C₄×C₈₉	—	4	16	no	—
272	C₂×C₄×C₈₉	—	4	16	no	—
273	C₂×C₄×C₈₉	—	4	8	no	—
274	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
281	C₂²×C₈₉	—	2	2	no	—
282	C₂²×C₈₉	—	2	2	no	—
283	C₂²×C₈₉	—	2	8	no	—
284	C₂²×C₈₉	—	2	8	no	—
285	C₂²×C₈₉	—	2	16	no	—
286	C₂²×C₈₉	—	4	32	no	—
287	C₂²×C₈₉	—	4	4	no	—
288	C₂²×C₈₉	—	4	4	no	—
289	C₂²×C₈₉	—	4	8	no	—
290	C₂²×C₈₉	—	4	2	no	—
291	C₂²×C₈₉	—	4	2	no	—
292	C₂²×C₈₉	—	4	2	no	—
293	C₂²×C₈₉	—	4	8	no	—
294	C₂²×C₈₉	#535	1	4	yes	—
295	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
296	C₂²×C₈₉	C₂²×#41	1	2	yes	—
297	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
298	C₂²×C₈₉	#585	1	2	yes	—
299	C₄×C₈₉	—	2	2	no	—
300	C₄×C₈₉	—	2	2	no	—
301	C₄×C₈₉	—	2	2	no	—
302	C₄×C₈₉	—	2	2	no	—
303	C₄×C₈₉	—	2	2	no	—
304	C₄×C₈₉	—	4	2	no	—
305	C₄×C₈₉	—	4	4	no	—
306	C₄×C₈₉	—	4	2	no	—
307	C₄×C₈₉	—	4	2	no	—
308	C₄×C₈₉	—	4	4	no	—
309	C₄×C₈₉	—	4	4	no	—
310	C₄×C₈₉	#525	1	4	yes	—
311	#35233	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
312	C₂×C₈₉	—	2	4	no	—
313	C₂×C₈₉	—	4	8	no	—
314	C₂×C₈₉	—	8	2	no	—
315	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
316	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
317	C₂×C₈₉	#3449	1	1	yes	characteristic
318	C₂×C₈₉	#3513	1	4	yes	—
319	C₂×#529	C₂×C₈₉	1	1	yes	characteristic, complemented
320	C₂²×C₄×D₄	C₂×C₈₉	1	1	yes	characteristic, complemented
321	#2849	C₂×C₈₉	1	4	yes	complemented
322	#2636	C₂×C₈₉	1	4	yes	complemented
323	#2598	C₂×C₈₉	1	4	yes	complemented
324	#2566	C₂×C₈₉	1	2	yes	complemented
325	#2544	C₂×C₈₉	1	2	yes	complemented
326	#2534	C₂×C₈₉	1	4	yes	complemented
327	#2503	C₂×C₈₉	1	1	yes	characteristic, complemented
328	#2501	C₂×C₈₉	1	1	yes	characteristic, complemented
329	#2435	C₂×C₈₉	1	1	yes	characteristic, complemented
330	#2433	C₂×C₈₉	1	1	yes	characteristic
331	#2328	C₂×C₈₉	1	2	yes	complemented
332	#2326	C₂×C₈₉	1	2	yes	complemented
333	#2283	C₂×C₈₉	1	1	yes	characteristic, complemented
334	C₈₉	#35233	1	1	yes	characteristic, direct product factor, Sylow
335	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
336	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
337	C₂×#127	C₂²×C₈₉	1	2	yes	—
338	#529	—	2	8	no	—
339	C₂×#126	C₂²×C₈₉	1	2	yes	complemented
340	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic, complemented
341	C₂×#124	C₂²×C₈₉	1	2	yes	—
342	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic, complemented
343	C₂×#123	C₂²×C₈₉	1	2	yes	—
344	C₂×#123	C₂²×C₈₉	1	4	yes	—
345	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
346	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
347	C₂×#120	C₂²×C₈₉	1	4	yes	—
348	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
349	C₂×C₄×D₄	—	2	8	no	—
350	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
351	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
352	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
353	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
354	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
355	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
356	C₂×#117	C₂²×C₈₉	1	2	yes	complemented
357	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
358	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
359	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
360	#400	C₂²×C₈₉	1	4	yes	complemented
361	#399	C₂²×C₈₉	1	4	yes	complemented
362	#396	C₂²×C₈₉	1	8	yes	complemented
363	#394	C₂²×C₈₉	1	4	yes	—
364	#394	C₂²×C₈₉	1	8	yes	complemented
365	#392	C₂²×C₈₉	1	4	yes	—
366	#391	C₂²×C₈₉	1	2	yes	complemented
367	#390	C₂²×C₈₉	1	2	yes	—
368	#390	C₂²×C₈₉	1	2	yes	—
369	#390	C₂²×C₈₉	1	2	yes	complemented
370	#388	C₂²×C₈₉	1	8	yes	complemented
371	#386	C₂²×C₈₉	1	8	yes	complemented
372	#386	C₂²×C₈₉	1	8	yes	complemented
373	#386	C₂²×C₈₉	1	8	yes	complemented
374	#385	C₂²×C₈₉	1	8	yes	complemented
375	#385	C₂²×C₈₉	1	8	yes	complemented
376	#384	C₂²×C₈₉	1	2	yes	complemented
377	#383	C₂²×C₈₉	1	2	yes	complemented
378	#382	C₂²×C₈₉	1	2	yes	complemented
379	#382	C₂²×C₈₉	1	2	yes	complemented
380	#381	C₂²×C₈₉	1	2	yes	—
381	#381	C₂²×C₈₉	1	4	yes	—
382	#380	C₂²×C₈₉	1	2	yes	—
383	#379	C₂²×C₈₉	1	4	yes	—
384	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
385	C₄×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
386	C₄×C₂²⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
387	#376	C₂²×C₈₉	1	2	yes	complemented
388	C₂⁵	C₂³×C₈₉	1	2	yes	—
389	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
390	C₂²×D₄	—	2	8	no	—
391	C₂²×D₄	—	2	4	no	—
392	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
393	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
394	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
395	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
396	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
397	C₂³×C₄	—	2	8	no	—
398	C₂³×C₄	—	4	8	no	—
399	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
400	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
401	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
402	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
403	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
404	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
405	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
406	#127	—	2	8	no	—
407	#126	—	2	8	no	—
408	#125	—	2	4	no	—
409	#124	—	2	8	no	—
410	#124	—	2	4	no	—
411	#123	—	2	8	no	—
412	#123	—	4	8	no	—
413	C₂²⋊Q₈	—	2	4	no	—
414	#121	—	2	4	no	—
415	#120	—	4	8	no	—
416	C₄×Q₈	—	2	4	no	—
417	C₄×D₄	—	2	8	no	—
418	C₄×D₄	—	2	8	no	—
419	C₄×D₄	—	2	8	no	—
420	C₄×D₄	—	2	8	no	—
421	C₄×D₄	—	2	8	no	—
422	C₄×D₄	—	2	4	no	—
423	#117	—	2	8	no	—
424	C₂×C₄⋊C₄	—	2	4	no	—
425	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
427	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
428	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
429	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
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	8	yes	—
434	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
435	C₂×C₂²⋊C₄	—	2	8	no	—
436	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
438	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
439	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
440	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
441	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
442	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
449	C₂×C₄²	D₄×C₈₉	1	8	yes	complemented
450	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
451	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
452	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
453	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
456	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
457	#95	C₂³×C₈₉	1	4	yes	—
458	#95	C₂³×C₈₉	1	4	yes	—
459	#95	C₂³×C₈₉	1	2	yes	—
460	#95	C₂³×C₈₉	1	2	yes	—
461	#95	C₂³×C₈₉	1	4	yes	—
462	#95	C₂³×C₈₉	1	8	yes	—
463	#95	C₂³×C₈₉	1	8	yes	—
464	C₂⁴	—	2	4	no	—
465	C₂⁴	—	2	8	no	—
466	C₂⁴	—	2	8	no	—
467	C₂⁴	—	4	4	no	—
468	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
469	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
470	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×D₄	—	2	32	no	—
474	C₂×D₄	—	2	8	no	—
475	C₂×D₄	—	2	4	no	—
476	C₂×D₄	—	2	8	no	—
477	C₂×D₄	—	2	16	no	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	4	8	no	—
484	C₂×D₄	—	4	8	no	—
485	C₂×D₄	#41×C₈₉	1	8	yes	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	4	no	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	2	8	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	8	no	—
492	C₂²×C₄	—	2	8	no	—
493	C₂²×C₄	—	4	8	no	—
494	C₂²×C₄	—	4	8	no	—
495	C₂²×C₄	—	4	2	no	—
496	C₂²×C₄	—	4	4	no	—
497	C₂²×C₄	—	4	32	no	—
498	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
499	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
500	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
501	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
502	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
503	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
505	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
506	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
508	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
509	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
510	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
512	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	4	no	—
516	C₄⋊C₄	—	2	4	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	4	8	no	—
522	C₄⋊C₄	—	4	4	no	—
523	C₂²⋊C₄	—	2	4	no	—
524	C₂²⋊C₄	—	2	16	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	8	no	—
528	C₂²⋊C₄	—	2	8	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	4	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	4	8	no	—
534	C₂²⋊C₄	—	4	8	no	—
535	C₂²⋊C₄	—	4	8	no	—
536	C₄²	—	2	4	no	—
537	C₄²	—	2	4	no	—
538	C₄²	—	2	4	no	—
539	C₄²	C₂×D₄×C₈₉	1	4	yes	—
540	C₄²	C₂×D₄×C₈₉	1	4	yes	—
541	C₄²	C₂×D₄×C₈₉	1	4	yes	—
542	C₄²	C₂×D₄×C₈₉	1	4	yes	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	16	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	2	4	no	—
548	C₂³	—	2	8	no	—
549	C₂³	—	2	8	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	4	no	—
552	C₂³	—	2	4	no	—
553	C₂³	—	2	4	no	—
554	C₂³	—	4	32	no	—
555	C₂³	—	4	8	no	—
556	C₂³	—	4	8	no	—
557	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
558	C₂³	C₂×#41×C₈₉	1	8	yes	—
559	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	—
560	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
561	Q₈	—	4	4	no	—
562	D₄	—	2	16	no	—
563	D₄	—	2	16	no	—
564	D₄	—	4	8	no	—
565	D₄	—	4	4	no	—
566	D₄	—	8	16	no	—
567	C₂×C₄	—	2	8	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	8	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	8	no	—
590	C₂×C₄	—	4	16	no	—
591	C₂×C₄	—	4	16	no	—
592	C₂×C₄	—	4	8	no	—
593	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
596	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
597	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
598	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
599	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	8	no	—
603	C₂²	—	2	8	no	—
604	C₂²	—	2	16	no	—
605	C₂²	—	4	32	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	8	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	4	8	no	—
613	C₂²	#535×C₈₉	1	4	yes	—
614	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
615	C₂²	C₂²×#41×C₈₉	1	2	yes	—
616	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
617	C₂²	#585×C₈₉	1	2	yes	—
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	4	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	4	no	—
628	C₄	—	4	4	no	—
629	C₄	#525×C₈₉	1	4	yes	—
630	C₂	—	2	4	no	—
631	C₂	—	4	8	no	—
632	C₂	—	8	2	no	—
633	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
634	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
635	C₂	C₈₉×#3449	1	1	yes	characteristic
636	C₂	C₈₉×#3513	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₈₉×#2433	C₂
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
1. C₂×#529×C₈₉	C₂	semidirect
2. C₂²×C₄×D₄×C₈₉	C₂	semidirect
3. C₈₉×#2849	C₂	semidirect
4. C₈₉×#2636	C₂	semidirect
5. C₈₉×#2598	C₂	semidirect
6. C₈₉×#2566	C₂	semidirect
7. C₈₉×#2544	C₂	semidirect
8. C₈₉×#2534	C₂	semidirect
9. C₈₉×#2503	C₂	semidirect
10. C₈₉×#2501	C₂	semidirect
11. C₈₉×#2435	C₂	semidirect
13. C₈₉×#2328	C₂	semidirect
14. C₈₉×#2326	C₂	semidirect
15. C₈₉×#2283	C₂	semidirect
20. C₂×#126×C₈₉	C₂²	semidirect
21. C₂×#125×C₈₉	C₂²	semidirect
23. C₂×#124×C₈₉	C₂²	semidirect
29. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
36. C₂×C₄×D₄×C₈₉	C₂²	semidirect
37. C₂×#117×C₈₉	C₂²	semidirect
41. #400×C₈₉	C₂²	semidirect
42. #399×C₈₉	C₂²	semidirect
43. #396×C₈₉	C₂²	semidirect
45. #394×C₈₉	C₂²	semidirect
47. #391×C₈₉	C₂²	semidirect
50. #390×C₈₉	C₂²	semidirect
51. #388×C₈₉	C₂²	semidirect
52. #386×C₈₉	C₂²	semidirect
53. #386×C₈₉	C₂²	semidirect
54. #386×C₈₉	C₂²	semidirect
55. #385×C₈₉	C₂²	semidirect
56. #385×C₈₉	C₂²	semidirect
57. #384×C₈₉	C₂²	semidirect
58. #383×C₈₉	C₂²	semidirect
59. #382×C₈₉	C₂²	semidirect
60. #382×C₈₉	C₂²	semidirect
65. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
68. #376×C₈₉	C₂²	semidirect
130. C₂×C₄²×C₈₉	D₄	semidirect
311. #35233	C₈₉	direct (N × H)
319. C₂×#529	C₂×C₈₉	semidirect
320. C₂²×C₄×D₄	C₂×C₈₉	semidirect
321. #2849	C₂×C₈₉	semidirect
322. #2636	C₂×C₈₉	semidirect
323. #2598	C₂×C₈₉	semidirect
324. #2566	C₂×C₈₉	semidirect
325. #2544	C₂×C₈₉	semidirect
326. #2534	C₂×C₈₉	semidirect
327. #2503	C₂×C₈₉	semidirect
328. #2501	C₂×C₈₉	semidirect
329. #2435	C₂×C₈₉	semidirect
331. #2328	C₂×C₈₉	semidirect
332. #2326	C₂×C₈₉	semidirect
333. #2283	C₂×C₈₉	semidirect
334. C₈₉	#35233	direct (N × H)
339. C₂×#126	C₂²×C₈₉	semidirect
340. C₂×#125	C₂²×C₈₉	semidirect
342. C₂×#124	C₂²×C₈₉	semidirect
348. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
355. C₂×C₄×D₄	C₂²×C₈₉	semidirect
356. C₂×#117	C₂²×C₈₉	semidirect
360. #400	C₂²×C₈₉	semidirect
361. #399	C₂²×C₈₉	semidirect
362. #396	C₂²×C₈₉	semidirect
364. #394	C₂²×C₈₉	semidirect
366. #391	C₂²×C₈₉	semidirect
369. #390	C₂²×C₈₉	semidirect
370. #388	C₂²×C₈₉	semidirect
371. #386	C₂²×C₈₉	semidirect
372. #386	C₂²×C₈₉	semidirect
373. #386	C₂²×C₈₉	semidirect
374. #385	C₂²×C₈₉	semidirect
375. #385	C₂²×C₈₉	semidirect
376. #384	C₂²×C₈₉	semidirect
377. #383	C₂²×C₈₉	semidirect
378. #382	C₂²×C₈₉	semidirect
379. #382	C₂²×C₈₉	semidirect
384. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
387. #376	C₂²×C₈₉	semidirect
449. C₂×C₄²	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₈₉	2
C₂²×C₄×D₄×C₈₉	2
C₈₉×#2849	2
C₈₉×#2636	2
C₈₉×#2598	2
C₈₉×#2566	2
C₈₉×#2544	2
C₈₉×#2534	2
C₈₉×#2503	2
C₈₉×#2501	2
C₈₉×#2435	2
C₈₉×#2433	2
C₈₉×#2328	2
C₈₉×#2326	2
C₈₉×#2283	2
#35233	89

Subgroups — C89×#35233 #36413081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35233 #36413081