Subgroups — C₈₉×#35227 #36413075

Order 22784.

2898 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#525×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2849	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2623	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2504	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2503	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2502	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2501	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2487	C₂	1	4	yes	maximal, complemented
10	C₈₉×#2486	C₂	1	4	yes	maximal, complemented
11	C₈₉×#2317	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2311	C₂	1	2	yes	maximal, complemented
13	C₈₉×#2280	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
17	#525×C₈₉	—	2	8	no	—
18	C₂×#123×C₈₉	C₂²	1	2	yes	—
19	C₂×#123×C₈₉	C₂²	1	8	yes	—
20	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#120×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
27	C₂×#117×C₈₉	C₂²	1	2	yes	—
28	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄⋊C₄×C₈₉	—	2	8	no	—
30	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
36	#400×C₈₉	C₂²	1	8	yes	complemented
37	#399×C₈₉	C₂²	1	8	yes	complemented
38	#398×C₈₉	C₂²	1	8	yes	complemented
39	#397×C₈₉	C₂²	1	8	yes	complemented
40	#391×C₈₉	C₂²	1	4	yes	complemented
41	#390×C₈₉	C₂²	1	4	yes	complemented
42	#389×C₈₉	C₂²	1	8	yes	complemented
43	#386×C₈₉	C₂²	1	8	yes	—
44	#386×C₈₉	C₂²	1	8	yes	—
45	#385×C₈₉	C₂²	1	8	yes	—
46	#385×C₈₉	C₂²	1	8	yes	—
47	#384×C₈₉	C₂²	1	4	yes	complemented
48	#382×C₈₉	C₂²	1	4	yes	complemented
49	#381×C₈₉	C₂²	1	8	yes	—
50	#381×C₈₉	C₂²	1	2	yes	—
51	#380×C₈₉	C₂²	1	2	yes	—
52	#380×C₈₉	C₂²	1	2	yes	—
53	#380×C₈₉	C₂²	1	8	yes	complemented
54	#379×C₈₉	C₂²	1	2	yes	—
55	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
56	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
57	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×#41×C₈₉	—	2	4	no	—
59	C₂×#41×C₈₉	—	2	8	no	—
60	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
62	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
63	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
65	C₂³×C₄×C₈₉	—	2	8	no	—
66	C₂³×C₄×C₈₉	—	2	8	no	—
67	C₂³×C₄×C₈₉	—	4	8	no	—
68	C₂³×C₄×C₈₉	D₄	1	8	yes	—
69	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
70	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
77	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
78	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
79	#123×C₈₉	—	2	8	no	—
80	#123×C₈₉	—	4	16	no	—
81	C₂²⋊Q₈×C₈₉	—	2	4	no	—
82	C₂²⋊Q₈×C₈₉	—	2	4	no	—
83	#121×C₈₉	—	2	4	no	—
84	#121×C₈₉	—	2	4	no	—
85	#120×C₈₉	—	2	8	no	—
86	C₄×D₄×C₈₉	—	2	8	no	—
87	C₄×D₄×C₈₉	—	2	8	no	—
88	#117×C₈₉	—	2	4	no	—
89	#117×C₈₉	—	2	8	no	—
90	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
91	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
92	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
93	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
94	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
95	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
115	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
116	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
117	#95×C₈₉	C₂³	1	4	yes	—
118	#95×C₈₉	C₂³	1	4	yes	—
119	#95×C₈₉	C₂³	1	16	yes	—
120	#95×C₈₉	C₂³	1	16	yes	—
121	C₂⁴×C₈₉	—	2	4	no	—
122	C₂⁴×C₈₉	—	2	4	no	—
123	C₂⁴×C₈₉	—	2	4	no	—
124	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
125	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
126	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
129	#41×C₈₉	—	4	16	no	—
130	C₂×Q₈×C₈₉	—	2	2	no	—
131	C₂×Q₈×C₈₉	—	2	4	no	—
132	C₂×D₄×C₈₉	—	2	4	no	—
133	C₂×D₄×C₈₉	—	2	4	no	—
134	C₂×D₄×C₈₉	—	2	4	no	—
135	C₂×D₄×C₈₉	—	2	8	no	—
136	C₂×D₄×C₈₉	—	2	8	no	—
137	C₂×D₄×C₈₉	—	2	8	no	—
138	C₂×D₄×C₈₉	—	2	2	no	—
139	C₂×D₄×C₈₉	—	2	4	no	—
140	C₂²×C₄×C₈₉	—	2	4	no	—
141	C₂²×C₄×C₈₉	—	2	32	no	—
142	C₂²×C₄×C₈₉	—	2	4	no	—
143	C₂²×C₄×C₈₉	—	2	4	no	—
144	C₂²×C₄×C₈₉	—	2	4	no	—
145	C₂²×C₄×C₈₉	—	2	4	no	—
146	C₂²×C₄×C₈₉	—	2	8	no	—
147	C₂²×C₄×C₈₉	—	2	8	no	—
148	C₂²×C₄×C₈₉	—	2	8	no	—
149	C₂²×C₄×C₈₉	—	2	16	no	—
150	C₂²×C₄×C₈₉	—	2	4	no	—
151	C₂²×C₄×C₈₉	—	2	8	no	—
152	C₂²×C₄×C₈₉	—	4	8	no	—
153	C₂²×C₄×C₈₉	—	4	8	no	—
154	C₂²×C₄×C₈₉	—	4	32	no	—
155	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₈₉	#41	1	8	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₄⋊C₄×C₈₉	—	2	8	no	—
172	C₄⋊C₄×C₈₉	—	2	8	no	—
173	C₄⋊C₄×C₈₉	—	2	8	no	—
174	C₄⋊C₄×C₈₉	—	2	8	no	—
175	C₄⋊C₄×C₈₉	—	2	2	no	—
176	C₄⋊C₄×C₈₉	—	2	2	no	—
177	C₄⋊C₄×C₈₉	—	2	4	no	—
178	C₄⋊C₄×C₈₉	—	2	4	no	—
179	C₄⋊C₄×C₈₉	—	2	8	no	—
180	C₄⋊C₄×C₈₉	—	4	16	no	—
181	C₂²⋊C₄×C₈₉	—	2	4	no	—
182	C₂²⋊C₄×C₈₉	—	2	4	no	—
183	C₂²⋊C₄×C₈₉	—	2	4	no	—
184	C₂²⋊C₄×C₈₉	—	2	8	no	—
185	C₂²⋊C₄×C₈₉	—	2	8	no	—
186	C₂²⋊C₄×C₈₉	—	2	8	no	—
187	C₂²⋊C₄×C₈₉	—	2	4	no	—
188	C₂²⋊C₄×C₈₉	—	2	4	no	—
189	C₂²⋊C₄×C₈₉	—	4	16	no	—
190	C₄²×C₈₉	—	2	4	no	—
191	C₄²×C₈₉	—	2	8	no	—
192	C₂³×C₈₉	—	2	32	no	—
193	C₂³×C₈₉	—	2	8	no	—
194	C₂³×C₈₉	—	2	8	no	—
195	C₂³×C₈₉	—	2	8	no	—
196	C₂³×C₈₉	—	2	2	no	—
197	C₂³×C₈₉	—	2	2	no	—
198	C₂³×C₈₉	—	2	2	no	—
199	C₂³×C₈₉	—	2	2	no	—
200	C₂³×C₈₉	—	2	2	no	—
201	C₂³×C₈₉	—	2	2	no	—
202	C₂³×C₈₉	—	2	2	no	—
203	C₂³×C₈₉	—	2	4	no	—
204	C₂³×C₈₉	—	2	4	no	—
205	C₂³×C₈₉	—	2	4	no	—
206	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
207	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
208	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
209	C₂³×C₈₉	C₂×#41	1	4	yes	—
210	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
211	C₂³×C₈₉	2₋¹⁺⁴	1	4	yes	—
212	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
213	Q₈×C₈₉	—	4	4	no	—
214	D₄×C₈₉	—	4	8	no	—
215	D₄×C₈₉	—	4	8	no	—
216	D₄×C₈₉	—	4	8	no	—
217	D₄×C₈₉	—	4	4	no	—
218	C₂×C₄×C₈₉	—	2	8	no	—
219	C₂×C₄×C₈₉	—	2	2	no	—
220	C₂×C₄×C₈₉	—	2	2	no	—
221	C₂×C₄×C₈₉	—	2	8	no	—
222	C₂×C₄×C₈₉	—	2	8	no	—
223	C₂×C₄×C₈₉	—	2	8	no	—
224	C₂×C₄×C₈₉	—	2	8	no	—
225	C₂×C₄×C₈₉	—	2	2	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	2	no	—
228	C₂×C₄×C₈₉	—	2	8	no	—
229	C₂×C₄×C₈₉	—	2	8	no	—
230	C₂×C₄×C₈₉	—	2	4	no	—
231	C₂×C₄×C₈₉	—	2	8	no	—
232	C₂×C₄×C₈₉	—	2	16	no	—
233	C₂×C₄×C₈₉	—	2	4	no	—
234	C₂×C₄×C₈₉	—	2	8	no	—
235	C₂×C₄×C₈₉	—	2	4	no	—
236	C₂×C₄×C₈₉	—	4	16	no	—
237	C₂×C₄×C₈₉	—	4	16	no	—
238	C₂×C₄×C₈₉	—	4	16	no	—
239	C₂×C₄×C₈₉	—	4	8	no	—
240	C₂×C₄×C₈₉	#123	1	16	yes	complemented
241	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
246	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
247	C₂²×C₈₉	—	2	32	no	—
248	C₂²×C₈₉	—	2	2	no	—
249	C₂²×C₈₉	—	2	2	no	—
250	C₂²×C₈₉	—	2	2	no	—
251	C₂²×C₈₉	—	2	2	no	—
252	C₂²×C₈₉	—	2	2	no	—
253	C₂²×C₈₉	—	2	2	no	—
254	C₂²×C₈₉	—	2	8	no	—
255	C₂²×C₈₉	—	2	8	no	—
256	C₂²×C₈₉	—	2	8	no	—
257	C₂²×C₈₉	—	4	2	no	—
258	C₂²×C₈₉	—	4	2	no	—
259	C₂²×C₈₉	—	4	2	no	—
260	C₂²×C₈₉	—	4	8	no	—
261	C₂²×C₈₉	C₂×#123	1	4	yes	—
262	C₂²×C₈₉	#535	1	4	yes	—
263	C₂²×C₈₉	#536	1	4	yes	—
264	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
265	C₂²×C₈₉	C₂²×#41	1	2	yes	—
266	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
267	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
268	C₄×C₈₉	—	2	2	no	—
269	C₄×C₈₉	—	2	2	no	—
270	C₄×C₈₉	—	4	2	no	—
271	C₄×C₈₉	—	4	8	no	—
272	C₄×C₈₉	—	4	2	no	—
273	C₄×C₈₉	—	4	2	no	—
274	C₄×C₈₉	—	4	8	no	—
275	C₄×C₈₉	C₂×#123	1	4	yes	—
276	C₄×C₈₉	C₂×#123	1	4	yes	—
277	#35227	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂×C₈₉	—	2	4	no	—
279	C₂×C₈₉	—	2	4	no	—
280	C₂×C₈₉	—	2	4	no	—
281	C₂×C₈₉	—	8	2	no	—
282	C₂×C₈₉	C₂²×#123	1	1	yes	characteristic
283	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
284	C₂×C₈₉	C₂×#536	1	1	yes	characteristic
285	C₂×C₈₉	#3492	1	4	yes	—
286	C₂×#525	C₂×C₈₉	1	1	yes	characteristic
287	C₂³×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic, complemented
288	#2849	C₂×C₈₉	1	4	yes	complemented
289	#2623	C₂×C₈₉	1	4	yes	complemented
290	#2504	C₂×C₈₉	1	2	yes	complemented
291	#2503	C₂×C₈₉	1	2	yes	complemented
292	#2502	C₂×C₈₉	1	2	yes	complemented
293	#2501	C₂×C₈₉	1	2	yes	complemented
294	#2487	C₂×C₈₉	1	4	yes	complemented
295	#2486	C₂×C₈₉	1	4	yes	complemented
296	#2317	C₂×C₈₉	1	2	yes	complemented
297	#2311	C₂×C₈₉	1	2	yes	complemented
298	#2280	C₂×C₈₉	1	1	yes	characteristic, complemented
299	C₈₉	#35227	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
301	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
302	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
303	#525	—	2	8	no	—
304	C₂×#123	C₂²×C₈₉	1	2	yes	—
305	C₂×#123	C₂²×C₈₉	1	8	yes	—
306	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
307	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
308	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
309	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
310	C₂×#120	C₂²×C₈₉	1	2	yes	—
311	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
312	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
313	C₂×#117	C₂²×C₈₉	1	2	yes	—
314	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
315	C₂²×C₄⋊C₄	—	2	8	no	—
316	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
317	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
318	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
319	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
320	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
321	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
322	#400	C₂²×C₈₉	1	8	yes	complemented
323	#399	C₂²×C₈₉	1	8	yes	complemented
324	#398	C₂²×C₈₉	1	8	yes	complemented
325	#397	C₂²×C₈₉	1	8	yes	complemented
326	#391	C₂²×C₈₉	1	4	yes	complemented
327	#390	C₂²×C₈₉	1	4	yes	complemented
328	#389	C₂²×C₈₉	1	8	yes	complemented
329	#386	C₂²×C₈₉	1	8	yes	—
330	#386	C₂²×C₈₉	1	8	yes	—
331	#385	C₂²×C₈₉	1	8	yes	—
332	#385	C₂²×C₈₉	1	8	yes	—
333	#384	C₂²×C₈₉	1	4	yes	complemented
334	#382	C₂²×C₈₉	1	4	yes	complemented
335	#381	C₂²×C₈₉	1	8	yes	—
336	#381	C₂²×C₈₉	1	2	yes	—
337	#380	C₂²×C₈₉	1	2	yes	—
338	#380	C₂²×C₈₉	1	2	yes	—
339	#380	C₂²×C₈₉	1	8	yes	complemented
340	#379	C₂²×C₈₉	1	2	yes	—
341	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
342	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
343	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
344	C₂×#41	—	2	4	no	—
345	C₂×#41	—	2	8	no	—
346	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
347	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
348	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
349	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
350	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
351	C₂³×C₄	—	2	8	no	—
352	C₂³×C₄	—	2	8	no	—
353	C₂³×C₄	—	4	8	no	—
354	C₂³×C₄	D₄×C₈₉	1	8	yes	—
355	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
356	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
357	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
362	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
363	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
364	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
365	#123	—	2	8	no	—
366	#123	—	4	16	no	—
367	C₂²⋊Q₈	—	2	4	no	—
368	C₂²⋊Q₈	—	2	4	no	—
369	#121	—	2	4	no	—
370	#121	—	2	4	no	—
371	#120	—	2	8	no	—
372	C₄×D₄	—	2	8	no	—
373	C₄×D₄	—	2	8	no	—
374	#117	—	2	4	no	—
375	#117	—	2	8	no	—
376	C₂×C₄⋊C₄	—	2	8	no	—
377	C₂×C₄⋊C₄	—	2	8	no	—
378	C₂×C₄⋊C₄	—	2	8	no	—
379	C₂×C₄⋊C₄	—	2	8	no	—
380	C₂×C₄⋊C₄	—	2	8	no	—
381	C₂×C₄⋊C₄	—	2	8	no	—
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
401	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
402	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
403	#95	C₂³×C₈₉	1	4	yes	—
404	#95	C₂³×C₈₉	1	4	yes	—
405	#95	C₂³×C₈₉	1	16	yes	—
406	#95	C₂³×C₈₉	1	16	yes	—
407	C₂⁴	—	2	4	no	—
408	C₂⁴	—	2	4	no	—
409	C₂⁴	—	2	4	no	—
410	C₂⁴	C₂×D₄×C₈₉	1	4	yes	—
411	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
412	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
413	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
414	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
415	#41	—	4	16	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	32	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	8	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	—	2	8	no	—
435	C₂²×C₄	—	2	16	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	8	no	—
438	C₂²×C₄	—	4	8	no	—
439	C₂²×C₄	—	4	8	no	—
440	C₂²×C₄	—	4	32	no	—
441	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
442	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
443	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
444	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
445	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
446	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
447	C₂²×C₄	#41×C₈₉	1	8	yes	—
448	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
449	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
451	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
452	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	—	2	8	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	4	16	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	4	16	no	—
476	C₄²	—	2	4	no	—
477	C₄²	—	2	8	no	—
478	C₂³	—	2	32	no	—
479	C₂³	—	2	8	no	—
480	C₂³	—	2	8	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	4	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	2	4	no	—
492	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
493	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
494	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
495	C₂³	C₂×#41×C₈₉	1	4	yes	—
496	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	—
497	C₂³	2₋¹⁺⁴×C₈₉	1	4	yes	—
498	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
499	Q₈	—	4	4	no	—
500	D₄	—	4	8	no	—
501	D₄	—	4	8	no	—
502	D₄	—	4	8	no	—
503	D₄	—	4	4	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	8	no	—
518	C₂×C₄	—	2	16	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	4	16	no	—
523	C₂×C₄	—	4	16	no	—
524	C₂×C₄	—	4	16	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	#123×C₈₉	1	16	yes	complemented
527	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
528	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
530	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
531	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
532	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
533	C₂²	—	2	32	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	8	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	2	8	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	4	8	no	—
547	C₂²	C₂×#123×C₈₉	1	4	yes	—
548	C₂²	#535×C₈₉	1	4	yes	—
549	C₂²	#536×C₈₉	1	4	yes	—
550	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
551	C₂²	C₂²×#41×C₈₉	1	2	yes	—
552	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
553	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	8	no	—
558	C₄	—	4	2	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	8	no	—
561	C₄	C₂×#123×C₈₉	1	4	yes	—
562	C₄	C₂×#123×C₈₉	1	4	yes	—
563	C₂	—	2	4	no	—
564	C₂	—	2	4	no	—
565	C₂	—	2	4	no	—
566	C₂	—	8	2	no	—
567	C₂	C₂²×#123×C₈₉	1	1	yes	characteristic
568	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
569	C₂	C₂×#536×C₈₉	1	1	yes	characteristic
570	C₂	C₈₉×#3492	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#525×C₈₉	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
2. C₂³×C₄⋊C₄×C₈₉	C₂	semidirect
3. C₈₉×#2849	C₂	semidirect
4. C₈₉×#2623	C₂	semidirect
5. C₈₉×#2504	C₂	semidirect
6. C₈₉×#2503	C₂	semidirect
7. C₈₉×#2502	C₂	semidirect
8. C₈₉×#2501	C₂	semidirect
9. C₈₉×#2487	C₂	semidirect
10. C₈₉×#2486	C₂	semidirect
11. C₈₉×#2317	C₂	semidirect
12. C₈₉×#2311	C₂	semidirect
13. C₈₉×#2280	C₂	semidirect
36. #400×C₈₉	C₂²	semidirect
37. #399×C₈₉	C₂²	semidirect
38. #398×C₈₉	C₂²	semidirect
39. #397×C₈₉	C₂²	semidirect
40. #391×C₈₉	C₂²	semidirect
41. #390×C₈₉	C₂²	semidirect
42. #389×C₈₉	C₂²	semidirect
47. #384×C₈₉	C₂²	semidirect
48. #382×C₈₉	C₂²	semidirect
53. #380×C₈₉	C₂²	semidirect
240. C₂×C₄×C₈₉	#123	semidirect
277. #35227	C₈₉	direct (N × H)
287. C₂³×C₄⋊C₄	C₂×C₈₉	semidirect
288. #2849	C₂×C₈₉	semidirect
289. #2623	C₂×C₈₉	semidirect
290. #2504	C₂×C₈₉	semidirect
291. #2503	C₂×C₈₉	semidirect
292. #2502	C₂×C₈₉	semidirect
293. #2501	C₂×C₈₉	semidirect
294. #2487	C₂×C₈₉	semidirect
295. #2486	C₂×C₈₉	semidirect
296. #2317	C₂×C₈₉	semidirect
297. #2311	C₂×C₈₉	semidirect
298. #2280	C₂×C₈₉	semidirect
299. C₈₉	#35227	direct (N × H)
322. #400	C₂²×C₈₉	semidirect
323. #399	C₂²×C₈₉	semidirect
324. #398	C₂²×C₈₉	semidirect
325. #397	C₂²×C₈₉	semidirect
326. #391	C₂²×C₈₉	semidirect
327. #390	C₂²×C₈₉	semidirect
328. #389	C₂²×C₈₉	semidirect
333. #384	C₂²×C₈₉	semidirect
334. #382	C₂²×C₈₉	semidirect
339. #380	C₂²×C₈₉	semidirect
526. C₂×C₄	#123×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#525×C₈₉	2
C₂³×C₄⋊C₄×C₈₉	2
C₈₉×#2849	2
C₈₉×#2623	2
C₈₉×#2504	2
C₈₉×#2503	2
C₈₉×#2502	2
C₈₉×#2501	2
C₈₉×#2487	2
C₈₉×#2486	2
C₈₉×#2317	2
C₈₉×#2311	2
C₈₉×#2280	2
#35227	89

Subgroups — C89×#35227 #36413075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35227 #36413075