Subgroups — C₈₉×#35247 #36413095

Order 22784.

2378 conjugacy classes of subgroups.

The subgroup lattice has 590 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
2	C₂²×C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2855	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2854	C₂	1	2	yes	maximal
5	C₈₉×#2637	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2592	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2584	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2582	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2538	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2486	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2413	C₂	1	4	yes	maximal, complemented
12	C₈₉×#2409	C₂	1	2	yes	maximal, complemented
13	C₈₉×#2323	C₂	1	4	yes	maximal, complemented
14	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	#529×C₈₉	—	2	8	no	—
17	C₂×#126×C₈₉	C₂²	1	2	yes	—
18	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#123×C₈₉	C₂²	1	2	yes	—
21	C₂×#123×C₈₉	C₂²	1	4	yes	—
22	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
23	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₈₉	—	2	8	no	—
26	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#117×C₈₉	C₂²	1	2	yes	—
34	C₂×#117×C₈₉	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
38	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
39	#398×C₈₉	C₂²	1	8	yes	—
40	#398×C₈₉	C₂²	1	4	yes	—
41	#397×C₈₉	C₂²	1	4	yes	—
42	#396×C₈₉	C₂²	1	8	yes	—
43	#395×C₈₉	C₂²	1	4	yes	—
44	#393×C₈₉	C₂²	1	4	yes	—
45	#391×C₈₉	C₂²	1	4	yes	—
46	#391×C₈₉	C₂²	1	4	yes	—
47	#390×C₈₉	C₂²	1	4	yes	—
48	#389×C₈₉	C₂²	1	4	yes	—
49	#389×C₈₉	C₂²	1	8	yes	—
50	#389×C₈₉	C₂²	1	4	yes	—
51	#389×C₈₉	C₂²	1	4	yes	—
52	#388×C₈₉	C₂²	1	8	yes	—
53	#388×C₈₉	C₂²	1	8	yes	—
54	#387×C₈₉	C₂²	1	8	yes	—
55	#386×C₈₉	C₂²	1	4	yes	—
56	#385×C₈₉	C₂²	1	4	yes	—
57	#384×C₈₉	C₂²	1	2	yes	—
58	#382×C₈₉	C₂²	1	2	yes	—
59	#381×C₈₉	C₂²	1	4	yes	—
60	#380×C₈₉	C₂²	1	4	yes	—
61	#380×C₈₉	C₂²	1	2	yes	—
62	#380×C₈₉	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
65	C₂²×Q₈×C₈₉	—	2	8	no	—
66	C₂²×Q₈×C₈₉	—	2	4	no	—
67	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
69	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
70	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
71	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
74	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
75	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
78	#126×C₈₉	—	2	8	no	—
79	#125×C₈₉	—	2	4	no	—
80	#124×C₈₉	—	2	4	no	—
81	#123×C₈₉	—	2	8	no	—
82	#123×C₈₉	—	4	8	no	—
83	C₂²⋊Q₈×C₈₉	—	2	16	no	—
84	C₂²⋊Q₈×C₈₉	—	2	4	no	—
85	#121×C₈₉	—	2	4	no	—
86	C₄×Q₈×C₈₉	—	2	8	no	—
87	C₄×Q₈×C₈₉	—	2	8	no	—
88	C₄×Q₈×C₈₉	—	2	8	no	—
89	C₄×Q₈×C₈₉	—	2	8	no	—
90	C₄×Q₈×C₈₉	—	2	4	no	—
91	C₄×D₄×C₈₉	—	2	8	no	—
92	C₄×D₄×C₈₉	—	2	4	no	—
93	#117×C₈₉	—	2	8	no	—
94	#117×C₈₉	—	4	4	no	—
95	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
96	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
112	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
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	8	yes	—
120	C₂×C₄²×C₈₉	—	2	8	no	—
121	C₂×C₄²×C₈₉	D₄	1	8	yes	—
122	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
123	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
124	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
125	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
126	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
130	#95×C₈₉	C₂³	1	4	yes	—
131	#95×C₈₉	C₂³	1	4	yes	—
132	#95×C₈₉	C₂³	1	8	yes	—
133	#95×C₈₉	C₂³	1	8	yes	—
134	#95×C₈₉	C₂³	1	8	yes	—
135	#95×C₈₉	C₂³	1	8	yes	—
136	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
137	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
138	C₂×Q₈×C₈₉	—	2	32	no	—
139	C₂×Q₈×C₈₉	—	2	4	no	—
140	C₂×Q₈×C₈₉	—	2	4	no	—
141	C₂×Q₈×C₈₉	—	2	2	no	—
142	C₂×Q₈×C₈₉	—	2	8	no	—
143	C₂×Q₈×C₈₉	—	2	4	no	—
144	C₂×Q₈×C₈₉	—	2	4	no	—
145	C₂×Q₈×C₈₉	#41	1	8	yes	—
146	C₂×D₄×C₈₉	—	2	4	no	—
147	C₂×D₄×C₈₉	—	2	4	no	—
148	C₂×D₄×C₈₉	—	2	4	no	—
149	C₂×D₄×C₈₉	—	2	2	no	—
150	C₂×D₄×C₈₉	—	2	4	no	—
151	C₂²×C₄×C₈₉	—	2	4	no	—
152	C₂²×C₄×C₈₉	—	2	2	no	—
153	C₂²×C₄×C₈₉	—	2	2	no	—
154	C₂²×C₄×C₈₉	—	2	8	no	—
155	C₂²×C₄×C₈₉	—	2	4	no	—
156	C₂²×C₄×C₈₉	—	2	4	no	—
157	C₂²×C₄×C₈₉	—	2	4	no	—
158	C₂²×C₄×C₈₉	—	2	4	no	—
159	C₂²×C₄×C₈₉	—	2	4	no	—
160	C₂²×C₄×C₈₉	—	2	4	no	—
161	C₂²×C₄×C₈₉	—	2	8	no	—
162	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
167	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₈₉	—	2	8	no	—
178	C₄⋊C₄×C₈₉	—	2	8	no	—
179	C₄⋊C₄×C₈₉	—	2	4	no	—
180	C₄⋊C₄×C₈₉	—	2	4	no	—
181	C₄⋊C₄×C₈₉	—	2	4	no	—
182	C₄⋊C₄×C₈₉	—	2	4	no	—
183	C₄⋊C₄×C₈₉	—	2	2	no	—
184	C₄⋊C₄×C₈₉	—	2	4	no	—
185	C₄⋊C₄×C₈₉	—	2	2	no	—
186	C₄⋊C₄×C₈₉	—	2	4	no	—
187	C₄⋊C₄×C₈₉	—	2	8	no	—
188	C₄⋊C₄×C₈₉	—	2	16	no	—
189	C₄⋊C₄×C₈₉	—	2	8	no	—
190	C₄⋊C₄×C₈₉	—	4	4	no	—
191	C₄⋊C₄×C₈₉	—	4	8	no	—
192	C₂²⋊C₄×C₈₉	—	2	4	no	—
193	C₂²⋊C₄×C₈₉	—	2	8	no	—
194	C₂²⋊C₄×C₈₉	—	2	4	no	—
195	C₂²⋊C₄×C₈₉	—	2	4	no	—
196	C₂²⋊C₄×C₈₉	—	2	4	no	—
197	C₂²⋊C₄×C₈₉	—	2	4	no	—
198	C₂²⋊C₄×C₈₉	—	2	16	no	—
199	C₂²⋊C₄×C₈₉	—	4	8	no	—
200	C₄²×C₈₉	—	2	8	no	—
201	C₄²×C₈₉	—	2	8	no	—
202	C₄²×C₈₉	—	2	4	no	—
203	C₄²×C₈₉	—	4	4	no	—
204	C₄²×C₈₉	C₂×D₄	1	4	yes	—
205	C₄²×C₈₉	C₂×D₄	1	4	yes	—
206	C₄²×C₈₉	C₂×D₄	1	4	yes	—
207	C₄²×C₈₉	C₂×D₄	1	4	yes	—
208	C₂³×C₈₉	—	2	2	no	—
209	C₂³×C₈₉	—	2	2	no	—
210	C₂³×C₈₉	—	2	2	no	—
211	C₂³×C₈₉	—	2	4	no	—
212	C₂³×C₈₉	—	2	2	no	—
213	C₂³×C₈₉	—	2	4	no	—
214	C₂³×C₈₉	—	2	2	no	—
215	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
216	C₂³×C₈₉	H(5,𝔽₂)	1	2	yes	—
217	C₂³×C₈₉	2₋¹⁺⁴	1	2	yes	—
218	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
219	Q₈×C₈₉	—	2	16	no	—
220	Q₈×C₈₉	—	2	16	no	—
221	Q₈×C₈₉	—	4	4	no	—
222	D₄×C₈₉	—	4	8	no	—
223	D₄×C₈₉	—	4	4	no	—
224	C₂×C₄×C₈₉	—	2	16	no	—
225	C₂×C₄×C₈₉	—	2	8	no	—
226	C₂×C₄×C₈₉	—	2	4	no	—
227	C₂×C₄×C₈₉	—	2	2	no	—
228	C₂×C₄×C₈₉	—	2	4	no	—
229	C₂×C₄×C₈₉	—	2	1	no	—
230	C₂×C₄×C₈₉	—	2	1	no	—
231	C₂×C₄×C₈₉	—	2	2	no	—
232	C₂×C₄×C₈₉	—	2	1	no	—
233	C₂×C₄×C₈₉	—	2	1	no	—
234	C₂×C₄×C₈₉	—	2	8	no	—
235	C₂×C₄×C₈₉	—	2	8	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	2	2	no	—
238	C₂×C₄×C₈₉	—	2	2	no	—
239	C₂×C₄×C₈₉	—	2	2	no	—
240	C₂×C₄×C₈₉	—	2	8	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	2	8	no	—
243	C₂×C₄×C₈₉	—	2	4	no	—
244	C₂×C₄×C₈₉	—	2	2	no	—
245	C₂×C₄×C₈₉	—	2	8	no	—
246	C₂×C₄×C₈₉	—	2	4	no	—
247	C₂×C₄×C₈₉	—	2	2	no	—
248	C₂×C₄×C₈₉	—	2	4	no	—
249	C₂×C₄×C₈₉	—	4	8	no	—
250	C₂×C₄×C₈₉	—	4	8	no	—
251	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
258	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
259	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
260	C₂²×C₈₉	—	2	2	no	—
261	C₂²×C₈₉	—	2	2	no	—
262	C₂²×C₈₉	—	2	4	no	—
263	C₂²×C₈₉	—	2	4	no	—
264	C₂²×C₈₉	—	4	2	no	—
265	C₂²×C₈₉	—	4	2	no	—
266	C₂²×C₈₉	—	4	2	no	—
267	C₂²×C₈₉	—	4	4	no	—
268	C₂²×C₈₉	—	4	4	no	—
269	C₂²×C₈₉	#537	1	4	yes	—
270	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
271	C₂²×C₈₉	C₂²×#41	1	2	yes	—
272	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
273	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
274	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
275	C₄×C₈₉	—	2	2	no	—
276	C₄×C₈₉	—	2	8	no	—
277	C₄×C₈₉	—	2	2	no	—
278	C₄×C₈₉	—	2	2	no	—
279	C₄×C₈₉	—	2	2	no	—
280	C₄×C₈₉	—	4	2	no	—
281	C₄×C₈₉	—	4	2	no	—
282	C₄×C₈₉	—	4	4	no	—
283	C₄×C₈₉	—	4	2	no	—
284	C₄×C₈₉	—	4	8	no	—
285	C₄×C₈₉	#525	1	4	yes	—
286	C₄×C₈₉	#536	1	4	yes	—
287	#35247	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂×C₈₉	—	2	4	no	—
289	C₂×C₈₉	—	8	2	no	—
290	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
291	C₂×C₈₉	C₂×#536	1	1	yes	characteristic
292	C₂×C₈₉	C₂×#537	1	1	yes	characteristic
293	C₂×C₈₉	#3522	1	2	yes	—
294	C₂×C₈₉	#3523	1	2	yes	—
295	C₂×#529	C₂×C₈₉	1	1	yes	characteristic
296	C₂²×C₄×Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
297	#2855	C₂×C₈₉	1	2	yes	complemented
298	#2854	C₂×C₈₉	1	2	yes	—
299	#2637	C₂×C₈₉	1	4	yes	complemented
300	#2592	C₂×C₈₉	1	2	yes	complemented
301	#2584	C₂×C₈₉	1	2	yes	complemented
302	#2582	C₂×C₈₉	1	2	yes	complemented
303	#2538	C₂×C₈₉	1	2	yes	complemented
304	#2486	C₂×C₈₉	1	2	yes	complemented
305	#2413	C₂×C₈₉	1	4	yes	complemented
306	#2409	C₂×C₈₉	1	2	yes	complemented
307	#2323	C₂×C₈₉	1	4	yes	complemented
308	#2285	C₂×C₈₉	1	1	yes	characteristic, complemented
309	C₈₉	#35247	1	1	yes	characteristic, direct product factor, Sylow
310	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
311	#529	—	2	8	no	—
312	C₂×#126	C₂²×C₈₉	1	2	yes	—
313	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
314	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
315	C₂×#123	C₂²×C₈₉	1	2	yes	—
316	C₂×#123	C₂²×C₈₉	1	4	yes	—
317	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
318	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
319	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
320	C₂×C₄×Q₈	—	2	8	no	—
321	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
322	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
323	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
324	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
325	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
326	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
327	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
328	C₂×#117	C₂²×C₈₉	1	2	yes	—
329	C₂×#117	C₂²×C₈₉	1	2	yes	—
330	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
331	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
332	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
333	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
334	#398	C₂²×C₈₉	1	8	yes	—
335	#398	C₂²×C₈₉	1	4	yes	—
336	#397	C₂²×C₈₉	1	4	yes	—
337	#396	C₂²×C₈₉	1	8	yes	—
338	#395	C₂²×C₈₉	1	4	yes	—
339	#393	C₂²×C₈₉	1	4	yes	—
340	#391	C₂²×C₈₉	1	4	yes	—
341	#391	C₂²×C₈₉	1	4	yes	—
342	#390	C₂²×C₈₉	1	4	yes	—
343	#389	C₂²×C₈₉	1	4	yes	—
344	#389	C₂²×C₈₉	1	8	yes	—
345	#389	C₂²×C₈₉	1	4	yes	—
346	#389	C₂²×C₈₉	1	4	yes	—
347	#388	C₂²×C₈₉	1	8	yes	—
348	#388	C₂²×C₈₉	1	8	yes	—
349	#387	C₂²×C₈₉	1	8	yes	—
350	#386	C₂²×C₈₉	1	4	yes	—
351	#385	C₂²×C₈₉	1	4	yes	—
352	#384	C₂²×C₈₉	1	2	yes	—
353	#382	C₂²×C₈₉	1	2	yes	—
354	#381	C₂²×C₈₉	1	4	yes	—
355	#380	C₂²×C₈₉	1	4	yes	—
356	#380	C₂²×C₈₉	1	2	yes	—
357	#380	C₂²×C₈₉	1	2	yes	—
358	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
359	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
360	C₂²×Q₈	—	2	8	no	—
361	C₂²×Q₈	—	2	4	no	—
362	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
363	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
364	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
365	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
366	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
368	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
369	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
370	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
371	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
372	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
373	#126	—	2	8	no	—
374	#125	—	2	4	no	—
375	#124	—	2	4	no	—
376	#123	—	2	8	no	—
377	#123	—	4	8	no	—
378	C₂²⋊Q₈	—	2	16	no	—
379	C₂²⋊Q₈	—	2	4	no	—
380	#121	—	2	4	no	—
381	C₄×Q₈	—	2	8	no	—
382	C₄×Q₈	—	2	8	no	—
383	C₄×Q₈	—	2	8	no	—
384	C₄×Q₈	—	2	8	no	—
385	C₄×Q₈	—	2	4	no	—
386	C₄×D₄	—	2	8	no	—
387	C₄×D₄	—	2	4	no	—
388	#117	—	2	8	no	—
389	#117	—	4	4	no	—
390	C₂×C₄⋊C₄	—	2	8	no	—
391	C₂×C₄⋊C₄	—	2	4	no	—
392	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
394	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
398	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
404	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
415	C₂×C₄²	—	2	8	no	—
416	C₂×C₄²	D₄×C₈₉	1	8	yes	—
417	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
418	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
419	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
420	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
421	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
425	#95	C₂³×C₈₉	1	4	yes	—
426	#95	C₂³×C₈₉	1	4	yes	—
427	#95	C₂³×C₈₉	1	8	yes	—
428	#95	C₂³×C₈₉	1	8	yes	—
429	#95	C₂³×C₈₉	1	8	yes	—
430	#95	C₂³×C₈₉	1	8	yes	—
431	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
432	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
433	C₂×Q₈	—	2	32	no	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	8	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	4	no	—
440	C₂×Q₈	#41×C₈₉	1	8	yes	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	8	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	8	no	—
457	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
458	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
459	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
462	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
464	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
467	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
468	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
470	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
472	C₄⋊C₄	—	2	8	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	8	no	—
483	C₄⋊C₄	—	2	16	no	—
484	C₄⋊C₄	—	2	8	no	—
485	C₄⋊C₄	—	4	4	no	—
486	C₄⋊C₄	—	4	8	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	16	no	—
494	C₂²⋊C₄	—	4	8	no	—
495	C₄²	—	2	8	no	—
496	C₄²	—	2	8	no	—
497	C₄²	—	2	4	no	—
498	C₄²	—	4	4	no	—
499	C₄²	C₂×D₄×C₈₉	1	4	yes	—
500	C₄²	C₂×D₄×C₈₉	1	4	yes	—
501	C₄²	C₂×D₄×C₈₉	1	4	yes	—
502	C₄²	C₂×D₄×C₈₉	1	4	yes	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
511	C₂³	H(5,𝔽₂)×C₈₉	1	2	yes	—
512	C₂³	2₋¹⁺⁴×C₈₉	1	2	yes	—
513	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
514	Q₈	—	2	16	no	—
515	Q₈	—	2	16	no	—
516	Q₈	—	4	4	no	—
517	D₄	—	4	8	no	—
518	D₄	—	4	4	no	—
519	C₂×C₄	—	2	16	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	8	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	8	no	—
546	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
547	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
549	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
552	C₂×C₄	C₂×#41×C₈₉	1	8	yes	—
553	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
554	C₂×C₄	2₋¹⁺⁴×C₈₉	1	4	yes	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	4	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	4	4	no	—
564	C₂²	#537×C₈₉	1	4	yes	—
565	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
566	C₂²	C₂²×#41×C₈₉	1	2	yes	—
567	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
568	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
569	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
570	C₄	—	2	2	no	—
571	C₄	—	2	8	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	4	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	8	no	—
580	C₄	#525×C₈₉	1	4	yes	—
581	C₄	#536×C₈₉	1	4	yes	—
582	C₂	—	2	4	no	—
583	C₂	—	8	2	no	—
584	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
585	C₂	C₂×#536×C₈₉	1	1	yes	characteristic
586	C₂	C₂×#537×C₈₉	1	1	yes	characteristic
587	C₂	C₈₉×#3522	1	2	yes	—
588	C₂	C₈₉×#3523	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#121×C₈₉	C₂²
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. C₂²×C₄×Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2855	C₂	semidirect
5. C₈₉×#2637	C₂	semidirect
6. C₈₉×#2592	C₂	semidirect
7. C₈₉×#2584	C₂	semidirect
8. C₈₉×#2582	C₂	semidirect
9. C₈₉×#2538	C₂	semidirect
10. C₈₉×#2486	C₂	semidirect
11. C₈₉×#2413	C₂	semidirect
12. C₈₉×#2409	C₂	semidirect
13. C₈₉×#2323	C₂	semidirect
14. C₈₉×#2285	C₂	semidirect
287. #35247	C₈₉	direct (N × H)
296. C₂²×C₄×Q₈	C₂×C₈₉	semidirect
297. #2855	C₂×C₈₉	semidirect
299. #2637	C₂×C₈₉	semidirect
300. #2592	C₂×C₈₉	semidirect
301. #2584	C₂×C₈₉	semidirect
302. #2582	C₂×C₈₉	semidirect
303. #2538	C₂×C₈₉	semidirect
304. #2486	C₂×C₈₉	semidirect
305. #2413	C₂×C₈₉	semidirect
306. #2409	C₂×C₈₉	semidirect
307. #2323	C₂×C₈₉	semidirect
308. #2285	C₂×C₈₉	semidirect
309. C₈₉	#35247	direct (N × H)

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×Q₈×C₈₉	2
C₈₉×#2855	2
C₈₉×#2854	2
C₈₉×#2637	2
C₈₉×#2592	2
C₈₉×#2584	2
C₈₉×#2582	2
C₈₉×#2538	2
C₈₉×#2486	2
C₈₉×#2413	2
C₈₉×#2409	2
C₈₉×#2323	2
C₈₉×#2285	2
#35247	89

Subgroups — C89×#35247 #36413095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35247 #36413095