Subgroups — C₈₉×#35172 #36413020

Order 22784.

2810 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2848	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2626	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2580	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2566	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2563	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2542	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2538	C₂	1	4	yes	maximal, complemented
10	C₈₉×#2317	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2309	C₂	1	2	yes	maximal
12	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
15	#533×C₈₉	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
18	C₂×#125×C₈₉	C₂²	1	2	yes	complemented
19	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	8	yes	complemented
22	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
23	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
24	C₂×C₄×D₄×C₈₉	—	2	8	no	—
25	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
29	C₂×#117×C₈₉	C₂²	1	2	yes	—
30	C₂×#117×C₈₉	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
33	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
34	#399×C₈₉	C₂²	1	8	yes	complemented
35	#396×C₈₉	C₂²	1	16	yes	complemented
36	#393×C₈₉	C₂²	1	8	yes	complemented
37	#390×C₈₉	C₂²	1	4	yes	—
38	#390×C₈₉	C₂²	1	4	yes	—
39	#390×C₈₉	C₂²	1	8	yes	—
40	#389×C₈₉	C₂²	1	8	yes	complemented
41	#385×C₈₉	C₂²	1	8	yes	—
42	#385×C₈₉	C₂²	1	16	yes	complemented
43	#384×C₈₉	C₂²	1	4	yes	complemented
44	#384×C₈₉	C₂²	1	2	yes	complemented
45	#382×C₈₉	C₂²	1	4	yes	complemented
46	#382×C₈₉	C₂²	1	2	yes	complemented
47	#380×C₈₉	C₂²	1	2	yes	—
48	#380×C₈₉	C₂²	1	2	yes	—
49	#380×C₈₉	C₂²	1	8	yes	complemented
50	#379×C₈₉	C₂²	1	8	yes	—
51	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
52	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	complemented
53	C₂⁵×C₈₉	C₂³	1	2	yes	—
54	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
55	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
56	C₂²×D₄×C₈₉	—	2	4	no	—
57	C₂²×D₄×C₈₉	—	2	4	no	—
58	C₂²×D₄×C₈₉	Q₈	1	8	yes	complemented
59	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
60	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
61	C₂³×C₄×C₈₉	—	2	8	no	—
62	C₂³×C₄×C₈₉	—	4	8	no	—
63	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
64	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
66	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
67	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₈₉	—	2	4	no	—
70	C₄⋊Q₈×C₈₉	—	2	4	no	—
71	#125×C₈₉	—	2	8	no	—
72	C₂²⋊Q₈×C₈₉	—	2	8	no	—
73	C₂²⋊Q₈×C₈₉	—	2	8	no	—
74	C₂²⋊Q₈×C₈₉	—	4	16	no	—
75	C₄×Q₈×C₈₉	—	2	8	no	—
76	C₄×Q₈×C₈₉	—	2	8	no	—
77	C₄×D₄×C₈₉	—	2	8	no	—
78	C₄×D₄×C₈₉	—	2	8	no	—
79	C₄×D₄×C₈₉	—	2	8	no	—
80	C₄×D₄×C₈₉	—	2	8	no	—
81	#117×C₈₉	—	2	8	no	—
82	#117×C₈₉	—	2	8	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
84	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
96	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
97	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
102	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
103	C₂×C₄²×C₈₉	D₄	1	8	yes	—
104	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
105	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
106	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
110	#95×C₈₉	C₂³	1	4	yes	—
111	#95×C₈₉	C₂³	1	4	yes	—
112	#95×C₈₉	C₂³	1	16	yes	—
113	C₂⁴×C₈₉	—	2	4	no	—
114	C₂⁴×C₈₉	—	2	4	no	—
115	C₂⁴×C₈₉	—	4	4	no	—
116	C₂⁴×C₈₉	—	4	4	no	—
117	C₂⁴×C₈₉	C₂×Q₈	1	8	yes	—
118	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
119	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂×Q₈×C₈₉	—	2	4	no	—
121	C₂×Q₈×C₈₉	—	2	4	no	—
122	C₂×Q₈×C₈₉	—	2	8	no	—
123	C₂×Q₈×C₈₉	—	4	4	no	—
124	C₂×D₄×C₈₉	—	2	32	no	—
125	C₂×D₄×C₈₉	—	2	4	no	—
126	C₂×D₄×C₈₉	—	2	4	no	—
127	C₂×D₄×C₈₉	—	2	4	no	—
128	C₂×D₄×C₈₉	—	2	4	no	—
129	C₂×D₄×C₈₉	C₂×Q₈	1	8	yes	—
130	C₂×D₄×C₈₉	C₂×Q₈	1	8	yes	—
131	C₂²×C₄×C₈₉	—	2	4	no	—
132	C₂²×C₄×C₈₉	—	2	4	no	—
133	C₂²×C₄×C₈₉	—	2	8	no	—
134	C₂²×C₄×C₈₉	—	2	8	no	—
135	C₂²×C₄×C₈₉	—	2	16	no	—
136	C₂²×C₄×C₈₉	—	4	8	no	—
137	C₂²×C₄×C₈₉	—	4	4	no	—
138	C₂²×C₄×C₈₉	—	4	2	no	—
139	C₂²×C₄×C₈₉	—	4	32	no	—
140	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
144	C₂²×C₄×C₈₉	#41	1	4	yes	—
145	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
146	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
150	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
151	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
155	C₄⋊C₄×C₈₉	—	2	4	no	—
156	C₄⋊C₄×C₈₉	—	2	4	no	—
157	C₄⋊C₄×C₈₉	—	2	4	no	—
158	C₄⋊C₄×C₈₉	—	2	4	no	—
159	C₄⋊C₄×C₈₉	—	2	8	no	—
160	C₄⋊C₄×C₈₉	—	2	8	no	—
161	C₄⋊C₄×C₈₉	—	2	4	no	—
162	C₄⋊C₄×C₈₉	—	2	4	no	—
163	C₄⋊C₄×C₈₉	—	2	8	no	—
164	C₄⋊C₄×C₈₉	—	2	8	no	—
165	C₄⋊C₄×C₈₉	—	4	4	no	—
166	C₄⋊C₄×C₈₉	—	4	16	no	—
167	C₂²⋊C₄×C₈₉	—	2	8	no	—
168	C₂²⋊C₄×C₈₉	—	2	8	no	—
169	C₂²⋊C₄×C₈₉	—	2	8	no	—
170	C₂²⋊C₄×C₈₉	—	2	8	no	—
171	C₂²⋊C₄×C₈₉	—	4	16	no	—
172	C₂²⋊C₄×C₈₉	—	4	16	no	—
173	C₄²×C₈₉	—	2	8	no	—
174	C₄²×C₈₉	—	2	8	no	—
175	C₄²×C₈₉	C₂×D₄	1	4	yes	—
176	C₄²×C₈₉	C₂×D₄	1	4	yes	—
177	C₄²×C₈₉	C₂×D₄	1	4	yes	—
178	C₄²×C₈₉	C₂×D₄	1	4	yes	—
179	C₂³×C₈₉	—	2	2	no	—
180	C₂³×C₈₉	—	2	2	no	—
181	C₂³×C₈₉	—	2	16	no	—
182	C₂³×C₈₉	—	2	4	no	—
183	C₂³×C₈₉	—	2	4	no	—
184	C₂³×C₈₉	—	2	8	no	—
185	C₂³×C₈₉	—	2	8	no	—
186	C₂³×C₈₉	—	4	16	no	—
187	C₂³×C₈₉	—	4	8	no	—
188	C₂³×C₈₉	—	4	8	no	—
189	C₂³×C₈₉	—	4	16	no	—
190	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
191	C₂³×C₈₉	C₂²×Q₈	1	8	yes	—
192	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
193	C₂³×C₈₉	C₂×#41	1	2	yes	—
194	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
195	Q₈×C₈₉	—	4	8	no	—
196	Q₈×C₈₉	—	4	8	no	—
197	D₄×C₈₉	—	2	16	no	—
198	D₄×C₈₉	—	2	16	no	—
199	C₂×C₄×C₈₉	—	2	2	no	—
200	C₂×C₄×C₈₉	—	2	2	no	—
201	C₂×C₄×C₈₉	—	2	2	no	—
202	C₂×C₄×C₈₉	—	2	8	no	—
203	C₂×C₄×C₈₉	—	2	4	no	—
204	C₂×C₄×C₈₉	—	2	4	no	—
205	C₂×C₄×C₈₉	—	2	4	no	—
206	C₂×C₄×C₈₉	—	2	8	no	—
207	C₂×C₄×C₈₉	—	2	8	no	—
208	C₂×C₄×C₈₉	—	2	8	no	—
209	C₂×C₄×C₈₉	—	2	8	no	—
210	C₂×C₄×C₈₉	—	2	8	no	—
211	C₂×C₄×C₈₉	—	2	8	no	—
212	C₂×C₄×C₈₉	—	4	4	no	—
213	C₂×C₄×C₈₉	—	4	16	no	—
214	C₂×C₄×C₈₉	—	4	8	no	—
215	C₂×C₄×C₈₉	—	4	16	no	—
216	C₂×C₄×C₈₉	C₂²⋊Q₈	1	16	yes	complemented
217	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
224	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
225	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
226	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
227	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
228	C₂²×C₈₉	—	2	2	no	—
229	C₂²×C₈₉	—	2	2	no	—
230	C₂²×C₈₉	—	2	4	no	—
231	C₂²×C₈₉	—	2	4	no	—
232	C₂²×C₈₉	—	2	8	no	—
233	C₂²×C₈₉	—	2	16	no	—
234	C₂²×C₈₉	—	4	32	no	—
235	C₂²×C₈₉	—	4	4	no	—
236	C₂²×C₈₉	—	4	4	no	—
237	C₂²×C₈₉	—	4	8	no	—
238	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
239	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
240	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
241	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
242	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
243	C₂²×C₈₉	#585	1	2	yes	—
244	C₄×C₈₉	—	2	2	no	—
245	C₄×C₈₉	—	2	2	no	—
246	C₄×C₈₉	—	2	2	no	—
247	C₄×C₈₉	—	4	4	no	—
248	C₄×C₈₉	—	4	4	no	—
249	C₄×C₈₉	—	4	4	no	—
250	C₄×C₈₉	—	4	8	no	—
251	C₄×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
252	C₄×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
253	C₄×C₈₉	#535	1	4	yes	—
254	#35172	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₈₉	—	2	4	no	—
256	C₂×C₈₉	—	4	8	no	—
257	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
259	C₂×C₈₉	#3449	1	1	yes	characteristic
260	C₂×C₈₉	#3501	1	2	yes	—
261	C₂×C₈₉	#3506	1	2	yes	—
262	C₂×#533	C₂×C₈₉	1	1	yes	characteristic, complemented
263	C₂²×C₄×D₄	C₂×C₈₉	1	1	yes	characteristic
264	#2848	C₂×C₈₉	1	4	yes	complemented
265	#2626	C₂×C₈₉	1	4	yes	complemented
266	#2580	C₂×C₈₉	1	4	yes	complemented
267	#2566	C₂×C₈₉	1	4	yes	complemented
268	#2563	C₂×C₈₉	1	2	yes	complemented
269	#2542	C₂×C₈₉	1	2	yes	complemented
270	#2538	C₂×C₈₉	1	4	yes	complemented
271	#2317	C₂×C₈₉	1	2	yes	complemented
272	#2309	C₂×C₈₉	1	2	yes	—
273	#2285	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₈₉	#35172	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
276	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
277	#533	—	2	8	no	—
278	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
279	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
280	C₂×#125	C₂²×C₈₉	1	2	yes	complemented
281	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
282	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
283	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	complemented
284	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	complemented
285	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	complemented
286	C₂×C₄×D₄	—	2	8	no	—
287	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
288	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
289	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
290	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
291	C₂×#117	C₂²×C₈₉	1	2	yes	—
292	C₂×#117	C₂²×C₈₉	1	2	yes	—
293	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
294	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
295	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
296	#399	C₂²×C₈₉	1	8	yes	complemented
297	#396	C₂²×C₈₉	1	16	yes	complemented
298	#393	C₂²×C₈₉	1	8	yes	complemented
299	#390	C₂²×C₈₉	1	4	yes	—
300	#390	C₂²×C₈₉	1	4	yes	—
301	#390	C₂²×C₈₉	1	8	yes	—
302	#389	C₂²×C₈₉	1	8	yes	complemented
303	#385	C₂²×C₈₉	1	8	yes	—
304	#385	C₂²×C₈₉	1	16	yes	complemented
305	#384	C₂²×C₈₉	1	4	yes	complemented
306	#384	C₂²×C₈₉	1	2	yes	complemented
307	#382	C₂²×C₈₉	1	4	yes	complemented
308	#382	C₂²×C₈₉	1	2	yes	complemented
309	#380	C₂²×C₈₉	1	2	yes	—
310	#380	C₂²×C₈₉	1	2	yes	—
311	#380	C₂²×C₈₉	1	8	yes	complemented
312	#379	C₂²×C₈₉	1	8	yes	—
313	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
314	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	complemented
315	C₂⁵	C₂³×C₈₉	1	2	yes	—
316	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
317	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
318	C₂²×D₄	—	2	4	no	—
319	C₂²×D₄	—	2	4	no	—
320	C₂²×D₄	Q₈×C₈₉	1	8	yes	complemented
321	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
322	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
323	C₂³×C₄	—	2	8	no	—
324	C₂³×C₄	—	4	8	no	—
325	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
326	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
328	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
329	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
330	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
331	C₄⋊Q₈	—	2	4	no	—
332	C₄⋊Q₈	—	2	4	no	—
333	#125	—	2	8	no	—
334	C₂²⋊Q₈	—	2	8	no	—
335	C₂²⋊Q₈	—	2	8	no	—
336	C₂²⋊Q₈	—	4	16	no	—
337	C₄×Q₈	—	2	8	no	—
338	C₄×Q₈	—	2	8	no	—
339	C₄×D₄	—	2	8	no	—
340	C₄×D₄	—	2	8	no	—
341	C₄×D₄	—	2	8	no	—
342	C₄×D₄	—	2	8	no	—
343	#117	—	2	8	no	—
344	#117	—	2	8	no	—
345	C₂×C₄⋊C₄	—	2	4	no	—
346	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
348	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
349	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
357	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
358	C₂×C₂²⋊C₄	—	2	8	no	—
359	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
365	C₂×C₄²	D₄×C₈₉	1	8	yes	—
366	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
367	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
368	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
372	#95	C₂³×C₈₉	1	4	yes	—
373	#95	C₂³×C₈₉	1	4	yes	—
374	#95	C₂³×C₈₉	1	16	yes	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	4	4	no	—
378	C₂⁴	—	4	4	no	—
379	C₂⁴	C₂×Q₈×C₈₉	1	8	yes	—
380	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
381	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
382	C₂×Q₈	—	2	4	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	2	8	no	—
385	C₂×Q₈	—	4	4	no	—
386	C₂×D₄	—	2	32	no	—
387	C₂×D₄	—	2	4	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	C₂×Q₈×C₈₉	1	8	yes	—
392	C₂×D₄	C₂×Q₈×C₈₉	1	8	yes	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	4	8	no	—
399	C₂²×C₄	—	4	4	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	—	4	32	no	—
402	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
404	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
405	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
406	C₂²×C₄	#41×C₈₉	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
408	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	8	no	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	4	16	no	—
428	C₄⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	2	8	no	—
432	C₂²⋊C₄	—	2	8	no	—
433	C₂²⋊C₄	—	4	16	no	—
434	C₂²⋊C₄	—	4	16	no	—
435	C₄²	—	2	8	no	—
436	C₄²	—	2	8	no	—
437	C₄²	C₂×D₄×C₈₉	1	4	yes	—
438	C₄²	C₂×D₄×C₈₉	1	4	yes	—
439	C₄²	C₂×D₄×C₈₉	1	4	yes	—
440	C₄²	C₂×D₄×C₈₉	1	4	yes	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	16	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	8	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	4	16	no	—
449	C₂³	—	4	8	no	—
450	C₂³	—	4	8	no	—
451	C₂³	—	4	16	no	—
452	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
453	C₂³	C₂²×Q₈×C₈₉	1	8	yes	—
454	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
455	C₂³	C₂×#41×C₈₉	1	2	yes	—
456	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
457	Q₈	—	4	8	no	—
458	Q₈	—	4	8	no	—
459	D₄	—	2	16	no	—
460	D₄	—	2	16	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	8	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	16	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	4	16	no	—
478	C₂×C₄	C₂²⋊Q₈×C₈₉	1	16	yes	complemented
479	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
483	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
485	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
486	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
487	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
488	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
489	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	2	8	no	—
495	C₂²	—	2	16	no	—
496	C₂²	—	4	32	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	8	no	—
500	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
501	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
502	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
503	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
504	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
505	C₂²	#585×C₈₉	1	2	yes	—
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	8	no	—
513	C₄	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
514	C₄	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
515	C₄	#535×C₈₉	1	4	yes	—
516	C₂	—	2	4	no	—
517	C₂	—	4	8	no	—
518	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
519	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
520	C₂	C₈₉×#3449	1	1	yes	characteristic
521	C₂	C₈₉×#3501	1	2	yes	—
522	C₂	C₈₉×#3506	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂³×D₄×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
1. C₂×#533×C₈₉	C₂	semidirect
3. C₈₉×#2848	C₂	semidirect
4. C₈₉×#2626	C₂	semidirect
5. C₈₉×#2580	C₂	semidirect
6. C₈₉×#2566	C₂	semidirect
7. C₈₉×#2563	C₂	semidirect
8. C₈₉×#2542	C₂	semidirect
9. C₈₉×#2538	C₂	semidirect
10. C₈₉×#2317	C₂	semidirect
12. C₈₉×#2285	C₂	semidirect
16. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
17. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
18. C₂×#125×C₈₉	C₂²	semidirect
21. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
22. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
23. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
34. #399×C₈₉	C₂²	semidirect
35. #396×C₈₉	C₂²	semidirect
36. #393×C₈₉	C₂²	semidirect
40. #389×C₈₉	C₂²	semidirect
42. #385×C₈₉	C₂²	semidirect
43. #384×C₈₉	C₂²	semidirect
44. #384×C₈₉	C₂²	semidirect
45. #382×C₈₉	C₂²	semidirect
46. #382×C₈₉	C₂²	semidirect
49. #380×C₈₉	C₂²	semidirect
51. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
52. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
58. C₂²×D₄×C₈₉	Q₈	semidirect
216. C₂×C₄×C₈₉	C₂²⋊Q₈	semidirect
254. #35172	C₈₉	direct (N × H)
262. C₂×#533	C₂×C₈₉	semidirect
264. #2848	C₂×C₈₉	semidirect
265. #2626	C₂×C₈₉	semidirect
266. #2580	C₂×C₈₉	semidirect
267. #2566	C₂×C₈₉	semidirect
268. #2563	C₂×C₈₉	semidirect
269. #2542	C₂×C₈₉	semidirect
270. #2538	C₂×C₈₉	semidirect
271. #2317	C₂×C₈₉	semidirect
273. #2285	C₂×C₈₉	semidirect
274. C₈₉	#35172	direct (N × H)
278. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
279. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
280. C₂×#125	C₂²×C₈₉	semidirect
283. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
284. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
285. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
296. #399	C₂²×C₈₉	semidirect
297. #396	C₂²×C₈₉	semidirect
298. #393	C₂²×C₈₉	semidirect
302. #389	C₂²×C₈₉	semidirect
304. #385	C₂²×C₈₉	semidirect
305. #384	C₂²×C₈₉	semidirect
306. #384	C₂²×C₈₉	semidirect
307. #382	C₂²×C₈₉	semidirect
308. #382	C₂²×C₈₉	semidirect
311. #380	C₂²×C₈₉	semidirect
313. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
314. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
320. C₂²×D₄	Q₈×C₈₉	semidirect
478. C₂×C₄	C₂²⋊Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₈₉	2
C₂²×C₄×D₄×C₈₉	2
C₈₉×#2848	2
C₈₉×#2626	2
C₈₉×#2580	2
C₈₉×#2566	2
C₈₉×#2563	2
C₈₉×#2542	2
C₈₉×#2538	2
C₈₉×#2317	2
C₈₉×#2309	2
C₈₉×#2285	2
#35172	89

Subgroups — C89×#35172 #36413020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35172 #36413020