Subgroups — C₈₉×#35512 #36413360

Order 22784.

5794 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#525×C₈₉	C₂	1	2	yes	maximal, complemented
3	C₂²×#121×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2408	C₂	1	8	yes	maximal, complemented
5	C₈₉×#2407	C₂	1	8	yes	maximal, complemented
6	C₈₉×#2405	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2403	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2311	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2309	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₈₉	—	2	16	no	—
11	C₂×H(5,𝔽₂)×C₈₉	—	2	8	no	—
12	C₂²×#41×C₈₉	C₂²	1	2	yes	—
13	C₂²×#41×C₈₉	C₂²	1	4	yes	complemented
14	C₂³×D₄×C₈₉	C₂²	1	4	yes	—
15	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₈₉	C₂²	1	4	yes	complemented
17	C₂³×D₄×C₈₉	C₂²	1	2	yes	complemented
18	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
19	#525×C₈₉	—	2	16	no	—
20	C₂×#123×C₈₉	C₂²	1	4	yes	complemented
21	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	complemented
22	C₂×#121×C₈₉	—	2	8	no	—
23	C₂×#121×C₈₉	C₂²	1	8	yes	complemented
24	C₂×#121×C₈₉	C₂²	1	4	yes	complemented
25	C₂×#120×C₈₉	C₂²	1	4	yes	complemented
26	C₂×#120×C₈₉	C₂²	1	8	yes	complemented
27	C₂×C₄×D₄×C₈₉	C₂²	1	8	yes	complemented
28	C₂×#117×C₈₉	C₂²	1	2	yes	complemented
29	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	complemented
31	#394×C₈₉	C₂²	1	16	yes	complemented
32	#392×C₈₉	C₂²	1	16	yes	complemented
33	#391×C₈₉	C₂²	1	4	yes	complemented
34	#390×C₈₉	C₂²	1	8	yes	complemented
35	#390×C₈₉	C₂²	1	4	yes	complemented
36	#386×C₈₉	C₂²	1	8	yes	complemented
37	#386×C₈₉	C₂²	1	8	yes	complemented
38	#385×C₈₉	C₂²	1	8	yes	complemented
39	#380×C₈₉	C₂²	1	8	yes	complemented
40	#379×C₈₉	C₂²	1	4	yes	complemented
41	#379×C₈₉	C₂²	1	8	yes	complemented
42	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
43	C₂⁵×C₈₉	C₂³	1	4	yes	—
44	C₂⁵×C₈₉	C₂³	1	2	yes	—
45	H(5,𝔽₂)×C₈₉	—	2	128	no	—
46	C₂×#41×C₈₉	—	2	16	no	—
47	C₂×#41×C₈₉	—	2	16	no	—
48	C₂×#41×C₈₉	—	2	8	no	—
49	C₂×#41×C₈₉	—	2	8	no	—
50	C₂×#41×C₈₉	—	2	16	no	—
51	C₂×#41×C₈₉	D₄	1	16	yes	complemented
52	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
53	C₂²×D₄×C₈₉	—	2	16	no	—
54	C₂²×D₄×C₈₉	—	2	32	no	—
55	C₂²×D₄×C₈₉	—	2	16	no	—
56	C₂²×D₄×C₈₉	—	2	16	no	—
57	C₂²×D₄×C₈₉	—	2	8	no	—
58	C₂²×D₄×C₈₉	—	2	8	no	—
59	C₂²×D₄×C₈₉	—	2	16	no	—
60	C₂²×D₄×C₈₉	—	4	8	no	—
61	C₂²×D₄×C₈₉	D₄	1	8	yes	complemented
62	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
63	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
64	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
65	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
66	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
67	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
68	C₂³×C₄×C₈₉	—	2	4	no	—
69	C₂³×C₄×C₈₉	—	2	8	no	—
70	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
74	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
75	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	#123×C₈₉	—	2	16	no	—
77	C₂²⋊Q₈×C₈₉	—	2	16	no	—
78	#121×C₈₉	—	2	32	no	—
79	#121×C₈₉	—	2	16	no	—
80	#120×C₈₉	—	2	16	no	—
81	#120×C₈₉	—	4	16	no	—
82	C₄×D₄×C₈₉	—	2	32	no	—
83	#117×C₈₉	—	2	8	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
85	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
89	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
95	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
96	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
97	#95×C₈₉	C₂³	1	16	yes	—
98	C₂⁴×C₈₉	—	2	2	no	—
99	C₂⁴×C₈₉	—	2	4	no	—
100	C₂⁴×C₈₉	—	2	8	no	—
101	C₂⁴×C₈₉	—	2	16	no	—
102	C₂⁴×C₈₉	—	2	8	no	—
103	C₂⁴×C₈₉	—	2	4	no	—
104	C₂⁴×C₈₉	—	4	2	no	—
105	C₂⁴×C₈₉	—	4	4	no	—
106	C₂⁴×C₈₉	—	4	4	no	—
107	C₂⁴×C₈₉	—	4	8	no	—
108	C₂⁴×C₈₉	—	4	8	no	—
109	C₂⁴×C₈₉	C₂×D₄	1	8	yes	—
110	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
111	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
112	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
113	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
114	#41×C₈₉	—	2	128	no	—
115	#41×C₈₉	—	2	64	no	—
116	C₂×Q₈×C₈₉	—	2	4	no	—
117	C₂×Q₈×C₈₉	—	2	4	no	—
118	C₂×Q₈×C₈₉	C₂×D₄	1	8	yes	complemented
119	C₂×D₄×C₈₉	—	2	128	no	—
120	C₂×D₄×C₈₉	—	2	128	no	—
121	C₂×D₄×C₈₉	—	2	16	no	—
122	C₂×D₄×C₈₉	—	2	16	no	—
123	C₂×D₄×C₈₉	—	2	8	no	—
124	C₂×D₄×C₈₉	—	2	8	no	—
125	C₂×D₄×C₈₉	—	2	16	no	—
126	C₂×D₄×C₈₉	—	2	16	no	—
127	C₂×D₄×C₈₉	—	2	16	no	—
128	C₂×D₄×C₈₉	—	2	16	no	—
129	C₂×D₄×C₈₉	—	2	32	no	—
130	C₂×D₄×C₈₉	—	2	16	no	—
131	C₂×D₄×C₈₉	—	2	8	no	—
132	C₂×D₄×C₈₉	—	2	4	no	—
133	C₂×D₄×C₈₉	—	2	8	no	—
134	C₂×D₄×C₈₉	—	2	4	no	—
135	C₂×D₄×C₈₉	—	4	8	no	—
136	C₂×D₄×C₈₉	—	4	8	no	—
137	C₂×D₄×C₈₉	—	4	32	no	—
138	C₂×D₄×C₈₉	C₂×D₄	1	16	yes	—
139	C₂×D₄×C₈₉	C₂×D₄	1	16	yes	—
140	C₂×D₄×C₈₉	C₂×D₄	1	8	yes	complemented
141	C₂²×C₄×C₈₉	—	2	8	no	—
142	C₂²×C₄×C₈₉	—	2	16	no	—
143	C₂²×C₄×C₈₉	—	2	8	no	—
144	C₂²×C₄×C₈₉	—	2	8	no	—
145	C₂²×C₄×C₈₉	—	2	2	no	—
146	C₂²×C₄×C₈₉	—	2	4	no	—
147	C₂²×C₄×C₈₉	—	2	4	no	—
148	C₂²×C₄×C₈₉	—	2	8	no	—
149	C₂²×C₄×C₈₉	—	2	8	no	—
150	C₂²×C₄×C₈₉	—	2	8	no	—
151	C₂²×C₄×C₈₉	—	4	4	no	—
152	C₂²×C₄×C₈₉	—	4	16	no	—
153	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
155	C₂²×C₄×C₈₉	C₂×D₄	1	16	yes	complemented
156	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
158	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
161	C₄⋊C₄×C₈₉	—	2	8	no	—
162	C₄⋊C₄×C₈₉	—	2	8	no	—
163	C₄⋊C₄×C₈₉	—	2	16	no	—
164	C₂²⋊C₄×C₈₉	—	2	16	no	—
165	C₂²⋊C₄×C₈₉	—	2	8	no	—
166	C₂²⋊C₄×C₈₉	—	2	16	no	—
167	C₂²⋊C₄×C₈₉	—	2	16	no	—
168	C₂²⋊C₄×C₈₉	—	4	8	no	—
169	C₂²⋊C₄×C₈₉	—	4	8	no	—
170	C₂²⋊C₄×C₈₉	—	4	16	no	—
171	C₄²×C₈₉	—	2	8	no	—
172	C₂³×C₈₉	—	2	2	no	—
173	C₂³×C₈₉	—	2	2	no	—
174	C₂³×C₈₉	—	2	8	no	—
175	C₂³×C₈₉	—	2	32	no	—
176	C₂³×C₈₉	—	2	4	no	—
177	C₂³×C₈₉	—	2	2	no	—
178	C₂³×C₈₉	—	2	4	no	—
179	C₂³×C₈₉	—	2	8	no	—
180	C₂³×C₈₉	—	2	4	no	—
181	C₂³×C₈₉	—	2	8	no	—
182	C₂³×C₈₉	—	2	8	no	—
183	C₂³×C₈₉	—	2	2	no	—
184	C₂³×C₈₉	—	2	8	no	—
185	C₂³×C₈₉	—	2	4	no	—
186	C₂³×C₈₉	—	2	8	no	—
187	C₂³×C₈₉	—	2	16	no	—
188	C₂³×C₈₉	—	4	8	no	—
189	C₂³×C₈₉	—	4	8	no	—
190	C₂³×C₈₉	—	4	16	no	—
191	C₂³×C₈₉	—	4	4	no	—
192	C₂³×C₈₉	—	4	4	no	—
193	C₂³×C₈₉	—	8	16	no	—
194	C₂³×C₈₉	—	8	8	no	—
195	C₂³×C₈₉	—	8	8	no	—
196	C₂³×C₈₉	—	8	32	no	—
197	C₂³×C₈₉	C₂²×D₄	1	8	yes	—
198	C₂³×C₈₉	C₂²×D₄	1	8	yes	—
199	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
200	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
201	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	complemented
202	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
203	Q₈×C₈₉	—	2	16	no	—
204	D₄×C₈₉	—	2	64	no	—
205	D₄×C₈₉	—	2	32	no	—
206	D₄×C₈₉	—	2	32	no	—
207	D₄×C₈₉	—	2	16	no	—
208	D₄×C₈₉	—	4	32	no	—
209	D₄×C₈₉	—	4	32	no	—
210	C₂×C₄×C₈₉	—	2	32	no	—
211	C₂×C₄×C₈₉	—	2	4	no	—
212	C₂×C₄×C₈₉	—	2	4	no	—
213	C₂×C₄×C₈₉	—	2	32	no	—
214	C₂×C₄×C₈₉	—	2	8	no	—
215	C₂×C₄×C₈₉	—	2	2	no	—
216	C₂×C₄×C₈₉	—	2	4	no	—
217	C₂×C₄×C₈₉	—	2	4	no	—
218	C₂×C₄×C₈₉	—	2	8	no	—
219	C₂×C₄×C₈₉	—	2	8	no	—
220	C₂×C₄×C₈₉	—	2	8	no	—
221	C₂×C₄×C₈₉	—	4	8	no	—
222	C₂×C₄×C₈₉	—	4	4	no	—
223	C₂×C₄×C₈₉	—	4	16	no	—
224	C₂×C₄×C₈₉	#120	1	16	yes	complemented
225	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
226	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
228	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
229	C₂²×C₈₉	—	2	2	no	—
230	C₂²×C₈₉	—	2	2	no	—
231	C₂²×C₈₉	—	2	8	no	—
232	C₂²×C₈₉	—	2	8	no	—
233	C₂²×C₈₉	—	2	8	no	—
234	C₂²×C₈₉	—	2	4	no	—
235	C₂²×C₈₉	—	2	8	no	—
236	C₂²×C₈₉	—	2	8	no	—
237	C₂²×C₈₉	—	2	8	no	—
238	C₂²×C₈₉	—	2	16	no	—
239	C₂²×C₈₉	—	4	16	no	—
240	C₂²×C₈₉	—	4	2	no	—
241	C₂²×C₈₉	—	4	4	no	—
242	C₂²×C₈₉	—	4	2	no	—
243	C₂²×C₈₉	—	4	4	no	—
244	C₂²×C₈₉	—	4	4	no	—
245	C₂²×C₈₉	—	4	8	no	—
246	C₂²×C₈₉	—	4	16	no	—
247	C₂²×C₈₉	—	8	16	no	—
248	C₂²×C₈₉	—	8	32	no	—
249	C₂²×C₈₉	C₂×#120	1	4	yes	—
250	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
251	C₂²×C₈₉	C₂³×D₄	1	2	yes	—
252	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂²×C₈₉	#585	1	2	yes	—
254	C₄×C₈₉	—	2	8	no	—
255	C₄×C₈₉	—	4	2	no	—
256	C₄×C₈₉	—	4	4	no	—
257	C₄×C₈₉	—	4	8	no	—
258	C₄×C₈₉	C₂×#120	1	8	yes	—
259	#35512	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂×C₈₉	—	2	4	no	—
261	C₂×C₈₉	—	4	4	no	—
262	C₂×C₈₉	—	4	8	no	—
263	C₂×C₈₉	—	4	8	no	—
264	C₂×C₈₉	C₂²×#120	1	1	yes	characteristic
265	C₂×C₈₉	#3484	1	1	yes	characteristic
266	C₂×C₈₉	#3485	1	1	yes	characteristic
267	C₂×C₈₉	#3488	1	4	yes	—
268	C₂²×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic, complemented
269	C₂×#525	C₂×C₈₉	1	2	yes	complemented
270	C₂²×#121	C₂×C₈₉	1	1	yes	characteristic, complemented
271	#2408	C₂×C₈₉	1	8	yes	complemented
272	#2407	C₂×C₈₉	1	8	yes	complemented
273	#2405	C₂×C₈₉	1	4	yes	complemented
274	#2403	C₂×C₈₉	1	4	yes	complemented
275	#2311	C₂×C₈₉	1	2	yes	complemented
276	#2309	C₂×C₈₉	1	1	yes	characteristic, complemented
277	C₈₉	#35512	1	1	yes	characteristic, direct product factor, Sylow
278	C₂×H(5,𝔽₂)	—	2	16	no	—
279	C₂×H(5,𝔽₂)	—	2	8	no	—
280	C₂²×#41	C₂²×C₈₉	1	2	yes	—
281	C₂²×#41	C₂²×C₈₉	1	4	yes	complemented
282	C₂³×D₄	C₂²×C₈₉	1	4	yes	—
283	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
284	C₂³×D₄	C₂²×C₈₉	1	4	yes	complemented
285	C₂³×D₄	C₂²×C₈₉	1	2	yes	complemented
286	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
287	#525	—	2	16	no	—
288	C₂×#123	C₂²×C₈₉	1	4	yes	complemented
289	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	complemented
290	C₂×#121	—	2	8	no	—
291	C₂×#121	C₂²×C₈₉	1	8	yes	complemented
292	C₂×#121	C₂²×C₈₉	1	4	yes	complemented
293	C₂×#120	C₂²×C₈₉	1	4	yes	complemented
294	C₂×#120	C₂²×C₈₉	1	8	yes	complemented
295	C₂×C₄×D₄	C₂²×C₈₉	1	8	yes	complemented
296	C₂×#117	C₂²×C₈₉	1	2	yes	complemented
297	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
298	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	complemented
299	#394	C₂²×C₈₉	1	16	yes	complemented
300	#392	C₂²×C₈₉	1	16	yes	complemented
301	#391	C₂²×C₈₉	1	4	yes	complemented
302	#390	C₂²×C₈₉	1	8	yes	complemented
303	#390	C₂²×C₈₉	1	4	yes	complemented
304	#386	C₂²×C₈₉	1	8	yes	complemented
305	#386	C₂²×C₈₉	1	8	yes	complemented
306	#385	C₂²×C₈₉	1	8	yes	complemented
307	#380	C₂²×C₈₉	1	8	yes	complemented
308	#379	C₂²×C₈₉	1	4	yes	complemented
309	#379	C₂²×C₈₉	1	8	yes	complemented
310	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
311	C₂⁵	C₂³×C₈₉	1	4	yes	—
312	C₂⁵	C₂³×C₈₉	1	2	yes	—
313	H(5,𝔽₂)	—	2	128	no	—
314	C₂×#41	—	2	16	no	—
315	C₂×#41	—	2	16	no	—
316	C₂×#41	—	2	8	no	—
317	C₂×#41	—	2	8	no	—
318	C₂×#41	—	2	16	no	—
319	C₂×#41	D₄×C₈₉	1	16	yes	complemented
320	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
321	C₂²×D₄	—	2	16	no	—
322	C₂²×D₄	—	2	32	no	—
323	C₂²×D₄	—	2	16	no	—
324	C₂²×D₄	—	2	16	no	—
325	C₂²×D₄	—	2	8	no	—
326	C₂²×D₄	—	2	8	no	—
327	C₂²×D₄	—	2	16	no	—
328	C₂²×D₄	—	4	8	no	—
329	C₂²×D₄	D₄×C₈₉	1	8	yes	complemented
330	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
331	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
332	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
333	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
334	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
335	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
336	C₂³×C₄	—	2	4	no	—
337	C₂³×C₄	—	2	8	no	—
338	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
340	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
341	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
342	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
343	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
344	#123	—	2	16	no	—
345	C₂²⋊Q₈	—	2	16	no	—
346	#121	—	2	32	no	—
347	#121	—	2	16	no	—
348	#120	—	2	16	no	—
349	#120	—	4	16	no	—
350	C₄×D₄	—	2	32	no	—
351	#117	—	2	8	no	—
352	C₂×C₄⋊C₄	—	2	4	no	—
353	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
355	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
356	C₂×C₂²⋊C₄	—	2	8	no	—
357	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
364	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
365	#95	C₂³×C₈₉	1	16	yes	—
366	C₂⁴	—	2	2	no	—
367	C₂⁴	—	2	4	no	—
368	C₂⁴	—	2	8	no	—
369	C₂⁴	—	2	16	no	—
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	2	4	no	—
372	C₂⁴	—	4	2	no	—
373	C₂⁴	—	4	4	no	—
374	C₂⁴	—	4	4	no	—
375	C₂⁴	—	4	8	no	—
376	C₂⁴	—	4	8	no	—
377	C₂⁴	C₂×D₄×C₈₉	1	8	yes	—
378	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
379	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
380	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
381	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
382	#41	—	2	128	no	—
383	#41	—	2	64	no	—
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	C₂×D₄×C₈₉	1	8	yes	complemented
387	C₂×D₄	—	2	128	no	—
388	C₂×D₄	—	2	128	no	—
389	C₂×D₄	—	2	16	no	—
390	C₂×D₄	—	2	16	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	8	no	—
393	C₂×D₄	—	2	16	no	—
394	C₂×D₄	—	2	16	no	—
395	C₂×D₄	—	2	16	no	—
396	C₂×D₄	—	2	16	no	—
397	C₂×D₄	—	2	32	no	—
398	C₂×D₄	—	2	16	no	—
399	C₂×D₄	—	2	8	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	4	8	no	—
404	C₂×D₄	—	4	8	no	—
405	C₂×D₄	—	4	32	no	—
406	C₂×D₄	C₂×D₄×C₈₉	1	16	yes	—
407	C₂×D₄	C₂×D₄×C₈₉	1	16	yes	—
408	C₂×D₄	C₂×D₄×C₈₉	1	8	yes	complemented
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	16	no	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	—	2	8	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	2	8	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	—	4	16	no	—
421	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
423	C₂²×C₄	C₂×D₄×C₈₉	1	16	yes	complemented
424	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
425	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
426	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
427	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	16	no	—
432	C₂²⋊C₄	—	2	16	no	—
433	C₂²⋊C₄	—	2	8	no	—
434	C₂²⋊C₄	—	2	16	no	—
435	C₂²⋊C₄	—	2	16	no	—
436	C₂²⋊C₄	—	4	8	no	—
437	C₂²⋊C₄	—	4	16	no	—
438	C₂²⋊C₄	—	4	8	no	—
439	C₄²	—	2	8	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	8	no	—
443	C₂³	—	2	32	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	8	no	—
450	C₂³	—	2	8	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	8	no	—
453	C₂³	—	2	4	no	—
454	C₂³	—	2	8	no	—
455	C₂³	—	2	16	no	—
456	C₂³	—	4	8	no	—
457	C₂³	—	4	8	no	—
458	C₂³	—	4	16	no	—
459	C₂³	—	4	4	no	—
460	C₂³	—	4	4	no	—
461	C₂³	—	8	16	no	—
462	C₂³	—	8	8	no	—
463	C₂³	—	8	8	no	—
464	C₂³	—	8	32	no	—
465	C₂³	C₂²×D₄×C₈₉	1	8	yes	—
466	C₂³	C₂²×D₄×C₈₉	1	8	yes	—
467	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
468	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
469	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	complemented
470	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
471	Q₈	—	2	16	no	—
472	D₄	—	2	64	no	—
473	D₄	—	2	32	no	—
474	D₄	—	2	32	no	—
475	D₄	—	2	16	no	—
476	D₄	—	4	32	no	—
477	D₄	—	4	32	no	—
478	C₂×C₄	—	2	32	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	32	no	—
482	C₂×C₄	—	2	8	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	8	no	—
488	C₂×C₄	—	2	8	no	—
489	C₂×C₄	—	4	8	no	—
490	C₂×C₄	—	4	4	no	—
491	C₂×C₄	—	4	16	no	—
492	C₂×C₄	#120×C₈₉	1	16	yes	complemented
493	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
494	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
495	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
496	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	8	no	—
504	C₂²	—	2	8	no	—
505	C₂²	—	2	8	no	—
506	C₂²	—	2	16	no	—
507	C₂²	—	4	16	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	8	no	—
514	C₂²	—	4	16	no	—
515	C₂²	—	8	16	no	—
516	C₂²	—	8	32	no	—
517	C₂²	C₂×#120×C₈₉	1	4	yes	—
518	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
519	C₂²	C₂³×D₄×C₈₉	1	2	yes	—
520	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
521	C₂²	#585×C₈₉	1	2	yes	—
522	C₄	—	2	8	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	8	no	—
526	C₄	C₂×#120×C₈₉	1	8	yes	—
527	C₂	—	2	4	no	—
528	C₂	—	4	4	no	—
529	C₂	—	4	8	no	—
530	C₂	—	4	8	no	—
531	C₂	C₂²×#120×C₈₉	1	1	yes	characteristic
532	C₂	C₈₉×#3484	1	1	yes	characteristic
533	C₂	C₈₉×#3485	1	1	yes	characteristic
534	C₂	C₈₉×#3488	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
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. C₂²×H(5,𝔽₂)×C₈₉	C₂	semidirect
2. C₂×#525×C₈₉	C₂	semidirect
3. C₂²×#121×C₈₉	C₂	semidirect
4. C₈₉×#2408	C₂	semidirect
5. C₈₉×#2407	C₂	semidirect
6. C₈₉×#2405	C₂	semidirect
7. C₈₉×#2403	C₂	semidirect
8. C₈₉×#2311	C₂	semidirect
9. C₈₉×#2309	C₂	semidirect
13. C₂²×#41×C₈₉	C₂²	semidirect
16. C₂³×D₄×C₈₉	C₂²	semidirect
17. C₂³×D₄×C₈₉	C₂²	semidirect
18. C₂⁴×C₄×C₈₉	C₂²	semidirect
20. C₂×#123×C₈₉	C₂²	semidirect
21. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
23. C₂×#121×C₈₉	C₂²	semidirect
24. C₂×#121×C₈₉	C₂²	semidirect
25. C₂×#120×C₈₉	C₂²	semidirect
26. C₂×#120×C₈₉	C₂²	semidirect
27. C₂×C₄×D₄×C₈₉	C₂²	semidirect
28. C₂×#117×C₈₉	C₂²	semidirect
29. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
30. C₂²×C₂²⋊C₄×C₈₉	C₂²	semidirect
31. #394×C₈₉	C₂²	semidirect
32. #392×C₈₉	C₂²	semidirect
33. #391×C₈₉	C₂²	semidirect
34. #390×C₈₉	C₂²	semidirect
35. #390×C₈₉	C₂²	semidirect
36. #386×C₈₉	C₂²	semidirect
37. #386×C₈₉	C₂²	semidirect
38. #385×C₈₉	C₂²	semidirect
39. #380×C₈₉	C₂²	semidirect
40. #379×C₈₉	C₂²	semidirect
41. #379×C₈₉	C₂²	semidirect
51. C₂×#41×C₈₉	D₄	semidirect
61. C₂²×D₄×C₈₉	D₄	semidirect
118. C₂×Q₈×C₈₉	C₂×D₄	semidirect
140. C₂×D₄×C₈₉	C₂×D₄	semidirect
155. C₂²×C₄×C₈₉	C₂×D₄	semidirect
201. C₂³×C₈₉	H(5,𝔽₂)	semidirect
224. C₂×C₄×C₈₉	#120	semidirect
259. #35512	C₈₉	direct (N × H)
268. C₂²×H(5,𝔽₂)	C₂×C₈₉	semidirect
269. C₂×#525	C₂×C₈₉	semidirect
270. C₂²×#121	C₂×C₈₉	semidirect
271. #2408	C₂×C₈₉	semidirect
272. #2407	C₂×C₈₉	semidirect
273. #2405	C₂×C₈₉	semidirect
274. #2403	C₂×C₈₉	semidirect
275. #2311	C₂×C₈₉	semidirect
276. #2309	C₂×C₈₉	semidirect
277. C₈₉	#35512	direct (N × H)
281. C₂²×#41	C₂²×C₈₉	semidirect
284. C₂³×D₄	C₂²×C₈₉	semidirect
285. C₂³×D₄	C₂²×C₈₉	semidirect
286. C₂⁴×C₄	C₂²×C₈₉	semidirect
288. C₂×#123	C₂²×C₈₉	semidirect
289. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
291. C₂×#121	C₂²×C₈₉	semidirect
292. C₂×#121	C₂²×C₈₉	semidirect
293. C₂×#120	C₂²×C₈₉	semidirect
294. C₂×#120	C₂²×C₈₉	semidirect
295. C₂×C₄×D₄	C₂²×C₈₉	semidirect
296. C₂×#117	C₂²×C₈₉	semidirect
297. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
298. C₂²×C₂²⋊C₄	C₂²×C₈₉	semidirect
299. #394	C₂²×C₈₉	semidirect
300. #392	C₂²×C₈₉	semidirect
301. #391	C₂²×C₈₉	semidirect
302. #390	C₂²×C₈₉	semidirect
303. #390	C₂²×C₈₉	semidirect
304. #386	C₂²×C₈₉	semidirect
305. #386	C₂²×C₈₉	semidirect
306. #385	C₂²×C₈₉	semidirect
307. #380	C₂²×C₈₉	semidirect
308. #379	C₂²×C₈₉	semidirect
309. #379	C₂²×C₈₉	semidirect
319. C₂×#41	D₄×C₈₉	semidirect
329. C₂²×D₄	D₄×C₈₉	semidirect
386. C₂×Q₈	C₂×D₄×C₈₉	semidirect
408. C₂×D₄	C₂×D₄×C₈₉	semidirect
423. C₂²×C₄	C₂×D₄×C₈₉	semidirect
469. C₂³	H(5,𝔽₂)×C₈₉	semidirect
492. C₂×C₄	#120×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₈₉	2
C₂×#525×C₈₉	2
C₂²×#121×C₈₉	2
C₈₉×#2408	2
C₈₉×#2407	2
C₈₉×#2405	2
C₈₉×#2403	2
C₈₉×#2311	2
C₈₉×#2309	2
#35512	89

Subgroups — C89×#35512 #36413360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35512 #36413360