Subgroups — C₈₉×#60818 #36438666

Order 22784.

2138 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	C₈₉×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3498	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3222	C₂	1	8	yes	maximal, complemented
4	C₈₉×#3210	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3208	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3152	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#492×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#3043	C₂	1	4	yes	maximal, complemented
11	C₈₉×#3009	C₂	1	4	yes	maximal, complemented
12	C₈₉×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	2	yes	—
14	C₂×#137×C₈₉	C₂²	1	4	yes	complemented
15	C₂²×SD₁₆×C₈₉	C₂²	1	4	yes	complemented
16	#557×C₈₉	—	2	2	no	—
17	#556×C₈₉	—	2	2	no	—
18	#552×C₈₉	—	2	4	no	—
19	#550×C₈₉	—	2	2	no	—
20	D₄²×C₈₉	—	2	4	no	—
21	D₄²×C₈₉	—	2	2	no	—
22	#541×C₈₉	C₂²	1	4	yes	complemented
23	#536×C₈₉	C₂²	1	2	yes	—
24	#535×C₈₉	C₂²	1	4	yes	—
25	#535×C₈₉	C₂²	1	2	yes	—
26	#533×C₈₉	C₂²	1	4	yes	complemented
27	#533×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	#532×C₈₉	C₂²	1	1	yes	characteristic
29	#532×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
31	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#127×C₈₉	C₂²	1	2	yes	—
33	#519×C₈₉	C₂²	1	2	yes	complemented
34	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
35	#497×C₈₉	C₂²	1	4	yes	complemented
36	#494×C₈₉	C₂²	1	2	yes	complemented
37	#492×C₈₉	C₂²	1	4	yes	complemented
38	#492×C₈₉	C₂²	1	4	yes	complemented
39	#492×C₈₉	C₂²	1	2	yes	complemented
40	#490×C₈₉	C₂²	1	4	yes	complemented
41	#488×C₈₉	C₂²	1	4	yes	complemented
42	#487×C₈₉	C₂²	1	2	yes	complemented
43	#486×C₈₉	C₂²	1	2	yes	complemented
44	#486×C₈₉	C₂²	1	4	yes	complemented
45	#464×C₈₉	C₂²	1	16	yes	complemented
46	#460×C₈₉	C₂²	1	16	yes	complemented
47	#452×C₈₉	C₂²	1	16	yes	complemented
48	#450×C₈₉	C₂²	1	16	yes	complemented
49	#433×C₈₉	C₂²	1	4	yes	complemented
50	#431×C₈₉	C₂²	1	4	yes	complemented
51	#424×C₈₉	C₂²	1	2	yes	complemented
52	#418×C₈₉	C₂²	1	4	yes	complemented
53	#416×C₈₉	C₂²	1	4	yes	complemented
54	#408×C₈₉	C₂²	1	2	yes	—
55	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	H(5,𝔽₂)×C₈₉	—	4	8	no	—
58	C₂×#41×C₈₉	—	2	2	no	—
59	C₂×#41×C₈₉	C₂³	1	2	yes	—
60	C₂×#41×C₈₉	C₂³	1	4	yes	—
61	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
63	C₂²×D₄×C₈₉	—	2	4	no	—
64	C₂²×D₄×C₈₉	—	2	2	no	—
65	C₂²×D₄×C₈₉	—	2	2	no	—
66	C₂²×D₄×C₈₉	—	2	4	no	—
67	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
68	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
69	#137×C₈₉	—	4	8	no	—
70	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	—
71	C₂×SD₁₆×C₈₉	—	4	8	no	—
72	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
73	C₂×SD₁₆×C₈₉	C₂³	1	16	yes	—
74	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₈₉	—	2	2	no	—
77	C₄⋊Q₈×C₈₉	—	2	2	no	—
78	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
82	#127×C₈₉	—	2	4	no	—
83	#127×C₈₉	—	2	2	no	—
84	#127×C₈₉	C₂³	1	4	yes	—
85	#127×C₈₉	C₂³	1	1	yes	characteristic
86	#127×C₈₉	C₂³	1	2	yes	—
87	#127×C₈₉	C₂³	1	2	yes	—
88	#126×C₈₉	—	2	8	no	—
89	#125×C₈₉	—	2	8	no	—
90	#125×C₈₉	—	2	4	no	—
91	#125×C₈₉	C₂³	1	2	yes	—
92	#124×C₈₉	—	2	2	no	—
93	#124×C₈₉	C₂³	1	4	yes	—
94	#124×C₈₉	C₂³	1	2	yes	—
95	#123×C₈₉	—	2	4	no	—
96	C₂²⋊Q₈×C₈₉	—	2	2	no	—
97	C₂²⋊Q₈×C₈₉	—	2	4	no	—
98	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
99	#121×C₈₉	—	2	2	no	—
100	#121×C₈₉	—	2	4	no	—
101	#121×C₈₉	—	2	4	no	—
102	#121×C₈₉	—	2	8	no	—
103	#120×C₈₉	—	2	4	no	—
104	#120×C₈₉	—	2	8	no	—
105	C₄×Q₈×C₈₉	—	2	4	no	—
106	C₄×Q₈×C₈₉	—	2	4	no	—
107	C₄×Q₈×C₈₉	—	2	2	no	—
108	C₄×Q₈×C₈₉	D₄	1	2	yes	—
109	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
110	C₄×D₄×C₈₉	—	2	4	no	—
111	C₄×D₄×C₈₉	—	2	2	no	—
112	C₄×D₄×C₈₉	D₄	1	4	yes	—
113	C₄×D₄×C₈₉	D₄	1	2	yes	—
114	#117×C₈₉	—	2	4	no	—
115	#117×C₈₉	—	2	4	no	—
116	#117×C₈₉	C₂³	1	2	yes	—
117	#117×C₈₉	C₂³	1	1	yes	characteristic
118	#117×C₈₉	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
122	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
123	#103×C₈₉	C₂³	1	8	yes	—
124	#102×C₈₉	C₂³	1	8	yes	—
125	#102×C₈₉	C₂³	1	16	yes	—
126	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
127	#97×C₈₉	C₂³	1	2	yes	—
128	C₄×C₈×C₈₉	C₂³	1	2	yes	—
129	C₄×C₈×C₈₉	C₂³	1	2	yes	—
130	C₄×C₈×C₈₉	C₂³	1	2	yes	—
131	C₂⁴×C₈₉	—	2	4	no	—
132	C₂⁴×C₈₉	—	2	2	no	—
133	#41×C₈₉	—	2	4	no	—
134	#41×C₈₉	—	2	8	no	—
135	#41×C₈₉	—	4	8	no	—
136	#41×C₈₉	—	4	8	no	—
137	C₂×Q₈×C₈₉	—	2	2	no	—
138	C₂×Q₈×C₈₉	—	2	2	no	—
139	C₂×Q₈×C₈₉	—	2	8	no	—
140	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
142	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
143	C₂×D₄×C₈₉	—	2	4	no	—
144	C₂×D₄×C₈₉	—	2	2	no	—
145	C₂×D₄×C₈₉	—	2	4	no	—
146	C₂×D₄×C₈₉	—	2	2	no	—
147	C₂×D₄×C₈₉	—	2	4	no	—
148	C₂×D₄×C₈₉	—	2	2	no	—
149	C₂×D₄×C₈₉	—	2	8	no	—
150	C₂×D₄×C₈₉	—	2	4	no	—
151	C₂×D₄×C₈₉	—	2	8	no	—
152	C₂×D₄×C₈₉	—	4	8	no	—
153	C₂×D₄×C₈₉	—	4	16	no	—
154	C₂×D₄×C₈₉	—	4	8	no	—
155	C₂×D₄×C₈₉	—	8	4	no	—
156	C₂×D₄×C₈₉	—	8	8	no	—
157	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
158	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
159	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
160	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₈₉	—	2	2	no	—
162	C₂²×C₄×C₈₉	—	2	4	no	—
163	C₂²×C₄×C₈₉	—	2	2	no	—
164	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₈₉	—	4	8	no	—
169	SD₁₆×C₈₉	—	4	16	no	—
170	SD₁₆×C₈₉	—	4	8	no	—
171	M₁₆×C₈₉	—	2	4	no	—
172	C₂×C₈×C₈₉	—	2	4	no	—
173	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
174	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₈₉	—	2	4	no	—
176	C₄⋊C₄×C₈₉	—	2	4	no	—
177	C₄⋊C₄×C₈₉	—	2	2	no	—
178	C₄⋊C₄×C₈₉	—	2	8	no	—
179	C₄⋊C₄×C₈₉	—	2	2	no	—
180	C₄⋊C₄×C₈₉	—	2	4	no	—
181	C₄⋊C₄×C₈₉	—	2	8	no	—
182	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
185	C₂²⋊C₄×C₈₉	—	2	2	no	—
186	C₂²⋊C₄×C₈₉	—	2	8	no	—
187	C₂²⋊C₄×C₈₉	—	2	8	no	—
188	C₂²⋊C₄×C₈₉	—	2	4	no	—
189	C₂²⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
190	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
191	C₄²×C₈₉	—	2	8	no	—
192	C₄²×C₈₉	—	2	2	no	—
193	C₄²×C₈₉	C₂×D₄	1	4	yes	—
194	C₄²×C₈₉	C₂×D₄	1	2	yes	—
195	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
196	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
197	C₄²×C₈₉	C₂⁴	1	2	yes	—
198	C₂³×C₈₉	—	2	4	no	—
199	C₂³×C₈₉	—	2	2	no	—
200	C₂³×C₈₉	—	4	8	no	—
201	C₂³×C₈₉	—	4	4	no	—
202	C₂³×C₈₉	—	8	4	no	—
203	C₂³×C₈₉	—	8	8	no	—
204	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
205	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₈₉	—	2	2	no	—
207	Q₈×C₈₉	—	2	8	no	—
208	Q₈×C₈₉	—	2	4	no	—
209	Q₈×C₈₉	—	4	4	no	—
210	Q₈×C₈₉	—	4	4	no	—
211	D₄×C₈₉	—	2	4	no	—
212	D₄×C₈₉	—	2	2	no	—
213	D₄×C₈₉	—	2	8	no	—
214	D₄×C₈₉	—	2	4	no	—
215	D₄×C₈₉	—	4	8	no	—
216	D₄×C₈₉	—	4	8	no	—
217	D₄×C₈₉	—	4	4	no	—
218	D₄×C₈₉	—	4	4	no	—
219	D₄×C₈₉	—	4	8	no	—
220	D₄×C₈₉	—	4	4	no	—
221	D₄×C₈₉	—	8	8	no	—
222	D₄×C₈₉	—	8	4	no	—
223	C₂×C₄×C₈₉	—	2	4	no	—
224	C₂×C₄×C₈₉	—	2	8	no	—
225	C₂×C₄×C₈₉	—	2	4	no	—
226	C₂×C₄×C₈₉	—	2	4	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	2	2	no	—
229	C₂×C₄×C₈₉	—	4	8	no	—
230	C₂×C₄×C₈₉	—	4	4	no	—
231	C₂×C₄×C₈₉	—	4	4	no	—
232	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
238	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
239	C₈×C₈₉	—	2	4	no	—
240	C₈×C₈₉	—	2	4	no	—
241	C₂²×C₈₉	—	2	2	no	—
242	C₂²×C₈₉	—	2	2	no	—
243	C₂²×C₈₉	—	2	4	no	—
244	C₂²×C₈₉	—	4	4	no	—
245	C₂²×C₈₉	—	4	2	no	—
246	C₂²×C₈₉	—	4	8	no	—
247	C₂²×C₈₉	—	4	4	no	—
248	C₂²×C₈₉	—	16	4	no	—
249	C₂²×C₈₉	—	16	8	no	—
250	C₂²×C₈₉	#535	1	2	yes	—
251	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₈₉	—	2	2	no	—
253	C₄×C₈₉	—	2	4	no	—
254	C₄×C₈₉	—	2	2	no	—
255	C₄×C₈₉	—	4	2	no	—
256	C₄×C₈₉	—	4	4	no	—
257	C₄×C₈₉	—	4	4	no	—
258	C₄×C₈₉	#535	1	2	yes	—
259	C₄×C₈₉	#577	1	4	yes	—
260	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
261	#60818	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂×C₈₉	—	2	2	no	—
263	C₂×C₈₉	—	4	2	no	—
264	C₂×C₈₉	—	8	4	no	—
265	C₂×C₈₉	—	8	2	no	—
266	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
267	C₂×C₈₉	C₂×#577	1	2	yes	—
268	#3509	C₂×C₈₉	1	1	yes	characteristic, complemented
269	#3498	C₂×C₈₉	1	1	yes	characteristic
270	#3222	C₂×C₈₉	1	8	yes	complemented
271	#3210	C₂×C₈₉	1	4	yes	complemented
272	#3208	C₂×C₈₉	1	4	yes	complemented
273	#3152	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₂×#492	C₂×C₈₉	1	1	yes	characteristic, complemented
275	#3138	C₂×C₈₉	1	1	yes	characteristic, complemented
276	#3131	C₂×C₈₉	1	1	yes	characteristic, complemented
277	#3043	C₂×C₈₉	1	4	yes	complemented
278	#3009	C₂×C₈₉	1	4	yes	complemented
279	#2922	C₂×C₈₉	1	1	yes	characteristic, complemented
280	C₈₉	#60818	1	1	yes	characteristic, direct product factor, Sylow
281	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	2	yes	—
282	C₂×#137	C₂²×C₈₉	1	4	yes	complemented
283	C₂²×SD₁₆	C₂²×C₈₉	1	4	yes	complemented
284	#557	—	2	2	no	—
285	#556	—	2	2	no	—
286	#552	—	2	4	no	—
287	#550	—	2	2	no	—
288	D₄²	—	2	4	no	—
289	D₄²	—	2	2	no	—
290	#541	C₂²×C₈₉	1	4	yes	complemented
291	#536	C₂²×C₈₉	1	2	yes	—
292	#535	C₂²×C₈₉	1	4	yes	—
293	#535	C₂²×C₈₉	1	2	yes	—
294	#533	C₂²×C₈₉	1	4	yes	complemented
295	#533	C₂²×C₈₉	1	1	yes	characteristic, complemented
296	#532	C₂²×C₈₉	1	1	yes	characteristic
297	#532	C₂²×C₈₉	1	2	yes	—
298	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
299	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
300	C₂×#127	C₂²×C₈₉	1	2	yes	—
301	#519	C₂²×C₈₉	1	2	yes	complemented
302	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
303	#497	C₂²×C₈₉	1	4	yes	complemented
304	#494	C₂²×C₈₉	1	2	yes	complemented
305	#492	C₂²×C₈₉	1	4	yes	complemented
306	#492	C₂²×C₈₉	1	4	yes	complemented
307	#492	C₂²×C₈₉	1	2	yes	complemented
308	#490	C₂²×C₈₉	1	4	yes	complemented
309	#488	C₂²×C₈₉	1	4	yes	complemented
310	#487	C₂²×C₈₉	1	2	yes	complemented
311	#486	C₂²×C₈₉	1	2	yes	complemented
312	#486	C₂²×C₈₉	1	4	yes	complemented
313	#464	C₂²×C₈₉	1	16	yes	complemented
314	#460	C₂²×C₈₉	1	16	yes	complemented
315	#452	C₂²×C₈₉	1	16	yes	complemented
316	#450	C₂²×C₈₉	1	16	yes	complemented
317	#433	C₂²×C₈₉	1	4	yes	complemented
318	#431	C₂²×C₈₉	1	4	yes	complemented
319	#424	C₂²×C₈₉	1	2	yes	complemented
320	#418	C₂²×C₈₉	1	4	yes	complemented
321	#416	C₂²×C₈₉	1	4	yes	complemented
322	#408	C₂²×C₈₉	1	2	yes	—
323	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic, complemented
324	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
325	H(5,𝔽₂)	—	4	8	no	—
326	C₂×#41	—	2	2	no	—
327	C₂×#41	C₂³×C₈₉	1	2	yes	—
328	C₂×#41	C₂³×C₈₉	1	4	yes	—
329	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
330	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
331	C₂²×D₄	—	2	4	no	—
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	4	no	—
335	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
336	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
337	#137	—	4	8	no	—
338	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	—
339	C₂×SD₁₆	—	4	8	no	—
340	C₂×SD₁₆	C₂³×C₈₉	1	16	yes	—
341	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
342	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
343	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
344	C₄⋊Q₈	—	2	2	no	—
345	C₄⋊Q₈	—	2	2	no	—
346	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
347	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
348	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
350	#127	—	2	4	no	—
351	#127	—	2	2	no	—
352	#127	C₂³×C₈₉	1	2	yes	—
353	#127	C₂³×C₈₉	1	1	yes	characteristic
354	#127	C₂³×C₈₉	1	4	yes	—
355	#127	C₂³×C₈₉	1	2	yes	—
356	#126	—	2	8	no	—
357	#125	—	2	8	no	—
358	#125	—	2	4	no	—
359	#125	C₂³×C₈₉	1	2	yes	—
360	#124	—	2	2	no	—
361	#124	C₂³×C₈₉	1	4	yes	—
362	#124	C₂³×C₈₉	1	2	yes	—
363	#123	—	2	4	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	4	no	—
366	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
367	#121	—	2	2	no	—
368	#121	—	2	4	no	—
369	#121	—	2	4	no	—
370	#121	—	2	8	no	—
371	#120	—	2	4	no	—
372	#120	—	2	8	no	—
373	C₄×Q₈	—	2	4	no	—
374	C₄×Q₈	—	2	4	no	—
375	C₄×Q₈	—	2	2	no	—
376	C₄×Q₈	D₄×C₈₉	1	2	yes	—
377	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
378	C₄×D₄	—	2	4	no	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	D₄×C₈₉	1	4	yes	—
381	C₄×D₄	D₄×C₈₉	1	2	yes	—
382	#117	—	2	4	no	—
383	#117	—	2	4	no	—
384	#117	C₂³×C₈₉	1	2	yes	—
385	#117	C₂³×C₈₉	1	1	yes	characteristic
386	#117	C₂³×C₈₉	1	2	yes	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
390	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
391	#103	C₂³×C₈₉	1	8	yes	—
392	#102	C₂³×C₈₉	1	8	yes	—
393	#102	C₂³×C₈₉	1	16	yes	—
394	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
395	#97	C₂³×C₈₉	1	2	yes	—
396	C₄×C₈	C₂³×C₈₉	1	2	yes	—
397	C₄×C₈	C₂³×C₈₉	1	2	yes	—
398	C₄×C₈	C₂³×C₈₉	1	2	yes	—
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	2	no	—
401	#41	—	2	4	no	—
402	#41	—	2	8	no	—
403	#41	—	4	8	no	—
404	#41	—	4	8	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	8	no	—
408	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
409	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
410	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	4	8	no	—
421	C₂×D₄	—	4	16	no	—
422	C₂×D₄	—	4	8	no	—
423	C₂×D₄	—	8	4	no	—
424	C₂×D₄	—	8	8	no	—
425	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
426	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
427	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
428	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
433	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
436	Q₁₆	—	4	8	no	—
437	SD₁₆	—	4	16	no	—
438	SD₁₆	—	4	8	no	—
439	M₁₆	—	2	4	no	—
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
442	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	8	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
451	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
452	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
453	C₂²⋊C₄	—	2	2	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	8	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
458	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
459	C₄²	—	2	8	no	—
460	C₄²	—	2	2	no	—
461	C₄²	C₂×D₄×C₈₉	1	4	yes	—
462	C₄²	C₂×D₄×C₈₉	1	2	yes	—
463	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
464	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
465	C₄²	C₂⁴×C₈₉	1	2	yes	—
466	C₂³	—	2	4	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	4	8	no	—
469	C₂³	—	4	4	no	—
470	C₂³	—	8	4	no	—
471	C₂³	—	8	8	no	—
472	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
473	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
474	Q₈	—	2	2	no	—
475	Q₈	—	2	8	no	—
476	Q₈	—	2	4	no	—
477	Q₈	—	4	4	no	—
478	Q₈	—	4	4	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	2	no	—
481	D₄	—	2	8	no	—
482	D₄	—	2	4	no	—
483	D₄	—	4	8	no	—
484	D₄	—	4	8	no	—
485	D₄	—	4	4	no	—
486	D₄	—	4	4	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	4	no	—
489	D₄	—	8	8	no	—
490	D₄	—	8	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	4	8	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
501	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
505	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
506	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
507	C₈	—	2	4	no	—
508	C₈	—	2	4	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	4	8	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	16	4	no	—
517	C₂²	—	16	8	no	—
518	C₂²	#535×C₈₉	1	2	yes	—
519	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
520	C₄	—	2	4	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	4	4	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	4	no	—
526	C₄	#535×C₈₉	1	2	yes	—
527	C₄	#577×C₈₉	1	4	yes	—
528	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
529	C₂	—	2	2	no	—
530	C₂	—	4	2	no	—
531	C₂	—	8	4	no	—
532	C₂	—	8	2	no	—
533	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
534	C₂	C₂×#577×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₈₉×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄×#41×C₈₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₈₉×#3509	C₂	semidirect
3. C₈₉×#3222	C₂	semidirect
4. C₈₉×#3210	C₂	semidirect
5. C₈₉×#3208	C₂	semidirect
6. C₈₉×#3152	C₂	semidirect
7. C₂×#492×C₈₉	C₂	semidirect
8. C₈₉×#3138	C₂	semidirect
9. C₈₉×#3131	C₂	semidirect
10. C₈₉×#3043	C₂	semidirect
11. C₈₉×#3009	C₂	semidirect
12. C₈₉×#2922	C₂	semidirect
14. C₂×#137×C₈₉	C₂²	semidirect
15. C₂²×SD₁₆×C₈₉	C₂²	semidirect
22. #541×C₈₉	C₂²	semidirect
26. #533×C₈₉	C₂²	semidirect
27. #533×C₈₉	C₂²	semidirect
30. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
33. #519×C₈₉	C₂²	semidirect
35. #497×C₈₉	C₂²	semidirect
36. #494×C₈₉	C₂²	semidirect
37. #492×C₈₉	C₂²	semidirect
38. #492×C₈₉	C₂²	semidirect
39. #492×C₈₉	C₂²	semidirect
40. #490×C₈₉	C₂²	semidirect
41. #488×C₈₉	C₂²	semidirect
42. #487×C₈₉	C₂²	semidirect
43. #486×C₈₉	C₂²	semidirect
44. #486×C₈₉	C₂²	semidirect
45. #464×C₈₉	C₂²	semidirect
46. #460×C₈₉	C₂²	semidirect
47. #452×C₈₉	C₂²	semidirect
48. #450×C₈₉	C₂²	semidirect
49. #433×C₈₉	C₂²	semidirect
50. #431×C₈₉	C₂²	semidirect
51. #424×C₈₉	C₂²	semidirect
52. #418×C₈₉	C₂²	semidirect
53. #416×C₈₉	C₂²	semidirect
55. C₄×M₁₆×C₈₉	C₂²	semidirect
56. C₂×C₄×C₈×C₈₉	C₂²	semidirect
261. #60818	C₈₉	direct (N × H)
268. #3509	C₂×C₈₉	semidirect
270. #3222	C₂×C₈₉	semidirect
271. #3210	C₂×C₈₉	semidirect
272. #3208	C₂×C₈₉	semidirect
273. #3152	C₂×C₈₉	semidirect
274. C₂×#492	C₂×C₈₉	semidirect
275. #3138	C₂×C₈₉	semidirect
276. #3131	C₂×C₈₉	semidirect
277. #3043	C₂×C₈₉	semidirect
278. #3009	C₂×C₈₉	semidirect
279. #2922	C₂×C₈₉	semidirect
280. C₈₉	#60818	direct (N × H)
282. C₂×#137	C₂²×C₈₉	semidirect
283. C₂²×SD₁₆	C₂²×C₈₉	semidirect
290. #541	C₂²×C₈₉	semidirect
294. #533	C₂²×C₈₉	semidirect
295. #533	C₂²×C₈₉	semidirect
298. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
301. #519	C₂²×C₈₉	semidirect
303. #497	C₂²×C₈₉	semidirect
304. #494	C₂²×C₈₉	semidirect
305. #492	C₂²×C₈₉	semidirect
306. #492	C₂²×C₈₉	semidirect
307. #492	C₂²×C₈₉	semidirect
308. #490	C₂²×C₈₉	semidirect
309. #488	C₂²×C₈₉	semidirect
310. #487	C₂²×C₈₉	semidirect
311. #486	C₂²×C₈₉	semidirect
312. #486	C₂²×C₈₉	semidirect
313. #464	C₂²×C₈₉	semidirect
314. #460	C₂²×C₈₉	semidirect
315. #452	C₂²×C₈₉	semidirect
316. #450	C₂²×C₈₉	semidirect
317. #433	C₂²×C₈₉	semidirect
318. #431	C₂²×C₈₉	semidirect
319. #424	C₂²×C₈₉	semidirect
320. #418	C₂²×C₈₉	semidirect
321. #416	C₂²×C₈₉	semidirect
323. C₄×M₁₆	C₂²×C₈₉	semidirect
324. C₂×C₄×C₈	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3509	2
C₈₉×#3498	2
C₈₉×#3222	2
C₈₉×#3210	2
C₈₉×#3208	2
C₈₉×#3152	2
C₂×#492×C₈₉	2
C₈₉×#3138	2
C₈₉×#3131	2
C₈₉×#3043	2
C₈₉×#3009	2
C₈₉×#2922	2
#60818	89

Subgroups — C89×#60818 #36438666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60818 #36438666