Subgroups — C₈₉×#25448 #36403296

Order 22784.

860 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3156	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3069	C₂	1	2	yes	maximal, complemented
3	C₈₉×#3067	C₂	1	1	yes	maximal, characteristic
4	C₂×#473×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2049	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2009	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1953	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1915	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	#575×C₈₉	C₂²	1	2	yes	—
13	C₂×#137×C₈₉	C₂²	1	2	yes	—
14	C₂×#137×C₈₉	C₂²	1	2	yes	—
15	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#568×C₈₉	C₂²	1	2	yes	—
18	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
19	#536×C₈₉	C₂²	1	2	yes	—
20	#497×C₈₉	—	2	2	no	—
21	#495×C₈₉	—	2	1	no	—
22	#494×C₈₉	—	2	1	no	—
23	#473×C₈₉	—	2	1	no	—
24	#473×C₈₉	—	2	2	no	—
25	#473×C₈₉	—	2	2	no	—
26	#473×C₈₉	—	2	4	no	—
27	#472×C₈₉	—	2	1	no	—
28	#471×C₈₉	—	2	2	no	—
29	#471×C₈₉	—	2	4	no	—
30	#444×C₈₉	—	2	2	no	—
31	#443×C₈₉	—	2	2	no	—
32	#430×C₈₉	C₂²	1	2	yes	—
33	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
35	#421×C₈₉	C₂²	1	2	yes	—
36	#419×C₈₉	C₂²	1	2	yes	—
37	#413×C₈₉	C₂²	1	2	yes	—
38	#413×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#101×C₈₉	C₂²	1	2	yes	—
40	C₂×#101×C₈₉	C₂²	1	2	yes	—
41	#410×C₈₉	C₂²	1	1	yes	characteristic
42	#405×C₈₉	C₂²	1	1	yes	characteristic
43	#405×C₈₉	C₂²	1	1	yes	characteristic
44	#344×C₈₉	C₂²	1	1	yes	characteristic
45	#344×C₈₉	C₂²	1	1	yes	characteristic
46	#344×C₈₉	C₂²	1	2	yes	—
47	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×#41×C₈₉	C₂³	1	2	yes	—
49	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
50	#137×C₈₉	—	2	2	no	—
51	#137×C₈₉	—	2	2	no	—
52	#137×C₈₉	—	2	4	no	—
53	#137×C₈₉	—	2	4	no	—
54	#137×C₈₉	—	4	2	no	—
55	#137×C₈₉	—	4	2	no	—
56	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
57	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
58	#135×C₈₉	—	2	1	no	—
59	#135×C₈₉	—	2	1	no	—
60	#135×C₈₉	—	2	1	no	—
61	#135×C₈₉	—	2	1	no	—
62	#135×C₈₉	—	4	2	no	—
63	C₂×Q₁₆×C₈₉	—	2	2	no	—
64	C₂×Q₁₆×C₈₉	—	2	2	no	—
65	C₂×Q₁₆×C₈₉	—	2	1	no	—
66	C₂×Q₁₆×C₈₉	—	2	1	no	—
67	C₂×Q₁₆×C₈₉	—	2	1	no	—
68	C₂×Q₁₆×C₈₉	—	2	1	no	—
69	C₂×Q₁₆×C₈₉	—	2	1	no	—
70	C₂×Q₁₆×C₈₉	—	2	1	no	—
71	C₂×Q₁₆×C₈₉	D₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₈₉	—	2	2	no	—
73	C₂×SD₁₆×C₈₉	—	2	2	no	—
74	C₂×SD₁₆×C₈₉	D₄	1	2	yes	—
75	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
76	#131×C₈₉	—	2	4	no	—
77	#131×C₈₉	—	2	4	no	—
78	#131×C₈₉	—	2	1	no	—
79	#131×C₈₉	—	2	1	no	—
80	#131×C₈₉	—	2	1	no	—
81	#131×C₈₉	—	2	1	no	—
82	C₂×M₁₆×C₈₉	—	2	1	no	—
83	C₂×M₁₆×C₈₉	—	4	2	no	—
84	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
86	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
89	C₂²×C₈×C₈₉	—	2	1	no	—
90	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₈₉	—	2	2	no	—
92	#124×C₈₉	—	2	2	no	—
93	#123×C₈₉	—	2	4	no	—
94	C₂²⋊Q₈×C₈₉	—	4	2	no	—
95	#117×C₈₉	C₂³	1	1	yes	characteristic
96	#117×C₈₉	C₂³	1	1	yes	characteristic
97	#108×C₈₉	—	2	4	no	—
98	#108×C₈₉	—	2	1	no	—
99	#108×C₈₉	—	2	1	no	—
100	#108×C₈₉	D₄	1	2	yes	—
101	#108×C₈₉	D₄	1	2	yes	—
102	#104×C₈₉	—	2	4	no	—
103	#103×C₈₉	—	4	2	no	—
104	#101×C₈₉	—	2	2	no	—
105	#101×C₈₉	—	2	4	no	—
106	#101×C₈₉	—	2	4	no	—
107	#101×C₈₉	D₄	1	2	yes	—
108	#100×C₈₉	—	2	2	no	—
109	#100×C₈₉	D₄	1	2	yes	—
110	#99×C₈₉	—	2	2	no	—
111	#97×C₈₉	—	2	1	no	—
112	#97×C₈₉	—	2	1	no	—
113	C₄×C₈×C₈₉	—	2	1	no	—
114	C₄×C₈×C₈₉	—	2	1	no	—
115	#41×C₈₉	—	2	2	no	—
116	#41×C₈₉	—	2	1	no	—
117	#41×C₈₉	—	2	2	no	—
118	#41×C₈₉	—	2	1	no	—
119	#41×C₈₉	—	2	2	no	—
120	#41×C₈₉	—	2	1	no	—
121	#41×C₈₉	—	2	2	no	—
122	#41×C₈₉	—	2	1	no	—
123	C₂×Q₈×C₈₉	—	2	2	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	2	2	no	—
126	C₂×Q₈×C₈₉	—	2	2	no	—
127	C₂×Q₈×C₈₉	—	2	2	no	—
128	C₂×Q₈×C₈₉	—	2	2	no	—
129	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
130	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₈₉	—	2	1	no	—
132	C₂×D₄×C₈₉	—	2	2	no	—
133	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₈₉	—	2	1	no	—
136	C₂²×C₄×C₈₉	—	2	2	no	—
137	C₂²×C₄×C₈₉	—	2	2	no	—
138	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₈₉	—	2	1	no	—
140	Q₁₆×C₈₉	—	2	1	no	—
141	Q₁₆×C₈₉	—	4	4	no	—
142	Q₁₆×C₈₉	—	4	4	no	—
143	Q₁₆×C₈₉	—	4	2	no	—
144	Q₁₆×C₈₉	—	4	1	no	—
145	Q₁₆×C₈₉	—	4	1	no	—
146	Q₁₆×C₈₉	—	4	1	no	—
147	Q₁₆×C₈₉	—	4	1	no	—
148	SD₁₆×C₈₉	—	2	2	no	—
149	SD₁₆×C₈₉	—	2	2	no	—
150	SD₁₆×C₈₉	—	4	4	no	—
151	SD₁₆×C₈₉	—	4	4	no	—
152	SD₁₆×C₈₉	—	4	2	no	—
153	SD₁₆×C₈₉	—	4	2	no	—
154	D₈×C₈₉	—	2	1	no	—
155	D₈×C₈₉	—	2	1	no	—
156	D₈×C₈₉	—	4	2	no	—
157	M₁₆×C₈₉	—	2	1	no	—
158	M₁₆×C₈₉	—	2	2	no	—
159	M₁₆×C₈₉	—	2	1	no	—
160	M₁₆×C₈₉	—	2	1	no	—
161	M₁₆×C₈₉	—	2	2	no	—
162	M₁₆×C₈₉	—	2	1	no	—
163	M₁₆×C₈₉	—	2	2	no	—
164	M₁₆×C₈₉	—	4	4	no	—
165	M₁₆×C₈₉	—	4	1	no	—
166	M₁₆×C₈₉	—	4	1	no	—
167	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
168	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
169	M₁₆×C₈₉	C₂×D₄	1	4	yes	—
170	C₂×C₈×C₈₉	—	2	1	no	—
171	C₂×C₈×C₈₉	—	2	1	no	—
172	C₂×C₈×C₈₉	—	2	2	no	—
173	C₂×C₈×C₈₉	—	2	1	no	—
174	C₂×C₈×C₈₉	—	2	2	no	—
175	C₂×C₈×C₈₉	—	4	4	no	—
176	C₂×C₈×C₈₉	—	4	1	no	—
177	C₂×C₈×C₈₉	—	4	1	no	—
178	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
183	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
184	C₄⋊C₄×C₈₉	—	2	1	no	—
185	C₄⋊C₄×C₈₉	—	2	1	no	—
186	C₄⋊C₄×C₈₉	—	4	4	no	—
187	C₂²⋊C₄×C₈₉	—	2	1	no	—
188	C₂²⋊C₄×C₈₉	—	4	4	no	—
189	C₂²⋊C₄×C₈₉	#41	1	2	yes	—
190	C₄²×C₈₉	—	2	1	no	—
191	C₄²×C₈₉	—	2	1	no	—
192	C₂³×C₈₉	—	2	1	no	—
193	C₂³×C₈₉	—	2	2	no	—
194	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
195	Q₈×C₈₉	—	2	2	no	—
196	Q₈×C₈₉	—	2	1	no	—
197	Q₈×C₈₉	—	2	2	no	—
198	Q₈×C₈₉	—	2	1	no	—
199	Q₈×C₈₉	—	4	2	no	—
200	Q₈×C₈₉	—	4	2	no	—
201	Q₈×C₈₉	—	4	2	no	—
202	Q₈×C₈₉	—	4	2	no	—
203	D₄×C₈₉	—	2	2	no	—
204	D₄×C₈₉	—	2	1	no	—
205	D₄×C₈₉	—	2	2	no	—
206	D₄×C₈₉	—	2	1	no	—
207	D₄×C₈₉	—	4	2	no	—
208	D₄×C₈₉	—	4	1	no	—
209	D₄×C₈₉	—	4	1	no	—
210	D₄×C₈₉	—	4	2	no	—
211	C₂×C₄×C₈₉	—	2	1	no	—
212	C₂×C₄×C₈₉	—	2	2	no	—
213	C₂×C₄×C₈₉	—	2	2	no	—
214	C₂×C₄×C₈₉	—	4	1	no	—
215	C₂×C₄×C₈₉	—	4	1	no	—
216	C₂×C₄×C₈₉	—	4	2	no	—
217	C₂×C₄×C₈₉	—	4	2	no	—
218	C₂×C₄×C₈₉	—	4	2	no	—
219	C₂×C₄×C₈₉	—	4	2	no	—
220	C₂×C₄×C₈₉	—	4	1	no	—
221	C₂×C₄×C₈₉	—	4	2	no	—
222	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
228	C₈×C₈₉	—	2	2	no	—
229	C₈×C₈₉	—	2	4	no	—
230	C₈×C₈₉	—	2	2	no	—
231	C₈×C₈₉	—	4	1	no	—
232	C₈×C₈₉	—	4	2	no	—
233	C₈×C₈₉	#121	1	1	yes	characteristic
234	C₈×C₈₉	#121	1	1	yes	characteristic
235	C₈×C₈₉	#121	1	1	yes	characteristic
236	C₈×C₈₉	#121	1	1	yes	characteristic
237	C₂²×C₈₉	—	4	1	no	—
238	C₂²×C₈₉	—	4	1	no	—
239	C₂²×C₈₉	—	4	2	no	—
240	C₂²×C₈₉	—	8	2	no	—
241	C₂²×C₈₉	—	8	1	no	—
242	C₂²×C₈₉	#532	1	1	yes	characteristic
243	C₂²×C₈₉	#546	1	1	yes	characteristic
244	C₂²×C₈₉	#550	1	1	yes	characteristic
245	C₄×C₈₉	—	4	2	no	—
246	C₄×C₈₉	—	4	1	no	—
247	C₄×C₈₉	—	4	2	no	—
248	C₄×C₈₉	—	8	1	no	—
249	C₄×C₈₉	—	8	2	no	—
250	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
251	C₄×C₈₉	#540	1	1	yes	characteristic
252	C₄×C₈₉	D₄²	1	1	yes	characteristic
253	C₄×C₈₉	D₄²	1	1	yes	characteristic
254	#25448	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₈₉	—	2	1	no	—
256	C₂×C₈₉	—	2	1	no	—
257	C₂×C₈₉	—	2	1	no	—
258	C₂×C₈₉	—	8	1	no	—
259	C₂×C₈₉	—	8	2	no	—
260	C₂×C₈₉	#2428	1	1	yes	characteristic
261	#3156	C₂×C₈₉	1	1	yes	characteristic, complemented
262	#3069	C₂×C₈₉	1	2	yes	complemented
263	#3067	C₂×C₈₉	1	1	yes	characteristic
264	C₂×#473	C₂×C₈₉	1	1	yes	characteristic, complemented
265	#2957	C₂×C₈₉	1	1	yes	characteristic, complemented
266	#2049	C₂×C₈₉	1	2	yes	complemented
267	#2009	C₂×C₈₉	1	2	yes	complemented
268	#1979	C₂×C₈₉	1	1	yes	characteristic, complemented
269	#1953	C₂×C₈₉	1	1	yes	characteristic, complemented
270	#1915	C₂×C₈₉	1	2	yes	complemented
271	#1757	C₂×C₈₉	1	1	yes	characteristic, complemented
272	C₈₉	#25448	1	1	yes	characteristic, direct product factor, Sylow
273	#575	C₂²×C₈₉	1	2	yes	—
274	C₂×#137	C₂²×C₈₉	1	2	yes	—
275	C₂×#137	C₂²×C₈₉	1	2	yes	—
276	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
277	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
278	#568	C₂²×C₈₉	1	2	yes	—
279	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
280	#536	C₂²×C₈₉	1	2	yes	—
281	#497	—	2	2	no	—
282	#495	—	2	1	no	—
283	#494	—	2	1	no	—
284	#473	—	2	1	no	—
285	#473	—	2	2	no	—
286	#473	—	2	2	no	—
287	#473	—	2	4	no	—
288	#472	—	2	1	no	—
289	#471	—	2	2	no	—
290	#471	—	2	4	no	—
291	#444	—	2	2	no	—
292	#443	—	2	2	no	—
293	#430	C₂²×C₈₉	1	2	yes	—
294	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
295	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
296	#421	C₂²×C₈₉	1	2	yes	—
297	#419	C₂²×C₈₉	1	2	yes	—
298	#413	C₂²×C₈₉	1	2	yes	—
299	#413	C₂²×C₈₉	1	1	yes	characteristic
300	C₂×#101	C₂²×C₈₉	1	2	yes	—
301	C₂×#101	C₂²×C₈₉	1	2	yes	—
302	#410	C₂²×C₈₉	1	1	yes	characteristic
303	#405	C₂²×C₈₉	1	1	yes	characteristic
304	#405	C₂²×C₈₉	1	1	yes	characteristic
305	#344	C₂²×C₈₉	1	1	yes	characteristic
306	#344	C₂²×C₈₉	1	1	yes	characteristic
307	#344	C₂²×C₈₉	1	2	yes	—
308	C₂×#41	C₂³×C₈₉	1	2	yes	—
309	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
310	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
311	#137	—	2	2	no	—
312	#137	—	2	2	no	—
313	#137	—	2	4	no	—
314	#137	—	2	4	no	—
315	#137	—	4	2	no	—
316	#137	—	4	2	no	—
317	AGL(1,ℤ/8)	—	2	2	no	—
318	AGL(1,ℤ/8)	—	2	2	no	—
319	#135	—	2	1	no	—
320	#135	—	2	1	no	—
321	#135	—	2	1	no	—
322	#135	—	2	1	no	—
323	#135	—	4	2	no	—
324	C₂×Q₁₆	—	2	2	no	—
325	C₂×Q₁₆	—	2	2	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	1	no	—
328	C₂×Q₁₆	—	2	1	no	—
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×Q₁₆	—	2	1	no	—
331	C₂×Q₁₆	—	2	1	no	—
332	C₂×Q₁₆	D₄×C₈₉	1	1	yes	characteristic
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×SD₁₆	—	2	2	no	—
335	C₂×SD₁₆	D₄×C₈₉	1	2	yes	—
336	C₂×D₈	D₄×C₈₉	1	1	yes	characteristic
337	#131	—	2	4	no	—
338	#131	—	2	4	no	—
339	#131	—	2	1	no	—
340	#131	—	2	1	no	—
341	#131	—	2	1	no	—
342	#131	—	2	1	no	—
343	C₂×M₁₆	—	2	1	no	—
344	C₂×M₁₆	—	4	2	no	—
345	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
346	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
347	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
348	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
349	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
350	C₂²×C₈	—	2	1	no	—
351	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
352	C₄⋊Q₈	—	2	2	no	—
353	#124	—	2	2	no	—
354	#123	—	2	4	no	—
355	C₂²⋊Q₈	—	4	2	no	—
356	#117	C₂³×C₈₉	1	1	yes	characteristic
357	#117	C₂³×C₈₉	1	1	yes	characteristic
358	#108	—	2	4	no	—
359	#108	—	2	1	no	—
360	#108	—	2	1	no	—
361	#108	D₄×C₈₉	1	2	yes	—
362	#108	D₄×C₈₉	1	2	yes	—
363	#104	—	2	4	no	—
364	#103	—	4	2	no	—
365	#101	—	2	2	no	—
366	#101	—	2	4	no	—
367	#101	—	2	4	no	—
368	#101	D₄×C₈₉	1	2	yes	—
369	#100	—	2	2	no	—
370	#100	D₄×C₈₉	1	2	yes	—
371	#99	—	2	2	no	—
372	#97	—	2	1	no	—
373	#97	—	2	1	no	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	—	2	1	no	—
376	#41	—	2	2	no	—
377	#41	—	2	1	no	—
378	#41	—	2	2	no	—
379	#41	—	2	1	no	—
380	#41	—	2	2	no	—
381	#41	—	2	1	no	—
382	#41	—	2	2	no	—
383	#41	—	2	1	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
391	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
395	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
400	Q₁₆	—	2	1	no	—
401	Q₁₆	—	2	1	no	—
402	Q₁₆	—	4	4	no	—
403	Q₁₆	—	4	4	no	—
404	Q₁₆	—	4	2	no	—
405	Q₁₆	—	4	1	no	—
406	Q₁₆	—	4	1	no	—
407	Q₁₆	—	4	1	no	—
408	Q₁₆	—	4	1	no	—
409	SD₁₆	—	2	2	no	—
410	SD₁₆	—	2	2	no	—
411	SD₁₆	—	4	4	no	—
412	SD₁₆	—	4	4	no	—
413	SD₁₆	—	4	2	no	—
414	SD₁₆	—	4	2	no	—
415	D₈	—	2	1	no	—
416	D₈	—	2	1	no	—
417	D₈	—	4	2	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	2	1	no	—
422	M₁₆	—	2	2	no	—
423	M₁₆	—	2	1	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	4	4	no	—
426	M₁₆	—	4	1	no	—
427	M₁₆	—	4	1	no	—
428	M₁₆	C₂×D₄×C₈₉	1	2	yes	—
429	M₁₆	C₂×D₄×C₈₉	1	2	yes	—
430	M₁₆	C₂×D₄×C₈₉	1	4	yes	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	4	4	no	—
437	C₂×C₈	—	4	1	no	—
438	C₂×C₈	—	4	1	no	—
439	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
440	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
441	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
442	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
443	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
444	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	4	4	no	—
448	C₂²⋊C₄	—	2	1	no	—
449	C₂²⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	#41×C₈₉	1	2	yes	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₂³	—	2	1	no	—
454	C₂³	—	2	2	no	—
455	C₂³	C₂×#41×C₈₉	1	1	yes	characteristic
456	Q₈	—	2	2	no	—
457	Q₈	—	2	1	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	2	1	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	4	2	no	—
462	Q₈	—	4	2	no	—
463	Q₈	—	4	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	2	1	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	1	no	—
468	D₄	—	4	2	no	—
469	D₄	—	4	1	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
487	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
488	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	4	no	—
491	C₈	—	2	2	no	—
492	C₈	—	4	1	no	—
493	C₈	—	4	2	no	—
494	C₈	#121×C₈₉	1	1	yes	characteristic
495	C₈	#121×C₈₉	1	1	yes	characteristic
496	C₈	#121×C₈₉	1	1	yes	characteristic
497	C₈	#121×C₈₉	1	1	yes	characteristic
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	8	2	no	—
502	C₂²	—	8	1	no	—
503	C₂²	#532×C₈₉	1	1	yes	characteristic
504	C₂²	#546×C₈₉	1	1	yes	characteristic
505	C₂²	#550×C₈₉	1	1	yes	characteristic
506	C₄	—	4	2	no	—
507	C₄	—	4	1	no	—
508	C₄	—	4	2	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	2	no	—
511	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
512	C₄	#540×C₈₉	1	1	yes	characteristic
513	C₄	D₄²×C₈₉	1	1	yes	characteristic
514	C₄	D₄²×C₈₉	1	1	yes	characteristic
515	C₂	—	2	1	no	—
516	C₂	—	2	1	no	—
517	C₂	—	2	1	no	—
518	C₂	—	8	1	no	—
519	C₂	—	8	2	no	—
520	C₂	C₈₉×#2428	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	#2428
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₈₉×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3156	C₂	semidirect
2. C₈₉×#3069	C₂	semidirect
4. C₂×#473×C₈₉	C₂	semidirect
5. C₈₉×#2957	C₂	semidirect
6. C₈₉×#2049	C₂	semidirect
7. C₈₉×#2009	C₂	semidirect
8. C₈₉×#1979	C₂	semidirect
9. C₈₉×#1953	C₂	semidirect
10. C₈₉×#1915	C₂	semidirect
11. C₈₉×#1757	C₂	semidirect
254. #25448	C₈₉	direct (N × H)
261. #3156	C₂×C₈₉	semidirect
262. #3069	C₂×C₈₉	semidirect
264. C₂×#473	C₂×C₈₉	semidirect
265. #2957	C₂×C₈₉	semidirect
266. #2049	C₂×C₈₉	semidirect
267. #2009	C₂×C₈₉	semidirect
268. #1979	C₂×C₈₉	semidirect
269. #1953	C₂×C₈₉	semidirect
270. #1915	C₂×C₈₉	semidirect
271. #1757	C₂×C₈₉	semidirect
272. C₈₉	#25448	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#3156	2
C₈₉×#3069	2
C₈₉×#3067	2
C₂×#473×C₈₉	2
C₈₉×#2957	2
C₈₉×#2049	2
C₈₉×#2009	2
C₈₉×#1979	2
C₈₉×#1953	2
C₈₉×#1915	2
C₈₉×#1757	2
#25448	89

Subgroups — C89×#25448 #36403296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25448 #36403296