Finite Group Databasebeta

← #36552708

Subgroups — #36552708

Order 22784.

1034 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151652C211yesmaximal, characteristic
2C22×#15667922C211yesmaximal, characteristic
3C4×#15667968C211yesmaximal, characteristic
4#26020128C212yesmaximal, complemented
5#26020125C212yesmaximal, complemented
6#26020030C211yesmaximal, characteristic
7#26020026C211yesmaximal, characteristic
8C8×C22⋊Dic89C211yesmaximal, characteristic
9#26019999C212yesmaximal, complemented
10#97×Dic89C212yesmaximal, complemented
11C89×#1752C211yesmaximal, characteristic
12C23×#10623494C2211yescharacteristic
13C2×#15667968C412yes—
14C22×C4×Dic89C2211yescharacteristic
15#19151652C418yescomplemented
16C2×#15667941C2212yes—
17C2×C22⋊#10623494C2211yescharacteristic
18C2×#15667923C2212yes—
19C2×#15667922C418yescomplemented
20C2×#15667922C2211yescharacteristic
21C2×#15667922C2211yescharacteristic
22C2×#15667922C2212yes—
23C2×#15667922C2211yescharacteristic
24C2×#15667922C2211yescharacteristic
25C2×C8×Dic89C2212yes—
26C2×C8×Dic89C2212yes—
27C4×#13083553C2212yes—
28#19151449C412yes—
29C22⋊C4×Dic89C412yes—
30C4×C22⋊Dic89C2211yescharacteristic
31#19151443C412yes—
32C42×Dic89C2212yes—
33#19151353C2212yes—
34#19151352C2212yes—
35#19151352C2212yes—
36C23×C8×C89C2211yescharacteristic
37C2×C22⋊C8×C89C2211yescharacteristic
38C2×#97×C89C2212yes—
39C4×C22⋊C4×C89C2211yescharacteristic
40#336×C89C2212yes—
41C23×Dic89C2×C411yescharacteristic
42C23×Dic89C2×C411yescharacteristic
43C22×#10623494—24no—
44C22×#10623494C2×C414yes—
45C22×#10623494C2312yes—
46C22×#10623494C2311yescharacteristic
47C22×#10623494C2311yescharacteristic
48#15667968—28no—
49C2×#13083553C2×C414yes—
50C2×C22⋊Dic89C2×C412yes—
51C2×C4×Dic89C2×C414yes—
52C2×C4×Dic89C2×C414yes—
53C2×C4×Dic89C2×C411yescharacteristic
54C2×C4×Dic89C2×C411yescharacteristic
55C2×C4×Dic89C2×C414yes—
56C2×C4×Dic89C2×C414yes—
57C2×C4×Dic89C2312yes—
58C2×C4×Dic89C2311yescharacteristic
59C2×C4×Dic89C2311yescharacteristic
60#15667941C2×C418yes—
61C22⋊#10623494C2×C414yes—
62#15667923—24no—
63#15667922C2×C418yes—
64#15667922C2×C414yes—
65#15667922C2×C414yes—
66#15667922C2×C414yes—
67#15667922C2×C414yes—
68C8×Dic89—24no—
69C8×Dic89—24no—
70#15667919C2×C412yes—
71#15667919C2×C414yes—
72C23×C4×C89C2311yescharacteristic
73C22×C8×C89—24no—
74C22×C8×C89C2×C414yes—
75C22×C8×C89C2312yes—
76C22×C8×C89C2311yescharacteristic
77C22×C8×C89C2311yescharacteristic
78C2×C22⋊C4×C89C2×C412yes—
79C2×C42×C89C2312yes—
80C22⋊C8×C89C2×C414yes—
81#97×C89—24no—
82#95×C89C2×C412yes—
83C22×Dic89—24no—
84C22×Dic89—42no—
85C22×Dic89C4214yes—
86C22×Dic89C4214yes—
87C22×Dic89C22×C411yescharacteristic
88C22×Dic89C22×C411yescharacteristic
89C22×Dic89C22×C411yescharacteristic
90C22×Dic89C22×C411yescharacteristic
91C22×Dic89C22×C414yes—
92C2×#10623494—22no—
93C2×#10623494—22no—
94C2×#10623494—22no—
95C2×#10623494—28no—
96C2×#10623494C22×C412yes—
97C2×#10623494C22×C414yes—
98C2×#10623494C22×C412yes—
99C2×#10623494#4114yes—
100#13083553—28no—
101C22⋊Dic89M1618yes—
102C4×Dic89—22no—
103C4×Dic89—22no—
104C4×Dic89—28no—
105C4×Dic89—44no—
106C4×Dic89C4214yes—
107C4×Dic89C4214yes—
108C4×Dic89C22×C412yes—
109C4×Dic89C22×C412yes—
110C4×Dic89C22×C412yes—
111C4×Dic89C22×C412yes—
112C24×C89C22×C411yescharacteristic
113C22×C4×C89—22no—
114C22×C4×C89—22no—
115C22×C4×C89C22×C414yes—
116C22×C4×C89C22×C411yescharacteristic
117C22×C4×C89C22×C412yes—
118C22×C4×C89C22×C412yes—
119C22×C4×C89C2411yescharacteristic
120C2×C8×C89—22no—
121C2×C8×C89—22no—
122C2×C8×C89—22no—
123C2×C8×C89—28no—
124C2×C8×C89C22×C414yes—
125C2×C8×C89C22×C412yes—
126C2×C8×C89C22×C412yes—
127C2×C8×C89#4114yes—
128C22⋊C4×C89—24no—
129C42×C89—22no—
130C42×C89—22no—
131C23×C89—22no—
132C23×C89—22no—
133C23×C89C2×C4212yes—
134C23×C89C2×M1614yes—
135C23×C89C23×C411yescharacteristic
136C2×C4×C89—22no—
137C2×C4×C89—22no—
138C2×C4×C89—22no—
139C2×C4×C89—24no—
140C2×C4×C89—24no—
141C2×C4×C89—24no—
142C2×C4×C89—24no—
143C2×C4×C89C2×C4212yes—
144C2×C4×C89C23×C411yescharacteristic
145C2×C4×C89C23×C411yescharacteristic
146C2×C4×C89C2×#4112yes—
147C2×C4×C89C2×#4112yes—
148C2×Dic89—24no—
149C2×Dic89—22no—
150C2×Dic89—22no—
151C2×Dic89—22no—
152C2×Dic89—24no—
153C2×Dic89—22no—
154C2×Dic89—28no—
155C2×Dic89—44no—
156C2×Dic89C2×C4212yes—
157C2×Dic89C2×C4212yes—
158C2×Dic89C2×C4212yes—
159C2×Dic89C2×C4212yes—
160C2×Dic89C2×M1618yes—
161C8×C89—22no—
162C8×C89—24no—
163C8×C89#11714yescomplemented
164#10623494—22no—
165#10623494—24no—
166#10623494#11714yescomplemented
167C22×C89—22no—
168C22×C89—28no—
169C22×C89—22no—
170C22×C89C4×M1614yes—
171C22×C89C22×C4211yescharacteristic
172C22×C89C4×#4112yes—
173C22×C89C22×M1612yes—
174C22×C89#56812yes—
175C4×C89—22no—
176C4×C89—44no—
177C4×C89C2×#11711yescharacteristic
178C4×C89C2×#11711yescharacteristic
179C4×C89C4×#4112yes—
180Dic89—22no—
181Dic89—22no—
182Dic89—22no—
183Dic89—22no—
184Dic89—28no—
185#20479—891nomaximal, Sylow
186C2×C89—24no—
187C2×C89C4×#11711yescharacteristic
188C2×C89C2×C4×M1611yescharacteristic
189C2×C89#287311yescharacteristic
190C2×C89#297014yes—
191C2×#432—891no—
192C22×#97—891no—
193C4×#117—891no—
194#1771—892no—
195#1771—892no—
196#1752—891no—
197#1752D8911yescharacteristic
198C8×C22⋊C4—891no—
199C8×C22⋊C4—891no—
200#1745—892no—
201C4×#97—892no—
202C89#2047911yescharacteristic, complemented, Sylow
203C23×C8—891no—
204C23×C8C2×D8911yescharacteristic
205C2×#117—892no—
206C22×C42—891no—
207#432—898no—
208C2×C4⋊C8—892no—
209C2×C22⋊C8—891no—
210C2×C22⋊C8C2×D8911yescharacteristic
211C2×#97—898no—
212C2×#97—891no—
213C2×#97—891no—
214C2×#97—891no—
215C2×#97—891no—
216C2×#97—892no—
217C2×#97—892no—
218C2×#97C2×D8912yes—
219C2×C4×C8—892no—
220C2×C4×C8—892no—
221C4×C4⋊C4—892no—
222C4×C4⋊C4—892no—
223C4×C22⋊C4—891no—
224C4×C22⋊C4—892no—
225C4×C22⋊C4C2×D8911yescharacteristic
226#376—892no—
227C43—892no—
228#336—892no—
229#336—892no—
230#336—892no—
231#336C2×D8912yes—
232C23×C4—891no—
233C23×C4—891no—
234C23×C4C22×D8911yescharacteristic
235C22×C8—24no—
236C22×C8—891no—
237C22×C8—894no—
238C22×C8—891no—
239C22×C8—892no—
240C22×C8—1784no—
241C22×C8C4×D8914yes—
242C22×C8C22×D8911yescharacteristic
243C22×C8C22×D8912yes—
244C22×C8C22×D8911yescharacteristic
245#117—1788no—
246C2×C4⋊C4—894no—
247C2×C22⋊C4—892no—
248C2×C22⋊C4C4×D8912yes—
249C2×C42—891no—
250C2×C42—891no—
251C2×C42—894no—
252C2×C42—894no—
253C2×C42—891no—
254C2×C42—891no—
255C2×C42—894no—
256C2×C42—894no—
257C2×C42—892no—
258C2×C42C22×D8912yes—
259C4⋊C8—898no—
260C22⋊C8—894no—
261C22⋊C8C4×D8914yes—
262#97—24no—
263#97—894no—
264#97—894no—
265#97—894no—
266#97—894no—
267#97—898no—
268#97—1784no—
269C4×C8—1784no—
270C4×C8—1784no—
271#95—894no—
272#95—892no—
273#95C4×D8912yes—
274C24C2×C4×D8911yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—894no—
278C22×C4—891no—
279C22×C4—894no—
280C22×C4—891no—
281C22×C4—891no—
282C22×C4—891no—
283C22×C4—894no—
284C22×C4—3562no—
285C22×C4—1784no—
286C22×C4C2×C4×D8914yes—
287C22×C4C2×C4×D8912yes—
288C22×C4C2×C4×D8912yes—
289C22×C4C2×C4×D8911yescharacteristic
290C22×C4C23×D8911yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—28no—
295C2×C8—892no—
296C2×C8—894no—
297C2×C8—892no—
298C2×C8—894no—
299C2×C8—1788no—
300C2×C8—1782no—
301C2×C8—1782no—
302C2×C8—1782no—
303C2×C8C2×C4×D8912yes—
304C2×C8C2×C4×D8914yes—
305C2×C8C2×C4×D8912yes—
306C2×C8#1308357314yes—
307C4⋊C4—1788no—
308C22⋊C4—24no—
309C22⋊C4—898no—
310C42—22no—
311C42—22no—
312C42—892no—
313C42—892no—
314C42—892no—
315C42—892no—
316C42—894no—
317C42—894no—
318C42—3564no—
319C42—1788no—
320C42—1782no—
321C42—1782no—
322C23—22no—
323C23—22no—
324C23C42×D8912yes—
325C23M16×D8914yes—
326C23C22×C4×D8911yescharacteristic
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—892no—
335C2×C4—892no—
336C2×C4—892no—
337C2×C4—892no—
338C2×C4—898no—
339C2×C4—3564no—
340C2×C4—1788no—
341C2×C4—1784no—
342C2×C4—1782no—
343C2×C4—1782no—
344C2×C4—1782no—
345C2×C4—1784no—
346C2×C4—1782no—
347C2×C4C42×D8912yes—
348C2×C4C22×C4×D8911yescharacteristic
349C2×C4C22×C4×D8911yescharacteristic
350C2×C4#41×D8912yes—
351C2×C4C2×#1308357312yes—
352C8—22no—
353C8—24no—
354C8—894no—
355C8—1782no—
356C8—1784no—
357C8#1566796814yescomplemented
358C22—22no—
359C22—22no—
360C22—28no—
361C22#1915155014yes—
362C22C2×C42×D8911yescharacteristic
363C22C4×#1308357312yes—
364C22C2×M16×D8912yes—
365C22#1915236312yes—
366C4—22no—
367C4—44no—
368C4—1782no—
369C4—1782no—
370C4—1782no—
371C4—1782no—
372C4—1788no—
373C4#117×D8911yescharacteristic
374C4C2×#1566796811yescharacteristic
375C4C4×#1308357312yes—
376C2—24no—
377C2C4×#1566796811yescharacteristic
378C2C2×#1915155011yescharacteristic
379C2#2602866811yescharacteristic
380C2#2602926214yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutatorC22×C89C22×C42
FittingC89×#1752C2
socleC23×C89C23×C4
prime radicalC24×C89C22×C4
cosocleC22×C4×C89C24
hypercenter#1752D89
hypocenterC89#20479
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
4. #26020128C2semidirect
5. #26020125C2semidirect
9. #26019999C2semidirect
10. #97×Dic89C2semidirect
15. #19151652C4semidirect
19. C2×#15667922C4semidirect
163. C8×C89#117semidirect
166. #10623494#117semidirect
202. C89#20479semidirect
357. C8#15667968semidirect

Maximal subgroups

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

subgroupindex
C2×#191516522
C22×#156679222
C4×#156679682
#260201282
#260201252
#260200302
#260200262
C8×C22⋊Dic892
#260199992
#97×Dic892
C89×#17522
#2047989