Finite Group Databasebeta

← #14418550

Subgroups — #14418550

Order 1920.

728 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754174C211yesmaximal, characteristic
2#11752417C211yesmaximal, characteristic
3C2×C3×C4×#170C211yesmaximal, characteristic
4#11752011C212yesmaximal
5#11751988C212yesmaximal
6#11751823C211yesmaximal, characteristic, complemented
7#11751789C211yesmaximal, characteristic, complemented
8#11751785C212yesmaximal, complemented
9#11751768C211yesmaximal, characteristic, complemented
10#11751758C212yesmaximal, complemented
11Dic6×#170C211yesmaximal, characteristic, complemented
12#10617261—31nomaximal, Hall
13C2×#6762C2212yes—
14C22×C3×#170C2211yescharacteristic
15C22×C3×#170C2211yescharacteristic
16#91813C411yescharacteristic
17C5×#924C411yescharacteristic
18C2×C3×C4×Dic5C2211yescharacteristic
19#91574C412yes—
20#91528C2212yes—
21#91468C411yescharacteristic
22#91378C412yes—
23#91376C411yescharacteristic
24#91333C2211yescharacteristic
25Dic5×Dic6C2211yescharacteristic
26#91323C2212yes—
27#91233C2212yes—
28C4⋊(C3⋊#170)C2211yescharacteristic
29C4⋊(C3⋊#170)C2211yescharacteristic
30C3×C4×#170C2211yescharacteristic
31C3×C4×#170C2211yescharacteristic
32C3×C4×#170C2211yescharacteristic
33C3×C4×#170C2211yescharacteristic
34#91177C2212yes—
35#91177C2212yes—
36Dic3⋊#170C2212yes—
37Dic3⋊#170C2212yes—
38#91169C2212yes—
39#91169C2212yes—
40Dic3×#170C2212yes—
41Dic3×#170C2212yes—
42#78445—51nomaximal, Hall
43#65956—31no—
44#65788—31no—
45Q8×#170—31no—
46#65758—31no—
47C2×C4×#170D311yescharacteristic
48#65709—31no—
49#65707—32no—
50#65703—32no—
51#65703—31no—
52#65696—32no—
53#65691—32no—
54C22×C3×C4×C5C2×C411yescharacteristic
55C2×Dic30C2×C411yescharacteristic
56C2×C5×Dic6C2×C411yescharacteristic
57C22×C3×Dic5C2×C411yescharacteristic
58C22×C3×Dic5C2311yescharacteristic
59C2×#1204C2×C412yes—
60#6762—24no—
61C2×C3⋊#170C2312yes—
62C2×C3⋊#170C2312yes—
63C2×C3×#170—22no—
64C2×C3×#170Q814yescomplemented
65C2×C3×#170C2311yescharacteristic
66C2×C3×#170C2311yescharacteristic
67C2×C3×#170C2311yescharacteristic
68C2×C3×#170C2311yescharacteristic
69#6719C2×C412yes—
70C4⋊Dic15C2×C411yescharacteristic
71#6712C2×C412yes—
72C5×#217C2×C412yes—
73C5×C4⋊Dic3C2×C411yescharacteristic
74C5×Dic3⋊C4C2×C412yes—
75C3×C4×Dic5C2×C411yescharacteristic
76C3×C4×Dic5C2×C411yescharacteristic
77C3×C4×Dic5C2311yescharacteristic
78C3×C4×Dic5C2311yescharacteristic
79Dic3⋊Dic5C2×C412yes—
80#6669C2×C412yes—
81#6668C2×C412yes—
82#6668C2×C412yes—
83#6668C2×C412yes—
84Dic3×Dic5C2312yes—
85Dic3×Dic5C2312yes—
86#5470—51no—
87C2×C3×C4×C8—51no—
88#5102—52no—
89#5101—52no—
90#4924—51no—
91#4679—52no—
92#4679—51no—
93#4679—51no—
94C8×Dic6—51no—
95C8×Dic6—52no—
96C8×Dic6—51no—
97C2×#704—32no—
98C22×#170C2×D311yescharacteristic
99C22×#170C2×D311yescharacteristic
100C5×C22⋊Q8—31no—
101Q8×Dic5—31no—
102#4090—31no—
103#4070—31no—
104C2×C4×Dic5C2×D311yescharacteristic
105#4035—32no—
106#4034—31no—
107#4033—32no—
108#4033—31no—
109#4024—32no—
110#4023—32no—
111#4004—32no—
112#4004—32no—
113#4004—32no—
114#4002—32no—
115#4002—32no—
116C4⋊#170—32no—
117C4⋊#170—32no—
118C4⋊#170—31no—
119C4⋊#170—31no—
120C4×#170—32no—
121C4×#170—32no—
122C4×#170C2×D311yescharacteristic
123C4×#170C2×D311yescharacteristic
124C4×#170C2×D311yescharacteristic
125C4×#170C2×D311yescharacteristic
126#2960—151noSylow
127C23×C3×C5C22×C411yescharacteristic
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5C22×C411yescharacteristic
130C2×C3×C4×C5C22×C411yescharacteristic
131C2×Dic15C22×C412yes—
132C2×Dic15C22×C412yes—
133Dic30—22no—
134C2×C5×Dic3C22×C412yes—
135C2×C5×Dic3C22×C412yes—
136C5×Dic6—22no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5C22×C411yescharacteristic
139C2×C3×Dic5C22×C411yescharacteristic
140C2×C3×Dic5C22×C411yescharacteristic
141C2×C3×Dic5C2×Q812yes—
142C2×C3×Dic5#4112yes—
143C2×C3×Dic5C2411yescharacteristic
144#1204—24no—
145C3⋊#170—22no—
146C3⋊#170—22no—
147C3×#170—21no—
148C3×#170—21no—
149C3×#170C2×Q812yes—
150C3×#170C2×Q812yes—
151C22×C3×C8—51no—
152C22×C3×C8—51no—
153C2×C3×C42—51no—
154#926—52no—
155#924—51no—
156#924—52no—
157#924AGL(1,𝔽5)11yescharacteristic
158C2×#208—52no—
159#870—52no—
160#869—51no—
161#869—51no—
162C4×Dic6—51no—
163C4×Dic6—51no—
164C4×Dic6—52no—
165C3×C4×C8—51no—
166C3×C4×C8—51no—
167C3×C4×C8—51no—
168C3×C4×C8—51no—
169#815—52no—
170#815—52no—
171Dic3⋊C8—52no—
172Dic3⋊C8—52no—
173Dic3⋊C8—52no—
174Dic3⋊C8—52no—
175C8×Dic3—52no—
176C8×Dic3—52no—
177C4⋊(C3⋊C8)—51no—
178C4⋊(C3⋊C8)—51no—
179C4⋊(C3⋊C8)—52no—
180C2×C5×Q8—31no—
181C22×C4×C5C4×D311yescharacteristic
182C22×Dic5C4×D311yescharacteristic
183C22×Dic5C22×D311yescharacteristic
184C2×Dic10—31no—
185C2×Dic10—32no—
186#704—64no—
187C2×#170—22no—
188C2×#170—32no—
189C2×#170—32no—
190C2×#170Dic614yescomplemented
191C2×#170C22×D311yescharacteristic
192C2×#170C22×D311yescharacteristic
193C2×#170C22×D311yescharacteristic
194C2×#170C22×D311yescharacteristic
195C5×C4⋊C4—32no—
196C5×C4⋊C4—31no—
197C5×C22⋊C4—32no—
198C22⋊Dic5—32no—
199C4⋊Dic5—31no—
200C4⋊Dic5—32no—
201#683—32no—
202#683—32no—
203#683—32no—
204#683—32no—
205#683—32no—
206C4×Dic5—32no—
207C4×Dic5—32no—
208C4×Dic5C4×D311yescharacteristic
209C4×Dic5C4×D311yescharacteristic
210C4×Dic5C22×D311yescharacteristic
211C4×Dic5C22×D311yescharacteristic
212#533—151no—
213#446—152no—
214#446—151no—
215#446—151no—
216C8×Q8—151no—
217C8×Q8—152no—
218C8×Q8—151no—
219#424—152no—
220#423—151no—
221#405—152no—
222C2×C4×C8—51no—
223C22×C3×C5—21no—
224C22×C3×C5—21no—
225C22×C3×C5C4×Q812yes—
226C22×C3×C5C23×C411yescharacteristic
227C3×C4×C5—21no—
228C3×C4×C5#13112yes—
229Dic15—22no—
230Dic15—22no—
231C3×Dic5—21no—
232C3×Dic5C4×Q812yes—
233C3×Dic5#13112yes—
234C3×Dic5C22×Q811yescharacteristic
235C3×Dic5C2×#4111yescharacteristic
236C5×Dic3—22no—
237C5×Dic3—22no—
238C22×C3×C4—51no—
239C22×C3×C4—51no—
240C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
241C2×Dic6—51no—
242C2×Dic6—52no—
243C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
244C2×C3×C8—51no—
245C2×C3×C8—51no—
246C2×C3×C8—54no—
247C2×C3×C8—51no—
248C2×C3×C8—51no—
249C2×C3×C8—102no—
250C3×C42—51no—
251C3×C42—51no—
252C3×C42—51no—
253C3×C42—51no—
254#217—52no—
255#217C2×AGL(1,𝔽5)12yes—
256C4⋊Dic3—51no—
257C4⋊Dic3—52no—
258C4⋊Dic3—52no—
259C4⋊Dic3—52no—
260C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
261Dic3⋊C4—52no—
262Dic3⋊C4—52no—
263Dic3⋊C4—52no—
264Dic3⋊C4C2×AGL(1,𝔽5)12yes—
265C4×Dic3—52no—
266C4×Dic3—52no—
267#208—104no—
268C2×C3⋊C8—52no—
269C2×C3⋊C8—52no—
270C23×C5C2×C4×D311yescharacteristic
271C5×Q8—62no—
272C2×C4×C5—22no—
273C2×C4×C5—32no—
274C2×C4×C5—32no—
275C2×C4×C5C2×C4×D311yescharacteristic
276C2×C4×C5C2×C4×D311yescharacteristic
277C2×Dic5—22no—
278C2×Dic5—32no—
279C2×Dic5—32no—
280C2×Dic5C2×Dic612yes—
281C2×Dic5C2×C4×D311yescharacteristic
282C2×Dic5C2×C4×D311yescharacteristic
283C2×Dic5C2×C4×D311yescharacteristic
284C2×Dic5#23512yes—
285C2×Dic5C23×D311yescharacteristic
286Dic10—62no—
287Dic10—64no—
288#170—21no—
289#170—21no—
290#170—62no—
291#170—62no—
292#170C2×Dic612yes—
293#170C2×Dic612yes—
294C2×M16—152no—
295C22×C8—51no—
296C22×C8—51no—
297C4⋊Q8—151no—
298C4⋊Q8—151no—
299#125—152no—
300C22⋊Q8—31no—
301C22⋊Q8—152no—
302C22⋊Q8—151no—
303C4×Q8—152no—
304C4×Q8—151no—
305C4×Q8—151no—
306#117—152no—
307C2×C42—51no—
308C4⋊C8—151no—
309C4⋊C8—151no—
310C4⋊C8—152no—
311C4⋊C8—152no—
312C4⋊C8—152no—
313C4⋊C8—152no—
314C4⋊C8—152no—
315#97—152no—
316#97—152no—
317C4×C8—51no—
318C4×C8—51no—
319C4×C8—51no—
320C4×C8—51no—
321C4×C8—152no—
322C4×C8—152no—
323C2×C3×C5—22no—
324C2×C3×C5C2×C4×Q811yescharacteristic
325C2×C3×C5C2×#13111yescharacteristic
326C2×C3×C5C2×#13111yescharacteristic
327C23×C3C22×AGL(1,𝔽5)11yescharacteristic
328C2×C3×C4—22no—
329C2×C3×C4—52no—
330C2×C3×C4—52no—
331C2×C3×C4—51no—
332C2×C3×C4—51no—
333C2×C3×C4—51no—
334C2×C3×C4—51no—
335C2×C3×C4—102no—
336C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
337C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
338C2×Dic3—52no—
339C2×Dic3—52no—
340C2×Dic3C22×AGL(1,𝔽5)12yes—
341C2×Dic3C22×AGL(1,𝔽5)12yes—
342Dic6—22no—
343Dic6—104no—
344Dic6—102no—
345C3×C8—52no—
346C3×C8—52no—
347C3×C8—101no—
348C3×C8—101no—
349C3⋊C8—102no—
350C3⋊C8—102no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5C4×Dic612yes—
354C22×C5C22×C4×D311yescharacteristic
355C4×C5—21no—
356C4×C5—62no—
357C4×C5—62no—
358C4×C5#90112yes—
359Dic5—21no—
360Dic5—62no—
361Dic5—62no—
362Dic5C4×Dic612yes—
363Dic5#90112yes—
364Dic5C22×Dic611yescharacteristic
365Dic5C2×#23511yescharacteristic
366C2×Q8—31no—
367C2×Q8—152no—
368C2×Q8—151no—
369C22×C4—51no—
370C22×C4—51no—
371C22×C4D3×AGL(1,𝔽5)11yescharacteristic
372M16—304no—
373C2×C8—51no—
374C2×C8—51no—
375C2×C8—54no—
376C2×C8—51no—
377C2×C8—51no—
378C2×C8—102no—
379C2×C8—152no—
380C2×C8—152no—
381C4⋊C4—32no—
382C4⋊C4—31no—
383C4⋊C4—152no—
384C4⋊C4—152no—
385C4⋊C4—152no—
386C4⋊C4—152no—
387C4⋊C4—152no—
388C4⋊C4—152no—
389C4⋊C4—151no—
390C22⋊C4—32no—
391C22⋊C4—152no—
392C42—51no—
393C42—51no—
394C42—51no—
395C42—51no—
396C42—152no—
397C42—152no—
398C3×C5#296011yescharacteristic, complemented, Hall
399C22×C3—21no—
400C22×C3—21no—
401C22×C3Q8×AGL(1,𝔽5)12yes—
402C22×C3C23×AGL(1,𝔽5)11yescharacteristic
403C3×C4—21no—
404C3×C4—52no—
405C3×C4—51no—
406C3×C4—51no—
407C3×C4—52no—
408C3×C4—101no—
409C3×C4#413112yes—
410Dic3—22no—
411Dic3—22no—
412Dic3—102no—
413Dic3—102no—
414C2×C5—22no—
415C2×C5C2×C4×Dic611yescharacteristic
416C2×C5C2×#90111yescharacteristic
417C2×C5C2×#90111yescharacteristic
418C23C2×D3×AGL(1,𝔽5)11yescharacteristic
419Q8—62no—
420Q8—304no—
421Q8—302no—
422C2×C4—22no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—52no—
426C2×C4—52no—
427C2×C4—51no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—102no—
432C2×C4—152no—
433C2×C4—152no—
434C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
435C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
436C8—52no—
437C8—52no—
438C8—101no—
439C8—101no—
440C8—302no—
441C8—302no—
442C2×C3—22no—
443C2×C3C2×#413111yescharacteristic
444C2×C3C2×Q8×AGL(1,𝔽5)11yescharacteristic
445C2×C3D5⋊#13111yescharacteristic
446C5#7844511yescharacteristic, complemented, Sylow
447C22—21no—
448C22—21no—
449C22AGL(1,𝔽5)×Dic612yes—
450C22C22×D3×AGL(1,𝔽5)11yescharacteristic
451C4—21no—
452C4—51no—
453C4—51no—
454C4—52no—
455C4—52no—
456C4—62no—
457C4—62no—
458C4—101no—
459C4—302no—
460C4—302no—
461C4#9192712yes—
462C3#1061726111yescharacteristic, complemented, Sylow
463C2—22no—
464C2C2×AGL(1,𝔽5)×Dic611yescharacteristic
465C2D5⋊#90111yescharacteristic
466C2C2×#9192711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radicalC23×C3×C5C22×C4
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2960
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11751823C2semidirect
7. #11751789C2semidirect
8. #11751785C2semidirect
9. #11751768C2semidirect
10. #11751758C2semidirect
11. Dic6×#170C2semidirect
64. C2×C3×#170Q8semidirect
190. C2×#170Dic6semidirect
398. C3×C5#2960semidirect
446. C5#78445semidirect
462. C3#10617261semidirect

Maximal subgroups

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

subgroupindex
#117541742
#117524172
C2×C3×C4×#1702
#117520112
#117519882
#117518232
#117517892
#117517852
#117517682
#117517582
Dic6×#1702
#106172613
#784455