Finite Group Databasebeta

← #14424610

Subgroups — #14424610

Order 1920.

1400 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92004C211yesmaximal, characteristic
2#11756285C211yesmaximal, characteristic, complemented
3C2×#91322C211yesmaximal, characteristic, complemented
4C22×#6662C211yesmaximal, characteristic, complemented
5C5×#5002C211yesmaximal, characteristic, complemented
6C3×#65293C211yesmaximal, characteristic, complemented
7#11748350C211yesmaximal, characteristic, complemented
8#11747521C214yesmaximal, complemented
9#11747496C214yesmaximal, complemented
10#10610568—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C22×Dic30C2211yescharacteristic
13C2×C5×#235C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C2×#6782—22no—
16C2×D15⋊Q8C2212yes—
17C2×#6769—21no—
18#92004—24no—
19#92004—24no—
20C2×#6768C2212yes—
21C2×D3×Dic10—21no—
22C3×C5×#117C2211yescharacteristic
23#91786C2214yes—
24#91775—22no—
25#91764—21no—
26#91758—21no—
27C22×C5×C3⋊C8C2211yescharacteristic
28C2×C3×#681C2211yescharacteristic
29#91322—24no—
30C2×#6663C2212yes—
31C2×#6662—24no—
32C2×#6662C2214yes—
33C2×#6659C2212yes—
34C5×#810C2212yes—
35C5×#809C2212yes—
36C3×#3942C2214yes—
37#90957C2214yes—
38#78961—51nomaximal, Hall
39C2×#4150—31no—
40C2×#4097—31no—
41C22×Dic20—31no—
42#65860—31no—
43C5×#416—31no—
44#65399—31no—
45#65293D311yescharacteristic, complemented
46#65097—34no—
47#65072—34no—
48C22×C3×C4×C5C2311yescharacteristic
49C2×#1220D412yes—
50C22×Dic15—21no—
51#6817—22no—
52#6817—22no—
53C2×D60D411yescharacteristic
54C2×C4×D15C2312yes—
55C2×Dic30—22no—
56C2×Dic30—22no—
57C2×Dic30—21no—
58C2×Dic30D411yescharacteristic
59C2×Dic30C2312yes—
60C2×C5×#67—21no—
61C5×#235—22no—
62C5×#235D414yes—
63C2×C5×D12C2311yescharacteristic
64C2×C4×C5×D3—21no—
65C2×C5×Dic6C2311yescharacteristic
66C22×C3×Dic5—21no—
67C2×C3×Dic10—21no—
68C2×C3×Dic10—22no—
69C2×C3×Dic10D414yes—
70C2×C3×Dic10C2312yes—
71C2×#1204—22no—
72C2×#1203—22no—
73C2×#1200—22no—
74#6782—216no—
75C2×D3×Dic5—22no—
76D15⋊Q8—24no—
77D15⋊Q8—24no—
78#6769—28no—
79#6768—24no—
80#6768—24no—
81D3×Dic10—28no—
82C3×C5×C4⋊C4C2312yes—
83C3×C5×C22⋊C4—21no—
84C3×C42×C5—21no—
85#6714—24no—
86C4⋊Dic15—24no—
87C2×C5×C3⋊C8—22no—
88C2×C5×C3⋊C8C2312yes—
89C3×#681—22no—
90C2×C3×#168C2312yes—
91#6663—24no—
92#6662—28no—
93#6659—24no—
94C2×#1007—51no—
95C22×#216—51no—
96C2×#909—51no—
97#5526—51no—
98C3×#419—51no—
99#5129—51no—
100#5002D511yescharacteristic, complemented
101#4809—54no—
102#4808—54no—
103C2×C5×#41—31no—
104#4150—64no—
105#4150—64no—
106C2×Q8×D5—32no—
107C2×#711—32no—
108C2×#711—62no—
109C2×#709—31no—
110C2×#709—61no—
111C22×Dic10—31no—
112C22×Dic10—61no—
113C22×Dic10C2×D311yescharacteristic
114C22×C5×C8—31no—
115C5×#117C2×D311yescharacteristic
116#4097—64no—
117C2×#689—32no—
118C2×#687—32no—
119C2×#681C2×D311yescharacteristic
120C2×Dic20—34no—
121C2×Dic20—64no—
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#103—32no—
127C5×#102—32no—
128#3952—34no—
129#3942C2×D314yes—
130#3011—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2411yescharacteristic
139#1220—24no—
140C2×Dic15—22no—
141C2×Dic15—42no—
142C2×Dic15C2×D412yes—
143D60—22no—
144C4×D15—22no—
145C4×D15—22no—
146Dic30—22no—
147Dic30—22no—
148Dic30—22no—
149Dic30—42no—
150C5×#67—24no—
151C2×C5×Dic3—21no—
152C5×D12—21no—
153C5×D12C2×D412yes—
154C4×C5×D3—24no—
155C5×Dic6—21no—
156C5×Dic6C2×D412yes—
157C2×C3×Dic5—24no—
158C2×C3×Dic5—22no—
159C3×Dic10—24no—
160C3×Dic10—22no—
161C3×Dic10C2×D414yes—
162#1204—28no—
163#1203—28no—
164#1200—28no—
165D3×Dic5—28no—
166C3×#168—22no—
167C5×C3⋊C8—22no—
168C22×C3×Q8—51no—
169#1007—104no—
170#1007—104no—
171C2×#239—52no—
172C2×#239—101no—
173C2×D3×Q8—52no—
174C2×D3×Q8—101no—
175C2×#235—51no—
176C2×#235—102no—
177C2×#235C2×D511yescharacteristic
178C22×Dic6—51no—
179C2×C3×M16—51no—
180C3×#117C2×D511yescharacteristic
181C2×#216—54no—
182C2×#216—104no—
183C22×C3⋊C8C2×D511yescharacteristic
184#909—104no—
185C2×Dic12—52no—
186C2×#204—52no—
187#897—54no—
188#886—102no—
189#875—101no—
190#869—101no—
191C3×#103—54no—
192(C2×D3)⋊C8—54no—
193#810C2×D512yes—
194#809C2×D512yes—
195C5×#41—34no—
196C5×#41—62no—
197C2×C5×Q8—31no—
198C2×C5×D4—31no—
199C2×C5×D4—61no—
200C22×C4×C5—61no—
201C22×C4×C5C22×D311yescharacteristic
202C2×C22⋊D5—32no—
203C2×C22⋊D5—62no—
204C22×Dic5—21no—
205C22×Dic5—62no—
206C22×Dic5—61no—
207Q8×D5—64no—
208Q8×D5—64no—
209#711—64no—
210#711—616no—
211#711—64no—
212#709—68no—
213#709—62no—
214#709—62no—
215C2×D20—31no—
216C2×C4×D5—32no—
217C2×C4×D5—62no—
218C2×Dic10—21no—
219C2×Dic10—22no—
220C2×Dic10—31no—
221C2×Dic10—32no—
222C2×Dic10—68no—
223C2×Dic10—62no—
224C2×Dic10—62no—
225C2×Dic10—62no—
226C2×Dic10—61no—
227C2×Dic10D1214yes—
228C2×Dic10C22×D312yes—
229C2×C5×C8—32no—
230C2×C5×C8—62no—
231C5×C4⋊C4C22×D312yes—
232C5×C22⋊C4—21no—
233C42×C5—21no—
234#689—64no—
235#687—64no—
236#685—64no—
237C4⋊Dic5—64no—
238#681—22no—
239C2×#168C22×D312yes—
240Dic20—68no—
241C2×2−1+4—151no—
242C2×#137—151no—
243C22×Q16—151no—
244#536—151no—
245#452—154no—
246#451—154no—
247#419—51no—
248#416—31no—
249#408—151no—
250C22×C3×C5—22no—
251C22×C3×C5#12012yes—
252C22×C3×C5C22×D411yescharacteristic
253C2×D15—22no—
254C2×D15—24no—
255C2×C5×D3—22no—
256C2×C5×D3—42no—
257C3×C4×C5—42no—
258C3×C4×C5#12012yes—
259C3×C4×C5C22×D411yescharacteristic
260C3×C4×C5C22×D411yescharacteristic
261Dic15—22no—
262Dic15—42no—
263C3×Dic5—24no—
264C5×Dic3—22no—
265C2×C3×Q8—54no—
266C2×C3×Q8—52no—
267C2×C3×Q8—102no—
268C2×C3×Q8—101no—
269C22×C3×C4—101no—
270C22×C3×C4C22×D511yescharacteristic
271C2×#67—21no—
272C2×#67—52no—
273C22×Dic3—101no—
274#239—108no—
275#239—104no—
276#239—104no—
277D3×Q8—104no—
278D3×Q8—108no—
279D3×Q8—104no—
280#235—22no—
281#235—1016no—
282#235—102no—
283#235—102no—
284#235D2014yes—
285C2×D12—51no—
286C2×D12—102no—
287C2×D12C22×D511yescharacteristic
288C2×C4×D3—21no—
289C2×C4×D3—52no—
290C2×C4×D3—102no—
291C2×C4×D3—102no—
292C2×Dic6—51no—
293C2×Dic6—52no—
294C2×Dic6—102no—
295C2×Dic6—102no—
296C2×Dic6—102no—
297C2×Dic6—101no—
298C2×Dic6C22×D511yescharacteristic
299C3×M16—102no—
300C2×C3×C8—52no—
301C3×C4⋊C4C22×D512yes—
302C3×C22⋊C4—21no—
303C3×C42—21no—
304#216—108no—
305(C2×D3)⋊C4—104no—
306C4⋊Dic3—104no—
307C2×C3⋊C8—22no—
308C2×C3⋊C8C22×D512yes—
309Dic12—104no—
310#204—104no—
311C23×C5—61no—
312C23×C5D3×D411yescharacteristic
313C22×D5—32no—
314C5×Q8—32no—
315C5×Q8—61no—
316C5×D4—32no—
317C5×D4—64no—
318C5×D4—61no—
319C2×C4×C5—22no—
320C2×C4×C5—61no—
321C2×C4×C5—64no—
322C2×C4×C5C2×D1212yes—
323C2×C4×C5D3×D412yes—
324C2×C4×C5D3×D411yescharacteristic
325C2×C4×C5C23×D311yescharacteristic
326C22⋊D5—68no—
327C22⋊D5—64no—
328C2×Dic5—24no—
329C2×Dic5—22no—
330C2×Dic5—32no—
331C2×Dic5—68no—
332C2×Dic5—62no—
333C2×Dic5—122no—
334D20—62no—
335C4×D5—68no—
336C4×D5—62no—
337C4×D5—62no—
338Dic10—24no—
339Dic10—22no—
340Dic10—68no—
341Dic10—62no—
342Dic10—62no—
343Dic10—62no—
344Dic10—122no—
345Dic10C2×D1214yes—
346C5×C8—62no—
347#168—22no—
3482−1+4—304no—
3492−1+4—304no—
350C2×#41—31no—
351C2×#41—151no—
352C2×#41—152no—
353C2×#41—301no—
354C2×#41—302no—
355C22×Q8—51no—
356C22×Q8—152no—
357C22×Q8—151no—
358C22×Q8—301no—
359#137—304no—
360C2×Q16—154no—
361C2×Q16—152no—
362C2×Q16—304no—
363C2×SD16—152no—
364C2×M16—51no—
365C22×C8—31no—
366C4⋊Q8—301no—
367#124—301no—
368#123—302no—
369C22⋊Q8—154no—
370#117D3×D511yescharacteristic
371#103—32no—
372#103—54no—
373#102—32no—
374C22⋊C8—154no—
375C2×C3×C5—22no—
376C2×C3×C5C2×#12011yescharacteristic
377C2×C3×C5#57812yes—
378D15—42no—
379C5×D3—42no—
380C23×C3D4×D511yescharacteristic
381C22×D3—21no—
382C22×D3—52no—
383C3×Q8—54no—
384C3×Q8—102no—
385C3×Q8—104no—
386C2×C3×C4—22no—
387C2×C3×C4—102no—
388C2×C3×C4—104no—
389C2×C3×C4C2×D2012yes—
390C2×C3×C4D4×D512yes—
391C2×C3×C4D4×D511yescharacteristic
392C2×C3×C4C23×D511yescharacteristic
393#67—24no—
394#67—104no—
395C2×Dic3—21no—
396C2×Dic3—52no—
397C2×Dic3—102no—
398C2×Dic3—202no—
399D12—21no—
400D12—108no—
401D12—102no—
402D12C2×D2012yes—
403C4×D3—24no—
404C4×D3—108no—
405C4×D3—108no—
406C4×D3—102no—
407C4×D3—102no—
408Dic6—21no—
409Dic6—108no—
410Dic6—102no—
411Dic6—102no—
412Dic6—102no—
413Dic6—202no—
414Dic6C2×D2012yes—
415C3×C8—102no—
416C3⋊C8—22no—
417C22×C5—22no—
418C22×C5—62no—
419C22×C5—122no—
420C22×C5#88212yes—
421C22×C5C2×D3×D411yescharacteristic
422C2×D5—62no—
423C2×D5—64no—
424C4×C5—42no—
425C4×C5—62no—
426C4×C5#88212yes—
427C4×C5C22×D1211yescharacteristic
428C4×C5C2×D3×D411yescharacteristic
429Dic5—24no—
430Dic5—62no—
431Dic5—122no—
432#41—34no—
433#41—62no—
434#41—308no—
435#41—3016no—
436#41—304no—
437#41—304no—
438#41—302no—
439#41—302no—
440C2×Q8—31no—
441C2×Q8—54no—
442C2×Q8—52no—
443C2×Q8—102no—
444C2×Q8—101no—
445C2×Q8—151no—
446C2×Q8—152no—
447C2×Q8—302no—
448C2×Q8—308no—
449C2×Q8—304no—
450C2×Q8—304no—
451C2×Q8—301no—
452C2×Q8—302no—
453C2×Q8—302no—
454C2×D4—31no—
455C2×D4—61no—
456C2×D4—152no—
457C2×D4—151no—
458C2×D4—302no—
459C22×C4—61no—
460C22×C4—101no—
461C22×C4—152no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4—301no—
465C22×C4C2×D3×D511yescharacteristic
466Q16—308no—
467Q16—304no—
468SD16—304no—
469M16—102no—
470C2×C8—32no—
471C2×C8—52no—
472C2×C8—62no—
473C4⋊C4—304no—
474C4⋊C4C2×D3×D512yes—
475C22⋊C4—21no—
476C22⋊C4—304no—
477C42—21no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3#402812yes—
481C22×C3C2×D4×D511yescharacteristic
482C2×D3—22no—
483C2×D3—42no—
484C2×D3—102no—
485C2×D3—104no—
486C3×C4—42no—
487C3×C4—104no—
488C3×C4#402812yes—
489C3×C4C22×D2011yescharacteristic
490C3×C4C2×D4×D511yescharacteristic
491Dic3—22no—
492Dic3—102no—
493Dic3—202no—
494C2×C5—22no—
495C2×C5—122no—
496C2×C5C2×#88211yescharacteristic
497C2×C5#577912yes—
498D5—122no—
499C23—61no—
500C23—152no—
501C23#679011yescharacteristic
502Q8—32no—
503Q8—54no—
504Q8—61no—
505Q8—104no—
506Q8—102no—
507Q8—308no—
508Q8—302no—
509Q8—302no—
510Q8—302no—
511Q8—602no—
512D4—32no—
513D4—64no—
514D4—61no—
515D4—308no—
516D4—304no—
517D4—302no—
518C2×C4—22no—
519C2×C4—61no—
520C2×C4—64no—
521C2×C4—104no—
522C2×C4—102no—
523C2×C4—152no—
524C2×C4—308no—
525C2×C4—308no—
526C2×C4—302no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4—602no—
530C2×C4D5×D1212yes—
531C2×C4D3×D2012yes—
532C2×C4#679011yescharacteristic
533C2×C4C22×D3×D511yescharacteristic
534C8—62no—
535C8—102no—
536C2×C3—22no—
537C2×C3C2×#402811yescharacteristic
538C2×C3#6610512yes—
539D3—42no—
540D3—202no—
541C5#7896111yescharacteristic, complemented, Sylow
542C22—22no—
543C22—62no—
544C22—122no—
545C22—304no—
546C22—302no—
547C22#9147612yes—
548C22C2×#679011yescharacteristic
549C4—42no—
550C4—62no—
551C4—104no—
552C4—302no—
553C4—602no—
554C4#9147612yes—
555C4C2×D5×D1211yescharacteristic
556C4C2×D3×D2011yescharacteristic
557C3#1061056811yescharacteristic, complemented, Sylow
558C2—22no—
559C2—122no—
560C2—602no—
561C2C2×#9147611yescharacteristic
562C2#1175535312yes—
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#92004C2
hypercenter#117D3×D5
hypocenterC3×C5#3011
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11756285C2semidirect
3. C2×#91322C2semidirect
4. C22×#6662C2semidirect
5. C5×#5002C2semidirect
6. C3×#65293C2semidirect
7. #11748350C2semidirect
8. #11747521C2semidirect
9. #11747496C2semidirect
45. #65293D3semidirect
100. #5002D5semidirect
478. C3×C5#3011semidirect
541. C5#78961semidirect
557. C3#10610568semidirect

Maximal subgroups

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

subgroupindex
C2×#920042
#117562852
C2×#913222
C22×#66622
C5×#50022
C3×#652932
#117483502
#117475212
#117474962
#106105683
#789615