Finite Group Databasebeta

← #14412235

Subgroups — #14412235

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92020C211yesmaximal, characteristic
2#11756985C211yesmaximal, characteristic, complemented
3C3×#66256C211yesmaximal, characteristic, complemented
4C2×#91175C214yesmaximal, complemented
5#11752008C214yesmaximal, complemented
6D5⋊#824C212yesmaximal, complemented
7#11751818C212yesmaximal, complemented
8#10617554—31nomaximal, Hall
9C2×#6819C411yescharacteristic
10C2×C5×#237C411yescharacteristic
11C2×C3×#711C2211yescharacteristic
12C22×#1204C2212yes—
13C2×#6782C2212yes—
14C2×#6780C412yes—
15C2×#6779C412yes—
16#92020—24no—
17#92020—24no—
18C2×#6771C2211yescharacteristic
19C2×D15⋊Q8C411yescharacteristic
20C2×D3×Dic10C411yescharacteristic
21C2×D5×Dic6C2211yescharacteristic
22C2×#6762C2214yes—
23#91990—24no—
24D5⋊#208C2212yes—
25C2×C3×#704C2214yes—
26C3×#4131—24no—
27C3×D5⋊M16C2212yes—
28#91175C2218yes—
29#91175C2214yes—
30#91174C2214yes—
31#83815—51nomaximal, Hall
32C2×#4150—31no—
33#66256—31no—
34#66256D311yescharacteristic, complemented
35D5⋊#101—32no—
36#65777—34no—
37#65761—32no—
38C2×#4005—34no—
39C2×C3×C5×D4C2×C411yescharacteristic
40C2×#1220C2×C412yes—
41C22×Dic15C2×C412yes—
42#6819—24no—
43C2×C4×D15C2×C411yescharacteristic
44C2×Dic30C2×C411yescharacteristic
45C2×C5×#67C2×C412yes—
46C22×C5×Dic3C2×C412yes—
47C5×#237—24no—
48C2×C4×C5×D3C2×C411yescharacteristic
49C2×C5×Dic6C2×C411yescharacteristic
50C2×C3×C22⋊D5C2×C412yes—
51C22×C3×Dic5C2312yes—
52C3×#711D414yes—
53C3×#711D414yes—
54C2×C3×C4×D5C2311yescharacteristic
55C2×C3×Dic10C2×C411yescharacteristic
56C2×#1204—24no—
57C2×#1204—24no—
58C2×#1204C2×C412yes—
59C2×#1204C2×C412yes—
60C2×#1204C2312yes—
61C2×#1204C2311yescharacteristic
62C2×#1203C2311yescharacteristic
63C2×#1202C2×C411yescharacteristic
64C2×#1201C2×C411yescharacteristic
65C2×#1200C2×C412yes—
66#6782—24no—
67#6782—24no—
68C2×D3×Dic5C2×C412yes—
69#6780—28no—
70#6779—28no—
71C2×D5×Dic3C2×C412yes—
72#6771—22no—
73#6771—22no—
74D15⋊Q8—24no—
75D3×Dic10—24no—
76D5×Dic6—22no—
77D5×Dic6—22no—
78#6762—24no—
79#6762C2314yes—
80C2×C3⋊#170—24no—
81#6758—22no—
82D5⋊(C3⋊C8)—22no—
83C3×#704—24no—
84C3×#704C2314yes—
85C2×C3×#170—24no—
86C3×#700—22no—
87C3×D5⋊C8—22no—
88C2×#1010—51no—
89C3×#568—51no—
90#5819—51no—
91C2×#824—54no—
92C2×#824—52no—
93#5132—54no—
94#5132—52no—
95C2×C5×#41—31no—
96#4150—64no—
97#4150—64no—
98C2×Q8×D5—31no—
99C2×Q8×D5—31no—
100C2×#711—31no—
101C2×#711—32no—
102C2×#711—32no—
103C2×#711C2×D311yescharacteristic
104C2×#709—32no—
105C2×#709—31no—
106C22×Dic10—32no—
107C22×Dic10—31no—
108C2×#704—34no—
109C2×#704C2×D314yes—
110#4131—24no—
111#4131—64no—
112D5⋊M16—32no—
113D5⋊M16C2×D312yes—
114#4013—34no—
115#4005—34no—
116#4005—38no—
117#2889—151noSylow
118C23×C3×C5C22×C412yes—
119C22×D15C22×C411yescharacteristic
120C22×C5×D3C22×C411yescharacteristic
121C22×C3×D5C22×C411yescharacteristic
122C3×C5×D4C22⋊C414yes—
123C2×C3×C4×C5C22×C411yescharacteristic
124#1220—24no—
125C2×Dic15—24no—
126C2×Dic15C22×C412yes—
127C2×Dic15C22×C411yescharacteristic
128C4×D15—22no—
129Dic30—22no—
130C5×#67—24no—
131C2×C5×Dic3—24no—
132C2×C5×Dic3C22×C412yes—
133C2×C5×Dic3C22×C411yescharacteristic
134C4×C5×D3—22no—
135C5×Dic6—22no—
136C3×C22⋊D5C22⋊C418yes—
137C2×C3×Dic5C22×C412yes—
138C2×C3×Dic5C2×D414yes—
139C2×C3×Dic5C2×D414yes—
140C2×C3×Dic5C2411yescharacteristic
141C3×C4×D5C2×D412yes—
142C3×C4×D5C2×D412yes—
143C3×Dic10C22⋊C414yes—
144#1204—24no—
145#1204—24no—
146#1204—22no—
147#1204—22no—
148#1204—21no—
149#1204—21no—
150#1203—21no—
151#1203—21no—
152#1202—22no—
153#1201—22no—
154#1200—24no—
155D3×Dic5—24no—
156D5×Dic3—24no—
157C3⋊#170—24no—
158C3×#170—24no—
159C2×C3×#41—51no—
160#1010—104no—
161#1010—104no—
162C2×D3×Q8—51no—
163C2×D3×Q8—51no—
164C2×#237—51no—
165C2×#237—52no—
166C2×#237—52no—
167C2×#237AGL(1,𝔽5)11yescharacteristic
168C2×#235—52no—
169C2×#235—51no—
170C22×Dic6—52no—
171C22×Dic6—51no—
172C3×#131—104no—
173C2×C3×M16—52no—
174C2×C3×M16—54no—
175#948—104no—
176C2×#208—52no—
177C2×#208—54no—
178#824—58no—
179#824—54no—
180#823—54no—
181C5×#41—64no—
182C2×C5×Q8—31no—
183C2×C5×D4—32no—
184C2×C5×D4C4×D311yescharacteristic
185C22×C4×C5—32no—
186C22×C4×C5—31no—
187C2×C22⋊D5—32no—
188C2×C22⋊D5—31no—
189C2×C22⋊D5—31no—
190C2×C22⋊D5C4×D312yes—
191C22×Dic5—32no—
192C22×Dic5—32no—
193C22×Dic5C22×D312yes—
194Q8×D5—62no—
195Q8×D5—62no—
196Q8×D5—64no—
197#711—64no—
198#711—68no—
199#711—64no—
200#711—64no—
201#711D1214yes—
202#711#6714yes—
203#709—68no—
204#709—62no—
205#709—62no—
206C2×D20—31no—
207C2×C4×D5—32no—
208C2×C4×D5—31no—
209C2×C4×D5—32no—
210C2×C4×D5C22×D311yescharacteristic
211C2×Dic10—31no—
212C2×Dic10—32no—
213C2×Dic10—32no—
214C2×Dic10—31no—
215C2×Dic10—32no—
216C2×Dic10—64no—
217C2×Dic10—64no—
218C2×Dic10—64no—
219C2×Dic10C4×D311yescharacteristic
220#704—24no—
221#704—34no—
222#704—64no—
223#704C22×D314yes—
224C2×#170—24no—
225C2×#170—64no—
226#700—22no—
227#700—62no—
228D5⋊C8—22no—
229D5⋊C8—62no—
230C2×2−1+4—151no—
231#568—51no—
232#568—151no—
233#413—152no—
234#413—154no—
235C2×#101—154no—
236C2×#101—152no—
237C22×C3×C5—42no—
238C22×C3×C5C2×C22⋊C414yes—
239C22×C3×C5C23×C411yescharacteristic
240C2×D15—21no—
241C2×D15—41no—
242C2×C5×D3—21no—
243C2×C5×D3—41no—
244C2×C3×D5—41no—
245C2×C3×D5C2×C22⋊C412yes—
246C3×C4×C5C2×C22⋊C412yes—
247Dic15—22no—
248Dic15—21no—
249C3×Dic5C2×C22⋊C414yes—
250C3×Dic5C22×D411yescharacteristic
251C3×Dic5C22×D411yescharacteristic
252C5×Dic3—22no—
253C5×Dic3—21no—
254C3×#41—54no—
255C3×#41—54no—
256C2×C3×Q8—51no—
257C2×C3×D4—52no—
258C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
259C22×C3×C4—52no—
260C22×C3×C4—51no—
261C2×#67—52no—
262C2×#67—51no—
263C2×#67—51no—
264C2×#67C2×AGL(1,𝔽5)12yes—
265C22×Dic3—52no—
266C22×Dic3—52no—
267C22×Dic3C2×AGL(1,𝔽5)12yes—
268D3×Q8—104no—
269D3×Q8—104no—
270#237—24no—
271#237—108no—
272#237—108no—
273#237—104no—
274#235—102no—
275#235—102no—
276#235—104no—
277#235—104no—
278C2×D12—51no—
279C2×C4×D3—52no—
280C2×C4×D3—51no—
281C2×C4×D3—52no—
282C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
283C2×Dic6—51no—
284C2×Dic6—51no—
285C2×Dic6—52no—
286C2×Dic6—52no—
287C2×Dic6—52no—
288C2×Dic6—102no—
289C2×Dic6—102no—
290C2×Dic6—104no—
291C2×Dic6—104no—
292C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
293C3×M16—54no—
294C3×M16—102no—
295C3×M16—104no—
296C2×C3×C8—102no—
297C2×C3×C8—104no—
298#208—54no—
299#208—102no—
300#208—104no—
301C2×C3⋊C8—102no—
302C2×C3⋊C8—104no—
303C23×C5—31no—
304C23×C5C2×C4×D312yes—
305C22×D5—31no—
306C22×D5C2×C4×D311yescharacteristic
307C5×Q8—62no—
308C5×D4—64no—
309C5×D4(C2×D3)⋊C414yes—
310C2×C4×C5—31no—
311C2×C4×C5—32no—
312C2×C4×C5—64no—
313C2×C4×C5—62no—
314C2×C4×C5C2×C4×D311yescharacteristic
315C22⋊D5—64no—
316C22⋊D5—62no—
317C22⋊D5—61no—
318C22⋊D5—61no—
319C22⋊D5(C2×D3)⋊C418yes—
320C2×Dic5—32no—
321C2×Dic5—31no—
322C2×Dic5—64no—
323C2×Dic5—64no—
324C2×Dic5C2×C4×D312yes—
325C2×Dic5C2×D1214yes—
326C2×Dic5C2×#6714yes—
327C2×Dic5C23×D311yescharacteristic
328D20—62no—
329C4×D5—64no—
330C4×D5—62no—
331C4×D5—64no—
332C4×D5C2×D1212yes—
333C4×D5C2×#6712yes—
334Dic10—62no—
335Dic10—64no—
336Dic10—64no—
337Dic10—61no—
338Dic10—62no—
339Dic10—61no—
340Dic10—62no—
341Dic10(C2×D3)⋊C414yes—
342#170—24no—
343#170—64no—
3442−1+4—304no—
3452−1+4—304no—
346C2×#41—31no—
347C2×#41—51no—
348C2×#41—151no—
349C2×#41—152no—
350C2×#41—152no—
351C2×#41—152no—
352C2×#41—151no—
353C22×Q8—152no—
354C22×Q8—151no—
355C22×Q8—151no—
356C22×Q8—151no—
357#131—104no—
358#131—304no—
359C2×M16—52no—
360C2×M16—54no—
361C2×M16—152no—
362C2×M16—154no—
363#101—154no—
364#101—158no—
365#100—154no—
366C2×C3×C5—21no—
367C2×C3×C5—24no—
368C2×C3×C5C22×C22⋊C411yescharacteristic
369D15—41no—
370C3×D5—22no—
371C5×D3—41no—
372C23×C3—51no—
373C23×C3C22×AGL(1,𝔽5)12yes—
374C22×D3—51no—
375C22×D3C22×AGL(1,𝔽5)11yescharacteristic
376C3×Q8—54no—
377C3×D4—58no—
378C3×D4C22⋊AGL(1,𝔽5)14yes—
379C2×C3×C4—54no—
380C2×C3×C4—54no—
381C2×C3×C4—52no—
382C2×C3×C4—51no—
383C2×C3×C4—52no—
384C2×C3×C4—52no—
385C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
386#67—24no—
387#67—102no—
388#67—102no—
389#67—104no—
390C2×Dic3—24no—
391C2×Dic3—52no—
392C2×Dic3—51no—
393C2×Dic3—104no—
394C2×Dic3—104no—
395C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
396C2×Dic3C22×AGL(1,𝔽5)12yes—
397D12—101no—
398D12—101no—
399C4×D3—22no—
400C4×D3—104no—
401C4×D3—104no—
402C4×D3—102no—
403Dic6—22no—
404Dic6—102no—
405Dic6—102no—
406Dic6—101no—
407Dic6—101no—
408Dic6—104no—
409Dic6—104no—
410Dic6—102no—
411C3×C8—104no—
412C3⋊C8—104no—
413C22×C5—42no—
414C22×C5—61no—
415C22×C5—121no—
416C22×C5C2×(C2×D3)⋊C414yes—
417C22×C5C22×C4×D311yescharacteristic
418C2×D5—41no—
419C2×D5—61no—
420C2×D5—121no—
421C2×D5C2×(C2×D3)⋊C412yes—
422C4×C5—61no—
423C4×C5—62no—
424C4×C5C2×(C2×D3)⋊C412yes—
425Dic5—62no—
426Dic5—61no—
427Dic5C2×(C2×D3)⋊C414yes—
428Dic5C22×D1211yescharacteristic
429Dic5C22×#6711yescharacteristic
430#41—54no—
431#41—54no—
432#41—64no—
433#41—302no—
434#41—302no—
435#41—304no—
436#41—304no—
437#41—308no—
438#41—308no—
439#41—304no—
440C2×Q8—31no—
441C2×Q8—51no—
442C2×Q8—151no—
443C2×Q8—152no—
444C2×Q8—152no—
445C2×Q8—152no—
446C2×Q8—151no—
447C2×Q8—304no—
448C2×Q8—304no—
449C2×Q8—302no—
450C2×Q8—302no—
451C2×Q8—304no—
452C2×Q8—304no—
453C2×D4—32no—
454C2×D4—52no—
455C2×D4—151no—
456C2×D4—151no—
457C2×D4—151no—
458C2×D4—152no—
459C2×D4D3×AGL(1,𝔽5)11yescharacteristic
460C22×C4—32no—
461C22×C4—31no—
462C22×C4—52no—
463C22×C4—51no—
464C22×C4—152no—
465C22×C4—152no—
466C22×C4—152no—
467C22×C4—152no—
468C22×C4—151no—
469M16—54no—
470M16—102no—
471M16—104no—
472M16—154no—
473M16—302no—
474M16—304no—
475C2×C8—102no—
476C2×C8—104no—
477C2×C8—302no—
478C2×C8—304no—
479C3×C5#288911yescharacteristic, complemented, Hall
480C22×C3—42no—
481C22×C3—52no—
482C22×C3—201no—
483C22×C3C2×C22⋊AGL(1,𝔽5)14yes—
484C22×C3C23×AGL(1,𝔽5)11yescharacteristic
485C2×D3—21no—
486C2×D3—41no—
487C2×D3—101no—
488C2×D3—201no—
489C3×C4—54no—
490C3×C4—51no—
491C3×C4—51no—
492C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
493Dic3—22no—
494Dic3—21no—
495Dic3—102no—
496Dic3—101no—
497C2×C5—21no—
498C2×C5—24no—
499C2×C5—121no—
500C2×C5C22×(C2×D3)⋊C411yescharacteristic
501D5—22no—
502D5—121no—
503C23—31no—
504C23—51no—
505C23—151no—
506C23C2×D3×AGL(1,𝔽5)12yes—
507Q8—54no—
508Q8—62no—
509Q8—301no—
510Q8—302no—
511Q8—301no—
512Q8—302no—
513Q8—304no—
514Q8—304no—
515Q8—302no—
516D4—58no—
517D4—64no—
518D4—301no—
519D4—301no—
520D4—302no—
521D4—302no—
522D4—304no—
523D4(C2×D3)⋊AGL(1,𝔽5)14yes—
524C2×C4—31no—
525C2×C4—32no—
526C2×C4—54no—
527C2×C4—54no—
528C2×C4—51no—
529C2×C4—52no—
530C2×C4—52no—
531C2×C4—52no—
532C2×C4—64no—
533C2×C4—62no—
534C2×C4—152no—
535C2×C4—151no—
536C2×C4—304no—
537C2×C4—304no—
538C2×C4—304no—
539C2×C4—304no—
540C2×C4—302no—
541C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
542C8—104no—
543C8—304no—
544C2×C3—21no—
545C2×C3—24no—
546C2×C3—102no—
547C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
548D3—41no—
549D3—201no—
550C5#8381511yescharacteristic, complemented, Sylow
551C22—42no—
552C22—52no—
553C22—61no—
554C22—121no—
555C22—201no—
556C22—301no—
557C22—601no—
558C22C2×(C2×D3)⋊AGL(1,𝔽5)14yes—
559C22C22×D3×AGL(1,𝔽5)11yescharacteristic
560C4—51no—
561C4—51no—
562C4—54no—
563C4—61no—
564C4—62no—
565C4—302no—
566C4—301no—
567C4C2×(C2×D3)⋊AGL(1,𝔽5)12yes—
568C3#1061755411yescharacteristic, complemented, Sylow
569C2—21no—
570C2—24no—
571C2—102no—
572C2—121no—
573C2—601no—
574C2C22×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2C22×(C2×D3)⋊AGL(1,𝔽5)
commutatorC22×C3×C5C23×C4
FrattiniC22C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC2×C3×C5C22×C22⋊C4
prime radicalC2×#92020C2
cosocleC2×C3×Dic5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2889
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11756985C2semidirect
3. C3×#66256C2semidirect
4. C2×#91175C2semidirect
5. #11752008C2semidirect
6. D5⋊#824C2semidirect
7. #11751818C2semidirect
34. #66256D3semidirect
479. C3×C5#2889semidirect
550. C5#83815semidirect
568. C3#10617554semidirect

Maximal subgroups

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

subgroupindex
C2×#920202
#117569852
C3×#662562
C2×#911752
#117520082
D5⋊#8242
#117518182
#106175543
#838155