Finite Group Databasebeta

← #14301123

Subgroups — #14301123

Order 1920.

748 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91344C211yesmaximal, characteristic, complemented
2C22×#6660C211yesmaximal, characteristic
3#11751089C211yesmaximal, characteristic
4C2×C5×#810C211yesmaximal, characteristic, complemented
5C2×#90968C211yesmaximal, characteristic
6C3×#64749C211yesmaximal, characteristic, complemented
7#11745968C211yesmaximal, characteristic, complemented
8#10601033—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×C5×Dic6C2211yescharacteristic
11C2×C3×C5×C4⋊C4C2211yescharacteristic
12C2×#6714—21no—
13C2×#6714—21no—
14C2×C4×Dic15C2211yescharacteristic
15C22×C5×C3⋊C8C2211yescharacteristic
16C2×C3×C4⋊Dic5C2211yescharacteristic
17C22×C3×#168C2211yescharacteristic
18C2×Dic3⋊Dic5—21no—
19C2×Dic3×Dic5—21no—
20#91344—22no—
21#91344—22no—
22C2×#6660—22no—
23C2×#6660—22no—
24C2×#6660C414yescomplemented
25C5×#810—22no—
26#90968—22no—
27#68909—51nomaximal, Hall
28C2×Q8×Dic5—31no—
29C22×#688—31no—
30C2×C5×#103—31no—
31C2×#3953—31no—
32#65277—31no—
33#64770—31no—
34#64749D311yescharacteristic, complemented
35C23×D15—21no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15D411yescharacteristic
39C22×Dic15D411yescharacteristic
40C2×D60—21no—
41C2×D60—21no—
42C2×D60—21no—
43C2×D60—21no—
44C2×D60—21no—
45C2×D60C2×C412yes—
46C22×C5×Dic3—21no—
47C2×C5×Dic6—21no—
48C2×C5×Dic6—21no—
49C2×C5×Dic6—21no—
50C2×C5×Dic6C2×C412yes—
51C2×C5×Dic6D412yes—
52C2×C5×Dic6D412yes—
53C22×C3×Dic5—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4—21no—
56#6714—42no—
57#6714—42no—
58C4×Dic15—21no—
59C4×Dic15—21no—
60C2×C5×C3⋊C8—21no—
61C2×C5×C3⋊C8—21no—
62C2×C5×C3⋊C8—21no—
63C3×C4⋊Dic5—21no—
64C3×C4⋊Dic5D412yes—
65C2×C3×#168—21no—
66C2×C3×#168—21no—
67C2×C3×#168C2×C412yes—
68Dic3⋊Dic5—24no—
69Dic3×Dic5—24no—
70#6660—28no—
71C22×#204—51no—
72C2×#887—51no—
73#4988—51no—
74C2×#810D511yescharacteristic, complemented
75C2×#809—51no—
76C3×#340—51no—
77#4550—51no—
78C22×C5×Q8—31no—
79C22×D20—31no—
80C22×C5×C8—31no—
81C2×C5×C4⋊C4C2×D311yescharacteristic
82Q8×Dic5—62no—
83Q8×Dic5—62no—
84C2×#688—34no—
85C2×#688—62no—
86C2×#688—62no—
87C2×#685—61no—
88C2×#685—61no—
89C2×C4⋊Dic5—61no—
90C2×C4⋊Dic5C2×D311yescharacteristic
91C2×C4×Dic5—31no—
92C2×C4×Dic5—61no—
93C22×#168C2×D311yescharacteristic
94C5×#103—62no—
95#3953—62no—
96#1880—151noSylow
97C23×C3×C5C2×D411yescharacteristic
98C22×D15—22no—
99C22×D15—41no—
100C22×D15—41no—
101C22×D15—41no—
102C22×D15—41no—
103C22×D15—41no—
104C22×D15—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5#4111yescharacteristic
113C2×C3×C4×C5#4111yescharacteristic
114C2×Dic15—21no—
115C2×Dic15—21no—
116C2×Dic15—21no—
117C2×Dic15—21no—
118C2×Dic15—21no—
119C2×Dic15SD1612yescomplemented
120D60—22no—
121D60—22no—
122D60—41no—
123D60—41no—
124C2×C5×Dic3—22no—
125C2×C5×Dic3—22no—
126C2×C5×Dic3—22no—
127C5×Dic6—22no—
128C5×Dic6—22no—
129C5×Dic6—22no—
130C5×Dic6C22⋊C414yescomplemented
131C2×C3×Dic5—22no—
132C2×C3×Dic5—41no—
133C2×C3×Dic5—41no—
134C3×#168—22no—
135C5×C3⋊C8—41no—
136C22×D12—51no—
137C22×Dic6C2×D511yescharacteristic
138C22×C3×C8—51no—
139C2×C3×C4⋊C4—51no—
140C2×C3×C4⋊C4C2×D511yescharacteristic
141C2×(C2×D3)⋊C4—101no—
142C2×(C2×D3)⋊C4—101no—
143C2×Dic3⋊C4—101no—
144C2×C4×Dic3—51no—
145C2×C4×Dic3—101no—
146C22×C3⋊C8C2×D511yescharacteristic
147C2×#204—54no—
148C2×#204—102no—
149C2×#204—102no—
150#887—102no—
151#887—102no—
152#810—22no—
153#809—102no—
154C23×D5—61no—
155C2×C5×Q8—32no—
156C2×C5×Q8—32no—
157C2×C5×Q8—32no—
158C2×C5×Q8—61no—
159C2×C5×Q8—61no—
160C2×C5×Q8—61no—
161C22×C4×C5—21no—
162C22×C4×C5—61no—
163C22×C4×C5C22×D311yescharacteristic
164C22×Dic5—21no—
165C22×Dic5—31no—
166C22×Dic5—31no—
167C2×D20—32no—
168C2×D20—61no—
169C2×D20—61no—
170C2×D20—61no—
171C2×D20—61no—
172C2×D20—61no—
173C2×C5×C8—61no—
174C2×C5×C8—61no—
175C2×C5×C8—61no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178#688—68no—
179#685—122no—
180#685—122no—
181C4⋊Dic5—21no—
182C4⋊Dic5—64no—
183C4⋊Dic5D1212yes—
184C4×Dic5—64no—
185C4×Dic5—61no—
186C4×Dic5—61no—
187C2×#168—21no—
188C2×#168—21no—
189C2×#168C4×D312yes—
190C22×SD16—151no—
191C2×C4×Q8—151no—
192C2×#103—31no—
193C2×#102—151no—
194#390—151no—
195#340—51no—
196#336—151no—
197C22×C3×C5C4×D411yescharacteristic
198C22×C3×C5#12011yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5C2×SD1611yescharacteristic
201C22×C3×C5#13511yescharacteristic
202C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
203C22×C3×C5#13711yescharacteristic
204C2×D15—42no—
205C2×D15—42no—
206C2×D15—42no—
207C2×D15—81no—
208C2×D15—81no—
209C2×D15—81no—
210C2×D15—81no—
211C3×C4×C5—81no—
212C3×C4×C5C2×C22⋊C411yescharacteristic
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5#12111yescharacteristic
215C3×C4×C5#12311yescharacteristic
216Dic15—22no—
217Dic15—41no—
218C3×Dic5—42no—
219C5×Dic3—24no—
220C23×D3—101no—
221C22×C3×C4—21no—
222C22×C3×C4—101no—
223C22×C3×C4C22×D511yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—51no—
226C22×Dic3—51no—
227C2×D12—52no—
228C2×D12—101no—
229C2×D12—101no—
230C2×D12—101no—
231C2×D12—101no—
232C2×D12—101no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6C4×D512yes—
237C2×Dic6D2012yes—
238C2×Dic6C22⋊D512yes—
239C2×C3×C8—52no—
240C2×C3×C8—101no—
241C2×C3×C8—101no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—52no—
245C3×C4⋊C4—101no—
246(C2×D3)⋊C4—202no—
247(C2×D3)⋊C4—202no—
248Dic3⋊C4—104no—
249C4×Dic3—104no—
250C4×Dic3—101no—
251C4×Dic3—101no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255#204—108no—
256C23×C5D3×D411yescharacteristic
257C22×D5—62no—
258C22×D5—121no—
259C22×D5—121no—
260C22×D5—121no—
261C22×D5—121no—
262C22×D5—121no—
263C22×D5—121no—
264C5×Q8—34no—
265C5×Q8—62no—
266C5×Q8—62no—
267C5×Q8—62no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—62no—
272C2×C4×C5—62no—
273C2×C4×C5—62no—
274C2×C4×C5C2×C4×D311yescharacteristic
275C2×C4×C5C2×D1211yescharacteristic
276C2×C4×C5#23511yescharacteristic
277C2×C4×C5D3×D411yescharacteristic
278C2×C4×C5D3×D411yescharacteristic
279C2×C4×C5#23911yescharacteristic
280C2×Dic5—22no—
281C2×Dic5—32no—
282C2×Dic5—41no—
283C2×Dic5—41no—
284C2×Dic5—61no—
285C2×Dic5—61no—
286C2×Dic5—61no—
287C2×Dic5—61no—
288C2×Dic5—61no—
289D20—62no—
290D20—62no—
291D20—121no—
292D20—121no—
293C5×C8—121no—
294#168—22no—
295C22×Q8—31no—
296C22×D4—151no—
297C2×SD16—154no—
298C2×SD16—302no—
299C2×SD16—302no—
300C22×C8—31no—
301C22×C8—51no—
302C4×Q8—302no—
303C4×Q8—302no—
304C2×C4⋊C4—51no—
305C2×C4⋊C4—301no—
306C2×C4⋊C4D3×D511yescharacteristic
307C2×C22⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—151no—
310C2×C42—301no—
311#103—62no—
312#102—302no—
313C2×C3×C5#38611yescharacteristic
314C2×C3×C5C4×SD1611yescharacteristic
315C2×C3×C5#44011yescharacteristic
316C2×C3×C5#44811yescharacteristic
317C2×C3×C5#44911yescharacteristic
318C2×C3×C5#46011yescharacteristic
319C2×C3×C5#46411yescharacteristic
320D15—82no—
321C23×C3D4×D511yescharacteristic
322C22×D3—102no—
323C22×D3—201no—
324C22×D3—201no—
325C22×D3—201no—
326C22×D3—201no—
327C22×D3—201no—
328C22×D3—201no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—102no—
333C2×C3×C4—201no—
334C2×C3×C4—201no—
335C2×C3×C4C2×C4×D511yescharacteristic
336C2×C3×C4C2×D2011yescharacteristic
337C2×C3×C4D4×D511yescharacteristic
338C2×C3×C4#71111yescharacteristic
339C2×C3×C4#71111yescharacteristic
340C2×C3×C4C2×C22⋊D511yescharacteristic
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—52no—
345C2×Dic3—101no—
346C2×Dic3—101no—
347C2×Dic3—101no—
348C2×Dic3—101no—
349C2×Dic3—101no—
350D12—102no—
351D12—102no—
352D12—201no—
353D12—201no—
354Dic6—22no—
355Dic6—22no—
356Dic6—22no—
357Dic6#68514yescomplemented
358C3×C8—102no—
359C3⋊C8—41no—
360C22×C5#88211yescharacteristic
361C22×C5#88411yescharacteristic
362C22×C5#89311yescharacteristic
363C22×C5#91111yescharacteristic
364C22×C5D3×SD1611yescharacteristic
365C22×C5#91611yescharacteristic
366C22×C5#91811yescharacteristic
367C2×D5—122no—
368C2×D5—122no—
369C2×D5—122no—
370C2×D5—241no—
371C2×D5—241no—
372C2×D5—241no—
373C2×D5—241no—
374C4×C5—64no—
375C4×C5—81no—
376C4×C5C4×D1211yescharacteristic
377C4×C5D3×C22⋊C411yescharacteristic
378C4×C5#89411yescharacteristic
379C4×C5#89511yescharacteristic
380Dic5—42no—
381Dic5—62no—
382Dic5—121no—
383C24—301no—
384C2×Q8—32no—
385C2×Q8—32no—
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×D4—152no—
391C2×D4—301no—
392C2×D4—301no—
393C2×D4—301no—
394C2×D4—301no—
395C2×D4—301no—
396C22×C4—21no—
397C22×C4—61no—
398C22×C4—101no—
399C22×C4—151no—
400C22×C4—151no—
401C22×C4C2×D3×D511yescharacteristic
402SD16—308no—
403C2×C8—52no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—101no—
408C2×C8—101no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—52no—
412C4⋊C4—101no—
413C4⋊C4—304no—
414C22⋊C4—602no—
415C22⋊C4—602no—
416C42—304no—
417C42—301no—
418C42—301no—
419C3×C5#188011yescharacteristic, complemented, Hall
420C22×C3#402711yescharacteristic
421C22×C3#402811yescharacteristic
422C22×C3D5×SD1611yescharacteristic
423C22×C3#406011yescharacteristic
424C22×C3#406411yescharacteristic
425C22×C3#406511yescharacteristic
426C22×C3#408511yescharacteristic
427C2×D3—202no—
428C2×D3—202no—
429C2×D3—202no—
430C2×D3—401no—
431C2×D3—401no—
432C2×D3—401no—
433C2×D3—401no—
434C3×C4—81no—
435C3×C4—202no—
436C3×C4#402711yescharacteristic
437C3×C4#403211yescharacteristic
438C3×C4C2×#68511yescharacteristic
439C3×C4#408511yescharacteristic
440Dic3—24no—
441Dic3—102no—
442Dic3—201no—
443C2×C5#466511yescharacteristic
444C2×C5#476211yescharacteristic
445C2×C5#478111yescharacteristic
446C2×C5#479811yescharacteristic
447C2×C5#480611yescharacteristic
448C2×C5#485211yescharacteristic
449C2×C5#488911yescharacteristic
450D5—242no—
451C23—302no—
452C23—601no—
453C23—601no—
454C23—601no—
455C23—601no—
456C23—601no—
457C23—601no—
458C23#679011yescharacteristic
459Q8—34no—
460Q8—62no—
461Q8—62no—
462Q8—62no—
463D4—302no—
464D4—302no—
465D4—601no—
466D4—601no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—102no—
474C2×C4—152no—
475C2×C4—201no—
476C2×C4—201no—
477C2×C4—301no—
478C2×C4—301no—
479C2×C4—301no—
480C2×C4—301no—
481C2×C4—301no—
482C2×C4#676811yescharacteristic
483C2×C4C4×D3×D511yescharacteristic
484C2×C4D5×D1211yescharacteristic
485C2×C4D3×D2011yescharacteristic
486C2×C4#678211yescharacteristic
487C2×C4D3×C22⋊D511yescharacteristic
488C8—102no—
489C8—121no—
490C2×C3#6507711yescharacteristic
491C2×C3#6509511yescharacteristic
492C2×C3#6510011yescharacteristic
493C2×C3#6510611yescharacteristic
494C2×C3#6524711yescharacteristic
495C2×C3#6545211yescharacteristic
496C2×C3#6547411yescharacteristic
497D3—402no—
498C5#6890911yescharacteristic, complemented, Sylow
499C22—602no—
500C22—602no—
501C22—602no—
502C22—1201no—
503C22—1201no—
504C22—1201no—
505C22—1201no—
506C22#9140611yescharacteristic
507C22#9147611yescharacteristic
508C22#9148711yescharacteristic
509C22#9149811yescharacteristic
510C22#9151211yescharacteristic
511C22#9152111yescharacteristic
512C22#9155811yescharacteristic
513C4—64no—
514C4—81no—
515C4—202no—
516C4—302no—
517C4—601no—
518C4#9140511yescharacteristic
519C4#9141611yescharacteristic
520C4#9142711yescharacteristic
521C4D3×#68511yescharacteristic
522C3#1060103311yescharacteristic, complemented, Sylow
523C2—1202no—
524C2#1174727911yescharacteristic
525C2#1174728011yescharacteristic
526C2#1174749711yescharacteristic
527C2#1174751811yescharacteristic
528C2#1174888311yescharacteristic
529C2#1174901611yescharacteristic
530C2#1174926111yescharacteristic
subgroupG/H
centerC23#6790
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC23×C3×C5C2×D4
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#1880
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91344C2semidirect
4. C2×C5×#810C2semidirect
6. C3×#64749C2semidirect
7. #11745968C2semidirect
24. C2×#6660C4semidirect
34. #64749D3semidirect
74. C2×#810D5semidirect
119. C2×Dic15SD16semidirect
130. C5×Dic6C22⋊C4semidirect
357. Dic6#685semidirect
419. C3×C5#1880semidirect
498. C5#68909semidirect
522. C3#10601033semidirect

Maximal subgroups

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

subgroupindex
C2×#913442
C22×#66602
#117510892
C2×C5×#8102
C2×#909682
C3×#647492
#117459682
#106010333
#689095