Finite Group Databasebeta

← C3×#10599702 #14301087

Subgroups — C3×#10599702 #14301087

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
1C22×C3×#677C211yesmaximal, characteristic
2C2×C3×#4033C211yesmaximal, characteristic, complemented
3C3×#65277C211yesmaximal, characteristic
4C2×C3×#3942C211yesmaximal, characteristic, complemented
5C2×C3×#3941C211yesmaximal, characteristic
6C3×C5×#340C211yesmaximal, characteristic, complemented
7C3×#64770C211yesmaximal, characteristic, complemented
8#10599702C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×D20C2211yescharacteristic
10C22×C3×Dic10C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×#685—21no—
15C2×C3×#685—21no—
16C2×C3×#683—21no—
17C2×C3×C4×Dic5—21no—
18C2×C3×C4×Dic5C2211yescharacteristic
19C22×C3×#168C2211yescharacteristic
20C2×C3×#677—22no—
21C2×C3×#677—22no—
22C2×C3×#677C414yescomplemented
23C3×#4033—22no—
24C3×#4033—22no—
25C3×#3942—22no—
26C3×#3941—22no—
27C3×#1880—51nomaximal, Hall
28C22×#677C2×C311yescharacteristic
29C2×#4033C2×C311yescharacteristic, complemented
30#65277C2×C311yescharacteristic
31C2×#3942C2×C311yescharacteristic, complemented
32C2×#3941C2×C311yescharacteristic
33C5×#340C2×C311yescharacteristic, complemented
34#64770C2×C311yescharacteristic, complemented
35C23×C3×D5—21no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5C2311yescharacteristic
39C22×C3×Dic5—21no—
40C22×C3×Dic5D411yescharacteristic
41C22×C3×Dic5D411yescharacteristic
42C2×C3×D20—21no—
43C2×C3×D20—21no—
44C2×C3×D20—21no—
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20C2×C412yes—
48C2×C3×Dic10—21no—
49C2×C3×Dic10—21no—
50C2×C3×Dic10—21no—
51C2×C3×Dic10C2×C412yes—
52C2×C3×Dic10D412yes—
53C2×C3×Dic10D412yes—
54C2×C3×C5×C8—21no—
55C2×C3×C5×C8—21no—
56C2×C3×C5×C8C2×C412yes—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4—21no—
60C3×C5×C4⋊C4D412yes—
61C3×#685—42no—
62C3×#685—42no—
63C3×#683—24no—
64C3×C4×Dic5—24no—
65C3×C4×Dic5—21no—
66C3×C4×Dic5—21no—
67C2×C3×#168—21no—
68C2×C3×#168—21no—
69C2×C3×#168—21no—
70C3×#677—28no—
71C22×C3×SD16—51no—
72C2×C3×C4×Q8—51no—
73C2×C3×#103—51no—
74C2×C3×#102—51no—
75C3×#390—51no—
76C3×#340D511yescharacteristic, complemented
77C3×#336—51no—
78C22×D20C22×C311yescharacteristic
79C22×Dic10C22×C311yescharacteristic
80C22×C5×C8C22×C311yescharacteristic
81C2×C5×C4⋊C4C22×C311yescharacteristic
82C2×C5×C4⋊C4C22×C311yescharacteristic
83C2×#685—21no—
84C2×#685—21no—
85C2×#683—21no—
86C2×C4×Dic5—21no—
87C2×C4×Dic5C22×C311yescharacteristic
88C22×#168C22×C311yescharacteristic
89C2×#677—22no—
90C2×#677—22no—
91C2×#677C3×C414yescomplemented
92#4033—22no—
93#4033—22no—
94#3942—22no—
95#3941—22no—
96#1880—51noSylow
97C23×C3×C5C2×D411yescharacteristic
98C22×C3×D5—22no—
99C22×C3×D5—41no—
100C22×C3×D5—41no—
101C22×C3×D5—41no—
102C22×C3×D5—41no—
103C22×C3×D5—41no—
104C22×C3×D5—41no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5C22×C411yescharacteristic
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5#4111yescharacteristic
116C2×C3×C4×C5#4111yescharacteristic
117C2×C3×Dic5—22no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—22no—
120C2×C3×Dic5—22no—
121C2×C3×Dic5—21no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5—21no—
124C2×C3×Dic5—21no—
125C2×C3×Dic5SD1612yescomplemented
126C3×D20—22no—
127C3×D20—22no—
128C3×D20—41no—
129C3×D20—41no—
130C3×Dic10—22no—
131C3×Dic10—22no—
132C3×Dic10—22no—
133C3×Dic10C22⋊C414yescomplemented
134C3×C5×C8—22no—
135C3×#168—41no—
136C22×C3×Q8—51no—
137C22×C3×D4—51no—
138C2×C3×SD16—54no—
139C2×C3×SD16—102no—
140C2×C3×SD16—102no—
141C22×C3×C8—51no—
142C22×C3×C8C2×D511yescharacteristic
143C3×C4×Q8—102no—
144C3×C4×Q8—102no—
145C2×C3×C4⋊C4—101no—
146C2×C3×C4⋊C4C2×D511yescharacteristic
147C2×C3×C4⋊C4C2×D511yescharacteristic
148C2×C3×C22⋊C4—101no—
149C2×C3×C22⋊C4—101no—
150C2×C3×C42—51no—
151C2×C3×C42—101no—
152C3×#103—102no—
153C3×#102—102no—
154C23×D5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5C23×C311yescharacteristic
158C22×Dic5—21no—
159C22×Dic5C3×D411yescharacteristic
160C22×Dic5C3×D411yescharacteristic
161C2×D20—21no—
162C2×D20—21no—
163C2×D20—21no—
164C2×D20—21no—
165C2×D20—21no—
166C2×D20C2×C3×C412yes—
167C2×Dic10—21no—
168C2×Dic10—21no—
169C2×Dic10—21no—
170C2×Dic10C2×C3×C412yes—
171C2×Dic10C3×D412yes—
172C2×Dic10C3×D412yes—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8C2×C3×C412yes—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4C3×D412yes—
180#685—42no—
181#685—42no—
182#683—24no—
183C4×Dic5—24no—
184C4×Dic5—21no—
185C4×Dic5—21no—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—21no—
189#677—28no—
190C22×SD16—51no—
191C2×C4×Q8—51no—
192C2×#103—51no—
193C2×#102—51no—
194#390—51no—
195#340C3×D511yescharacteristic, complemented
196#336—51no—
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×C3×D5—42no—
205C2×C3×D5—42no—
206C2×C3×D5—42no—
207C2×C3×D5—81no—
208C2×C3×D5—81no—
209C2×C3×D5—81no—
210C2×C3×D5—81no—
211C3×C4×C5—42no—
212C3×C4×C5—81no—
213C3×C4×C5C2×C22⋊C411yescharacteristic
214C3×C4×C5C4×D411yescharacteristic
215C3×C4×C5#12111yescharacteristic
216C3×C4×C5#12311yescharacteristic
217C3×Dic5—24no—
218C3×Dic5—22no—
219C3×Dic5—41no—
220C24×C3—101no—
221C2×C3×Q8—52no—
222C2×C3×Q8—52no—
223C2×C3×Q8—52no—
224C2×C3×Q8—101no—
225C2×C3×Q8—101no—
226C2×C3×Q8—101no—
227C2×C3×D4—52no—
228C2×C3×D4—101no—
229C2×C3×D4—101no—
230C2×C3×D4—101no—
231C2×C3×D4—101no—
232C2×C3×D4—101no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—51no—
236C22×C3×C4—51no—
237C22×C3×C4—101no—
238C22×C3×C4C22×D511yescharacteristic
239C3×SD16—108no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—101no—
243C2×C3×C8—101no—
244C2×C3×C8—101no—
245C2×C3×C8C4×D512yes—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—104no—
250C3×C4⋊C4D2012yes—
251C3×C22⋊C4—202no—
252C3×C22⋊C4—202no—
253C3×C42—104no—
254C3×C42—101no—
255C3×C42—101no—
256C23×C5C2×C3×D411yescharacteristic
257C22×D5—22no—
258C22×D5—41no—
259C22×D5—41no—
260C22×D5—41no—
261C22×D5—41no—
262C22×D5—41no—
263C22×D5—41no—
264C2×C4×C5—22no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5C22×C3×C411yescharacteristic
271C2×C4×C5C2×C3×D411yescharacteristic
272C2×C4×C5C2×C3×D411yescharacteristic
273C2×C4×C5C2×C3×D411yescharacteristic
274C2×C4×C5C3×#4111yescharacteristic
275C2×C4×C5C3×#4111yescharacteristic
276C2×Dic5—22no—
277C2×Dic5—21no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—21no—
281C2×Dic5—21no—
282C2×Dic5—21no—
283C2×Dic5—21no—
284C2×Dic5C3×SD1612yescomplemented
285D20—22no—
286D20—22no—
287D20—41no—
288D20—41no—
289Dic10—22no—
290Dic10—22no—
291Dic10—22no—
292Dic10C3×C22⋊C414yescomplemented
293C5×C8—22no—
294#168—41no—
295C22×Q8—51no—
296C22×D4—51no—
297C2×SD16—54no—
298C2×SD16—102no—
299C2×SD16—102no—
300C22×C8—51no—
301C22×C8C2×C3×D511yescharacteristic
302C4×Q8—102no—
303C4×Q8—102no—
304C2×C4⋊C4—101no—
305C2×C4⋊C4C2×C3×D511yescharacteristic
306C2×C4⋊C4C2×C3×D511yescharacteristic
307C2×C22⋊C4—101no—
308C2×C22⋊C4—101no—
309C2×C42—51no—
310C2×C42—101no—
311#103—102no—
312#102—102no—
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
320C3×D5—82no—
321C23×C3—102no—
322C23×C3—201no—
323C23×C3—201no—
324C23×C3—201no—
325C23×C3—201no—
326C23×C3—201no—
327C23×C3—201no—
328C23×C3D4×D511yescharacteristic
329C3×Q8—54no—
330C3×Q8—102no—
331C3×Q8—102no—
332C3×Q8—102no—
333C3×D4—102no—
334C3×D4—102no—
335C3×D4—201no—
336C3×D4—201no—
337C2×C3×C4—22no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—52no—
344C2×C3×C4—101no—
345C2×C3×C4—102no—
346C2×C3×C4—102no—
347C2×C3×C4—102no—
348C2×C3×C4—101no—
349C2×C3×C4—101no—
350C2×C3×C4—101no—
351C2×C3×C4—101no—
352C2×C3×C4C2×C4×D511yescharacteristic
353C2×C3×C4C2×D2011yescharacteristic
354C2×C3×C4#70911yescharacteristic
355C2×C3×C4D4×D511yescharacteristic
356C2×C3×C4D4×D511yescharacteristic
357C2×C3×C4#71311yescharacteristic
358C3×C8—22no—
359C3×C8—201no—
360C22×C5C3×C4×D411yescharacteristic
361C22×C5C3×#12011yescharacteristic
362C22×C5C3×#12111yescharacteristic
363C22×C5C2×C3×SD1611yescharacteristic
364C22×C5C3×#13511yescharacteristic
365C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
366C22×C5C3×#13711yescharacteristic
367C2×D5—42no—
368C2×D5—42no—
369C2×D5—42no—
370C2×D5—81no—
371C2×D5—81no—
372C2×D5—81no—
373C2×D5—81no—
374C4×C5—42no—
375C4×C5—81no—
376C4×C5C2×C3×C22⋊C411yescharacteristic
377C4×C5C3×C4×D411yescharacteristic
378C4×C5C3×#12111yescharacteristic
379C4×C5C3×#12311yescharacteristic
380Dic5—24no—
381Dic5—22no—
382Dic5—41no—
383C24—101no—
384C2×Q8—52no—
385C2×Q8—52no—
386C2×Q8—52no—
387C2×Q8—101no—
388C2×Q8—101no—
389C2×Q8—101no—
390C2×D4—52no—
391C2×D4—101no—
392C2×D4—101no—
393C2×D4—101no—
394C2×D4—101no—
395C2×D4—101no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—51no—
399C22×C4—51no—
400C22×C4—101no—
401C22×C4C22×C3×D511yescharacteristic
402SD16—108no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—101no—
406C2×C8—101no—
407C2×C8—101no—
408C2×C8C3×C4×D512yes—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—104no—
413C4⋊C4C3×D2012yes—
414C22⋊C4—202no—
415C22⋊C4—202no—
416C42—104no—
417C42—101no—
418C42—101no—
419C3×C5#188011yescharacteristic, complemented, Hall
420C22×C3—202no—
421C22×C3—202no—
422C22×C3—202no—
423C22×C3—401no—
424C22×C3—401no—
425C22×C3—401no—
426C22×C3—401no—
427C22×C3#402811yescharacteristic
428C22×C3#403011yescharacteristic
429C22×C3#403911yescharacteristic
430C22×C3#405711yescharacteristic
431C22×C3D5×SD1611yescharacteristic
432C22×C3#406211yescharacteristic
433C22×C3#406411yescharacteristic
434C3×C4—42no—
435C3×C4—81no—
436C3×C4—102no—
437C3×C4—104no—
438C3×C4—201no—
439C3×C4C4×D2011yescharacteristic
440C3×C4D5×C22⋊C411yescharacteristic
441C3×C4#404011yescharacteristic
442C3×C4#404111yescharacteristic
443C2×C5C3×#38611yescharacteristic
444C2×C5C3×C4×SD1611yescharacteristic
445C2×C5C3×#44011yescharacteristic
446C2×C5C3×#44811yescharacteristic
447C2×C5C3×#44911yescharacteristic
448C2×C5C3×#46011yescharacteristic
449C2×C5C3×#46411yescharacteristic
450D5—82no—
451C23—102no—
452C23—201no—
453C23—201no—
454C23—201no—
455C23—201no—
456C23—201no—
457C23—201no—
458C23C3×D4×D511yescharacteristic
459Q8—54no—
460Q8—102no—
461Q8—102no—
462Q8—102no—
463D4—102no—
464D4—102no—
465D4—201no—
466D4—201no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—52no—
474C2×C4—101no—
475C2×C4—102no—
476C2×C4—102no—
477C2×C4—102no—
478C2×C4—101no—
479C2×C4—101no—
480C2×C4—101no—
481C2×C4—101no—
482C2×C4C2×C3×C4×D511yescharacteristic
483C2×C4C2×C3×D2011yescharacteristic
484C2×C4C3×#70911yescharacteristic
485C2×C4C3×D4×D511yescharacteristic
486C2×C4C3×D4×D511yescharacteristic
487C2×C4C3×#71311yescharacteristic
488C8—22no—
489C8—201no—
490C2×C3—402no—
491C2×C3#6495411yescharacteristic
492C2×C3#6505111yescharacteristic
493C2×C3#6507011yescharacteristic
494C2×C3#6508711yescharacteristic
495C2×C3#6509511yescharacteristic
496C2×C3#6514111yescharacteristic
497C2×C3#6517811yescharacteristic
498C5C3×#188011yescharacteristic, complemented, Sylow
499C22—202no—
500C22—202no—
501C22—202no—
502C22—401no—
503C22—401no—
504C22—401no—
505C22—401no—
506C22C3×#402811yescharacteristic
507C22C3×#403011yescharacteristic
508C22C3×#403911yescharacteristic
509C22C3×#405711yescharacteristic
510C22C3×D5×SD1611yescharacteristic
511C22C3×#406211yescharacteristic
512C22C3×#406411yescharacteristic
513C4—42no—
514C4—81no—
515C4—102no—
516C4—104no—
517C4—201no—
518C4C3×C4×D2011yescharacteristic
519C4C3×D5×C22⋊C411yescharacteristic
520C4C3×#404011yescharacteristic
521C4C3×#404111yescharacteristic
522C3#1059970211yescharacteristic, direct product factor, Sylow
523C2—402no—
524C2C3×#6495411yescharacteristic
525C2C3×#6505111yescharacteristic
526C2C3×#6507011yescharacteristic
527C2C3×#6508711yescharacteristic
528C2C3×#6509511yescharacteristic
529C2C3×#6514111yescharacteristic
530C2C3×#6517811yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#340C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
cosocleC22×C4×C5C23×C3
hypercenterC3×#340D5
hypocenterC5C3×#1880
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#340

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4033C2semidirect
4. C2×C3×#3942C2semidirect
6. C3×C5×#340C2semidirect
7. C3×#64770C2semidirect
8. #10599702C3direct (N × H)
22. C2×C3×#677C4semidirect
29. C2×#4033C2×C3semidirect
31. C2×#3942C2×C3semidirect
33. C5×#340C2×C3semidirect
34. #64770C2×C3semidirect
76. C3×#340D5semidirect
91. C2×#677C3×C4semidirect
125. C2×C3×Dic5SD16semidirect
133. C3×Dic10C22⋊C4semidirect
195. #340C3×D5semidirect
284. C2×Dic5C3×SD16semidirect
292. Dic10C3×C22⋊C4semidirect
419. C3×C5#1880semidirect
498. C5C3×#1880semidirect
522. C3#10599702direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6772
C2×C3×#40332
C3×#652772
C2×C3×#39422
C2×C3×#39412
C3×C5×#3402
C3×#647702
#105997023
C3×#18805