Finite Group Databasebeta

← C3×#10604525 #14301765

Subgroups — C3×#10604525 #14301765

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D5⋊C8C211yesmaximal, characteristic
2C3×D5×C22⋊Q8C211yesmaximal, characteristic
3C2×C3×#4009C212yesmaximal, complemented
4C3×#65765C211yesmaximal, characteristic
5C3×#64874C212yesmaximal, complemented
6#10604525C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C22×C3×#170—21no—
11C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
12C2×C3×D5⋊C8—21no—
13C2×C3×D5⋊C8—24no—
14C2×C3×D5⋊C8C2212yes—
15C3×C5×C22⋊Q8C411yescharacteristic, complemented
16C3×#4092C411yescharacteristic, complemented
17C3×#4070C411yescharacteristic, complemented
18C3×#4043C411yescharacteristic, complemented
19C3×#4042—21no—
20C3×D5×C4⋊C4—21no—
21C3×D5×C4⋊C4C2211yescharacteristic
22C3×D5×C22⋊C4—21no—
23C3×#4024—21no—
24C3×#4009—24no—
25C3×(C2×AGL(1,𝔽5))⋊C4—21no—
26C3×#1892—51nomaximal, Hall
27C22×D5⋊C8C2×C311yescharacteristic
28D5×C22⋊Q8C2×C311yescharacteristic
29C2×#4009C2×C312yescomplemented
30#65765C2×C311yescharacteristic
31#64874C2×C312yescomplemented
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—22no—
34C2×C3×C5×Q8C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×C3×Dic5D411yescharacteristic
37C3×Q8×D5—22no—
38C3×Q8×D5—22no—
39C2×C3×C4×D5—22no—
40C2×C3×C4×D5—22no—
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5C2×C411yescharacteristic
44C2×C3×C4×D5D411yescharacteristic
45C2×C3×C4×D5D411yescharacteristic
46C2×C3×C4×D5C2311yescharacteristic
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C411yescharacteristic
49C3×C22⋊AGL(1,𝔽5)—42no—
50C2×C3×#170—24no—
51C2×C3×#170—22no—
52C3×C4⋊AGL(1,𝔽5)—22no—
53C3×C4⋊AGL(1,𝔽5)—22no—
54C3×D5⋊C8—24no—
55C3×D5⋊C8—22no—
56C3×D5⋊C8D414yes—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4C2×C411yescharacteristic
59C3×C5×C22⋊C4—21no—
60C3×C22⋊Dic5—21no—
61C3×#685—21no—
62C3×#685—21no—
63C3×C4⋊Dic5—21no—
64C3×C4⋊Dic5C2×C411yescharacteristic
65C3×#683—21no—
66C3×#683—21no—
67C3×#683—21no—
68C23×C3×C8—51no—
69C2×C3×C22⋊Q8—51no—
70C2×C3×#103—52no—
71C3×#380—51no—
72C3×#340—52no—
73C2×Q8×D5C22×C311yescharacteristic
74C22×C4×D5C22×C311yescharacteristic
75C2×C22⋊AGL(1,𝔽5)—21no—
76C22×#170—21no—
77C2×C4⋊AGL(1,𝔽5)C22×C312yes—
78C2×D5⋊C8—21no—
79C2×D5⋊C8—24no—
80C2×D5⋊C8C22×C312yes—
81C5×C22⋊Q8C3×C411yescharacteristic, complemented
82#4092C3×C411yescharacteristic, complemented
83#4070C3×C411yescharacteristic, complemented
84#4043C3×C411yescharacteristic, complemented
85#4042—21no—
86D5×C4⋊C4—21no—
87D5×C4⋊C4C22×C311yescharacteristic
88D5×C22⋊C4—21no—
89#4024—21no—
90#4009—24no—
91(C2×AGL(1,𝔽5))⋊C4—21no—
92#1892—51noSylow
93C23×C3×C5C22⋊C411yescharacteristic
94C22×C3×D5—22no—
95C22×C3×D5—22no—
96C22×C3×D5C22⋊C411yescharacteristic
97C22×C3×D5SD1612yes—
98C22×C3×D5Q1612yes—
99C22×C3×D5C2×D411yescharacteristic
100C2×C3×AGL(1,𝔽5)—42no—
101C2×C3×AGL(1,𝔽5)—42no—
102C2×C3×AGL(1,𝔽5)—42no—
103C3×C5×Q8—22no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5C22⋊C411yescharacteristic
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×C3×Dic5—21no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5C22⋊C411yescharacteristic
114C2×C3×Dic5C2×D411yescharacteristic
115C3×C4×D5—22no—
116C3×C4×D5—22no—
117C3×C4×D5—21no—
118C3×C4×D5—21no—
119C3×C4×D5—42no—
120C3×C4×D5—42no—
121C3×C4×D5C2×D412yes—
122C3×C4×D5#4112yes—
123C3×Dic10—22no—
124C3×Dic10—41no—
125C3×Dic10—41no—
126C3×#170—24no—
127C22×C3×Q8—51no—
128C23×C3×C4—51no—
129C22×C3×C8—52no—
130C22×C3×C8—104no—
131C22×C3×C8—101no—
132C22×C3×C8—101no—
133C3×C22⋊Q8—51no—
134C3×C22⋊Q8—51no—
135C3×C22⋊Q8—51no—
136C3×C22⋊Q8—101no—
137C3×C22⋊Q8—101no—
138C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
139C2×C3×C4⋊C4—51no—
140C2×C3×C4⋊C4—52no—
141C2×C3×C4⋊C4—101no—
142C2×C3×C22⋊C4—101no—
143C2×C3×C22⋊C4—101no—
144C3×#103—104no—
145C3×#95—101no—
146C23×D5C3×D411yescharacteristic
147C22×AGL(1,𝔽5)—22no—
148C2×C5×Q8C2×C3×C411yescharacteristic
149C22×C4×C5C2×C3×C411yescharacteristic
150C22×Dic5C3×D411yescharacteristic
151Q8×D5—22no—
152Q8×D5—22no—
153C2×C4×D5—22no—
154C2×C4×D5—22no—
155C2×C4×D5—21no—
156C2×C4×D5—21no—
157C2×C4×D5C2×C3×C411yescharacteristic
158C2×C4×D5C3×D411yescharacteristic
159C2×C4×D5C3×D411yescharacteristic
160C2×C4×D5C23×C311yescharacteristic
161C2×Dic10—21no—
162C2×Dic10C2×C3×C411yescharacteristic
163C22⋊AGL(1,𝔽5)—42no—
164C2×#170—24no—
165C2×#170—22no—
166C4⋊AGL(1,𝔽5)—22no—
167C4⋊AGL(1,𝔽5)—22no—
168D5⋊C8—24no—
169D5⋊C8—22no—
170D5⋊C8C3×D414yes—
171C5×C4⋊C4—21no—
172C5×C4⋊C4C2×C3×C411yescharacteristic
173C5×C22⋊C4—21no—
174C22⋊Dic5—21no—
175#685—21no—
176#685—21no—
177C4⋊Dic5—21no—
178C4⋊Dic5C2×C3×C411yescharacteristic
179#683—21no—
180#683—21no—
181#683—21no—
182C23×C8—51no—
183C2×C22⋊Q8—51no—
184C2×#103—52no—
185#380—51no—
186#340—52no—
187C22×C3×C5—21no—
188C22×C3×C5—21no—
189C22×C3×C5#10312yescomplemented
190C22×C3×C5C2×C22⋊C411yescharacteristic
191C2×C3×D5—21no—
192C2×C3×D5—21no—
193C2×C3×D5—24no—
194C2×C3×D5—24no—
195C2×C3×D5#10312yescomplemented
196C2×C3×D5#12112yes—
197C2×C3×D5C2×SD1611yescharacteristic
198C2×C3×D5C2×Q1611yescharacteristic
199C2×C3×D5#13512yes—
200C3×AGL(1,𝔽5)—82no—
201C3×C4×C5—21no—
202C3×C4×C5—41no—
203C3×C4×C5—41no—
204C3×C4×C5C4×D412yes—
205C3×Dic5—21no—
206C3×Dic5—41no—
207C3×Dic5—41no—
208C3×Dic5#12011yescharacteristic
209C3×Dic5#12311yescharacteristic
210C24×C3—51no—
211C2×C3×Q8—51no—
212C2×C3×Q8—101no—
213C2×C3×Q8—102no—
214C2×C3×Q8—102no—
215C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
216C22×C3×C4—51no—
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—51no—
221C22×C3×C4—102no—
222C22×C3×C4—102no—
223C22×C3×C4—101no—
224C22×C3×C4—101no—
225C22×C3×C4—102no—
226C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
227C2×C3×C8—54no—
228C2×C3×C8—102no—
229C2×C3×C8—102no—
230C2×C3×C8—104no—
231C2×C3×C8—104no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—51no—
234C3×C4⋊C4—101no—
235C3×C4⋊C4—101no—
236C3×C4⋊C4—101no—
237C3×C4⋊C4—102no—
238C3×C4⋊C4—102no—
239C3×C4⋊C4—101no—
240C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
241C3×C22⋊C4—21no—
242C3×C22⋊C4—101no—
243C3×C22⋊C4—101no—
244C3×C22⋊C4—101no—
245C3×C22⋊C4—202no—
246C23×C5C3×C22⋊C411yescharacteristic
247C22×D5—22no—
248C22×D5—22no—
249C22×D5C3×C22⋊C411yescharacteristic
250C22×D5C3×SD1612yes—
251C22×D5C3×Q1612yes—
252C22×D5C2×C3×D411yescharacteristic
253C2×AGL(1,𝔽5)—42no—
254C2×AGL(1,𝔽5)—42no—
255C2×AGL(1,𝔽5)—42no—
256C5×Q8—22no—
257C2×C4×C5—22no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5C3×C22⋊C411yescharacteristic
261C2×C4×C5C22×C3×C411yescharacteristic
262C2×Dic5—21no—
263C2×Dic5—21no—
264C2×Dic5—21no—
265C2×Dic5—21no—
266C2×Dic5C3×C22⋊C411yescharacteristic
267C2×Dic5C2×C3×D411yescharacteristic
268C4×D5—22no—
269C4×D5—22no—
270C4×D5—21no—
271C4×D5—21no—
272C4×D5—42no—
273C4×D5—42no—
274C4×D5C2×C3×D412yes—
275C4×D5C3×#4112yes—
276Dic10—22no—
277Dic10—41no—
278Dic10—41no—
279#170—24no—
280C22×Q8—51no—
281C23×C4—51no—
282C22×C8—52no—
283C22×C8—104no—
284C22×C8—101no—
285C22×C8—101no—
286C22⋊Q8—51no—
287C22⋊Q8—51no—
288C22⋊Q8—51no—
289C22⋊Q8—101no—
290C22⋊Q8—101no—
291C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
292C2×C4⋊C4—51no—
293C2×C4⋊C4—52no—
294C2×C4⋊C4—101no—
295C2×C22⋊C4—101no—
296C2×C22⋊C4—101no—
297#103—104no—
298#95—101no—
299C2×C3×C5—22no—
300C2×C3×C5#38611yescharacteristic
301C2×C3×C5C2×#10311yescharacteristic
302C2×C3×C5#41611yescharacteristic
303C3×D5—22no—
304C3×D5#46512yes—
305C3×D5#46712yes—
306C23×C3—51no—
307C23×C3—51no—
308C23×C3—52no—
309C23×C3—52no—
310C23×C3—102no—
311C23×C3—102no—
312C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
313C3×Q8—22no—
314C3×Q8—102no—
315C3×Q8—201no—
316C3×Q8—201no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—51no—
321C2×C3×C4—52no—
322C2×C3×C4—52no—
323C2×C3×C4—51no—
324C2×C3×C4—101no—
325C2×C3×C4—101no—
326C2×C3×C4—102no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—102no—
330C2×C3×C4—101no—
331C2×C3×C4—101no—
332C2×C3×C4—202no—
333C2×C3×C4—202no—
334C2×C3×C4—202no—
335C2×C3×C4—202no—
336C2×C3×C4—202no—
337C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
338C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
339C3×C8—104no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5C3×#10312yescomplemented
343C22×C5C2×C3×C22⋊C411yescharacteristic
344C2×D5—21no—
345C2×D5—21no—
346C2×D5—24no—
347C2×D5—24no—
348C2×D5C3×#10312yescomplemented
349C2×D5C3×#12112yes—
350C2×D5C2×C3×SD1611yescharacteristic
351C2×D5C2×C3×Q1611yescharacteristic
352C2×D5C3×#13512yes—
353AGL(1,𝔽5)—82no—
354C4×C5—21no—
355C4×C5—41no—
356C4×C5—41no—
357C4×C5C3×C4×D412yes—
358Dic5—21no—
359Dic5—41no—
360Dic5—41no—
361Dic5C3×#12011yescharacteristic
362Dic5C3×#12311yescharacteristic
363C24—51no—
364C2×Q8—51no—
365C2×Q8—101no—
366C2×Q8—102no—
367C2×Q8—102no—
368C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
369C22×C4—51no—
370C22×C4—51no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—102no—
375C22×C4—102no—
376C22×C4—101no—
377C22×C4—101no—
378C22×C4—102no—
379C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
380C2×C8—54no—
381C2×C8—102no—
382C2×C8—102no—
383C2×C8—104no—
384C2×C8—104no—
385C4⋊C4—21no—
386C4⋊C4—51no—
387C4⋊C4—101no—
388C4⋊C4—101no—
389C4⋊C4—101no—
390C4⋊C4—102no—
391C4⋊C4—102no—
392C4⋊C4—101no—
393C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
394C22⋊C4—21no—
395C22⋊C4—101no—
396C22⋊C4—101no—
397C22⋊C4—101no—
398C22⋊C4—202no—
399C3×C5#189211yescharacteristic, complemented, Hall
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—52no—
403C22×C3—52no—
404C22×C3—51no—
405C22×C3—51no—
406C22×C3—52no—
407C22×C3—104no—
408C22×C3—104no—
409C22×C3—101no—
410C22×C3—101no—
411C22×C3#400912yescomplemented
412C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
413C3×C4—21no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—51no—
417C3×C4—51no—
418C3×C4—101no—
419C3×C4—201no—
420C3×C4—201no—
421C3×C4—402no—
422C3×C4D4×AGL(1,𝔽5)12yes—
423C2×C5—22no—
424C2×C5C3×#38611yescharacteristic
425C2×C5C2×C3×#10311yescharacteristic
426C2×C5C3×#41611yescharacteristic
427D5—22no—
428D5C3×#46512yes—
429D5C3×#46712yes—
430C23—51no—
431C23—52no—
432C23—52no—
433C23—51no—
434C23—102no—
435C23—102no—
436C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
437Q8—22no—
438Q8—102no—
439Q8—201no—
440Q8—201no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—51no—
445C2×C4—52no—
446C2×C4—52no—
447C2×C4—51no—
448C2×C4—101no—
449C2×C4—101no—
450C2×C4—102no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—102no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—202no—
457C2×C4—202no—
458C2×C4—202no—
459C2×C4—202no—
460C2×C4—202no—
461C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
462C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
463C8—104no—
464C2×C3—22no—
465C2×C3—52no—
466C2×C3—52no—
467C2×C3—102no—
468C2×C3#6577911yescharacteristic
469C2×C3C2×#400911yescharacteristic
470C2×C3#6579411yescharacteristic
471C5C3×#189211yescharacteristic, complemented, Sylow
472C22—21no—
473C22—21no—
474C22—52no—
475C22—51no—
476C22—51no—
477C22—52no—
478C22—52no—
479C22—104no—
480C22—104no—
481C22—101no—
482C22—101no—
483C22C3×#400912yescomplemented
484C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
485C4—21no—
486C4—41no—
487C4—41no—
488C4—51no—
489C4—51no—
490C4—101no—
491C4—201no—
492C4—201no—
493C4—402no—
494C4C3×D4×AGL(1,𝔽5)12yes—
495C3#1060452511yescharacteristic, direct product factor, Sylow
496C2—22no—
497C2—52no—
498C2—52no—
499C2—102no—
500C2C3×#6577911yescharacteristic
501C2C2×C3×#400911yescharacteristic
502C2C3×#6579411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C22⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C4×D5C23×C3
hypercenterC3×C22⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1892
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×C3×#4009C2semidirect
5. C3×#64874C2semidirect
6. #10604525C3direct (N × H)
15. C3×C5×C22⋊Q8C4semidirect
16. C3×#4092C4semidirect
17. C3×#4070C4semidirect
18. C3×#4043C4semidirect
29. C2×#4009C2×C3semidirect
31. #64874C2×C3semidirect
81. C5×C22⋊Q8C3×C4semidirect
82. #4092C3×C4semidirect
83. #4070C3×C4semidirect
84. #4043C3×C4semidirect
138. C3×C22⋊Q8AGL(1,𝔽5)semidirect
189. C22×C3×C5#103semidirect
195. C2×C3×D5#103semidirect
291. C22⋊Q8C3×AGL(1,𝔽5)semidirect
342. C22×C5C3×#103semidirect
348. C2×D5C3×#103semidirect
399. C3×C5#1892semidirect
411. C22×C3#4009semidirect
471. C5C3×#1892semidirect
483. C22C3×#4009semidirect
495. C3#10604525direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D5⋊C82
C3×D5×C22⋊Q82
C2×C3×#40092
C3×#657652
C3×#648742
#106045253
C3×#18925