Finite Group Databasebeta

← C3×#10604213 #14313208

Subgroups — C3×#10604213 #14313208

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65827C211yesmaximal, characteristic
2C2×C3×#4013C211yesmaximal, characteristic, complemented
3C2×C3×#4011C211yesmaximal, characteristic, complemented
4C3×#64865C212yesmaximal, complemented
5C3×#64864C212yesmaximal, complemented
6#10604213C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C22⋊D5C411yescharacteristic, complemented
8C22×C3×C22⋊D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C2×C3×#704—21no—
11C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
12C3×#4076—21no—
13C3×#4076—21no—
14C3×#4032—21no—
15C3×#4032C411yescharacteristic, complemented
16C3×#4032C2211yescharacteristic
17C3×#4028—21no—
18C3×#4028C411yescharacteristic, complemented
19C3×#4028C2211yescharacteristic
20C3×#4013—21no—
21C3×#4013C2212yes—
22C3×#4011—21no—
23C3×#4011C2212yes—
24C3×#2121—51nomaximal, Hall
25#65827C2×C311yescharacteristic
26C2×#4013C2×C311yescharacteristic, complemented
27C2×#4011C2×C311yescharacteristic, complemented
28#64865C2×C312yescomplemented
29#64864C2×C312yescomplemented
30C24×C3×C5C2×C411yescharacteristic
31C23×C3×D5C2×C411yescharacteristic, complemented
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—41no—
34C22×C3×C4×C5—21no—
35C2×C3×C22⋊D5—21no—
36C2×C3×C22⋊D5—21no—
37C2×C3×C22⋊D5—41no—
38C2×C3×C22⋊D5—41no—
39C2×C3×C22⋊D5—41no—
40C2×C3×C22⋊D5—41no—
41C2×C3×C22⋊D5C2×C411yescharacteristic
42C2×C3×C22⋊D5D411yescharacteristic
43C2×C3×C22⋊D5D411yescharacteristic
44C2×C3×C22⋊D5C2311yescharacteristic
45C22×C3×Dic5C2×C411yescharacteristic, complemented
46C22×C3×Dic5D411yescharacteristic
47C2×C3×D20—21no—
48C2×C3×D20—21no—
49C3×C22⋊AGL(1,𝔽5)—22no—
50C3×C22⋊AGL(1,𝔽5)—41no—
51C3×#704—22no—
52C3×#704—41no—
53C2×C3×#170—41no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
56C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
57C3×#685—21no—
58C3×#685—21no—
59C3×#685—21no—
60C3×#685—21no—
61C3×C4⋊Dic5—21no—
62C3×C4⋊Dic5—21no—
63C3×#534—51no—
64C2×C3×#100—51no—
65C2×C3×#99—51no—
66C3×#352—52no—
67C3×AΣL(1,𝔽16)—52no—
68C22×C22⋊D5C3×C411yescharacteristic, complemented
69C22×C22⋊D5C22×C311yescharacteristic
70C2×C22⋊AGL(1,𝔽5)—21no—
71C2×#704—21no—
72C2×C5×C22⋊C4C3×C411yescharacteristic, complemented
73#4076—21no—
74#4076—21no—
75#4032—21no—
76#4032C3×C411yescharacteristic, complemented
77#4032C22×C311yescharacteristic
78#4028—21no—
79#4028C3×C411yescharacteristic, complemented
80#4028C22×C311yescharacteristic
81#4013—21no—
82#4013C22×C312yes—
83#4011—21no—
84#4011C22×C312yes—
85#2121—51noSylow
86C23×C3×C5—21no—
87C23×C3×C5—21no—
88C23×C3×C5—81no—
89C23×C3×C5C22⋊C411yescharacteristic
90C23×C3×C5C22⋊C411yescharacteristic
91C23×C3×C5C22×C411yescharacteristic
92C22×C3×D5—21no—
93C22×C3×D5—21no—
94C22×C3×D5—21no—
95C22×C3×D5—41no—
96C22×C3×D5—81no—
97C22×C3×D5—81no—
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×C3×AGL(1,𝔽5)—42no—
101C2×C3×AGL(1,𝔽5)—82no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—81no—
105C3×C22⋊D5—41no—
106C3×C22⋊D5—41no—
107C3×C22⋊D5—41no—
108C3×C22⋊D5—41no—
109C3×C22⋊D5—42no—
110C3×C22⋊D5—41no—
111C3×C22⋊D5—41no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—21no—
115C2×C3×Dic5—41no—
116C2×C3×Dic5C22⋊C411yescharacteristic
117C2×C3×Dic5C2×D411yescharacteristic
118C3×D20—81no—
119C3×D20—81no—
120C3×#170—42no—
121C22×C3×D4—51no—
122C22×C3×D4—51no—
123C2×C3×M16—101no—
124C3×#123—51no—
125C3×#123—51no—
126C3×#123—101no—
127C3×#121—101no—
128C3×#121—101no—
129C3×#120—51no—
130C3×#120—51no—
131C3×#120—101no—
132C2×C3×C22⋊C4—101no—
133C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
134C3×#100—52no—
135C3×#100—101no—
136C3×#99—52no—
137C3×#99—101no—
138C24×C5C2×C3×C411yescharacteristic
139C23×D5C2×C3×C411yescharacteristic, complemented
140C23×D5C3×D411yescharacteristic
141C22×AGL(1,𝔽5)—41no—
142C22×C4×C5—21no—
143C2×C22⋊D5—21no—
144C2×C22⋊D5—21no—
145C2×C22⋊D5—41no—
146C2×C22⋊D5—41no—
147C2×C22⋊D5—41no—
148C2×C22⋊D5—41no—
149C2×C22⋊D5C2×C3×C411yescharacteristic
150C2×C22⋊D5C3×D411yescharacteristic
151C2×C22⋊D5C3×D411yescharacteristic
152C2×C22⋊D5C23×C311yescharacteristic
153C22×Dic5C2×C3×C411yescharacteristic, complemented
154C22×Dic5C3×D411yescharacteristic
155C2×D20—21no—
156C2×D20—21no—
157C22⋊AGL(1,𝔽5)—22no—
158C22⋊AGL(1,𝔽5)—41no—
159#704—22no—
160#704—41no—
161C2×#170—41no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4C2×C3×C411yescharacteristic, complemented
164C5×C22⋊C4C2×C3×C411yescharacteristic, complemented
165#685—21no—
166#685—21no—
167#685—21no—
168#685—21no—
169C4⋊Dic5—21no—
170C4⋊Dic5—21no—
171#534—51no—
172C2×#100—51no—
173C2×#99—51no—
174#352—52no—
175AΣL(1,𝔽16)—52no—
176C22×C3×C5—21no—
177C22×C3×C5—21no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5—81no—
182C22×C3×C5—81no—
183C22×C3×C5#9911yescharacteristic
184C22×C3×C5#9911yescharacteristic
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C2×C3×D5—21no—
187C2×C3×D5—21no—
188C2×C3×D5—42no—
189C2×C3×D5—41no—
190C2×C3×D5—41no—
191C2×C3×D5—41no—
192C2×C3×D5—41no—
193C2×C3×D5—41no—
194C2×C3×D5—81no—
195C2×C3×D5—82no—
196C3×AGL(1,𝔽5)—82no—
197C3×C4×C5—41no—
198C3×C4×C5—41no—
199C3×Dic5—21no—
200C3×Dic5—21no—
201C3×Dic5—41no—
202C24×C3—51no—
203C24×C3—51no—
204C24×C3C2×AGL(1,𝔽5)11yescharacteristic
205C2×C3×D4—51no—
206C2×C3×D4—51no—
207C2×C3×D4—51no—
208C2×C3×D4—51no—
209C2×C3×D4—101no—
210C2×C3×D4—101no—
211C2×C3×D4—101no—
212C2×C3×D4—101no—
213C2×C3×D4—201no—
214C2×C3×D4—201no—
215C2×C3×D4—201no—
216C2×C3×D4—201no—
217C22×C3×C4—21no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—201no—
221C3×M16—102no—
222C3×M16—201no—
223C2×C3×C8—201no—
224C3×C4⋊C4—101no—
225C3×C4⋊C4—101no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—101no—
228C3×C22⋊C4—101no—
229C3×C22⋊C4—101no—
230C3×C22⋊C4—101no—
231C3×C22⋊C4—102no—
232C3×C22⋊C4—201no—
233C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
234C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
235C23×C5—21no—
236C23×C5—21no—
237C23×C5—81no—
238C23×C5C3×C22⋊C411yescharacteristic
239C23×C5C3×C22⋊C411yescharacteristic
240C23×C5C22×C3×C411yescharacteristic
241C22×D5—21no—
242C22×D5—21no—
243C22×D5—21no—
244C22×D5—41no—
245C22×D5—81no—
246C22×D5—81no—
247C22×D5C3×C22⋊C411yescharacteristic
248C22×D5C2×C3×D411yescharacteristic
249C2×AGL(1,𝔽5)—42no—
250C2×AGL(1,𝔽5)—82no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—81no—
254C22⋊D5—41no—
255C22⋊D5—41no—
256C22⋊D5—41no—
257C22⋊D5—41no—
258C22⋊D5—42no—
259C22⋊D5—41no—
260C22⋊D5—41no—
261C2×Dic5—21no—
262C2×Dic5—21no—
263C2×Dic5—21no—
264C2×Dic5—41no—
265C2×Dic5C3×C22⋊C411yescharacteristic
266C2×Dic5C2×C3×D411yescharacteristic
267D20—81no—
268D20—81no—
269#170—42no—
270C22×D4—51no—
271C22×D4—51no—
272C2×M16—101no—
273#123—51no—
274#123—51no—
275#123—101no—
276#121—101no—
277#121—101no—
278#120—51no—
279#120—51no—
280#120—101no—
281C2×C22⋊C4—101no—
282C2×C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
283#100—52no—
284#100—101no—
285#99—52no—
286#99—101no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5—41no—
291C2×C3×C5—41no—
292C2×C3×C5C2×#9911yescharacteristic
293C3×D5—41no—
294C3×D5—41no—
295C3×D5—81no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—51no—
299C23×C3—51no—
300C23×C3—81no—
301C23×C3—101no—
302C23×C3—101no—
303C23×C3—101no—
304C23×C3—201no—
305C23×C3—401no—
306C23×C3—401no—
307C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
308C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
309C23×C3C22×AGL(1,𝔽5)11yescharacteristic
310C3×D4—201no—
311C3×D4—201no—
312C3×D4—201no—
313C3×D4—201no—
314C3×D4—202no—
315C3×D4—201no—
316C3×D4—201no—
317C3×D4—401no—
318C3×D4—401no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—51no—
322C2×C3×C4—51no—
323C2×C3×C4—81no—
324C2×C3×C4—101no—
325C2×C3×C4—101no—
326C2×C3×C4—101no—
327C2×C3×C4—201no—
328C2×C3×C4—202no—
329C2×C3×C4—402no—
330C3×C8—202no—
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—41no—
334C22×C5—41no—
335C22×C5—41no—
336C22×C5—81no—
337C22×C5—81no—
338C22×C5C3×#9911yescharacteristic
339C22×C5C3×#9911yescharacteristic
340C22×C5C2×C3×C22⋊C411yescharacteristic
341C2×D5—21no—
342C2×D5—21no—
343C2×D5—42no—
344C2×D5—41no—
345C2×D5—41no—
346C2×D5—41no—
347C2×D5—41no—
348C2×D5—41no—
349C2×D5—81no—
350C2×D5—82no—
351AGL(1,𝔽5)—82no—
352C4×C5—41no—
353C4×C5—41no—
354Dic5—21no—
355Dic5—21no—
356Dic5—41no—
357C24—51no—
358C24—51no—
359C24C2×C3×AGL(1,𝔽5)11yescharacteristic
360C2×D4—51no—
361C2×D4—51no—
362C2×D4—51no—
363C2×D4—51no—
364C2×D4—101no—
365C2×D4—101no—
366C2×D4—101no—
367C2×D4—101no—
368C2×D4—201no—
369C2×D4—201no—
370C2×D4—201no—
371C2×D4—201no—
372C22×C4—21no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—201no—
376M16—102no—
377M16—201no—
378C2×C8—201no—
379C4⋊C4—101no—
380C4⋊C4—101no—
381C22⋊C4—21no—
382C22⋊C4—101no—
383C22⋊C4—101no—
384C22⋊C4—101no—
385C22⋊C4—101no—
386C22⋊C4—102no—
387C22⋊C4—201no—
388C22⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
389C22⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
390C3×C5#212111yescharacteristic, complemented, Hall
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—101no—
399C22×C3—101no—
400C22×C3—201no—
401C22×C3—201no—
402C22×C3—202no—
403C22×C3—201no—
404C22×C3—201no—
405C22×C3—201no—
406C22×C3—401no—
407C22×C3—402no—
408C22×C3#399911yescharacteristic
409C22×C3#399911yescharacteristic
410C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—101no—
414C3×C4—101no—
415C3×C4—201no—
416C3×C4—402no—
417C2×C5—21no—
418C2×C5—21no—
419C2×C5—21no—
420C2×C5—41no—
421C2×C5—41no—
422C2×C5C2×C3×#9911yescharacteristic
423D5—41no—
424D5—41no—
425D5—81no—
426C23—21no—
427C23—21no—
428C23—51no—
429C23—51no—
430C23—81no—
431C23—101no—
432C23—101no—
433C23—101no—
434C23—201no—
435C23—401no—
436C23—401no—
437C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
438C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
439C23C22×C3×AGL(1,𝔽5)11yescharacteristic
440D4—201no—
441D4—201no—
442D4—201no—
443D4—201no—
444D4—202no—
445D4—201no—
446D4—201no—
447D4—401no—
448D4—401no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—51no—
452C2×C4—51no—
453C2×C4—81no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—101no—
457C2×C4—201no—
458C2×C4—202no—
459C2×C4—402no—
460C8—202no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—201no—
467C2×C3—201no—
468C2×C3—401no—
469C2×C3C2×#399911yescharacteristic
470C5C3×#212111yescharacteristic, complemented, Sylow
471C22—21no—
472C22—21no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—81no—
477C22—81no—
478C22—101no—
479C22—101no—
480C22—201no—
481C22—201no—
482C22—202no—
483C22—201no—
484C22—201no—
485C22—201no—
486C22—401no—
487C22—402no—
488C22C3×#399911yescharacteristic
489C22C3×#399911yescharacteristic
490C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
491C4—41no—
492C4—41no—
493C4—101no—
494C4—101no—
495C4—201no—
496C4—402no—
497C3#1060421311yescharacteristic, direct product factor, Sylow
498C2—21no—
499C2—21no—
500C2—21no—
501C2—41no—
502C2—41no—
503C2—201no—
504C2—201no—
505C2—401no—
506C2C2×C3×#399911yescharacteristic
subgroupG/H
centerC2×C3C2×#3999
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C22⋊C4C4
socleC2×C3×C5C2×#99
prime radicalC3×#65827C2
cosocleC2×C22⋊D5C23×C3
hypercenterC2×C3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C3×#2121
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C24×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4013C2semidirect
3. C2×C3×#4011C2semidirect
4. C3×#64865C2semidirect
5. C3×#64864C2semidirect
6. #10604213C3direct (N × H)
7. C22×C3×C22⋊D5C4semidirect
11. C2×C3×C5×C22⋊C4C4semidirect
15. C3×#4032C4semidirect
18. C3×#4028C4semidirect
26. C2×#4013C2×C3semidirect
27. C2×#4011C2×C3semidirect
28. #64865C2×C3semidirect
29. #64864C2×C3semidirect
31. C23×C3×D5C2×C4semidirect
45. C22×C3×Dic5C2×C4semidirect
55. C3×C5×C22⋊C4C2×C4semidirect
56. C3×C5×C22⋊C4C2×C4semidirect
68. C22×C22⋊D5C3×C4semidirect
72. C2×C5×C22⋊C4C3×C4semidirect
76. #4032C3×C4semidirect
79. #4028C3×C4semidirect
133. C2×C3×C22⋊C4AGL(1,𝔽5)semidirect
139. C23×D5C2×C3×C4semidirect
153. C22×Dic5C2×C3×C4semidirect
163. C5×C22⋊C4C2×C3×C4semidirect
164. C5×C22⋊C4C2×C3×C4semidirect
233. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
234. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
282. C2×C22⋊C4C3×AGL(1,𝔽5)semidirect
388. C22⋊C4C2×C3×AGL(1,𝔽5)semidirect
389. C22⋊C4C2×C3×AGL(1,𝔽5)semidirect
390. C3×C5#2121semidirect
470. C5C3×#2121semidirect
497. C3#10604213direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658272
C2×C3×#40132
C2×C3×#40112
C3×#648652
C3×#648642
#106042133
C3×#21215