Finite Group Databasebeta

← C3×#10604522 #14301998

Subgroups — C3×#10604522 #14301998

Order 1920.

716 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
1C2×C3×#4130C211yesmaximal, characteristic, complemented
2C3×#65961C211yesmaximal, characteristic
3C2×C3×#4010C212yesmaximal, complemented
4C3×#65756C211yesmaximal, characteristic, complemented
5C3×#64875C212yesmaximal, complemented
6#10604522C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C3×#4130—24no—
11C2×C3×C4⋊AGL(1,𝔽5)—21no—
12C2×C3×C4×AGL(1,𝔽5)C2212yes—
13C2×C3×#700C2212yes—
14C3×C5×C22⋊Q8C411yescharacteristic, complemented
15C3×#4092C411yescharacteristic, complemented
16C3×#4076C411yescharacteristic, complemented
17C3×#4041C411yescharacteristic, complemented
18C3×#4040—21no—
19C3×#4039—21no—
20C3×#4038C2211yescharacteristic
21C3×#4028—21no—
22C3×#4027—21no—
23C3×C22⋊#170—21no—
24C3×#4010—24no—
25C3×#4003—21no—
26C3×#1897—51nomaximal, Hall
27C2×#4130C2×C311yescharacteristic, complemented
28#65961C2×C311yescharacteristic
29C2×#4010C2×C312yescomplemented
30#65756C2×C311yescharacteristic, complemented
31#64875C2×C312yescomplemented
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—22no—
34C2×C3×C5×Q8C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic
36C2×C3×C22⋊D5—21no—
37C22×C3×Dic5D411yescharacteristic
38C3×#713—22no—
39C3×#713—22no—
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C2×C3×D20C2×C411yescharacteristic
43C2×C3×C4×D5—22no—
44C2×C3×C4×D5—22no—
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5C2×C411yescharacteristic
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×C4×D5D411yescharacteristic
49C2×C3×C4×D5C2311yescharacteristic
50C3×C22⋊AGL(1,𝔽5)—24no—
51C2×C3×#170—22no—
52C3×C4⋊AGL(1,𝔽5)—24no—
53C3×C4×AGL(1,𝔽5)—22no—
54C3×C4×AGL(1,𝔽5)D414yescomplemented
55C3×#700—22no—
56C3×#700—22no—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4C2×C411yescharacteristic
59C3×C5×C22⋊C4—21no—
60C3×#685—21no—
61C3×#685—21no—
62C3×#685—21no—
63C3×#685—21no—
64C3×C4⋊Dic5C2×C411yescharacteristic
65C3×#683—21no—
66C3×C4×Dic5—21no—
67C3×#525—51no—
68C2×C3×#117—51no—
69C2×C3×#104—52no—
70C3×#407—51no—
71C3×#339—52no—
72C2×#713C22×C311yescharacteristic
73C22×C4×D5C22×C311yescharacteristic
74C2×C22⋊AGL(1,𝔽5)—21no—
75#4130—24no—
76C2×C4⋊AGL(1,𝔽5)—21no—
77C2×C4×AGL(1,𝔽5)C22×C312yes—
78C2×#700C22×C312yes—
79C5×C22⋊Q8C3×C411yescharacteristic, complemented
80#4092C3×C411yescharacteristic, complemented
81#4076C3×C411yescharacteristic, complemented
82#4041C3×C411yescharacteristic, complemented
83#4040—21no—
84#4039—21no—
85#4038C22×C311yescharacteristic
86#4028—21no—
87#4027—21no—
88C22⋊#170—21no—
89#4010—24no—
90#4003—21no—
91#1897—51noSylow
92C23×C3×C5C22⋊C411yescharacteristic
93C22×C3×D5—21no—
94C22×C3×D5—21no—
95C22×C3×D5—21no—
96C22×C3×D5—41no—
97C22×C3×D5—41no—
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×C3×AGL(1,𝔽5)—24no—
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
109C3×C22⋊D5—42no—
110C2×C3×Dic5—22no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5C22⋊C411yescharacteristic
113C2×C3×Dic5C2×D411yescharacteristic
114C3×D20—22no—
115C3×D20—42no—
116C3×D20—41no—
117C3×D20—41no—
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—22no—
121C3×C4×D5—22no—
122C3×C4×D5—42no—
123C3×C4×D5C2×D412yes—
124C3×C4×D5#4112yes—
125C3×#170—42no—
126C2×C3×#41—51no—
127C23×C3×C4—51no—
128C2×C3×M16—52no—
129C3×#123—101no—
130C3×C22⋊Q8—51no—
131C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
132C3×#121—51no—
133C3×#121—51no—
134C3×#120—101no—
135C3×C4×D4—101no—
136C3×C4×D4—101no—
137C3×#117—51no—
138C3×#117—104no—
139C2×C3×C4⋊C4—101no—
140C2×C3×C22⋊C4—101no—
141C2×C3×C42—52no—
142C3×#104—104no—
143C3×C22⋊C8—101no—
144C3×C22⋊C8—101no—
145C23×D5C3×D411yescharacteristic
146C22×AGL(1,𝔽5)—22no—
147C2×C5×Q8C2×C3×C411yescharacteristic
148C22×C4×C5C2×C3×C411yescharacteristic
149C2×C22⋊D5—21no—
150C22×Dic5C3×D411yescharacteristic
151#713—22no—
152#713—22no—
153C2×D20—21no—
154C2×D20—21no—
155C2×D20C2×C3×C411yescharacteristic
156C2×C4×D5—22no—
157C2×C4×D5—22no—
158C2×C4×D5—21no—
159C2×C4×D5C2×C3×C411yescharacteristic
160C2×C4×D5C3×D411yescharacteristic
161C2×C4×D5C3×D411yescharacteristic
162C2×C4×D5C23×C311yescharacteristic
163C22⋊AGL(1,𝔽5)—24no—
164C2×#170—22no—
165C4⋊AGL(1,𝔽5)—24no—
166C4×AGL(1,𝔽5)—22no—
167C4×AGL(1,𝔽5)C3×D414yescomplemented
168#700—22no—
169#700—22no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4C2×C3×C411yescharacteristic
172C5×C22⋊C4—21no—
173#685—21no—
174#685—21no—
175#685—21no—
176#685—21no—
177C4⋊Dic5C2×C3×C411yescharacteristic
178#683—21no—
179C4×Dic5—21no—
180#525—51no—
181C2×#117—51no—
182C2×#104—52no—
183#407—51no—
184#339—52no—
185C22×C3×C5—21no—
186C22×C3×C5—21no—
187C22×C3×C5—21no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C2×C3×D5—21no—
190C2×C3×D5—21no—
191C2×C3×D5—21no—
192C2×C3×D5—21no—
193C2×C3×D5—21no—
194C2×C3×D5—42no—
195C2×C3×D5—42no—
196C2×C3×D5—41no—
197C2×C3×D5—41no—
198C2×C3×D5—41no—
199C2×C3×D5#12011yescharacteristic
200C2×C3×D5#12311yescharacteristic
201C3×AGL(1,𝔽5)—44no—
202C3×C4×C5—21no—
203C3×C4×C5—41no—
204C3×C4×C5—41no—
205C3×C4×C5C4×D412yes—
206C3×Dic5—21no—
207C3×Dic5—41no—
208C3×Dic5#12112yes—
209C24×C3—51no—
210C3×#41—102no—
211C3×#41—102no—
212C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
213C2×C3×D4—51no—
214C2×C3×D4—101no—
215C2×C3×D4—101no—
216C2×C3×D4—101no—
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4—102no—
223C22×C3×C4—102no—
224C22×C3×C4—101no—
225C22×C3×C4—102no—
226C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
227C3×M16—102no—
228C3×M16—102no—
229C2×C3×C8—102no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—51no—
232C3×C4⋊C4—101no—
233C3×C4⋊C4—104no—
234C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
235C3×C22⋊C4—21no—
236C3×C22⋊C4—101no—
237C3×C22⋊C4—101no—
238C3×C22⋊C4—101no—
239C3×C22⋊C4—104no—
240C3×C22⋊C4—101no—
241C3×C42—54no—
242C3×C42—101no—
243C3×C42—102no—
244C23×C5C3×C22⋊C411yescharacteristic
245C22×D5—21no—
246C22×D5—21no—
247C22×D5—21no—
248C22×D5—41no—
249C22×D5—41no—
250C22×D5C3×C22⋊C411yescharacteristic
251C22×D5C2×C3×D411yescharacteristic
252C2×AGL(1,𝔽5)—24no—
253C2×AGL(1,𝔽5)—42no—
254C2×AGL(1,𝔽5)—42no—
255C5×Q8—22no—
256C2×C4×C5—22no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5C3×C22⋊C411yescharacteristic
260C2×C4×C5C22×C3×C411yescharacteristic
261C22⋊D5—42no—
262C2×Dic5—22no—
263C2×Dic5—21no—
264C2×Dic5C3×C22⋊C411yescharacteristic
265C2×Dic5C2×C3×D411yescharacteristic
266D20—22no—
267D20—42no—
268D20—41no—
269D20—41no—
270C4×D5—21no—
271C4×D5—21no—
272C4×D5—22no—
273C4×D5—22no—
274C4×D5—42no—
275C4×D5C2×C3×D412yes—
276C4×D5C3×#4112yes—
277#170—42no—
278C2×#41—51no—
279C23×C4—51no—
280C2×M16—52no—
281#123—101no—
282C22⋊Q8—51no—
283C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
284#121—51no—
285#121—51no—
286#120—101no—
287C4×D4—101no—
288C4×D4—101no—
289#117—51no—
290#117—104no—
291C2×C4⋊C4—101no—
292C2×C22⋊C4—101no—
293C2×C42—52no—
294#104—104no—
295C22⋊C8—101no—
296C22⋊C8—101no—
297C2×C3×C5—41no—
298C2×C3×C5#38611yescharacteristic
299C2×C3×C5#42111yescharacteristic
300C2×C3×C5#42111yescharacteristic
301C3×D5—21no—
302C3×D5—21no—
303C3×D5—41no—
304C3×D5—81no—
305C23×C3—51no—
306C23×C3—51no—
307C23×C3—101no—
308C23×C3—101no—
309C23×C3—101no—
310C23×C3—201no—
311C23×C3—201no—
312C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
313C3×Q8—22no—
314C3×D4—102no—
315C3×D4—201no—
316C3×D4—201no—
317C3×D4—202no—
318C3×D4—202no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—51no—
323C2×C3×C4—52no—
324C2×C3×C4—52no—
325C2×C3×C4—51no—
326C2×C3×C4—102no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4—104no—
332C2×C3×C4—101no—
333C2×C3×C4—202no—
334C2×C3×C4—202no—
335C2×C3×C4—202no—
336C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
337C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
338C3×C8—202no—
339C22×C5—21no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5C2×C3×C22⋊C411yescharacteristic
343C2×D5—21no—
344C2×D5—21no—
345C2×D5—21no—
346C2×D5—21no—
347C2×D5—21no—
348C2×D5—42no—
349C2×D5—42no—
350C2×D5—41no—
351C2×D5—41no—
352C2×D5—41no—
353C2×D5C3×#12011yescharacteristic
354C2×D5C3×#12311yescharacteristic
355AGL(1,𝔽5)—44no—
356C4×C5—21no—
357C4×C5—41no—
358C4×C5—41no—
359C4×C5C3×C4×D412yes—
360Dic5—21no—
361Dic5—41no—
362Dic5C3×#12112yes—
363C24—51no—
364#41—102no—
365#41—102no—
366C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
367C2×D4—51no—
368C2×D4—101no—
369C2×D4—101no—
370C2×D4—101no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—102no—
377C22×C4—102no—
378C22×C4—101no—
379C22×C4—102no—
380C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
381M16—102no—
382M16—102no—
383C2×C8—102no—
384C4⋊C4—21no—
385C4⋊C4—51no—
386C4⋊C4—101no—
387C4⋊C4—104no—
388C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—101no—
391C22⋊C4—101no—
392C22⋊C4—101no—
393C22⋊C4—104no—
394C22⋊C4—101no—
395C42—54no—
396C42—101no—
397C42—102no—
398C3×C5#189711yescharacteristic, complemented, Hall
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—51no—
403C22×C3—51no—
404C22×C3—101no—
405C22×C3—101no—
406C22×C3—101no—
407C22×C3—101no—
408C22×C3—101no—
409C22×C3—202no—
410C22×C3—202no—
411C22×C3—201no—
412C22×C3—201no—
413C22×C3—201no—
414C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
415C3×C4—21no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—52no—
419C3×C4—101no—
420C3×C4—204no—
421C3×C4—201no—
422C3×C4D4×AGL(1,𝔽5)12yes—
423C2×C5—41no—
424C2×C5C3×#38611yescharacteristic
425C2×C5C3×#42111yescharacteristic
426C2×C5C3×#42111yescharacteristic
427D5—21no—
428D5—21no—
429D5—41no—
430D5—81no—
431C23—51no—
432C23—51no—
433C23—101no—
434C23—101no—
435C23—101no—
436C23—201no—
437C23—201no—
438C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
439Q8—22no—
440D4—102no—
441D4—201no—
442D4—201no—
443D4—202no—
444D4—202no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—51no—
449C2×C4—52no—
450C2×C4—52no—
451C2×C4—51no—
452C2×C4—102no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—104no—
458C2×C4—101no—
459C2×C4—202no—
460C2×C4—202no—
461C2×C4—202no—
462C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
463C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
464C8—202no—
465C2×C3—41no—
466C2×C3—101no—
467C2×C3—101no—
468C2×C3—201no—
469C2×C3—401no—
470C2×C3#6577911yescharacteristic
471C2×C3#6578411yescharacteristic
472C2×C3#6579511yescharacteristic
473C5C3×#189711yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—21no—
477C22—51no—
478C22—51no—
479C22—101no—
480C22—101no—
481C22—101no—
482C22—101no—
483C22—101no—
484C22—202no—
485C22—202no—
486C22—201no—
487C22—201no—
488C22—201no—
489C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
490C4—21no—
491C4—41no—
492C4—41no—
493C4—52no—
494C4—101no—
495C4—204no—
496C4—201no—
497C4C3×D4×AGL(1,𝔽5)12yes—
498C3#1060452211yescharacteristic, direct product factor, Sylow
499C2—41no—
500C2—101no—
501C2—101no—
502C2—201no—
503C2—401no—
504C2C3×#6577911yescharacteristic
505C2C3×#6578411yescharacteristic
506C2C3×#6579511yescharacteristic
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 radicalC3×#65961C2
cosocleC2×C4×D5C23×C3
hypercenterC3×C22⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1897
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
1. C2×C3×#4130C2semidirect
3. C2×C3×#4010C2semidirect
4. C3×#65756C2semidirect
5. C3×#64875C2semidirect
6. #10604522C3direct (N × H)
14. C3×C5×C22⋊Q8C4semidirect
15. C3×#4092C4semidirect
16. C3×#4076C4semidirect
17. C3×#4041C4semidirect
27. C2×#4130C2×C3semidirect
29. C2×#4010C2×C3semidirect
30. #65756C2×C3semidirect
31. #64875C2×C3semidirect
54. C3×C4×AGL(1,𝔽5)D4semidirect
79. C5×C22⋊Q8C3×C4semidirect
80. #4092C3×C4semidirect
81. #4076C3×C4semidirect
82. #4041C3×C4semidirect
131. C3×C22⋊Q8AGL(1,𝔽5)semidirect
167. C4×AGL(1,𝔽5)C3×D4semidirect
283. C22⋊Q8C3×AGL(1,𝔽5)semidirect
398. C3×C5#1897semidirect
473. C5C3×#1897semidirect
498. C3#10604522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41302
C3×#659612
C2×C3×#40102
C3×#657562
C3×#648752
#106045223
C3×#18975