Finite Group Databasebeta

← C3×#10604952 #14302149

Subgroups — C3×#10604952 #14302149

Order 1920.

780 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#704C211yesmaximal, characteristic, complemented
2C2×C3×#4085C211yesmaximal, characteristic
3C2×C3×#4013C212yesmaximal, complemented
4C3×#65798C211yesmaximal, characteristic, complemented
5C3×#64919C212yesmaximal, complemented
6#10604952C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C411yescharacteristic
8C22×C3×C22⋊D5C411yescharacteristic
9C22×C3×C22⋊D5C2211yescharacteristic
10C23×C3×Dic5C2211yescharacteristic
11C2×C3×#704—21no—
12C2×C3×#704—24no—
13C2×C3×#704C2212yes—
14C2×C3×#704C2212yes—
15C22×C3×#170—21no—
16C2×C3×C22⋊Dic5C411yescharacteristic
17C3×#4085—21no—
18C3×#4085—21no—
19C3×#4085—21no—
20C3×#4085—21no—
21C2×C3×#685C411yescharacteristic
22C2×C3×#683C2211yescharacteristic
23C3×#4013—24no—
24C3×C22⋊#170—21no—
25C3×C22⋊#170—21no—
26C3×#1900—51nomaximal, Hall
27C22×#704C2×C311yescharacteristic, complemented
28C2×#4085C2×C311yescharacteristic
29C2×#4013C2×C312yescomplemented
30#65798C2×C311yescharacteristic, complemented
31#64919C2×C312yescomplemented
32C24×C3×C5C2×C411yescharacteristic
33C24×C3×C5C2×C411yescharacteristic
34C23×C3×D5C2×C411yescharacteristic
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—42no—
38C22×C3×C4×C5C2×C411yescharacteristic
39C2×C3×C22⋊D5—21no—
40C2×C3×C22⋊D5—21no—
41C2×C3×C22⋊D5—21no—
42C2×C3×C22⋊D5—21no—
43C2×C3×C22⋊D5—22no—
44C2×C3×C22⋊D5—22no—
45C2×C3×C22⋊D5—42no—
46C22×C3×Dic5—22no—
47C22×C3×Dic5—22no—
48C22×C3×Dic5C2×C411yescharacteristic
49C22×C3×Dic5C2×C411yescharacteristic
50C22×C3×Dic5D411yescharacteristic
51C22×C3×Dic5D411yescharacteristic
52C22×C3×Dic5D411yescharacteristic
53C22×C3×Dic5D411yescharacteristic
54C22×C3×Dic5C2311yescharacteristic
55C3×#704—24no—
56C3×#704—22no—
57C3×#704—22no—
58C3×#704—22no—
59C3×#704D414yescomplemented
60C2×C3×#170—24no—
61C2×C3×#170—22no—
62C2×C3×#170—22no—
63C3×C22⋊Dic5—21no—
64C3×C22⋊Dic5—21no—
65C3×#685—21no—
66C3×#685—21no—
67C3×#683—21no—
68C3×#683—21no—
69C22×C3×M16—51no—
70C2×C3×#121—51no—
71C2×C3×#100—52no—
72C3×#407—51no—
73C3×#343—52no—
74C22×C5×D4C3×C411yescharacteristic
75C22×C22⋊D5C3×C411yescharacteristic
76C22×C22⋊D5C22×C311yescharacteristic
77C23×Dic5C22×C311yescharacteristic
78C2×#704—21no—
79C2×#704—24no—
80C2×#704C22×C312yes—
81C2×#704C22×C312yes—
82C22×#170—21no—
83C2×C22⋊Dic5C3×C411yescharacteristic
84#4085—21no—
85#4085—21no—
86#4085—21no—
87#4085—21no—
88C2×#685C3×C411yescharacteristic
89C2×#683C22×C311yescharacteristic
90#4013—24no—
91C22⋊#170—21no—
92C22⋊#170—21no—
93#1900—51noSylow
94C23×C3×C5—22no—
95C23×C3×C5—22no—
96C23×C3×C5—21no—
97C23×C3×C5—22no—
98C23×C3×C5—41no—
99C23×C3×C5—41no—
100C23×C3×C5—41no—
101C23×C3×C5C22⋊C411yescharacteristic
102C23×C3×C5C22⋊C411yescharacteristic
103C23×C3×C5C22×C411yescharacteristic
104C22×C3×D5—21no—
105C22×C3×D5—22no—
106C22×C3×D5—41no—
107C22×C3×D5—41no—
108C3×C5×D4—42no—
109C3×C5×D4—42no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—41no—
112C3×C22⋊D5—42no—
113C3×C22⋊D5—42no—
114C3×C22⋊D5—41no—
115C3×C22⋊D5—42no—
116C3×C22⋊D5—41no—
117C2×C3×Dic5—22no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—21no—
120C2×C3×Dic5—22no—
121C2×C3×Dic5—22no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5—21no—
124C2×C3×Dic5—21no—
125C2×C3×Dic5—41no—
126C2×C3×Dic5C22⋊C411yescharacteristic
127C2×C3×Dic5C22⋊C411yescharacteristic
128C2×C3×Dic5C2×D412yes—
129C2×C3×Dic5C2×D411yescharacteristic
130C2×C3×Dic5C2×D411yescharacteristic
131C2×C3×Dic5#4112yes—
132C3×#170—24no—
133C3×#170—42no—
134C22×C3×D4—51no—
135C22×C3×D4—51no—
136C22×C3×D4AGL(1,𝔽5)11yescharacteristic
137C23×C3×C4—51no—
138C2×C3×M16—52no—
139C2×C3×M16—52no—
140C2×C3×M16—104no—
141C2×C3×M16—101no—
142C22×C3×C8—101no—
143C3×#121—101no—
144C3×#121—101no—
145C3×#121—101no—
146C3×#121—101no—
147C2×C3×C4⋊C4—51no—
148C2×C3×C22⋊C4—51no—
149C2×C3×C22⋊C4—51no—
150C3×#100—104no—
151C3×C22⋊C8—101no—
152C3×C22⋊C8—101no—
153C24×C5C2×C3×C411yescharacteristic
154C24×C5C2×C3×C411yescharacteristic
155C23×D5C2×C3×C411yescharacteristic
156C2×C5×D4—21no—
157C2×C5×D4—21no—
158C2×C5×D4—42no—
159C22×C4×C5C2×C3×C411yescharacteristic
160C2×C22⋊D5—21no—
161C2×C22⋊D5—21no—
162C2×C22⋊D5—22no—
163C2×C22⋊D5—21no—
164C2×C22⋊D5—21no—
165C2×C22⋊D5—22no—
166C2×C22⋊D5—42no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5C2×C3×C411yescharacteristic
170C22×Dic5C2×C3×C411yescharacteristic
171C22×Dic5C3×D411yescharacteristic
172C22×Dic5C3×D411yescharacteristic
173C22×Dic5C3×D411yescharacteristic
174C22×Dic5C3×D411yescharacteristic
175C22×Dic5C23×C311yescharacteristic
176#704—24no—
177#704—22no—
178#704—22no—
179#704—22no—
180#704C3×D414yescomplemented
181C2×#170—24no—
182C2×#170—22no—
183C2×#170—22no—
184C22⋊Dic5—21no—
185C22⋊Dic5—21no—
186#685—21no—
187#685—21no—
188#683—21no—
189#683—21no—
190C22×M16—51no—
191C2×#121—51no—
192C2×#100—52no—
193#407—51no—
194#343—52no—
195C22×C3×C5—21no—
196C22×C3×C5—21no—
197C22×C3×C5—21no—
198C22×C3×C5—21no—
199C22×C3×C5—21no—
200C22×C3×C5—21no—
201C22×C3×C5—21no—
202C22×C3×C5—44no—
203C22×C3×C5—41no—
204C22×C3×C5—41no—
205C22×C3×C5—41no—
206C22×C3×C5—82no—
207C22×C3×C5#10012yescomplemented
208C22×C3×C5C2×C22⋊C411yescharacteristic
209C22×C3×C5C4×D412yes—
210C2×C3×D5—41no—
211C2×C3×D5—41no—
212C2×C3×D5—41no—
213C2×C3×D5—82no—
214C3×C4×C5—41no—
215C3×Dic5—21no—
216C3×Dic5—21no—
217C3×Dic5—41no—
218C3×Dic5#12011yescharacteristic
219C3×Dic5#12112yes—
220C3×Dic5#12311yescharacteristic
221C24×C3—51no—
222C24×C3C2×AGL(1,𝔽5)11yescharacteristic
223C24×C3C2×AGL(1,𝔽5)11yescharacteristic
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—42no—
227C2×C3×D4—102no—
228C2×C3×D4—101no—
229C2×C3×D4—101no—
230C2×C3×D4—102no—
231C2×C3×D4—101no—
232C2×C3×D4—101no—
233C2×C3×D4—202no—
234C22×C3×C4—51no—
235C22×C3×C4—51no—
236C22×C3×C4—51no—
237C22×C3×C4—51no—
238C22×C3×C4—51no—
239C22×C3×C4—51no—
240C22×C3×C4—51no—
241C22×C3×C4—102no—
242C22×C3×C4—102no—
243C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
244C3×M16—54no—
245C3×M16—102no—
246C3×M16—104no—
247C3×M16—102no—
248C3×M16—102no—
249C2×C3×C8—102no—
250C2×C3×C8—104no—
251C2×C3×C8—102no—
252C3×C4⋊C4—101no—
253C3×C4⋊C4—101no—
254C3×C22⋊C4—101no—
255C3×C22⋊C4—101no—
256C3×C22⋊C4—101no—
257C3×C22⋊C4—101no—
258C23×C5—22no—
259C23×C5—22no—
260C23×C5—21no—
261C23×C5—22no—
262C23×C5—41no—
263C23×C5—41no—
264C23×C5—41no—
265C23×C5C3×C22⋊C411yescharacteristic
266C23×C5C3×C22⋊C411yescharacteristic
267C23×C5C22×C3×C411yescharacteristic
268C22×D5—21no—
269C22×D5—22no—
270C22×D5—41no—
271C22×D5—41no—
272C5×D4—42no—
273C5×D4—42no—
274C2×C4×C5—21no—
275C2×C4×C5—41no—
276C22⋊D5—42no—
277C22⋊D5—42no—
278C22⋊D5—41no—
279C22⋊D5—42no—
280C22⋊D5—41no—
281C2×Dic5—22no—
282C2×Dic5—21no—
283C2×Dic5—21no—
284C2×Dic5—22no—
285C2×Dic5—22no—
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5—21no—
289C2×Dic5—41no—
290C2×Dic5C3×C22⋊C411yescharacteristic
291C2×Dic5C3×C22⋊C411yescharacteristic
292C2×Dic5C2×C3×D412yes—
293C2×Dic5C2×C3×D411yescharacteristic
294C2×Dic5C2×C3×D411yescharacteristic
295C2×Dic5C3×#4112yes—
296#170—24no—
297#170—42no—
298C22×D4—51no—
299C22×D4—51no—
300C22×D4C3×AGL(1,𝔽5)11yescharacteristic
301C23×C4—51no—
302C2×M16—52no—
303C2×M16—52no—
304C2×M16—104no—
305C2×M16—101no—
306C22×C8—101no—
307#121—101no—
308#121—101no—
309#121—101no—
310#121—101no—
311C2×C4⋊C4—51no—
312C2×C22⋊C4—51no—
313C2×C22⋊C4—51no—
314#100—104no—
315C22⋊C8—101no—
316C22⋊C8—101no—
317C2×C3×C5—22no—
318C2×C3×C5—22no—
319C2×C3×C5—41no—
320C2×C3×C5—81no—
321C2×C3×C5#38611yescharacteristic
322C2×C3×C5C2×#10011yescharacteristic
323C2×C3×C5#41311yescharacteristic
324C3×D5—81no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—101no—
333C23×C3—102no—
334C23×C3—201no—
335C23×C3—201no—
336C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
337C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
338C23×C3C22×AGL(1,𝔽5)11yescharacteristic
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—201no—
342C3×D4—202no—
343C3×D4—201no—
344C3×D4—202no—
345C3×D4—202no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—52no—
349C2×C3×C4—52no—
350C2×C3×C4—51no—
351C2×C3×C4—51no—
352C2×C3×C4—51no—
353C2×C3×C4—51no—
354C2×C3×C4—101no—
355C2×C3×C4—101no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—101no—
359C2×C3×C4—101no—
360C2×C3×C4—102no—
361C2×C3×C4—101no—
362C2×C3×C4—201no—
363C3×C8—104no—
364C3×C8—202no—
365C22×C5—21no—
366C22×C5—21no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5—21no—
370C22×C5—21no—
371C22×C5—21no—
372C22×C5—44no—
373C22×C5—41no—
374C22×C5—41no—
375C22×C5—41no—
376C22×C5—82no—
377C22×C5C3×#10012yescomplemented
378C22×C5C2×C3×C22⋊C411yescharacteristic
379C22×C5C3×C4×D412yes—
380C2×D5—41no—
381C2×D5—41no—
382C2×D5—41no—
383C2×D5—82no—
384C4×C5—41no—
385Dic5—21no—
386Dic5—21no—
387Dic5—41no—
388Dic5C3×#12011yescharacteristic
389Dic5C3×#12112yes—
390Dic5C3×#12311yescharacteristic
391C24—51no—
392C24C2×C3×AGL(1,𝔽5)11yescharacteristic
393C24C2×C3×AGL(1,𝔽5)11yescharacteristic
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—42no—
397C2×D4—102no—
398C2×D4—101no—
399C2×D4—101no—
400C2×D4—102no—
401C2×D4—101no—
402C2×D4—101no—
403C2×D4—202no—
404C22×C4—51no—
405C22×C4—51no—
406C22×C4—51no—
407C22×C4—51no—
408C22×C4—51no—
409C22×C4—51no—
410C22×C4—51no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
414M16—54no—
415M16—102no—
416M16—104no—
417M16—102no—
418M16—102no—
419C2×C8—102no—
420C2×C8—104no—
421C2×C8—102no—
422C4⋊C4—101no—
423C4⋊C4—101no—
424C22⋊C4—101no—
425C22⋊C4—101no—
426C22⋊C4—101no—
427C22⋊C4—101no—
428C3×C5#190011yescharacteristic, complemented, Hall
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—44no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—82no—
441C22×C3—201no—
442C22×C3—201no—
443C22×C3—201no—
444C22×C3—402no—
445C22×C3#401312yescomplemented
446C22×C3D4×AGL(1,𝔽5)12yes—
447C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
448C3×C4—41no—
449C3×C4—51no—
450C3×C4—52no—
451C3×C4—51no—
452C3×C4—101no—
453C3×C4—101no—
454C3×C4—201no—
455C2×C5—22no—
456C2×C5—22no—
457C2×C5—41no—
458C2×C5—81no—
459C2×C5C3×#38611yescharacteristic
460C2×C5C2×C3×#10011yescharacteristic
461C2×C5C3×#41311yescharacteristic
462D5—81no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—101no—
471C23—102no—
472C23—201no—
473C23—201no—
474C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
475C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
476C23C22×C3×AGL(1,𝔽5)11yescharacteristic
477D4—42no—
478D4—42no—
479D4—201no—
480D4—202no—
481D4—201no—
482D4—202no—
483D4—202no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—52no—
487C2×C4—52no—
488C2×C4—51no—
489C2×C4—51no—
490C2×C4—51no—
491C2×C4—51no—
492C2×C4—101no—
493C2×C4—101no—
494C2×C4—102no—
495C2×C4—102no—
496C2×C4—101no—
497C2×C4—101no—
498C2×C4—102no—
499C2×C4—101no—
500C2×C4—201no—
501C8—104no—
502C8—202no—
503C2×C3—22no—
504C2×C3—22no—
505C2×C3—41no—
506C2×C3—81no—
507C2×C3—401no—
508C2×C3#6577711yescharacteristic
509C2×C3#6577911yescharacteristic
510C2×C3C2×#401311yescharacteristic
511C5C3×#190011yescharacteristic, complemented, Sylow
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—44no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—82no—
524C22—201no—
525C22—201no—
526C22—201no—
527C22—402no—
528C22C3×#401312yescomplemented
529C22C3×D4×AGL(1,𝔽5)12yes—
530C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
531C4—41no—
532C4—51no—
533C4—52no—
534C4—51no—
535C4—101no—
536C4—101no—
537C4—201no—
538C3#1060495211yescharacteristic, direct product factor, Sylow
539C2—22no—
540C2—22no—
541C2—41no—
542C2—81no—
543C2—401no—
544C2C3×#6577711yescharacteristic
545C2C3×#6577911yescharacteristic
546C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#4085C2
cosocleC22×Dic5C23×C3
hypercenterC22×C3×D4AGL(1,𝔽5)
hypocenterC5C3×#1900
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#704C2semidirect
3. C2×C3×#4013C2semidirect
4. C3×#65798C2semidirect
5. C3×#64919C2semidirect
6. #10604952C3direct (N × H)
27. C22×#704C2×C3semidirect
29. C2×#4013C2×C3semidirect
30. #65798C2×C3semidirect
31. #64919C2×C3semidirect
59. C3×#704D4semidirect
180. #704C3×D4semidirect
207. C22×C3×C5#100semidirect
377. C22×C5C3×#100semidirect
428. C3×C5#1900semidirect
445. C22×C3#4013semidirect
511. C5C3×#1900semidirect
528. C22C3×#4013semidirect
538. C3#10604952direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7042
C2×C3×#40852
C2×C3×#40132
C3×#657982
C3×#649192
#106049523
C3×#19005