Finite Group Databasebeta

← C3×C4×#4007 #14295701

Subgroups — C3×C4×#4007 #14295701

Order 1920.

792 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×D4×D5C211yesmaximal, characteristic
2C2×C3×#4007C212yesmaximal
3C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×C4×D5⋊C8C211yesmaximal, characteristic, complemented
5C3×#64874C212yesmaximal, complemented
6C4×#4007C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×D4×D5C2211yescharacteristic
8C22×C3×C4×D5—21no—
9C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
10C2×C3×C4×AGL(1,𝔽5)—21no—
11C2×C3×D5⋊C8C2212yes—
12C3×C4×C5×D4C411yescharacteristic, complemented
13C3×D4×Dic5C411yescharacteristic, complemented
14C3×C4×C22⋊D5—21no—
15C3×#4039C411yescharacteristic, complemented
16C3×D5×C4⋊C4C2211yescharacteristic
17C3×#4027—21no—
18C3×D5×C22⋊C4—21no—
19C3×C4×D20C411yescharacteristic, complemented
20C3×C42×D5C2211yescharacteristic
21C3×#4007C418yesdirect product factor
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×C4×#170—21no—
24C3×C4×#170—21no—
25C3×C4×#102—51nomaximal, Hall
26C4×D4×D5C2×C311yescharacteristic
27C2×#4007C2×C312yes—
28C4×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
29C4×D5⋊C8C2×C311yescharacteristic, complemented
30#64874C2×C312yescomplemented
31C23×C3×D5—21no—
32C22×C3×AGL(1,𝔽5)—22no—
33C2×C3×C5×D4C2×C411yescharacteristic
34C22×C3×C4×C5—21no—
35C2×C3×C22⋊D5—21no—
36C22×C3×Dic5—21no—
37C3×D4×D5—22no—
38C3×D4×D5C2×C414yes—
39C2×C3×D20C2×C411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—42no—
44C2×C3×C4×D5—42no—
45C2×C3×C4×D5D411yescharacteristic
46C2×C3×C4×D5D411yescharacteristic
47C2×C3×C4×D5C2311yescharacteristic
48C2×C3×#170—22no—
49C3×C4⋊AGL(1,𝔽5)—22no—
50C3×C4⋊AGL(1,𝔽5)C2×C414yescomplemented
51C3×C4×AGL(1,𝔽5)—42no—
52C3×D5⋊C8—22no—
53C3×D5⋊C8C2×C414yescomplemented
54C3×C5×C4⋊C4C2×C411yescharacteristic
55C3×C5×C22⋊C4—21no—
56C3×C42×C5C2×C411yescharacteristic
57C3×C22⋊Dic5—21no—
58C3×#685—21no—
59C3×#685—21no—
60C3×C4⋊Dic5C2×C411yescharacteristic
61C3×#683—21no—
62C3×C4×Dic5C2×C411yescharacteristic
63C3×C4×Dic5D411yescharacteristic
64C3×C4×Dic5D411yescharacteristic
65C2×C3×C4×D4—51no—
66C2×C3×#102—52no—
67C2×C3×C4×C8—51no—
68C3×C4×C4⋊C4—51no—
69C3×#340—52no—
70C2×D4×D5C22×C311yescharacteristic
71C22×C4×D5—21no—
72C2×C4⋊AGL(1,𝔽5)C22×C312yes—
73C2×C4×AGL(1,𝔽5)—21no—
74C2×D5⋊C8C22×C312yes—
75C4×C5×D4C3×C411yescharacteristic, complemented
76D4×Dic5C3×C411yescharacteristic, complemented
77C4×C22⋊D5—21no—
78#4039C3×C411yescharacteristic, complemented
79D5×C4⋊C4C22×C311yescharacteristic
80#4027—21no—
81D5×C22⋊C4—21no—
82C4×D20C3×C411yescharacteristic, complemented
83C42×D5C22×C311yescharacteristic
84#4007C3×C418yesdirect product factor
85(C2×AGL(1,𝔽5))⋊C4—21no—
86C4×#170—21no—
87C4×#170—21no—
88C4×#102—51noSylow
89C23×C3×C5—21no—
90C22×C3×D5—21no—
91C22×C3×D5—24no—
92C22×C3×D5—41no—
93C22×C3×D5—41no—
94C22×C3×D5—42no—
95C22×C3×D5C2×D411yescharacteristic
96C2×C3×AGL(1,𝔽5)—24no—
97C2×C3×AGL(1,𝔽5)—42no—
98C2×C3×AGL(1,𝔽5)—42no—
99C3×C5×D4—21no—
100C3×C5×D4C4212yescomplemented
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—42no—
103C2×C3×C4×C5C22⋊C411yescharacteristic
104C2×C3×C4×C5C22⋊C411yescharacteristic
105C2×C3×C4×C5C22×C411yescharacteristic
106C3×C22⋊D5—24no—
107C2×C3×Dic5—21no—
108C2×C3×Dic5—42no—
109C2×C3×Dic5C22⋊C411yescharacteristic
110C2×C3×Dic5C22⋊C411yescharacteristic
111C2×C3×Dic5C2×D411yescharacteristic
112C3×D20—21no—
113C3×D20C4212yescomplemented
114C3×C4×D5—22no—
115C3×C4×D5—41no—
116C3×C4×D5—41no—
117C3×C4×D5—42no—
118C3×C4×D5—42no—
119C3×C4×D5D812yes—
120C3×C4×D5SD1612yes—
121C3×C4×D5C22×C412yes—
122C3×C4×D5#4112yes—
123C3×#170—24no—
124C22×C3×D4—51no—
125C23×C3×C4—101no—
126C22×C3×C8—52no—
127C3×C4×D4—51no—
128C3×C4×D4—51no—
129C3×C4×D4—51no—
130C3×C4×D4—101no—
131C3×C4×D4—101no—
132C3×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
133C2×C3×C4⋊C4—52no—
134C2×C3×C4⋊C4—51no—
135C2×C3×C22⋊C4—101no—
136C2×C3×C42—51no—
137C2×C3×C42—101no—
138C3×#102—58no—
139C3×C4×C8—101no—
140C3×C4×C8—101no—
141C3×#95—101no—
142C23×D5—21no—
143C22×AGL(1,𝔽5)—22no—
144C2×C5×D4C2×C3×C411yescharacteristic
145C22×C4×C5—21no—
146C2×C22⋊D5—21no—
147C22×Dic5—21no—
148D4×D5—22no—
149D4×D5C2×C3×C414yes—
150C2×D20C2×C3×C411yescharacteristic
151C2×C4×D5—21no—
152C2×C4×D5—21no—
153C2×C4×D5—21no—
154C2×C4×D5—42no—
155C2×C4×D5—42no—
156C2×C4×D5C3×D411yescharacteristic
157C2×C4×D5C3×D411yescharacteristic
158C2×C4×D5C23×C311yescharacteristic
159C2×#170—22no—
160C4⋊AGL(1,𝔽5)—22no—
161C4⋊AGL(1,𝔽5)C2×C3×C414yescomplemented
162C4×AGL(1,𝔽5)—42no—
163D5⋊C8—22no—
164D5⋊C8C2×C3×C414yescomplemented
165C5×C4⋊C4C2×C3×C411yescharacteristic
166C5×C22⋊C4—21no—
167C42×C5C2×C3×C411yescharacteristic
168C22⋊Dic5—21no—
169#685—21no—
170#685—21no—
171C4⋊Dic5C2×C3×C411yescharacteristic
172#683—21no—
173C4×Dic5C2×C3×C411yescharacteristic
174C4×Dic5C3×D411yescharacteristic
175C4×Dic5C3×D411yescharacteristic
176C2×C4×D4—51no—
177C2×#102—52no—
178C2×C4×C8—51no—
179C4×C4⋊C4—51no—
180#340—52no—
181C22×C3×C5—22no—
182C22×C3×C5—41no—
183C22×C3×C5—41no—
184C22×C3×C5C2×C22⋊C411yescharacteristic
185C2×C3×D5—22no—
186C2×C3×D5—41no—
187C2×C3×D5—41no—
188C2×C3×D5—44no—
189C2×C3×D5—44no—
190C2×C3×D5C4×D412yes—
191C2×C3×D5C2×D811yescharacteristic
192C2×C3×D5C2×SD1611yescharacteristic
193C2×C3×D5#13512yes—
194C3×AGL(1,𝔽5)—44no—
195C3×C4×C5—21no—
196C3×C4×C5—41no—
197C3×C4×C5#10212yescomplemented
198C3×C4×C5C2×C4211yescharacteristic
199C3×C4×C5#11711yescharacteristic
200C3×Dic5—21no—
201C3×Dic5—41no—
202C3×Dic5#10212yescomplemented
203C3×Dic5C4×D412yes—
204C24×C3—101no—
205C2×C3×D4—54no—
206C2×C3×D4—51no—
207C2×C3×D4—102no—
208C2×C3×D4—101no—
209C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
210C22×C3×C4—21no—
211C22×C3×C4—51no—
212C22×C3×C4—51no—
213C22×C3×C4—51no—
214C22×C3×C4—101no—
215C22×C3×C4—101no—
216C22×C3×C4—101no—
217C22×C3×C4—102no—
218C22×C3×C4—101no—
219C22×C3×C4—202no—
220C22×C3×C4—202no—
221C2×C3×C8—54no—
222C2×C3×C8—102no—
223C2×C3×C8—102no—
224C3×C4⋊C4—54no—
225C3×C4⋊C4—51no—
226C3×C4⋊C4—101no—
227C3×C4⋊C4—102no—
228C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
229C3×C22⋊C4—21no—
230C3×C22⋊C4—101no—
231C3×C22⋊C4—101no—
232C3×C22⋊C4—101no—
233C3×C42—51no—
234C3×C42—51no—
235C3×C42—51no—
236C3×C42—202no—
237C3×C42C2×AGL(1,𝔽5)11yescharacteristic
238C23×C5—21no—
239C22×D5—21no—
240C22×D5—24no—
241C22×D5—41no—
242C22×D5—41no—
243C22×D5—42no—
244C22×D5C2×C3×D411yescharacteristic
245C2×AGL(1,𝔽5)—24no—
246C2×AGL(1,𝔽5)—42no—
247C2×AGL(1,𝔽5)—42no—
248C5×D4—21no—
249C5×D4C3×C4212yescomplemented
250C2×C4×C5—21no—
251C2×C4×C5—42no—
252C2×C4×C5C3×C22⋊C411yescharacteristic
253C2×C4×C5C3×C22⋊C411yescharacteristic
254C2×C4×C5C22×C3×C411yescharacteristic
255C22⋊D5—24no—
256C2×Dic5—21no—
257C2×Dic5—42no—
258C2×Dic5C3×C22⋊C411yescharacteristic
259C2×Dic5C3×C22⋊C411yescharacteristic
260C2×Dic5C2×C3×D411yescharacteristic
261D20—21no—
262D20C3×C4212yescomplemented
263C4×D5—22no—
264C4×D5—41no—
265C4×D5—41no—
266C4×D5—42no—
267C4×D5—42no—
268C4×D5C3×D812yes—
269C4×D5C3×SD1612yes—
270C4×D5C22×C3×C412yes—
271C4×D5C3×#4112yes—
272#170—24no—
273C22×D4—51no—
274C23×C4—101no—
275C22×C8—52no—
276C4×D4—51no—
277C4×D4—51no—
278C4×D4—51no—
279C4×D4—101no—
280C4×D4—101no—
281C4×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
282C2×C4⋊C4—52no—
283C2×C4⋊C4—51no—
284C2×C22⋊C4—101no—
285C2×C42—51no—
286C2×C42—101no—
287#102—58no—
288C4×C8—101no—
289C4×C8—101no—
290#95—101no—
291C2×C3×C5—42no—
292C2×C3×C5C4×C22⋊C411yescharacteristic
293C2×C3×C5C2×#10211yescharacteristic
294C2×C3×C5#41611yescharacteristic
295C3×D5—42no—
296C3×D5C4×D812yes—
297C3×D5C4×SD1612yes—
298C23×C3—21no—
299C23×C3—51no—
300C23×C3—104no—
301C23×C3—101no—
302C23×C3—201no—
303C23×C3—201no—
304C23×C3—202no—
305C3×D4—21no—
306C3×D4—52no—
307C3×D4—104no—
308C3×D4—101no—
309C3×D4C4×AGL(1,𝔽5)12yescomplemented
310C2×C3×C4—21no—
311C2×C3×C4—42no—
312C2×C3×C4—51no—
313C2×C3×C4—51no—
314C2×C3×C4—52no—
315C2×C3×C4—52no—
316C2×C3×C4—52no—
317C2×C3×C4—52no—
318C2×C3×C4—51no—
319C2×C3×C4—102no—
320C2×C3×C4—104no—
321C2×C3×C4—101no—
322C2×C3×C4—202no—
323C2×C3×C4—201no—
324C2×C3×C4—201no—
325C2×C3×C4—202no—
326C2×C3×C4—202no—
327C2×C3×C4—202no—
328C2×C3×C4—202no—
329C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
330C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
331C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
332C3×C8—104no—
333C22×C5—22no—
334C22×C5—41no—
335C22×C5—41no—
336C22×C5C2×C3×C22⋊C411yescharacteristic
337C2×D5—22no—
338C2×D5—41no—
339C2×D5—41no—
340C2×D5—44no—
341C2×D5—44no—
342C2×D5C3×C4×D412yes—
343C2×D5C2×C3×D811yescharacteristic
344C2×D5C2×C3×SD1611yescharacteristic
345C2×D5C3×#13512yes—
346AGL(1,𝔽5)—44no—
347C4×C5—21no—
348C4×C5—41no—
349C4×C5C3×#10212yescomplemented
350C4×C5C2×C3×C4211yescharacteristic
351C4×C5C3×#11711yescharacteristic
352Dic5—21no—
353Dic5—41no—
354Dic5C3×#10212yescomplemented
355Dic5C3×C4×D412yes—
356C24—101no—
357C2×D4—54no—
358C2×D4—51no—
359C2×D4—102no—
360C2×D4—101no—
361C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
362C22×C4—21no—
363C22×C4—51no—
364C22×C4—51no—
365C22×C4—51no—
366C22×C4—101no—
367C22×C4—101no—
368C22×C4—101no—
369C22×C4—102no—
370C22×C4—101no—
371C22×C4—202no—
372C22×C4—202no—
373C2×C8—54no—
374C2×C8—102no—
375C2×C8—102no—
376C4⋊C4—54no—
377C4⋊C4—51no—
378C4⋊C4—101no—
379C4⋊C4—102no—
380C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
381C22⋊C4—21no—
382C22⋊C4—101no—
383C22⋊C4—101no—
384C22⋊C4—101no—
385C42—51no—
386C42—51no—
387C42—51no—
388C42—202no—
389C42C2×C3×AGL(1,𝔽5)11yescharacteristic
390C3×C5C4×#10211yescharacteristic, complemented, Hall
391C22×C3—22no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—52no—
395C22×C3—52no—
396C22×C3—51no—
397C22×C3—51no—
398C22×C3—102no—
399C22×C3—204no—
400C22×C3—204no—
401C22×C3—201no—
402C22×C3—201no—
403C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
404C3×C4—21no—
405C3×C4—41no—
406C3×C4—52no—
407C3×C4—52no—
408C3×C4—101no—
409C3×C4—204no—
410C3×C4—201no—
411C3×C4#400712yesdirect product factor
412C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
413C3×C4#413011yescharacteristic
414C2×C5—42no—
415C2×C5C3×C4×C22⋊C411yescharacteristic
416C2×C5C2×C3×#10211yescharacteristic
417C2×C5C3×#41611yescharacteristic
418D5—42no—
419D5C3×C4×D812yes—
420D5C3×C4×SD1612yes—
421C23—21no—
422C23—51no—
423C23—104no—
424C23—101no—
425C23—201no—
426C23—201no—
427C23—202no—
428D4—21no—
429D4—52no—
430D4—104no—
431D4—101no—
432D4C3×C4×AGL(1,𝔽5)12yescomplemented
433C2×C4—21no—
434C2×C4—42no—
435C2×C4—51no—
436C2×C4—51no—
437C2×C4—52no—
438C2×C4—52no—
439C2×C4—52no—
440C2×C4—52no—
441C2×C4—51no—
442C2×C4—102no—
443C2×C4—104no—
444C2×C4—101no—
445C2×C4—202no—
446C2×C4—201no—
447C2×C4—201no—
448C2×C4—202no—
449C2×C4—202no—
450C2×C4—202no—
451C2×C4—202no—
452C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
453C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
454C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
455C8—104no—
456C2×C3—42no—
457C2×C3—52no—
458C2×C3—52no—
459C2×C3—202no—
460C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
461C2×C3C2×#400711yescharacteristic
462C2×C3#6579411yescharacteristic
463C5C3×C4×#10211yescharacteristic, complemented, Sylow
464C22—22no—
465C22—41no—
466C22—41no—
467C22—52no—
468C22—51no—
469C22—51no—
470C22—52no—
471C22—102no—
472C22—204no—
473C22—204no—
474C22—201no—
475C22—201no—
476C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
477C4—21no—
478C4—41no—
479C4—52no—
480C4—52no—
481C4—101no—
482C4—204no—
483C4—201no—
484C4C3×#400712yesdirect product factor
485C4C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
486C4C3×#413011yescharacteristic
487C3C4×#400711yescharacteristic, direct product factor, Sylow
488C2—42no—
489C2—52no—
490C2—52no—
491C2—202no—
492C2C3×C4×C22⋊AGL(1,𝔽5)11yescharacteristic
493C2C2×C3×#400711yescharacteristic
494C2C3×#6579411yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊AGL(1,𝔽5)
commutatorC4×C5C2×C3×C42
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×D4×D5C22
cosocleC2×C4×D5C23×C3
hypercenterC3×C4×D4AGL(1,𝔽5)
hypocenterC5C3×C4×#102
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C3×C4×C4⋊AGL(1,𝔽5)C2semidirect
4. C3×C4×D5⋊C8C2semidirect
5. C3×#64874C2semidirect
6. C4×#4007C3direct (N × H)
12. C3×C4×C5×D4C4semidirect
13. C3×D4×Dic5C4semidirect
15. C3×#4039C4semidirect
19. C3×C4×D20C4semidirect
21. C3×#4007C4direct (N × H)
28. C4×C4⋊AGL(1,𝔽5)C2×C3semidirect
29. C4×D5⋊C8C2×C3semidirect
30. #64874C2×C3semidirect
50. C3×C4⋊AGL(1,𝔽5)C2×C4semidirect
53. C3×D5⋊C8C2×C4semidirect
75. C4×C5×D4C3×C4semidirect
76. D4×Dic5C3×C4semidirect
78. #4039C3×C4semidirect
82. C4×D20C3×C4semidirect
84. #4007C3×C4direct (N × H)
100. C3×C5×D4C42semidirect
113. C3×D20C42semidirect
132. C3×C4×D4AGL(1,𝔽5)semidirect
161. C4⋊AGL(1,𝔽5)C2×C3×C4semidirect
164. D5⋊C8C2×C3×C4semidirect
197. C3×C4×C5#102semidirect
202. C3×Dic5#102semidirect
249. C5×D4C3×C42semidirect
262. D20C3×C42semidirect
281. C4×D4C3×AGL(1,𝔽5)semidirect
309. C3×D4C4×AGL(1,𝔽5)semidirect
349. C4×C5C3×#102semidirect
354. Dic5C3×#102semidirect
390. C3×C5C4×#102semidirect
411. C3×C4#4007direct (N × H)
432. D4C3×C4×AGL(1,𝔽5)semidirect
463. C5C3×C4×#102semidirect
484. C4C3×#4007direct (N × H)
487. C3C4×#4007direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×D4×D52
C2×C3×#40072
C3×C4×C4⋊AGL(1,𝔽5)2
C3×C4×D5⋊C82
C3×#648742
C4×#40073
C3×C4×#1025