Finite Group Databasebeta

← C3×C4⋊C4×D20 #14327014

Subgroups — C3×C4⋊C4×D20 #14327014

Order 1920.

1552 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×C4⋊C4C212yesmaximal, complemented
2C2×C3×C4×D20C212yesmaximal
3C2×C3×C4×D20C211yesmaximal, characteristic
4C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#65276C212yesmaximal, complemented
6C3×#65224C211yesmaximal, characteristic, complemented
7C3×#65217C212yesmaximal, complemented
8C3×#64957C214yesmaximal, complemented
9C4⋊C4×D20C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×D20C2211yescharacteristic
11C22×C3×C4×D5C2214yes—
12C22×C3×C4×D5C2212yes—
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C42×C5C2212yes—
16C2×C3×C42×C5C2211yescharacteristic
17C2×C3×#685C2214yes—
18C2×C3×#685C2212yes—
19C2×C3×C4⋊Dic5C2212yes—
20C2×C3×C4⋊Dic5C2212yes—
21C2×C3×C4⋊Dic5C2212yes—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×#683C2214yes—
24C3×D5×C4⋊C4—28no—
25C3×C4×D20—28no—
26C3×C4×D20C418yescomplemented
27C3×C5×#95C2212yes—
28C3×#3963C2214yes—
29C3×D4×C4⋊C4—51nomaximal, Hall
30C2×D5×C4⋊C4C2×C312yescomplemented
31C2×C4×D20C2×C312yes—
32C2×C4×D20C2×C311yescharacteristic
33C4×C5×C4⋊C4C2×C311yescharacteristic, complemented
34#65276C2×C312yescomplemented
35#65224C2×C311yescharacteristic, complemented
36#65217C2×C312yescomplemented
37#64957C2×C314yescomplemented
38C23×C3×D5C2312yes—
39C22×C3×C4×C5C2312yes—
40C22×C3×C4×C5C2311yescharacteristic
41C22×C3×C4×C5C2312yes—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×C4×C5C2311yescharacteristic
44C22×C3×Dic5C2314yes—
45C22×C3×Dic5C2312yes—
46C2×C3×D20C2×C414yes—
47C2×C3×D20D414yes—
48C2×C3×D20Q814yes—
49C2×C3×C4×D5—24no—
50C2×C3×C4×D5—24no—
51C2×C3×C4×D5—28no—
52C2×C3×C4×D5—28no—
53C2×C3×C4×D5—44no—
54C2×C3×C4×D5C2×C418yescomplemented
55C3×C5×C4⋊C4—22no—
56C3×C5×C4⋊C4D414yescomplemented
57C3×C42×C5—24no—
58C3×C42×C5C2×C414yescomplemented
59C3×#685—28no—
60C3×#685C2×C418yescomplemented
61C3×C4⋊Dic5—24no—
62C3×C4⋊Dic5—24no—
63C3×C4⋊Dic5—42no—
64C3×C4⋊Dic5C2×C414yescomplemented
65C3×#683—28no—
66C2×C3×C4×D4—51no—
67C2×C3×C4×D4—52no—
68C22×C3×C4⋊C4—52no—
69C3×#389—52no—
70C3×#385—54no—
71C3×#384—51no—
72C3×#380—52no—
73C3×C4×C4⋊C4D511yescharacteristic, complemented
74C22×D20C22×C311yescharacteristic
75C22×C4×D5C22×C314yes—
76C22×C4×D5C22×C312yes—
77C2×C5×C4⋊C4C22×C311yescharacteristic
78C2×C5×C4⋊C4C22×C311yescharacteristic
79C2×C42×C5C22×C312yes—
80C2×C42×C5C22×C311yescharacteristic
81C2×#685C22×C314yes—
82C2×#685C22×C312yes—
83C2×C4⋊Dic5C22×C312yes—
84C2×C4⋊Dic5C22×C312yes—
85C2×C4⋊Dic5C22×C312yes—
86C2×C4⋊Dic5C22×C311yescharacteristic
87C2×#683C22×C314yes—
88D5×C4⋊C4—28no—
89C4×D20—28no—
90C4×D20C3×C418yescomplemented
91C5×#95C22×C312yes—
92#3963C22×C314yes—
93D4×C4⋊C4—51noSylow
94C23×C3×C5C2411yescharacteristic
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5—22no—
98C22×C3×D5—22no—
99C22×C3×D5C22×C414yes—
100C22×C3×D5C2×D414yes—
101C22×C3×D5C2×Q814yes—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5C22×C412yes—
110C2×C3×C4×C5C22×C412yes—
111C2×C3×C4×C5C22×C412yes—
112C2×C3×C4×C5C2×D414yes—
113C2×C3×C4×C5C2×D412yes—
114C2×C3×C4×C5C2×D412yes—
115C2×C3×C4×C5C2×Q812yes—
116C2×C3×C4×C5#4112yes—
117C2×C3×Dic5—22no—
118C2×C3×Dic5—22no—
119C2×C3×Dic5—24no—
120C2×C3×Dic5—24no—
121C2×C3×Dic5—24no—
122C2×C3×Dic5C22×C414yes—
123C3×D20C4⋊C4116yesdirect product factor
124C3×C4×D5—216no—
125C3×C4×D5—416no—
126C22×C3×D4—51no—
127C23×C3×C4—52no—
128C23×C3×C4—54no—
129C3×C4×D4—58no—
130C3×C4×D4—108no—
131C2×C3×C4⋊C4—51no—
132C2×C3×C4⋊C4—52no—
133C2×C3×C4⋊C4—52no—
134C2×C3×C4⋊C4—54no—
135C2×C3×C4⋊C4—52no—
136C2×C3×C4⋊C4—108no—
137C2×C3×C4⋊C4C2×D511yescharacteristic
138C2×C3×C4⋊C4C2×D511yescharacteristic
139C2×C3×C22⋊C4—52no—
140C2×C3×C22⋊C4—54no—
141C2×C3×C42C2×D512yes—
142C2×C3×C42C2×D511yescharacteristic
143C3×#95—54no—
144C3×#95C2×D512yes—
145C23×D5C23×C312yes—
146C22×C4×C5C23×C312yes—
147C22×C4×C5C23×C311yescharacteristic
148C22×C4×C5C23×C312yes—
149C22×C4×C5C23×C311yescharacteristic
150C22×C4×C5C23×C311yescharacteristic
151C22×Dic5C23×C314yes—
152C22×Dic5C23×C312yes—
153C2×D20C2×C3×C414yes—
154C2×D20C3×D414yes—
155C2×D20C3×Q814yes—
156C2×C4×D5—24no—
157C2×C4×D5—24no—
158C2×C4×D5—28no—
159C2×C4×D5—28no—
160C2×C4×D5—44no—
161C2×C4×D5C2×C3×C418yescomplemented
162C5×C4⋊C4—22no—
163C5×C4⋊C4C3×D414yescomplemented
164C42×C5—24no—
165C42×C5C2×C3×C414yescomplemented
166#685—28no—
167#685C2×C3×C418yescomplemented
168C4⋊Dic5—24no—
169C4⋊Dic5—24no—
170C4⋊Dic5—42no—
171C4⋊Dic5C2×C3×C414yescomplemented
172#683—28no—
173C2×C4×D4—51no—
174C2×C4×D4—52no—
175C22×C4⋊C4—52no—
176#389—52no—
177#385—54no—
178#384—51no—
179#380—52no—
180C4×C4⋊C4C3×D511yescharacteristic, complemented
181C22×C3×C5C23×C411yescharacteristic
182C22×C3×C5C22×D411yescharacteristic
183C22×C3×C5C22×D411yescharacteristic
184C22×C3×C5C22×Q811yescharacteristic
185C22×C3×C5C2×#4111yescharacteristic
186C22×C3×C5H(5,𝔽2)11yescharacteristic
187C22×C3×C52−1+411yescharacteristic
188C2×C3×D5—24no—
189C2×C3×D5—24no—
190C2×C3×D5—24no—
191C2×C3×D5—24no—
192C2×C3×D5—24no—
193C2×C3×D5—24no—
194C2×C3×D5C2×C4⋊C418yescomplemented
195C3×C4×C5—22no—
196C3×C4×C5—24no—
197C3×C4×C5—42no—
198C3×C4×C5C2×C4⋊C414yescomplemented
199C3×C4×C5C4×D414yescomplemented
200C3×Dic5—24no—
201C3×Dic5—44no—
202C24×C3—52no—
203C2×C3×D4—54no—
204C2×C3×D4—54no—
205C2×C3×D4—54no—
206C22×C3×C4—52no—
207C22×C3×C4—58no—
208C22×C3×C4—54no—
209C22×C3×C4—108no—
210C22×C3×C4—104no—
211C22×C3×C4—104no—
212C22×C3×C4—108no—
213C22×C3×C4—204no—
214C22×C3×C4C22×D512yes—
215C22×C3×C4C22×D511yescharacteristic
216C22×C3×C4C22×D512yes—
217C22×C3×C4C22×D511yescharacteristic
218C22×C3×C4C22×D511yescharacteristic
219C3×C4⋊C4—22no—
220C3×C4⋊C4—54no—
221C3×C4⋊C4—108no—
222C3×C4⋊C4—104no—
223C3×C4⋊C4—104no—
224C3×C4⋊C4—202no—
225C3×C4⋊C4D2014yesdirect product factor
226C3×C22⋊C4—58no—
227C3×C22⋊C4—108no—
228C3×C42—24no—
229C3×C42C4×D514yescomplemented
230C23×C5C24×C311yescharacteristic
231C22×D5—22no—
232C22×D5—22no—
233C22×D5—22no—
234C22×D5—22no—
235C22×D5C22×C3×C414yes—
236C22×D5C2×C3×D414yes—
237C22×D5C2×C3×Q814yes—
238C2×C4×C5—22no—
239C2×C4×C5—21no—
240C2×C4×C5—22no—
241C2×C4×C5—21no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5C22×C3×C412yes—
246C2×C4×C5C22×C3×C412yes—
247C2×C4×C5C22×C3×C412yes—
248C2×C4×C5C2×C3×D414yes—
249C2×C4×C5C2×C3×D412yes—
250C2×C4×C5C2×C3×D412yes—
251C2×C4×C5C2×C3×Q812yes—
252C2×C4×C5C3×#4112yes—
253C2×Dic5—22no—
254C2×Dic5—22no—
255C2×Dic5—24no—
256C2×Dic5—24no—
257C2×Dic5—24no—
258C2×Dic5C22×C3×C414yes—
259D20C3×C4⋊C4116yesdirect product factor
260C4×D5—216no—
261C4×D5—416no—
262C22×D4—51no—
263C23×C4—52no—
264C23×C4—54no—
265C4×D4—58no—
266C4×D4—108no—
267C2×C4⋊C4—51no—
268C2×C4⋊C4—52no—
269C2×C4⋊C4—54no—
270C2×C4⋊C4—52no—
271C2×C4⋊C4—52no—
272C2×C4⋊C4—108no—
273C2×C4⋊C4C2×C3×D511yescharacteristic
274C2×C4⋊C4C2×C3×D511yescharacteristic
275C2×C22⋊C4—52no—
276C2×C22⋊C4—54no—
277C2×C42C2×C3×D512yes—
278C2×C42C2×C3×D511yescharacteristic
279#95—54no—
280#95C2×C3×D512yes—
281C2×C3×C5C22×C4⋊C411yescharacteristic
282C2×C3×C5C2×C4×D411yescharacteristic
283C2×C3×C5#52011yescharacteristic
284C2×C3×C5D4211yescharacteristic
285C2×C3×C5#54711yescharacteristic
286C2×C3×C5D4×Q811yescharacteristic
287C2×C3×C5#55411yescharacteristic
288C3×D5—28no—
289C23×C3—54no—
290C23×C3—54no—
291C23×C3—54no—
292C23×C3—102no—
293C23×C3—102no—
294C23×C3—102no—
295C23×C3—102no—
296C23×C3C23×D511yescharacteristic
297C3×D4—516no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—54no—
306C2×C3×C4—102no—
307C2×C3×C4—102no—
308C2×C3×C4—1016no—
309C2×C3×C4—104no—
310C2×C3×C4—104no—
311C2×C3×C4—104no—
312C2×C3×C4—2016no—
313C2×C3×C4C2×C4×D512yes—
314C2×C3×C4C2×C4×D512yes—
315C2×C3×C4C2×C4×D512yes—
316C2×C3×C4C2×D2014yes—
317C2×C3×C4C2×D2012yes—
318C2×C3×C4#70912yes—
319C2×C3×C4D4×D512yes—
320C2×C3×C4Q8×D512yes—
321C22×C5C23×C3×C411yescharacteristic
322C22×C5C22×C3×D411yescharacteristic
323C22×C5C22×C3×D411yescharacteristic
324C22×C5C22×C3×Q811yescharacteristic
325C22×C5C2×C3×#4111yescharacteristic
326C22×C5C3×H(5,𝔽2)11yescharacteristic
327C22×C5C3×2−1+411yescharacteristic
328C2×D5—24no—
329C2×D5—24no—
330C2×D5—24no—
331C2×D5—24no—
332C2×D5—24no—
333C2×D5—24no—
334C2×D5C2×C3×C4⋊C418yescomplemented
335C4×C5—22no—
336C4×C5—24no—
337C4×C5—42no—
338C4×C5C2×C3×C4⋊C414yescomplemented
339C4×C5C3×C4×D414yescomplemented
340Dic5—24no—
341Dic5—44no—
342C24—52no—
343C2×D4—54no—
344C2×D4—54no—
345C2×D4—54no—
346C22×C4—52no—
347C22×C4—58no—
348C22×C4—54no—
349C22×C4—108no—
350C22×C4—104no—
351C22×C4—104no—
352C22×C4—108no—
353C22×C4—204no—
354C22×C4C22×C3×D512yes—
355C22×C4C22×C3×D511yescharacteristic
356C22×C4C22×C3×D512yes—
357C22×C4C22×C3×D511yescharacteristic
358C22×C4C22×C3×D511yescharacteristic
359C4⋊C4—22no—
360C4⋊C4—54no—
361C4⋊C4—108no—
362C4⋊C4—104no—
363C4⋊C4—104no—
364C4⋊C4—202no—
365C4⋊C4C3×D2014yesdirect product factor
366C22⋊C4—58no—
367C22⋊C4—108no—
368C42—24no—
369C42C3×C4×D514yescomplemented
370C3×C5D4×C4⋊C411yescharacteristic, complemented, Hall
371C22×C3—58no—
372C22×C3—104no—
373C22×C3—104no—
374C22×C3—104no—
375C22×C3—104no—
376C22×C3—104no—
377C22×C3—104no—
378C22×C3C22×C4×D511yescharacteristic
379C22×C3C22×D2011yescharacteristic
380C22×C3C2×#70911yescharacteristic
381C22×C3C2×D4×D511yescharacteristic
382C22×C3C2×Q8×D511yescharacteristic
383C22×C3#414911yescharacteristic
384C22×C3#415011yescharacteristic
385C3×C4—22no—
386C3×C4—24no—
387C3×C4—42no—
388C3×C4—104no—
389C3×C4—204no—
390C3×C4C4×D2014yescomplemented
391C3×C4D5×C4⋊C414yescomplemented
392C2×C5C22×C3×C4⋊C411yescharacteristic
393C2×C5C2×C3×C4×D411yescharacteristic
394C2×C5C3×#52011yescharacteristic
395C2×C5C3×D4211yescharacteristic
396C2×C5C3×#54711yescharacteristic
397C2×C5C3×D4×Q811yescharacteristic
398C2×C5C3×#55411yescharacteristic
399D5—28no—
400C23—54no—
401C23—54no—
402C23—54no—
403C23—102no—
404C23—102no—
405C23—102no—
406C23—102no—
407C23C23×C3×D511yescharacteristic
408D4—516no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—54no—
417C2×C4—102no—
418C2×C4—102no—
419C2×C4—1016no—
420C2×C4—104no—
421C2×C4—104no—
422C2×C4—104no—
423C2×C4—2016no—
424C2×C4C2×C3×C4×D512yes—
425C2×C4C2×C3×C4×D512yes—
426C2×C4C2×C3×C4×D512yes—
427C2×C4C2×C3×D2014yes—
428C2×C4C2×C3×D2012yes—
429C2×C4C3×#70912yes—
430C2×C4C3×D4×D512yes—
431C2×C4C3×Q8×D512yes—
432C2×C3—108no—
433C2×C3C2×C4×D2011yescharacteristic
434C2×C3C2×D5×C4⋊C411yescharacteristic
435C2×C3#6585711yescharacteristic
436C2×C3D4×D2011yescharacteristic
437C2×C3#6588511yescharacteristic
438C2×C3Q8×D2011yescharacteristic
439C2×C3#6591311yescharacteristic
440C5C3×D4×C4⋊C411yescharacteristic, complemented, Sylow
441C22—58no—
442C22—104no—
443C22—104no—
444C22—104no—
445C22—104no—
446C22—104no—
447C22—104no—
448C22C22×C3×C4×D511yescharacteristic
449C22C22×C3×D2011yescharacteristic
450C22C2×C3×#70911yescharacteristic
451C22C2×C3×D4×D511yescharacteristic
452C22C2×C3×Q8×D511yescharacteristic
453C22C3×#414911yescharacteristic
454C22C3×#415011yescharacteristic
455C4—22no—
456C4—24no—
457C4—42no—
458C4—104no—
459C4—204no—
460C4C3×C4×D2014yescomplemented
461C4C3×D5×C4⋊C414yescomplemented
462C3C4⋊C4×D2011yescharacteristic, direct product factor, Sylow
463C2—108no—
464C2C2×C3×C4×D2011yescharacteristic
465C2C2×C3×D5×C4⋊C411yescharacteristic
466C2C3×#6585711yescharacteristic
467C2C3×D4×D2011yescharacteristic
468C2C3×#6588511yescharacteristic
469C2C3×Q8×D2011yescharacteristic
470C2C3×#6591311yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C4×C5×C4⋊C4C2
socleC23×C3×C5C24
prime radicalC22×C3×D20C22
cosocleC23×C5C24×C3
hypercenterC3×C4×C4⋊C4D5
hypocenterC5C3×D4×C4⋊C4
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D5×C4⋊C4C2semidirect
4. C3×C4×C5×C4⋊C4C2semidirect
5. C3×#65276C2semidirect
6. C3×#65224C2semidirect
7. C3×#65217C2semidirect
8. C3×#64957C2semidirect
9. C4⋊C4×D20C3direct (N × H)
26. C3×C4×D20C4semidirect
30. C2×D5×C4⋊C4C2×C3semidirect
33. C4×C5×C4⋊C4C2×C3semidirect
34. #65276C2×C3semidirect
35. #65224C2×C3semidirect
36. #65217C2×C3semidirect
37. #64957C2×C3semidirect
54. C2×C3×C4×D5C2×C4semidirect
56. C3×C5×C4⋊C4D4semidirect
58. C3×C42×C5C2×C4semidirect
60. C3×#685C2×C4semidirect
64. C3×C4⋊Dic5C2×C4semidirect
73. C3×C4×C4⋊C4D5semidirect
90. C4×D20C3×C4semidirect
123. C3×D20C4⋊C4direct (N × H)
161. C2×C4×D5C2×C3×C4semidirect
163. C5×C4⋊C4C3×D4semidirect
165. C42×C5C2×C3×C4semidirect
167. #685C2×C3×C4semidirect
171. C4⋊Dic5C2×C3×C4semidirect
180. C4×C4⋊C4C3×D5semidirect
194. C2×C3×D5C2×C4⋊C4semidirect
198. C3×C4×C5C2×C4⋊C4semidirect
199. C3×C4×C5C4×D4semidirect
225. C3×C4⋊C4D20direct (N × H)
229. C3×C42C4×D5semidirect
259. D20C3×C4⋊C4direct (N × H)
334. C2×D5C2×C3×C4⋊C4semidirect
338. C4×C5C2×C3×C4⋊C4semidirect
339. C4×C5C3×C4×D4semidirect
365. C4⋊C4C3×D20direct (N × H)
369. C42C3×C4×D5semidirect
370. C3×C5D4×C4⋊C4semidirect
390. C3×C4C4×D20semidirect
391. C3×C4D5×C4⋊C4semidirect
440. C5C3×D4×C4⋊C4semidirect
460. C4C3×C4×D20semidirect
461. C4C3×D5×C4⋊C4semidirect
462. C3C4⋊C4×D20direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5×C4⋊C42
C2×C3×C4×D202
C3×C4×C5×C4⋊C42
C3×#652762
C3×#652242
C3×#652172
C3×#649572
C4⋊C4×D203
C3×D4×C4⋊C45