Finite Group Databasebeta

← C3×C4×#4008 #14295583

Subgroups — C3×C4×#4008 #14295583

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#711C211yesmaximal, characteristic
2C2×C3×#4008C212yesmaximal
3C3×C42×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×C4×#700C211yesmaximal, characteristic, complemented
5C3×#64875C212yesmaximal, complemented
6C4×#4008C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711C2211yescharacteristic
8C2×C3×C4×AGL(1,𝔽5)—21no—
9C2×C3×C4×AGL(1,𝔽5)—21no—
10C2×C3×C4×AGL(1,𝔽5)C2212yes—
11C2×C3×#700C2212yes—
12C3×C4×C5×D4C411yescharacteristic, complemented
13C3×D4×Dic5C411yescharacteristic, complemented
14C3×C4×C22⋊D5—21no—
15C2×C3×C4×Dic5—21no—
16C3×#4038C2211yescharacteristic
17C3×#4033C411yescharacteristic, complemented
18C3×#4027—21no—
19C3×#4023—21no—
20C3×C42×D5C2211yescharacteristic
21C3×C4×Dic10C411yescharacteristic, complemented
22C3×#4008C418yesdirect product factor
23C3×#4002—21no—
24C3×C4×#170—21no—
25C3×C4×#104—51nomaximal, Hall
26C4×#711C2×C311yescharacteristic
27C2×#4008C2×C312yes—
28C42×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
29C4×#700C2×C311yescharacteristic, complemented
30#64875C2×C312yescomplemented
31C22×C3×AGL(1,𝔽5)—22no—
32C2×C3×C5×D4C2×C411yescharacteristic
33C22×C3×C4×C5—21no—
34C2×C3×C22⋊D5—21no—
35C22×C3×Dic5—21no—
36C22×C3×Dic5—21no—
37C3×#711—22no—
38C3×#711C2×C414yes—
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5C2311yescharacteristic
42C2×C3×Dic10C2×C411yescharacteristic
43C2×C3×#170—22no—
44C3×C4×AGL(1,𝔽5)—24no—
45C3×C4×AGL(1,𝔽5)—24no—
46C3×C4×AGL(1,𝔽5)—22no—
47C3×C4×AGL(1,𝔽5)C2×C414yescomplemented
48C3×#700—22no—
49C3×#700C2×C414yescomplemented
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C3×C5×C22⋊C4—21no—
52C3×C42×C5C2×C411yescharacteristic
53C3×C22⋊Dic5—21no—
54C3×#685—21no—
55C3×C4⋊Dic5C2×C411yescharacteristic
56C3×#683—21no—
57C3×#683—21no—
58C3×C4×Dic5—21no—
59C3×C4×Dic5—21no—
60C3×C4×Dic5—21no—
61C3×C4×Dic5C2×C411yescharacteristic
62C3×C4×Dic5D411yescharacteristic
63C3×C4×Dic5D411yescharacteristic
64C3×C4×#41—51no—
65C2×C3×#104—52no—
66C3×C4×M16—51no—
67C3×C43—51no—
68C3×#339—52no—
69C2×#711C22×C311yescharacteristic
70C2×C4×AGL(1,𝔽5)—21no—
71C2×C4×AGL(1,𝔽5)—21no—
72C2×C4×AGL(1,𝔽5)C22×C312yes—
73C2×#700C22×C312yes—
74C4×C5×D4C3×C411yescharacteristic, complemented
75D4×Dic5C3×C411yescharacteristic, complemented
76C4×C22⋊D5—21no—
77C2×C4×Dic5—21no—
78#4038C22×C311yescharacteristic
79#4033C3×C411yescharacteristic, complemented
80#4027—21no—
81#4023—21no—
82C42×D5C22×C311yescharacteristic
83C4×Dic10C3×C411yescharacteristic, complemented
84#4008C3×C418yesdirect product factor
85#4002—21no—
86C4×#170—21no—
87C4×#104—51noSylow
88C23×C3×C5—21no—
89C22×C3×D5C2×D411yescharacteristic
90C2×C3×AGL(1,𝔽5)—24no—
91C2×C3×AGL(1,𝔽5)—24no—
92C2×C3×AGL(1,𝔽5)—24no—
93C3×C5×D4—21no—
94C3×C5×D4C4212yescomplemented
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—42no—
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C22×C411yescharacteristic
100C3×C22⋊D5—24no—
101C2×C3×Dic5—21no—
102C2×C3×Dic5—24no—
103C2×C3×Dic5—21no—
104C2×C3×Dic5—42no—
105C2×C3×Dic5C22⋊C411yescharacteristic
106C2×C3×Dic5C22⋊C411yescharacteristic
107C2×C3×Dic5C2×D411yescharacteristic
108C3×C4×D5—22no—
109C3×C4×D5—22no—
110C3×C4×D5C22×C412yes—
111C3×C4×D5#4112yes—
112C3×Dic10—21no—
113C3×Dic10C4212yescomplemented
114C3×#170—24no—
115C2×C3×#41—51no—
116C2×C3×M16—52no—
117C3×C4×Q8—51no—
118C3×C4×Q8—51no—
119C3×C4×D4—51no—
120C3×C4×D4—101no—
121C3×C4×D4—101no—
122C3×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
123C3×#117—51no—
124C3×#117—101no—
125C2×C3×C42—51no—
126C2×C3×C42—52no—
127C2×C3×C42—101no—
128C2×C3×C42—101no—
129C2×C3×C42—101no—
130C3×#104—58no—
131C3×#97—101no—
132C3×C4×C8—101no—
133C22×AGL(1,𝔽5)—22no—
134C2×C5×D4C2×C3×C411yescharacteristic
135C22×C4×C5—21no—
136C2×C22⋊D5—21no—
137C22×Dic5—21no—
138C22×Dic5—21no—
139#711—22no—
140#711C2×C3×C414yes—
141C2×C4×D5C3×D411yescharacteristic
142C2×C4×D5C3×D411yescharacteristic
143C2×C4×D5C23×C311yescharacteristic
144C2×Dic10C2×C3×C411yescharacteristic
145C2×#170—22no—
146C4×AGL(1,𝔽5)—24no—
147C4×AGL(1,𝔽5)—24no—
148C4×AGL(1,𝔽5)—22no—
149C4×AGL(1,𝔽5)C2×C3×C414yescomplemented
150#700—22no—
151#700C2×C3×C414yescomplemented
152C5×C4⋊C4C2×C3×C411yescharacteristic
153C5×C22⋊C4—21no—
154C42×C5C2×C3×C411yescharacteristic
155C22⋊Dic5—21no—
156#685—21no—
157C4⋊Dic5C2×C3×C411yescharacteristic
158#683—21no—
159#683—21no—
160C4×Dic5—21no—
161C4×Dic5—21no—
162C4×Dic5—21no—
163C4×Dic5C2×C3×C411yescharacteristic
164C4×Dic5C3×D411yescharacteristic
165C4×Dic5C3×D411yescharacteristic
166C4×#41—51no—
167C2×#104—52no—
168C4×M16—51no—
169C43—51no—
170#339—52no—
171C22×C3×C5—22no—
172C22×C3×C5—41no—
173C22×C3×C5—41no—
174C22×C3×C5C2×C22⋊C411yescharacteristic
175C2×C3×D5—21no—
176C2×C3×D5—21no—
177C2×C3×D5C4×D412yes—
178C3×AGL(1,𝔽5)—28no—
179C3×C4×C5—21no—
180C3×C4×C5—41no—
181C3×C4×C5#10412yescomplemented
182C3×C4×C5C2×C4211yescharacteristic
183C3×C4×C5#11711yescharacteristic
184C3×Dic5—22no—
185C3×Dic5—21no—
186C3×Dic5—41no—
187C3×Dic5#10412yescomplemented
188C3×Dic5C4×D412yes—
189C3×#41—54no—
190C3×#41—102no—
191C2×C3×Q8—51no—
192C2×C3×D4—101no—
193C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
194C22×C3×C4—21no—
195C22×C3×C4—51no—
196C22×C3×C4—51no—
197C22×C3×C4—51no—
198C22×C3×C4—101no—
199C22×C3×C4—101no—
200C22×C3×C4—102no—
201C3×M16—54no—
202C3×M16—102no—
203C2×C3×C8—102no—
204C3×C4⋊C4—51no—
205C3×C4⋊C4—101no—
206C3×C4⋊C4—101no—
207C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
208C3×C22⋊C4—21no—
209C3×C22⋊C4—101no—
210C3×C22⋊C4—101no—
211C3×C42—51no—
212C3×C42—51no—
213C3×C42—51no—
214C3×C42—54no—
215C3×C42—101no—
216C3×C42—101no—
217C3×C42—102no—
218C3×C42—104no—
219C3×C42—104no—
220C3×C42—101no—
221C3×C42C2×AGL(1,𝔽5)11yescharacteristic
222C23×C5—21no—
223C22×D5C2×C3×D411yescharacteristic
224C2×AGL(1,𝔽5)—24no—
225C2×AGL(1,𝔽5)—24no—
226C2×AGL(1,𝔽5)—24no—
227C5×D4—21no—
228C5×D4C3×C4212yescomplemented
229C2×C4×C5—21no—
230C2×C4×C5—42no—
231C2×C4×C5C3×C22⋊C411yescharacteristic
232C2×C4×C5C3×C22⋊C411yescharacteristic
233C2×C4×C5C22×C3×C411yescharacteristic
234C22⋊D5—24no—
235C2×Dic5—21no—
236C2×Dic5—24no—
237C2×Dic5—21no—
238C2×Dic5—42no—
239C2×Dic5C3×C22⋊C411yescharacteristic
240C2×Dic5C3×C22⋊C411yescharacteristic
241C2×Dic5C2×C3×D411yescharacteristic
242C4×D5—22no—
243C4×D5—22no—
244C4×D5C22×C3×C412yes—
245C4×D5C3×#4112yes—
246Dic10—21no—
247Dic10C3×C4212yescomplemented
248#170—24no—
249C2×#41—51no—
250C2×M16—52no—
251C4×Q8—51no—
252C4×Q8—51no—
253C4×D4—51no—
254C4×D4—101no—
255C4×D4—101no—
256C4×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
257#117—51no—
258#117—101no—
259C2×C42—51no—
260C2×C42—52no—
261C2×C42—101no—
262C2×C42—101no—
263C2×C42—101no—
264#104—58no—
265#97—101no—
266C4×C8—101no—
267C2×C3×C5—42no—
268C2×C3×C5C4×C22⋊C411yescharacteristic
269C2×C3×C5C2×#10411yescharacteristic
270C2×C3×C5C2×#10411yescharacteristic
271C3×D5—22no—
272C23×C3—21no—
273C23×C3—51no—
274C3×Q8—52no—
275C3×Q8—101no—
276C3×D4—21no—
277C3×D4—104no—
278C3×D4C4×AGL(1,𝔽5)12yescomplemented
279C2×C3×C4—21no—
280C2×C3×C4—42no—
281C2×C3×C4—51no—
282C2×C3×C4—51no—
283C2×C3×C4—52no—
284C2×C3×C4—52no—
285C2×C3×C4—51no—
286C2×C3×C4—102no—
287C2×C3×C4—104no—
288C2×C3×C4—104no—
289C2×C3×C4—104no—
290C2×C3×C4—104no—
291C2×C3×C4—101no—
292C2×C3×C4—102no—
293C2×C3×C4—101no—
294C2×C3×C4—202no—
295C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
296C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
297C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
298C3×C8—104no—
299C22×C5—22no—
300C22×C5—41no—
301C22×C5—41no—
302C22×C5C2×C3×C22⋊C411yescharacteristic
303C2×D5—21no—
304C2×D5—21no—
305C2×D5C3×C4×D412yes—
306AGL(1,𝔽5)—28no—
307C4×C5—21no—
308C4×C5—41no—
309C4×C5C3×#10412yescomplemented
310C4×C5C2×C3×C4211yescharacteristic
311C4×C5C3×#11711yescharacteristic
312Dic5—22no—
313Dic5—21no—
314Dic5—41no—
315Dic5C3×#10412yescomplemented
316Dic5C3×C4×D412yes—
317#41—54no—
318#41—102no—
319C2×Q8—51no—
320C2×D4—101no—
321C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
322C22×C4—21no—
323C22×C4—51no—
324C22×C4—51no—
325C22×C4—51no—
326C22×C4—101no—
327C22×C4—101no—
328C22×C4—102no—
329M16—54no—
330M16—102no—
331C2×C8—102no—
332C4⋊C4—51no—
333C4⋊C4—101no—
334C4⋊C4—101no—
335C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
336C22⋊C4—21no—
337C22⋊C4—101no—
338C22⋊C4—101no—
339C42—51no—
340C42—51no—
341C42—51no—
342C42—54no—
343C42—101no—
344C42—101no—
345C42—102no—
346C42—104no—
347C42—104no—
348C42—101no—
349C42C2×C3×AGL(1,𝔽5)11yescharacteristic
350C3×C5C4×#10411yescharacteristic, complemented, Hall
351C22×C3—22no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—52no—
355C22×C3—101no—
356C22×C3—101no—
357C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
358C3×C4—21no—
359C3×C4—41no—
360C3×C4—52no—
361C3×C4—52no—
362C3×C4—108no—
363C3×C4—102no—
364C3×C4—101no—
365C3×C4—201no—
366C3×C4#400812yesdirect product factor
367C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
368C3×C4#413011yescharacteristic
369C2×C5—42no—
370C2×C5C3×C4×C22⋊C411yescharacteristic
371C2×C5C2×C3×#10411yescharacteristic
372C2×C5C2×C3×#10411yescharacteristic
373D5—22no—
374C23—21no—
375C23—51no—
376Q8—52no—
377Q8—101no—
378D4—21no—
379D4—104no—
380D4C3×C4×AGL(1,𝔽5)12yescomplemented
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—51no—
384C2×C4—51no—
385C2×C4—52no—
386C2×C4—52no—
387C2×C4—51no—
388C2×C4—102no—
389C2×C4—104no—
390C2×C4—104no—
391C2×C4—104no—
392C2×C4—104no—
393C2×C4—101no—
394C2×C4—102no—
395C2×C4—101no—
396C2×C4—202no—
397C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
398C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
399C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
400C8—104no—
401C2×C3—42no—
402C2×C3—102no—
403C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
404C2×C3C2×#400811yescharacteristic
405C2×C3D5⋊#10411yescharacteristic
406C5C3×C4×#10411yescharacteristic, complemented, Sylow
407C22—22no—
408C22—41no—
409C22—41no—
410C22—52no—
411C22—101no—
412C22—101no—
413C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
414C4—21no—
415C4—41no—
416C4—52no—
417C4—52no—
418C4—108no—
419C4—102no—
420C4—101no—
421C4—201no—
422C4C3×#400812yesdirect product factor
423C4C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
424C4C3×#413011yescharacteristic
425C3C4×#400811yescharacteristic, direct product factor, Sylow
426C2—42no—
427C2—102no—
428C2C3×C4×C22⋊AGL(1,𝔽5)11yescharacteristic
429C2C2×C3×#400811yescharacteristic
430C2C3×D5⋊#10411yescharacteristic
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×#711C22
cosocleC2×C4×D5C23×C3
hypercenterC3×C4×D4AGL(1,𝔽5)
hypocenterC5C3×C4×#104
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×C42×AGL(1,𝔽5)C2semidirect
4. C3×C4×#700C2semidirect
5. C3×#64875C2semidirect
6. C4×#4008C3direct (N × H)
12. C3×C4×C5×D4C4semidirect
13. C3×D4×Dic5C4semidirect
17. C3×#4033C4semidirect
21. C3×C4×Dic10C4semidirect
22. C3×#4008C4direct (N × H)
28. C42×AGL(1,𝔽5)C2×C3semidirect
29. C4×#700C2×C3semidirect
30. #64875C2×C3semidirect
47. C3×C4×AGL(1,𝔽5)C2×C4semidirect
49. C3×#700C2×C4semidirect
74. C4×C5×D4C3×C4semidirect
75. D4×Dic5C3×C4semidirect
79. #4033C3×C4semidirect
83. C4×Dic10C3×C4semidirect
84. #4008C3×C4direct (N × H)
94. C3×C5×D4C42semidirect
113. C3×Dic10C42semidirect
122. C3×C4×D4AGL(1,𝔽5)semidirect
149. C4×AGL(1,𝔽5)C2×C3×C4semidirect
151. #700C2×C3×C4semidirect
181. C3×C4×C5#104semidirect
187. C3×Dic5#104semidirect
228. C5×D4C3×C42semidirect
247. Dic10C3×C42semidirect
256. C4×D4C3×AGL(1,𝔽5)semidirect
278. C3×D4C4×AGL(1,𝔽5)semidirect
309. C4×C5C3×#104semidirect
315. Dic5C3×#104semidirect
350. C3×C5C4×#104semidirect
366. C3×C4#4008direct (N × H)
380. D4C3×C4×AGL(1,𝔽5)semidirect
406. C5C3×C4×#104semidirect
422. C4C3×#4008direct (N × H)
425. C3C4×#4008direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#7112
C2×C3×#40082
C3×C42×AGL(1,𝔽5)2
C3×C4×#7002
C3×#648752
C4×#40083
C3×C4×#1045