Finite Group Databasebeta

← C3×#10604455 #14297823

Subgroups — C3×#10604455 #14297823

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#713C211yesmaximal, characteristic
2C2×C3×#4010C212yesmaximal
3C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×#65681C211yesmaximal, characteristic, complemented
5C3×#64875C212yesmaximal, complemented
6#10604455C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×C4⋊AGL(1,𝔽5)—21no—
9C2×C3×C4×AGL(1,𝔽5)C2212yes—
10C2×C3×#700C2212yes—
11C3×C4×C5×Q8C411yescharacteristic, complemented
12C3×Q8×Dic5C411yescharacteristic, complemented
13C3×#4039—21no—
14C3×#4039C411yescharacteristic, complemented
15C3×#4038—21no—
16C3×#4038C2211yescharacteristic
17C3×C4×D20—21no—
18C3×C4×D20C411yescharacteristic, complemented
19C3×C42×D5—21no—
20C3×C42×D5C2211yescharacteristic
21C3×#4010—24no—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×C4⋊#170—21no—
24C3×C4⋊#170—21no—
25C3×#1806—51nomaximal, Hall
26C4×#713C2×C311yescharacteristic
27C2×#4010C2×C312yes—
28C4×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
29#65681C2×C311yescharacteristic, complemented
30#64875C2×C312yescomplemented
31C22×C3×AGL(1,𝔽5)—22no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C3×#713—21no—
34C3×#713—21no—
35C3×#713D412yes—
36C3×#713Q812yes—
37C2×C3×D20—21no—
38C2×C3×D20C2×C411yescharacteristic
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3×#170—22no—
46C3×C4⋊AGL(1,𝔽5)—24no—
47C3×C4×AGL(1,𝔽5)—22no—
48C3×C4×AGL(1,𝔽5)—22no—
49C3×#700—22no—
50C3×#700—22no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4C2×C411yescharacteristic
53C3×C42×C5—21no—
54C3×C42×C5C2×C411yescharacteristic
55C3×#685—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×C4⋊Dic5—21no—
59C3×C4⋊Dic5C2×C411yescharacteristic
60C3×C4×Dic5—21no—
61C3×C4×Dic5—21no—
62C3×C4×Dic5C2×C411yescharacteristic
63C3×C4×Dic5D411yescharacteristic
64C3×C4×Dic5D411yescharacteristic
65C3×C4×#41—51no—
66C3×#423—51no—
67C2×C3×#104—52no—
68C3×C4×C4⋊C4—51no—
69C3×#339—52no—
70C2×#713C22×C311yescharacteristic
71C2×C4⋊AGL(1,𝔽5)—21no—
72C2×C4×AGL(1,𝔽5)C22×C312yes—
73C2×#700C22×C312yes—
74C4×C5×Q8C3×C411yescharacteristic, complemented
75Q8×Dic5C3×C411yescharacteristic, complemented
76#4039—21no—
77#4039C3×C411yescharacteristic, complemented
78#4038—21no—
79#4038C22×C311yescharacteristic
80C4×D20—21no—
81C4×D20C3×C411yescharacteristic, complemented
82C42×D5—21no—
83C42×D5C22×C311yescharacteristic
84#4010—24no—
85(C2×AGL(1,𝔽5))⋊C4—21no—
86C4⋊#170—21no—
87C4⋊#170—21no—
88#1806—51noSylow
89C22×C3×D5—21no—
90C22×C3×D5C2×D411yescharacteristic
91C2×C3×AGL(1,𝔽5)—24no—
92C2×C3×AGL(1,𝔽5)—42no—
93C2×C3×AGL(1,𝔽5)—42no—
94C3×C5×Q8—21no—
95C3×C5×Q8C4⋊C412yescomplemented
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C2×C3×C4×C5C22×C411yescharacteristic
101C2×C3×Dic5—21no—
102C2×C3×Dic5C22⋊C411yescharacteristic
103C2×C3×Dic5C22⋊C411yescharacteristic
104C2×C3×Dic5C2×D411yescharacteristic
105C3×D20—22no—
106C3×D20—22no—
107C3×D20—21no—
108C3×D20C4⋊C412yescomplemented
109C3×C4×D5—22no—
110C3×C4×D5—22no—
111C3×C4×D5—22no—
112C3×C4×D5—41no—
113C3×C4×D5—42no—
114C3×C4×D5—42no—
115C3×C4×D5C2×D411yescharacteristic
116C3×C4×D5C2×Q811yescharacteristic
117C3×C4×D5#4111yescharacteristic
118C3×C4×D5#4111yescharacteristic
119C3×#170—42no—
120C2×C3×#41—51no—
121C2×C3×M16—52no—
122C3×C4×Q8—51no—
123C3×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
124C3×C4×D4—51no—
125C3×C4×D4—51no—
126C3×C4×D4—101no—
127C3×C4×D4—101no—
128C3×#117—51no—
129C3×#117—101no—
130C2×C3×C4⋊C4—101no—
131C2×C3×C42—51no—
132C2×C3×C42—52no—
133C2×C3×C42—101no—
134C3×C4⋊C8—101no—
135C3×C4⋊C8—101no—
136C3×#104—104no—
137C3×#95—101no—
138C22×AGL(1,𝔽5)—22no—
139C2×C5×Q8C2×C3×C411yescharacteristic
140#713—21no—
141#713—21no—
142#713C3×D412yes—
143#713C3×Q812yes—
144C2×D20—21no—
145C2×D20C2×C3×C411yescharacteristic
146C2×C4×D5—21no—
147C2×C4×D5—21no—
148C2×C4×D5—21no—
149C2×C4×D5C3×D411yescharacteristic
150C2×C4×D5C3×D411yescharacteristic
151C2×C4×D5C23×C311yescharacteristic
152C2×#170—22no—
153C4⋊AGL(1,𝔽5)—24no—
154C4×AGL(1,𝔽5)—22no—
155C4×AGL(1,𝔽5)—22no—
156#700—22no—
157#700—22no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4C2×C3×C411yescharacteristic
160C42×C5—21no—
161C42×C5C2×C3×C411yescharacteristic
162#685—21no—
163#685—21no—
164#685—21no—
165C4⋊Dic5—21no—
166C4⋊Dic5C2×C3×C411yescharacteristic
167C4×Dic5—21no—
168C4×Dic5—21no—
169C4×Dic5C2×C3×C411yescharacteristic
170C4×Dic5C3×D411yescharacteristic
171C4×Dic5C3×D411yescharacteristic
172C4×#41—51no—
173#423—51no—
174C2×#104—52no—
175C4×C4⋊C4—51no—
176#339—52no—
177C22×C3×C5C2×C22⋊C411yescharacteristic
178C2×C3×D5—21no—
179C2×C3×D5—21no—
180C2×C3×D5—22no—
181C2×C3×D5—41no—
182C2×C3×D5—41no—
183C2×C3×D5#12111yescharacteristic
184C2×C3×D5C22⋊Q811yescharacteristic
185C3×AGL(1,𝔽5)—44no—
186C3×C4×C5—22no—
187C3×C4×C5—21no—
188C3×C4×C5—41no—
189C3×C4×C5#10412yescomplemented
190C3×C4×C5C2×C4⋊C411yescharacteristic
191C3×C4×C5#11711yescharacteristic
192C3×Dic5—21no—
193C3×Dic5—21no—
194C3×Dic5—41no—
195C3×Dic5#12111yescharacteristic
196C3×Dic5C22⋊Q811yescharacteristic
197C3×#41—52no—
198C3×#41—52no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
202C2×C3×D4—51no—
203C2×C3×D4—101no—
204C22×C3×C4—51no—
205C22×C3×C4—51no—
206C22×C3×C4—51no—
207C22×C3×C4—101no—
208C22×C3×C4—101no—
209C22×C3×C4—102no—
210C22×C3×C4—101no—
211C3×M16—102no—
212C3×M16—102no—
213C2×C3×C8—102no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—51no—
216C3×C4⋊C4—101no—
217C3×C4⋊C4—104no—
218C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
219C3×C22⋊C4—101no—
220C3×C22⋊C4—101no—
221C3×C22⋊C4—101no—
222C3×C42—21no—
223C3×C42—51no—
224C3×C42—51no—
225C3×C42—51no—
226C3×C42—101no—
227C3×C42—101no—
228C3×C42—102no—
229C3×C42—102no—
230C3×C42C2×AGL(1,𝔽5)11yescharacteristic
231C22×D5—21no—
232C22×D5C2×C3×D411yescharacteristic
233C2×AGL(1,𝔽5)—24no—
234C2×AGL(1,𝔽5)—42no—
235C2×AGL(1,𝔽5)—42no—
236C5×Q8—21no—
237C5×Q8C3×C4⋊C412yescomplemented
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5C3×C22⋊C411yescharacteristic
241C2×C4×C5C3×C22⋊C411yescharacteristic
242C2×C4×C5C22×C3×C411yescharacteristic
243C2×Dic5—21no—
244C2×Dic5C3×C22⋊C411yescharacteristic
245C2×Dic5C3×C22⋊C411yescharacteristic
246C2×Dic5C2×C3×D411yescharacteristic
247D20—22no—
248D20—22no—
249D20—21no—
250D20C3×C4⋊C412yescomplemented
251C4×D5—22no—
252C4×D5—22no—
253C4×D5—22no—
254C4×D5—41no—
255C4×D5—42no—
256C4×D5—42no—
257C4×D5C2×C3×D411yescharacteristic
258C4×D5C2×C3×Q811yescharacteristic
259C4×D5C3×#4111yescharacteristic
260C4×D5C3×#4111yescharacteristic
261#170—42no—
262C2×#41—51no—
263C2×M16—52no—
264C4×Q8—51no—
265C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
266C4×D4—51no—
267C4×D4—51no—
268C4×D4—101no—
269C4×D4—101no—
270#117—51no—
271#117—101no—
272C2×C4⋊C4—101no—
273C2×C42—51no—
274C2×C42—52no—
275C2×C42—101no—
276C4⋊C8—101no—
277C4⋊C8—101no—
278#104—104no—
279#95—101no—
280C2×C3×C5#38011yescharacteristic
281C2×C3×C5C2×#10411yescharacteristic
282C2×C3×C5#42111yescharacteristic
283C3×D5—21no—
284C3×D5—21no—
285C3×D5—42no—
286C23×C3—51no—
287C23×C3—101no—
288C3×Q8—21no—
289C3×Q8C4⋊AGL(1,𝔽5)12yescomplemented
290C3×D4—52no—
291C3×D4—102no—
292C3×D4—102no—
293C3×D4—101no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—51no—
297C2×C3×C4—51no—
298C2×C3×C4—51no—
299C2×C3×C4—51no—
300C2×C3×C4—51no—
301C2×C3×C4—51no—
302C2×C3×C4—51no—
303C2×C3×C4—102no—
304C2×C3×C4—102no—
305C2×C3×C4—102no—
306C2×C3×C4—104no—
307C2×C3×C4—101no—
308C2×C3×C4—201no—
309C2×C3×C4—202no—
310C2×C3×C4—202no—
311C2×C3×C4—202no—
312C2×C3×C4—202no—
313C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
314C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
315C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
316C3×C8—202no—
317C22×C5C2×C3×C22⋊C411yescharacteristic
318C2×D5—21no—
319C2×D5—21no—
320C2×D5—22no—
321C2×D5—41no—
322C2×D5—41no—
323C2×D5C3×#12111yescharacteristic
324C2×D5C3×C22⋊Q811yescharacteristic
325AGL(1,𝔽5)—44no—
326C4×C5—22no—
327C4×C5—21no—
328C4×C5—41no—
329C4×C5C3×#10412yescomplemented
330C4×C5C2×C3×C4⋊C411yescharacteristic
331C4×C5C3×#11711yescharacteristic
332Dic5—21no—
333Dic5—21no—
334Dic5—41no—
335Dic5C3×#12111yescharacteristic
336Dic5C3×C22⋊Q811yescharacteristic
337#41—52no—
338#41—52no—
339#41—101no—
340#41—101no—
341C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
342C2×D4—51no—
343C2×D4—101no—
344C22×C4—51no—
345C22×C4—51no—
346C22×C4—51no—
347C22×C4—101no—
348C22×C4—101no—
349C22×C4—102no—
350C22×C4—101no—
351M16—102no—
352M16—102no—
353C2×C8—102no—
354C4⋊C4—21no—
355C4⋊C4—51no—
356C4⋊C4—101no—
357C4⋊C4—104no—
358C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
359C22⋊C4—101no—
360C22⋊C4—101no—
361C22⋊C4—101no—
362C42—21no—
363C42—51no—
364C42—51no—
365C42—51no—
366C42—101no—
367C42—101no—
368C42—102no—
369C42—102no—
370C42C2×C3×AGL(1,𝔽5)11yescharacteristic
371C3×C5#180611yescharacteristic, complemented, Hall
372C22×C3—51no—
373C22×C3—51no—
374C22×C3—101no—
375C22×C3—101no—
376C22×C3—102no—
377C22×C3—201no—
378C22×C3—201no—
379C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
380C3×C4—21no—
381C3×C4—22no—
382C3×C4—41no—
383C3×C4—51no—
384C3×C4—51no—
385C3×C4—101no—
386C3×C4—101no—
387C3×C4—204no—
388C3×C4—201no—
389C3×C4#401012yescomplemented
390C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
391C3×C4#413011yescharacteristic
392C2×C5C3×#38011yescharacteristic
393C2×C5C2×C3×#10411yescharacteristic
394C2×C5C3×#42111yescharacteristic
395D5—21no—
396D5—21no—
397D5—42no—
398C23—51no—
399C23—101no—
400Q8—21no—
401Q8C3×C4⋊AGL(1,𝔽5)12yescomplemented
402D4—52no—
403D4—102no—
404D4—102no—
405D4—101no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—51no—
409C2×C4—51no—
410C2×C4—51no—
411C2×C4—51no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—102no—
416C2×C4—102no—
417C2×C4—102no—
418C2×C4—104no—
419C2×C4—101no—
420C2×C4—201no—
421C2×C4—202no—
422C2×C4—202no—
423C2×C4—202no—
424C2×C4—202no—
425C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
426C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
427C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
428C8—202no—
429C2×C3—101no—
430C2×C3—101no—
431C2×C3—202no—
432C2×C3#6576511yescharacteristic
433C2×C3C2×#401011yescharacteristic
434C2×C3#6579511yescharacteristic
435C5C3×#180611yescharacteristic, complemented, Sylow
436C22—51no—
437C22—51no—
438C22—101no—
439C22—101no—
440C22—102no—
441C22—201no—
442C22—201no—
443C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
444C4—22no—
445C4—21no—
446C4—41no—
447C4—51no—
448C4—51no—
449C4—101no—
450C4—101no—
451C4—204no—
452C4—201no—
453C4C3×#401012yescomplemented
454C4C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
455C4C3×#413011yescharacteristic
456C3#1060445511yescharacteristic, direct product factor, Sylow
457C2—101no—
458C2—101no—
459C2—202no—
460C2C3×#6576511yescharacteristic
461C2C2×C3×#401011yescharacteristic
462C2C3×#6579511yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#713C22
cosocleC2×C4×D5C23×C3
hypercenterC3×C4×Q8AGL(1,𝔽5)
hypocenterC5C3×#1806
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C3×C4×C4⋊AGL(1,𝔽5)C2semidirect
4. C3×#65681C2semidirect
5. C3×#64875C2semidirect
6. #10604455C3direct (N × H)
11. C3×C4×C5×Q8C4semidirect
12. C3×Q8×Dic5C4semidirect
14. C3×#4039C4semidirect
18. C3×C4×D20C4semidirect
28. C4×C4⋊AGL(1,𝔽5)C2×C3semidirect
29. #65681C2×C3semidirect
30. #64875C2×C3semidirect
74. C4×C5×Q8C3×C4semidirect
75. Q8×Dic5C3×C4semidirect
77. #4039C3×C4semidirect
81. C4×D20C3×C4semidirect
95. C3×C5×Q8C4⋊C4semidirect
108. C3×D20C4⋊C4semidirect
123. C3×C4×Q8AGL(1,𝔽5)semidirect
189. C3×C4×C5#104semidirect
237. C5×Q8C3×C4⋊C4semidirect
250. D20C3×C4⋊C4semidirect
265. C4×Q8C3×AGL(1,𝔽5)semidirect
289. C3×Q8C4⋊AGL(1,𝔽5)semidirect
329. C4×C5C3×#104semidirect
371. C3×C5#1806semidirect
389. C3×C4#4010semidirect
401. Q8C3×C4⋊AGL(1,𝔽5)semidirect
435. C5C3×#1806semidirect
453. C4C3×#4010semidirect
456. C3#10604455direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#7132
C2×C3×#40102
C3×C4×C4⋊AGL(1,𝔽5)2
C3×#656812
C3×#648752
#106044553
C3×#18065