Finite Group Databasebeta

← C3×#10604422 #14297821

Subgroups — C3×#10604422 #14297821

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×#711C211yesmaximal, characteristic
2C2×C3×#4008C212yesmaximal
3C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×#65682C211yesmaximal, characteristic, complemented
5C3×#64875C212yesmaximal, complemented
6#10604422C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711C2211yescharacteristic
8C2×C3×C4⋊AGL(1,𝔽5)—21no—
9C2×C3×C4×AGL(1,𝔽5)C2212yes—
10C2×C3×#700C2212yes—
11C3×C4×C5×D4C411yescharacteristic, complemented
12C3×D4×Dic5C411yescharacteristic, complemented
13C3×C4×C22⋊D5—21no—
14C2×C3×C4×Dic5—21no—
15C3×#4038C2211yescharacteristic
16C3×#4033C411yescharacteristic, complemented
17C3×#4027—21no—
18C3×#4023—21no—
19C3×C42×D5C2211yescharacteristic
20C3×C4×Dic10C411yescharacteristic, complemented
21C3×#4008—24no—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×#4004—21no—
24C3×#4004—21no—
25C3×#1806—51nomaximal, Hall
26C4×#711C2×C311yescharacteristic
27C2×#4008C2×C312yes—
28C4×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
29#65682C2×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—21no—
38C3×#711—21no—
39C3×#711D412yes—
40C3×#711Q812yes—
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5C2311yescharacteristic
44C2×C3×Dic10C2×C411yescharacteristic
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⋊C4C2×C411yescharacteristic
52C3×C5×C22⋊C4—21no—
53C3×C42×C5C2×C411yescharacteristic
54C3×C22⋊Dic5—21no—
55C3×#685—21no—
56C3×C4⋊Dic5C2×C411yescharacteristic
57C3×#683—21no—
58C3×#683—21no—
59C3×C4×Dic5—21no—
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×#711C22×C311yescharacteristic
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#4008—24no—
85(C2×AGL(1,𝔽5))⋊C4—21no—
86#4004—21no—
87#4004—21no—
88#1806—51noSylow
89C23×C3×C5—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×D4—21no—
95C3×C5×D4C4⋊C412yescomplemented
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—42no—
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C2×C3×C4×C5C22×C411yescharacteristic
101C3×C22⋊D5—22no—
102C3×C22⋊D5—22no—
103C2×C3×Dic5—21no—
104C2×C3×Dic5—22no—
105C2×C3×Dic5—22no—
106C2×C3×Dic5—21no—
107C2×C3×Dic5—42no—
108C2×C3×Dic5C22⋊C411yescharacteristic
109C2×C3×Dic5C22⋊C411yescharacteristic
110C2×C3×Dic5C2×D411yescharacteristic
111C3×C4×D5—22no—
112C3×C4×D5—41no—
113C3×C4×D5C2×D411yescharacteristic
114C3×C4×D5C2×Q811yescharacteristic
115C3×C4×D5#4111yescharacteristic
116C3×C4×D5#4111yescharacteristic
117C3×Dic10—21no—
118C3×Dic10C4⋊C412yescomplemented
119C3×#170—42no—
120C2×C3×#41—51no—
121C2×C3×M16—52no—
122C3×C4×Q8—51no—
123C3×C4×Q8—51no—
124C3×C4×D4—51no—
125C3×C4×D4—101no—
126C3×C4×D4—101no—
127C3×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
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×D4C2×C3×C411yescharacteristic
140C22×C4×C5—21no—
141C2×C22⋊D5—21no—
142C22×Dic5—21no—
143C22×Dic5—21no—
144#711—21no—
145#711—21no—
146#711C3×D412yes—
147#711C3×Q812yes—
148C2×C4×D5C3×D411yescharacteristic
149C2×C4×D5C3×D411yescharacteristic
150C2×C4×D5C23×C311yescharacteristic
151C2×Dic10C2×C3×C411yescharacteristic
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⋊C4C2×C3×C411yescharacteristic
159C5×C22⋊C4—21no—
160C42×C5C2×C3×C411yescharacteristic
161C22⋊Dic5—21no—
162#685—21no—
163C4⋊Dic5C2×C3×C411yescharacteristic
164#683—21no—
165#683—21no—
166C4×Dic5—21no—
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×C5—22no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5C2×C22⋊C411yescharacteristic
181C2×C3×D5—21no—
182C2×C3×D5—21no—
183C2×C3×D5#12111yescharacteristic
184C2×C3×D5C22⋊Q811yescharacteristic
185C3×AGL(1,𝔽5)—44no—
186C3×C4×C5—21no—
187C3×C4×C5—21no—
188C3×C4×C5—41no—
189C3×C4×C5C2×C4⋊C411yescharacteristic
190C3×C4×C5#11711yescharacteristic
191C3×Dic5—22no—
192C3×Dic5—21no—
193C3×Dic5—41no—
194C3×Dic5#10412yescomplemented
195C3×Dic5#12111yescharacteristic
196C3×Dic5C22⋊Q811yescharacteristic
197C3×#41—52no—
198C3×#41—52no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8—51no—
202C2×C3×D4—101no—
203C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
204C22×C3×C4—21no—
205C22×C3×C4—51no—
206C22×C3×C4—51no—
207C22×C3×C4—51no—
208C22×C3×C4—101no—
209C22×C3×C4—101no—
210C22×C3×C4—102no—
211C3×M16—102no—
212C3×M16—102no—
213C2×C3×C8—102no—
214C3×C4⋊C4—51no—
215C3×C4⋊C4—101no—
216C3×C4⋊C4—104no—
217C3×C4⋊C4—101no—
218C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
219C3×C22⋊C4—21no—
220C3×C22⋊C4—101no—
221C3×C22⋊C4—101no—
222C3×C42—51no—
223C3×C42—51no—
224C3×C42—51no—
225C3×C42—101no—
226C3×C42—101no—
227C3×C42—102no—
228C3×C42—102no—
229C3×C42—101no—
230C3×C42C2×AGL(1,𝔽5)11yescharacteristic
231C23×C5—21no—
232C22×D5C2×C3×D411yescharacteristic
233C2×AGL(1,𝔽5)—24no—
234C2×AGL(1,𝔽5)—42no—
235C2×AGL(1,𝔽5)—42no—
236C5×D4—21no—
237C5×D4C3×C4⋊C412yescomplemented
238C2×C4×C5—21no—
239C2×C4×C5—42no—
240C2×C4×C5C3×C22⋊C411yescharacteristic
241C2×C4×C5C3×C22⋊C411yescharacteristic
242C2×C4×C5C22×C3×C411yescharacteristic
243C22⋊D5—22no—
244C22⋊D5—22no—
245C2×Dic5—21no—
246C2×Dic5—22no—
247C2×Dic5—22no—
248C2×Dic5—21no—
249C2×Dic5—42no—
250C2×Dic5C3×C22⋊C411yescharacteristic
251C2×Dic5C3×C22⋊C411yescharacteristic
252C2×Dic5C2×C3×D411yescharacteristic
253C4×D5—22no—
254C4×D5—41no—
255C4×D5C2×C3×D411yescharacteristic
256C4×D5C2×C3×Q811yescharacteristic
257C4×D5C3×#4111yescharacteristic
258C4×D5C3×#4111yescharacteristic
259Dic10—21no—
260Dic10C3×C4⋊C412yescomplemented
261#170—42no—
262C2×#41—51no—
263C2×M16—52no—
264C4×Q8—51no—
265C4×Q8—51no—
266C4×D4—51no—
267C4×D4—101no—
268C4×D4—101no—
269C4×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
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—42no—
281C2×C3×C5#38011yescharacteristic
282C2×C3×C5C2×#10411yescharacteristic
283C2×C3×C5#42111yescharacteristic
284C3×D5—21no—
285C3×D5—21no—
286C23×C3—21no—
287C23×C3—51no—
288C3×Q8—52no—
289C3×Q8—101no—
290C3×D4—21no—
291C3×D4—102no—
292C3×D4—102no—
293C3×D4C4⋊AGL(1,𝔽5)12yescomplemented
294C2×C3×C4—21no—
295C2×C3×C4—42no—
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—101no—
307C2×C3×C4—104no—
308C2×C3×C4—101no—
309C2×C3×C4—201no—
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×C5—22no—
318C22×C5—41no—
319C22×C5—41no—
320C22×C5C2×C3×C22⋊C411yescharacteristic
321C2×D5—21no—
322C2×D5—21no—
323C2×D5C3×#12111yescharacteristic
324C2×D5C3×C22⋊Q811yescharacteristic
325AGL(1,𝔽5)—44no—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—41no—
329C4×C5C2×C3×C4⋊C411yescharacteristic
330C4×C5C3×#11711yescharacteristic
331Dic5—22no—
332Dic5—21no—
333Dic5—41no—
334Dic5C3×#10412yescomplemented
335Dic5C3×#12111yescharacteristic
336Dic5C3×C22⋊Q811yescharacteristic
337#41—52no—
338#41—52no—
339#41—101no—
340#41—101no—
341C2×Q8—51no—
342C2×D4—101no—
343C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
344C22×C4—21no—
345C22×C4—51no—
346C22×C4—51no—
347C22×C4—51no—
348C22×C4—101no—
349C22×C4—101no—
350C22×C4—102no—
351M16—102no—
352M16—102no—
353C2×C8—102no—
354C4⋊C4—51no—
355C4⋊C4—101no—
356C4⋊C4—104no—
357C4⋊C4—101no—
358C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
359C22⋊C4—21no—
360C22⋊C4—101no—
361C22⋊C4—101no—
362C42—51no—
363C42—51no—
364C42—51no—
365C42—101no—
366C42—101no—
367C42—102no—
368C42—102no—
369C42—101no—
370C42C2×C3×AGL(1,𝔽5)11yescharacteristic
371C3×C5#180611yescharacteristic, complemented, Hall
372C22×C3—22no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—51no—
376C22×C3—51no—
377C22×C3—101no—
378C22×C3—101no—
379C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—41no—
383C3×C4—52no—
384C3×C4—51no—
385C3×C4—51no—
386C3×C4—102no—
387C3×C4—101no—
388C3×C4—204no—
389C3×C4—201no—
390C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
391C3×C4#413011yescharacteristic
392C2×C5—42no—
393C2×C5C3×#38011yescharacteristic
394C2×C5C2×C3×#10411yescharacteristic
395C2×C5C3×#42111yescharacteristic
396D5—21no—
397D5—21no—
398C23—21no—
399C23—51no—
400Q8—52no—
401Q8—101no—
402D4—21no—
403D4—102no—
404D4—102no—
405D4C3×C4⋊AGL(1,𝔽5)12yescomplemented
406C2×C4—21no—
407C2×C4—42no—
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—101no—
419C2×C4—104no—
420C2×C4—101no—
421C2×C4—201no—
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—42no—
430C2×C3—101no—
431C2×C3—101no—
432C2×C3#6576511yescharacteristic
433C2×C3#6576911yescharacteristic
434C2×C3D5⋊#10411yescharacteristic
435C5C3×#180611yescharacteristic, complemented, Sylow
436C22—22no—
437C22—41no—
438C22—41no—
439C22—51no—
440C22—51no—
441C22—101no—
442C22—101no—
443C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—51no—
448C4—51no—
449C4—52no—
450C4—102no—
451C4—101no—
452C4—204no—
453C4—201no—
454C4C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
455C4C3×#413011yescharacteristic
456C3#1060442211yescharacteristic, direct product factor, Sylow
457C2—42no—
458C2—101no—
459C2—101no—
460C2C3×#6576511yescharacteristic
461C2C3×#6576911yescharacteristic
462C2C3×D5⋊#10411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
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×#1806
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ 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×#65682C2semidirect
5. C3×#64875C2semidirect
6. #10604422C3direct (N × H)
11. C3×C4×C5×D4C4semidirect
12. C3×D4×Dic5C4semidirect
16. C3×#4033C4semidirect
20. C3×C4×Dic10C4semidirect
28. C4×C4⋊AGL(1,𝔽5)C2×C3semidirect
29. #65682C2×C3semidirect
30. #64875C2×C3semidirect
74. C4×C5×D4C3×C4semidirect
75. D4×Dic5C3×C4semidirect
79. #4033C3×C4semidirect
83. C4×Dic10C3×C4semidirect
95. C3×C5×D4C4⋊C4semidirect
118. C3×Dic10C4⋊C4semidirect
127. C3×C4×D4AGL(1,𝔽5)semidirect
194. C3×Dic5#104semidirect
237. C5×D4C3×C4⋊C4semidirect
260. Dic10C3×C4⋊C4semidirect
269. C4×D4C3×AGL(1,𝔽5)semidirect
293. C3×D4C4⋊AGL(1,𝔽5)semidirect
334. Dic5C3×#104semidirect
371. C3×C5#1806semidirect
405. D4C3×C4⋊AGL(1,𝔽5)semidirect
435. C5C3×#1806semidirect
456. C3#10604422direct (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×C4×C4⋊AGL(1,𝔽5)2
C3×#656822
C3×#648752
#106044223
C3×#18065