Finite Group Databasebeta

← C3×#10604281 #14294490

Subgroups — C3×#10604281 #14294490

Order 1920.

808 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×#4130C211yesmaximal, characteristic
2C2×C3×D5⋊M16C211yesmaximal, characteristic
3C2×C3×#4038C211yesmaximal, characteristic
4C3×#64875C214yesmaximal, complemented
5#10604281C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C4×D5C2211yescharacteristic
7C2×C3×C22⋊AGL(1,𝔽5)—21no—
8C2×C3×#704—21no—
9C3×#4130—22no—
10C3×#4130C414yescomplemented
11C2×C3×C4⋊AGL(1,𝔽5)—21no—
12C2×C3×C4×AGL(1,𝔽5)C2212yes—
13C3×D5⋊M16—22no—
14C3×D5⋊M16C414yescomplemented
15C2×C3×#700C2212yes—
16C2×C3×D5⋊C8—21no—
17C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
18C2×C3×#685—21no—
19C2×C3×C4⋊Dic5C411yescharacteristic, complemented
20C2×C3×C4×Dic5—21no—
21C3×#4038—21no—
22C3×#4038—21no—
23C3×#4038C412yescomplemented
24C3×#4038C2212yes—
25C3×#1733—51nomaximal, Hall
26C2×#4130C2×C311yescharacteristic
27C2×D5⋊M16C2×C311yescharacteristic
28C2×#4038C2×C311yescharacteristic
29#64875C2×C314yescomplemented
30C23×C3×D5D411yescharacteristic
31C22×C3×AGL(1,𝔽5)—22no—
32C22×C3×C4×C5—21no—
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×C3×Dic5—21no—
35C22×C3×Dic5D411yescharacteristic
36C2×C3×C4×D5C2×C411yescharacteristic
37C2×C3×C4×D5C2×C412yes—
38C2×C3×C4×D5C2×C412yes—
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D412yes—
42C2×C3×C4×D5Q812yes—
43C2×C3×C4×D5C2311yescharacteristic
44C3×C22⋊AGL(1,𝔽5)—24no—
45C3×#704—24no—
46C2×C3×#170—22no—
47C3×C4⋊AGL(1,𝔽5)—24no—
48C3×C4×AGL(1,𝔽5)—22no—
49C3×C4×AGL(1,𝔽5)C2×C414yescomplemented
50C3×#700—22no—
51C3×#700C2×C414yescomplemented
52C3×D5⋊C8—24no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4C2×C412yes—
55C3×#685—22no—
56C3×#685—22no—
57C3×C4⋊Dic5—21no—
58C3×C4⋊Dic5C2×C412yes—
59C3×C4×Dic5—22no—
60C3×C4×Dic5—22no—
61C22×C3×M16—51no—
62C2×C3×#117—51no—
63C2×C3×#117—51no—
64C3×#339—54no—
65C22×C4×D5C22×C311yescharacteristic
66C2×C22⋊AGL(1,𝔽5)—21no—
67C2×#704—21no—
68#4130—22no—
69#4130C3×C414yescomplemented
70C2×C4⋊AGL(1,𝔽5)—21no—
71C2×C4×AGL(1,𝔽5)C22×C312yes—
72D5⋊M16—22no—
73D5⋊M16C3×C414yescomplemented
74C2×#700C22×C312yes—
75C2×D5⋊C8—21no—
76C2×C5×C4⋊C4C3×C411yescharacteristic, complemented
77C2×#685—21no—
78C2×C4⋊Dic5C3×C411yescharacteristic, complemented
79C2×C4×Dic5—21no—
80#4038—21no—
81#4038—21no—
82#4038C3×C412yescomplemented
83#4038C22×C312yes—
84#1733—51noSylow
85C23×C3×C5C22⋊C411yescharacteristic
86C22×C3×D5—41no—
87C22×C3×D5—41no—
88C22×C3×D5C22⋊C411yescharacteristic
89C22×C3×D5C22⋊C412yes—
90C22×C3×D5C4⋊C412yes—
91C22×C3×D5C2×D411yescharacteristic
92C2×C3×AGL(1,𝔽5)—24no—
93C2×C3×AGL(1,𝔽5)—42no—
94C2×C3×AGL(1,𝔽5)—42no—
95C2×C3×C4×C5—22no—
96C2×C3×C4×C5—42no—
97C2×C3×C4×C5C4212yes—
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C4⋊C412yes—
100C2×C3×C4×C5C22×C411yescharacteristic
101C2×C3×Dic5—22no—
102C2×C3×Dic5—42no—
103C2×C3×Dic5C22⋊C411yescharacteristic
104C2×C3×Dic5C22⋊C412yes—
105C2×C3×Dic5C4⋊C412yes—
106C2×C3×Dic5C2×D411yescharacteristic
107C3×C4×D5C4212yes—
108C3×C4×D5C22⋊C412yes—
109C3×C4×D5C22⋊C412yes—
110C3×C4×D5C4⋊C412yes—
111C3×C4×D5C4⋊C412yes—
112C3×C4×D5C4⋊C412yes—
113C3×C4×D5C22×C411yescharacteristic
114C3×C4×D5C22×C411yescharacteristic
115C3×C4×D5C2×D411yescharacteristic
116C3×C4×D5C2×Q811yescharacteristic
117C3×#170—24no—
118C23×C3×C4—51no—
119C2×C3×M16—52no—
120C2×C3×M16—54no—
121C2×C3×M16—102no—
122C2×C3×M16—101no—
123C22×C3×C8—101no—
124C3×#117—54no—
125C3×#117—52no—
126C3×#117—52no—
127C3×#117—101no—
128C3×#117—101no—
129C3×#117—102no—
130C2×C3×C4⋊C4—51no—
131C2×C3×C4⋊C4—101no—
132C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
133C2×C3×C22⋊C4—101no—
134C2×C3×C22⋊C4—101no—
135C2×C3×C42—52no—
136C2×C3×C42—101no—
137C23×D5C3×D411yescharacteristic
138C22×AGL(1,𝔽5)—22no—
139C22×C4×C5—21no—
140C22×C4×C5C2×C3×C411yescharacteristic
141C22×Dic5—21no—
142C22×Dic5C3×D411yescharacteristic
143C2×C4×D5C2×C3×C411yescharacteristic
144C2×C4×D5C2×C3×C412yes—
145C2×C4×D5C2×C3×C412yes—
146C2×C4×D5C3×D411yescharacteristic
147C2×C4×D5C3×D411yescharacteristic
148C2×C4×D5C3×D412yes—
149C2×C4×D5C3×Q812yes—
150C2×C4×D5C23×C311yescharacteristic
151C22⋊AGL(1,𝔽5)—24no—
152#704—24no—
153C2×#170—22no—
154C4⋊AGL(1,𝔽5)—24no—
155C4×AGL(1,𝔽5)—22no—
156C4×AGL(1,𝔽5)C2×C3×C414yescomplemented
157#700—22no—
158#700C2×C3×C414yescomplemented
159D5⋊C8—24no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4C2×C3×C412yes—
162#685—22no—
163#685—22no—
164C4⋊Dic5—21no—
165C4⋊Dic5C2×C3×C412yes—
166C4×Dic5—22no—
167C4×Dic5—22no—
168C22×M16—51no—
169C2×#117—51no—
170C2×#117—51no—
171#339—54no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5#9512yes—
175C22×C3×C5C2×C22⋊C411yescharacteristic
176C2×C3×D5—21no—
177C2×C3×D5—21no—
178C2×C3×D5—21no—
179C2×C3×D5—21no—
180C2×C3×D5—42no—
181C2×C3×D5—42no—
182C2×C3×D5#9512yes—
183C2×C3×D5C2×C22⋊C411yescharacteristic
184C2×C3×D5C2×C4⋊C411yescharacteristic
185C3×AGL(1,𝔽5)—44no—
186C3×C4×C5—42no—
187C3×C4×C5#9512yes—
188C3×C4×C5C2×C4211yescharacteristic
189C3×C4×C5C2×C4⋊C411yescharacteristic
190C3×Dic5—42no—
191C3×Dic5#9512yes—
192C3×Dic5C2×C22⋊C411yescharacteristic
193C3×Dic5C2×C4⋊C411yescharacteristic
194C24×C3—51no—
195C22×C3×C4—21no—
196C22×C3×C4—51no—
197C22×C3×C4—52no—
198C22×C3×C4—52no—
199C22×C3×C4—52no—
200C22×C3×C4—52no—
201C22×C3×C4—51no—
202C22×C3×C4—51no—
203C22×C3×C4—51no—
204C22×C3×C4—51no—
205C22×C3×C4—102no—
206C22×C3×C4—101no—
207C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
208C3×M16—54no—
209C3×M16—102no—
210C3×M16—104no—
211C2×C3×C8—104no—
212C2×C3×C8—102no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—52no—
215C3×C4⋊C4—104no—
216C3×C4⋊C4—101no—
217C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
218C3×C22⋊C4—102no—
219C3×C22⋊C4—102no—
220C3×C22⋊C4—104no—
221C3×C42—54no—
222C3×C42—102no—
223C3×C42—102no—
224C3×C42—102no—
225C23×C5C3×C22⋊C411yescharacteristic
226C22×D5—41no—
227C22×D5—41no—
228C22×D5C3×C22⋊C411yescharacteristic
229C22×D5C3×C22⋊C412yes—
230C22×D5C3×C4⋊C412yes—
231C22×D5C2×C3×D411yescharacteristic
232C2×AGL(1,𝔽5)—24no—
233C2×AGL(1,𝔽5)—42no—
234C2×AGL(1,𝔽5)—42no—
235C2×C4×C5—22no—
236C2×C4×C5—42no—
237C2×C4×C5C3×C4212yes—
238C2×C4×C5C3×C22⋊C411yescharacteristic
239C2×C4×C5C3×C4⋊C412yes—
240C2×C4×C5C22×C3×C411yescharacteristic
241C2×Dic5—22no—
242C2×Dic5—42no—
243C2×Dic5C3×C22⋊C411yescharacteristic
244C2×Dic5C3×C22⋊C412yes—
245C2×Dic5C3×C4⋊C412yes—
246C2×Dic5C2×C3×D411yescharacteristic
247C4×D5C3×C4212yes—
248C4×D5C3×C22⋊C412yes—
249C4×D5C3×C22⋊C412yes—
250C4×D5C3×C4⋊C412yes—
251C4×D5C3×C4⋊C412yes—
252C4×D5C3×C4⋊C412yes—
253C4×D5C22×C3×C411yescharacteristic
254C4×D5C22×C3×C411yescharacteristic
255C4×D5C2×C3×D411yescharacteristic
256C4×D5C2×C3×Q811yescharacteristic
257#170—24no—
258C23×C4—51no—
259C2×M16—52no—
260C2×M16—54no—
261C2×M16—102no—
262C2×M16—101no—
263C22×C8—101no—
264#117—54no—
265#117—52no—
266#117—52no—
267#117—101no—
268#117—101no—
269#117—102no—
270C2×C4⋊C4—51no—
271C2×C4⋊C4—101no—
272C2×C4⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
273C2×C22⋊C4—101no—
274C2×C22⋊C4—101no—
275C2×C42—52no—
276C2×C42—101no—
277C2×C3×C5—22no—
278C2×C3×C5C2×#9511yescharacteristic
279C2×C3×C5#42111yescharacteristic
280C2×C3×C5#42111yescharacteristic
281C3×D5—21no—
282C3×D5—22no—
283C3×D5—21no—
284C23×C3—51no—
285C23×C3—51no—
286C23×C3—52no—
287C23×C3—52no—
288C23×C3—201no—
289C23×C3—201no—
290C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
291C2×C3×C4—22no—
292C2×C3×C4—42no—
293C2×C3×C4—51no—
294C2×C3×C4—51no—
295C2×C3×C4—52no—
296C2×C3×C4—52no—
297C2×C3×C4—52no—
298C2×C3×C4—51no—
299C2×C3×C4—51no—
300C2×C3×C4—52no—
301C2×C3×C4—51no—
302C2×C3×C4—51no—
303C2×C3×C4—52no—
304C2×C3×C4—52no—
305C2×C3×C4—52no—
306C2×C3×C4—52no—
307C2×C3×C4—104no—
308C2×C3×C4—102no—
309C2×C3×C4—202no—
310C2×C3×C4—202no—
311C2×C3×C4—202no—
312C2×C3×C4C4×AGL(1,𝔽5)12yes—
313C2×C3×C4C4⋊AGL(1,𝔽5)12yes—
314C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
315C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
316C3×C8—104no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5C3×#9512yes—
320C22×C5C2×C3×C22⋊C411yescharacteristic
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—42no—
326C2×D5—42no—
327C2×D5C3×#9512yes—
328C2×D5C2×C3×C22⋊C411yescharacteristic
329C2×D5C2×C3×C4⋊C411yescharacteristic
330AGL(1,𝔽5)—44no—
331C4×C5—42no—
332C4×C5C3×#9512yes—
333C4×C5C2×C3×C4211yescharacteristic
334C4×C5C2×C3×C4⋊C411yescharacteristic
335Dic5—42no—
336Dic5C3×#9512yes—
337Dic5C2×C3×C22⋊C411yescharacteristic
338Dic5C2×C3×C4⋊C411yescharacteristic
339C24—51no—
340C22×C4—21no—
341C22×C4—51no—
342C22×C4—52no—
343C22×C4—52no—
344C22×C4—52no—
345C22×C4—52no—
346C22×C4—51no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—102no—
351C22×C4—101no—
352C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
353M16—54no—
354M16—102no—
355M16—104no—
356C2×C8—104no—
357C2×C8—102no—
358C4⋊C4—21no—
359C4⋊C4—52no—
360C4⋊C4—104no—
361C4⋊C4—101no—
362C4⋊C4C2×C3×AGL(1,𝔽5)12yes—
363C22⋊C4—102no—
364C22⋊C4—102no—
365C22⋊C4—104no—
366C42—54no—
367C42—102no—
368C42—102no—
369C42—102no—
370C3×C5#173311yescharacteristic, complemented, Hall
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—52no—
374C22×C3—51no—
375C22×C3—51no—
376C22×C3—101no—
377C22×C3—101no—
378C22×C3—101no—
379C22×C3—101no—
380C22×C3—202no—
381C22×C3—202no—
382C22×C3(C2×AGL(1,𝔽5))⋊C412yes—
383C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
384C3×C4—42no—
385C3×C4—52no—
386C3×C4—51no—
387C3×C4—51no—
388C3×C4—204no—
389C3×C4—202no—
390C3×C4(C2×AGL(1,𝔽5))⋊C412yes—
391C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
392C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
393C2×C5—22no—
394C2×C5C2×C3×#9511yescharacteristic
395C2×C5C3×#42111yescharacteristic
396C2×C5C3×#42111yescharacteristic
397D5—21no—
398D5—22no—
399D5—21no—
400C23—51no—
401C23—51no—
402C23—52no—
403C23—52no—
404C23—201no—
405C23—201no—
406C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—52no—
410C2×C4—52no—
411C2×C4—51no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—52no—
416C2×C4—51no—
417C2×C4—52no—
418C2×C4—51no—
419C2×C4—52no—
420C2×C4—52no—
421C2×C4—52no—
422C2×C4—52no—
423C2×C4—104no—
424C2×C4—102no—
425C2×C4—202no—
426C2×C4—202no—
427C2×C4—202no—
428C2×C4C3×C4×AGL(1,𝔽5)12yes—
429C2×C4C3×C4⋊AGL(1,𝔽5)12yes—
430C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
431C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
432C8—104no—
433C2×C3—22no—
434C2×C3—101no—
435C2×C3—102no—
436C2×C3—101no—
437C2×C3C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
438C2×C3#6576911yescharacteristic
439C2×C3#6578411yescharacteristic
440C5C3×#173311yescharacteristic, complemented, Sylow
441C22—21no—
442C22—21no—
443C22—51no—
444C22—52no—
445C22—51no—
446C22—101no—
447C22—101no—
448C22—101no—
449C22—101no—
450C22—202no—
451C22—202no—
452C22C3×(C2×AGL(1,𝔽5))⋊C412yes—
453C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
454C4—42no—
455C4—51no—
456C4—52no—
457C4—51no—
458C4—204no—
459C4—202no—
460C4C3×(C2×AGL(1,𝔽5))⋊C412yes—
461C4C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
462C4C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
463C3#1060428111yescharacteristic, direct product factor, Sylow
464C2—22no—
465C2—101no—
466C2—102no—
467C2—101no—
468C2C2×C3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
469C2C3×#6576911yescharacteristic
470C2C3×#6578411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC4×C5C2×C3×C42
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C4⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C4×D5C23×C3
hypercenterC2×C3×C4⋊C4AGL(1,𝔽5)
hypocenterC5C3×#1733
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#64875C2semidirect
5. #10604281C3direct (N × H)
10. C3×#4130C4semidirect
14. C3×D5⋊M16C4semidirect
17. C2×C3×C5×C4⋊C4C4semidirect
19. C2×C3×C4⋊Dic5C4semidirect
23. C3×#4038C4semidirect
29. #64875C2×C3semidirect
49. C3×C4×AGL(1,𝔽5)C2×C4semidirect
51. C3×#700C2×C4semidirect
69. #4130C3×C4semidirect
73. D5⋊M16C3×C4semidirect
76. C2×C5×C4⋊C4C3×C4semidirect
78. C2×C4⋊Dic5C3×C4semidirect
82. #4038C3×C4semidirect
132. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
156. C4×AGL(1,𝔽5)C2×C3×C4semidirect
158. #700C2×C3×C4semidirect
272. C2×C4⋊C4C3×AGL(1,𝔽5)semidirect
370. C3×C5#1733semidirect
440. C5C3×#1733semidirect
463. C3#10604281direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41302
C2×C3×D5⋊M162
C2×C3×#40382
C3×#648752
#106042813
C3×#17335