Finite Group Databasebeta

← C3×#10602641 #14302990

Subgroups — C3×#10602641 #14302990

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#534C211yesmaximal, characteristic
2C3×#66132C211yesmaximal, characteristic, complemented
3C2×C3×#3966C211yesmaximal, characteristic, complemented
4C2×C3×#3966C212yesmaximal, complemented
5C3×#64747C212yesmaximal, complemented
6#10602641C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C2211yescharacteristic
8C22×C3×C5×D4C2211yescharacteristic
9C3×C5×#123—21no—
10C3×C5×#123C412yescomplemented
11C3×C5×#121—22no—
12C3×C5×#120—21no—
13C3×C5×#120C412yescomplemented
14C2×C3×C5×C22⋊C4C2211yescharacteristic
15C2×C3×C22⋊Dic5—21no—
16C2×C3×C22⋊Dic5C2212yes—
17C2×C3×C22⋊Dic5C2212yes—
18C3×D4×Dic5—24no—
19C3×#4072—21no—
20C3×#3966—22no—
21C3×#3966—24no—
22C3×#1913—51nomaximal, Hall
23C5×#534C2×C311yescharacteristic
24#66132C2×C311yescharacteristic, complemented
25C2×#3966C2×C311yescharacteristic, complemented
26C2×#3966C2×C312yescomplemented
27#64747C2×C312yescomplemented
28C24×C3×C5C2×C411yescharacteristic, complemented
29C24×C3×C5D411yescharacteristic
30C24×C3×C5C2311yescharacteristic
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—22no—
33C2×C3×C5×D4—22no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—22no—
37C2×C3×C5×D4—22no—
38C2×C3×C5×D4C2×C412yes—
39C2×C3×C5×D4D412yes—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5C2×C411yescharacteristic, complemented
42C22×C3×C4×C5D411yescharacteristic
43C22×C3×Dic5—22no—
44C22×C3×Dic5—22no—
45C22×C3×Dic5—22no—
46C3×C5×C4⋊C4—22no—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—21no—
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4C2×C412yescomplemented
51C3×C22⋊Dic5—22no—
52C3×C22⋊Dic5—22no—
53C3×C22⋊Dic5—22no—
54C3×C22⋊Dic5—22no—
55C3×C22⋊Dic5D414yescomplemented
56C3×C4⋊Dic5—22no—
57C3×C4×Dic5—22no—
58C3×#534D511yescharacteristic
59C3×#518—51no—
60C2×C3×#99—52no—
61C2×C3×#99—51no—
62C3×#342—52no—
63C22×C5×D4C22×C311yescharacteristic
64C22×C5×D4C22×C311yescharacteristic
65C5×#123—21no—
66C5×#123C3×C412yescomplemented
67C5×#121—22no—
68C5×#120—21no—
69C5×#120C3×C412yescomplemented
70C2×C5×C22⋊C4C22×C311yescharacteristic
71C2×C22⋊Dic5—21no—
72C2×C22⋊Dic5C22×C312yes—
73C2×C22⋊Dic5C22×C312yes—
74D4×Dic5—24no—
75#4072—21no—
76#3966—22no—
77#3966—24no—
78#1913—51noSylow
79C23×C3×C5—21no—
80C23×C3×C5—21no—
81C23×C3×C5—21no—
82C23×C3×C5—22no—
83C23×C3×C5—81no—
84C23×C3×C5—81no—
85C23×C3×C5—81no—
86C23×C3×C5C22⋊C412yes—
87C23×C3×C5C22×C411yescharacteristic
88C23×C3×C5C2×D411yescharacteristic
89C23×C3×C5C2×D411yescharacteristic
90C23×C3×C5C2×D412yes—
91C23×C3×C5#4112yes—
92C3×C5×D4—24no—
93C3×C5×D4—24no—
94C3×C5×D4—41no—
95C3×C5×D4—42no—
96C3×C5×D4—41no—
97C3×C5×D4—82no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—81no—
104C2×C3×C4×C5C22⋊C412yes—
105C2×C3×Dic5—24no—
106C2×C3×Dic5—42no—
107C2×C3×Dic5—82no—
108C2×C3×Dic5—82no—
109C2×C3×Dic5—82no—
110C22×C3×D4C2×D511yescharacteristic
111C22×C3×D4C2×D511yescharacteristic
112C3×#123—21no—
113C3×#123Dic512yescomplemented
114C3×#121—22no—
115C3×#120—21no—
116C3×#120Dic512yescomplemented
117C3×C4×D4—104no—
118C3×#117—101no—
119C2×C3×C22⋊C4—52no—
120C2×C3×C22⋊C4—52no—
121C2×C3×C22⋊C4—101no—
122C2×C3×C22⋊C4C2×D511yescharacteristic
123C3×#99—104no—
124C3×#99—102no—
125C24×C5C2×C3×C411yescharacteristic, complemented
126C24×C5C3×D411yescharacteristic
127C24×C5C23×C311yescharacteristic
128C2×C5×D4—22no—
129C2×C5×D4—22no—
130C2×C5×D4—21no—
131C2×C5×D4—21no—
132C2×C5×D4—22no—
133C2×C5×D4—22no—
134C2×C5×D4—22no—
135C2×C5×D4C2×C3×C412yes—
136C2×C5×D4C3×D412yes—
137C22×C4×C5—21no—
138C22×C4×C5C2×C3×C411yescharacteristic, complemented
139C22×C4×C5C3×D411yescharacteristic
140C22×Dic5—22no—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C5×C4⋊C4—22no—
144C5×C22⋊C4—22no—
145C5×C22⋊C4—21no—
146C5×C22⋊C4—22no—
147C5×C22⋊C4C2×C3×C412yescomplemented
148C22⋊Dic5—22no—
149C22⋊Dic5—22no—
150C22⋊Dic5—22no—
151C22⋊Dic5—22no—
152C22⋊Dic5C3×D414yescomplemented
153C4⋊Dic5—22no—
154C4×Dic5—22no—
155#534C3×D511yescharacteristic
156#518—51no—
157C2×#99—52no—
158C2×#99—51no—
159#342—52no—
160C22×C3×C5—22no—
161C22×C3×C5—42no—
162C22×C3×C5—42no—
163C22×C3×C5—41no—
164C22×C3×C5—41no—
165C22×C3×C5—41no—
166C22×C3×C5—42no—
167C22×C3×C5—42no—
168C22×C3×C5—42no—
169C22×C3×C5—41no—
170C22×C3×C5—81no—
171C22×C3×C5—82no—
172C22×C3×C5C2×C22⋊C411yescharacteristic
173C22×C3×C5C4×D412yes—
174C22×C3×C5#12011yescharacteristic
175C22×C3×C5#12112yes—
176C22×C3×C5#12311yescharacteristic
177C3×C4×C5—22no—
178C3×C4×C5—42no—
179C3×C4×C5—41no—
180C3×Dic5—44no—
181C3×Dic5—82no—
182C24×C3C2×Dic511yescharacteristic, complemented
183C24×C3C22⋊D511yescharacteristic
184C24×C3C22×D511yescharacteristic
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4C2×Dic512yes—
193C2×C3×D4C22⋊D512yes—
194C22×C3×C4—21no—
195C22×C3×C4—102no—
196C22×C3×C4—102no—
197C22×C3×C4—102no—
198C22×C3×C4C2×Dic511yescharacteristic, complemented
199C22×C3×C4C22⋊D511yescharacteristic
200C3×C4⋊C4—22no—
201C3×C4⋊C4—102no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—21no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—54no—
206C3×C22⋊C4—102no—
207C3×C22⋊C4—102no—
208C3×C22⋊C4—102no—
209C3×C22⋊C4—102no—
210C3×C22⋊C4C2×Dic512yescomplemented
211C3×C42—102no—
212C23×C5—21no—
213C23×C5—21no—
214C23×C5—21no—
215C23×C5—22no—
216C23×C5—81no—
217C23×C5—81no—
218C23×C5—81no—
219C23×C5C3×C22⋊C412yes—
220C23×C5C22×C3×C411yescharacteristic
221C23×C5C2×C3×D412yes—
222C23×C5C2×C3×D411yescharacteristic
223C23×C5C2×C3×D411yescharacteristic
224C23×C5C3×#4112yes—
225C5×D4—24no—
226C5×D4—24no—
227C5×D4—41no—
228C5×D4—42no—
229C5×D4—41no—
230C5×D4—82no—
231C2×C4×C5—21no—
232C2×C4×C5—21no—
233C2×C4×C5—22no—
234C2×C4×C5—21no—
235C2×C4×C5—22no—
236C2×C4×C5—81no—
237C2×C4×C5C3×C22⋊C412yes—
238C2×Dic5—24no—
239C2×Dic5—42no—
240C2×Dic5—82no—
241C2×Dic5—82no—
242C2×Dic5—82no—
243C22×D4C2×C3×D511yescharacteristic
244C22×D4C2×C3×D511yescharacteristic
245#123—21no—
246#123C3×Dic512yescomplemented
247#121—22no—
248#120—21no—
249#120C3×Dic512yescomplemented
250C4×D4—104no—
251#117—101no—
252C2×C22⋊C4—52no—
253C2×C22⋊C4—52no—
254C2×C22⋊C4—101no—
255C2×C22⋊C4C2×C3×D511yescharacteristic
256#99—104no—
257#99—102no—
258C2×C3×C5—21no—
259C2×C3×C5—22no—
260C2×C3×C5—21no—
261C2×C3×C5—22no—
262C2×C3×C5—21no—
263C2×C3×C5—42no—
264C2×C3×C5—81no—
265C2×C3×C5#38611yescharacteristic
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—81no—
271C23×C3—81no—
272C23×C3—81no—
273C23×C3C22⋊Dic512yes—
274C23×C3D4×D512yes—
275C23×C3#71112yes—
276C23×C3C22×Dic511yescharacteristic
277C23×C3C2×C22⋊D511yescharacteristic
278C23×C3C2×C22⋊D511yescharacteristic
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—41no—
282C3×D4—41no—
283C3×D4—42no—
284C3×D4—82no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—81no—
291C2×C3×C4—104no—
292C2×C3×C4—202no—
293C2×C3×C4—402no—
294C2×C3×C4—402no—
295C2×C3×C4—402no—
296C2×C3×C4C22⋊Dic512yes—
297C22×C5—22no—
298C22×C5—42no—
299C22×C5—42no—
300C22×C5—41no—
301C22×C5—41no—
302C22×C5—41no—
303C22×C5—42no—
304C22×C5—42no—
305C22×C5—42no—
306C22×C5—41no—
307C22×C5—81no—
308C22×C5—82no—
309C22×C5C2×C3×C22⋊C411yescharacteristic
310C22×C5C3×C4×D412yes—
311C22×C5C3×#12011yescharacteristic
312C22×C5C3×#12112yes—
313C22×C5C3×#12311yescharacteristic
314C4×C5—22no—
315C4×C5—42no—
316C4×C5—41no—
317Dic5—44no—
318Dic5—82no—
319C24C2×C3×Dic511yescharacteristic, complemented
320C24C3×C22⋊D511yescharacteristic
321C24C22×C3×D511yescharacteristic
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—22no—
329C2×D4C2×C3×Dic512yes—
330C2×D4C3×C22⋊D512yes—
331C22×C4—21no—
332C22×C4—102no—
333C22×C4—102no—
334C22×C4—102no—
335C22×C4C2×C3×Dic511yescharacteristic, complemented
336C22×C4C3×C22⋊D511yescharacteristic
337C4⋊C4—22no—
338C4⋊C4—102no—
339C22⋊C4—22no—
340C22⋊C4—21no—
341C22⋊C4—22no—
342C22⋊C4—54no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4C2×C3×Dic512yescomplemented
348C42—102no—
349C3×C5#191311yescharacteristic, complemented, Hall
350C22×C3—22no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—81no—
361C22×C3—82no—
362C22×C3D4×Dic512yes—
363C22×C3#408111yescharacteristic
364C22×C3#408311yescharacteristic
365C22×C3#408512yes—
366C22×C3C2×C22⋊Dic511yescharacteristic
367C3×C4—22no—
368C3×C4—42no—
369C3×C4—41no—
370C3×C4—204no—
371C3×C4—402no—
372C2×C5—21no—
373C2×C5—22no—
374C2×C5—21no—
375C2×C5—22no—
376C2×C5—21no—
377C2×C5—42no—
378C2×C5—81no—
379C2×C5C3×#38611yescharacteristic
380C23—21no—
381C23—21no—
382C23—21no—
383C23—22no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23C3×C22⋊Dic512yes—
388C23C3×D4×D512yes—
389C23C3×#71112yes—
390C23C22×C3×Dic511yescharacteristic
391C23C2×C3×C22⋊D511yescharacteristic
392C23C2×C3×C22⋊D511yescharacteristic
393D4—24no—
394D4—24no—
395D4—41no—
396D4—41no—
397D4—42no—
398D4—82no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—81no—
405C2×C4—104no—
406C2×C4—202no—
407C2×C4—402no—
408C2×C4—402no—
409C2×C4—402no—
410C2×C4C3×C22⋊Dic512yes—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—22no—
414C2×C3—22no—
415C2×C3—21no—
416C2×C3—42no—
417C2×C3—81no—
418C2×C3#6550911yescharacteristic
419C5C3×#191311yescharacteristic, complemented, Sylow
420C22—22no—
421C22—41no—
422C22—42no—
423C22—42no—
424C22—41no—
425C22—41no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—81no—
431C22—82no—
432C22C3×D4×Dic512yes—
433C22C3×#408111yescharacteristic
434C22C3×#408311yescharacteristic
435C22C3×#408512yes—
436C22C2×C3×C22⋊Dic511yescharacteristic
437C4—22no—
438C4—42no—
439C4—41no—
440C4—204no—
441C4—402no—
442C3#1060264111yescharacteristic, direct product factor, Sylow
443C2—21no—
444C2—21no—
445C2—22no—
446C2—22no—
447C2—21no—
448C2—42no—
449C2—81no—
450C2C3×#6550911yescharacteristic
subgroupG/H
centerC2×C3#65509
commutatorC23×C5C22×C3×C4
FrattiniC24C22×C3×D5
Fitting, prime radicalC3×C5×#534C2
socleC2×C3×C5#386
cosocleC24×C5C23×C3
hypercenterC3×#534D5
hypocenterC5C3×#1913
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C24×C3 ◁ C3×#534

Complements

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

NH ≅ G/Ntype
2. C3×#66132C2semidirect
3. C2×C3×#3966C2semidirect
4. C2×C3×#3966C2semidirect
5. C3×#64747C2semidirect
6. #10602641C3direct (N × H)
10. C3×C5×#123C4semidirect
13. C3×C5×#120C4semidirect
24. #66132C2×C3semidirect
25. C2×#3966C2×C3semidirect
26. C2×#3966C2×C3semidirect
27. #64747C2×C3semidirect
28. C24×C3×C5C2×C4semidirect
41. C22×C3×C4×C5C2×C4semidirect
50. C3×C5×C22⋊C4C2×C4semidirect
55. C3×C22⋊Dic5D4semidirect
66. C5×#123C3×C4semidirect
69. C5×#120C3×C4semidirect
113. C3×#123Dic5semidirect
116. C3×#120Dic5semidirect
125. C24×C5C2×C3×C4semidirect
138. C22×C4×C5C2×C3×C4semidirect
147. C5×C22⋊C4C2×C3×C4semidirect
152. C22⋊Dic5C3×D4semidirect
182. C24×C3C2×Dic5semidirect
198. C22×C3×C4C2×Dic5semidirect
210. C3×C22⋊C4C2×Dic5semidirect
246. #123C3×Dic5semidirect
249. #120C3×Dic5semidirect
319. C24C2×C3×Dic5semidirect
335. C22×C4C2×C3×Dic5semidirect
347. C22⋊C4C2×C3×Dic5semidirect
349. C3×C5#1913semidirect
419. C5C3×#1913semidirect
442. C3#10602641direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5342
C3×#661322
C2×C3×#39662
C3×#647472
#106026413
C3×#19135