Finite Group Databasebeta

← C3×#10600853 #14313406

Subgroups — C3×#10600853 #14313406

Order 1920.

516 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66166C211yesmaximal, characteristic
2C3×C5×#410C211yesmaximal, characteristic, complemented
3C3×#65529C211yesmaximal, characteristic, complemented
4C3×#64694C212yesmaximal, complemented
5C3×#64693C212yesmaximal, complemented
6#10600853C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41C2211yescharacteristic
8C2×C3×#709C411yescharacteristic, complemented
9C3×C5×#117—21no—
10C3×#4093C411yescharacteristic, complemented
11C3×#4090C411yescharacteristic, complemented
12C3×#4086C2211yescharacteristic
13C3×#4085—21no—
14C3×#4085—21no—
15C3×#4082C2211yescharacteristic
16C3×C4×C22⋊D5—21no—
17C3×C4×C22⋊D5—21no—
18C3×#4072—21no—
19C2×C3×C4×Dic5C411yescharacteristic, complemented
20C3×C5×#99—21no—
21C3×C5×#99C2212yes—
22C3×#3966—21no—
23C3×#3966C2212yes—
24C3×#2126—51nomaximal, Hall
25#66166C2×C311yescharacteristic
26C5×#410C2×C311yescharacteristic, complemented
27#65529C2×C311yescharacteristic, complemented
28#64694C2×C312yescomplemented
29#64693C2×C312yescomplemented
30C3×C5×#41—41no—
31C3×C5×#41—41no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C2×C3×C5×D4D411yescharacteristic
34C2×C3×C5×D4D411yescharacteristic
35C2×C3×C5×D4C2311yescharacteristic
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5D411yescharacteristic
39C2×C3×C22⋊D5—21no—
40C2×C3×C22⋊D5—21no—
41C2×C3×C22⋊D5—21no—
42C22×C3×Dic5—21no—
43C3×#709—41no—
44C3×#709—41no—
45C2×C3×D20C2×C411yescharacteristic
46C2×C3×C4×D5—21no—
47C2×C3×Dic10C2×C411yescharacteristic
48C3×C5×C4⋊C4—41no—
49C3×C5×C22⋊C4—22no—
50C3×C42×C5—41no—
51C3×C22⋊Dic5—21no—
52C3×C22⋊Dic5—21no—
53C3×C22⋊Dic5—22no—
54C3×#685—21no—
55C3×#685—21no—
56C3×C4⋊Dic5—41no—
57C3×#683—21no—
58C3×#683—21no—
59C3×C4×Dic5—21no—
60C3×C4×Dic5—41no—
61C3×C4×Dic5C2×C411yescharacteristic
62C3×C4×Dic5C2×C411yescharacteristic
63C3×#532—51no—
64C3×#410—51no—
65C3×#410D511yescharacteristic, complemented
66C3×#354—52no—
67C3×#353—52no—
68C2×C5×#41C22×C311yescharacteristic
69C2×#709C3×C411yescharacteristic, complemented
70C5×#117—21no—
71#4093C3×C411yescharacteristic, complemented
72#4090C3×C411yescharacteristic, complemented
73#4086C22×C311yescharacteristic
74#4085—21no—
75#4085—21no—
76#4082C22×C311yescharacteristic
77C4×C22⋊D5—21no—
78C4×C22⋊D5—21no—
79#4072—21no—
80C2×C4×Dic5C3×C411yescharacteristic, complemented
81C5×#99—21no—
82C5×#99C22×C312yes—
83#3966—21no—
84#3966C22×C312yes—
85#2126—51noSylow
86C23×C3×C5C22⋊C411yescharacteristic
87C23×C3×C5C2×D411yescharacteristic
88C23×C3×C5C2×D411yescharacteristic
89C22×C3×D5—21no—
90C3×C5×Q8—41no—
91C3×C5×D4—41no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—42no—
101C2×C3×C4×C5C22⋊C411yescharacteristic
102C2×C3×C4×C5C22⋊C411yescharacteristic
103C2×C3×C4×C5C22⋊C411yescharacteristic
104C2×C3×C4×C5C22×C411yescharacteristic
105C3×C22⋊D5—42no—
106C3×C22⋊D5—42no—
107C3×C22⋊D5—42no—
108C2×C3×Dic5—21no—
109C2×C3×Dic5—21no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5—42no—
112C2×C3×Dic5—42no—
113C3×D20—41no—
114C3×C4×D5—41no—
115C3×C4×D5—41no—
116C3×Dic10—41no—
117C2×C3×#41—51no—
118C2×C3×#41C2×D511yescharacteristic
119C3×C4⋊Q8—51no—
120C3×#127—51no—
121C3×#124—51no—
122C3×#124—51no—
123C3×#121—101no—
124C3×#121—101no—
125C3×C4×D4—101no—
126C3×C4×D4—101no—
127C3×#117—21no—
128C3×#117—101no—
129C2×C3×C42—51no—
130C3×#99—21no—
131C3×#99—52no—
132C3×#99—101no—
133C3×#99C2×D512yes—
134C5×#41—41no—
135C5×#41—41no—
136C2×C5×Q8C2×C3×C411yescharacteristic
137C2×C5×D4C3×D411yescharacteristic
138C2×C5×D4C3×D411yescharacteristic
139C2×C5×D4C23×C311yescharacteristic
140C22×C4×C5C2×C3×C411yescharacteristic
141C22×C4×C5C3×D411yescharacteristic
142C22×C4×C5C3×D411yescharacteristic
143C2×C22⋊D5—21no—
144C2×C22⋊D5—21no—
145C2×C22⋊D5—21no—
146C22×Dic5—21no—
147#709—41no—
148#709—41no—
149C2×D20C2×C3×C411yescharacteristic
150C2×C4×D5—21no—
151C2×Dic10C2×C3×C411yescharacteristic
152C5×C4⋊C4—41no—
153C5×C22⋊C4—22no—
154C42×C5—41no—
155C22⋊Dic5—21no—
156C22⋊Dic5—21no—
157C22⋊Dic5—22no—
158#685—21no—
159#685—21no—
160C4⋊Dic5—41no—
161#683—21no—
162#683—21no—
163C4×Dic5—21no—
164C4×Dic5—41no—
165C4×Dic5C2×C3×C411yescharacteristic
166C4×Dic5C2×C3×C411yescharacteristic
167#532—51no—
168#410—51no—
169#410C3×D511yescharacteristic, complemented
170#354—52no—
171#353—52no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5—41no—
176C22×C3×C5—41no—
177C22×C3×C5—41no—
178C22×C3×C5C2×C22⋊C411yescharacteristic
179C2×C3×D5—41no—
180C2×C3×D5—41no—
181C2×C3×D5—41no—
182C3×C4×C5—21no—
183C3×C4×C5—21no—
184C3×C4×C5—21no—
185C3×C4×C5—82no—
186C3×C4×C5#9911yescharacteristic
187C3×C4×C5#9911yescharacteristic
188C3×Dic5—22no—
189C3×Dic5—41no—
190C3×Dic5—41no—
191C3×Dic5—82no—
192C3×#41—41no—
193C3×#41—41no—
194C3×#41—201no—
195C3×#41—201no—
196C2×C3×Q8—51no—
197C2×C3×Q8C4×D511yescharacteristic
198C2×C3×D4—51no—
199C2×C3×D4—101no—
200C2×C3×D4—101no—
201C2×C3×D4—101no—
202C2×C3×D4D2011yescharacteristic
203C2×C3×D4C22⋊D511yescharacteristic
204C2×C3×D4C22×D511yescharacteristic
205C22×C3×C4—101no—
206C22×C3×C4—101no—
207C22×C3×C4C4×D511yescharacteristic
208C22×C3×C4D2011yescharacteristic
209C22×C3×C4C22⋊D511yescharacteristic
210C3×C4⋊C4—41no—
211C3×C4⋊C4—101no—
212C3×C4⋊C4—101no—
213C3×C4⋊C4—201no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—101no—
216C3×C22⋊C4—101no—
217C3×C22⋊C4—101no—
218C3×C22⋊C4—101no—
219C3×C22⋊C4—102no—
220C3×C42—41no—
221C3×C42—51no—
222C3×C42—51no—
223C3×C42—101no—
224C3×C42—201no—
225C23×C5C3×C22⋊C411yescharacteristic
226C23×C5C2×C3×D411yescharacteristic
227C23×C5C2×C3×D411yescharacteristic
228C22×D5—21no—
229C5×Q8—41no—
230C5×D4—41no—
231C5×D4—41no—
232C5×D4—41no—
233C2×C4×C5—21no—
234C2×C4×C5—21no—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—21no—
239C2×C4×C5—42no—
240C2×C4×C5C3×C22⋊C411yescharacteristic
241C2×C4×C5C3×C22⋊C411yescharacteristic
242C2×C4×C5C3×C22⋊C411yescharacteristic
243C2×C4×C5C22×C3×C411yescharacteristic
244C22⋊D5—42no—
245C22⋊D5—42no—
246C22⋊D5—42no—
247C2×Dic5—21no—
248C2×Dic5—21no—
249C2×Dic5—21no—
250C2×Dic5—42no—
251C2×Dic5—42no—
252D20—41no—
253C4×D5—41no—
254C4×D5—41no—
255Dic10—41no—
256C2×#41—51no—
257C2×#41C2×C3×D511yescharacteristic
258C4⋊Q8—51no—
259#127—51no—
260#124—51no—
261#124—51no—
262#121—101no—
263#121—101no—
264C4×D4—101no—
265C4×D4—101no—
266#117—21no—
267#117—101no—
268C2×C42—51no—
269#99—21no—
270#99—52no—
271#99—101no—
272#99C2×C3×D512yes—
273C2×C3×C5—21no—
274C2×C3×C5—41no—
275C2×C3×C5—41no—
276C2×C3×C5—41no—
277C2×C3×C5C2×#9911yescharacteristic
278C3×D5—81no—
279C23×C3—101no—
280C23×C3#68511yescharacteristic
281C23×C3C2×D2011yescharacteristic
282C23×C3C2×C22⋊D511yescharacteristic
283C3×Q8—41no—
284C3×Q8—201no—
285C3×D4—41no—
286C3×D4—41no—
287C3×D4—41no—
288C3×D4—201no—
289C3×D4—202no—
290C3×D4—202no—
291C3×D4—202no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—42no—
299C2×C3×C4—101no—
300C2×C3×C4—101no—
301C2×C3×C4—101no—
302C2×C3×C4—202no—
303C2×C3×C4—201no—
304C2×C3×C4—201no—
305C2×C3×C4—202no—
306C2×C3×C4#68511yescharacteristic
307C2×C3×C4#68511yescharacteristic
308C2×C3×C4#68511yescharacteristic
309C2×C3×C4C2×C4×D511yescharacteristic
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—41no—
314C22×C5—41no—
315C22×C5—41no—
316C22×C5C2×C3×C22⋊C411yescharacteristic
317C2×D5—41no—
318C2×D5—41no—
319C2×D5—41no—
320C4×C5—21no—
321C4×C5—21no—
322C4×C5—21no—
323C4×C5—82no—
324C4×C5C3×#9911yescharacteristic
325C4×C5C3×#9911yescharacteristic
326Dic5—22no—
327Dic5—41no—
328Dic5—41no—
329Dic5—82no—
330#41—41no—
331#41—41no—
332#41—201no—
333#41—201no—
334C2×Q8—51no—
335C2×Q8C3×C4×D511yescharacteristic
336C2×D4—51no—
337C2×D4—101no—
338C2×D4—101no—
339C2×D4—101no—
340C2×D4C3×D2011yescharacteristic
341C2×D4C3×C22⋊D511yescharacteristic
342C2×D4C22×C3×D511yescharacteristic
343C22×C4—101no—
344C22×C4—101no—
345C22×C4C3×C4×D511yescharacteristic
346C22×C4C3×D2011yescharacteristic
347C22×C4C3×C22⋊D511yescharacteristic
348C4⋊C4—41no—
349C4⋊C4—101no—
350C4⋊C4—101no—
351C4⋊C4—201no—
352C22⋊C4—22no—
353C22⋊C4—101no—
354C22⋊C4—101no—
355C22⋊C4—101no—
356C22⋊C4—101no—
357C22⋊C4—102no—
358C42—41no—
359C42—51no—
360C42—51no—
361C42—101no—
362C42—201no—
363C3×C5#212611yescharacteristic, complemented, Hall
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—201no—
371C22×C3—201no—
372C22×C3—201no—
373C22×C3C2×#68511yescharacteristic
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—82no—
378C3×C4—102no—
379C3×C4—201no—
380C3×C4—201no—
381C3×C4—402no—
382C3×C4#393811yescharacteristic
383C3×C4#393811yescharacteristic
384C2×C5—21no—
385C2×C5—41no—
386C2×C5—41no—
387C2×C5—41no—
388C2×C5C2×C3×#9911yescharacteristic
389D5—81no—
390C23—101no—
391C23C3×#68511yescharacteristic
392C23C2×C3×D2011yescharacteristic
393C23C2×C3×C22⋊D511yescharacteristic
394Q8—41no—
395Q8—201no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—201no—
400D4—202no—
401D4—202no—
402D4—202no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—42no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—202no—
414C2×C4—201no—
415C2×C4—201no—
416C2×C4—202no—
417C2×C4C3×#68511yescharacteristic
418C2×C4C3×#68511yescharacteristic
419C2×C4C3×#68511yescharacteristic
420C2×C4C2×C3×C4×D511yescharacteristic
421C2×C3—21no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—401no—
426C2×C3C2×#393811yescharacteristic
427C5C3×#212611yescharacteristic, complemented, Sylow
428C22—21no—
429C22—21no—
430C22—21no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—201no—
435C22—201no—
436C22—201no—
437C22C2×C3×#68511yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—102no—
443C4—201no—
444C4—201no—
445C4—402no—
446C4C3×#393811yescharacteristic
447C4C3×#393811yescharacteristic
448C3#1060085311yescharacteristic, direct product factor, Sylow
449C2—21no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2—401no—
454C2C2×C3×#393811yescharacteristic
subgroupG/H
centerC3×C4#3938
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×D4C22×C3×D5
FittingC3×C5×#410C2
socleC2×C3×C5C2×#99
prime radicalC3×#66166C2
cosocleC2×C5×D4C23×C3
hypercenterC3×#410D5
hypocenterC5C3×#2126
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
2. C3×C5×#410C2semidirect
3. C3×#65529C2semidirect
4. C3×#64694C2semidirect
5. C3×#64693C2semidirect
6. #10600853C3direct (N × H)
8. C2×C3×#709C4semidirect
10. C3×#4093C4semidirect
11. C3×#4090C4semidirect
19. C2×C3×C4×Dic5C4semidirect
26. C5×#410C2×C3semidirect
27. #65529C2×C3semidirect
28. #64694C2×C3semidirect
29. #64693C2×C3semidirect
65. C3×#410D5semidirect
69. C2×#709C3×C4semidirect
71. #4093C3×C4semidirect
72. #4090C3×C4semidirect
80. C2×C4×Dic5C3×C4semidirect
169. #410C3×D5semidirect
363. C3×C5#2126semidirect
427. C5C3×#2126semidirect
448. C3#10600853direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661662
C3×C5×#4102
C3×#655292
C3×#646942
C3×#646932
#106008533
C3×#21265