Finite Group Databasebeta

← C3×#10600355 #14299785

Subgroups — C3×#10600355 #14299785

Order 1920.

572 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65838C211yesmaximal, characteristic
2C3×C5×#382C211yesmaximal, characteristic, complemented
3C3×#65255C211yesmaximal, characteristic
4C3×#65253C211yesmaximal, characteristic, complemented
5C2×C3×#3937C211yesmaximal, characteristic
6C3×#64750C211yesmaximal, characteristic, complemented
7C3×#64743C211yesmaximal, characteristic, complemented
8#10600355C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×#709C2211yescharacteristic
10C2×C3×C5×C4⋊C4C2211yescharacteristic
11C2×C3×C5×C4⋊C4C2211yescharacteristic
12C2×C3×C42×C5C2211yescharacteristic
13C3×C4×C22⋊D5—22no—
14C3×#4072C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17C3×#4039—21no—
18C3×#4038—21no—
19C3×D5×C4⋊C4—21no—
20C3×#4033—21no—
21C3×C5×#95—22no—
22C3×#3942—21no—
23C3×#3941—21no—
24C3×#3940—21no—
25C3×#3939—21no—
26C3×#3937C414yescomplemented
27C3×#1860—51nomaximal, Hall
28#65838C2×C311yescharacteristic
29C5×#382C2×C311yescharacteristic, complemented
30#65255C2×C311yescharacteristic
31#65253C2×C311yescharacteristic, complemented
32C2×#3937C2×C311yescharacteristic
33#64750C2×C311yescharacteristic, complemented
34#64743C2×C311yescharacteristic, complemented
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5D412yes—
38C22×C3×C4×C5C2311yescharacteristic
39C2×C3×C22⋊D5—21no—
40C3×#709—21no—
41C3×#709—21no—
42C3×#709C2×C412yes—
43C3×#709Q812yes—
44C2×C3×D20D411yescharacteristic
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5—22no—
47C2×C3×Dic10D411yescharacteristic
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C42×C5—21no—
53C3×C42×C5C2×C412yescomplemented
54C3×C22⋊Dic5—21no—
55C3×#685—22no—
56C3×C4⋊Dic5D411yescharacteristic
57C3×C4⋊Dic5D411yescharacteristic
58C3×#683—22no—
59C3×C4×Dic5—22no—
60C3×C4×Dic5—21no—
61C3×#681—21no—
62C3×#681—21no—
63C3×#681—21no—
64C3×#681C2×C412yescomplemented
65C2×C3×#168—21no—
66C2×C3×#168—21no—
67C3×#520—51no—
68C3×#428—51no—
69C2×C3×#104—51no—
70C3×#417—51no—
71C3×#382D511yescharacteristic, complemented
72C3×#343—51no—
73C3×#339—51no—
74C2×#709C22×C311yescharacteristic
75C2×C5×C4⋊C4C22×C311yescharacteristic
76C2×C5×C4⋊C4C22×C311yescharacteristic
77C2×C42×C5C22×C311yescharacteristic
78C4×C22⋊D5—22no—
79#4072C22×C311yescharacteristic
80C2×#681C22×C311yescharacteristic
81C2×#681C22×C311yescharacteristic
82#4039—21no—
83#4038—21no—
84D5×C4⋊C4—21no—
85#4033—21no—
86C5×#95—22no—
87#3942—21no—
88#3941—21no—
89#3940—21no—
90#3939—21no—
91#3937C3×C414yescomplemented
92#1860—51noSylow
93C23×C3×C5C2×D411yescharacteristic
94C22×C3×D5—21no—
95C2×C3×C4×C5—22no—
96C2×C3×C4×C5—22no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—42no—
100C2×C3×C4×C5—42no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5C22×C411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×Q811yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C3×C22⋊D5—22no—
109C3×C22⋊D5—22no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C3×D20—21no—
114C3×D20C4⋊C412yes—
115C3×C4×D5—22no—
116C3×C4×D5—22no—
117C3×C4×D5—82no—
118C3×Dic10—21no—
119C3×Dic10C4⋊C412yes—
120C3×#168—22no—
121C3×#168—41no—
122C2×C3×#41—51no—
123C2×C3×M16—51no—
124C2×C3×M16—51no—
125C3×C4×Q8—101no—
126C3×C4×D4—101no—
127C3×C4×D4—102no—
128C3×#117—51no—
129C3×#117—101no—
130C2×C3×C4⋊C4—101no—
131C2×C3×C4⋊C4C2×D511yescharacteristic
132C2×C3×C4⋊C4C2×D511yescharacteristic
133C2×C3×C42C2×D511yescharacteristic
134C3×#107—101no—
135C3×#106—101no—
136C3×#104—54no—
137C3×#103—101no—
138C3×#102—101no—
139C3×#95—22no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5C3×D412yes—
143C22×C4×C5C23×C311yescharacteristic
144C2×C22⋊D5—21no—
145#709—21no—
146#709—21no—
147#709C2×C3×C412yes—
148#709C3×Q812yes—
149C2×D20C3×D411yescharacteristic
150C2×C4×D5—21no—
151C2×C4×D5—22no—
152C2×Dic10C3×D411yescharacteristic
153C5×C4⋊C4—21no—
154C5×C4⋊C4—21no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4—21no—
157C42×C5—21no—
158C42×C5C2×C3×C412yescomplemented
159C22⋊Dic5—21no—
160#685—22no—
161C4⋊Dic5C3×D411yescharacteristic
162C4⋊Dic5C3×D411yescharacteristic
163#683—22no—
164C4×Dic5—22no—
165C4×Dic5—21no—
166#681—21no—
167#681—21no—
168#681—21no—
169#681C2×C3×C412yescomplemented
170C2×#168—21no—
171C2×#168—21no—
172#520—51no—
173#428—51no—
174C2×#104—51no—
175#417—51no—
176#382C3×D511yescharacteristic, complemented
177#343—51no—
178#339—51no—
179C22×C3×C5—21no—
180C22×C3×C5—21no—
181C22×C3×C5C4×D411yescharacteristic
182C22×C3×C5#12011yescharacteristic
183C22×C3×C5C22⋊Q811yescharacteristic
184C2×C3×D5—22no—
185C2×C3×D5—41no—
186C2×C3×D5—41no—
187C3×C4×C5—42no—
188C3×C4×C5—42no—
189C3×C4×C5—81no—
190C3×C4×C5C2×C4⋊C411yescharacteristic
191C3×C4×C5C4×D411yescharacteristic
192C3×C4×C5C22⋊Q811yescharacteristic
193C3×C4×C5#12311yescharacteristic
194C3×Dic5—22no—
195C3×Dic5—41no—
196C3×Dic5—41no—
197C3×#41—52no—
198C3×#41—52no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8—51no—
202C2×C3×D4—51no—
203C2×C3×D4—101no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—102no—
207C22×C3×C4—101no—
208C22×C3×C4C22⋊D512yes—
209C22×C3×C4C22×D511yescharacteristic
210C3×M16—52no—
211C3×M16—101no—
212C3×M16—101no—
213C3×M16—101no—
214C2×C3×C8—101no—
215C2×C3×C8—101no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—51no—
221C3×C4⋊C4—51no—
222C3×C4⋊C4—102no—
223C3×C22⋊C4—102no—
224C3×C22⋊C4—101no—
225C3×C42—21no—
226C3×C42—102no—
227C3×C42—101no—
228C3×C42C4×D512yescomplemented
229C23×C5C2×C3×D411yescharacteristic
230C22×D5—21no—
231C2×C4×C5—22no—
232C2×C4×C5—22no—
233C2×C4×C5—41no—
234C2×C4×C5—41no—
235C2×C4×C5—42no—
236C2×C4×C5—42no—
237C2×C4×C5—41no—
238C2×C4×C5C22×C3×C411yescharacteristic
239C2×C4×C5C2×C3×D411yescharacteristic
240C2×C4×C5C2×C3×D411yescharacteristic
241C2×C4×C5C2×C3×Q811yescharacteristic
242C2×C4×C5C3×#4111yescharacteristic
243C2×C4×C5C3×#4111yescharacteristic
244C22⋊D5—22no—
245C22⋊D5—22no—
246C2×Dic5—21no—
247C2×Dic5—21no—
248C2×Dic5—21no—
249D20—21no—
250D20C3×C4⋊C412yes—
251C4×D5—22no—
252C4×D5—22no—
253C4×D5—82no—
254Dic10—21no—
255Dic10C3×C4⋊C412yes—
256#168—22no—
257#168—41no—
258C2×#41—51no—
259C2×M16—51no—
260C2×M16—51no—
261C4×Q8—101no—
262C4×D4—101no—
263C4×D4—102no—
264#117—51no—
265#117—101no—
266C2×C4⋊C4—101no—
267C2×C4⋊C4C2×C3×D511yescharacteristic
268C2×C4⋊C4C2×C3×D511yescharacteristic
269C2×C42C2×C3×D511yescharacteristic
270#107—101no—
271#106—101no—
272#104—54no—
273#103—101no—
274#102—101no—
275#95—22no—
276C2×C3×C5—21no—
277C2×C3×C5—21no—
278C2×C3×C5#38511yescharacteristic
279C2×C3×C5#45411yescharacteristic
280C2×C3×C5#45511yescharacteristic
281C3×D5—42no—
282C23×C3—101no—
283C23×C3C2×C22⋊D511yescharacteristic
284C3×Q8—52no—
285C3×Q8—101no—
286C3×D4—52no—
287C3×D4—102no—
288C3×D4—102no—
289C3×D4—101no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—101no—
298C2×C3×C4—102no—
299C2×C3×C4—102no—
300C2×C3×C4—101no—
301C2×C3×C4—101no—
302C2×C3×C4—402no—
303C2×C3×C4C2×C4×D511yescharacteristic
304C2×C3×C4#70911yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4D4×D511yescharacteristic
307C2×C3×C4Q8×D511yescharacteristic
308C2×C3×C4#71311yescharacteristic
309C3×C8—102no—
310C3×C8—201no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5C3×C4×D411yescharacteristic
314C22×C5C3×#12011yescharacteristic
315C22×C5C3×C22⋊Q811yescharacteristic
316C2×D5—22no—
317C2×D5—41no—
318C2×D5—41no—
319C4×C5—42no—
320C4×C5—42no—
321C4×C5—81no—
322C4×C5C2×C3×C4⋊C411yescharacteristic
323C4×C5C3×C4×D411yescharacteristic
324C4×C5C3×C22⋊Q811yescharacteristic
325C4×C5C3×#12311yescharacteristic
326Dic5—22no—
327Dic5—41no—
328Dic5—41no—
329#41—52no—
330#41—52no—
331#41—101no—
332#41—101no—
333C2×Q8—51no—
334C2×D4—51no—
335C2×D4—101no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—102no—
339C22×C4—101no—
340C22×C4C3×C22⋊D512yes—
341C22×C4C22×C3×D511yescharacteristic
342M16—52no—
343M16—101no—
344M16—101no—
345M16—101no—
346C2×C8—101no—
347C2×C8—101no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—51no—
353C4⋊C4—51no—
354C4⋊C4—102no—
355C22⋊C4—102no—
356C22⋊C4—101no—
357C42—21no—
358C42—102no—
359C42—101no—
360C42C3×C4×D512yescomplemented
361C3×C5#186011yescharacteristic, complemented, Hall
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—102no—
365C22×C3—201no—
366C22×C3—201no—
367C22×C3C4×C22⋊D511yescharacteristic
368C22×C3#408311yescharacteristic
369C22×C3#409211yescharacteristic
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—81no—
373C3×C4—102no—
374C3×C4—201no—
375C3×C4—201no—
376C3×C4D5×C4⋊C411yescharacteristic
377C3×C4#403911yescharacteristic
378C3×C4#404011yescharacteristic
379C3×C4#404211yescharacteristic
380C2×C5—21no—
381C2×C5—21no—
382C2×C5C3×#38511yescharacteristic
383C2×C5C3×#45411yescharacteristic
384C2×C5C3×#45511yescharacteristic
385D5—42no—
386C23—101no—
387C23C2×C3×C22⋊D511yescharacteristic
388Q8—52no—
389Q8—101no—
390D4—52no—
391D4—102no—
392D4—102no—
393D4—101no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—101no—
402C2×C4—102no—
403C2×C4—102no—
404C2×C4—101no—
405C2×C4—101no—
406C2×C4—402no—
407C2×C4C2×C3×C4×D511yescharacteristic
408C2×C4C3×#70911yescharacteristic
409C2×C4C3×D4×D511yescharacteristic
410C2×C4C3×D4×D511yescharacteristic
411C2×C4C3×Q8×D511yescharacteristic
412C2×C4C3×#71311yescharacteristic
413C8—102no—
414C8—201no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—202no—
418C2×C3#6527811yescharacteristic
419C2×C3#6535011yescharacteristic
420C2×C3#6535111yescharacteristic
421C5C3×#186011yescharacteristic, complemented, Sylow
422C22—21no—
423C22—21no—
424C22—102no—
425C22—201no—
426C22—201no—
427C22C3×C4×C22⋊D511yescharacteristic
428C22C3×#408311yescharacteristic
429C22C3×#409211yescharacteristic
430C4—42no—
431C4—42no—
432C4—81no—
433C4—102no—
434C4—201no—
435C4—201no—
436C4C3×D5×C4⋊C411yescharacteristic
437C4C3×#403911yescharacteristic
438C4C3×#404011yescharacteristic
439C4C3×#404211yescharacteristic
440C3#1060035511yescharacteristic, direct product factor, Sylow
441C2—21no—
442C2—21no—
443C2—202no—
444C2C3×#6527811yescharacteristic
445C2C3×#6535011yescharacteristic
446C2C3×#6535111yescharacteristic
subgroupG/H
centerC22×C3#4083
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#382C2
socleC22×C3×C5#120
prime radicalC2×C3×#709C22
cosocleC22×C4×C5C23×C3
hypercenterC3×#382D5
hypocenterC5C3×#1860
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#382

Complements

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

NH ≅ G/Ntype
2. C3×C5×#382C2semidirect
4. C3×#65253C2semidirect
6. C3×#64750C2semidirect
7. C3×#64743C2semidirect
8. #10600355C3direct (N × H)
26. C3×#3937C4semidirect
29. C5×#382C2×C3semidirect
31. #65253C2×C3semidirect
33. #64750C2×C3semidirect
34. #64743C2×C3semidirect
53. C3×C42×C5C2×C4semidirect
64. C3×#681C2×C4semidirect
71. C3×#382D5semidirect
91. #3937C3×C4semidirect
158. C42×C5C2×C3×C4semidirect
169. #681C2×C3×C4semidirect
176. #382C3×D5semidirect
228. C3×C42C4×D5semidirect
360. C42C3×C4×D5semidirect
361. C3×C5#1860semidirect
421. C5C3×#1860semidirect
440. C3#10600355direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658382
C3×C5×#3822
C3×#652552
C3×#652532
C2×C3×#39372
C3×#647502
C3×#647432
#106003553
C3×#18605