Finite Group Databasebeta

← C3×#10601020 #14305876

Subgroups — C3×#10601020 #14305876

Order 1920.

688 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
1C2×C3×#4093C211yesmaximal, characteristic
2C2×C3×C5×#103C211yesmaximal, characteristic, complemented
3C2×C3×#3967C211yesmaximal, characteristic, complemented
4C2×C3×#3953C211yesmaximal, characteristic
5C2×C3×#3947C211yesmaximal, characteristic, complemented
6C3×#65265C211yesmaximal, characteristic, complemented
7C2×C3×#3941C211yesmaximal, characteristic
8#10601020C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C3×#4093—22no—
14C3×#4093C414yescomplemented
15C2×C3×#685—22no—
16C2×C3×C4⋊Dic5C2211yescharacteristic
17C2×C3×#683—22no—
18C2×C3×C4×Dic5C2211yescharacteristic
19C22×C3×#168C2211yescharacteristic
20C3×C5×#103—22no—
21C3×#3967—22no—
22C3×#3953—22no—
23C3×#3947—22no—
24C3×#3941—22no—
25C3×#1959—51nomaximal, Hall
26C2×#4093C2×C311yescharacteristic
27C2×C5×#103C2×C311yescharacteristic, complemented
28C2×#3967C2×C311yescharacteristic, complemented
29C2×#3953C2×C311yescharacteristic
30C2×#3947C2×C311yescharacteristic, complemented
31#65265C2×C311yescharacteristic, complemented
32C2×#3941C2×C311yescharacteristic
33C23×C3×D5—21no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8C2×C412yes—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×Dic5—21no—
44C22×C3×Dic5D412yes—
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20—21no—
48C2×C3×D20—21no—
49C2×C3×D20—21no—
50C2×C3×D20C2×C412yes—
51C2×C3×C5×C8—21no—
52C2×C3×C5×C8—21no—
53C2×C3×C5×C8D412yes—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4—21no—
56C3×#685—28no—
57C3×C4⋊Dic5—21no—
58C3×C4⋊Dic5—21no—
59C3×#683—44no—
60C3×C4×Dic5C2×C412yes—
61C3×C4×Dic5D412yes—
62C2×C3×#168—21no—
63C2×C3×#168—21no—
64C2×C3×#168D412yes—
65C2×C3×#124—51no—
66C2×C3×#103—51no—
67C2×C3×#103D511yescharacteristic, complemented
68C2×C3×#102—51no—
69C2×C3×#102—51no—
70C2×C3×#97—51no—
71C3×#384—51no—
72C22×C5×Q8C22×C311yescharacteristic
73C22×D20C22×C311yescharacteristic
74C22×C5×C8C22×C311yescharacteristic
75C2×C5×C4⋊C4C22×C311yescharacteristic
76#4093—22no—
77#4093C3×C414yescomplemented
78C2×#685—22no—
79C2×C4⋊Dic5C22×C311yescharacteristic
80C2×#683—22no—
81C2×C4×Dic5C22×C311yescharacteristic
82C22×#168C22×C311yescharacteristic
83C5×#103—22no—
84#3967—22no—
85#3953—22no—
86#3947—22no—
87#3941—22no—
88#1959—51noSylow
89C23×C3×C5C2×D411yescharacteristic
90C22×C3×D5—22no—
91C22×C3×D5—41no—
92C22×C3×D5—41no—
93C22×C3×D5—42no—
94C22×C3×D5—42no—
95C3×C5×Q8—22no—
96C3×C5×Q8—22no—
97C3×C5×Q8—41no—
98C3×C5×Q8—41no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—42no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×Dic5—22no—
111C2×C3×Dic5—22no—
112C2×C3×Dic5—41no—
113C2×C3×Dic5—42no—
114C2×C3×Dic5C22⋊C414yes—
115C3×D20—22no—
116C3×D20—22no—
117C3×D20—41no—
118C3×D20—41no—
119C3×C5×C8—22no—
120C3×#168—22no—
121C22×C3×Q8C2×D511yescharacteristic
122C22×C3×D4—51no—
123C22×C3×C8—51no—
124C22×C3×C8C2×D511yescharacteristic
125C3×#124—54no—
126C3×#124—102no—
127C2×C3×C4⋊C4—51no—
128C2×C3×C4⋊C4—102no—
129C2×C3×C4⋊C4C2×D511yescharacteristic
130C2×C3×C22⋊C4—102no—
131C2×C3×C42—51no—
132C3×#103—22no—
133C3×#103—102no—
134C3×#102—102no—
135C3×#102—102no—
136C3×#97—102no—
137C23×D5—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8C2×C3×C412yes—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5C23×C311yescharacteristic
147C22×Dic5—21no—
148C22×Dic5C3×D412yes—
149C2×D20—21no—
150C2×D20—21no—
151C2×D20—21no—
152C2×D20—21no—
153C2×D20—21no—
154C2×D20C2×C3×C412yes—
155C2×C5×C8—21no—
156C2×C5×C8—21no—
157C2×C5×C8C3×D412yes—
158C5×C4⋊C4—21no—
159C5×C4⋊C4—21no—
160#685—28no—
161C4⋊Dic5—21no—
162C4⋊Dic5—21no—
163#683—44no—
164C4×Dic5C2×C3×C412yes—
165C4×Dic5C3×D412yes—
166C2×#168—21no—
167C2×#168—21no—
168C2×#168C3×D412yes—
169C2×#124—51no—
170C2×#103—51no—
171C2×#103C3×D511yescharacteristic, complemented
172C2×#102—51no—
173C2×#102—51no—
174C2×#97—51no—
175#384—51no—
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C22×C3×C5#12411yescharacteristic
178C22×C3×C5#12711yescharacteristic
179C22×C3×C5AGL(1,ℤ/8)12yes—
180C22×C3×C5#13712yes—
181C2×C3×D5—42no—
182C2×C3×D5—44no—
183C2×C3×D5—82no—
184C2×C3×D5—81no—
185C2×C3×D5—81no—
186C3×C4×C5—42no—
187C3×C4×C5—81no—
188C3×C4×C5C4×D411yescharacteristic
189C3×C4×C5C4×D411yescharacteristic
190C3×C4×C5#12111yescharacteristic
191C3×C4×C5#12111yescharacteristic
192C3×Dic5—24no—
193C3×Dic5—81no—
194C24×C3—101no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C4×D512yes—
201C2×C3×D4—52no—
202C2×C3×D4—101no—
203C2×C3×D4—101no—
204C2×C3×D4—101no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—52no—
210C22×C3×C4—101no—
211C22×C3×C4C22×D511yescharacteristic
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—52no—
215C2×C3×C8—101no—
216C2×C3×C8—101no—
217C2×C3×C8C22⋊D512yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—101no—
221C3×C4⋊C4—101no—
222C3×C4⋊C4—204no—
223C3×C22⋊C4—108no—
224C3×C42—52no—
225C3×C42—52no—
226C23×C5C2×C3×D411yescharacteristic
227C22×D5—22no—
228C22×D5—41no—
229C22×D5—41no—
230C22×D5—42no—
231C22×D5—42no—
232C5×Q8—22no—
233C5×Q8—22no—
234C5×Q8—41no—
235C5×Q8—41no—
236C2×C4×C5—22no—
237C2×C4×C5—41no—
238C2×C4×C5—41no—
239C2×C4×C5—41no—
240C2×C4×C5—42no—
241C2×C4×C5C22×C3×C411yescharacteristic
242C2×C4×C5C2×C3×D411yescharacteristic
243C2×C4×C5C2×C3×D411yescharacteristic
244C2×C4×C5C2×C3×D411yescharacteristic
245C2×C4×C5C3×#4111yescharacteristic
246C2×C4×C5C3×#4111yescharacteristic
247C2×Dic5—22no—
248C2×Dic5—22no—
249C2×Dic5—41no—
250C2×Dic5—42no—
251C2×Dic5C3×C22⋊C414yes—
252D20—22no—
253D20—22no—
254D20—41no—
255D20—41no—
256C5×C8—22no—
257#168—22no—
258C22×Q8C2×C3×D511yescharacteristic
259C22×D4—51no—
260C22×C8—51no—
261C22×C8C2×C3×D511yescharacteristic
262#124—54no—
263#124—102no—
264C2×C4⋊C4—51no—
265C2×C4⋊C4—102no—
266C2×C4⋊C4C2×C3×D511yescharacteristic
267C2×C22⋊C4—102no—
268C2×C42—51no—
269#103—22no—
270#103—102no—
271#102—102no—
272#102—102no—
273#97—102no—
274C2×C3×C5#39011yescharacteristic
275C2×C3×C5#41712yes—
276C2×C3×C5#48912yes—
277C2×C3×C5#49611yescharacteristic
278C2×C3×C5#49711yescharacteristic
279C3×D5—82no—
280C23×C3—102no—
281C23×C3—201no—
282C23×C3—202no—
283C23×C3—202no—
284C23×C3—201no—
285C23×C3D4×D511yescharacteristic
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—41no—
289C3×Q8—41no—
290C3×D4—102no—
291C3×D4—102no—
292C3×D4—201no—
293C3×D4—201no—
294C2×C3×C4—22no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—54no—
300C2×C3×C4—102no—
301C2×C3×C4—102no—
302C2×C3×C4—202no—
303C2×C3×C4—201no—
304C2×C3×C4C2×C4×D511yescharacteristic
305C2×C3×C4#70911yescharacteristic
306C2×C3×C4D4×D511yescharacteristic
307C2×C3×C4D4×D511yescharacteristic
308C2×C3×C4#71111yescharacteristic
309C2×C3×C4C2×C22⋊D511yescharacteristic
310C3×C8—22no—
311C3×C8—102no—
312C22×C5C2×C3×C22⋊C411yescharacteristic
313C22×C5C3×#12411yescharacteristic
314C22×C5C3×#12711yescharacteristic
315C22×C5C3×AGL(1,ℤ/8)12yes—
316C22×C5C3×#13712yes—
317C2×D5—42no—
318C2×D5—44no—
319C2×D5—82no—
320C2×D5—81no—
321C2×D5—81no—
322C4×C5—42no—
323C4×C5—81no—
324C4×C5C3×C4×D411yescharacteristic
325C4×C5C3×C4×D411yescharacteristic
326C4×C5C3×#12111yescharacteristic
327C4×C5C3×#12111yescharacteristic
328Dic5—24no—
329Dic5—81no—
330C24—101no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8C3×C4×D512yes—
337C2×D4—52no—
338C2×D4—101no—
339C2×D4—101no—
340C2×D4—101no—
341C2×D4—101no—
342C2×D4—101no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—52no—
346C22×C4—101no—
347C22×C4C22×C3×D511yescharacteristic
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—52no—
351C2×C8—101no—
352C2×C8—101no—
353C2×C8C3×C22⋊D512yes—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—101no—
357C4⋊C4—101no—
358C4⋊C4—204no—
359C22⋊C4—108no—
360C42—52no—
361C42—52no—
362C3×C5#195911yescharacteristic, complemented, Hall
363C22×C3—204no—
364C22×C3—202no—
365C22×C3—402no—
366C22×C3—401no—
367C22×C3—401no—
368C22×C3D5×C22⋊C411yescharacteristic
369C22×C3#403111yescharacteristic
370C22×C3#406012yes—
371C22×C3#406412yes—
372C22×C3#408611yescharacteristic
373C3×C4—42no—
374C3×C4—81no—
375C3×C4—104no—
376C3×C4—401no—
377C3×C4#402711yescharacteristic
378C3×C4#403011yescharacteristic
379C3×C4C4×C22⋊D511yescharacteristic
380C3×C4#408511yescharacteristic
381C2×C5C3×#39011yescharacteristic
382C2×C5C3×#41712yes—
383C2×C5C3×#48912yes—
384C2×C5C3×#49611yescharacteristic
385C2×C5C3×#49711yescharacteristic
386D5—82no—
387C23—102no—
388C23—201no—
389C23—201no—
390C23—202no—
391C23—202no—
392C23C3×D4×D511yescharacteristic
393Q8—22no—
394Q8—22no—
395Q8—41no—
396Q8—41no—
397D4—102no—
398D4—102no—
399D4—201no—
400D4—201no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—54no—
407C2×C4—102no—
408C2×C4—102no—
409C2×C4—201no—
410C2×C4—202no—
411C2×C4C2×C3×C4×D511yescharacteristic
412C2×C4C3×#70911yescharacteristic
413C2×C4C3×D4×D511yescharacteristic
414C2×C4C3×D4×D511yescharacteristic
415C2×C4C3×#71111yescharacteristic
416C2×C4C2×C3×C22⋊D511yescharacteristic
417C8—22no—
418C8—102no—
419C2×C3—402no—
420C2×C3#6508412yes—
421C2×C3#6509212yes—
422C2×C3#6524611yescharacteristic
423C2×C3#6546511yescharacteristic
424C2×C3#6547911yescharacteristic
425C5C3×#195911yescharacteristic, complemented, Sylow
426C22—204no—
427C22—202no—
428C22—402no—
429C22—401no—
430C22—401no—
431C22C3×D5×C22⋊C411yescharacteristic
432C22C3×#403111yescharacteristic
433C22C3×#406012yes—
434C22C3×#406412yes—
435C22C3×#408611yescharacteristic
436C4—42no—
437C4—81no—
438C4—104no—
439C4—401no—
440C4C3×#402711yescharacteristic
441C4C3×#403011yescharacteristic
442C4C3×C4×C22⋊D511yescharacteristic
443C4C3×#408511yescharacteristic
444C3#1060102011yescharacteristic, direct product factor, Sylow
445C2—402no—
446C2C3×#6508412yes—
447C2C3×#6509212yes—
448C2C3×#6524611yescharacteristic
449C2C3×#6546511yescharacteristic
450C2C3×#6547911yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC2×C3×C5×#103C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
cosocleC22×C4×C5C23×C3
hypercenterC2×C3×#103D5
hypocenterC5C3×#1959
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#103C2semidirect
3. C2×C3×#3967C2semidirect
5. C2×C3×#3947C2semidirect
6. C3×#65265C2semidirect
8. #10601020C3direct (N × H)
14. C3×#4093C4semidirect
27. C2×C5×#103C2×C3semidirect
28. C2×#3967C2×C3semidirect
30. C2×#3947C2×C3semidirect
31. #65265C2×C3semidirect
67. C2×C3×#103D5semidirect
77. #4093C3×C4semidirect
171. C2×#103C3×D5semidirect
362. C3×C5#1959semidirect
425. C5C3×#1959semidirect
444. C3#10601020direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40932
C2×C3×C5×#1032
C2×C3×#39672
C2×C3×#39532
C2×C3×#39472
C3×#652652
C2×C3×#39412
#106010203
C3×#19595