Finite Group Databasebeta

← C3×#10599090 #14288390

Subgroups — C3×#10599090 #14288390

Order 1920.

508 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65931C211yesmaximal, characteristic
2C3×#65322C211yesmaximal, characteristic, complemented
3C3×#65025C211yesmaximal, characteristic, complemented
4C3×#64950C211yesmaximal, characteristic, complemented
5C3×C5×#327C211yesmaximal, characteristic, complemented
6C3×#64689C211yesmaximal, characteristic, complemented
7C3×#64671C211yesmaximal, characteristic, complemented
8#10599090C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×#711—21no—
10C3×C5×#121C2211yescharacteristic
11C3×#4085—21no—
12C3×#4085—21no—
13C3×#4076C2211yescharacteristic
14C2×C3×C4⋊Dic5—21no—
15C2×C3×C4⋊Dic5C2211yescharacteristic
16C2×C3×#683C2211yescharacteristic
17C3×#4042—21no—
18C3×#4030—21no—
19C3×#4024—21no—
20C3×C5×C22⋊C8C2211yescharacteristic
21C3×C5×#95C2211yescharacteristic
22C3×#3964—21no—
23C3×#3963—21no—
24C3×C22⋊#168C2211yescharacteristic
25C3×#3953—21no—
26C3×#3950—21no—
27C3×#3939—21no—
28C3×#1610—51nomaximal, Hall
29#65931C2×C311yescharacteristic
30#65322C2×C311yescharacteristic, complemented
31#65025C2×C311yescharacteristic, complemented
32#64950C2×C311yescharacteristic, complemented
33C5×#327C2×C311yescharacteristic, complemented
34#64689C2×C311yescharacteristic, complemented
35#64671C2×C311yescharacteristic, complemented
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4D411yescharacteristic
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5C2311yescharacteristic
40C2×C3×C22⋊D5—21no—
41C2×C3×C22⋊D5—21no—
42C2×C3×C22⋊D5—21no—
43C22×C3×Dic5—21no—
44C22×C3×Dic5—21no—
45C22×C3×Dic5D411yescharacteristic
46C22×C3×Dic5D411yescharacteristic
47C3×#711—42no—
48C3×#711—42no—
49C2×C3×D20D411yescharacteristic
50C2×C3×C4×D5—21no—
51C2×C3×Dic10—21no—
52C2×C3×C5×C8—21no—
53C3×C5×C4⋊C4D411yescharacteristic
54C3×C5×C22⋊C4—21no—
55C3×C22⋊Dic5—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×C4⋊Dic5—21no—
59C3×C4⋊Dic5—21no—
60C3×C4⋊Dic5—42no—
61C3×C4⋊Dic5D411yescharacteristic
62C3×#683—21no—
63C3×#683—21no—
64C3×#683—21no—
65C2×C3×#168—21no—
66C3×#537—51no—
67C3×#480—51no—
68C3×#480—51no—
69C3×#398—51no—
70C3×#327—51no—
71C3×#327D511yescharacteristic, complemented
72C3×#324—51no—
73C2×#711—21no—
74C5×#121C22×C311yescharacteristic
75#4085—21no—
76#4085—21no—
77#4076C22×C311yescharacteristic
78C2×C4⋊Dic5—21no—
79C2×C4⋊Dic5C22×C311yescharacteristic
80C2×#683C22×C311yescharacteristic
81#4042—21no—
82#4030—21no—
83#4024—21no—
84C5×C22⋊C8C22×C311yescharacteristic
85C5×#95C22×C311yescharacteristic
86#3964—21no—
87#3963—21no—
88C22⋊#168C22×C311yescharacteristic
89#3953—21no—
90#3950—21no—
91#3939—21no—
92#1610—51noSylow
93C23×C3×C5—21no—
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5—21no—
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C3×C5×D4—41no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C3×C22⋊D5—42no—
107C3×C22⋊D5—41no—
108C3×C22⋊D5—41no—
109C3×C22⋊D5—81no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—42no—
115C2×C3×Dic5—41no—
116C2×C3×Dic5—41no—
117C2×C3×Dic5—41no—
118C2×C3×Dic5—41no—
119C2×C3×Dic5D812yes—
120C3×D20—41no—
121C3×C4×D5—42no—
122C3×Dic10—42no—
123C3×C5×C8—41no—
124C3×#168—41no—
125C2×C3×#41—101no—
126C3×C22⋊Q8—101no—
127C3×C22⋊Q8—101no—
128C3×#121—51no—
129C3×#121—101no—
130C3×#121—101no—
131C3×#121—101no—
132C3×#121C2×D511yescharacteristic
133C2×C3×C4⋊C4—51no—
134C2×C3×C4⋊C4—51no—
135C2×C3×C4⋊C4—101no—
136C3×#107—101no—
137C3×#107—101no—
138C3×#102—101no—
139C3×#102—101no—
140C3×C22⋊C8—51no—
141C3×C22⋊C8C2×D511yescharacteristic
142C3×#95—101no—
143C3×#95C2×D511yescharacteristic
144C2×C5×D4—21no—
145C2×C5×D4C3×D411yescharacteristic
146C22×C4×C5—21no—
147C22×C4×C5C23×C311yescharacteristic
148C2×C22⋊D5—21no—
149C2×C22⋊D5—21no—
150C2×C22⋊D5—21no—
151C22×Dic5—21no—
152C22×Dic5—21no—
153C22×Dic5C3×D411yescharacteristic
154C22×Dic5C3×D411yescharacteristic
155#711—42no—
156#711—42no—
157C2×D20C3×D411yescharacteristic
158C2×C4×D5—21no—
159C2×Dic10—21no—
160C2×C5×C8—21no—
161C5×C4⋊C4C3×D411yescharacteristic
162C5×C22⋊C4—21no—
163C22⋊Dic5—21no—
164#685—21no—
165#685—21no—
166C4⋊Dic5—21no—
167C4⋊Dic5—21no—
168C4⋊Dic5—42no—
169C4⋊Dic5C3×D411yescharacteristic
170#683—21no—
171#683—21no—
172#683—21no—
173C2×#168—21no—
174#537—51no—
175#480—51no—
176#480—51no—
177#398—51no—
178#327—51no—
179#327C3×D511yescharacteristic, complemented
180#324—51no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5—41no—
184C22×C3×C5—41no—
185C22×C3×C5—41no—
186C22×C3×C5#12011yescharacteristic
187C22×C3×C5C2×D811yescharacteristic
188C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
189C2×C3×D5—41no—
190C2×C3×D5—41no—
191C2×C3×D5—41no—
192C3×C4×C5—21no—
193C3×C4×C5—21no—
194C3×C4×C5—42no—
195C3×C4×C5—41no—
196C3×Dic5—22no—
197C3×Dic5—41no—
198C3×Dic5—41no—
199C3×Dic5—81no—
200C3×#41—202no—
201C3×#41—202no—
202C2×C3×Q8—101no—
203C2×C3×D4—21no—
204C2×C3×D4—51no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C2×C3×D4D2011yescharacteristic
209C22×C3×C4—21no—
210C22×C3×C4—51no—
211C22×C3×C4—51no—
212C22×C3×C4—101no—
213C22×C3×C4—101no—
214C22×C3×C4—101no—
215C22×C3×C4C22×D511yescharacteristic
216C2×C3×C8—21no—
217C2×C3×C8—101no—
218C3×C4⋊C4—51no—
219C3×C4⋊C4—101no—
220C3×C4⋊C4—101no—
221C3×C4⋊C4—101no—
222C3×C4⋊C4—101no—
223C3×C4⋊C4—101no—
224C3×C4⋊C4—202no—
225C3×C4⋊C4D2011yescharacteristic
226C3×C22⋊C4—21no—
227C3×C22⋊C4—101no—
228C3×C22⋊C4—101no—
229C3×C22⋊C4—101no—
230C23×C5—21no—
231C23×C5C2×C3×D411yescharacteristic
232C22×D5—21no—
233C5×D4—41no—
234C5×D4—41no—
235C5×D4—41no—
236C2×C4×C5—22no—
237C2×C4×C5—21no—
238C2×C4×C5—41no—
239C2×C4×C5—41no—
240C2×C4×C5—41no—
241C2×C4×C5C2×C3×D411yescharacteristic
242C2×C4×C5C2×C3×D411yescharacteristic
243C22⋊D5—42no—
244C22⋊D5—41no—
245C22⋊D5—41no—
246C22⋊D5—81no—
247C2×Dic5—21no—
248C2×Dic5—21no—
249C2×Dic5—21no—
250C2×Dic5—21no—
251C2×Dic5—42no—
252C2×Dic5—41no—
253C2×Dic5—41no—
254C2×Dic5—41no—
255C2×Dic5—41no—
256C2×Dic5C3×D812yes—
257D20—41no—
258C4×D5—42no—
259Dic10—42no—
260C5×C8—41no—
261#168—41no—
262C2×#41—101no—
263C22⋊Q8—101no—
264C22⋊Q8—101no—
265#121—51no—
266#121—101no—
267#121—101no—
268#121—101no—
269#121C2×C3×D511yescharacteristic
270C2×C4⋊C4—51no—
271C2×C4⋊C4—51no—
272C2×C4⋊C4—101no—
273#107—101no—
274#107—101no—
275#102—101no—
276#102—101no—
277C22⋊C8—51no—
278C22⋊C8C2×C3×D511yescharacteristic
279#95—101no—
280#95C2×C3×D511yescharacteristic
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5—81no—
284C2×C3×C5#44711yescharacteristic
285C2×C3×C5#45611yescharacteristic
286C2×C3×C5#45811yescharacteristic
287C3×D5—81no—
288C23×C3—21no—
289C23×C3—101no—
290C23×C3D4×D511yescharacteristic
291C3×Q8—202no—
292C3×D4—41no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—201no—
296C3×D4—202no—
297C3×D4—201no—
298C3×D4—201no—
299C3×D4—401no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—52no—
306C2×C3×C4—101no—
307C2×C3×C4—101no—
308C2×C3×C4—101no—
309C2×C3×C4—101no—
310C2×C3×C4—202no—
311C2×C3×C4—202no—
312C2×C3×C4—201no—
313C2×C3×C4—201no—
314C2×C3×C4—201no—
315C2×C3×C4—201no—
316C2×C3×C4C2×D2011yescharacteristic
317C2×C3×C4D4×D511yescharacteristic
318C3×C8—41no—
319C3×C8—201no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—41no—
323C22×C5—41no—
324C22×C5—41no—
325C22×C5C3×#12011yescharacteristic
326C22×C5C2×C3×D811yescharacteristic
327C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
328C2×D5—41no—
329C2×D5—41no—
330C2×D5—41no—
331C4×C5—21no—
332C4×C5—21no—
333C4×C5—42no—
334C4×C5—41no—
335Dic5—22no—
336Dic5—41no—
337Dic5—41no—
338Dic5—81no—
339#41—202no—
340#41—202no—
341C2×Q8—101no—
342C2×D4—21no—
343C2×D4—51no—
344C2×D4—101no—
345C2×D4—101no—
346C2×D4—101no—
347C2×D4C3×D2011yescharacteristic
348C22×C4—21no—
349C22×C4—51no—
350C22×C4—51no—
351C22×C4—101no—
352C22×C4—101no—
353C22×C4—101no—
354C22×C4C22×C3×D511yescharacteristic
355C2×C8—21no—
356C2×C8—101no—
357C4⋊C4—51no—
358C4⋊C4—101no—
359C4⋊C4—101no—
360C4⋊C4—101no—
361C4⋊C4—101no—
362C4⋊C4—101no—
363C4⋊C4—202no—
364C4⋊C4C3×D2011yescharacteristic
365C22⋊C4—21no—
366C22⋊C4—101no—
367C22⋊C4—101no—
368C22⋊C4—101no—
369C3×C5#161011yescharacteristic, complemented, Hall
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—201no—
376C22×C3—201no—
377C22×C3—201no—
378C22×C3#402811yescharacteristic
379C22×C3D5×D811yescharacteristic
380C22×C3#406011yescharacteristic
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—42no—
384C3×C4—41no—
385C3×C4—102no—
386C3×C4—201no—
387C3×C4—201no—
388C3×C4—401no—
389C2×C5—21no—
390C2×C5—21no—
391C2×C5—81no—
392C2×C5C3×#44711yescharacteristic
393C2×C5C3×#45611yescharacteristic
394C2×C5C3×#45811yescharacteristic
395D5—81no—
396C23—21no—
397C23—101no—
398C23C3×D4×D511yescharacteristic
399Q8—202no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—201no—
404D4—202no—
405D4—201no—
406D4—201no—
407D4—401no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—52no—
414C2×C4—101no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—101no—
418C2×C4—202no—
419C2×C4—202no—
420C2×C4—201no—
421C2×C4—201no—
422C2×C4—201no—
423C2×C4—201no—
424C2×C4C2×C3×D2011yescharacteristic
425C2×C4C3×D4×D511yescharacteristic
426C8—41no—
427C8—201no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—81no—
431C2×C3—401no—
432C2×C3#6503111yescharacteristic
433C2×C3#6506211yescharacteristic
434C2×C3#6511511yescharacteristic
435C5C3×#161011yescharacteristic, complemented, Sylow
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—201no—
442C22—201no—
443C22—201no—
444C22C3×#402811yescharacteristic
445C22C3×D5×D811yescharacteristic
446C22C3×#406011yescharacteristic
447C4—21no—
448C4—21no—
449C4—42no—
450C4—41no—
451C4—102no—
452C4—201no—
453C4—201no—
454C4—401no—
455C3#1059909011yescharacteristic, direct product factor, Sylow
456C2—21no—
457C2—21no—
458C2—81no—
459C2—401no—
460C2C3×#6503111yescharacteristic
461C2C3×#6506211yescharacteristic
462C2C3×#6511511yescharacteristic
subgroupG/H
centerC22×C3#4028
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#327C2
socleC22×C3×C5#120
prime radicalC3×#65931C2
hypercenterC3×#327D5
hypocenterC5C3×#1610
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#327

Complements

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

NH ≅ G/Ntype
2. C3×#65322C2semidirect
3. C3×#65025C2semidirect
4. C3×#64950C2semidirect
5. C3×C5×#327C2semidirect
6. C3×#64689C2semidirect
7. C3×#64671C2semidirect
8. #10599090C3direct (N × H)
30. #65322C2×C3semidirect
31. #65025C2×C3semidirect
32. #64950C2×C3semidirect
33. C5×#327C2×C3semidirect
34. #64689C2×C3semidirect
35. #64671C2×C3semidirect
71. C3×#327D5semidirect
179. #327C3×D5semidirect
369. C3×C5#1610semidirect
435. C5C3×#1610semidirect
455. C3#10599090direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659312
C3×#653222
C3×#650252
C3×#649502
C3×C5×#3272
C3×#646892
C3×#646712
#105990903
C3×#16105