Finite Group Databasebeta

← C3×#10601766 #14288177

Subgroups — C3×#10601766 #14288177

Order 1920.

520 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65951C211yesmaximal, characteristic
2C3×C5×#448C211yesmaximal, characteristic, complemented
3C3×#65520C211yesmaximal, characteristic, complemented
4C3×#65517C211yesmaximal, characteristic, complemented
5C3×#64757C211yesmaximal, characteristic, complemented
6C3×#64688C211yesmaximal, characteristic, complemented
7C3×#64687C211yesmaximal, characteristic, complemented
8#10601766C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×C3×C5×SD16—21no—
11C3×C5×#121C2211yescharacteristic
12C2×C3×#689—21no—
13C3×#4085—21no—
14C3×#4083—21no—
15C3×D4×Dic5—21no—
16C2×C3×#685—21no—
17C2×C3×#685C2211yescharacteristic
18C3×#4070C2211yescharacteristic
19C3×#4044—21no—
20C3×#4032—21no—
21C3×#4031—21no—
22C3×C5×#103—21no—
23C3×C5×C22⋊C8C2211yescharacteristic
24C3×#3967—21no—
25C3×#3963—21no—
26C3×#3963C2211yescharacteristic
27C3×C22⋊#168C2211yescharacteristic
28C3×#1606—51nomaximal, Hall
29#65951C2×C311yescharacteristic
30C5×#448C2×C311yescharacteristic, complemented
31#65520C2×C311yescharacteristic, complemented
32#65517C2×C311yescharacteristic, complemented
33#64757C2×C311yescharacteristic, complemented
34#64688C2×C311yescharacteristic, complemented
35#64687C2×C311yescharacteristic, complemented
36C23×C3×D5D411yescharacteristic
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8—42no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4D411yescharacteristic
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C2×C3×C22⋊D5—21no—
45C2×C3×C22⋊D5—21no—
46C22×C3×Dic5—21no—
47C22×C3×Dic5—21no—
48C22×C3×Dic5D411yescharacteristic
49C2×C3×Dic10D411yescharacteristic
50C3×C5×SD16—42no—
51C2×C3×C5×C8—21no—
52C3×C5×C4⋊C4D411yescharacteristic
53C3×C5×C22⋊C4—21no—
54C3×C22⋊Dic5—21no—
55C3×C22⋊Dic5—21no—
56C3×#689—42no—
57C3×#685—21no—
58C3×#685—21no—
59C3×#685—21no—
60C3×#685—21no—
61C3×#685—42no—
62C3×C4⋊Dic5—21no—
63C3×C4⋊Dic5D411yescharacteristic
64C3×#683—21no—
65C3×C4×Dic5—21no—
66C2×C3×#168—21no—
67C3×#538—51no—
68C3×#451—51no—
69C3×#448D511yescharacteristic, complemented
70C3×#393—51no—
71C3×#328—51no—
72C3×#327—51no—
73C3×#323—51no—
74C22×C5×Q8C22×C311yescharacteristic
75C2×C5×SD16—21no—
76C5×#121C22×C311yescharacteristic
77C2×#689—21no—
78#4085—21no—
79#4083—21no—
80D4×Dic5—21no—
81C2×#685—21no—
82C2×#685C22×C311yescharacteristic
83#4070C22×C311yescharacteristic
84#4044—21no—
85#4032—21no—
86#4031—21no—
87C5×#103—21no—
88C5×C22⋊C8C22×C311yescharacteristic
89#3967—21no—
90#3963—21no—
91#3963C22×C311yescharacteristic
92C22⋊#168C22×C311yescharacteristic
93#1606—51noSylow
94C23×C3×C5—21no—
95C23×C3×C5C2×D411yescharacteristic
96C22×C3×D5—21no—
97C22×C3×D5—21no—
98C22×C3×D5—21no—
99C22×C3×D5—41no—
100C22×C3×D5—41no—
101C3×C5×Q8—22no—
102C3×C5×Q8—22no—
103C3×C5×Q8—41no—
104C3×C5×Q8—41no—
105C3×C5×D4—41no—
106C3×C5×D4—41no—
107C3×C5×D4—41no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C3×C22⋊D5—81no—
116C3×C22⋊D5—81no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—21no—
120C2×C3×Dic5—41no—
121C2×C3×Dic5—41no—
122C2×C3×Dic5—41no—
123C2×C3×Dic5—41no—
124C2×C3×Dic5—81no—
125C3×Dic10—41no—
126C3×C5×C8—41no—
127C3×#168—41no—
128C22×C3×Q8C2×D511yescharacteristic
129C2×C3×Q16—101no—
130C2×C3×SD16—21no—
131C3×#126—101no—
132C3×#124—101no—
133C3×#123—101no—
134C3×C22⋊Q8—51no—
135C3×#121—101no—
136C3×#121C2×D511yescharacteristic
137C3×#120—101no—
138C3×C4×D4—101no—
139C2×C3×C22⋊C4—51no—
140C2×C3×C22⋊C4—101no—
141C3×#103—21no—
142C3×#103—101no—
143C3×C22⋊C8—51no—
144C3×C22⋊C8C2×D511yescharacteristic
145C3×#95—51no—
146C3×#95—101no—
147C23×D5C3×D411yescharacteristic
148C2×C5×Q8—21no—
149C2×C5×Q8—21no—
150C2×C5×Q8—42no—
151C2×C5×D4—21no—
152C2×C5×D4C3×D411yescharacteristic
153C22×C4×C5—21no—
154C22×C4×C5C23×C311yescharacteristic
155C2×C22⋊D5—21no—
156C2×C22⋊D5—21no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5C3×D411yescharacteristic
160C2×Dic10C3×D411yescharacteristic
161C5×SD16—42no—
162C2×C5×C8—21no—
163C5×C4⋊C4C3×D411yescharacteristic
164C5×C22⋊C4—21no—
165C22⋊Dic5—21no—
166C22⋊Dic5—21no—
167#689—42no—
168#685—21no—
169#685—21no—
170#685—21no—
171#685—21no—
172#685—42no—
173C4⋊Dic5—21no—
174C4⋊Dic5C3×D411yescharacteristic
175#683—21no—
176C4×Dic5—21no—
177C2×#168—21no—
178#538—51no—
179#451—51no—
180#448C3×D511yescharacteristic, complemented
181#393—51no—
182#328—51no—
183#327—51no—
184#323—51no—
185C22×C3×C5—21no—
186C22×C3×C5—21no—
187C22×C3×C5—41no—
188C22×C3×C5—41no—
189C22×C3×C5—41no—
190C22×C3×C5#12011yescharacteristic
191C22×C3×C5#13511yescharacteristic
192C22×C3×C5#13711yescharacteristic
193C2×C3×D5—42no—
194C2×C3×D5—41no—
195C2×C3×D5—41no—
196C2×C3×D5—41no—
197C2×C3×D5—81no—
198C3×C4×C5—21no—
199C3×C4×C5—21no—
200C3×C4×C5—41no—
201C3×C4×C5—42no—
202C3×Dic5—41no—
203C3×Dic5—41no—
204C3×Dic5—41no—
205C3×Dic5—81no—
206C24×C3—51no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—42no—
210C2×C3×Q8—51no—
211C2×C3×D4—21no—
212C2×C3×D4—101no—
213C2×C3×D4—101no—
214C2×C3×D4C22⋊D511yescharacteristic
215C22×C3×C4—21no—
216C22×C3×C4—51no—
217C22×C3×C4—101no—
218C22×C3×C4—101no—
219C22×C3×C4C22×D511yescharacteristic
220C3×Q16—202no—
221C3×SD16—42no—
222C2×C3×C8—21no—
223C2×C3×C8—101no—
224C3×C4⋊C4—51no—
225C3×C4⋊C4—101no—
226C3×C4⋊C4—101no—
227C3×C4⋊C4C22⋊D511yescharacteristic
228C3×C22⋊C4—21no—
229C3×C22⋊C4—101no—
230C3×C22⋊C4—101no—
231C3×C22⋊C4—101no—
232C3×C22⋊C4—101no—
233C3×C22⋊C4—101no—
234C3×C22⋊C4—101no—
235C3×C22⋊C4—202no—
236C3×C42—101no—
237C23×C5—21no—
238C23×C5C2×C3×D411yescharacteristic
239C22×D5—21no—
240C22×D5—21no—
241C22×D5—21no—
242C22×D5—41no—
243C22×D5—41no—
244C5×Q8—22no—
245C5×Q8—22no—
246C5×Q8—41no—
247C5×Q8—41no—
248C5×D4—41no—
249C5×D4—41no—
250C5×D4—41no—
251C2×C4×C5—22no—
252C2×C4×C5—21no—
253C2×C4×C5—41no—
254C2×C4×C5—41no—
255C2×C4×C5—41no—
256C2×C4×C5C2×C3×D411yescharacteristic
257C2×C4×C5C2×C3×D411yescharacteristic
258C22⋊D5—81no—
259C22⋊D5—81no—
260C2×Dic5—21no—
261C2×Dic5—21no—
262C2×Dic5—21no—
263C2×Dic5—41no—
264C2×Dic5—41no—
265C2×Dic5—41no—
266C2×Dic5—41no—
267C2×Dic5—81no—
268Dic10—41no—
269C5×C8—41no—
270#168—41no—
271C22×Q8C2×C3×D511yescharacteristic
272C2×Q16—101no—
273C2×SD16—21no—
274#126—101no—
275#124—101no—
276#123—101no—
277C22⋊Q8—51no—
278#121—101no—
279#121C2×C3×D511yescharacteristic
280#120—101no—
281C4×D4—101no—
282C2×C22⋊C4—51no—
283C2×C22⋊C4—101no—
284#103—21no—
285#103—101no—
286C22⋊C8—51no—
287C22⋊C8C2×C3×D511yescharacteristic
288#95—51no—
289#95—101no—
290C2×C3×C5—21no—
291C2×C3×C5—21no—
292C2×C3×C5—81no—
293C2×C3×C5#45211yescharacteristic
294C2×C3×C5#45511yescharacteristic
295C2×C3×C5B(4,𝔽2)11yescharacteristic
296C3×D5—81no—
297C23×C3—21no—
298C23×C3—101no—
299C23×C3—101no—
300C23×C3—101no—
301C23×C3—201no—
302C23×C3—201no—
303C23×C3D4×D511yescharacteristic
304C3×Q8—22no—
305C3×Q8—22no—
306C3×Q8—41no—
307C3×Q8—41no—
308C3×Q8—201no—
309C3×D4—41no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—401no—
313C3×D4—401no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—101no—
320C2×C3×C4—101no—
321C2×C3×C4—101no—
322C2×C3×C4—201no—
323C2×C3×C4—201no—
324C2×C3×C4—201no—
325C2×C3×C4—201no—
326C2×C3×C4—401no—
327C2×C3×C4D4×D511yescharacteristic
328C2×C3×C4C2×C22⋊D511yescharacteristic
329C3×C8—41no—
330C3×C8—201no—
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—41no—
334C22×C5—41no—
335C22×C5—41no—
336C22×C5C3×#12011yescharacteristic
337C22×C5C3×#13511yescharacteristic
338C22×C5C3×#13711yescharacteristic
339C2×D5—42no—
340C2×D5—41no—
341C2×D5—41no—
342C2×D5—41no—
343C2×D5—81no—
344C4×C5—21no—
345C4×C5—21no—
346C4×C5—41no—
347C4×C5—42no—
348Dic5—41no—
349Dic5—41no—
350Dic5—41no—
351Dic5—81no—
352C24—51no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—42no—
356C2×Q8—51no—
357C2×D4—21no—
358C2×D4—101no—
359C2×D4—101no—
360C2×D4C3×C22⋊D511yescharacteristic
361C22×C4—21no—
362C22×C4—51no—
363C22×C4—101no—
364C22×C4—101no—
365C22×C4C22×C3×D511yescharacteristic
366Q16—202no—
367SD16—42no—
368C2×C8—21no—
369C2×C8—101no—
370C4⋊C4—51no—
371C4⋊C4—101no—
372C4⋊C4—101no—
373C4⋊C4C3×C22⋊D511yescharacteristic
374C22⋊C4—21no—
375C22⋊C4—101no—
376C22⋊C4—101no—
377C22⋊C4—101no—
378C22⋊C4—101no—
379C22⋊C4—101no—
380C22⋊C4—101no—
381C22⋊C4—202no—
382C42—101no—
383C3×C5#160611yescharacteristic, complemented, Hall
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—202no—
390C22×C3—201no—
391C22×C3—201no—
392C22×C3—201no—
393C22×C3—401no—
394C22×C3#406111yescharacteristic
395C22×C3#406211yescharacteristic
396C22×C3#408311yescharacteristic
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—42no—
400C3×C4—41no—
401C3×C4—201no—
402C3×C4—201no—
403C3×C4—201no—
404C3×C4—401no—
405C2×C5—21no—
406C2×C5—21no—
407C2×C5—81no—
408C2×C5C3×#45211yescharacteristic
409C2×C5C3×#45511yescharacteristic
410C2×C5C3×B(4,𝔽2)11yescharacteristic
411D5—81no—
412C23—21no—
413C23—101no—
414C23—101no—
415C23—101no—
416C23—201no—
417C23—201no—
418C23C3×D4×D511yescharacteristic
419Q8—22no—
420Q8—22no—
421Q8—41no—
422Q8—41no—
423Q8—201no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—401no—
428D4—401no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—101no—
435C2×C4—101no—
436C2×C4—101no—
437C2×C4—201no—
438C2×C4—201no—
439C2×C4—201no—
440C2×C4—201no—
441C2×C4—401no—
442C2×C4C3×D4×D511yescharacteristic
443C2×C4C2×C3×C22⋊D511yescharacteristic
444C8—41no—
445C8—201no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—81no—
449C2×C3—401no—
450C2×C3#6532111yescharacteristic
451C2×C3#6546211yescharacteristic
452C2×C3#6549111yescharacteristic
453C5C3×#160611yescharacteristic, complemented, Sylow
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—202no—
460C22—201no—
461C22—201no—
462C22—201no—
463C22—401no—
464C22C3×#406111yescharacteristic
465C22C3×#406211yescharacteristic
466C22C3×#408311yescharacteristic
467C4—21no—
468C4—21no—
469C4—42no—
470C4—41no—
471C4—201no—
472C4—201no—
473C4—201no—
474C4—401no—
475C3#1060176611yescharacteristic, direct product factor, Sylow
476C2—21no—
477C2—21no—
478C2—81no—
479C2—401no—
480C2C3×#6532111yescharacteristic
481C2C3×#6546211yescharacteristic
482C2C3×#6549111yescharacteristic
subgroupG/H
centerC22×C3#4083
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#448C2
socleC22×C3×C5#120
prime radicalC3×#65951C2
hypercenterC3×#448D5
hypocenterC5C3×#1606
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#448

Complements

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

NH ≅ G/Ntype
2. C3×C5×#448C2semidirect
3. C3×#65520C2semidirect
4. C3×#65517C2semidirect
5. C3×#64757C2semidirect
6. C3×#64688C2semidirect
7. C3×#64687C2semidirect
8. #10601766C3direct (N × H)
30. C5×#448C2×C3semidirect
31. #65520C2×C3semidirect
32. #65517C2×C3semidirect
33. #64757C2×C3semidirect
34. #64688C2×C3semidirect
35. #64687C2×C3semidirect
69. C3×#448D5semidirect
180. #448C3×D5semidirect
383. C3×C5#1606semidirect
453. C5C3×#1606semidirect
475. C3#10601766direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659512
C3×C5×#4482
C3×#655202
C3×#655172
C3×#647572
C3×#646882
C3×#646872
#106017663
C3×#16065