Finite Group Databasebeta

← C3×#10599946 #14316686

Subgroups — C3×#10599946 #14316686

Order 1920.

516 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 82 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×#66013C211yesmaximal, characteristic, complemented
2C3×#65350C211yesmaximal, characteristic, complemented
3C3×#65031C211yesmaximal, characteristic, complemented
4C3×#65030C211yesmaximal, characteristic
5C3×C5×#354C211yesmaximal, characteristic, complemented
6C3×#64692C211yesmaximal, characteristic, complemented
7C3×#64691C211yesmaximal, characteristic, complemented
8#10599946C311yesmaximal, characteristic, direct product factor, Hall
9C3×#4144C2211yescharacteristic
10C3×C5×#124C2211yescharacteristic
11C3×#4092—21no—
12C3×#4088—21no—
13C3×#4083C2211yescharacteristic
14C3×#4081C2211yescharacteristic
15C3×#4032—21no—
16C3×#4029—21no—
17C3×#4028—21no—
18C3×#4027—21no—
19C3×D5×C22⋊C4—21no—
20C3×#4025—21no—
21C3×#4018—21no—
22C3×C5×#99C2211yescharacteristic
23C3×C5×#99C2211yescharacteristic
24C3×#3965C2211yescharacteristic
25C3×#3938—21no—
26C3×#3938—21no—
27C3×#3937—21no—
28C3×#2197—51nomaximal, Hall
29#66013C2×C311yescharacteristic, complemented
30#65350C2×C311yescharacteristic, complemented
31#65031C2×C311yescharacteristic, complemented
32#65030C2×C311yescharacteristic
33C5×#354C2×C311yescharacteristic, complemented
34#64692C2×C311yescharacteristic, complemented
35#64691C2×C311yescharacteristic, complemented
36C23×C3×D5D411yescharacteristic
37C2×C3×C5×Q8D411yescharacteristic, complemented
38C2×C3×C5×D4C2311yescharacteristic
39C2×C3×C22⋊D5—21no—
40C2×C3×C22⋊D5—21no—
41C2×C3×C22⋊D5—21no—
42C22×C3×Dic5D411yescharacteristic, complemented
43C3×#711—41no—
44C3×D4×D5—41no—
45C3×#709—21no—
46C2×C3×D20—41no—
47C2×C3×C4×D5—21no—
48C2×C3×C4×D5—21no—
49C3×C5×C22⋊C4—21no—
50C3×C5×C22⋊C4—21no—
51C3×C5×C22⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C42×C5D411yescharacteristic, complemented
54C3×C22⋊Dic5—21no—
55C3×C22⋊Dic5D411yescharacteristic
56C3×C22⋊Dic5D411yescharacteristic, complemented
57C3×#689—41no—
58C3×#688—41no—
59C3×#685—21no—
60C3×#685—21no—
61C3×#685—21no—
62C3×#685—41no—
63C3×#685—41no—
64C3×C4⋊Dic5—21no—
65C3×C4⋊Dic5—41no—
66C3×#683—21no—
67C3×#683—21no—
68C3×#683—21no—
69C3×C4×Dic5—21no—
70C3×#681—21no—
71C3×#551—51no—
72C3×#458—51no—
73C3×B(4,𝔽2)—51no—
74C3×#455—51no—
75C3×#354D511yescharacteristic, complemented
76C3×#352—51no—
77C3×AΣL(1,𝔽16)—51no—
78#4144C22×C311yescharacteristic
79C5×#124C22×C311yescharacteristic
80#4092—21no—
81#4088—21no—
82#4083C22×C311yescharacteristic
83#4081C22×C311yescharacteristic
84#4032—21no—
85#4029—21no—
86#4028—21no—
87#4027—21no—
88D5×C22⋊C4—21no—
89#4025—21no—
90#4018—21no—
91C5×#99C22×C311yescharacteristic
92C5×#99C22×C311yescharacteristic
93#3965C22×C311yescharacteristic
94#3938—21no—
95#3938—21no—
96#3937—21no—
97#2197—51noSylow
98C23×C3×C5C2×D411yescharacteristic
99C23×C3×C5C2×D411yescharacteristic
100C22×C3×D5—21no—
101C22×C3×D5—21no—
102C22×C3×D5—21no—
103C22×C3×D5—42no—
104C22×C3×D5—41no—
105C3×C5×Q8—41no—
106C3×C5×D4—41no—
107C2×C3×C4×C5—21no—
108C2×C3×C4×C5—21no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5C2×D411yescharacteristic
112C3×C22⋊D5—42no—
113C3×C22⋊D5—41no—
114C3×C22⋊D5—41no—
115C3×C22⋊D5—41no—
116C2×C3×Dic5—21no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—21no—
120C2×C3×Dic5—21no—
121C2×C3×Dic5—41no—
122C3×D20—21no—
123C3×D20—82no—
124C3×C4×D5—41no—
125C3×C4×D5—42no—
126C3×C4×D5—81no—
127C3×Dic10—21no—
128C3×#168—41no—
129C3×H(5,𝔽2)—51no—
130C3×#137—101no—
131C3×#126—101no—
132C3×#124C2×D511yescharacteristic
133C3×#123—51no—
134C3×#123—101no—
135C3×#123—101no—
136C3×C22⋊Q8—101no—
137C3×#120—51no—
138C3×#120—101no—
139C3×C4×D4—101no—
140C3×#117—101no—
141C2×C3×C22⋊C4—101no—
142C3×#104—101no—
143C3×#100—51no—
144C3×#99—101no—
145C3×#99—101no—
146C3×#99C2×D511yescharacteristic
147C3×#99C2×D511yescharacteristic
148C23×D5C3×D411yescharacteristic
149C2×C5×Q8C3×D411yescharacteristic, complemented
150C2×C5×D4C23×C311yescharacteristic
151C2×C22⋊D5—21no—
152C2×C22⋊D5—21no—
153C2×C22⋊D5—21no—
154C22×Dic5C3×D411yescharacteristic, complemented
155#711—41no—
156D4×D5—41no—
157#709—21no—
158C2×D20—41no—
159C2×C4×D5—21no—
160C2×C4×D5—21no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4—21no—
164C5×C22⋊C4—21no—
165C42×C5C3×D411yescharacteristic, complemented
166C22⋊Dic5—21no—
167C22⋊Dic5C3×D411yescharacteristic
168C22⋊Dic5C3×D411yescharacteristic, complemented
169#689—41no—
170#688—41no—
171#685—21no—
172#685—21no—
173#685—21no—
174#685—41no—
175#685—41no—
176C4⋊Dic5—21no—
177C4⋊Dic5—41no—
178#683—21no—
179#683—21no—
180#683—21no—
181C4×Dic5—21no—
182#681—21no—
183#551—51no—
184#458—51no—
185B(4,𝔽2)—51no—
186#455—51no—
187#354C3×D511yescharacteristic, complemented
188#352—51no—
189AΣL(1,𝔽16)—51no—
190C22×C3×C5—21no—
191C22×C3×C5—21no—
192C22×C3×C5—41no—
193C22×C3×C5—41no—
194C22×C3×C5#12011yescharacteristic
195C2×C3×D5—22no—
196C2×C3×D5—41no—
197C2×C3×D5—41no—
198C2×C3×D5—42no—
199C2×C3×D5—41no—
200C2×C3×D5—81no—
201C2×C3×D5—81no—
202C2×C3×D5—81no—
203C3×C4×C5—21no—
204C3×C4×C5—41no—
205C3×C4×C5—41no—
206C3×C4×C5—81no—
207C3×C4×C5—81no—
208C3×Dic5—41no—
209C3×Dic5—41no—
210C3×Dic5—41no—
211C3×Dic5—41no—
212C24×C3—51no—
213C3×#41—101no—
214C3×#41—201no—
215C2×C3×Q8D2011yescharacteristic, complemented
216C2×C3×D4—101no—
217C2×C3×D4—101no—
218C2×C3×D4—101no—
219C2×C3×D4—201no—
220C2×C3×D4—201no—
221C2×C3×D4C22×D511yescharacteristic
222C22×C3×C4—51no—
223C22×C3×C4—101no—
224C22×C3×C4—101no—
225C3×Q16—201no—
226C3×SD16—201no—
227C3×M16—101no—
228C3×C4⋊C4—101no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4—101no—
232C3×C4⋊C4—201no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—51no—
238C3×C22⋊C4—51no—
239C3×C22⋊C4—101no—
240C3×C22⋊C4—101no—
241C3×C22⋊C4—101no—
242C3×C22⋊C4—101no—
243C3×C22⋊C4—201no—
244C3×C22⋊C4—201no—
245C3×C42—101no—
246C3×C42D2011yescharacteristic, complemented
247C23×C5C2×C3×D411yescharacteristic
248C23×C5C2×C3×D411yescharacteristic
249C22×D5—21no—
250C22×D5—21no—
251C22×D5—21no—
252C22×D5—42no—
253C22×D5—41no—
254C5×Q8—41no—
255C5×D4—41no—
256C2×C4×C5—21no—
257C2×C4×C5—21no—
258C2×C4×C5—41no—
259C2×C4×C5—41no—
260C2×C4×C5C2×C3×D411yescharacteristic
261C22⋊D5—42no—
262C22⋊D5—41no—
263C22⋊D5—41no—
264C22⋊D5—41no—
265C2×Dic5—21no—
266C2×Dic5—21no—
267C2×Dic5—21no—
268C2×Dic5—21no—
269C2×Dic5—21no—
270C2×Dic5—41no—
271D20—21no—
272D20—82no—
273C4×D5—41no—
274C4×D5—42no—
275C4×D5—81no—
276Dic10—21no—
277#168—41no—
278H(5,𝔽2)—51no—
279#137—101no—
280#126—101no—
281#124C2×C3×D511yescharacteristic
282#123—51no—
283#123—101no—
284#123—101no—
285C22⋊Q8—101no—
286#120—51no—
287#120—101no—
288C4×D4—101no—
289#117—101no—
290C2×C22⋊C4—101no—
291#104—101no—
292#100—51no—
293#99—101no—
294#99—101no—
295#99C2×C3×D511yescharacteristic
296#99C2×C3×D511yescharacteristic
297C2×C3×C5—21no—
298C2×C3×C5—41no—
299C2×C3×C5—41no—
300C2×C3×C5B(4,𝔽2)11yescharacteristic
301C3×D5—42no—
302C3×D5—81no—
303C23×C3—101no—
304C23×C3—101no—
305C23×C3—101no—
306C23×C3—201no—
307C23×C3—202no—
308C23×C3D4×D511yescharacteristic
309C23×C3D4×D511yescharacteristic
310C3×Q8—41no—
311C3×Q8—101no—
312C3×D4—41no—
313C3×D4—101no—
314C3×D4—201no—
315C3×D4—201no—
316C3×D4—202no—
317C3×D4—201no—
318C3×D4—402no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—101no—
324C2×C3×C4—101no—
325C2×C3×C4—101no—
326C2×C3×C4—101no—
327C2×C3×C4—101no—
328C2×C3×C4—201no—
329C2×C3×C4—202no—
330C2×C3×C4—201no—
331C2×C3×C4—401no—
332C2×C3×C4C2×D2011yescharacteristic
333C3×C8—201no—
334C22×C5—21no—
335C22×C5—21no—
336C22×C5—41no—
337C22×C5—41no—
338C22×C5C3×#12011yescharacteristic
339C2×D5—22no—
340C2×D5—41no—
341C2×D5—41no—
342C2×D5—42no—
343C2×D5—41no—
344C2×D5—81no—
345C2×D5—81no—
346C2×D5—81no—
347C4×C5—21no—
348C4×C5—41no—
349C4×C5—41no—
350C4×C5—81no—
351C4×C5—81no—
352Dic5—41no—
353Dic5—41no—
354Dic5—41no—
355Dic5—41no—
356C24—51no—
357#41—101no—
358#41—201no—
359C2×Q8C3×D2011yescharacteristic, complemented
360C2×D4—101no—
361C2×D4—101no—
362C2×D4—101no—
363C2×D4—201no—
364C2×D4—201no—
365C2×D4C22×C3×D511yescharacteristic
366C22×C4—51no—
367C22×C4—101no—
368C22×C4—101no—
369Q16—201no—
370SD16—201no—
371M16—101no—
372C4⋊C4—101no—
373C4⋊C4—101no—
374C4⋊C4—101no—
375C4⋊C4—101no—
376C4⋊C4—201no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—21no—
381C22⋊C4—51no—
382C22⋊C4—51no—
383C22⋊C4—101no—
384C22⋊C4—101no—
385C22⋊C4—101no—
386C22⋊C4—101no—
387C22⋊C4—201no—
388C22⋊C4—201no—
389C42—101no—
390C42C3×D2011yescharacteristic, complemented
391C3×C5#219711yescharacteristic, complemented, Hall
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—102no—
397C22×C3—201no—
398C22×C3—202no—
399C22×C3—201no—
400C22×C3—201no—
401C22×C3—401no—
402C22×C3—401no—
403C22×C3—401no—
404C22×C3#402811yescharacteristic
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—81no—
409C3×C4—81no—
410C3×C4—201no—
411C3×C4—201no—
412C3×C4—201no—
413C3×C4—201no—
414C2×C5—21no—
415C2×C5—41no—
416C2×C5—41no—
417C2×C5C3×B(4,𝔽2)11yescharacteristic
418D5—42no—
419D5—81no—
420C23—101no—
421C23—101no—
422C23—101no—
423C23—201no—
424C23—202no—
425C23C3×D4×D511yescharacteristic
426C23C3×D4×D511yescharacteristic
427Q8—41no—
428Q8—101no—
429D4—41no—
430D4—101no—
431D4—201no—
432D4—201no—
433D4—202no—
434D4—201no—
435D4—402no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—101no—
441C2×C4—101no—
442C2×C4—101no—
443C2×C4—101no—
444C2×C4—101no—
445C2×C4—201no—
446C2×C4—202no—
447C2×C4—201no—
448C2×C4—401no—
449C2×C4C2×C3×D2011yescharacteristic
450C8—201no—
451C2×C3—21no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—202no—
455C2×C3—401no—
456C2×C3#6503011yescharacteristic
457C5C3×#219711yescharacteristic, complemented, Sylow
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—102no—
463C22—201no—
464C22—202no—
465C22—201no—
466C22—201no—
467C22—401no—
468C22—401no—
469C22—401no—
470C22C3×#402811yescharacteristic
471C4—21no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—201no—
477C4—201no—
478C4—201no—
479C4—201no—
480C3#1059994611yescharacteristic, direct product factor, Sylow
481C2—21no—
482C2—41no—
483C2—41no—
484C2—202no—
485C2—401no—
486C2C3×#6503011yescharacteristic
subgroupG/H
centerC2×C3#65030
commutator, cosocleC2×C5×D4C23×C3
FrattiniC2×D4C22×C3×D5
FittingC3×C5×#354C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC3×#65030C2
hypercenterC3×#354D5
hypocenterC5C3×#2197
LCSG ▷ C2×C5×D4 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#354

Complements

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

NH ≅ G/Ntype
1. C3×#66013C2semidirect
2. C3×#65350C2semidirect
3. C3×#65031C2semidirect
5. C3×C5×#354C2semidirect
6. C3×#64692C2semidirect
7. C3×#64691C2semidirect
8. #10599946C3direct (N × H)
29. #66013C2×C3semidirect
30. #65350C2×C3semidirect
31. #65031C2×C3semidirect
33. C5×#354C2×C3semidirect
34. #64692C2×C3semidirect
35. #64691C2×C3semidirect
37. C2×C3×C5×Q8D4semidirect
42. C22×C3×Dic5D4semidirect
53. C3×C42×C5D4semidirect
56. C3×C22⋊Dic5D4semidirect
75. C3×#354D5semidirect
149. C2×C5×Q8C3×D4semidirect
154. C22×Dic5C3×D4semidirect
165. C42×C5C3×D4semidirect
168. C22⋊Dic5C3×D4semidirect
187. #354C3×D5semidirect
215. C2×C3×Q8D20semidirect
246. C3×C42D20semidirect
359. C2×Q8C3×D20semidirect
390. C42C3×D20semidirect
391. C3×C5#2197semidirect
457. C5C3×#2197semidirect
480. C3#10599946direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660132
C3×#653502
C3×#650312
C3×#650302
C3×C5×#3542
C3×#646922
C3×#646912
#105999463
C3×#21975