Finite Group Databasebeta

← C3×#10601204 #14302927

Subgroups — C3×#10601204 #14302927

Order 1920.

628 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66159C211yesmaximal, characteristic
2C3×#65850C211yesmaximal, characteristic, complemented
3C3×C5×#421C211yesmaximal, characteristic
4C3×#65525C211yesmaximal, characteristic
5C2×C3×#3956C211yesmaximal, characteristic, complemented
6C3×#64777C211yesmaximal, characteristic, complemented
7C3×#64751C211yesmaximal, characteristic, complemented
8#10601204C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×Dic10C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C3×#4090C412yescomplemented
14C3×#4082C412yescomplemented
15C3×#4081—22no—
16C3×#4072C2211yescharacteristic
17C3×#4072C2211yescharacteristic
18C3×#4070—22no—
19C2×C3×#681C2211yescharacteristic
20C3×#4033—22no—
21C3×#4023—22no—
22C3×C4×Dic10—22no—
23C3×C5×#104—22no—
24C3×#3969—22no—
25C3×#3956—22no—
26C3×#1912—51nomaximal, Hall
27#66159C2×C311yescharacteristic
28#65850C2×C311yescharacteristic, complemented
29C5×#421C2×C311yescharacteristic
30#65525C2×C311yescharacteristic
31C2×#3956C2×C311yescharacteristic, complemented
32#64777C2×C311yescharacteristic, complemented
33#64751C2×C311yescharacteristic, complemented
34C3×C5×#41—21no—
35C3×C5×#41—21no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C2×C3×C5×Q8C2×C411yescharacteristic
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4C2×C411yescharacteristic
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×Dic5D412yes—
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—21no—
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C412yes—
49C2×C3×Dic10D412yes—
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C2×C3×C5×C8—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C42×C5D412yes—
56C3×C22⋊Dic5—21no—
57C3×C22⋊Dic5—21no—
58C3×C22⋊Dic5—24no—
59C3×C4⋊Dic5—21no—
60C3×C4⋊Dic5—21no—
61C3×#683—24no—
62C3×#683—24no—
63C3×C4×Dic5—24no—
64C3×C4×Dic5C2×C412yes—
65C3×C4×Dic5D412yes—
66C3×#681—21no—
67C3×#681—21no—
68C2×C3×#168—21no—
69C3×#536—51no—
70C3×#519—51no—
71C3×#421—51no—
72C3×#421D511yescharacteristic
73C2×C3×#101—51no—
74C3×H(3,ℤ/4)—51no—
75C3×H(3,ℤ/4)—51no—
76C2×C5×#41C22×C311yescharacteristic
77C22×Dic10C22×C311yescharacteristic
78C2×C5×M16C22×C311yescharacteristic
79C5×#117C22×C311yescharacteristic
80#4090C3×C412yescomplemented
81#4082C3×C412yescomplemented
82#4081—22no—
83#4072C22×C311yescharacteristic
84#4072C22×C311yescharacteristic
85#4070—22no—
86C2×#681C22×C311yescharacteristic
87#4033—22no—
88#4023—22no—
89C4×Dic10—22no—
90C5×#104—22no—
91#3969—22no—
92#3956—22no—
93#1912—51noSylow
94C23×C3×C5—21no—
95C23×C3×C5C2×D411yescharacteristic
96C3×C5×Q8—21no—
97C3×C5×Q8—21no—
98C3×C5×D4—21no—
99C3×C5×D4—21no—
100C3×C5×D4—41no—
101C3×C5×D4—41no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5C22×C411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×C3×C4×C5#4111yescharacteristic
112C2×C3×Dic5—22no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5—22no—
115C2×C3×Dic5—22no—
116C2×C3×Dic5C22⋊C414yes—
117C3×Dic10—22no—
118C3×Dic10—22no—
119C3×Dic10—22no—
120C3×Dic10—22no—
121C3×C5×C8—41no—
122C3×#168—41no—
123C2×C3×#41C2×D511yescharacteristic
124C22×C3×Q8—51no—
125C2×C3×M16—51no—
126C2×C3×M16C2×D511yescharacteristic
127C3×C4⋊Q8—52no—
128C3×#124—52no—
129C3×#123—102no—
130C3×C22⋊Q8—102no—
131C3×C4×Q8—102no—
132C3×C4×Q8—102no—
133C3×#117—51no—
134C3×#117—51no—
135C3×#117—102no—
136C3×#117C2×D511yescharacteristic
137C3×#104—22no—
138C3×#104—102no—
139C3×#101—102no—
140C5×#41—21no—
141C5×#41—21no—
142C5×#41—21no—
143C5×#41—21no—
144C2×C5×Q8C2×C3×C411yescharacteristic
145C2×C5×D4—21no—
146C2×C5×D4C2×C3×C411yescharacteristic
147C22×C4×C5—21no—
148C22×C4×C5C23×C311yescharacteristic
149C22×Dic5C3×D412yes—
150C2×Dic10—21no—
151C2×Dic10—21no—
152C2×Dic10—21no—
153C2×Dic10—21no—
154C2×Dic10C2×C3×C412yes—
155C2×Dic10C3×D412yes—
156C5×M16—21no—
157C5×M16—21no—
158C2×C5×C8—21no—
159C5×C4⋊C4—21no—
160C5×C22⋊C4—21no—
161C42×C5C3×D412yes—
162C22⋊Dic5—21no—
163C22⋊Dic5—21no—
164C22⋊Dic5—24no—
165C4⋊Dic5—21no—
166C4⋊Dic5—21no—
167#683—24no—
168#683—24no—
169C4×Dic5—24no—
170C4×Dic5C2×C3×C412yes—
171C4×Dic5C3×D412yes—
172#681—21no—
173#681—21no—
174C2×#168—21no—
175#536—51no—
176#519—51no—
177#421—51no—
178#421C3×D511yescharacteristic
179C2×#101—51no—
180H(3,ℤ/4)—51no—
181H(3,ℤ/4)—51no—
182C22×C3×C5—41no—
183C22×C3×C5—42no—
184C22×C3×C5—41no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C22×C3×C5#12011yescharacteristic
187C22×C3×C5#12311yescharacteristic
188C3×C4×C5—42no—
189C3×C4×C5—41no—
190C3×C4×C5C4×D411yescharacteristic
191C3×C4×C5C4×D411yescharacteristic
192C3×C4×C5#12111yescharacteristic
193C3×C4×C5#12111yescharacteristic
194C3×Dic5—24no—
195C3×Dic5—42no—
196C3×Dic5—42no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—21no—
200C3×#41—21no—
201C2×C3×Q8—52no—
202C2×C3×Q8—52no—
203C2×C3×Q8—101no—
204C2×C3×Q8—101no—
205C2×C3×Q8—101no—
206C2×C3×Q8—101no—
207C2×C3×Q8C4×D511yescharacteristic
208C2×C3×D4—21no—
209C2×C3×D4C4×D511yescharacteristic
210C22×C3×C4—21no—
211C22×C3×C4—52no—
212C22×C3×C4C22×D511yescharacteristic
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—101no—
216C3×M16—101no—
217C2×C3×C8—21no—
218C2×C3×C8—101no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—104no—
221C3×C4⋊C4—104no—
222C3×C4⋊C4—101no—
223C3×C4⋊C4—101no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—104no—
226C3×C22⋊C4—101no—
227C3×C22⋊C4—101no—
228C3×C42—52no—
229C3×C42—52no—
230C3×C42—104no—
231C3×C42C22⋊D512yes—
232C23×C5—21no—
233C23×C5C2×C3×D411yescharacteristic
234C5×Q8—21no—
235C5×Q8—21no—
236C5×D4—21no—
237C5×D4—21no—
238C5×D4—41no—
239C5×D4—41no—
240C2×C4×C5—21no—
241C2×C4×C5—22no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5C22×C3×C411yescharacteristic
245C2×C4×C5C2×C3×D411yescharacteristic
246C2×C4×C5C2×C3×D411yescharacteristic
247C2×C4×C5C2×C3×D411yescharacteristic
248C2×C4×C5C3×#4111yescharacteristic
249C2×C4×C5C3×#4111yescharacteristic
250C2×Dic5—22no—
251C2×Dic5—22no—
252C2×Dic5—22no—
253C2×Dic5—22no—
254C2×Dic5C3×C22⋊C414yes—
255Dic10—22no—
256Dic10—22no—
257Dic10—22no—
258Dic10—22no—
259C5×C8—41no—
260#168—41no—
261C2×#41C2×C3×D511yescharacteristic
262C22×Q8—51no—
263C2×M16—51no—
264C2×M16C2×C3×D511yescharacteristic
265C4⋊Q8—52no—
266#124—52no—
267#123—102no—
268C22⋊Q8—102no—
269C4×Q8—102no—
270C4×Q8—102no—
271#117—51no—
272#117—51no—
273#117—102no—
274#117C2×C3×D511yescharacteristic
275#104—22no—
276#104—102no—
277#101—102no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5—81no—
282C2×C3×C5#38611yescharacteristic
283C23×C3—21no—
284C23×C3D4×D511yescharacteristic
285C3×Q8—21no—
286C3×Q8—21no—
287C3×Q8—102no—
288C3×Q8—102no—
289C3×Q8—102no—
290C3×Q8—102no—
291C3×D4—21no—
292C3×D4—21no—
293C3×D4—41no—
294C3×D4—41no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—54no—
300C2×C3×C4—102no—
301C2×C3×C4—102no—
302C2×C3×C4—102no—
303C2×C3×C4—102no—
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—41no—
311C3×C8—201no—
312C22×C5—41no—
313C22×C5—42no—
314C22×C5—41no—
315C22×C5C2×C3×C22⋊C411yescharacteristic
316C22×C5C3×#12011yescharacteristic
317C22×C5C3×#12311yescharacteristic
318C4×C5—42no—
319C4×C5—41no—
320C4×C5C3×C4×D411yescharacteristic
321C4×C5C3×C4×D411yescharacteristic
322C4×C5C3×#12111yescharacteristic
323C4×C5C3×#12111yescharacteristic
324Dic5—24no—
325Dic5—42no—
326Dic5—42no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331C2×Q8—52no—
332C2×Q8—52no—
333C2×Q8—101no—
334C2×Q8—101no—
335C2×Q8—101no—
336C2×Q8—101no—
337C2×Q8C3×C4×D511yescharacteristic
338C2×D4—21no—
339C2×D4C3×C4×D511yescharacteristic
340C22×C4—21no—
341C22×C4—52no—
342C22×C4C22×C3×D511yescharacteristic
343M16—21no—
344M16—21no—
345M16—101no—
346M16—101no—
347C2×C8—21no—
348C2×C8—101no—
349C4⋊C4—21no—
350C4⋊C4—104no—
351C4⋊C4—104no—
352C4⋊C4—101no—
353C4⋊C4—101no—
354C22⋊C4—21no—
355C22⋊C4—104no—
356C22⋊C4—101no—
357C22⋊C4—101no—
358C42—52no—
359C42—52no—
360C42—104no—
361C42C3×C22⋊D512yes—
362C3×C5#191211yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3—42no—
365C22×C3—41no—
366C22×C3D5×C22⋊C411yescharacteristic
367C22×C3#402911yescharacteristic
368C22×C3#408311yescharacteristic
369C3×C4—42no—
370C3×C4—41no—
371C3×C4—104no—
372C3×C4—202no—
373C3×C4—202no—
374C3×C4#402711yescharacteristic
375C3×C4#403011yescharacteristic
376C3×C4C4×C22⋊D511yescharacteristic
377C3×C4#408411yescharacteristic
378C2×C5—21no—
379C2×C5—21no—
380C2×C5—21no—
381C2×C5—81no—
382C2×C5C3×#38611yescharacteristic
383C23—21no—
384C23C3×D4×D511yescharacteristic
385Q8—21no—
386Q8—21no—
387Q8—102no—
388Q8—102no—
389Q8—102no—
390Q8—102no—
391D4—21no—
392D4—21no—
393D4—41no—
394D4—41no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—54no—
400C2×C4—102no—
401C2×C4—102no—
402C2×C4—102no—
403C2×C4—102no—
404C2×C4C2×C3×C4×D511yescharacteristic
405C2×C4C3×#70911yescharacteristic
406C2×C4C3×D4×D511yescharacteristic
407C2×C4C3×D4×D511yescharacteristic
408C2×C4C3×#71111yescharacteristic
409C2×C4C2×C3×C22⋊D511yescharacteristic
410C8—41no—
411C8—201no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—81no—
416C2×C3#6524511yescharacteristic
417C5C3×#191211yescharacteristic, complemented, Sylow
418C22—41no—
419C22—42no—
420C22—41no—
421C22C3×D5×C22⋊C411yescharacteristic
422C22C3×#402911yescharacteristic
423C22C3×#408311yescharacteristic
424C4—42no—
425C4—41no—
426C4—104no—
427C4—202no—
428C4—202no—
429C4C3×#402711yescharacteristic
430C4C3×#403011yescharacteristic
431C4C3×C4×C22⋊D511yescharacteristic
432C4C3×#408411yescharacteristic
433C3#1060120411yescharacteristic, direct product factor, Sylow
434C2—21no—
435C2—21no—
436C2—21no—
437C2—81no—
438C2C3×#6524511yescharacteristic
subgroupG/H
centerC2×C3#65245
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#421C2
socleC2×C3×C5#386
prime radicalC2×C3×C5×#41C22
cosocleC22×C4×C5C23×C3
hypercenterC3×#421D5
hypocenterC5C3×#1912
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#421

Complements

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

NH ≅ G/Ntype
2. C3×#65850C2semidirect
5. C2×C3×#3956C2semidirect
6. C3×#64777C2semidirect
7. C3×#64751C2semidirect
8. #10601204C3direct (N × H)
13. C3×#4090C4semidirect
14. C3×#4082C4semidirect
28. #65850C2×C3semidirect
31. C2×#3956C2×C3semidirect
32. #64777C2×C3semidirect
33. #64751C2×C3semidirect
80. #4090C3×C4semidirect
81. #4082C3×C4semidirect
362. C3×C5#1912semidirect
417. C5C3×#1912semidirect
433. C3#10601204direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661592
C3×#658502
C3×C5×#4212
C3×#655252
C2×C3×#39562
C3×#647772
C3×#647512
#106012043
C3×#19125