Finite Group Databasebeta

← C3×#10602907 #14302568

Subgroups — C3×#10602907 #14302568

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic
2C3×#41×Dic5C211yesmaximal, characteristic, complemented
3C3×#65529C211yesmaximal, characteristic, complemented
4C3×#65525C211yesmaximal, characteristic, complemented
5C2×C3×#3969C211yesmaximal, characteristic, complemented
6C3×#64779C211yesmaximal, characteristic, complemented
7C3×#64775C211yesmaximal, characteristic, complemented
8#10602907C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×#41C2211yescharacteristic
11C3×C5×#137—21no—
12C3×C5×#137C412yescomplemented
13C3×C5×AGL(1,ℤ/8)—21no—
14C3×C5×AGL(1,ℤ/8)C412yescomplemented
15C3×C5×#135—22no—
16C2×C3×C5×M16C2211yescharacteristic
17C3×Q8×Dic5—21no—
18C3×D4×Dic5—21no—
19C3×D4×Dic5—22no—
20C3×#4072—21no—
21C3×#4072C2211yescharacteristic
22C3×#4072C2211yescharacteristic
23C2×C3×C4×Dic5—21no—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×C3×#681C2211yescharacteristic
26C3×#3969—22no—
27C3×#3969—21no—
28C3×#3969—21no—
29C3×#3966—22no—
30C3×#1906—51nomaximal, Hall
31C5×#575C2×C311yescharacteristic
32#41×Dic5C2×C311yescharacteristic, complemented
33#65529C2×C311yescharacteristic, complemented
34#65525C2×C311yescharacteristic, complemented
35C2×#3969C2×C311yescharacteristic, complemented
36#64779C2×C311yescharacteristic, complemented
37#64775C2×C311yescharacteristic, complemented
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41C2×C412yes—
45C3×C5×#41D412yes—
46C2×C3×C5×Q8C2×C411yescharacteristic, complemented
47C2×C3×C5×Q8D411yescharacteristic
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4C2×C411yescharacteristic, complemented
51C2×C3×C5×D4D411yescharacteristic
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5—21no—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×Dic5—22no—
56C22×C3×Dic5—21no—
57C3×C5×Q16—22no—
58C3×C5×SD16—22no—
59C3×C5×SD16—22no—
60C3×C5×D8—22no—
61C3×C5×M16—21no—
62C3×C5×M16C2×C412yescomplemented
63C2×C3×C5×C8—21no—
64C3×C22⋊Dic5—22no—
65C3×C22⋊Dic5—21no—
66C3×C22⋊Dic5D412yes—
67C3×C4⋊Dic5—22no—
68C3×C4⋊Dic5—21no—
69C3×C4⋊Dic5—21no—
70C3×C4×Dic5—22no—
71C3×C4×Dic5—21no—
72C3×C4×Dic5—21no—
73C3×C4×Dic5—21no—
74C3×C4×Dic5D411yescharacteristic
75C3×C4×Dic5D411yescharacteristic
76C3×#681—21no—
77C3×#681—21no—
78C2×C3×#168—21no—
79C3×#575D511yescharacteristic
80C3×C4×#41—51no—
81C3×#421—51no—
82C2×C3×#104—51no—
83C3×#410—51no—
84C3×#344—51no—
85C3×#339—51no—
86C2×C5×#41C22×C311yescharacteristic
87C2×C5×#41C22×C311yescharacteristic
88C5×#137—21no—
89C5×#137C3×C412yescomplemented
90C5×AGL(1,ℤ/8)—21no—
91C5×AGL(1,ℤ/8)C3×C412yescomplemented
92C5×#135—22no—
93C2×C5×M16C22×C311yescharacteristic
94Q8×Dic5—21no—
95D4×Dic5—21no—
96D4×Dic5—22no—
97#4072—21no—
98#4072C22×C311yescharacteristic
99#4072C22×C311yescharacteristic
100C2×C4×Dic5—21no—
101C2×C4×Dic5C22×C311yescharacteristic
102C2×#681C22×C311yescharacteristic
103#3969—22no—
104#3969—21no—
105#3969—21no—
106#3966—22no—
107#1906—51noSylow
108C23×C3×C5—21no—
109C23×C3×C5—21no—
110C23×C3×C5C2×D411yescharacteristic
111C3×C5×Q8—21no—
112C3×C5×Q8—21no—
113C3×C5×Q8—21no—
114C3×C5×Q8C22⋊C412yes—
115C3×C5×D4—22no—
116C3×C5×D4—22no—
117C3×C5×D4—21no—
118C3×C5×D4—21no—
119C3×C5×D4—21no—
120C3×C5×D4—41no—
121C3×C5×D4—41no—
122C3×C5×D4C22⋊C412yes—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—41no—
128C2×C3×C4×C5—41no—
129C2×C3×C4×C5C22×C411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5#4111yescharacteristic
134C2×C3×C4×C5#4111yescharacteristic
135C2×C3×Dic5—21no—
136C2×C3×Dic5—22no—
137C2×C3×Dic5—21no—
138C2×C3×Dic5—42no—
139C2×C3×Dic5—44no—
140C2×C3×Dic5—41no—
141C3×C5×C8—22no—
142C3×#168—41no—
143C2×C3×#41C2×D511yescharacteristic
144C2×C3×#41C2×D511yescharacteristic
145C3×#137—21no—
146C3×#137Dic512yescomplemented
147C3×AGL(1,ℤ/8)—21no—
148C3×AGL(1,ℤ/8)Dic512yescomplemented
149C3×#135—22no—
150C2×C3×M16—51no—
151C2×C3×M16C2×D511yescharacteristic
152C3×C4×Q8—101no—
153C3×C4×D4—102no—
154C3×C4×D4—101no—
155C3×#117—51no—
156C3×#117—51no—
157C3×#117—101no—
158C2×C3×C42—51no—
159C2×C3×C42—101no—
160C3×#104—101no—
161C3×#104—101no—
162C3×#104—102no—
163C3×#99—102no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41—21no—
167C5×#41—21no—
168C5×#41—21no—
169C5×#41—21no—
170C5×#41C2×C3×C412yes—
171C5×#41C3×D412yes—
172C2×C5×Q8C2×C3×C411yescharacteristic, complemented
173C2×C5×Q8C3×D411yescharacteristic
174C2×C5×D4—21no—
175C2×C5×D4—21no—
176C2×C5×D4C2×C3×C411yescharacteristic, complemented
177C2×C5×D4C3×D411yescharacteristic
178C22×C4×C5—21no—
179C22×C4×C5—21no—
180C22×C4×C5C23×C311yescharacteristic
181C22×Dic5—22no—
182C22×Dic5—21no—
183C5×Q16—22no—
184C5×SD16—22no—
185C5×SD16—22no—
186C5×D8—22no—
187C5×M16—21no—
188C5×M16C2×C3×C412yescomplemented
189C2×C5×C8—21no—
190C22⋊Dic5—22no—
191C22⋊Dic5—21no—
192C22⋊Dic5C3×D412yes—
193C4⋊Dic5—22no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4×Dic5—22no—
197C4×Dic5—21no—
198C4×Dic5—21no—
199C4×Dic5—21no—
200C4×Dic5C3×D411yescharacteristic
201C4×Dic5C3×D411yescharacteristic
202#681—21no—
203#681—21no—
204C2×#168—21no—
205#575C3×D511yescharacteristic
206C4×#41—51no—
207#421—51no—
208C2×#104—51no—
209#410—51no—
210#344—51no—
211#339—51no—
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—42no—
215C22×C3×C5—41no—
216C22×C3×C5—81no—
217C22×C3×C5C4×D411yescharacteristic
218C22×C3×C5#12011yescharacteristic
219C22×C3×C5#12111yescharacteristic
220C3×C4×C5—22no—
221C3×C4×C5—41no—
222C3×C4×C5C2×C22⋊C411yescharacteristic
223C3×C4×C5C4×D411yescharacteristic
224C3×C4×C5#12111yescharacteristic
225C3×C4×C5#12311yescharacteristic
226C3×Dic5—22no—
227C3×Dic5—42no—
228C3×Dic5—41no—
229C3×Dic5—81no—
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41—21no—
236C3×#41C2×Dic512yes—
237C3×#41C22⋊D512yes—
238C2×C3×Q8C2×Dic511yescharacteristic, complemented
239C2×C3×Q8C22⋊D511yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4C2×Dic511yescharacteristic, complemented
243C2×C3×D4C22⋊D511yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4—102no—
247C22×C3×C4—101no—
248C22×C3×C4C22×D511yescharacteristic
249C3×Q16—22no—
250C3×SD16—22no—
251C3×SD16—22no—
252C3×D8—22no—
253C3×M16—21no—
254C3×M16—101no—
255C3×M16—101no—
256C3×M16C2×Dic512yescomplemented
257C2×C3×C8—21no—
258C2×C3×C8—101no—
259C3×C4⋊C4—102no—
260C3×C4⋊C4—101no—
261C3×C4⋊C4—101no—
262C3×C22⋊C4—52no—
263C3×C22⋊C4—102no—
264C3×C22⋊C4—101no—
265C3×C42—51no—
266C3×C42—51no—
267C3×C42—102no—
268C3×C42—101no—
269C3×C42—101no—
270C3×C42—101no—
271C23×C5—21no—
272C23×C5—21no—
273C23×C5C2×C3×D411yescharacteristic
274C5×Q8—21no—
275C5×Q8—21no—
276C5×Q8—21no—
277C5×Q8C3×C22⋊C412yes—
278C5×D4—22no—
279C5×D4—22no—
280C5×D4—21no—
281C5×D4—21no—
282C5×D4—21no—
283C5×D4—41no—
284C5×D4—41no—
285C5×D4C3×C22⋊C412yes—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—21no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5C22×C3×C411yescharacteristic
293C2×C4×C5C2×C3×D411yescharacteristic
294C2×C4×C5C2×C3×D411yescharacteristic
295C2×C4×C5C2×C3×D411yescharacteristic
296C2×C4×C5C3×#4111yescharacteristic
297C2×C4×C5C3×#4111yescharacteristic
298C2×Dic5—21no—
299C2×Dic5—22no—
300C2×Dic5—21no—
301C2×Dic5—42no—
302C2×Dic5—44no—
303C2×Dic5—41no—
304C5×C8—22no—
305#168—41no—
306C2×#41C2×C3×D511yescharacteristic
307C2×#41C2×C3×D511yescharacteristic
308#137—21no—
309#137C3×Dic512yescomplemented
310AGL(1,ℤ/8)—21no—
311AGL(1,ℤ/8)C3×Dic512yescomplemented
312#135—22no—
313C2×M16—51no—
314C2×M16C2×C3×D511yescharacteristic
315C4×Q8—101no—
316C4×D4—102no—
317C4×D4—101no—
318#117—51no—
319#117—51no—
320#117—101no—
321C2×C42—51no—
322C2×C42—101no—
323#104—101no—
324#104—101no—
325#104—102no—
326#99—102no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—42no—
331C2×C3×C5—81no—
332C2×C3×C5#38611yescharacteristic
333C23×C3—21no—
334C23×C3—21no—
335C23×C3D4×D511yescharacteristic
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8C22⋊Dic512yes—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—21no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4C22⋊Dic512yes—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—101no—
355C2×C3×C4—101no—
356C2×C3×C4—102no—
357C2×C3×C4—202no—
358C2×C3×C4—204no—
359C2×C3×C4—201no—
360C2×C3×C4D4×D511yescharacteristic
361C2×C3×C4#71111yescharacteristic
362C2×C3×C4#71111yescharacteristic
363C2×C3×C4C22×Dic511yescharacteristic
364C2×C3×C4C2×C22⋊D511yescharacteristic
365C2×C3×C4C2×C22⋊D511yescharacteristic
366C3×C8—22no—
367C3×C8—201no—
368C22×C5—22no—
369C22×C5—41no—
370C22×C5—42no—
371C22×C5—41no—
372C22×C5—81no—
373C22×C5C3×C4×D411yescharacteristic
374C22×C5C3×#12011yescharacteristic
375C22×C5C3×#12111yescharacteristic
376C4×C5—22no—
377C4×C5—41no—
378C4×C5C2×C3×C22⋊C411yescharacteristic
379C4×C5C3×C4×D411yescharacteristic
380C4×C5C3×#12111yescharacteristic
381C4×C5C3×#12311yescharacteristic
382Dic5—22no—
383Dic5—42no—
384Dic5—41no—
385Dic5—81no—
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41—21no—
392#41C2×C3×Dic512yes—
393#41C3×C22⋊D512yes—
394C2×Q8C2×C3×Dic511yescharacteristic, complemented
395C2×Q8C3×C22⋊D511yescharacteristic
396C2×D4—21no—
397C2×D4—21no—
398C2×D4C2×C3×Dic511yescharacteristic, complemented
399C2×D4C3×C22⋊D511yescharacteristic
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—102no—
403C22×C4—101no—
404C22×C4C22×C3×D511yescharacteristic
405Q16—22no—
406SD16—22no—
407SD16—22no—
408D8—22no—
409M16—21no—
410M16—101no—
411M16—101no—
412M16C2×C3×Dic512yescomplemented
413C2×C8—21no—
414C2×C8—101no—
415C4⋊C4—102no—
416C4⋊C4—101no—
417C4⋊C4—101no—
418C22⋊C4—52no—
419C22⋊C4—102no—
420C22⋊C4—101no—
421C42—51no—
422C42—51no—
423C42—102no—
424C42—101no—
425C42—101no—
426C42—101no—
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—22no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3—41no—
432C22×C3—81no—
433C22×C3D4×Dic511yescharacteristic
434C22×C3#408311yescharacteristic
435C22×C3#408511yescharacteristic
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—102no—
439C3×C4—201no—
440C3×C4—202no—
441C3×C4—401no—
442C3×C4D4×Dic511yescharacteristic
443C3×C4#408111yescharacteristic
444C3×C4#408511yescharacteristic
445C3×C4C2×C22⋊Dic511yescharacteristic
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—42no—
450C2×C5—81no—
451C2×C5C3×#38611yescharacteristic
452C23—21no—
453C23—21no—
454C23C3×D4×D511yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—21no—
458Q8C3×C22⋊Dic512yes—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—41no—
465D4—41no—
466D4C3×C22⋊Dic512yes—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—101no—
474C2×C4—101no—
475C2×C4—102no—
476C2×C4—202no—
477C2×C4—204no—
478C2×C4—201no—
479C2×C4C3×D4×D511yescharacteristic
480C2×C4C3×#71111yescharacteristic
481C2×C4C3×#71111yescharacteristic
482C2×C4C22×C3×Dic511yescharacteristic
483C2×C4C2×C3×C22⋊D511yescharacteristic
484C2×C4C2×C3×C22⋊D511yescharacteristic
485C8—22no—
486C8—201no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—42no—
491C2×C3—81no—
492C2×C3#6550911yescharacteristic
493C5C3×#190611yescharacteristic, complemented, Sylow
494C22—22no—
495C22—41no—
496C22—42no—
497C22—41no—
498C22—81no—
499C22C3×D4×Dic511yescharacteristic
500C22C3×#408311yescharacteristic
501C22C3×#408511yescharacteristic
502C4—22no—
503C4—41no—
504C4—102no—
505C4—201no—
506C4—202no—
507C4—401no—
508C4C3×D4×Dic511yescharacteristic
509C4C3×#408111yescharacteristic
510C4C3×#408511yescharacteristic
511C4C2×C3×C22⋊Dic511yescharacteristic
512C3#1060290711yescharacteristic, direct product factor, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—81no—
518C2C3×#6550911yescharacteristic
subgroupG/H
centerC3×C4#4085
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
Fitting, prime radicalC3×C5×#575C2
socleC2×C3×C5#386
cosocleC22×C4×C5C23×C3
hypercenterC3×#575D5
hypocenterC5C3×#1906
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#575

Complements

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

NH ≅ G/Ntype
2. C3×#41×Dic5C2semidirect
3. C3×#65529C2semidirect
4. C3×#65525C2semidirect
5. C2×C3×#3969C2semidirect
6. C3×#64779C2semidirect
7. C3×#64775C2semidirect
8. #10602907C3direct (N × H)
12. C3×C5×#137C4semidirect
14. C3×C5×AGL(1,ℤ/8)C4semidirect
32. #41×Dic5C2×C3semidirect
33. #65529C2×C3semidirect
34. #65525C2×C3semidirect
35. C2×#3969C2×C3semidirect
36. #64779C2×C3semidirect
37. #64775C2×C3semidirect
46. C2×C3×C5×Q8C2×C4semidirect
50. C2×C3×C5×D4C2×C4semidirect
62. C3×C5×M16C2×C4semidirect
89. C5×#137C3×C4semidirect
91. C5×AGL(1,ℤ/8)C3×C4semidirect
146. C3×#137Dic5semidirect
148. C3×AGL(1,ℤ/8)Dic5semidirect
172. C2×C5×Q8C2×C3×C4semidirect
176. C2×C5×D4C2×C3×C4semidirect
188. C5×M16C2×C3×C4semidirect
238. C2×C3×Q8C2×Dic5semidirect
242. C2×C3×D4C2×Dic5semidirect
256. C3×M16C2×Dic5semidirect
309. #137C3×Dic5semidirect
311. AGL(1,ℤ/8)C3×Dic5semidirect
394. C2×Q8C2×C3×Dic5semidirect
398. C2×D4C2×C3×Dic5semidirect
412. M16C2×C3×Dic5semidirect
427. C3×C5#1906semidirect
493. C5C3×#1906semidirect
512. C3#10602907direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5752
C3×#41×Dic52
C3×#655292
C3×#655252
C2×C3×#39692
C3×#647792
C3×#647752
#106029073
C3×#19065