Finite Group Databasebeta

← C3×#10601148 #14297808

Subgroups — C3×#10601148 #14297808

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×Dic5C211yesmaximal, characteristic
2C2×C3×C5×#104C211yesmaximal, characteristic
3C2×C3×#3969C211yesmaximal, characteristic
4C3×#65407C211yesmaximal, characteristic, complemented
5C3×C4×#683C211yesmaximal, characteristic, complemented
6C3×#64775C211yesmaximal, characteristic, complemented
7C3×#64743C211yesmaximal, characteristic, complemented
8#10601148C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12C3×Q8×Dic5C411yescharacteristic, complemented
13C3×Q8×Dic5C411yescharacteristic, complemented
14C3×D4×Dic5—21no—
15C3×D4×Dic5—21no—
16C3×D4×Dic5C411yescharacteristic, complemented
17C3×D4×Dic5C411yescharacteristic, complemented
18C3×#4072—21no—
19C3×#4072C2211yescharacteristic
20C2×C3×#683—21no—
21C2×C3×C4×Dic5—21no—
22C2×C3×C4×Dic5C2211yescharacteristic
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×C3×#681C2211yescharacteristic
25C3×C5×#104—22no—
26C3×#3969—22no—
27C3×#3963—21no—
28C3×#3946—21no—
29C3×#3946—21no—
30C3×#1806—51nomaximal, Hall
31#41×Dic5C2×C311yescharacteristic
32C2×C5×#104C2×C311yescharacteristic
33C2×#3969C2×C311yescharacteristic
34#65407C2×C311yescharacteristic, complemented
35C4×#683C2×C311yescharacteristic, complemented
36#64775C2×C311yescharacteristic, complemented
37#64743C2×C311yescharacteristic, complemented
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41D412yes—
41C3×C5×#41Q812yes—
42C2×C3×C5×Q8C2×C411yescharacteristic
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4C2×C411yescharacteristic
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5C2311yescharacteristic
48C22×C3×Dic5—21no—
49C22×C3×Dic5—21no—
50C22×C3×Dic5—21no—
51C22×C3×Dic5D411yescharacteristic
52C22×C3×Dic5D411yescharacteristic
53C3×C5×M16—21no—
54C3×C5×M16—21no—
55C2×C3×C5×C8—21no—
56C3×C42×C5—21no—
57C3×C42×C5—21no—
58C3×C22⋊Dic5—21no—
59C3×C22⋊Dic5—21no—
60C3×C22⋊Dic5—21no—
61C3×C4⋊Dic5—21no—
62C3×C4⋊Dic5—21no—
63C3×C4⋊Dic5C2×C411yescharacteristic
64C3×C4⋊Dic5C2×C411yescharacteristic
65C3×#683—24no—
66C3×C4×Dic5—21no—
67C3×C4×Dic5—21no—
68C3×C4×Dic5—21no—
69C3×C4×Dic5—21no—
70C3×C4×Dic5—21no—
71C3×C4×Dic5C2×C411yescharacteristic
72C3×C4×Dic5C2×C411yescharacteristic
73C3×C4×Dic5D411yescharacteristic
74C3×C4×Dic5D411yescharacteristic
75C3×#681—21no—
76C3×#681—21no—
77C2×C3×#168—21no—
78C3×C4×#41—51no—
79C3×#423—51no—
80C2×C3×#104—51no—
81C2×C3×#104D511yescharacteristic
82C3×C4×C4⋊C4—51no—
83C3×#339—51no—
84C3×#339—51no—
85C2×C5×#41C22×C311yescharacteristic
86C2×C5×M16C22×C311yescharacteristic
87C2×C42×C5C22×C311yescharacteristic
88Q8×Dic5C3×C411yescharacteristic, complemented
89Q8×Dic5C3×C411yescharacteristic, complemented
90D4×Dic5—21no—
91D4×Dic5—21no—
92D4×Dic5C3×C411yescharacteristic, complemented
93D4×Dic5C3×C411yescharacteristic, complemented
94#4072—21no—
95#4072C22×C311yescharacteristic
96C2×#683—21no—
97C2×C4×Dic5—21no—
98C2×C4×Dic5C22×C311yescharacteristic
99C2×C4×Dic5C22×C311yescharacteristic
100C2×#681C22×C311yescharacteristic
101C5×#104—22no—
102#3969—22no—
103#3963—21no—
104#3946—21no—
105#3946—21no—
106#1806—51noSylow
107C23×C3×C5—21no—
108C23×C3×C5C2×D411yescharacteristic
109C3×C5×Q8—21no—
110C3×C5×Q8C4⋊C412yescomplemented
111C3×C5×D4—22no—
112C3×C5×D4—22no—
113C3×C5×D4—21no—
114C3×C5×D4C4⋊C412yescomplemented
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—41no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5C22×C411yescharacteristic
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×Q811yescharacteristic
125C2×C3×C4×C5#4111yescharacteristic
126C2×C3×C4×C5#4111yescharacteristic
127C2×C3×Dic5—22no—
128C2×C3×Dic5—22no—
129C2×C3×Dic5—21no—
130C2×C3×Dic5—21no—
131C2×C3×Dic5—41no—
132C2×C3×Dic5—42no—
133C2×C3×Dic5—41no—
134C2×C3×Dic5—41no—
135C2×C3×Dic5—42no—
136C2×C3×Dic5C22⋊C411yescharacteristic
137C2×C3×Dic5C22⋊C411yescharacteristic
138C2×C3×Dic5C22⋊C411yescharacteristic
139C2×C3×Dic5C22⋊C411yescharacteristic
140C3×C5×C8—41no—
141C3×#168—41no—
142C2×C3×#41C2×D511yescharacteristic
143C2×C3×M16—51no—
144C2×C3×M16C2×D511yescharacteristic
145C3×C4×Q8—51no—
146C3×C4×Q8—51no—
147C3×C4×D4—51no—
148C3×C4×D4—51no—
149C3×C4×D4—101no—
150C3×C4×D4—101no—
151C3×#117—51no—
152C3×#117—101no—
153C2×C3×C4⋊C4—101no—
154C2×C3×C42—51no—
155C2×C3×C42—51no—
156C2×C3×C42—101no—
157C2×C3×C42C2×D511yescharacteristic
158C3×C4⋊C8—101no—
159C3×C4⋊C8—101no—
160C3×#104—22no—
161C3×#104—102no—
162C3×#95—101no—
163C5×#41—21no—
164C5×#41—21no—
165C5×#41C3×D412yes—
166C5×#41C3×Q812yes—
167C2×C5×Q8C2×C3×C411yescharacteristic
168C2×C5×D4—21no—
169C2×C5×D4C2×C3×C411yescharacteristic
170C22×C4×C5—21no—
171C22×C4×C5—21no—
172C22×C4×C5C23×C311yescharacteristic
173C22×Dic5—21no—
174C22×Dic5—21no—
175C22×Dic5—21no—
176C22×Dic5C3×D411yescharacteristic
177C22×Dic5C3×D411yescharacteristic
178C5×M16—21no—
179C5×M16—21no—
180C2×C5×C8—21no—
181C42×C5—21no—
182C42×C5—21no—
183C22⋊Dic5—21no—
184C22⋊Dic5—21no—
185C22⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5C2×C3×C411yescharacteristic
189C4⋊Dic5C2×C3×C411yescharacteristic
190#683—24no—
191C4×Dic5—21no—
192C4×Dic5—21no—
193C4×Dic5—21no—
194C4×Dic5—21no—
195C4×Dic5—21no—
196C4×Dic5C2×C3×C411yescharacteristic
197C4×Dic5C2×C3×C411yescharacteristic
198C4×Dic5C3×D411yescharacteristic
199C4×Dic5C3×D411yescharacteristic
200#681—21no—
201#681—21no—
202C2×#168—21no—
203C4×#41—51no—
204#423—51no—
205C2×#104—51no—
206C2×#104C3×D511yescharacteristic
207C4×C4⋊C4—51no—
208#339—51no—
209#339—51no—
210C22×C3×C5—21no—
211C22×C3×C5—21no—
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—41no—
215C22×C3×C5C2×C22⋊C411yescharacteristic
216C22×C3×C5#12111yescharacteristic
217C22×C3×C5C22⋊Q811yescharacteristic
218C3×C4×C5—22no—
219C3×C4×C5—42no—
220C3×C4×C5C2×C4⋊C411yescharacteristic
221C3×C4×C5#11711yescharacteristic
222C3×C4×C5#12111yescharacteristic
223C3×C4×C5C22⋊Q811yescharacteristic
224C3×Dic5—21no—
225C3×Dic5—21no—
226C3×Dic5—21no—
227C3×Dic5—42no—
228C3×Dic5—41no—
229C3×Dic5—41no—
230C3×Dic5#10412yescomplemented
231C3×#41—21no—
232C3×#41—21no—
233C3×#41Dic1012yes—
234C3×#41C22⋊D512yes—
235C2×C3×Q8C4×D511yescharacteristic
236C2×C3×D4—21no—
237C2×C3×D4C4×D511yescharacteristic
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—51no—
241C22×C3×C4—51no—
242C22×C3×C4—101no—
243C22×C3×C4—101no—
244C22×C3×C4—101no—
245C22×C3×C4C22×D511yescharacteristic
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—101no—
249C3×M16—101no—
250C2×C3×C8—21no—
251C2×C3×C8—101no—
252C3×C4⋊C4—51no—
253C3×C4⋊C4—51no—
254C3×C4⋊C4—101no—
255C3×C4⋊C4—101no—
256C3×C4⋊C4—104no—
257C3×C22⋊C4—101no—
258C3×C22⋊C4—101no—
259C3×C22⋊C4—101no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—51no—
263C3×C42—51no—
264C3×C42—51no—
265C3×C42—51no—
266C3×C42—101no—
267C3×C42—101no—
268C3×C42—101no—
269C3×C42—101no—
270C3×C42—101no—
271C23×C5—21no—
272C23×C5C2×C3×D411yescharacteristic
273C5×Q8—21no—
274C5×Q8C3×C4⋊C412yescomplemented
275C5×D4—22no—
276C5×D4—22no—
277C5×D4—21no—
278C5×D4C3×C4⋊C412yescomplemented
279C2×C4×C5—21no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5C22×C3×C411yescharacteristic
286C2×C4×C5C2×C3×D411yescharacteristic
287C2×C4×C5C2×C3×D411yescharacteristic
288C2×C4×C5C2×C3×Q811yescharacteristic
289C2×C4×C5C3×#4111yescharacteristic
290C2×C4×C5C3×#4111yescharacteristic
291C2×Dic5—22no—
292C2×Dic5—22no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—41no—
296C2×Dic5—42no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—42no—
300C2×Dic5C3×C22⋊C411yescharacteristic
301C2×Dic5C3×C22⋊C411yescharacteristic
302C2×Dic5C3×C22⋊C411yescharacteristic
303C2×Dic5C3×C22⋊C411yescharacteristic
304C5×C8—41no—
305#168—41no—
306C2×#41C2×C3×D511yescharacteristic
307C2×M16—51no—
308C2×M16C2×C3×D511yescharacteristic
309C4×Q8—51no—
310C4×Q8—51no—
311C4×D4—51no—
312C4×D4—51no—
313C4×D4—101no—
314C4×D4—101no—
315#117—51no—
316#117—101no—
317C2×C4⋊C4—101no—
318C2×C42—51no—
319C2×C42—51no—
320C2×C42—101no—
321C2×C42C2×C3×D511yescharacteristic
322C4⋊C8—101no—
323C4⋊C8—101no—
324#104—22no—
325#104—102no—
326#95—101no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—42no—
330C2×C3×C5#38011yescharacteristic
331C2×C3×C5C2×#10411yescharacteristic
332C2×C3×C5#42111yescharacteristic
333C23×C3—21no—
334C23×C3D4×D511yescharacteristic
335C3×Q8—21no—
336C3×Q8#68312yescomplemented
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—21no—
340C3×D4#68312yescomplemented
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—51no—
348C2×C3×C4—51no—
349C2×C3×C4—51no—
350C2×C3×C4—51no—
351C2×C3×C4—102no—
352C2×C3×C4—102no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—202no—
356C2×C3×C4—201no—
357C2×C3×C4—202no—
358C2×C3×C4—201no—
359C2×C3×C4—201no—
360C2×C3×C4C2×Dic1011yescharacteristic
361C2×C3×C4C2×C4×D511yescharacteristic
362C2×C3×C4D4×D511yescharacteristic
363C2×C3×C4#71111yescharacteristic
364C2×C3×C4#71111yescharacteristic
365C2×C3×C4C2×C22⋊D511yescharacteristic
366C3×C8—41no—
367C3×C8—201no—
368C22×C5—21no—
369C22×C5—21no—
370C22×C5—22no—
371C22×C5—41no—
372C22×C5—41no—
373C22×C5C2×C3×C22⋊C411yescharacteristic
374C22×C5C3×#12111yescharacteristic
375C22×C5C3×C22⋊Q811yescharacteristic
376C4×C5—22no—
377C4×C5—42no—
378C4×C5C2×C3×C4⋊C411yescharacteristic
379C4×C5C3×#11711yescharacteristic
380C4×C5C3×#12111yescharacteristic
381C4×C5C3×C22⋊Q811yescharacteristic
382Dic5—21no—
383Dic5—21no—
384Dic5—21no—
385Dic5—42no—
386Dic5—41no—
387Dic5—41no—
388Dic5C3×#10412yescomplemented
389#41—21no—
390#41—21no—
391#41C3×Dic1012yes—
392#41C3×C22⋊D512yes—
393C2×Q8C3×C4×D511yescharacteristic
394C2×D4—21no—
395C2×D4C3×C4×D511yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—51no—
399C22×C4—51no—
400C22×C4—101no—
401C22×C4—101no—
402C22×C4—101no—
403C22×C4C22×C3×D511yescharacteristic
404M16—21no—
405M16—21no—
406M16—101no—
407M16—101no—
408C2×C8—21no—
409C2×C8—101no—
410C4⋊C4—51no—
411C4⋊C4—51no—
412C4⋊C4—101no—
413C4⋊C4—101no—
414C4⋊C4—104no—
415C22⋊C4—101no—
416C22⋊C4—101no—
417C22⋊C4—101no—
418C42—21no—
419C42—21no—
420C42—51no—
421C42—51no—
422C42—51no—
423C42—51no—
424C42—101no—
425C42—101no—
426C42—101no—
427C42—101no—
428C42—101no—
429C3×C5#180611yescharacteristic, complemented, Hall
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3#402411yescharacteristic
436C22×C3D5×C22⋊C411yescharacteristic
437C22×C3#408511yescharacteristic
438C3×C4—22no—
439C3×C4—42no—
440C3×C4—52no—
441C3×C4—101no—
442C3×C4—101no—
443C3×C4—101no—
444C3×C4—202no—
445C3×C4—201no—
446C3×C4—201no—
447C3×C4#402311yescharacteristic
448C3×C4#402411yescharacteristic
449C3×C4C2×#68311yescharacteristic
450C3×C4#408511yescharacteristic
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—42no—
454C2×C5C3×#38011yescharacteristic
455C2×C5C2×C3×#10411yescharacteristic
456C2×C5C3×#42111yescharacteristic
457C23—21no—
458C23C3×D4×D511yescharacteristic
459Q8—21no—
460Q8C3×#68312yescomplemented
461D4—22no—
462D4—22no—
463D4—21no—
464D4C3×#68312yescomplemented
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—51no—
472C2×C4—51no—
473C2×C4—51no—
474C2×C4—51no—
475C2×C4—102no—
476C2×C4—102no—
477C2×C4—101no—
478C2×C4—101no—
479C2×C4—202no—
480C2×C4—201no—
481C2×C4—202no—
482C2×C4—201no—
483C2×C4—201no—
484C2×C4C2×C3×Dic1011yescharacteristic
485C2×C4C2×C3×C4×D511yescharacteristic
486C2×C4C3×D4×D511yescharacteristic
487C2×C4C3×#71111yescharacteristic
488C2×C4C3×#71111yescharacteristic
489C2×C4C2×C3×C22⋊D511yescharacteristic
490C8—41no—
491C8—201no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—42no—
495C2×C3D5×#10411yescharacteristic
496C2×C3#6511011yescharacteristic
497C2×C3#6523111yescharacteristic
498C5C3×#180611yescharacteristic, complemented, Sylow
499C22—21no—
500C22—21no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22C3×#402411yescharacteristic
505C22C3×D5×C22⋊C411yescharacteristic
506C22C3×#408511yescharacteristic
507C4—22no—
508C4—42no—
509C4—52no—
510C4—101no—
511C4—101no—
512C4—101no—
513C4—202no—
514C4—201no—
515C4—201no—
516C4C3×#402311yescharacteristic
517C4C3×#402411yescharacteristic
518C4C2×C3×#68311yescharacteristic
519C4C3×#408511yescharacteristic
520C3#1060114811yescharacteristic, direct product factor, Sylow
521C2—21no—
522C2—21no—
523C2—42no—
524C2C3×D5×#10411yescharacteristic
525C2C3×#6511011yescharacteristic
526C2C3×#6523111yescharacteristic
subgroupG/H
centerC2×C3×C4D4×D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC2×C3×C5×#104C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×C5×#41C22
cosocleC22×C4×C5C23×C3
hypercenterC2×C3×#104D5
hypocenterC5C3×#1806
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#104

Complements

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

NH ≅ G/Ntype
4. C3×#65407C2semidirect
5. C3×C4×#683C2semidirect
6. C3×#64775C2semidirect
7. C3×#64743C2semidirect
8. #10601148C3direct (N × H)
12. C3×Q8×Dic5C4semidirect
13. C3×Q8×Dic5C4semidirect
16. C3×D4×Dic5C4semidirect
17. C3×D4×Dic5C4semidirect
34. #65407C2×C3semidirect
35. C4×#683C2×C3semidirect
36. #64775C2×C3semidirect
37. #64743C2×C3semidirect
88. Q8×Dic5C3×C4semidirect
89. Q8×Dic5C3×C4semidirect
92. D4×Dic5C3×C4semidirect
93. D4×Dic5C3×C4semidirect
110. C3×C5×Q8C4⋊C4semidirect
114. C3×C5×D4C4⋊C4semidirect
230. C3×Dic5#104semidirect
274. C5×Q8C3×C4⋊C4semidirect
278. C5×D4C3×C4⋊C4semidirect
336. C3×Q8#683semidirect
340. C3×D4#683semidirect
388. Dic5C3×#104semidirect
429. C3×C5#1806semidirect
460. Q8C3×#683semidirect
464. D4C3×#683semidirect
498. C5C3×#1806semidirect
520. C3#10601148direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×Dic52
C2×C3×C5×#1042
C2×C3×#39692
C3×#654072
C3×C4×#6832
C3×#647752
C3×#647432
#106011483
C3×#18065