Finite Group Databasebeta

← C3×#10610972 #14430685

Subgroups — C3×#10610972 #14430685

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65856C211yesmaximal, characteristic
2C2×C3×#4036C211yesmaximal, characteristic, complemented
3C3×C5×#428C211yesmaximal, characteristic, complemented
4C3×#65422C211yesmaximal, characteristic
5C3×#65407C211yesmaximal, characteristic, complemented
6C3×#65284C211yesmaximal, characteristic, complemented
7C3×#65255C211yesmaximal, characteristic
8C3×#65180C212yesmaximal, complemented
9C3×#65166C212yesmaximal, complemented
10C3×#65160C212yesmaximal, complemented
11C3×#65146C212yesmaximal, complemented
12#10610972C311yesmaximal, characteristic, direct product factor, Hall
13C2×C3×#709C2211yescharacteristic
14C2×C3×C5×M16C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊Dic5—21no—
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×C3×#683—21no—
20C2×C3×C4×Dic5C2211yescharacteristic
21C2×C3×#681C2211yescharacteristic
22C3×#4039C2211yescharacteristic
23C3×#4039C2211yescharacteristic
24C3×#4036—21no—
25C3×#4036—21no—
26C3×#4036C2212yes—
27C3×#4036C2212yes—
28C3×#4033C2211yescharacteristic
29C3×#4033C2211yescharacteristic
30C3×#4027—21no—
31C3×#4027—21no—
32C3×#4018—21no—
33C3×C42×D5—21no—
34C3×C5×#107C2212yes—
35C3×C5×#106C2212yes—
36C3×#3953C2212yes—
37C3×#3948C2212yes—
38C3×#3946C2212yes—
39C3×#3946C2212yes—
40C3×#3942C2212yes—
41C3×#3941C2212yes—
42C3×#3940C2212yes—
43C3×#3939C2212yes—
44C3×#3080—51nomaximal, Hall
45#65856C2×C311yescharacteristic
46C2×#4036C2×C311yescharacteristic, complemented
47C5×#428C2×C311yescharacteristic, complemented
48#65422C2×C311yescharacteristic
49#65407C2×C311yescharacteristic, complemented
50#65284C2×C311yescharacteristic, complemented
51#65255C2×C311yescharacteristic
52#65180C2×C312yescomplemented
53#65166C2×C312yescomplemented
54#65160C2×C312yescomplemented
55#65146C2×C312yescomplemented
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C2×C3×C22⋊D5—21no—
59C22×C3×Dic5—21no—
60C22×C3×Dic5D411yescharacteristic
61C22×C3×Dic5D411yescharacteristic
62C3×#709—22no—
63C3×#709Q814yes—
64C2×C3×D20C2311yescharacteristic
65C2×C3×C4×D5—21no—
66C2×C3×C4×D5—21no—
67C2×C3×C4×D5—21no—
68C2×C3×Dic10C2311yescharacteristic
69C3×C5×M16—22no—
70C2×C3×C5×C8C2312yes—
71C3×C5×C4⋊C4—21no—
72C3×C5×C4⋊C4C2311yescharacteristic
73C3×C5×C4⋊C4C2311yescharacteristic
74C3×C5×C4⋊C4C2312yes—
75C3×C5×C22⋊C4—21no—
76C3×C42×C5—21no—
77C3×#685—21no—
78C3×#685—21no—
79C3×C4⋊Dic5—21no—
80C3×C4⋊Dic5—22no—
81C3×C4⋊Dic5—22no—
82C3×C4⋊Dic5C2312yes—
83C3×#683—21no—
84C3×#683—21no—
85C3×#683—22no—
86C3×#683—22no—
87C3×C4×Dic5—21no—
88C3×C4×Dic5—21no—
89C3×C4×Dic5D411yescharacteristic
90C3×C4×Dic5D411yescharacteristic
91C3×C4×Dic5C2311yescharacteristic
92C3×C4×Dic5C2311yescharacteristic
93C3×#681—22no—
94C2×C3×#168C2312yes—
95C2×C3×#125—51no—
96C3×C4×#41—51no—
97C3×#479—52no—
98C3×#479—52no—
99C3×#478—52no—
100C3×#478—52no—
101C3×#428—51no—
102C3×#428D511yescharacteristic, complemented
103C3×#423—51no—
104C3×#417—51no—
105C3×#417—51no—
106C2×#709C22×C311yescharacteristic
107C2×C5×M16C22×C311yescharacteristic
108C5×#117C22×C311yescharacteristic
109C2×C5×C4⋊C4C22×C311yescharacteristic
110C2×C4⋊Dic5—21no—
111C2×C4⋊Dic5C22×C311yescharacteristic
112C2×#683—21no—
113C2×C4×Dic5C22×C311yescharacteristic
114C2×#681C22×C311yescharacteristic
115#4039C22×C311yescharacteristic
116#4039C22×C311yescharacteristic
117#4036—21no—
118#4036—21no—
119#4036C22×C312yes—
120#4036C22×C312yes—
121#4033C22×C311yescharacteristic
122#4033C22×C311yescharacteristic
123#4027—21no—
124#4027—21no—
125#4018—21no—
126C42×D5—21no—
127C5×#107C22×C312yes—
128C5×#106C22×C312yes—
129#3953C22×C312yes—
130#3948C22×C312yes—
131#3946C22×C312yes—
132#3946C22×C312yes—
133#3942C22×C312yes—
134#3941C22×C312yes—
135#3940C22×C312yes—
136#3939C22×C312yes—
137#3080—51noSylow
138C23×C3×C5C2×D411yescharacteristic
139C22×C3×D5—21no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—42no—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×Q812yes—
146C2×C3×C4×C5#4112yes—
147C2×C3×C4×C5C2411yescharacteristic
148C3×C22⋊D5—24no—
149C2×C3×Dic5—21no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—41no—
152C2×C3×Dic5—41no—
153C2×C3×Dic5—42no—
154C2×C3×Dic5C2×D411yescharacteristic
155C2×C3×Dic5C2×D411yescharacteristic
156C2×C3×Dic5C2×D411yescharacteristic
157C2×C3×Dic5C2×D411yescharacteristic
158C3×D20—21no—
159C3×D20C2×Q812yes—
160C3×C4×D5—24no—
161C3×C4×D5—42no—
162C3×C4×D5—42no—
163C3×Dic10—21no—
164C3×Dic10C2×Q812yes—
165C3×C5×C8—22no—
166C3×#168—22no—
167C2×C3×#41—51no—
168C2×C3×M16—51no—
169C2×C3×M16C2×D511yescharacteristic
170C3×#125—52no—
171C3×#125—52no—
172C3×#125—101no—
173C3×#125—101no—
174C3×C4×Q8—51no—
175C3×C4×Q8—51no—
176C3×C4×D4—51no—
177C3×C4×D4—51no—
178C3×C4×D4—101no—
179C3×C4×D4—101no—
180C3×#117—101no—
181C3×#117C2×D511yescharacteristic
182C2×C3×C4⋊C4—51no—
183C2×C3×C4⋊C4—101no—
184C2×C3×C4⋊C4—101no—
185C2×C3×C4⋊C4C2×D511yescharacteristic
186C2×C3×C42—51no—
187C2×C3×C42—101no—
188C3×#107—52no—
189C3×#107C2×D512yes—
190C3×#106—52no—
191C3×#106C2×D512yes—
192C3×C4⋊C8—52no—
193C3×C4⋊C8—52no—
194C3×#103—52no—
195C3×#103—52no—
196C3×#102—52no—
197C3×#102—52no—
198C22×C4×C5—21no—
199C22×C4×C5C23×C311yescharacteristic
200C2×C22⋊D5—21no—
201C22×Dic5—21no—
202C22×Dic5C3×D411yescharacteristic
203C22×Dic5C3×D411yescharacteristic
204#709—22no—
205#709C3×Q814yes—
206C2×D20C23×C311yescharacteristic
207C2×C4×D5—21no—
208C2×C4×D5—21no—
209C2×C4×D5—21no—
210C2×Dic10C23×C311yescharacteristic
211C5×M16—22no—
212C2×C5×C8C23×C312yes—
213C5×C4⋊C4—21no—
214C5×C4⋊C4C23×C312yes—
215C5×C4⋊C4C23×C311yescharacteristic
216C5×C4⋊C4C23×C311yescharacteristic
217C5×C22⋊C4—21no—
218C42×C5—21no—
219#685—21no—
220#685—21no—
221C4⋊Dic5—21no—
222C4⋊Dic5—22no—
223C4⋊Dic5—22no—
224C4⋊Dic5C23×C312yes—
225#683—21no—
226#683—21no—
227#683—22no—
228#683—22no—
229C4×Dic5—21no—
230C4×Dic5—21no—
231C4×Dic5C3×D411yescharacteristic
232C4×Dic5C3×D411yescharacteristic
233C4×Dic5C23×C311yescharacteristic
234C4×Dic5C23×C311yescharacteristic
235#681—22no—
236C2×#168C23×C312yes—
237C2×#125—51no—
238C4×#41—51no—
239#479—52no—
240#479—52no—
241#478—52no—
242#478—52no—
243#428—51no—
244#428C3×D511yescharacteristic, complemented
245#423—51no—
246#417—51no—
247#417—51no—
248C22×C3×C5—21no—
249C22×C3×C5—21no—
250C22×C3×C5C22⋊Q812yes—
251C22×C3×C5C22×D411yescharacteristic
252C2×C3×D5—22no—
253C2×C3×D5—41no—
254C2×C3×D5—41no—
255C3×C4×C5—41no—
256C3×C4×C5—41no—
257C3×C4×C5—42no—
258C3×C4×C5C22⋊Q812yes—
259C3×C4×C5C22×Q811yescharacteristic
260C3×C4×C5C2×#4111yescharacteristic
261C3×Dic5—22no—
262C3×Dic5—21no—
263C3×Dic5—21no—
264C3×Dic5—21no—
265C3×Dic5—21no—
266C3×Dic5—42no—
267C3×#41—54no—
268C3×#41—102no—
269C2×C3×Q8—51no—
270C2×C3×D4—51no—
271C2×C3×D4—101no—
272C22×C3×C4—21no—
273C22×C3×C4—51no—
274C22×C3×C4—51no—
275C22×C3×C4—101no—
276C22×C3×C4—101no—
277C22×C3×C4—101no—
278C22×C3×C4—101no—
279C22×C3×C4C22×D511yescharacteristic
280C3×M16—22no—
281C3×M16—102no—
282C2×C3×C8—52no—
283C2×C3×C8C22×D512yes—
284C3×C4⋊C4—21no—
285C3×C4⋊C4—52no—
286C3×C4⋊C4—102no—
287C3×C4⋊C4—102no—
288C3×C4⋊C4—101no—
289C3×C4⋊C4—101no—
290C3×C4⋊C4—102no—
291C3×C4⋊C4—102no—
292C3×C4⋊C4—101no—
293C3×C4⋊C4C22×D512yes—
294C3×C4⋊C4C22×D511yescharacteristic
295C3×C4⋊C4C22×D511yescharacteristic
296C3×C22⋊C4—21no—
297C3×C22⋊C4—101no—
298C3×C22⋊C4—101no—
299C3×C42—21no—
300C3×C42—51no—
301C3×C42—51no—
302C3×C42—51no—
303C3×C42—51no—
304C3×C42—101no—
305C3×C42—101no—
306C23×C5C2×C3×D411yescharacteristic
307C22×D5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5—22no—
311C2×C4×C5—42no—
312C2×C4×C5C2×C3×D411yescharacteristic
313C2×C4×C5C2×C3×Q812yes—
314C2×C4×C5C3×#4112yes—
315C2×C4×C5C24×C311yescharacteristic
316C22⋊D5—24no—
317C2×Dic5—21no—
318C2×Dic5—22no—
319C2×Dic5—41no—
320C2×Dic5—41no—
321C2×Dic5—42no—
322C2×Dic5C2×C3×D411yescharacteristic
323C2×Dic5C2×C3×D411yescharacteristic
324C2×Dic5C2×C3×D411yescharacteristic
325C2×Dic5C2×C3×D411yescharacteristic
326D20—21no—
327D20C2×C3×Q812yes—
328C4×D5—24no—
329C4×D5—42no—
330C4×D5—42no—
331Dic10—21no—
332Dic10C2×C3×Q812yes—
333C5×C8—22no—
334#168—22no—
335C2×#41—51no—
336C2×M16—51no—
337C2×M16C2×C3×D511yescharacteristic
338#125—52no—
339#125—52no—
340#125—101no—
341#125—101no—
342C4×Q8—51no—
343C4×Q8—51no—
344C4×D4—51no—
345C4×D4—51no—
346C4×D4—101no—
347C4×D4—101no—
348#117—101no—
349#117C2×C3×D511yescharacteristic
350C2×C4⋊C4—51no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4C2×C3×D511yescharacteristic
354C2×C42—51no—
355C2×C42—101no—
356#107—52no—
357#107C2×C3×D512yes—
358#106—52no—
359#106C2×C3×D512yes—
360C4⋊C8—52no—
361C4⋊C8—52no—
362#103—52no—
363#103—52no—
364#102—52no—
365#102—52no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C22⋊Q811yescharacteristic
368C2×C3×C5#57511yescharacteristic
369C2×C3×C5#57511yescharacteristic
370C3×D5—42no—
371C23×C3—101no—
372C23×C3D4×D511yescharacteristic
373C3×Q8—52no—
374C3×Q8—101no—
375C3×D4—52no—
376C3×D4—104no—
377C3×D4—101no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—42no—
382C2×C3×C4—51no—
383C2×C3×C4—51no—
384C2×C3×C4—51no—
385C2×C3×C4—51no—
386C2×C3×C4—101no—
387C2×C3×C4—104no—
388C2×C3×C4—102no—
389C2×C3×C4—202no—
390C2×C3×C4—202no—
391C2×C3×C4—201no—
392C2×C3×C4—201no—
393C2×C3×C4—202no—
394C2×C3×C4#70912yes—
395C2×C3×C4D4×D511yescharacteristic
396C2×C3×C4Q8×D512yes—
397C2×C3×C4C23×D511yescharacteristic
398C3×C8—22no—
399C3×C8—102no—
400C22×C5—21no—
401C22×C5—21no—
402C22×C5C3×C22⋊Q812yes—
403C22×C5C22×C3×D411yescharacteristic
404C2×D5—22no—
405C2×D5—41no—
406C2×D5—41no—
407C4×C5—41no—
408C4×C5—41no—
409C4×C5—42no—
410C4×C5C3×C22⋊Q812yes—
411C4×C5C22×C3×Q811yescharacteristic
412C4×C5C2×C3×#4111yescharacteristic
413Dic5—21no—
414Dic5—21no—
415Dic5—21no—
416Dic5—21no—
417Dic5—22no—
418Dic5—42no—
419#41—54no—
420#41—102no—
421C2×Q8—51no—
422C2×D4—51no—
423C2×D4—101no—
424C22×C4—21no—
425C22×C4—51no—
426C22×C4—51no—
427C22×C4—101no—
428C22×C4—101no—
429C22×C4—101no—
430C22×C4—101no—
431C22×C4C22×C3×D511yescharacteristic
432M16—22no—
433M16—102no—
434C2×C8—52no—
435C2×C8C22×C3×D512yes—
436C4⋊C4—21no—
437C4⋊C4—52no—
438C4⋊C4—102no—
439C4⋊C4—102no—
440C4⋊C4—102no—
441C4⋊C4—102no—
442C4⋊C4—101no—
443C4⋊C4—101no—
444C4⋊C4—101no—
445C4⋊C4C22×C3×D512yes—
446C4⋊C4C22×C3×D511yescharacteristic
447C4⋊C4C22×C3×D511yescharacteristic
448C22⋊C4—21no—
449C22⋊C4—101no—
450C22⋊C4—101no—
451C42—21no—
452C42—51no—
453C42—51no—
454C42—51no—
455C42—51no—
456C42—101no—
457C42—101no—
458C3×C5#308011yescharacteristic, complemented, Hall
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—102no—
462C22×C3—201no—
463C22×C3—201no—
464C22×C3#404212yes—
465C22×C3C2×D4×D511yescharacteristic
466C3×C4—42no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—102no—
470C3×C4—101no—
471C3×C4—101no—
472C3×C4—101no—
473C3×C4—101no—
474C3×C4—202no—
475C3×C4#404212yes—
476C3×C4C2×#70911yescharacteristic
477C3×C4C2×Q8×D511yescharacteristic
478C2×C5—22no—
479C2×C5C2×C3×C22⋊Q811yescharacteristic
480C2×C5C3×#57511yescharacteristic
481C2×C5C3×#57511yescharacteristic
482D5—42no—
483C23—101no—
484C23C3×D4×D511yescharacteristic
485Q8—52no—
486Q8—101no—
487D4—52no—
488D4—104no—
489D4—101no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—51no—
495C2×C4—51no—
496C2×C4—51no—
497C2×C4—51no—
498C2×C4—101no—
499C2×C4—104no—
500C2×C4—102no—
501C2×C4—202no—
502C2×C4—202no—
503C2×C4—201no—
504C2×C4—201no—
505C2×C4—202no—
506C2×C4C3×#70912yes—
507C2×C4C3×D4×D511yescharacteristic
508C2×C4C3×Q8×D512yes—
509C2×C4C23×C3×D511yescharacteristic
510C8—22no—
511C8—102no—
512C2×C3—22no—
513C2×C3—202no—
514C2×C3C2×#404211yescharacteristic
515C2×C3#6610711yescharacteristic
516C2×C3#6611111yescharacteristic
517C5C3×#308011yescharacteristic, complemented, Sylow
518C22—21no—
519C22—21no—
520C22—102no—
521C22—201no—
522C22—201no—
523C22C3×#404212yes—
524C22C2×C3×D4×D511yescharacteristic
525C4—42no—
526C4—41no—
527C4—41no—
528C4—101no—
529C4—101no—
530C4—101no—
531C4—101no—
532C4—102no—
533C4—202no—
534C4C3×#404212yes—
535C4C2×C3×#70911yescharacteristic
536C4C2×C3×Q8×D511yescharacteristic
537C3#1061097211yescharacteristic, direct product factor, Sylow
538C2—22no—
539C2—202no—
540C2C2×C3×#404211yescharacteristic
541C2C3×#6610711yescharacteristic
542C2C3×#6611111yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#428C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
hypercenterC3×#428D5
hypocenterC5C3×#3080
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#428

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4036C2semidirect
3. C3×C5×#428C2semidirect
5. C3×#65407C2semidirect
6. C3×#65284C2semidirect
8. C3×#65180C2semidirect
9. C3×#65166C2semidirect
10. C3×#65160C2semidirect
11. C3×#65146C2semidirect
12. #10610972C3direct (N × H)
46. C2×#4036C2×C3semidirect
47. C5×#428C2×C3semidirect
49. #65407C2×C3semidirect
50. #65284C2×C3semidirect
52. #65180C2×C3semidirect
53. #65166C2×C3semidirect
54. #65160C2×C3semidirect
55. #65146C2×C3semidirect
102. C3×#428D5semidirect
244. #428C3×D5semidirect
458. C3×C5#3080semidirect
517. C5C3×#3080semidirect
537. C3#10610972direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658562
C2×C3×#40362
C3×C5×#4282
C3×#654222
C3×#654072
C3×#652842
C3×#652552
C3×#651802
C3×#651662
C3×#651602
C3×#651462
#106109723
C3×#30805