Finite Group Databasebeta

← C2×C3×#65096 #14427607

Subgroups — C2×C3×#65096 #14427607

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4092C211yesmaximal, characteristic, complemented
2C22×C3×#688C211yesmaximal, characteristic, complemented
3C22×C3×C8×D5C211yesmaximal, characteristic, complemented
4C2×C3×#4041C211yesmaximal, characteristic
5C2×C3×C5×#103C211yesmaximal, characteristic, complemented
6C2×C3×#3953C211yesmaximal, characteristic, complemented
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C3×#65096C218yesmaximal, direct product factor
9C2×#65096C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×D20—21no—
12C22×C3×D20C2211yescharacteristic
13C22×C3×C4×D5C2211yescharacteristic
14C22×C3×C5×C8C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C3×#4092—22no—
17C3×#4092C2214yescomplemented
18C2×C3×#688—24no—
19C2×C3×#688C2214yescomplemented
20C2×C3×#685—21no—
21C2×C3×#685—21no—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×#683—21no—
24C22×C3×#168C2211yescharacteristic
25C2×C3×C8×D5—24no—
26C2×C3×C8×D5C2214yescomplemented
27C3×#4041—22no—
28C3×#4041C2214yes—
29C3×C5×#103C2214yescomplemented
30C3×#3953C2214yescomplemented
31C3×#3940C2214yescomplemented
32C2×C3×#465—51nomaximal, Hall
33C2×#4092C2×C311yescharacteristic, complemented
34C22×#688C2×C311yescharacteristic, complemented
35C22×C8×D5C2×C311yescharacteristic, complemented
36C2×#4041C2×C311yescharacteristic
37C2×C5×#103C2×C311yescharacteristic, complemented
38C2×#3953C2×C311yescharacteristic, complemented
39C2×#3940C2×C311yescharacteristic, complemented
40#65096C2×C318yesdirect product factor
41C23×C3×D5—21no—
42C23×C3×D5D411yescharacteristic
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8C2312yes—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×Dic5—21no—
51C22×C3×Dic5D411yescharacteristic
52C2×C3×D20—22no—
53C2×C3×D20—22no—
54C2×C3×D20—24no—
55C2×C3×D20—21no—
56C2×C3×D20—44no—
57C2×C3×D20C2312yes—
58C2×C3×C4×D5—22no—
59C2×C3×C4×D5—22no—
60C2×C3×C4×D5D412yes—
61C2×C3×C4×D5C2312yes—
62C2×C3×C5×C8—22no—
63C2×C3×C5×C8C2312yes—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4C2312yes—
66C3×#688—28no—
67C3×#685—24no—
68C3×#685—24no—
69C3×C4⋊Dic5—21no—
70C3×C4⋊Dic5C2312yes—
71C3×#683—24no—
72C2×C3×#168D414yescomplemented
73C2×C3×#168C2312yes—
74C3×C8×D5—28no—
75C22×C3×SD16—51no—
76C23×C3×C8—51no—
77C2×C3×C22⋊Q8—51no—
78C2×C3×#121—51no—
79C3×#465—58no—
80C2×C3×#106—51no—
81C2×C3×#103D511yescharacteristic, complemented
82C2×C3×#102—51no—
83C22×C5×Q8C22×C311yescharacteristic
84C22×D20—21no—
85C22×D20C22×C311yescharacteristic
86C22×C4×D5C22×C311yescharacteristic
87C22×C5×C8C22×C311yescharacteristic
88C2×C5×C4⋊C4C22×C311yescharacteristic
89#4092—22no—
90#4092C22×C314yescomplemented
91C2×#688—24no—
92C2×#688C22×C314yescomplemented
93C2×#685—21no—
94C2×#685—21no—
95C2×C4⋊Dic5C22×C311yescharacteristic
96C2×#683—21no—
97C22×#168C22×C311yescharacteristic
98C2×C8×D5—24no—
99C2×C8×D5C22×C314yescomplemented
100#4041—22no—
101#4041C22×C314yes—
102C5×#103C22×C314yescomplemented
103#3953C22×C314yescomplemented
104#3940C22×C314yescomplemented
105C2×#465—51noSylow
106C23×C3×C5C2×D411yescharacteristic
107C22×C3×D5—22no—
108C22×C3×D5—22no—
109C22×C3×D5—22no—
110C22×C3×D5—42no—
111C22×C3×D5—42no—
112C22×C3×D5—42no—
113C22×C3×D5SD1614yescomplemented
114C22×C3×D5C2×D412yes—
115C3×C5×Q8—22no—
116C3×C5×Q8—22no—
117C3×C5×Q8—42no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—42no—
121C2×C3×C4×C5—42no—
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5C2×D412yes—
124C2×C3×C4×C5#4112yes—
125C2×C3×C4×C5C2411yescharacteristic
126C2×C3×Dic5—22no—
127C2×C3×Dic5—22no—
128C2×C3×Dic5—42no—
129C2×C3×Dic5C2×D412yes—
130C3×D20—22no—
131C3×D20—22no—
132C3×D20—48no—
133C3×D20—42no—
134C3×C4×D5—24no—
135C3×C4×D5—22no—
136C3×C4×D5—22no—
137C3×C5×C8—22no—
138C3×#168C2×D414yescomplemented
139C22×C3×Q8C2×D511yescharacteristic
140C22×C3×D4—51no—
141C22×C3×D4—101no—
142C23×C3×C4—51no—
143C2×C3×SD16—54no—
144C2×C3×SD16—104no—
145C22×C3×C8—51no—
146C22×C3×C8—54no—
147C22×C3×C8—104no—
148C22×C3×C8C2×D511yescharacteristic
149C3×C22⋊Q8—54no—
150C3×C22⋊Q8—102no—
151C3×#121—54no—
152C3×#121—102no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C4⋊C4—101no—
155C2×C3×C4⋊C4C2×D511yescharacteristic
156C2×C3×C22⋊C4—101no—
157C2×C3×C22⋊C4—101no—
158C3×#106—54no—
159C3×#103C2×D514yescomplemented
160C3×#102—54no—
161C23×D5—21no—
162C23×D5C3×D411yescharacteristic
163C2×C5×Q8—21no—
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×Q8C23×C312yes—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5C23×C311yescharacteristic
170C22×Dic5—21no—
171C22×Dic5C3×D411yescharacteristic
172C2×D20—22no—
173C2×D20—22no—
174C2×D20—24no—
175C2×D20—21no—
176C2×D20—44no—
177C2×D20C23×C312yes—
178C2×C4×D5—22no—
179C2×C4×D5—22no—
180C2×C4×D5C3×D412yes—
181C2×C4×D5C23×C312yes—
182C2×C5×C8—22no—
183C2×C5×C8C23×C312yes—
184C5×C4⋊C4—21no—
185C5×C4⋊C4C23×C312yes—
186#688—28no—
187#685—24no—
188#685—24no—
189C4⋊Dic5—21no—
190C4⋊Dic5C23×C312yes—
191#683—24no—
192C2×#168C3×D414yescomplemented
193C2×#168C23×C312yes—
194C8×D5—28no—
195C22×SD16—51no—
196C23×C8—51no—
197C2×C22⋊Q8—51no—
198C2×#121—51no—
199#465—58no—
200C2×#106—51no—
201C2×#103C3×D511yescharacteristic, complemented
202C2×#102—51no—
203C22×C3×C5#12112yes—
204C22×C3×C5C2×SD1612yes—
205C22×C3×C5#13512yes—
206C22×C3×C5C22×D411yescharacteristic
207C2×C3×D5—22no—
208C2×C3×D5—22no—
209C2×C3×D5—28no—
210C2×C3×D5—42no—
211C2×C3×D5—42no—
212C2×C3×D5—42no—
213C2×C3×D5—84no—
214C2×C3×D5C2×SD1614yescomplemented
215C3×C4×C5—42no—
216C3×C4×C5—42no—
217C3×C4×C5#12112yes—
218C3×C4×C5C22×D411yescharacteristic
219C3×C4×C5C2×#4111yescharacteristic
220C3×Dic5—22no—
221C3×Dic5—42no—
222C24×C3—51no—
223C24×C3—101no—
224C2×C3×Q8—21no—
225C2×C3×Q8—22no—
226C2×C3×Q8—22no—
227C2×C3×Q8C22×D512yes—
228C2×C3×D4—52no—
229C2×C3×D4—104no—
230C2×C3×D4—102no—
231C2×C3×D4—102no—
232C2×C3×D4—101no—
233C2×C3×D4—204no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—51no—
237C22×C3×C4—52no—
238C22×C3×C4—52no—
239C22×C3×C4—102no—
240C22×C3×C4—102no—
241C22×C3×C4—101no—
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C2×C3×C8—22no—
245C2×C3×C8—54no—
246C2×C3×C8—52no—
247C2×C3×C8—108no—
248C2×C3×C8C22×D512yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—52no—
251C3×C4⋊C4—104no—
252C3×C4⋊C4—101no—
253C3×C4⋊C4C22×D512yes—
254C3×C22⋊C4—104no—
255C3×C22⋊C4—104no—
256C23×C5C2×C3×D411yescharacteristic
257C22×D5—22no—
258C22×D5—22no—
259C22×D5—22no—
260C22×D5—42no—
261C22×D5—42no—
262C22×D5—42no—
263C22×D5C3×SD1614yescomplemented
264C22×D5C2×C3×D412yes—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8—42no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5C2×C3×D411yescharacteristic
273C2×C4×C5C2×C3×D412yes—
274C2×C4×C5C3×#4112yes—
275C2×C4×C5C24×C311yescharacteristic
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—42no—
279C2×Dic5C2×C3×D412yes—
280D20—22no—
281D20—22no—
282D20—48no—
283D20—42no—
284C4×D5—24no—
285C4×D5—22no—
286C4×D5—22no—
287C5×C8—22no—
288#168C2×C3×D414yescomplemented
289C22×Q8C2×C3×D511yescharacteristic
290C22×D4—51no—
291C22×D4—101no—
292C23×C4—51no—
293C2×SD16—54no—
294C2×SD16—104no—
295C22×C8—51no—
296C22×C8—54no—
297C22×C8—104no—
298C22×C8C2×C3×D511yescharacteristic
299C22⋊Q8—54no—
300C22⋊Q8—102no—
301#121—54no—
302#121—102no—
303C2×C4⋊C4—51no—
304C2×C4⋊C4—101no—
305C2×C4⋊C4C2×C3×D511yescharacteristic
306C2×C22⋊C4—101no—
307C2×C22⋊C4—101no—
308#106—54no—
309#103C2×C3×D514yescomplemented
310#102—54no—
311C2×C3×C5#46514yescomplemented
312C2×C3×C5C2×#12111yescharacteristic
313C2×C3×C5C22×SD1611yescharacteristic
314C2×C3×C5C2×#13511yescharacteristic
315C3×D5—24no—
316C3×D5—82no—
317C23×C3—52no—
318C23×C3—54no—
319C23×C3—102no—
320C23×C3—102no—
321C23×C3—102no—
322C23×C3—202no—
323C23×C3—202no—
324C23×C3—202no—
325C23×C3D4×D511yescharacteristic
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—42no—
329C3×D4—102no—
330C3×D4—102no—
331C3×D4—208no—
332C3×D4—202no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—52no—
338C2×C3×C4—102no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4—104no—
342C2×C3×C4—102no—
343C2×C3×C4—202no—
344C2×C3×C4#70912yes—
345C2×C3×C4D4×D512yes—
346C2×C3×C4D4×D511yescharacteristic
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—22no—
349C3×C8—54no—
350C22×C5C3×#12112yes—
351C22×C5C2×C3×SD1612yes—
352C22×C5C3×#13512yes—
353C22×C5C22×C3×D411yescharacteristic
354C2×D5—22no—
355C2×D5—28no—
356C2×D5—22no—
357C2×D5—42no—
358C2×D5—42no—
359C2×D5—42no—
360C2×D5—84no—
361C2×D5C2×C3×SD1614yescomplemented
362C4×C5—42no—
363C4×C5—42no—
364C4×C5C3×#12112yes—
365C4×C5C22×C3×D411yescharacteristic
366C4×C5C2×C3×#4111yescharacteristic
367Dic5—22no—
368Dic5—42no—
369C24—51no—
370C24—101no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C22×C3×D512yes—
375C2×D4—52no—
376C2×D4—104no—
377C2×D4—102no—
378C2×D4—102no—
379C2×D4—101no—
380C2×D4—204no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—51no—
384C22×C4—52no—
385C22×C4—52no—
386C22×C4—102no—
387C22×C4—102no—
388C22×C4—101no—
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391C2×C8—22no—
392C2×C8—54no—
393C2×C8—52no—
394C2×C8—108no—
395C2×C8C22×C3×D512yes—
396C4⋊C4—21no—
397C4⋊C4—52no—
398C4⋊C4—104no—
399C4⋊C4—101no—
400C4⋊C4C22×C3×D512yes—
401C22⋊C4—104no—
402C22⋊C4—104no—
403C3×C5C2×#46511yescharacteristic, complemented, Hall
404C22×C3—54no—
405C22×C3—102no—
406C22×C3—102no—
407C22×C3—108no—
408C22×C3—202no—
409C22×C3—202no—
410C22×C3—202no—
411C22×C3—404no—
412C22×C3#403012yes—
413C22×C3D5×SD1612yes—
414C22×C3#406512yes—
415C22×C3C2×D4×D511yescharacteristic
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—102no—
419C3×C4—202no—
420C3×C4#403012yes—
421C3×C4C2×#70911yescharacteristic
422C3×C4C2×D4×D511yescharacteristic
423C2×C5C3×#46514yescomplemented
424C2×C5C2×C3×#12111yescharacteristic
425C2×C5C22×C3×SD1611yescharacteristic
426C2×C5C2×C3×#13511yescharacteristic
427D5—24no—
428D5—82no—
429C23—52no—
430C23—54no—
431C23—102no—
432C23—102no—
433C23—102no—
434C23—202no—
435C23—202no—
436C23—202no—
437C23C3×D4×D511yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—42no—
441D4—102no—
442D4—102no—
443D4—208no—
444D4—202no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—52no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—104no—
454C2×C4—102no—
455C2×C4—202no—
456C2×C4C3×#70912yes—
457C2×C4C3×D4×D512yes—
458C2×C4C3×D4×D511yescharacteristic
459C2×C4C23×C3×D511yescharacteristic
460C8—22no—
461C8—54no—
462C2×C3—104no—
463C2×C3—402no—
464C2×C3#6509614yesdirect product factor
465C2×C3C2×#403011yescharacteristic
466C2×C3C2×D5×SD1611yescharacteristic
467C2×C3C2×#406511yescharacteristic
468C5C2×C3×#46511yescharacteristic, complemented, Sylow
469C22—54no—
470C22—102no—
471C22—108no—
472C22—102no—
473C22—202no—
474C22—202no—
475C22—202no—
476C22—404no—
477C22C3×#403012yes—
478C22C3×D5×SD1612yes—
479C22C3×#406512yes—
480C22C2×C3×D4×D511yescharacteristic
481C4—42no—
482C4—42no—
483C4—102no—
484C4—202no—
485C4C3×#403012yes—
486C4C2×C3×#70911yescharacteristic
487C4C2×C3×D4×D511yescharacteristic
488C3C2×#6509611yescharacteristic, direct product factor, Sylow
489C2—104no—
490C2—402no—
491C2C3×#6509614yesdirect product factor
492C2C2×C3×#403011yescharacteristic
493C2C2×C3×D5×SD1611yescharacteristic
494C2C2×C3×#406511yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#103C2
socleC23×C3×C5C2×D4
prime radicalC2×C3×#4041C2
hypercenterC2×C3×#103D5
hypocenterC5C2×C3×#465
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4092C2semidirect
2. C22×C3×#688C2semidirect
3. C22×C3×C8×D5C2semidirect
5. C2×C3×C5×#103C2semidirect
6. C2×C3×#3953C2semidirect
7. C2×C3×#3940C2semidirect
8. C3×#65096C2direct (N × H)
9. C2×#65096C3direct (N × H)
17. C3×#4092C22semidirect
19. C2×C3×#688C22semidirect
26. C2×C3×C8×D5C22semidirect
29. C3×C5×#103C22semidirect
30. C3×#3953C22semidirect
31. C3×#3940C22semidirect
33. C2×#4092C2×C3semidirect
34. C22×#688C2×C3semidirect
35. C22×C8×D5C2×C3semidirect
37. C2×C5×#103C2×C3semidirect
38. C2×#3953C2×C3semidirect
39. C2×#3940C2×C3semidirect
40. #65096C2×C3direct (N × H)
72. C2×C3×#168D4semidirect
81. C2×C3×#103D5semidirect
90. #4092C22×C3semidirect
92. C2×#688C22×C3semidirect
99. C2×C8×D5C22×C3semidirect
102. C5×#103C22×C3semidirect
103. #3953C22×C3semidirect
104. #3940C22×C3semidirect
113. C22×C3×D5SD16semidirect
138. C3×#168C2×D4semidirect
159. C3×#103C2×D5semidirect
192. C2×#168C3×D4semidirect
201. C2×#103C3×D5semidirect
214. C2×C3×D5C2×SD16semidirect
263. C22×D5C3×SD16semidirect
288. #168C2×C3×D4semidirect
309. #103C2×C3×D5semidirect
311. C2×C3×C5#465semidirect
361. C2×D5C2×C3×SD16semidirect
403. C3×C5C2×#465semidirect
423. C2×C5C3×#465semidirect
464. C2×C3#65096direct (N × H)
468. C5C2×C3×#465semidirect
488. C3C2×#65096direct (N × H)
491. C2C3×#65096direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40922
C22×C3×#6882
C22×C3×C8×D52
C2×C3×#40412
C2×C3×C5×#1032
C2×C3×#39532
C2×C3×#39402
C3×#650962
C2×#650963
C2×C3×#4655