Finite Group Databasebeta

← C2×C3×#65337 #14427604

Subgroups — C2×C3×#65337 #14427604

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×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×C3×#688C211yesmaximal, characteristic, complemented
3C2×C3×#4076C211yesmaximal, characteristic
4C23×C3×#168C211yesmaximal, characteristic, complemented
5C2×C3×#3967C211yesmaximal, characteristic, complemented
6C3×#65337C218yesmaximal, direct product factor
7C2×C3×#3941C211yesmaximal, characteristic, complemented
8C2×C3×#3940C211yesmaximal, characteristic, complemented
9C2×#65337C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C23×C3×C4×C5C2211yescharacteristic
12C22×C3×C22⋊D5—21no—
13C22×C3×D20C2211yescharacteristic
14C3×C5×C22⋊Q8—22no—
15C3×C5×C22⋊Q8C2214yescomplemented
16C2×C3×C5×C4⋊C4—21no—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4—21no—
19C2×C3×#688—24no—
20C2×C3×#688C2214yescomplemented
21C3×#4076—22no—
22C3×#4076C2214yes—
23C2×C3×#685—21no—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C22×C3×#168—24no—
26C22×C3×#168C2211yescharacteristic
27C22×C3×#168C2211yescharacteristic
28C22×C3×#168C2214yescomplemented
29C3×#3967C2214yescomplemented
30C3×#3941C2214yescomplemented
31C3×#3940C2214yescomplemented
32C2×C3×#465—51nomaximal, Hall
33C2×C5×C22⋊Q8C2×C311yescharacteristic, complemented
34C22×#688C2×C311yescharacteristic, complemented
35C2×#4076C2×C311yescharacteristic
36C23×#168C2×C311yescharacteristic, complemented
37C2×#3967C2×C311yescharacteristic, complemented
38#65337C2×C318yesdirect product factor
39C2×#3941C2×C311yescharacteristic, complemented
40C2×#3940C2×C311yescharacteristic, complemented
41C24×C3×C5D411yescharacteristic
42C23×C3×D5—21no—
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8C2312yes—
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5—22no—
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5—21no—
51C22×C3×C4×C5D411yescharacteristic
52C22×C3×C4×C5D412yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×C3×C22⋊D5—24no—
56C2×C3×C22⋊D5—44no—
57C22×C3×Dic5—21no—
58C2×C3×D20—22no—
59C2×C3×D20—22no—
60C2×C3×D20—21no—
61C2×C3×D20C2312yes—
62C3×C5×C4⋊C4—24no—
63C3×C5×C4⋊C4—21no—
64C3×C5×C4⋊C4C2312yes—
65C3×C5×C22⋊C4—24no—
66C3×#688—28no—
67C3×#685—24no—
68C3×C4⋊Dic5—21no—
69C3×C4⋊Dic5C2312yes—
70C2×C3×#168—22no—
71C2×C3×#168—28no—
72C2×C3×#168D414yescomplemented
73C2×C3×#168C2312yes—
74C2×C3×#168C2312yes—
75C22×C3×SD16—51no—
76C23×C3×C8—51no—
77C2×C3×C22⋊Q8D511yescharacteristic, complemented
78C2×C3×#121—51no—
79C3×#465—58no—
80C2×C3×#106—51no—
81C2×C3×#103—51no—
82C2×C3×#102—51no—
83C22×C5×Q8C22×C311yescharacteristic
84C23×C4×C5C22×C311yescharacteristic
85C22×C22⋊D5—21no—
86C22×D20C22×C311yescharacteristic
87C5×C22⋊Q8—22no—
88C5×C22⋊Q8C22×C314yescomplemented
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4C22×C311yescharacteristic
91C2×C5×C22⋊C4—21no—
92C2×#688—24no—
93C2×#688C22×C314yescomplemented
94#4076—22no—
95#4076C22×C314yes—
96C2×#685—21no—
97C2×C4⋊Dic5C22×C311yescharacteristic
98C22×#168—24no—
99C22×#168C22×C311yescharacteristic
100C22×#168C22×C311yescharacteristic
101C22×#168C22×C314yescomplemented
102#3967C22×C314yescomplemented
103#3941C22×C314yescomplemented
104#3940C22×C314yescomplemented
105C2×#465—51noSylow
106C23×C3×C5—22no—
107C23×C3×C5—22no—
108C23×C3×C5SD1614yescomplemented
109C23×C3×C5C2×D412yes—
110C23×C3×C5C2×D411yescharacteristic
111C22×C3×D5—22no—
112C22×C3×D5—42no—
113C22×C3×D5—42no—
114C22×C3×D5—42no—
115C3×C5×Q8—22no—
116C3×C5×Q8—22no—
117C3×C5×Q8—42no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—24no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—42no—
125C2×C3×C4×C5—42no—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D411yescharacteristic
128C2×C3×C4×C5C2×D412yes—
129C2×C3×C4×C5#4112yes—
130C2×C3×C4×C5C2411yescharacteristic
131C3×C22⋊D5—48no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—42no—
134C3×D20—22no—
135C3×D20—22no—
136C3×D20—42no—
137C3×#168—22no—
138C3×#168C2×D414yescomplemented
139C22×C3×Q8C2×D511yescharacteristic
140C22×C3×D4—51no—
141C22×C3×D4—101no—
142C23×C3×C4C2×D511yescharacteristic
143C2×C3×SD16—54no—
144C2×C3×SD16—104no—
145C22×C3×C8—51no—
146C22×C3×C8—51no—
147C22×C3×C8—54no—
148C22×C3×C8—104no—
149C3×C22⋊Q8—22no—
150C3×C22⋊Q8C2×D514yescomplemented
151C3×#121—54no—
152C3×#121—102no—
153C2×C3×C4⋊C4—21no—
154C2×C3×C4⋊C4—51no—
155C2×C3×C4⋊C4C2×D511yescharacteristic
156C2×C3×C22⋊C4—21no—
157C2×C3×C22⋊C4—101no—
158C3×#106—54no—
159C3×#103—54no—
160C3×#102—54no—
161C24×C5C3×D411yescharacteristic
162C23×D5—21no—
163C2×C5×Q8—21no—
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×Q8C23×C312yes—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—21no—
170C22×C4×C5—21no—
171C22×C4×C5C3×D411yescharacteristic
172C22×C4×C5C3×D412yes—
173C22×C4×C5C23×C312yes—
174C22×C4×C5C23×C311yescharacteristic
175C2×C22⋊D5—24no—
176C2×C22⋊D5—44no—
177C22×Dic5—21no—
178C2×D20—22no—
179C2×D20—22no—
180C2×D20—21no—
181C2×D20C23×C312yes—
182C5×C4⋊C4—24no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4C23×C312yes—
185C5×C22⋊C4—24no—
186#688—28no—
187#685—24no—
188C4⋊Dic5—21no—
189C4⋊Dic5C23×C312yes—
190C2×#168—22no—
191C2×#168—28no—
192C2×#168C3×D414yescomplemented
193C2×#168C23×C312yes—
194C2×#168C23×C312yes—
195C22×SD16—51no—
196C23×C8—51no—
197C2×C22⋊Q8C3×D511yescharacteristic, complemented
198C2×#121—51no—
199#465—58no—
200C2×#106—51no—
201C2×#103—51no—
202C2×#102—51no—
203C22×C3×C5—22no—
204C22×C3×C5—22no—
205C22×C3×C5—28no—
206C22×C3×C5#12112yes—
207C22×C3×C5C2×SD1612yes—
208C22×C3×C5C2×SD1614yescomplemented
209C22×C3×C5#13512yes—
210C22×C3×C5C22×D411yescharacteristic
211C2×C3×D5—42no—
212C2×C3×D5—42no—
213C2×C3×D5—42no—
214C2×C3×D5—84no—
215C3×C4×C5—22no—
216C3×C4×C5—42no—
217C3×C4×C5—42no—
218C3×C4×C5#12112yes—
219C3×C4×C5C22×D411yescharacteristic
220C3×C4×C5C2×#4111yescharacteristic
221C3×Dic5—42no—
222C24×C3—101no—
223C24×C3C22⋊D511yescharacteristic
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—22no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—101no—
239C22×C3×C4C22⋊D512yes—
240C22×C3×C4C22⋊D511yescharacteristic
241C22×C3×C4C22×D512yes—
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C2×C3×C8—54no—
245C2×C3×C8—52no—
246C2×C3×C8—52no—
247C2×C3×C8—108no—
248C2×C3×C8—102no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—52no—
252C3×C4⋊C4—101no—
253C3×C4⋊C4C22×D512yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—104no—
256C23×C5—22no—
257C23×C5—22no—
258C23×C5C3×SD1614yescomplemented
259C23×C5C2×C3×D412yes—
260C23×C5C2×C3×D411yescharacteristic
261C22×D5—22no—
262C22×D5—42no—
263C22×D5—42no—
264C22×D5—42no—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8—42no—
268C2×C4×C5—22no—
269C2×C4×C5—24no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5C2×C3×D412yes—
277C2×C4×C5C2×C3×D412yes—
278C2×C4×C5C2×C3×D411yescharacteristic
279C2×C4×C5C3×#4112yes—
280C2×C4×C5C24×C311yescharacteristic
281C22⋊D5—48no—
282C2×Dic5—22no—
283C2×Dic5—42no—
284D20—22no—
285D20—22no—
286D20—42no—
287#168—22no—
288#168C2×C3×D414yescomplemented
289C22×Q8C2×C3×D511yescharacteristic
290C22×D4—51no—
291C22×D4—101no—
292C23×C4C2×C3×D511yescharacteristic
293C2×SD16—54no—
294C2×SD16—104no—
295C22×C8—51no—
296C22×C8—51no—
297C22×C8—54no—
298C22×C8—104no—
299C22⋊Q8—22no—
300C22⋊Q8C2×C3×D514yescomplemented
301#121—54no—
302#121—102no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—51no—
305C2×C4⋊C4C2×C3×D511yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—101no—
308#106—54no—
309#103—54no—
310#102—54no—
311C2×C3×C5—24no—
312C2×C3×C5#46514yescomplemented
313C2×C3×C5C2×#12111yescharacteristic
314C2×C3×C5C22×SD1611yescharacteristic
315C2×C3×C5C2×#13511yescharacteristic
316C3×D5—82no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—102no—
320C23×C3—202no—
321C23×C3—202no—
322C23×C3—202no—
323C23×C3#68814yescomplemented
324C23×C3C2×C22⋊D512yes—
325C23×C3C2×C22⋊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—22no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—102no—
342C2×C3×C4—202no—
343C2×C3×C4D4×D512yes—
344C2×C3×C4#71112yes—
345C2×C3×C4C2×C22⋊D512yes—
346C2×C3×C4C2×C22⋊D511yescharacteristic
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—54no—
349C3×C8—102no—
350C22×C5—22no—
351C22×C5—28no—
352C22×C5—22no—
353C22×C5C3×#12112yes—
354C22×C5C2×C3×SD1612yes—
355C22×C5C2×C3×SD1614yescomplemented
356C22×C5C3×#13512yes—
357C22×C5C22×C3×D411yescharacteristic
358C2×D5—42no—
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—84no—
362C4×C5—22no—
363C4×C5—42no—
364C4×C5—42no—
365C4×C5C3×#12112yes—
366C4×C5C22×C3×D411yescharacteristic
367C4×C5C2×C3×#4111yescharacteristic
368Dic5—42no—
369C24—101no—
370C24C3×C22⋊D511yescharacteristic
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—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—101no—
386C22×C4C3×C22⋊D511yescharacteristic
387C22×C4C3×C22⋊D512yes—
388C22×C4C22×C3×D512yes—
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391C2×C8—54no—
392C2×C8—52no—
393C2×C8—52no—
394C2×C8—108no—
395C2×C8—102no—
396C4⋊C4—24no—
397C4⋊C4—21no—
398C4⋊C4—52no—
399C4⋊C4—101no—
400C4⋊C4C22×C3×D512yes—
401C22⋊C4—24no—
402C22⋊C4—104no—
403C3×C5C2×#46511yescharacteristic, complemented, Hall
404C22×C3—28no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—202no—
408C22×C3—202no—
409C22×C3—202no—
410C22×C3—404no—
411C22×C3#408512yes—
412C22×C3C2×#68812yes—
413C22×C3C2×#68814yescomplemented
414C22×C3#409612yes—
415C22×C3C22×C22⋊D511yescharacteristic
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—202no—
420C3×C4#408512yes—
421C3×C4C2×D4×D511yescharacteristic
422C3×C4C2×#71111yescharacteristic
423C2×C5—24no—
424C2×C5C3×#46514yescomplemented
425C2×C5C2×C3×#12111yescharacteristic
426C2×C5C22×C3×SD1611yescharacteristic
427C2×C5C2×C3×#13511yescharacteristic
428D5—82no—
429C23—22no—
430C23—22no—
431C23—102no—
432C23—202no—
433C23—202no—
434C23—202no—
435C23C3×#68814yescomplemented
436C23C2×C3×C22⋊D512yes—
437C23C2×C3×C22⋊D511yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—42no—
441D4—102no—
442D4—102no—
443D4—208no—
444D4—202no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—102no—
454C2×C4—202no—
455C2×C4C3×D4×D512yes—
456C2×C4C3×#71112yes—
457C2×C4C2×C3×C22⋊D512yes—
458C2×C4C2×C3×C22⋊D511yescharacteristic
459C2×C4C23×C3×D511yescharacteristic
460C8—54no—
461C8—102no—
462C2×C3—24no—
463C2×C3—402no—
464C2×C3#6533714yesdirect product factor
465C2×C3C2×#408511yescharacteristic
466C2×C3C22×#68811yescharacteristic
467C2×C3C2×#409611yescharacteristic
468C5C2×C3×#46511yescharacteristic, complemented, Sylow
469C22—22no—
470C22—22no—
471C22—28no—
472C22—202no—
473C22—202no—
474C22—202no—
475C22—404no—
476C22C3×#408512yes—
477C22C2×C3×#68812yes—
478C22C2×C3×#68814yescomplemented
479C22C3×#409612yes—
480C22C22×C3×C22⋊D511yescharacteristic
481C4—22no—
482C4—42no—
483C4—42no—
484C4—202no—
485C4C3×#408512yes—
486C4C2×C3×D4×D511yescharacteristic
487C4C2×C3×#71111yescharacteristic
488C3C2×#6533711yescharacteristic, direct product factor, Sylow
489C2—24no—
490C2—402no—
491C2C3×#6533714yesdirect product factor
492C2C2×C3×#408511yescharacteristic
493C2C22×C3×#68811yescharacteristic
494C2C2×C3×#409611yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×C22⋊Q8C2
socleC23×C3×C5C2×D4
prime radicalC2×C3×#4076C2
hypercenterC2×C3×C22⋊Q8D5
hypocenterC5C2×C3×#465
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×C22⋊Q8C2semidirect
2. C22×C3×#688C2semidirect
4. C23×C3×#168C2semidirect
5. C2×C3×#3967C2semidirect
6. C3×#65337C2direct (N × H)
7. C2×C3×#3941C2semidirect
8. C2×C3×#3940C2semidirect
9. C2×#65337C3direct (N × H)
15. C3×C5×C22⋊Q8C22semidirect
20. C2×C3×#688C22semidirect
28. C22×C3×#168C22semidirect
29. C3×#3967C22semidirect
30. C3×#3941C22semidirect
31. C3×#3940C22semidirect
33. C2×C5×C22⋊Q8C2×C3semidirect
34. C22×#688C2×C3semidirect
36. C23×#168C2×C3semidirect
37. C2×#3967C2×C3semidirect
38. #65337C2×C3direct (N × H)
39. C2×#3941C2×C3semidirect
40. C2×#3940C2×C3semidirect
72. C2×C3×#168D4semidirect
77. C2×C3×C22⋊Q8D5semidirect
88. C5×C22⋊Q8C22×C3semidirect
93. C2×#688C22×C3semidirect
101. C22×#168C22×C3semidirect
102. #3967C22×C3semidirect
103. #3941C22×C3semidirect
104. #3940C22×C3semidirect
108. C23×C3×C5SD16semidirect
138. C3×#168C2×D4semidirect
150. C3×C22⋊Q8C2×D5semidirect
192. C2×#168C3×D4semidirect
197. C2×C22⋊Q8C3×D5semidirect
208. C22×C3×C5C2×SD16semidirect
258. C23×C5C3×SD16semidirect
288. #168C2×C3×D4semidirect
300. C22⋊Q8C2×C3×D5semidirect
312. C2×C3×C5#465semidirect
323. C23×C3#688semidirect
355. C22×C5C2×C3×SD16semidirect
403. C3×C5C2×#465semidirect
413. C22×C3C2×#688semidirect
424. C2×C5C3×#465semidirect
435. C23C3×#688semidirect
464. C2×C3#65337direct (N × H)
468. C5C2×C3×#465semidirect
478. C22C2×C3×#688semidirect
488. C3C2×#65337direct (N × H)
491. C2C3×#65337direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×C22⋊Q82
C22×C3×#6882
C2×C3×#40762
C23×C3×#1682
C2×C3×#39672
C3×#653372
C2×C3×#39412
C2×C3×#39402
C2×#653373
C2×C3×#4655