Finite Group Databasebeta

← C2×C3×#64688 #14280738

Subgroups — C2×C3×#64688 #14280738

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#685C211yesmaximal, characteristic
2C2×C3×C5×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#168C211yesmaximal, characteristic, complemented
4C3×#64688C214yesmaximal, direct product factor
5C2×#64688C311yesmaximal, characteristic, direct product factor, Hall
6C24×C3×D5C411yescharacteristic
7C23×C3×C4×C5C2211yescharacteristic
8C23×C3×Dic5C411yescharacteristic
9C22×C3×C5×C8—21no—
10C2×C3×#685—21no—
11C2×C3×#685—21no—
12C2×C3×#685—21no—
13C2×C3×#685—21no—
14C2×C3×#685C412yes—
15C2×C3×#685C2212yes—
16C22×C3×#168—21no—
17C3×C5×C22⋊C8—21no—
18C3×C5×C22⋊C8C2212yescomplemented
19C3×C22⋊#168—21no—
20C3×C22⋊#168C2212yescomplemented
21C2×C3×#323—51nomaximal, Hall
22C22×#685C2×C311yescharacteristic
23C2×C5×C22⋊C8C2×C311yescharacteristic, complemented
24C2×C22⋊#168C2×C311yescharacteristic, complemented
25#64688C2×C314yesdirect product factor
26C24×C3×C5C2×C411yescharacteristic
27C23×C3×D5—21no—
28C23×C3×D5—21no—
29C23×C3×D5—22no—
30C23×C3×D5—42no—
31C23×C3×D5—42no—
32C23×C3×D5C814yescomplemented
33C23×C3×D5C2×C412yes—
34C22×C3×C4×C5—42no—
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×C3×C4×C5D411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5D411yescharacteristic
39C22×C3×C4×C5D411yescharacteristic
40C22×C3×C4×C5C2311yescharacteristic
41C22×C3×Dic5—21no—
42C22×C3×Dic5—21no—
43C22×C3×Dic5—42no—
44C22×C3×Dic5C2×C412yes—
45C2×C3×C5×C8—22no—
46C2×C3×C5×C8—42no—
47C3×#685—22no—
48C3×#685—22no—
49C3×#685—22no—
50C3×#685—22no—
51C2×C3×#168—22no—
52C2×C3×#168—42no—
53C22×C3×C22⋊C4—51no—
54C2×C3×C22⋊C8—51no—
55C2×C3×C22⋊C8D511yescharacteristic, complemented
56C3×#323—54no—
57C24×D5C3×C411yescharacteristic
58C23×C4×C5C22×C311yescharacteristic
59C23×Dic5C3×C411yescharacteristic
60C22×C5×C8—21no—
61C2×#685—21no—
62C2×#685—21no—
63C2×#685—21no—
64C2×#685—21no—
65C2×#685C3×C412yes—
66C2×#685C22×C312yes—
67C22×#168—21no—
68C5×C22⋊C8—21no—
69C5×C22⋊C8C22×C312yescomplemented
70C22⋊#168—21no—
71C22⋊#168C22×C312yescomplemented
72C2×#323—51noSylow
73C23×C3×C5—22no—
74C23×C3×C5—22no—
75C23×C3×C5C22⋊C411yescharacteristic
76C23×C3×C5C22⋊C411yescharacteristic
77C23×C3×C5C2×C812yes—
78C23×C3×C5M1612yes—
79C23×C3×C5C22×C411yescharacteristic
80C22×C3×D5—24no—
81C22×C3×D5—24no—
82C22×C3×D5—22no—
83C22×C3×D5—22no—
84C22×C3×D5—48no—
85C22×C3×D5—42no—
86C22×C3×D5—42no—
87C22×C3×D5—42no—
88C22×C3×D5—42no—
89C22×C3×D5—48no—
90C22×C3×D5—42no—
91C22×C3×D5—42no—
92C22×C3×D5C2×C814yescomplemented
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C2×C3×C4×C5C22⋊C411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×Dic5—22no—
104C2×C3×Dic5—42no—
105C2×C3×Dic5—42no—
106C2×C3×Dic5—42no—
107C3×C5×C8—42no—
108C3×#168—42no—
109C25×C3—51no—
110C23×C3×C4—51no—
111C23×C3×C4C2×D511yescharacteristic
112C22×C3×C8—21no—
113C22×C3×C8—101no—
114C2×C3×C22⋊C4—52no—
115C2×C3×C22⋊C4—52no—
116C2×C3×C22⋊C4—101no—
117C2×C3×C22⋊C4—101no—
118C2×C3×C22⋊C4—101no—
119C2×C3×C22⋊C4—101no—
120C3×C22⋊C8—21no—
121C3×C22⋊C8—52no—
122C3×C22⋊C8—101no—
123C3×C22⋊C8C2×D512yescomplemented
124C24×C5C2×C3×C411yescharacteristic
125C23×D5—21no—
126C23×D5—21no—
127C23×D5—22no—
128C23×D5—42no—
129C23×D5—42no—
130C23×D5C3×C814yescomplemented
131C23×D5C2×C3×C412yes—
132C22×C4×C5—42no—
133C22×C4×C5C2×C3×C411yescharacteristic
134C22×C4×C5C3×D411yescharacteristic
135C22×C4×C5C3×D411yescharacteristic
136C22×C4×C5C3×D411yescharacteristic
137C22×C4×C5C3×D411yescharacteristic
138C22×C4×C5C23×C311yescharacteristic
139C22×Dic5—21no—
140C22×Dic5—21no—
141C22×Dic5—42no—
142C22×Dic5C2×C3×C412yes—
143C2×C5×C8—22no—
144C2×C5×C8—42no—
145#685—22no—
146#685—22no—
147#685—22no—
148#685—22no—
149C2×#168—22no—
150C2×#168—42no—
151C22×C22⋊C4—51no—
152C2×C22⋊C8—51no—
153C2×C22⋊C8C3×D511yescharacteristic, complemented
154#323—54no—
155C22×C3×C5—24no—
156C22×C3×C5—24no—
157C22×C3×C5—21no—
158C22×C3×C5—21no—
159C22×C3×C5—21no—
160C22×C3×C5—21no—
161C22×C3×C5C22⋊C812yes—
162C22×C3×C5C22⋊C812yes—
163C22×C3×C5#9912yes—
164C22×C3×C5#10012yes—
165C22×C3×C5C2×C22⋊C411yescharacteristic
166C22×C3×C5C22×C811yescharacteristic
167C22×C3×C5C2×M1611yescharacteristic
168C2×C3×D5—24no—
169C2×C3×D5—48no—
170C2×C3×D5—44no—
171C2×C3×D5—48no—
172C2×C3×D5—42no—
173C2×C3×D5—42no—
174C2×C3×D5—44no—
175C3×C4×C5—21no—
176C3×C4×C5—21no—
177C3×C4×C5—21no—
178C3×C4×C5—21no—
179C3×Dic5—42no—
180C24×C3—54no—
181C24×C3—52no—
182C24×C3—102no—
183C24×C3—101no—
184C24×C3—101no—
185C24×C3—202no—
186C24×C3—202no—
187C24×C3C4×D511yescharacteristic
188C22×C3×C4—42no—
189C22×C3×C4—52no—
190C22×C3×C4—101no—
191C22×C3×C4—101no—
192C22×C3×C4—202no—
193C22×C3×C4C4×D511yescharacteristic
194C22×C3×C4D2011yescharacteristic
195C22×C3×C4D2011yescharacteristic
196C22×C3×C4C22⋊D511yescharacteristic
197C22×C3×C4C22⋊D511yescharacteristic
198C22×C3×C4C22×D511yescharacteristic
199C2×C3×C8—22no—
200C2×C3×C8—42no—
201C2×C3×C8—102no—
202C2×C3×C8—202no—
203C3×C22⋊C4—102no—
204C3×C22⋊C4—102no—
205C3×C22⋊C4—102no—
206C3×C22⋊C4—102no—
207C23×C5—22no—
208C23×C5—22no—
209C23×C5C3×C22⋊C411yescharacteristic
210C23×C5C3×C22⋊C411yescharacteristic
211C23×C5C2×C3×C812yes—
212C23×C5C3×M1612yes—
213C23×C5C22×C3×C411yescharacteristic
214C22×D5—24no—
215C22×D5—24no—
216C22×D5—22no—
217C22×D5—22no—
218C22×D5—42no—
219C22×D5—42no—
220C22×D5—48no—
221C22×D5—42no—
222C22×D5—42no—
223C22×D5—48no—
224C22×D5—42no—
225C22×D5—42no—
226C22×D5C2×C3×C814yescomplemented
227C2×C4×C5—22no—
228C2×C4×C5—22no—
229C2×C4×C5—41no—
230C2×C4×C5—41no—
231C2×C4×C5—41no—
232C2×C4×C5—41no—
233C2×C4×C5C3×C22⋊C411yescharacteristic
234C2×C4×C5C3×C22⋊C411yescharacteristic
235C2×C4×C5C2×C3×D411yescharacteristic
236C2×C4×C5C2×C3×D411yescharacteristic
237C2×Dic5—22no—
238C2×Dic5—42no—
239C2×Dic5—42no—
240C2×Dic5—42no—
241C5×C8—42no—
242#168—42no—
243C25—51no—
244C23×C4—51no—
245C23×C4C2×C3×D511yescharacteristic
246C22×C8—21no—
247C22×C8—101no—
248C2×C22⋊C4—52no—
249C2×C22⋊C4—52no—
250C2×C22⋊C4—101no—
251C2×C22⋊C4—101no—
252C2×C22⋊C4—101no—
253C2×C22⋊C4—101no—
254C22⋊C8—21no—
255C22⋊C8—52no—
256C22⋊C8—101no—
257C22⋊C8C2×C3×D512yescomplemented
258C2×C3×C5—22no—
259C2×C3×C5—21no—
260C2×C3×C5—21no—
261C2×C3×C5#32314yescomplemented
262C2×C3×C5C2×C22⋊C811yescharacteristic
263C2×C3×C5C2×#9911yescharacteristic
264C2×C3×C5C2×#10011yescharacteristic
265C3×D5—44no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—54no—
269C23×C3—104no—
270C23×C3—104no—
271C23×C3—102no—
272C23×C3—102no—
273C23×C3—202no—
274C23×C3—208no—
275C23×C3—202no—
276C23×C3—202no—
277C23×C3—208no—
278C23×C3—202no—
279C23×C3—202no—
280C23×C3—202no—
281C23×C3C8×D512yes—
282C23×C3#67612yes—
283C23×C3#68511yescharacteristic
284C23×C3#68511yescharacteristic
285C23×C3C2×C4×D511yescharacteristic
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—102no—
293C2×C3×C4—202no—
294C2×C3×C4—202no—
295C2×C3×C4—202no—
296C2×C3×C4#68511yescharacteristic
297C2×C3×C4#68511yescharacteristic
298C2×C3×C4C2×D2011yescharacteristic
299C2×C3×C4C2×C22⋊D511yescharacteristic
300C3×C8—42no—
301C3×C8—202no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—21no—
305C22×C5—21no—
306C22×C5—24no—
307C22×C5—24no—
308C22×C5C3×C22⋊C812yes—
309C22×C5C3×C22⋊C812yes—
310C22×C5C3×#9912yes—
311C22×C5C3×#10012yes—
312C22×C5C2×C3×C22⋊C411yescharacteristic
313C22×C5C22×C3×C811yescharacteristic
314C22×C5C2×C3×M1611yescharacteristic
315C2×D5—24no—
316C2×D5—48no—
317C2×D5—44no—
318C2×D5—42no—
319C2×D5—42no—
320C2×D5—48no—
321C2×D5—44no—
322C4×C5—21no—
323C4×C5—21no—
324C4×C5—21no—
325C4×C5—21no—
326Dic5—42no—
327C24—54no—
328C24—52no—
329C24—102no—
330C24—101no—
331C24—101no—
332C24—202no—
333C24—202no—
334C24C3×C4×D511yescharacteristic
335C22×C4—42no—
336C22×C4—52no—
337C22×C4—101no—
338C22×C4—101no—
339C22×C4—202no—
340C22×C4C3×C4×D511yescharacteristic
341C22×C4C3×D2011yescharacteristic
342C22×C4C3×D2011yescharacteristic
343C22×C4C3×C22⋊D511yescharacteristic
344C22×C4C3×C22⋊D511yescharacteristic
345C22×C4C22×C3×D511yescharacteristic
346C2×C8—22no—
347C2×C8—42no—
348C2×C8—102no—
349C2×C8—202no—
350C22⋊C4—102no—
351C22⋊C4—102no—
352C22⋊C4—102no—
353C22⋊C4—102no—
354C3×C5C2×#32311yescharacteristic, complemented, Hall
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—24no—
358C22×C3—24no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—104no—
362C22×C3—204no—
363C22×C3—208no—
364C22×C3—204no—
365C22×C3—202no—
366C22×C3—208no—
367C22×C3—202no—
368C22×C3#393812yes—
369C22×C3#395212yes—
370C22×C3#395212yes—
371C22×C3#395512yes—
372C22×C3C2×C8×D511yescharacteristic
373C22×C3C2×#67611yescharacteristic
374C22×C3C2×#68511yescharacteristic
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—202no—
380C2×C5—22no—
381C2×C5—21no—
382C2×C5—21no—
383C2×C5C3×#32314yescomplemented
384C2×C5C2×C3×C22⋊C811yescharacteristic
385C2×C5C2×C3×#9911yescharacteristic
386C2×C5C2×C3×#10011yescharacteristic
387D5—44no—
388C23—22no—
389C23—22no—
390C23—54no—
391C23—104no—
392C23—104no—
393C23—102no—
394C23—102no—
395C23—202no—
396C23—202no—
397C23—208no—
398C23—202no—
399C23—202no—
400C23—208no—
401C23—202no—
402C23—202no—
403C23C3×C8×D512yes—
404C23C3×#67612yes—
405C23C3×#68511yescharacteristic
406C23C3×#68511yescharacteristic
407C23C2×C3×C4×D511yescharacteristic
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—102no—
415C2×C4—202no—
416C2×C4—202no—
417C2×C4—202no—
418C2×C4C3×#68511yescharacteristic
419C2×C4C3×#68511yescharacteristic
420C2×C4C2×C3×D2011yescharacteristic
421C2×C4C2×C3×C22⋊D511yescharacteristic
422C8—42no—
423C8—202no—
424C2×C3—22no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—204no—
428C2×C3#6468814yesdirect product factor
429C2×C3C2×#393811yescharacteristic
430C2×C3C2×#395211yescharacteristic
431C2×C3C2×#395511yescharacteristic
432C5C2×C3×#32311yescharacteristic, complemented, Sylow
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—24no—
439C22—104no—
440C22—204no—
441C22—208no—
442C22—204no—
443C22—202no—
444C22—202no—
445C22—208no—
446C22C3×#393812yes—
447C22C3×#395212yes—
448C22C3×#395212yes—
449C22C3×#395512yes—
450C22C2×C3×C8×D511yescharacteristic
451C22C2×C3×#67611yescharacteristic
452C22C2×C3×#68511yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—202no—
458C3C2×#6468811yescharacteristic, direct product factor, Sylow
459C2—22no—
460C2—21no—
461C2—21no—
462C2—204no—
463C2C3×#6468814yesdirect product factor
464C2C2×C3×#393811yescharacteristic
465C2C2×C3×#395211yescharacteristic
466C2C2×C3×#395511yescharacteristic
subgroupG/H
centerC23×C3#685
commutatorC22×C5C22×C3×C8
FrattiniC22×C4C22×C3×D5
FittingC2×C3×C5×C22⋊C8C2
socleC23×C3×C5C22⋊C4
prime radicalC24×C3×D5C4
cosocleC22×C4×C5C23×C3
hypercenterC2×C3×C22⋊C8D5
hypocenterC5C2×C3×#323
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×C22⋊C8C2semidirect
3. C2×C3×C22⋊#168C2semidirect
4. C3×#64688C2direct (N × H)
5. C2×#64688C3direct (N × H)
18. C3×C5×C22⋊C8C22semidirect
20. C3×C22⋊#168C22semidirect
23. C2×C5×C22⋊C8C2×C3semidirect
24. C2×C22⋊#168C2×C3semidirect
25. #64688C2×C3direct (N × H)
32. C23×C3×D5C8semidirect
55. C2×C3×C22⋊C8D5semidirect
69. C5×C22⋊C8C22×C3semidirect
71. C22⋊#168C22×C3semidirect
92. C22×C3×D5C2×C8semidirect
123. C3×C22⋊C8C2×D5semidirect
130. C23×D5C3×C8semidirect
153. C2×C22⋊C8C3×D5semidirect
226. C22×D5C2×C3×C8semidirect
257. C22⋊C8C2×C3×D5semidirect
261. C2×C3×C5#323semidirect
354. C3×C5C2×#323semidirect
383. C2×C5C3×#323semidirect
428. C2×C3#64688direct (N × H)
432. C5C2×C3×#323semidirect
458. C3C2×#64688direct (N × H)
463. C2C3×#64688direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6852
C2×C3×C5×C22⋊C82
C2×C3×C22⋊#1682
C3×#646882
C2×#646883
C2×C3×#3235