Finite Group Databasebeta

← C2×C3×C4×#688 #14416053

Subgroups — C2×C3×C4×#688 #14416053

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C22×C3×#688C211yesmaximal, characteristic
3C2×C3×C4×D20C211yesmaximal, characteristic
4C2×C3×#3967C211yesmaximal, characteristic, complemented
5C3×C4×#688C218yesmaximal, direct product factor
6C2×C3×#3941C211yesmaximal, characteristic
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C2×C3×C4×#168C211yesmaximal, characteristic, complemented
9C2×C4×#688C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×C4×D5—21no—
13C3×C4×C5×Q8—22no—
14C3×C4×C5×Q8C2214yescomplemented
15C2×C3×C5×C4⋊C4—21no—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C42×C5—21no—
18C2×C3×C42×C5C2211yescharacteristic
19C2×C3×#688C418yesdirect product factor
20C2×C3×#688C2214yes—
21C2×C3×#685—21no—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C22×C3×#168C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C3×C4×D20—22no—
26C3×C4×D20C2214yes—
27C3×#3967C2214yescomplemented
28C3×#3941C2214yes—
29C3×#3940C2214yescomplemented
30C3×C4×#168C2214yescomplemented
31C2×C3×C4×SD16—51nomaximal, Hall
32C2×C4×C5×Q8C2×C311yescharacteristic, complemented
33C22×#688C2×C311yescharacteristic
34C2×C4×D20C2×C311yescharacteristic
35C2×#3967C2×C311yescharacteristic, complemented
36C4×#688C2×C318yesdirect product factor
37C2×#3941C2×C311yescharacteristic
38C2×#3940C2×C311yescharacteristic, complemented
39C2×C4×#168C2×C311yescharacteristic, complemented
40C23×C3×D5—21no—
41C2×C3×C5×Q8—21no—
42C2×C3×C5×Q8—22no—
43C2×C3×C5×Q8C2×C414yescomplemented
44C2×C3×C5×Q8C2312yes—
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5D411yescharacteristic
48C22×C3×C4×C5D411yescharacteristic
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×Dic5—21no—
51C2×C3×D20—21no—
52C2×C3×D20—22no—
53C2×C3×D20C2×C414yes—
54C2×C3×D20C2312yes—
55C2×C3×C4×D5—24no—
56C2×C3×C4×D5—44no—
57C3×C5×C4⋊C4—24no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4C2312yes—
60C3×C42×C5—24no—
61C3×C42×C5D412yes—
62C3×C42×C5C2312yes—
63C3×#688C2×C4116yesdirect product factor
64C3×#685—24no—
65C3×C4⋊Dic5—21no—
66C3×C4⋊Dic5C2312yes—
67C2×C3×#168—22no—
68C2×C3×#168C2×C414yescomplemented
69C2×C3×#168C2312yes—
70C2×C3×#168C2312yes—
71C22×C3×SD16—51no—
72C2×C3×C4×Q8D511yescharacteristic, complemented
73C2×C3×C4×D4—51no—
74C3×C4×SD16—58no—
75C2×C3×#106—51no—
76C2×C3×#103—51no—
77C2×C3×#102—51no—
78C2×C3×C4×C8—51no—
79C22×C5×Q8C22×C311yescharacteristic
80C22×D20C22×C311yescharacteristic
81C22×C4×D5—21no—
82C4×C5×Q8—22no—
83C4×C5×Q8C22×C314yescomplemented
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4C22×C311yescharacteristic
86C2×C42×C5—21no—
87C2×C42×C5C22×C311yescharacteristic
88C2×#688C3×C418yesdirect product factor
89C2×#688C22×C314yes—
90C2×#685—21no—
91C2×C4⋊Dic5C22×C311yescharacteristic
92C22×#168C22×C311yescharacteristic
93C22×#168C22×C311yescharacteristic
94C4×D20—22no—
95C4×D20C22×C314yes—
96#3967C22×C314yescomplemented
97#3941C22×C314yes—
98#3940C22×C314yescomplemented
99C4×#168C22×C314yescomplemented
100C2×C4×SD16—51noSylow
101C23×C3×C5C2×D411yescharacteristic
102C22×C3×D5—24no—
103C22×C3×D5—22no—
104C22×C3×D5—42no—
105C22×C3×D5—42no—
106C3×C5×Q8—24no—
107C3×C5×Q8—22no—
108C3×C5×Q8C22×C414yescomplemented
109C2×C3×C4×C5—24no—
110C2×C3×C4×C5—22no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—42no—
114C2×C3×C4×C5SD1614yescomplemented
115C2×C3×C4×C5C22×C412yes—
116C2×C3×C4×C5C2×D412yes—
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D412yes—
119C2×C3×C4×C5#4112yes—
120C2×C3×C4×C5C2411yescharacteristic
121C2×C3×Dic5—22no—
122C2×C3×Dic5—42no—
123C3×D20—24no—
124C3×D20—22no—
125C3×D20C22×C414yes—
126C3×C4×D5—48no—
127C3×#168—22no—
128C3×#168C22×C414yescomplemented
129C22×C3×Q8C2×D511yescharacteristic
130C22×C3×D4—51no—
131C23×C3×C4—101no—
132C2×C3×SD16—58no—
133C2×C3×SD16—54no—
134C22×C3×C8—51no—
135C22×C3×C8—51no—
136C3×C4×Q8—22no—
137C3×C4×Q8C2×D514yescomplemented
138C3×C4×D4—54no—
139C3×C4×D4—102no—
140C2×C3×C4⋊C4—21no—
141C2×C3×C4⋊C4—51no—
142C2×C3×C4⋊C4C2×D511yescharacteristic
143C2×C3×C22⋊C4—101no—
144C2×C3×C42—21no—
145C2×C3×C42C2×D511yescharacteristic
146C3×#106—54no—
147C3×#103—54no—
148C3×#102—54no—
149C3×C4×C8—54no—
150C23×D5—21no—
151C2×C5×Q8—21no—
152C2×C5×Q8—22no—
153C2×C5×Q8C2×C3×C414yescomplemented
154C2×C5×Q8C23×C312yes—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5C3×D411yescharacteristic
158C22×C4×C5C3×D411yescharacteristic
159C22×C4×C5C23×C311yescharacteristic
160C22×Dic5—21no—
161C2×D20—21no—
162C2×D20—22no—
163C2×D20C2×C3×C414yes—
164C2×D20C23×C312yes—
165C2×C4×D5—24no—
166C2×C4×D5—44no—
167C5×C4⋊C4—24no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4C23×C312yes—
170C42×C5—24no—
171C42×C5C3×D412yes—
172C42×C5C23×C312yes—
173#688C2×C3×C4116yesdirect product factor
174#685—24no—
175C4⋊Dic5—21no—
176C4⋊Dic5C23×C312yes—
177C2×#168—22no—
178C2×#168C2×C3×C414yescomplemented
179C2×#168C23×C312yes—
180C2×#168C23×C312yes—
181C22×SD16—51no—
182C2×C4×Q8C3×D511yescharacteristic, complemented
183C2×C4×D4—51no—
184C4×SD16—58no—
185C2×#106—51no—
186C2×#103—51no—
187C2×#102—51no—
188C2×C4×C8—51no—
189C22×C3×C5C4×D412yes—
190C22×C3×C5C2×SD1612yes—
191C22×C3×C5#13512yes—
192C22×C3×C5C22×D411yescharacteristic
193C2×C3×D5—24no—
194C2×C3×D5—42no—
195C2×C3×D5—42no—
196C2×C3×D5—48no—
197C3×C4×C5—24no—
198C3×C4×C5—22no—
199C3×C4×C5—42no—
200C3×C4×C5C4×D412yes—
201C3×C4×C5C2×SD1614yescomplemented
202C3×C4×C5C23×C411yescharacteristic
203C3×C4×C5C2×#4111yescharacteristic
204C3×Dic5—42no—
205C24×C3—101no—
206C2×C3×Q8—22no—
207C2×C3×Q8—21no—
208C2×C3×Q8C4×D514yescomplemented
209C2×C3×Q8C22×D512yes—
210C2×C3×D4—54no—
211C2×C3×D4—52no—
212C2×C3×D4—102no—
213C2×C3×D4—101no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—101no—
217C22×C3×C4—104no—
218C22×C3×C4—204no—
219C22×C3×C4C22⋊D511yescharacteristic
220C22×C3×C4C22⋊D511yescharacteristic
221C22×C3×C4C22×D511yescharacteristic
222C3×SD16—516no—
223C2×C3×C8—54no—
224C2×C3×C8—52no—
225C2×C3×C8—52no—
226C2×C3×C8—102no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—52no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4C22×D512yes—
232C3×C22⋊C4—104no—
233C3×C42—24no—
234C3×C42C22⋊D512yes—
235C3×C42C22×D512yes—
236C23×C5C2×C3×D411yescharacteristic
237C22×D5—24no—
238C22×D5—22no—
239C22×D5—42no—
240C22×D5—42no—
241C5×Q8—24no—
242C5×Q8—22no—
243C5×Q8C22×C3×C414yescomplemented
244C2×C4×C5—24no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—42no—
249C2×C4×C5C3×SD1614yescomplemented
250C2×C4×C5C22×C3×C412yes—
251C2×C4×C5C2×C3×D412yes—
252C2×C4×C5C2×C3×D411yescharacteristic
253C2×C4×C5C2×C3×D412yes—
254C2×C4×C5C3×#4112yes—
255C2×C4×C5C24×C311yescharacteristic
256C2×Dic5—22no—
257C2×Dic5—42no—
258D20—24no—
259D20—22no—
260D20C22×C3×C414yes—
261C4×D5—48no—
262#168—22no—
263#168C22×C3×C414yescomplemented
264C22×Q8C2×C3×D511yescharacteristic
265C22×D4—51no—
266C23×C4—101no—
267C2×SD16—58no—
268C2×SD16—54no—
269C22×C8—51no—
270C22×C8—51no—
271C4×Q8—22no—
272C4×Q8C2×C3×D514yescomplemented
273C4×D4—54no—
274C4×D4—102no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—51no—
277C2×C4⋊C4C2×C3×D511yescharacteristic
278C2×C22⋊C4—101no—
279C2×C42—21no—
280C2×C42C2×C3×D511yescharacteristic
281#106—54no—
282#103—54no—
283#102—54no—
284C4×C8—54no—
285C2×C3×C5C4×SD1614yescomplemented
286C2×C3×C5C2×C4×D411yescharacteristic
287C2×C3×C5C22×SD1611yescharacteristic
288C2×C3×C5C2×#13511yescharacteristic
289C3×D5—44no—
290C23×C3—102no—
291C23×C3—104no—
292C23×C3—202no—
293C23×C3—202no—
294C23×C3C2×C22⋊D511yescharacteristic
295C3×Q8—22no—
296C3×Q8—24no—
297C3×Q8C2×C4×D514yescomplemented
298C3×D4—54no—
299C3×D4—102no—
300C3×D4—104no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—42no—
306C2×C3×C4—102no—
307C2×C3×C4—202no—
308C2×C3×C4—208no—
309C2×C3×C4#68814yesdirect product factor
310C2×C3×C4C2×C4×D512yes—
311C2×C3×C4#70912yes—
312C2×C3×C4C2×C22⋊D512yes—
313C2×C3×C4C2×C22⋊D511yescharacteristic
314C2×C3×C4C2×C22⋊D512yes—
315C2×C3×C4C23×D511yescharacteristic
316C3×C8—54no—
317C3×C8—102no—
318C22×C5C3×C4×D412yes—
319C22×C5C2×C3×SD1612yes—
320C22×C5C3×#13512yes—
321C22×C5C22×C3×D411yescharacteristic
322C2×D5—24no—
323C2×D5—42no—
324C2×D5—42no—
325C2×D5—48no—
326C4×C5—22no—
327C4×C5—24no—
328C4×C5—42no—
329C4×C5C3×C4×D412yes—
330C4×C5C2×C3×SD1614yescomplemented
331C4×C5C23×C3×C411yescharacteristic
332C4×C5C2×C3×#4111yescharacteristic
333Dic5—42no—
334C24—101no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8C3×C4×D514yescomplemented
338C2×Q8C22×C3×D512yes—
339C2×D4—54no—
340C2×D4—52no—
341C2×D4—102no—
342C2×D4—101no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—101no—
346C22×C4—104no—
347C22×C4—204no—
348C22×C4C3×C22⋊D511yescharacteristic
349C22×C4C3×C22⋊D511yescharacteristic
350C22×C4C22×C3×D511yescharacteristic
351SD16—516no—
352C2×C8—54no—
353C2×C8—52no—
354C2×C8—52no—
355C2×C8—102no—
356C4⋊C4—24no—
357C4⋊C4—21no—
358C4⋊C4—52no—
359C4⋊C4—101no—
360C4⋊C4C22×C3×D512yes—
361C22⋊C4—104no—
362C42—24no—
363C42C3×C22⋊D512yes—
364C42C22×C3×D512yes—
365C3×C5C2×C4×SD1611yescharacteristic, complemented, Hall
366C22×C3—104no—
367C22×C3—208no—
368C22×C3—202no—
369C22×C3—202no—
370C22×C3C4×C22⋊D512yes—
371C22×C3C2×#68812yes—
372C22×C3#409612yes—
373C22×C3C22×C22⋊D511yescharacteristic
374C3×C4—24no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—202no—
378C3×C4C4×C22⋊D512yes—
379C3×C4C2×#68814yesdirect product factor
380C3×C4C22×C4×D511yescharacteristic
381C3×C4C2×#70911yescharacteristic
382C2×C5C3×C4×SD1614yescomplemented
383C2×C5C2×C3×C4×D411yescharacteristic
384C2×C5C22×C3×SD1611yescharacteristic
385C2×C5C2×C3×#13511yescharacteristic
386D5—44no—
387C23—102no—
388C23—104no—
389C23—202no—
390C23—202no—
391C23C2×C3×C22⋊D511yescharacteristic
392Q8—22no—
393Q8—24no—
394Q8C2×C3×C4×D514yescomplemented
395D4—54no—
396D4—102no—
397D4—104no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—102no—
404C2×C4—202no—
405C2×C4—208no—
406C2×C4C3×#68814yesdirect product factor
407C2×C4C2×C3×C4×D512yes—
408C2×C4C3×#70912yes—
409C2×C4C2×C3×C22⋊D512yes—
410C2×C4C2×C3×C22⋊D512yes—
411C2×C4C2×C3×C22⋊D511yescharacteristic
412C2×C4C23×C3×D511yescharacteristic
413C8—54no—
414C8—102no—
415C2×C3—204no—
416C2×C3C4×#68814yesdirect product factor
417C2×C3C2×C4×C22⋊D511yescharacteristic
418C2×C3C22×#68811yescharacteristic
419C2×C3C2×#409611yescharacteristic
420C5C2×C3×C4×SD1611yescharacteristic, complemented, Sylow
421C22—104no—
422C22—208no—
423C22—202no—
424C22—202no—
425C22C3×C4×C22⋊D512yes—
426C22C2×C3×#68812yes—
427C22C3×#409612yes—
428C22C22×C3×C22⋊D511yescharacteristic
429C4—24no—
430C4—22no—
431C4—42no—
432C4—202no—
433C4C3×C4×C22⋊D512yes—
434C4C2×C3×#68814yesdirect product factor
435C4C22×C3×C4×D511yescharacteristic
436C4C2×C3×#70911yescharacteristic
437C3C2×C4×#68811yescharacteristic, direct product factor, Sylow
438C2—204no—
439C2C3×C4×#68814yesdirect product factor
440C2C2×C3×C4×C22⋊D511yescharacteristic
441C2C22×C3×#68811yescharacteristic
442C2C2×C3×#409611yescharacteristic
subgroupG/H
centerC22×C3×C4C22⋊D5
commutatorC4×C5C23×C3×C4
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
cosocleC2×C4×C5C24×C3
hypercenterC2×C3×C4×Q8D5
hypocenterC5C2×C3×C4×SD16
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
4. C2×C3×#3967C2semidirect
5. C3×C4×#688C2direct (N × H)
7. C2×C3×#3940C2semidirect
8. C2×C3×C4×#168C2semidirect
9. C2×C4×#688C3direct (N × H)
14. C3×C4×C5×Q8C22semidirect
19. C2×C3×#688C4direct (N × H)
27. C3×#3967C22semidirect
29. C3×#3940C22semidirect
30. C3×C4×#168C22semidirect
32. C2×C4×C5×Q8C2×C3semidirect
35. C2×#3967C2×C3semidirect
36. C4×#688C2×C3direct (N × H)
38. C2×#3940C2×C3semidirect
39. C2×C4×#168C2×C3semidirect
43. C2×C3×C5×Q8C2×C4semidirect
63. C3×#688C2×C4direct (N × H)
68. C2×C3×#168C2×C4semidirect
72. C2×C3×C4×Q8D5semidirect
83. C4×C5×Q8C22×C3semidirect
88. C2×#688C3×C4direct (N × H)
96. #3967C22×C3semidirect
98. #3940C22×C3semidirect
99. C4×#168C22×C3semidirect
108. C3×C5×Q8C22×C4semidirect
114. C2×C3×C4×C5SD16semidirect
128. C3×#168C22×C4semidirect
137. C3×C4×Q8C2×D5semidirect
153. C2×C5×Q8C2×C3×C4semidirect
173. #688C2×C3×C4direct (N × H)
178. C2×#168C2×C3×C4semidirect
182. C2×C4×Q8C3×D5semidirect
201. C3×C4×C5C2×SD16semidirect
208. C2×C3×Q8C4×D5semidirect
243. C5×Q8C22×C3×C4semidirect
249. C2×C4×C5C3×SD16semidirect
263. #168C22×C3×C4semidirect
272. C4×Q8C2×C3×D5semidirect
285. C2×C3×C5C4×SD16semidirect
297. C3×Q8C2×C4×D5semidirect
309. C2×C3×C4#688direct (N × H)
330. C4×C5C2×C3×SD16semidirect
337. C2×Q8C3×C4×D5semidirect
365. C3×C5C2×C4×SD16semidirect
379. C3×C4C2×#688direct (N × H)
382. C2×C5C3×C4×SD16semidirect
394. Q8C2×C3×C4×D5semidirect
406. C2×C4C3×#688direct (N × H)
416. C2×C3C4×#688direct (N × H)
420. C5C2×C3×C4×SD16semidirect
434. C4C2×C3×#688direct (N × H)
437. C3C2×C4×#688direct (N × H)
439. C2C3×C4×#688direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C22×C3×#6882
C2×C3×C4×D202
C2×C3×#39672
C3×C4×#6882
C2×C3×#39412
C2×C3×#39402
C2×C3×C4×#1682
C2×C4×#6883
C2×C3×C4×SD165