Finite Group Databasebeta

← C2×C3×#65303 #14416475

Subgroups — C2×C3×#65303 #14416475

Order 1920.

1216 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
2C22×C3×#686C211yesmaximal, characteristic
3C2×C3×C4×D20C211yesmaximal, characteristic, complemented
4C2×C3×#3964C211yesmaximal, characteristic, complemented
5C3×#65303C218yesmaximal, direct product factor
6C2×C3×#3941C211yesmaximal, characteristic, complemented
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C2×C3×#3935C211yesmaximal, characteristic, complemented
9C2×#65303C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C23×C3×C4×C5—21no—
12C22×C3×D20C2211yescharacteristic
13C22×C3×C4×D5—21no—
14C3×C4×C5×D4—22no—
15C3×C4×C5×D4C2214yescomplemented
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C22⋊C4—21no—
18C2×C3×C42×C5C2211yescharacteristic
19C2×C3×#686C418yescomplemented
20C2×C3×#686C2214yes—
21C2×C3×#685—21no—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C22×C3×#168C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C3×C4×D20—22no—
26C3×C4×D20C2214yescomplemented
27C3×#3964C2214yescomplemented
28C3×#3941C2214yescomplemented
29C3×#3940C2214yescomplemented
30C3×#3935C2214yescomplemented
31C2×C3×#442—51nomaximal, Hall
32C2×C4×C5×D4C2×C311yescharacteristic, complemented
33C22×#686C2×C311yescharacteristic
34C2×C4×D20C2×C311yescharacteristic, complemented
35C2×#3964C2×C311yescharacteristic, complemented
36#65303C2×C318yesdirect product factor
37C2×#3941C2×C311yescharacteristic, complemented
38C2×#3940C2×C311yescharacteristic, complemented
39C2×#3935C2×C311yescharacteristic, complemented
40C24×C3×C5—21no—
41C23×C3×D5—21no—
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4—22no—
44C2×C3×C5×D4C2×C414yescomplemented
45C2×C3×C5×D4C2312yes—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5—44no—
49C22×C3×C4×C5D411yescharacteristic
50C22×C3×C4×C5D411yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×Dic5—21no—
53C2×C3×D20—21no—
54C2×C3×D20—22no—
55C2×C3×D20C2×C414yescomplemented
56C2×C3×D20C2312yes—
57C2×C3×C4×D5—24no—
58C2×C3×C4×D5—44no—
59C3×C5×C4⋊C4—21no—
60C3×C5×C4⋊C4C2312yes—
61C3×C5×C22⋊C4—24no—
62C3×C42×C5D412yes—
63C3×C42×C5C2312yes—
64C3×#686C2×C4116yescomplemented
65C3×#685—24no—
66C3×C4⋊Dic5—21no—
67C3×C4⋊Dic5C2312yes—
68C2×C3×#168—22no—
69C2×C3×#168C2×C414yescomplemented
70C2×C3×#168C2312yes—
71C2×C3×#168C2312yes—
72C22×C3×D8—51no—
73C2×C3×C4×D4—51no—
74C2×C3×C4×D4D511yescharacteristic, complemented
75C3×#442—58no—
76C2×C3×#106—51no—
77C2×C3×#102—51no—
78C2×C3×#102—51no—
79C2×C3×#97—51no—
80C22×C5×D4C22×C311yescharacteristic
81C23×C4×C5—21no—
82C22×D20C22×C311yescharacteristic
83C22×C4×D5—21no—
84C4×C5×D4—22no—
85C4×C5×D4C22×C314yescomplemented
86C2×C5×C4⋊C4C22×C311yescharacteristic
87C2×C5×C22⋊C4—21no—
88C2×C42×C5C22×C311yescharacteristic
89C2×#686C3×C418yescomplemented
90C2×#686C22×C314yes—
91C2×#685—21no—
92C2×C4⋊Dic5C22×C311yescharacteristic
93C22×#168C22×C311yescharacteristic
94C22×#168C22×C311yescharacteristic
95C4×D20—22no—
96C4×D20C22×C314yescomplemented
97#3964C22×C314yescomplemented
98#3941C22×C314yescomplemented
99#3940C22×C314yescomplemented
100#3935C22×C314yescomplemented
101C2×#442—51noSylow
102C23×C3×C5—24no—
103C23×C3×C5—22no—
104C23×C3×C5—42no—
105C23×C3×C5—42no—
106C23×C3×C5C2×D411yescharacteristic
107C22×C3×D5—24no—
108C22×C3×D5—22no—
109C22×C3×D5—42no—
110C22×C3×D5—42no—
111C3×C5×D4—24no—
112C3×C5×D4—22no—
113C3×C5×D4C22×C414yescomplemented
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—48no—
118C2×C3×C4×C5—42no—
119C2×C3×C4×C5C22×C412yes—
120C2×C3×C4×C5C2×D412yes—
121C2×C3×C4×C5C2×D412yes—
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5#4112yes—
124C2×C3×C4×C5C2411yescharacteristic
125C2×C3×Dic5—22no—
126C2×C3×Dic5—42no—
127C3×D20—24no—
128C3×D20—22no—
129C3×D20C22×C414yescomplemented
130C3×C4×D5—48no—
131C3×#168—22no—
132C3×#168C22×C414yescomplemented
133C22×C3×D4—51no—
134C22×C3×D4C2×D511yescharacteristic
135C23×C3×C4—21no—
136C23×C3×C4—101no—
137C2×C3×D8—58no—
138C2×C3×D8—54no—
139C22×C3×C8—51no—
140C22×C3×C8—51no—
141C3×C4×D4—22no—
142C3×C4×D4—54no—
143C3×C4×D4—102no—
144C3×C4×D4C2×D514yescomplemented
145C2×C3×C4⋊C4—51no—
146C2×C3×C4⋊C4C2×D511yescharacteristic
147C2×C3×C22⋊C4—21no—
148C2×C3×C22⋊C4—101no—
149C2×C3×C42C2×D511yescharacteristic
150C3×#106—54no—
151C3×#102—54no—
152C3×#102—54no—
153C3×#97—54no—
154C24×C5—21no—
155C23×D5—21no—
156C2×C5×D4—21no—
157C2×C5×D4—22no—
158C2×C5×D4C2×C3×C414yescomplemented
159C2×C5×D4C23×C312yes—
160C22×C4×C5—21no—
161C22×C4×C5—24no—
162C22×C4×C5—44no—
163C22×C4×C5C3×D411yescharacteristic
164C22×C4×C5C3×D411yescharacteristic
165C22×C4×C5C23×C311yescharacteristic
166C22×Dic5—21no—
167C2×D20—21no—
168C2×D20—22no—
169C2×D20C2×C3×C414yescomplemented
170C2×D20C23×C312yes—
171C2×C4×D5—24no—
172C2×C4×D5—44no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4C23×C312yes—
175C5×C22⋊C4—24no—
176C42×C5C3×D412yes—
177C42×C5C23×C312yes—
178#686C2×C3×C4116yescomplemented
179#685—24no—
180C4⋊Dic5—21no—
181C4⋊Dic5C23×C312yes—
182C2×#168—22no—
183C2×#168C2×C3×C414yescomplemented
184C2×#168C23×C312yes—
185C2×#168C23×C312yes—
186C22×D8—51no—
187C2×C4×D4—51no—
188C2×C4×D4C3×D511yescharacteristic, complemented
189#442—58no—
190C2×#106—51no—
191C2×#102—51no—
192C2×#102—51no—
193C2×#97—51no—
194C22×C3×C5—24no—
195C22×C3×C5—42no—
196C22×C3×C5—42no—
197C22×C3×C5—48no—
198C22×C3×C5C4×D412yes—
199C22×C3×C5AGL(1,ℤ/8)12yes—
200C22×C3×C5AGL(1,ℤ/8)12yes—
201C22×C3×C5C22×D411yescharacteristic
202C2×C3×D5—24no—
203C2×C3×D5—42no—
204C2×C3×D5—42no—
205C2×C3×D5—48no—
206C3×C4×C5—22no—
207C3×C4×C5—22no—
208C3×C4×C5—42no—
209C3×C4×C5C4×D412yes—
210C3×C4×C5C23×C411yescharacteristic
211C3×C4×C5C2×#4111yescharacteristic
212C3×Dic5—42no—
213C24×C3—21no—
214C24×C3—101no—
215C2×C3×D4—22no—
216C2×C3×D4—21no—
217C2×C3×D4—54no—
218C2×C3×D4—52no—
219C2×C3×D4—102no—
220C2×C3×D4—101no—
221C2×C3×D4C4×D514yescomplemented
222C2×C3×D4C22×D512yes—
223C22×C3×C4—21no—
224C22×C3×C4—24no—
225C22×C3×C4—44no—
226C22×C3×C4—101no—
227C22×C3×C4—104no—
228C22×C3×C4—204no—
229C22×C3×C4C22⋊D511yescharacteristic
230C22×C3×C4C22⋊D511yescharacteristic
231C22×C3×C4C22×D511yescharacteristic
232C3×D8—516no—
233C2×C3×C8—54no—
234C2×C3×C8—52no—
235C2×C3×C8—52no—
236C2×C3×C8—102no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—52no—
239C3×C4⋊C4—101no—
240C3×C4⋊C4C22×D512yes—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—104no—
243C3×C42C22⋊D512yes—
244C3×C42C22×D512yes—
245C23×C5—24no—
246C23×C5—22no—
247C23×C5—42no—
248C23×C5—42no—
249C23×C5C2×C3×D411yescharacteristic
250C22×D5—24no—
251C22×D5—22no—
252C22×D5—42no—
253C22×D5—42no—
254C5×D4—24no—
255C5×D4—22no—
256C5×D4C22×C3×C414yescomplemented
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—48no—
261C2×C4×C5—42no—
262C2×C4×C5C22×C3×C412yes—
263C2×C4×C5C2×C3×D412yes—
264C2×C4×C5C2×C3×D412yes—
265C2×C4×C5C2×C3×D411yescharacteristic
266C2×C4×C5C3×#4112yes—
267C2×C4×C5C24×C311yescharacteristic
268C2×Dic5—22no—
269C2×Dic5—42no—
270D20—24no—
271D20—22no—
272D20C22×C3×C414yescomplemented
273C4×D5—48no—
274#168—22no—
275#168C22×C3×C414yescomplemented
276C22×D4—51no—
277C22×D4C2×C3×D511yescharacteristic
278C23×C4—21no—
279C23×C4—101no—
280C2×D8—58no—
281C2×D8—54no—
282C22×C8—51no—
283C22×C8—51no—
284C4×D4—22no—
285C4×D4—54no—
286C4×D4—102no—
287C4×D4C2×C3×D514yescomplemented
288C2×C4⋊C4—51no—
289C2×C4⋊C4C2×C3×D511yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—101no—
292C2×C42C2×C3×D511yescharacteristic
293#106—54no—
294#102—54no—
295#102—54no—
296#97—54no—
297C2×C3×C5—44no—
298C2×C3×C5#44214yescomplemented
299C2×C3×C5C2×C4×D411yescharacteristic
300C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
301C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
302C3×D5—44no—
303C23×C3—22no—
304C23×C3—24no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—102no—
308C23×C3—104no—
309C23×C3—202no—
310C23×C3—202no—
311C23×C3C2×C22⋊D511yescharacteristic
312C3×D4—22no—
313C3×D4—24no—
314C3×D4—54no—
315C3×D4—102no—
316C3×D4—104no—
317C3×D4C2×C4×D514yescomplemented
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—48no—
322C2×C3×C4—42no—
323C2×C3×C4—102no—
324C2×C3×C4—208no—
325C2×C3×C4—202no—
326C2×C3×C4C2×C4×D512yes—
327C2×C3×C4#70912yes—
328C2×C3×C4C2×C22⋊D512yes—
329C2×C3×C4C2×C22⋊D512yes—
330C2×C3×C4C2×C22⋊D511yescharacteristic
331C2×C3×C4C23×D511yescharacteristic
332C3×C8—54no—
333C3×C8—102no—
334C22×C5—24no—
335C22×C5—42no—
336C22×C5—42no—
337C22×C5—48no—
338C22×C5C3×C4×D412yes—
339C22×C5C3×AGL(1,ℤ/8)12yes—
340C22×C5C3×AGL(1,ℤ/8)12yes—
341C22×C5C22×C3×D411yescharacteristic
342C2×D5—24no—
343C2×D5—42no—
344C2×D5—42no—
345C2×D5—48no—
346C4×C5—22no—
347C4×C5—22no—
348C4×C5—42no—
349C4×C5C3×C4×D412yes—
350C4×C5C23×C3×C411yescharacteristic
351C4×C5C2×C3×#4111yescharacteristic
352Dic5—42no—
353C24—21no—
354C24—101no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—54no—
358C2×D4—52no—
359C2×D4—102no—
360C2×D4—101no—
361C2×D4C3×C4×D514yescomplemented
362C2×D4C22×C3×D512yes—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—44no—
366C22×C4—101no—
367C22×C4—104no—
368C22×C4—204no—
369C22×C4C3×C22⋊D511yescharacteristic
370C22×C4C3×C22⋊D511yescharacteristic
371C22×C4C22×C3×D511yescharacteristic
372D8—516no—
373C2×C8—54no—
374C2×C8—52no—
375C2×C8—52no—
376C2×C8—102no—
377C4⋊C4—21no—
378C4⋊C4—52no—
379C4⋊C4—101no—
380C4⋊C4C22×C3×D512yes—
381C22⋊C4—24no—
382C22⋊C4—104no—
383C42C3×C22⋊D512yes—
384C42C22×C3×D512yes—
385C3×C5C2×#44211yescharacteristic, complemented, Hall
386C22×C3—24no—
387C22×C3—48no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—104no—
391C22×C3—208no—
392C22×C3—202no—
393C22×C3—202no—
394C22×C3C4×C22⋊D512yes—
395C22×C3#407812yes—
396C22×C3#409512yes—
397C22×C3C22×C22⋊D511yescharacteristic
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—42no—
401C3×C4—202no—
402C3×C4C4×C22⋊D512yes—
403C3×C4C22×C4×D511yescharacteristic
404C3×C4C2×#70911yescharacteristic
405C2×C5—44no—
406C2×C5C3×#44214yescomplemented
407C2×C5C2×C3×C4×D411yescharacteristic
408C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
409C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
410D5—44no—
411C23—22no—
412C23—24no—
413C23—42no—
414C23—42no—
415C23—102no—
416C23—104no—
417C23—202no—
418C23—202no—
419C23C2×C3×C22⋊D511yescharacteristic
420D4—22no—
421D4—24no—
422D4—54no—
423D4—102no—
424D4—104no—
425D4C2×C3×C4×D514yescomplemented
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—48no—
430C2×C4—42no—
431C2×C4—102no—
432C2×C4—208no—
433C2×C4—202no—
434C2×C4C2×C3×C4×D512yes—
435C2×C4C3×#70912yes—
436C2×C4C2×C3×C22⋊D512yes—
437C2×C4C2×C3×C22⋊D512yes—
438C2×C4C2×C3×C22⋊D511yescharacteristic
439C2×C4C23×C3×D511yescharacteristic
440C8—54no—
441C8—102no—
442C2×C3—44no—
443C2×C3—204no—
444C2×C3#6530314yesdirect product factor
445C2×C3C2×C4×C22⋊D511yescharacteristic
446C2×C3C2×#407811yescharacteristic
447C2×C3C2×#409511yescharacteristic
448C5C2×C3×#44211yescharacteristic, complemented, Sylow
449C22—24no—
450C22—48no—
451C22—42no—
452C22—42no—
453C22—104no—
454C22—208no—
455C22—202no—
456C22—202no—
457C22C3×C4×C22⋊D512yes—
458C22C3×#407812yes—
459C22C3×#409512yes—
460C22C22×C3×C22⋊D511yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—202no—
465C4C3×C4×C22⋊D512yes—
466C4C22×C3×C4×D511yescharacteristic
467C4C2×C3×#70911yescharacteristic
468C3C2×#6530311yescharacteristic, direct product factor, Sylow
469C2—44no—
470C2—204no—
471C2C3×#6530314yesdirect product factor
472C2C2×C3×C4×C22⋊D511yescharacteristic
473C2C2×C3×#407811yescharacteristic
474C2C2×C3×#409511yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutatorC4×C5C23×C3×C4
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C4×C5×D4C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×#686C2
cosocleC2×C4×C5C24×C3
hypercenterC2×C3×C4×D4D5
hypocenterC5C2×C3×#442
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×D4C2semidirect
3. C2×C3×C4×D20C2semidirect
4. C2×C3×#3964C2semidirect
5. C3×#65303C2direct (N × H)
6. C2×C3×#3941C2semidirect
7. C2×C3×#3940C2semidirect
8. C2×C3×#3935C2semidirect
9. C2×#65303C3direct (N × H)
15. C3×C4×C5×D4C22semidirect
19. C2×C3×#686C4semidirect
26. C3×C4×D20C22semidirect
27. C3×#3964C22semidirect
28. C3×#3941C22semidirect
29. C3×#3940C22semidirect
30. C3×#3935C22semidirect
32. C2×C4×C5×D4C2×C3semidirect
34. C2×C4×D20C2×C3semidirect
35. C2×#3964C2×C3semidirect
36. #65303C2×C3direct (N × H)
37. C2×#3941C2×C3semidirect
38. C2×#3940C2×C3semidirect
39. C2×#3935C2×C3semidirect
44. C2×C3×C5×D4C2×C4semidirect
55. C2×C3×D20C2×C4semidirect
64. C3×#686C2×C4semidirect
69. C2×C3×#168C2×C4semidirect
74. C2×C3×C4×D4D5semidirect
85. C4×C5×D4C22×C3semidirect
89. C2×#686C3×C4semidirect
96. C4×D20C22×C3semidirect
97. #3964C22×C3semidirect
98. #3941C22×C3semidirect
99. #3940C22×C3semidirect
100. #3935C22×C3semidirect
113. C3×C5×D4C22×C4semidirect
129. C3×D20C22×C4semidirect
132. C3×#168C22×C4semidirect
144. C3×C4×D4C2×D5semidirect
158. C2×C5×D4C2×C3×C4semidirect
169. C2×D20C2×C3×C4semidirect
178. #686C2×C3×C4semidirect
183. C2×#168C2×C3×C4semidirect
188. C2×C4×D4C3×D5semidirect
221. C2×C3×D4C4×D5semidirect
256. C5×D4C22×C3×C4semidirect
272. D20C22×C3×C4semidirect
275. #168C22×C3×C4semidirect
287. C4×D4C2×C3×D5semidirect
298. C2×C3×C5#442semidirect
317. C3×D4C2×C4×D5semidirect
361. C2×D4C3×C4×D5semidirect
385. C3×C5C2×#442semidirect
406. C2×C5C3×#442semidirect
425. D4C2×C3×C4×D5semidirect
444. C2×C3#65303direct (N × H)
448. C5C2×C3×#442semidirect
468. C3C2×#65303direct (N × H)
471. C2C3×#65303direct (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×D42
C22×C3×#6862
C2×C3×C4×D202
C2×C3×#39642
C3×#653032
C2×C3×#39412
C2×C3×#39402
C2×C3×#39352
C2×#653033
C2×C3×#4425