Finite Group Databasebeta

← C3×#10617328 #14331139

Subgroups — C3×#10617328 #14331139

Order 1920.

1040 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
1C3×D5×#124C211yesmaximal, characteristic
2C3×(C2×AGL(1,𝔽5))⋊Q8C212yesmaximal, complemented
3C3×#65779C212yesmaximal
4C3×#65700C214yesmaximal, complemented
5C3×#65699C214yesmaximal, complemented
6C3×#65689C212yesmaximal, complemented
7#10617328C311yesmaximal, characteristic, direct product factor, Hall
8C23×C3×AGL(1,𝔽5)C2212yes—
9C2×C3×Q8×D5C2211yescharacteristic
10C2×C3×D4×D5C2211yescharacteristic
11C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
12C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
13C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
14C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
15C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
16C2×C3×C4×AGL(1,𝔽5)C2212yes—
17C3×C5×#124C411yescharacteristic, complemented
18C3×#4093C411yescharacteristic, complemented
19C3×#4082C411yescharacteristic, complemented
20C3×#4031C414yescomplemented
21C3×D5×C22⋊C4C2214yes—
22C3×#4021C411yescharacteristic, complemented
23C3×C42×D5C2211yescharacteristic
24C3×(C2×AGL(1,𝔽5))⋊C4C2214yes—
25C3×(C2×AGL(1,𝔽5))⋊C4C2214yes—
26C3×(C2×AGL(1,𝔽5))⋊C4C2214yes—
27C3×#2379—51nomaximal, Hall
28D5×#124C2×C311yescharacteristic
29(C2×AGL(1,𝔽5))⋊Q8C2×C312yescomplemented
30#65779C2×C312yes—
31#65700C2×C314yescomplemented
32#65699C2×C314yescomplemented
33#65689C2×C312yescomplemented
34C23×C3×D5C2312yes—
35C22×C3×AGL(1,𝔽5)—42no—
36C22×C3×AGL(1,𝔽5)—44no—
37C22×C3×AGL(1,𝔽5)C2314yes—
38C22×C3×AGL(1,𝔽5)C2314yes—
39C2×C3×C5×Q8C2×C411yescharacteristic, complemented
40C2×C3×C5×D4C2×C411yescharacteristic
41C2×C3×C22⋊D5C2×C412yes—
42C3×Q8×D5—41no—
43C3×Q8×D5—41no—
44C3×D4×D5—41no—
45C3×D4×D5—41no—
46C2×C3×D20C2×C411yescharacteristic
47C2×C3×C4×D5C2312yes—
48C2×C3×C4×D5C2312yes—
49C2×C3×C4×D5C2311yescharacteristic
50C2×C3×Dic10C2×C411yescharacteristic, complemented
51C2×C3×Dic10C2×C412yescomplemented
52C3×C22⋊AGL(1,𝔽5)—42no—
53C3×C22⋊AGL(1,𝔽5)—44no—
54C3×C4⋊AGL(1,𝔽5)—42no—
55C3×C4⋊AGL(1,𝔽5)—42no—
56C3×C4⋊AGL(1,𝔽5)—42no—
57C3×C4×AGL(1,𝔽5)—42no—
58C3×C5×C22⋊C4C2×C414yescomplemented
59C3×C42×C5C2×C411yescharacteristic, complemented
60C3×C22⋊Dic5C2×C414yescomplemented
61C3×#685C2×C414yescomplemented
62C3×#685C2×C414yescomplemented
63C3×C4×Dic5C2×C411yescharacteristic, complemented
64C3×C4×Dic5C2×C412yescomplemented
65C2×C3×#124—51no—
66C3×#391—52no—
67C3×#388—54no—
68C3×#386—52no—
69C3×#385—54no—
70C3×#382—52no—
71C23×AGL(1,𝔽5)C22×C312yes—
72C2×Q8×D5C22×C311yescharacteristic
73C2×D4×D5C22×C311yescharacteristic
74C2×C22⋊AGL(1,𝔽5)C22×C314yes—
75C2×C22⋊AGL(1,𝔽5)C22×C312yes—
76C2×C4⋊AGL(1,𝔽5)C22×C312yes—
77C2×C4⋊AGL(1,𝔽5)C22×C312yes—
78C2×C4⋊AGL(1,𝔽5)C22×C312yes—
79C2×C4×AGL(1,𝔽5)C22×C312yes—
80C5×#124C3×C411yescharacteristic, complemented
81#4093C3×C411yescharacteristic, complemented
82#4082C3×C411yescharacteristic, complemented
83#4031C3×C414yescomplemented
84D5×C22⋊C4C22×C314yes—
85#4021C3×C411yescharacteristic, complemented
86C42×D5C22×C311yescharacteristic
87(C2×AGL(1,𝔽5))⋊C4C22×C314yes—
88(C2×AGL(1,𝔽5))⋊C4C22×C314yes—
89(C2×AGL(1,𝔽5))⋊C4C22×C314yes—
90#2379—51noSylow
91C23×C3×C5C22×C412yes—
92C22×C3×D5—22no—
93C22×C3×D5—24no—
94C22×C3×D5—22no—
95C22×C3×D5—24no—
96C22×C3×D5C22×C412yes—
97C22×C3×D5C2411yescharacteristic
98C2×C3×AGL(1,𝔽5)—24no—
99C2×C3×AGL(1,𝔽5)—24no—
100C2×C3×AGL(1,𝔽5)—24no—
101C2×C3×AGL(1,𝔽5)—24no—
102C2×C3×AGL(1,𝔽5)—24no—
103C2×C3×AGL(1,𝔽5)—24no—
104C2×C3×AGL(1,𝔽5)—48no—
105C3×C5×Q8—41no—
106C3×C5×D4—41no—
107C2×C3×C4×C5C22×C412yes—
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×C3×C4×C5C22×C412yes—
110C3×C22⋊D5—42no—
111C2×C3×Dic5C22×C412yes—
112C2×C3×Dic5C22×C411yescharacteristic
113C2×C3×Dic5C22×C412yes—
114C3×D20—41no—
115C3×C4×D5—22no—
116C3×C4×D5—22no—
117C3×C4×D5—22no—
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—22no—
121C3×Dic10—41no—
122C3×Dic10—42no—
123C22×C3×Q8—51no—
124C22×C3×D4—51no—
125C23×C3×C4—52no—
126C3×#124—51no—
127C3×#124—51no—
128C3×#124—54no—
129C3×#124—51no—
130C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
131C2×C3×C4⋊C4—52no—
132C2×C3×C4⋊C4—52no—
133C2×C3×C4⋊C4—52no—
134C2×C3×C22⋊C4—54no—
135C2×C3×C22⋊C4—54no—
136C2×C3×C22⋊C4—52no—
137C2×C3×C42—51no—
138C2×C3×C42—52no—
139C3×#95—54no—
140C3×#95—54no—
141C3×#95—54no—
142C23×D5C23×C312yes—
143C22×AGL(1,𝔽5)—42no—
144C22×AGL(1,𝔽5)—44no—
145C22×AGL(1,𝔽5)C23×C314yes—
146C22×AGL(1,𝔽5)C23×C314yes—
147C2×C5×Q8C2×C3×C411yescharacteristic, complemented
148C2×C5×D4C2×C3×C411yescharacteristic
149C2×C22⋊D5C2×C3×C412yes—
150Q8×D5—41no—
151Q8×D5—41no—
152D4×D5—41no—
153D4×D5—41no—
154C2×D20C2×C3×C411yescharacteristic
155C2×C4×D5C23×C312yes—
156C2×C4×D5C23×C312yes—
157C2×C4×D5C23×C311yescharacteristic
158C2×Dic10C2×C3×C411yescharacteristic, complemented
159C2×Dic10C2×C3×C412yescomplemented
160C22⋊AGL(1,𝔽5)—42no—
161C22⋊AGL(1,𝔽5)—44no—
162C4⋊AGL(1,𝔽5)—42no—
163C4⋊AGL(1,𝔽5)—42no—
164C4⋊AGL(1,𝔽5)—42no—
165C4×AGL(1,𝔽5)—42no—
166C5×C22⋊C4C2×C3×C414yescomplemented
167C42×C5C2×C3×C411yescharacteristic, complemented
168C22⋊Dic5C2×C3×C414yescomplemented
169#685C2×C3×C414yescomplemented
170#685C2×C3×C414yescomplemented
171C4×Dic5C2×C3×C412yescomplemented
172C4×Dic5C2×C3×C411yescharacteristic, complemented
173C2×#124—51no—
174#391—52no—
175#388—54no—
176#386—52no—
177#385—54no—
178#382—52no—
179C22×C3×C5—22no—
180C22×C3×C5—24no—
181C22×C3×C5C23×C411yescharacteristic
182C2×C3×D5—22no—
183C2×C3×D5—24no—
184C2×C3×D5—42no—
185C2×C3×D5—44no—
186C2×C3×D5—42no—
187C2×C3×D5H(5,𝔽2)12yes—
188C2×C3×D5H(5,𝔽2)11yescharacteristic
189C2×C3×D5H(5,𝔽2)11yescharacteristic
190C2×C3×D52−1+412yes—
191C3×AGL(1,𝔽5)—44no—
192C3×AGL(1,𝔽5)—44no—
193C3×C4×C5—21no—
194C3×C4×C5—22no—
195C3×C4×C5—22no—
196C3×Dic5—22no—
197C3×Dic5—21no—
198C3×Dic5—22no—
199C24×C3—52no—
200C2×C3×Q8—52no—
201C2×C3×Q8—51no—
202C2×C3×Q8—201no—
203C2×C3×Q8—201no—
204C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
205C2×C3×D4—52no—
206C2×C3×D4—51no—
207C2×C3×D4—201no—
208C2×C3×D4—201no—
209C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
210C22×C3×C4—51no—
211C22×C3×C4—52no—
212C22×C3×C4—52no—
213C22×C3×C4—54no—
214C22×C3×C4—54no—
215C22×C3×C4—204no—
216C22×C3×C4—202no—
217C3×C4⋊C4—202no—
218C3×C4⋊C4—202no—
219C3×C4⋊C4—202no—
220C3×C22⋊C4—54no—
221C3×C22⋊C4—54no—
222C3×C22⋊C4—54no—
223C3×C22⋊C4—202no—
224C3×C22⋊C4—204no—
225C3×C22⋊C4C2×AGL(1,𝔽5)14yescomplemented
226C3×C42—51no—
227C3×C42—52no—
228C3×C42—202no—
229C3×C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
230C23×C5C22×C3×C412yes—
231C22×D5—22no—
232C22×D5—24no—
233C22×D5—22no—
234C22×D5—24no—
235C22×D5C22×C3×C412yes—
236C22×D5C24×C311yescharacteristic
237C2×AGL(1,𝔽5)—24no—
238C2×AGL(1,𝔽5)—24no—
239C2×AGL(1,𝔽5)—24no—
240C2×AGL(1,𝔽5)—24no—
241C2×AGL(1,𝔽5)—24no—
242C2×AGL(1,𝔽5)—24no—
243C2×AGL(1,𝔽5)—48no—
244C5×Q8—41no—
245C5×D4—41no—
246C2×C4×C5C22×C3×C412yes—
247C2×C4×C5C22×C3×C411yescharacteristic
248C2×C4×C5C22×C3×C412yes—
249C22⋊D5—42no—
250C2×Dic5C22×C3×C412yes—
251C2×Dic5C22×C3×C411yescharacteristic
252C2×Dic5C22×C3×C412yes—
253D20—41no—
254C4×D5—22no—
255C4×D5—22no—
256C4×D5—22no—
257C4×D5—21no—
258C4×D5—21no—
259C4×D5—22no—
260Dic10—41no—
261Dic10—42no—
262C22×Q8—51no—
263C22×D4—51no—
264C23×C4—52no—
265#124—51no—
266#124—51no—
267#124—54no—
268#124—51no—
269#124C3×AGL(1,𝔽5)11yescharacteristic, complemented
270C2×C4⋊C4—52no—
271C2×C4⋊C4—52no—
272C2×C4⋊C4—52no—
273C2×C22⋊C4—54no—
274C2×C22⋊C4—54no—
275C2×C22⋊C4—52no—
276C2×C42—51no—
277C2×C42—52no—
278#95—54no—
279#95—54no—
280#95—54no—
281C2×C3×C5—42no—
282C2×C3×C5#51811yescharacteristic
283C2×C3×C5#52012yes—
284C3×D5—42no—
285C3×D5H(3,𝔽4)11yescharacteristic
286C3×D5#56212yes—
287C3×D5#56311yescharacteristic
288C23×C3—51no—
289C23×C3—52no—
290C23×C3—104no—
291C23×C3—102no—
292C23×C3—102no—
293C23×C3—104no—
294C23×C3C22×AGL(1,𝔽5)12yes—
295C3×Q8—41no—
296C3×Q8—201no—
297C3×Q8—202no—
298C3×D4—41no—
299C3×D4—201no—
300C3×D4—202no—
301C2×C3×C4—52no—
302C2×C3×C4—51no—
303C2×C3×C4—52no—
304C2×C3×C4—102no—
305C2×C3×C4—102no—
306C2×C3×C4—102no—
307C2×C3×C4—102no—
308C2×C3×C4—101no—
309C2×C3×C4—101no—
310C2×C3×C4—104no—
311C2×C3×C4—104no—
312C2×C3×C4—104no—
313C2×C3×C4—104no—
314C2×C3×C4—104no—
315C2×C3×C4—104no—
316C2×C3×C4—208no—
317C2×C3×C4C22×AGL(1,𝔽5)12yes—
318C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
319C2×C3×C4C22×AGL(1,𝔽5)12yes—
320C22×C5—22no—
321C22×C5—24no—
322C22×C5C23×C3×C411yescharacteristic
323C2×D5—22no—
324C2×D5—24no—
325C2×D5—42no—
326C2×D5—44no—
327C2×D5—42no—
328C2×D5C3×H(5,𝔽2)12yes—
329C2×D5C3×H(5,𝔽2)11yescharacteristic
330C2×D5C3×H(5,𝔽2)11yescharacteristic
331C2×D5C3×2−1+412yes—
332AGL(1,𝔽5)—44no—
333AGL(1,𝔽5)—44no—
334C4×C5—21no—
335C4×C5—22no—
336C4×C5—22no—
337Dic5—22no—
338Dic5—21no—
339Dic5—22no—
340C24—52no—
341C2×Q8—52no—
342C2×Q8—51no—
343C2×Q8—201no—
344C2×Q8—201no—
345C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
346C2×D4—52no—
347C2×D4—51no—
348C2×D4—201no—
349C2×D4—201no—
350C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
351C22×C4—51no—
352C22×C4—52no—
353C22×C4—52no—
354C22×C4—54no—
355C22×C4—54no—
356C22×C4—204no—
357C22×C4—202no—
358C4⋊C4—202no—
359C4⋊C4—202no—
360C4⋊C4—202no—
361C22⋊C4—54no—
362C22⋊C4—54no—
363C22⋊C4—54no—
364C22⋊C4—202no—
365C22⋊C4—204no—
366C22⋊C4C2×C3×AGL(1,𝔽5)14yescomplemented
367C42—51no—
368C42—52no—
369C42—202no—
370C42C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
371C3×C5#237911yescharacteristic, complemented, Hall
372C22×C3—22no—
373C22×C3—24no—
374C22×C3—51no—
375C22×C3—51no—
376C22×C3—52no—
377C22×C3—52no—
378C22×C3—102no—
379C22×C3—104no—
380C22×C3—202no—
381C22×C3—204no—
382C22×C3—202no—
383C22×C3C23×AGL(1,𝔽5)11yescharacteristic
384C3×C4—21no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—101no—
388C3×C4—102no—
389C3×C4—102no—
390C3×C4—204no—
391C3×C4—204no—
392C2×C5—42no—
393C2×C5C3×#51811yescharacteristic
394C2×C5C3×#52012yes—
395D5—42no—
396D5C3×H(3,𝔽4)11yescharacteristic
397D5C3×#56212yes—
398D5C3×#56311yescharacteristic
399C23—51no—
400C23—52no—
401C23—104no—
402C23—102no—
403C23—102no—
404C23—104no—
405C23C22×C3×AGL(1,𝔽5)12yes—
406Q8—41no—
407Q8—201no—
408Q8—202no—
409D4—41no—
410D4—201no—
411D4—202no—
412C2×C4—52no—
413C2×C4—51no—
414C2×C4—52no—
415C2×C4—102no—
416C2×C4—102no—
417C2×C4—102no—
418C2×C4—101no—
419C2×C4—101no—
420C2×C4—102no—
421C2×C4—104no—
422C2×C4—104no—
423C2×C4—104no—
424C2×C4—104no—
425C2×C4—104no—
426C2×C4—104no—
427C2×C4—208no—
428C2×C4C22×C3×AGL(1,𝔽5)12yes—
429C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
430C2×C4C22×C3×AGL(1,𝔽5)12yes—
431C2×C3—42no—
432C2×C3—51no—
433C2×C3—52no—
434C2×C3—51no—
435C2×C3—202no—
436C2×C3#6625811yescharacteristic
437C2×C3#6626612yes—
438C5C3×#237911yescharacteristic, complemented, Sylow
439C22—22no—
440C22—24no—
441C22—51no—
442C22—51no—
443C22—52no—
444C22—52no—
445C22—102no—
446C22—104no—
447C22—202no—
448C22—204no—
449C22—202no—
450C22C23×C3×AGL(1,𝔽5)11yescharacteristic
451C4—21no—
452C4—22no—
453C4—22no—
454C4—101no—
455C4—102no—
456C4—102no—
457C4—204no—
458C4—204no—
459C3#1061732811yescharacteristic, direct product factor, Sylow
460C2—42no—
461C2—51no—
462C2—52no—
463C2—51no—
464C2—202no—
465C2C3×#6625811yescharacteristic
466C2C3×#6626612yes—
subgroupG/H
centerC22×C3C23×AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC3×C5×#124C4
socleC22×C3×C5C23×C4
prime radicalC2×C3×D4×D5C22
cosocleC22×D5C24×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#2379
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C3×(C2×AGL(1,𝔽5))⋊Q8C2semidirect
4. C3×#65700C2semidirect
5. C3×#65699C2semidirect
6. C3×#65689C2semidirect
7. #10617328C3direct (N × H)
17. C3×C5×#124C4semidirect
18. C3×#4093C4semidirect
19. C3×#4082C4semidirect
20. C3×#4031C4semidirect
22. C3×#4021C4semidirect
29. (C2×AGL(1,𝔽5))⋊Q8C2×C3semidirect
31. #65700C2×C3semidirect
32. #65699C2×C3semidirect
33. #65689C2×C3semidirect
39. C2×C3×C5×Q8C2×C4semidirect
50. C2×C3×Dic10C2×C4semidirect
51. C2×C3×Dic10C2×C4semidirect
58. C3×C5×C22⋊C4C2×C4semidirect
59. C3×C42×C5C2×C4semidirect
60. C3×C22⋊Dic5C2×C4semidirect
61. C3×#685C2×C4semidirect
62. C3×#685C2×C4semidirect
63. C3×C4×Dic5C2×C4semidirect
64. C3×C4×Dic5C2×C4semidirect
80. C5×#124C3×C4semidirect
81. #4093C3×C4semidirect
82. #4082C3×C4semidirect
83. #4031C3×C4semidirect
85. #4021C3×C4semidirect
130. C3×#124AGL(1,𝔽5)semidirect
147. C2×C5×Q8C2×C3×C4semidirect
158. C2×Dic10C2×C3×C4semidirect
159. C2×Dic10C2×C3×C4semidirect
166. C5×C22⋊C4C2×C3×C4semidirect
167. C42×C5C2×C3×C4semidirect
168. C22⋊Dic5C2×C3×C4semidirect
169. #685C2×C3×C4semidirect
170. #685C2×C3×C4semidirect
171. C4×Dic5C2×C3×C4semidirect
172. C4×Dic5C2×C3×C4semidirect
204. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
225. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
229. C3×C42C2×AGL(1,𝔽5)semidirect
269. #124C3×AGL(1,𝔽5)semidirect
345. C2×Q8C2×C3×AGL(1,𝔽5)semidirect
366. C22⋊C4C2×C3×AGL(1,𝔽5)semidirect
370. C42C2×C3×AGL(1,𝔽5)semidirect
371. C3×C5#2379semidirect
438. C5C3×#2379semidirect
459. C3#10617328direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×#1242
C3×(C2×AGL(1,𝔽5))⋊Q82
C3×#657792
C3×#657002
C3×#656992
C3×#656892
#106173283
C3×#23795