Finite Group Databasebeta

← C3×#10617130 #14322719

Subgroups — C3×#10617130 #14322719

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×#117C211yesmaximal, characteristic
2C3×#65765C212yesmaximal
3C3×#65711C214yesmaximal, complemented
4C3×#65700C214yesmaximal
5C3×#65689C211yesmaximal, characteristic, complemented
6C3×#65688C211yesmaximal, characteristic, complemented
7C3×C4×C4⋊AGL(1,𝔽5)C212yesmaximal, complemented
8#10617130C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
11C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
12C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
13C2×C3×C4⋊AGL(1,𝔽5)C2214yes—
14C2×C3×C4×AGL(1,𝔽5)C2214yes—
15C3×C5×#117C411yescharacteristic, complemented
16C3×#4072C411yescharacteristic, complemented
17C3×#4038C412yescomplemented
18C3×D5×C4⋊C4C2212yes—
19C3×D5×C22⋊C4C2212yes—
20C3×#4023C412yescomplemented
21C3×#4018C411yescharacteristic, complemented
22C3×#4018C411yescharacteristic, complemented
23C3×C42×D5C2211yescharacteristic
24C3×C42×D5C2211yescharacteristic
25C3×(C2×AGL(1,𝔽5))⋊C4C2214yes—
26C3×(C2×AGL(1,𝔽5))⋊C4C2214yes—
27C3×(C2×AGL(1,𝔽5))⋊C4C2214yes—
28C3×#2314—51nomaximal, Hall
29D5×#117C2×C311yescharacteristic
30#65765C2×C312yes—
31#65711C2×C314yescomplemented
32#65700C2×C314yes—
33#65689C2×C311yescharacteristic, complemented
34#65688C2×C311yescharacteristic, complemented
35C4×C4⋊AGL(1,𝔽5)C2×C312yescomplemented
36C23×C3×D5C2311yescharacteristic
37C22×C3×AGL(1,𝔽5)C2314yes—
38C22×C3×AGL(1,𝔽5)C2314yes—
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×Dic5C2×C411yescharacteristic
41C2×C3×C4×D5—22no—
42C2×C3×C4×D5—22no—
43C2×C3×C4×D5C2×C411yescharacteristic
44C2×C3×C4×D5C2×C411yescharacteristic
45C2×C3×C4×D5C2312yes—
46C2×C3×C4×D5C2312yes—
47C2×C3×C4×D5C2311yescharacteristic
48C2×C3×C4×D5C2311yescharacteristic
49C3×C22⋊AGL(1,𝔽5)—44no—
50C3×C4⋊AGL(1,𝔽5)—24no—
51C3×C4⋊AGL(1,𝔽5)—24no—
52C3×C4⋊AGL(1,𝔽5)—44no—
53C3×C4×AGL(1,𝔽5)—44no—
54C3×C5×C4⋊C4C2×C412yes—
55C3×C5×C22⋊C4C2×C412yes—
56C3×C42×C5C2×C411yescharacteristic
57C3×C42×C5C2×C411yescharacteristic
58C3×C22⋊Dic5C2×C412yes—
59C3×#685C2×C412yes—
60C3×#685C2×C412yes—
61C3×C4⋊Dic5C2×C412yes—
62C3×#683C2×C412yes—
63C3×#683C2×C412yes—
64C3×C4×Dic5C2×C411yescharacteristic
65C3×C4×Dic5C2×C411yescharacteristic
66C3×C4×Dic5C2×C412yes—
67C3×C4×Dic5C2×C412yes—
68C2×C3×#117—51no—
69C3×#388—54no—
70C3×#387—54no—
71C3×#384—51no—
72C3×#382—51no—
73C3×#380—52no—
74C3×C4×C4⋊C4—52no—
75C22×C4×D5C22×C311yescharacteristic
76C2×C22⋊AGL(1,𝔽5)C22×C314yes—
77C2×C4⋊AGL(1,𝔽5)C22×C312yes—
78C2×C4⋊AGL(1,𝔽5)C22×C312yes—
79C2×C4⋊AGL(1,𝔽5)C22×C314yes—
80C2×C4×AGL(1,𝔽5)C22×C314yes—
81C5×#117C3×C411yescharacteristic, complemented
82#4072C3×C411yescharacteristic, complemented
83#4038C3×C412yescomplemented
84D5×C4⋊C4C22×C312yes—
85D5×C22⋊C4C22×C312yes—
86#4023C3×C412yescomplemented
87#4018C3×C411yescharacteristic, complemented
88#4018C3×C411yescharacteristic, complemented
89C42×D5C22×C311yescharacteristic
90C42×D5C22×C311yescharacteristic
91(C2×AGL(1,𝔽5))⋊C4C22×C314yes—
92(C2×AGL(1,𝔽5))⋊C4C22×C314yes—
93(C2×AGL(1,𝔽5))⋊C4C22×C314yes—
94#2314—51noSylow
95C23×C3×C5C22×C411yescharacteristic
96C22×C3×D5—21no—
97C22×C3×D5—21no—
98C22×C3×D5—22no—
99C22×C3×D5—22no—
100C22×C3×D5C22×C411yescharacteristic
101C22×C3×D5C2411yescharacteristic
102C2×C3×AGL(1,𝔽5)—24no—
103C2×C3×AGL(1,𝔽5)—24no—
104C2×C3×AGL(1,𝔽5)—24no—
105C2×C3×AGL(1,𝔽5)—24no—
106C2×C3×AGL(1,𝔽5)—24no—
107C2×C3×AGL(1,𝔽5)—24no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5C22×C412yes—
110C2×C3×C4×C5C22×C412yes—
111C2×C3×C4×C5C22×C411yescharacteristic
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×C3×Dic5—22no—
114C2×C3×Dic5C22×C412yes—
115C2×C3×Dic5C22×C412yes—
116C2×C3×Dic5C22×C411yescharacteristic
117C2×C3×Dic5C22×C411yescharacteristic
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—22no—
121C3×C4×D5—22no—
122C3×C4×D5—22no—
123C3×C4×D5—22no—
124C3×C4×D5—22no—
125C3×C4×D5—22no—
126C3×C4×D5#4112yes—
127C3×C4×D5#4112yes—
128C23×C3×C4—51no—
129C3×#117—51no—
130C3×#117—52no—
131C3×#117—52no—
132C3×#117—51no—
133C3×#117—51no—
134C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
135C2×C3×C4⋊C4—52no—
136C2×C3×C4⋊C4—52no—
137C2×C3×C4⋊C4—52no—
138C2×C3×C4⋊C4—54no—
139C2×C3×C22⋊C4—52no—
140C2×C3×C22⋊C4—54no—
141C2×C3×C42—51no—
142C2×C3×C42—51no—
143C2×C3×C42—54no—
144C3×#95—54no—
145C3×#95—54no—
146C3×#95—54no—
147C23×D5C23×C311yescharacteristic
148C22×AGL(1,𝔽5)C23×C314yes—
149C22×AGL(1,𝔽5)C23×C314yes—
150C22×C4×C5C2×C3×C411yescharacteristic
151C22×Dic5C2×C3×C411yescharacteristic
152C2×C4×D5—22no—
153C2×C4×D5—22no—
154C2×C4×D5C2×C3×C411yescharacteristic
155C2×C4×D5C2×C3×C411yescharacteristic
156C2×C4×D5C23×C312yes—
157C2×C4×D5C23×C312yes—
158C2×C4×D5C23×C311yescharacteristic
159C2×C4×D5C23×C311yescharacteristic
160C22⋊AGL(1,𝔽5)—44no—
161C4⋊AGL(1,𝔽5)—24no—
162C4⋊AGL(1,𝔽5)—24no—
163C4⋊AGL(1,𝔽5)—44no—
164C4×AGL(1,𝔽5)—44no—
165C5×C4⋊C4C2×C3×C412yes—
166C5×C22⋊C4C2×C3×C412yes—
167C42×C5C2×C3×C411yescharacteristic
168C42×C5C2×C3×C411yescharacteristic
169C22⋊Dic5C2×C3×C412yes—
170#685C2×C3×C412yes—
171#685C2×C3×C412yes—
172C4⋊Dic5C2×C3×C412yes—
173#683C2×C3×C412yes—
174#683C2×C3×C412yes—
175C4×Dic5C2×C3×C411yescharacteristic
176C4×Dic5C2×C3×C411yescharacteristic
177C4×Dic5C2×C3×C412yes—
178C4×Dic5C2×C3×C412yes—
179C2×#117—51no—
180#388—54no—
181#387—54no—
182#384—51no—
183#382—51no—
184#380—52no—
185C4×C4⋊C4—52no—
186C22×C3×C5—21no—
187C22×C3×C5—22no—
188C22×C3×C5C23×C411yescharacteristic
189C2×C3×D5—21no—
190C2×C3×D5—22no—
191C2×C3×D5—41no—
192C2×C3×D5—41no—
193C2×C3×D5—42no—
194C2×C3×D5C2×#4111yescharacteristic
195C2×C3×D5C2×#4111yescharacteristic
196C2×C3×D5H(5,𝔽2)12yes—
197C2×C3×D52−1+412yes—
198C3×AGL(1,𝔽5)—44no—
199C3×AGL(1,𝔽5)—44no—
200C3×C4×C5—21no—
201C3×C4×C5—22no—
202C3×C4×C5—22no—
203C3×C4×C5#11712yes—
204C3×Dic5—21no—
205C3×Dic5—22no—
206C3×Dic5—22no—
207C3×Dic5#11712yes—
208C24×C3—51no—
209C22×C3×C4—51no—
210C22×C3×C4—51no—
211C22×C3×C4—51no—
212C22×C3×C4—52no—
213C22×C3×C4—52no—
214C22×C3×C4—51no—
215C22×C3×C4—51no—
216C22×C3×C4—54no—
217C22×C3×C4—54no—
218C22×C3×C4—102no—
219C22×C3×C4—102no—
220C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
221C3×C4⋊C4—52no—
222C3×C4⋊C4—52no—
223C3×C4⋊C4—52no—
224C3×C4⋊C4—104no—
225C3×C4⋊C4—104no—
226C3×C4⋊C4—204no—
227C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
228C3×C22⋊C4—52no—
229C3×C22⋊C4—52no—
230C3×C22⋊C4—52no—
231C3×C22⋊C4—204no—
232C3×C22⋊C4C2×AGL(1,𝔽5)12yes—
233C3×C42—52no—
234C3×C42—52no—
235C3×C42—51no—
236C3×C42—51no—
237C3×C42—204no—
238C3×C42C2×AGL(1,𝔽5)11yescharacteristic
239C3×C42C2×AGL(1,𝔽5)11yescharacteristic
240C23×C5C22×C3×C411yescharacteristic
241C22×D5—21no—
242C22×D5—21no—
243C22×D5—22no—
244C22×D5—22no—
245C22×D5C22×C3×C411yescharacteristic
246C22×D5C24×C311yescharacteristic
247C2×AGL(1,𝔽5)—24no—
248C2×AGL(1,𝔽5)—24no—
249C2×AGL(1,𝔽5)—24no—
250C2×AGL(1,𝔽5)—24no—
251C2×AGL(1,𝔽5)—24no—
252C2×AGL(1,𝔽5)—24no—
253C2×C4×C5—22no—
254C2×C4×C5C22×C3×C412yes—
255C2×C4×C5C22×C3×C412yes—
256C2×C4×C5C22×C3×C411yescharacteristic
257C2×C4×C5C22×C3×C411yescharacteristic
258C2×Dic5—22no—
259C2×Dic5C22×C3×C412yes—
260C2×Dic5C22×C3×C412yes—
261C2×Dic5C22×C3×C411yescharacteristic
262C2×Dic5C22×C3×C411yescharacteristic
263C4×D5—21no—
264C4×D5—21no—
265C4×D5—22no—
266C4×D5—22no—
267C4×D5—22no—
268C4×D5—22no—
269C4×D5—22no—
270C4×D5—22no—
271C4×D5C3×#4112yes—
272C4×D5C3×#4112yes—
273C23×C4—51no—
274#117—51no—
275#117—52no—
276#117—52no—
277#117—51no—
278#117—51no—
279#117C3×AGL(1,𝔽5)11yescharacteristic, complemented
280C2×C4⋊C4—52no—
281C2×C4⋊C4—52no—
282C2×C4⋊C4—52no—
283C2×C4⋊C4—54no—
284C2×C22⋊C4—52no—
285C2×C22⋊C4—54no—
286C2×C42—51no—
287C2×C42—51no—
288C2×C42—54no—
289#95—54no—
290#95—54no—
291#95—54no—
292C2×C3×C5—41no—
293C2×C3×C5C2×#11711yescharacteristic
294C2×C3×C5#52012yes—
295C3×D5—41no—
296C3×D5#54011yescharacteristic
297C3×D5#54111yescharacteristic
298C3×D5#54212yes—
299C23×C3—51no—
300C23×C3—51no—
301C23×C3—102no—
302C23×C3—101no—
303C23×C3—101no—
304C23×C3—102no—
305C23×C3C22×AGL(1,𝔽5)11yescharacteristic
306C2×C3×C4—22no—
307C2×C3×C4—51no—
308C2×C3×C4—52no—
309C2×C3×C4—52no—
310C2×C3×C4—52no—
311C2×C3×C4—52no—
312C2×C3×C4—51no—
313C2×C3×C4—102no—
314C2×C3×C4—102no—
315C2×C3×C4—102no—
316C2×C3×C4—102no—
317C2×C3×C4—102no—
318C2×C3×C4—101no—
319C2×C3×C4—101no—
320C2×C3×C4—102no—
321C2×C3×C4—102no—
322C2×C3×C4—104no—
323C2×C3×C4—104no—
324C2×C3×C4—104no—
325C2×C3×C4—104no—
326C2×C3×C4—104no—
327C2×C3×C4—104no—
328C2×C3×C4C22×AGL(1,𝔽5)12yes—
329C2×C3×C4C22×AGL(1,𝔽5)12yes—
330C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
331C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
332C22×C5—21no—
333C22×C5—22no—
334C22×C5C23×C3×C411yescharacteristic
335C2×D5—21no—
336C2×D5—22no—
337C2×D5—41no—
338C2×D5—41no—
339C2×D5—42no—
340C2×D5C2×C3×#4111yescharacteristic
341C2×D5C2×C3×#4111yescharacteristic
342C2×D5C3×H(5,𝔽2)12yes—
343C2×D5C3×2−1+412yes—
344AGL(1,𝔽5)—44no—
345AGL(1,𝔽5)—44no—
346C4×C5—21no—
347C4×C5—22no—
348C4×C5—22no—
349C4×C5C3×#11712yes—
350Dic5—21no—
351Dic5—22no—
352Dic5—22no—
353Dic5C3×#11712yes—
354C24—51no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—51no—
358C22×C4—52no—
359C22×C4—52no—
360C22×C4—51no—
361C22×C4—51no—
362C22×C4—54no—
363C22×C4—54no—
364C22×C4—102no—
365C22×C4—102no—
366C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
367C4⋊C4—52no—
368C4⋊C4—52no—
369C4⋊C4—52no—
370C4⋊C4—104no—
371C4⋊C4—104no—
372C4⋊C4—204no—
373C4⋊C4C2×C3×AGL(1,𝔽5)12yes—
374C22⋊C4—52no—
375C22⋊C4—52no—
376C22⋊C4—52no—
377C22⋊C4—204no—
378C22⋊C4C2×C3×AGL(1,𝔽5)12yes—
379C42—52no—
380C42—52no—
381C42—51no—
382C42—51no—
383C42—204no—
384C42C2×C3×AGL(1,𝔽5)11yescharacteristic
385C42C2×C3×AGL(1,𝔽5)11yescharacteristic
386C3×C5#231411yescharacteristic, complemented, Hall
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—51no—
390C22×C3—51no—
391C22×C3—52no—
392C22×C3—52no—
393C22×C3—101no—
394C22×C3—102no—
395C22×C3—201no—
396C22×C3—202no—
397C22×C3—201no—
398C22×C3C23×AGL(1,𝔽5)11yescharacteristic
399C3×C4—21no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—52no—
403C3×C4—101no—
404C3×C4—102no—
405C3×C4—102no—
406C3×C4—204no—
407C3×C4—204no—
408C3×C4#413012yes—
409C2×C5—41no—
410C2×C5C2×C3×#11711yescharacteristic
411C2×C5C3×#52012yes—
412D5—41no—
413D5C3×#54011yescharacteristic
414D5C3×#54111yescharacteristic
415D5C3×#54212yes—
416C23—51no—
417C23—51no—
418C23—102no—
419C23—101no—
420C23—101no—
421C23—102no—
422C23C22×C3×AGL(1,𝔽5)11yescharacteristic
423C2×C4—22no—
424C2×C4—51no—
425C2×C4—52no—
426C2×C4—52no—
427C2×C4—52no—
428C2×C4—52no—
429C2×C4—51no—
430C2×C4—102no—
431C2×C4—102no—
432C2×C4—102no—
433C2×C4—101no—
434C2×C4—101no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—102no—
438C2×C4—102no—
439C2×C4—104no—
440C2×C4—104no—
441C2×C4—104no—
442C2×C4—104no—
443C2×C4—104no—
444C2×C4—104no—
445C2×C4C22×C3×AGL(1,𝔽5)12yes—
446C2×C4C22×C3×AGL(1,𝔽5)12yes—
447C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
448C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
449C2×C3—41no—
450C2×C3—51no—
451C2×C3—52no—
452C2×C3—51no—
453C2×C3—201no—
454C2×C3C2×#413011yescharacteristic
455C2×C3#6626612yes—
456C5C3×#231411yescharacteristic, complemented, Sylow
457C22—21no—
458C22—22no—
459C22—51no—
460C22—51no—
461C22—52no—
462C22—52no—
463C22—101no—
464C22—102no—
465C22—201no—
466C22—202no—
467C22—201no—
468C22C23×C3×AGL(1,𝔽5)11yescharacteristic
469C4—21no—
470C4—22no—
471C4—22no—
472C4—52no—
473C4—101no—
474C4—102no—
475C4—102no—
476C4—204no—
477C4—204no—
478C4C3×#413012yes—
479C3#1061713011yescharacteristic, direct product factor, Sylow
480C2—41no—
481C2—51no—
482C2—52no—
483C2—51no—
484C2—201no—
485C2C2×C3×#413011yescharacteristic
486C2C3×#6626612yes—
subgroupG/H
centerC22×C3C23×AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC22×C3×C5C23×C4
prime radicalC23×C3×D5C23
cosocleC22×D5C24×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#2314
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#65711C2semidirect
5. C3×#65689C2semidirect
6. C3×#65688C2semidirect
7. C3×C4×C4⋊AGL(1,𝔽5)C2semidirect
8. #10617130C3direct (N × H)
15. C3×C5×#117C4semidirect
16. C3×#4072C4semidirect
17. C3×#4038C4semidirect
20. C3×#4023C4semidirect
21. C3×#4018C4semidirect
22. C3×#4018C4semidirect
31. #65711C2×C3semidirect
33. #65689C2×C3semidirect
34. #65688C2×C3semidirect
35. C4×C4⋊AGL(1,𝔽5)C2×C3semidirect
81. C5×#117C3×C4semidirect
82. #4072C3×C4semidirect
83. #4038C3×C4semidirect
86. #4023C3×C4semidirect
87. #4018C3×C4semidirect
88. #4018C3×C4semidirect
134. C3×#117AGL(1,𝔽5)semidirect
279. #117C3×AGL(1,𝔽5)semidirect
386. C3×C5#2314semidirect
456. C5C3×#2314semidirect
479. C3#10617130direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×#1172
C3×#657652
C3×#657112
C3×#657002
C3×#656892
C3×#656882
C3×C4×C4⋊AGL(1,𝔽5)2
#106171303
C3×#23145