Finite Group Databasebeta

← C3×#10617070 #14322599

Subgroups — C3×#10617070 #14322599

Order 1920.

1344 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×C22⋊C4C211yesmaximal, characteristic
2C3×#65800C212yesmaximal
3C3×#65765C212yesmaximal, complemented
4C3×C4×C22⋊AGL(1,𝔽5)C212yesmaximal, complemented
5C3×#65700C218yesmaximal, complemented
6#10617070C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×D5C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C22×C3×C4×D5C2211yescharacteristic
10C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
11C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
12C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
13C2×C3×C4⋊AGL(1,𝔽5)C2214yescomplemented
14C2×C3×C4×AGL(1,𝔽5)C2214yescomplemented
15C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
16C2×C3×C22⋊Dic5C411yescharacteristic, complemented
17C2×C3×#685C411yescharacteristic, complemented
18C2×C3×#685C411yescharacteristic, complemented
19C3×D5×C22⋊C4C414yescomplemented
20C3×D5×C22⋊C4C2214yes—
21C3×(C2×AGL(1,𝔽5))⋊C4C2214yescomplemented
22C3×(C2×AGL(1,𝔽5))⋊C4C2214yescomplemented
23C3×#2312—51nomaximal, Hall
24C2×D5×C22⋊C4C2×C311yescharacteristic
25#65800C2×C312yes—
26#65765C2×C312yescomplemented
27C4×C22⋊AGL(1,𝔽5)C2×C312yescomplemented
28#65700C2×C318yescomplemented
29C24×C3×C5C2×C411yescharacteristic
30C23×C3×D5—24no—
31C23×C3×D5—42no—
32C23×C3×D5—42no—
33C23×C3×D5C2×C412yes—
34C23×C3×D5C2×C411yescharacteristic
35C23×C3×D5C2312yes—
36C23×C3×D5C2311yescharacteristic
37C22×C3×AGL(1,𝔽5)C2318yes—
38C22×C3×C4×C5C2×C411yescharacteristic
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×Dic5C2×C411yescharacteristic
41C22×C3×Dic5C2×C411yescharacteristic
42C2×C3×C4×D5—42no—
43C2×C3×C4×D5—42no—
44C2×C3×C4×D5C2×C412yes—
45C2×C3×C4×D5C2×C412yes—
46C2×C3×C4×D5C2312yes—
47C2×C3×C4×D5C2312yes—
48C3×C22⋊AGL(1,𝔽5)—28no—
49C3×C22⋊AGL(1,𝔽5)—44no—
50C3×C22⋊AGL(1,𝔽5)—44no—
51C3×C4⋊AGL(1,𝔽5)—44no—
52C3×C4×AGL(1,𝔽5)—44no—
53C3×C5×C22⋊C4C2×C414yes—
54C3×C22⋊Dic5C2×C414yes—
55C3×#685C2×C414yes—
56C3×#685C2×C414yes—
57C22×C3×C22⋊C4—51no—
58C3×#388—58no—
59C3×#380—52no—
60C3×#379—52no—
61C3×C4×C22⋊C4—52no—
62C24×D5C22×C311yescharacteristic
63C22×C4×D5C22×C311yescharacteristic
64C22×C4×D5C22×C311yescharacteristic
65C2×C22⋊AGL(1,𝔽5)C22×C314yes—
66C2×C22⋊AGL(1,𝔽5)C22×C314yes—
67C2×C22⋊AGL(1,𝔽5)C22×C314yes—
68C2×C4⋊AGL(1,𝔽5)C22×C314yescomplemented
69C2×C4×AGL(1,𝔽5)C22×C314yescomplemented
70C2×C5×C22⋊C4C3×C411yescharacteristic, complemented
71C2×C22⋊Dic5C3×C411yescharacteristic, complemented
72C2×#685C3×C411yescharacteristic, complemented
73C2×#685C3×C411yescharacteristic, complemented
74D5×C22⋊C4C3×C414yescomplemented
75D5×C22⋊C4C22×C314yes—
76(C2×AGL(1,𝔽5))⋊C4C22×C314yescomplemented
77(C2×AGL(1,𝔽5))⋊C4C22×C314yescomplemented
78#2312—51noSylow
79C23×C3×C5—22no—
80C23×C3×C5—41no—
81C23×C3×C5—41no—
82C23×C3×C5C22×C412yes—
83C23×C3×C5C22×C411yescharacteristic
84C22×C3×D5—22no—
85C22×C3×D5—22no—
86C22×C3×D5—22no—
87C22×C3×D5—24no—
88C22×C3×D5—24no—
89C22×C3×D5—24no—
90C22×C3×D5—24no—
91C22×C3×D5—42no—
92C22×C3×D5—42no—
93C22×C3×D5—41no—
94C22×C3×D5—41no—
95C22×C3×D5—416no—
96C22×C3×D5C22×C412yes—
97C22×C3×D5C22×C411yescharacteristic
98C22×C3×D5#4114yes—
99C22×C3×D5C2411yescharacteristic
100C2×C3×AGL(1,𝔽5)—28no—
101C2×C3×AGL(1,𝔽5)—28no—
102C2×C3×AGL(1,𝔽5)—28no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5C22×C412yes—
106C2×C3×C4×C5C22×C412yes—
107C2×C3×Dic5—41no—
108C2×C3×Dic5—41no—
109C2×C3×Dic5C22×C412yes—
110C2×C3×Dic5C22×C412yes—
111C3×C4×D5—24no—
112C3×C4×D5—24no—
113C3×C4×D5—42no—
114C3×C4×D5—42no—
115C25×C3—51no—
116C23×C3×C4—51no—
117C23×C3×C4—51no—
118C2×C3×C4⋊C4—54no—
119C2×C3×C22⋊C4—51no—
120C2×C3×C22⋊C4—51no—
121C2×C3×C22⋊C4—51no—
122C2×C3×C22⋊C4—54no—
123C2×C3×C22⋊C4—54no—
124C2×C3×C22⋊C4—54no—
125C2×C3×C22⋊C4—54no—
126C2×C3×C22⋊C4—54no—
127C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
128C2×C3×C42—54no—
129C3×#95—54no—
130C3×#95—54no—
131C24×C5C2×C3×C411yescharacteristic
132C23×D5—24no—
133C23×D5—42no—
134C23×D5—42no—
135C23×D5C2×C3×C412yes—
136C23×D5C2×C3×C411yescharacteristic
137C23×D5C23×C312yes—
138C23×D5C23×C311yescharacteristic
139C22×AGL(1,𝔽5)C23×C318yes—
140C22×C4×C5C2×C3×C411yescharacteristic
141C22×C4×C5C2×C3×C411yescharacteristic
142C22×Dic5C2×C3×C411yescharacteristic
143C22×Dic5C2×C3×C411yescharacteristic
144C2×C4×D5—42no—
145C2×C4×D5—42no—
146C2×C4×D5C2×C3×C412yes—
147C2×C4×D5C2×C3×C412yes—
148C2×C4×D5C23×C312yes—
149C2×C4×D5C23×C312yes—
150C22⋊AGL(1,𝔽5)—28no—
151C22⋊AGL(1,𝔽5)—44no—
152C22⋊AGL(1,𝔽5)—44no—
153C4⋊AGL(1,𝔽5)—44no—
154C4×AGL(1,𝔽5)—44no—
155C5×C22⋊C4C2×C3×C414yes—
156C22⋊Dic5C2×C3×C414yes—
157#685C2×C3×C414yes—
158#685C2×C3×C414yes—
159C22×C22⋊C4—51no—
160#388—58no—
161#380—52no—
162#379—52no—
163C4×C22⋊C4—52no—
164C22×C3×C5—21no—
165C22×C3×C5—21no—
166C22×C3×C5—22no—
167C22×C3×C5—22no—
168C22×C3×C5—22no—
169C22×C3×C5—44no—
170C22×C3×C5#11712yes—
171C22×C3×C5C23×C411yescharacteristic
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—28no—
175C2×C3×D5—22no—
176C2×C3×D5—22no—
177C2×C3×D5—22no—
178C2×C3×D5—48no—
179C2×C3×D5—44no—
180C2×C3×D5—48no—
181C2×C3×D5#11712yes—
182C2×C3×D5C2×#4112yes—
183C2×C3×D5H(5,𝔽2)12yes—
184C2×C3×D5H(5,𝔽2)12yes—
185C3×AGL(1,𝔽5)—48no—
186C3×C4×C5—22no—
187C3×C4×C5—22no—
188C3×Dic5—22no—
189C3×Dic5—22no—
190C24×C3—51no—
191C24×C3—52no—
192C24×C3—52no—
193C24×C3—51no—
194C24×C3—104no—
195C24×C3—202no—
196C24×C3—202no—
197C24×C3C2×AGL(1,𝔽5)11yescharacteristic
198C22×C3×C4—51no—
199C22×C3×C4—51no—
200C22×C3×C4—52no—
201C22×C3×C4—52no—
202C22×C3×C4—52no—
203C22×C3×C4—52no—
204C22×C3×C4—58no—
205C22×C3×C4—202no—
206C22×C3×C4—202no—
207C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
208C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
209C3×C4⋊C4—204no—
210C3×C22⋊C4—54no—
211C3×C22⋊C4—54no—
212C3×C22⋊C4—54no—
213C3×C22⋊C4—108no—
214C3×C22⋊C4—204no—
215C3×C22⋊C4—204no—
216C3×C22⋊C4C2×AGL(1,𝔽5)14yes—
217C3×C42—204no—
218C23×C5—22no—
219C23×C5—41no—
220C23×C5—41no—
221C23×C5C22×C3×C412yes—
222C23×C5C22×C3×C411yescharacteristic
223C22×D5—22no—
224C22×D5—22no—
225C22×D5—22no—
226C22×D5—24no—
227C22×D5—24no—
228C22×D5—24no—
229C22×D5—24no—
230C22×D5—42no—
231C22×D5—42no—
232C22×D5—41no—
233C22×D5—41no—
234C22×D5—416no—
235C22×D5C22×C3×C412yes—
236C22×D5C22×C3×C411yescharacteristic
237C22×D5C3×#4114yes—
238C22×D5C24×C311yescharacteristic
239C2×AGL(1,𝔽5)—28no—
240C2×AGL(1,𝔽5)—28no—
241C2×AGL(1,𝔽5)—28no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5C22×C3×C412yes—
245C2×C4×C5C22×C3×C412yes—
246C2×Dic5—41no—
247C2×Dic5—41no—
248C2×Dic5C22×C3×C412yes—
249C2×Dic5C22×C3×C412yes—
250C4×D5—24no—
251C4×D5—24no—
252C4×D5—42no—
253C4×D5—42no—
254C25—51no—
255C23×C4—51no—
256C23×C4—51no—
257C2×C4⋊C4—54no—
258C2×C22⋊C4—51no—
259C2×C22⋊C4—51no—
260C2×C22⋊C4—51no—
261C2×C22⋊C4—54no—
262C2×C22⋊C4—54no—
263C2×C22⋊C4—54no—
264C2×C22⋊C4—54no—
265C2×C22⋊C4—54no—
266C2×C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
267C2×C42—54no—
268#95—54no—
269#95—54no—
270C2×C3×C5—22no—
271C2×C3×C5—42no—
272C2×C3×C5C2×#11711yescharacteristic
273C2×C3×C5#51811yescharacteristic
274C2×C3×C5#51811yescharacteristic
275C3×D5—22no—
276C3×D5—42no—
277C3×D5#53814yes—
278C23×C3—22no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—51no—
282C23×C3—54no—
283C23×C3—51no—
284C23×C3—52no—
285C23×C3—102no—
286C23×C3—102no—
287C23×C3—104no—
288C23×C3—104no—
289C23×C3—104no—
290C23×C3—104no—
291C23×C3—102no—
292C23×C3—201no—
293C23×C3—201no—
294C23×C3—202no—
295C23×C3—202no—
296C23×C3—2016no—
297C23×C3C22×AGL(1,𝔽5)12yes—
298C23×C3C22×AGL(1,𝔽5)11yescharacteristic
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—52no—
302C2×C3×C4—52no—
303C2×C3×C4—104no—
304C2×C3×C4—104no—
305C2×C3×C4—108no—
306C2×C3×C4—108no—
307C2×C3×C4—108no—
308C2×C3×C4—202no—
309C2×C3×C4—202no—
310C2×C3×C4—201no—
311C2×C3×C4—201no—
312C2×C3×C4C22×AGL(1,𝔽5)12yes—
313C2×C3×C4C22×AGL(1,𝔽5)12yes—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—22no—
317C22×C5—22no—
318C22×C5—22no—
319C22×C5—44no—
320C22×C5C3×#11712yes—
321C22×C5C23×C3×C411yescharacteristic
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—28no—
325C2×D5—22no—
326C2×D5—22no—
327C2×D5—22no—
328C2×D5—48no—
329C2×D5—44no—
330C2×D5—48no—
331C2×D5C3×#11712yes—
332C2×D5C2×C3×#4112yes—
333C2×D5C3×H(5,𝔽2)12yes—
334C2×D5C3×H(5,𝔽2)12yes—
335AGL(1,𝔽5)—48no—
336C4×C5—22no—
337C4×C5—22no—
338Dic5—22no—
339Dic5—22no—
340C24—51no—
341C24—52no—
342C24—52no—
343C24—51no—
344C24—104no—
345C24—202no—
346C24—202no—
347C24C2×C3×AGL(1,𝔽5)11yescharacteristic
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—52no—
351C22×C4—52no—
352C22×C4—52no—
353C22×C4—52no—
354C22×C4—58no—
355C22×C4—202no—
356C22×C4—202no—
357C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
358C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
359C4⋊C4—204no—
360C22⋊C4—54no—
361C22⋊C4—54no—
362C22⋊C4—54no—
363C22⋊C4—108no—
364C22⋊C4—204no—
365C22⋊C4—204no—
366C22⋊C4C2×C3×AGL(1,𝔽5)14yes—
367C42—204no—
368C3×C5#231211yescharacteristic, complemented, Hall
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—44no—
375C22×C3—52no—
376C22×C3—52no—
377C22×C3—52no—
378C22×C3—52no—
379C22×C3—108no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3—102no—
383C22×C3—102no—
384C22×C3—102no—
385C22×C3—204no—
386C22×C3—208no—
387C22×C3—208no—
388C22×C3#413012yes—
389C22×C3C23×AGL(1,𝔽5)11yescharacteristic
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—102no—
393C3×C4—102no—
394C3×C4—208no—
395C2×C5—22no—
396C2×C5—42no—
397C2×C5C2×C3×#11711yescharacteristic
398C2×C5C3×#51811yescharacteristic
399C2×C5C3×#51811yescharacteristic
400D5—22no—
401D5—42no—
402D5C3×#53814yes—
403C23—22no—
404C23—41no—
405C23—41no—
406C23—51no—
407C23—54no—
408C23—51no—
409C23—52no—
410C23—102no—
411C23—102no—
412C23—104no—
413C23—104no—
414C23—104no—
415C23—104no—
416C23—102no—
417C23—201no—
418C23—201no—
419C23—202no—
420C23—202no—
421C23—2016no—
422C23C22×C3×AGL(1,𝔽5)12yes—
423C23C22×C3×AGL(1,𝔽5)11yescharacteristic
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—52no—
427C2×C4—52no—
428C2×C4—104no—
429C2×C4—104no—
430C2×C4—108no—
431C2×C4—108no—
432C2×C4—108no—
433C2×C4—202no—
434C2×C4—202no—
435C2×C4—201no—
436C2×C4—201no—
437C2×C4C22×C3×AGL(1,𝔽5)12yes—
438C2×C4C22×C3×AGL(1,𝔽5)12yes—
439C2×C3—22no—
440C2×C3—42no—
441C2×C3—54no—
442C2×C3—102no—
443C2×C3—202no—
444C2×C3C2×#413011yescharacteristic
445C2×C3#6625811yescharacteristic
446C2×C3#6625811yescharacteristic
447C5C3×#231211yescharacteristic, complemented, Sylow
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—44no—
454C22—52no—
455C22—52no—
456C22—52no—
457C22—52no—
458C22—108no—
459C22—101no—
460C22—101no—
461C22—102no—
462C22—102no—
463C22—102no—
464C22—204no—
465C22—208no—
466C22—208no—
467C22C3×#413012yes—
468C22C23×C3×AGL(1,𝔽5)11yescharacteristic
469C4—22no—
470C4—22no—
471C4—102no—
472C4—102no—
473C4—208no—
474C3#1061707011yescharacteristic, direct product factor, Sylow
475C2—22no—
476C2—42no—
477C2—54no—
478C2—102no—
479C2—202no—
480C2C2×C3×#413011yescharacteristic
481C2C3×#6625811yescharacteristic
482C2C3×#6625811yescharacteristic
subgroupG/H
centerC22×C3C23×AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C22⋊C4C4
socleC22×C3×C5C23×C4
prime radicalC24×C3×D5C22
cosocleC22×D5C24×C3
hypercenterC2×C3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C3×#2312
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#65765C2semidirect
4. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
5. C3×#65700C2semidirect
6. #10617070C3direct (N × H)
13. C2×C3×C4⋊AGL(1,𝔽5)C22semidirect
14. C2×C3×C4×AGL(1,𝔽5)C22semidirect
15. C2×C3×C5×C22⋊C4C4semidirect
16. C2×C3×C22⋊Dic5C4semidirect
17. C2×C3×#685C4semidirect
18. C2×C3×#685C4semidirect
19. C3×D5×C22⋊C4C4semidirect
21. C3×(C2×AGL(1,𝔽5))⋊C4C22semidirect
22. C3×(C2×AGL(1,𝔽5))⋊C4C22semidirect
26. #65765C2×C3semidirect
27. C4×C22⋊AGL(1,𝔽5)C2×C3semidirect
28. #65700C2×C3semidirect
68. C2×C4⋊AGL(1,𝔽5)C22×C3semidirect
69. C2×C4×AGL(1,𝔽5)C22×C3semidirect
70. C2×C5×C22⋊C4C3×C4semidirect
71. C2×C22⋊Dic5C3×C4semidirect
72. C2×#685C3×C4semidirect
73. C2×#685C3×C4semidirect
74. D5×C22⋊C4C3×C4semidirect
76. (C2×AGL(1,𝔽5))⋊C4C22×C3semidirect
77. (C2×AGL(1,𝔽5))⋊C4C22×C3semidirect
127. C2×C3×C22⋊C4AGL(1,𝔽5)semidirect
266. C2×C22⋊C4C3×AGL(1,𝔽5)semidirect
368. C3×C5#2312semidirect
447. C5C3×#2312semidirect
474. C3#10617070direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5×C22⋊C42
C3×#658002
C3×#657652
C3×C4×C22⋊AGL(1,𝔽5)2
C3×#657002
#106170703
C3×#23125