Finite Group Databasebeta

← C3×#10604524 #14300646

Subgroups — C3×#10604524 #14300646

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4130C211yesmaximal, characteristic
2C3×D5×C22⋊Q8C211yesmaximal, characteristic
3C2×C3×#4009C212yesmaximal, complemented
4C3×D5⋊(C22⋊C8)C211yesmaximal, characteristic
5C3×#64874C212yesmaximal, complemented
6#10604524C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C3×#4130—24no—
11C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
12C2×C3×C4×AGL(1,𝔽5)—21no—
13C2×C3×D5⋊C8C2212yes—
14C3×C5×C22⋊Q8C411yescharacteristic, complemented
15C3×#4092C411yescharacteristic, complemented
16C3×#4070C411yescharacteristic, complemented
17C3×#4043C411yescharacteristic, complemented
18C3×#4042—21no—
19C3×D5×C4⋊C4—21no—
20C3×D5×C4⋊C4C2211yescharacteristic
21C3×D5×C22⋊C4—21no—
22C3×#4024—21no—
23C3×C22⋊#170—21no—
24C3×#4009—24no—
25C3×#4003—21no—
26C3×#1873—51nomaximal, Hall
27C2×#4130C2×C311yescharacteristic
28D5×C22⋊Q8C2×C311yescharacteristic
29C2×#4009C2×C312yescomplemented
30D5⋊(C22⋊C8)C2×C311yescharacteristic
31#64874C2×C312yescomplemented
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—22no—
34C2×C3×C5×Q8C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×C3×Dic5D411yescharacteristic
37C3×Q8×D5—22no—
38C3×Q8×D5—22no—
39C2×C3×C4×D5—22no—
40C2×C3×C4×D5—22no—
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5C2×C411yescharacteristic
44C2×C3×C4×D5D411yescharacteristic
45C2×C3×C4×D5D411yescharacteristic
46C2×C3×C4×D5C2311yescharacteristic
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C411yescharacteristic
49C3×C22⋊AGL(1,𝔽5)—24no—
50C2×C3×#170—22no—
51C3×C4⋊AGL(1,𝔽5)—22no—
52C3×C4⋊AGL(1,𝔽5)D414yes—
53C3×C4×AGL(1,𝔽5)—24no—
54C3×D5⋊C8—22no—
55C3×D5⋊C8—22no—
56C3×C5×C4⋊C4—21no—
57C3×C5×C4⋊C4C2×C411yescharacteristic
58C3×C5×C22⋊C4—21no—
59C3×C22⋊Dic5—21no—
60C3×#685—21no—
61C3×#685—21no—
62C3×C4⋊Dic5—21no—
63C3×C4⋊Dic5C2×C411yescharacteristic
64C3×#683—21no—
65C3×#683—21no—
66C3×#683—21no—
67C2×C3×C22⋊Q8—51no—
68C2×C3×#117—51no—
69C2×C3×#103—52no—
70C2×C3×C22⋊C8—51no—
71C3×#340—52no—
72C2×Q8×D5C22×C311yescharacteristic
73C22×C4×D5C22×C311yescharacteristic
74C2×C22⋊AGL(1,𝔽5)—21no—
75#4130—24no—
76C2×C4⋊AGL(1,𝔽5)C22×C312yes—
77C2×C4×AGL(1,𝔽5)—21no—
78C2×D5⋊C8C22×C312yes—
79C5×C22⋊Q8C3×C411yescharacteristic, complemented
80#4092C3×C411yescharacteristic, complemented
81#4070C3×C411yescharacteristic, complemented
82#4043C3×C411yescharacteristic, complemented
83#4042—21no—
84D5×C4⋊C4—21no—
85D5×C4⋊C4C22×C311yescharacteristic
86D5×C22⋊C4—21no—
87#4024—21no—
88C22⋊#170—21no—
89#4009—24no—
90#4003—21no—
91#1873—51noSylow
92C23×C3×C5C22⋊C411yescharacteristic
93C22×C3×D5—21no—
94C22×C3×D5—21no—
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5C22⋊C411yescharacteristic
98C22×C3×D5C2×D411yescharacteristic
99C2×C3×AGL(1,𝔽5)—24no—
100C2×C3×AGL(1,𝔽5)—42no—
101C2×C3×AGL(1,𝔽5)—42no—
102C3×C5×Q8—22no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5C22⋊C411yescharacteristic
107C2×C3×C4×C5C22×C411yescharacteristic
108C2×C3×Dic5—22no—
109C2×C3×Dic5—21no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5C22⋊C411yescharacteristic
112C2×C3×Dic5C2×D411yescharacteristic
113C3×C4×D5—21no—
114C3×C4×D5—21no—
115C3×C4×D5—22no—
116C3×C4×D5—22no—
117C3×C4×D5—41no—
118C3×C4×D5—41no—
119C3×C4×D5—42no—
120C3×C4×D5C2×D412yes—
121C3×C4×D5#4112yes—
122C3×Dic10—22no—
123C3×Dic10—42no—
124C3×#170—42no—
125C22×C3×Q8—51no—
126C23×C3×C4—51no—
127C22×C3×C8—52no—
128C3×C22⋊Q8—51no—
129C3×C22⋊Q8—51no—
130C3×C22⋊Q8—51no—
131C3×C22⋊Q8—101no—
132C3×C22⋊Q8—101no—
133C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
134C3×#117—104no—
135C2×C3×C4⋊C4—52no—
136C2×C3×C4⋊C4—51no—
137C2×C3×C4⋊C4—101no—
138C2×C3×C22⋊C4—101no—
139C2×C3×C22⋊C4—101no—
140C2×C3×C42—101no—
141C3×#103—104no—
142C3×C22⋊C8—101no—
143C3×C22⋊C8—101no—
144C23×D5C3×D411yescharacteristic
145C22×AGL(1,𝔽5)—22no—
146C2×C5×Q8C2×C3×C411yescharacteristic
147C22×C4×C5C2×C3×C411yescharacteristic
148C22×Dic5C3×D411yescharacteristic
149Q8×D5—22no—
150Q8×D5—22no—
151C2×C4×D5—22no—
152C2×C4×D5—22no—
153C2×C4×D5—21no—
154C2×C4×D5—21no—
155C2×C4×D5C2×C3×C411yescharacteristic
156C2×C4×D5C3×D411yescharacteristic
157C2×C4×D5C3×D411yescharacteristic
158C2×C4×D5C23×C311yescharacteristic
159C2×Dic10—21no—
160C2×Dic10C2×C3×C411yescharacteristic
161C22⋊AGL(1,𝔽5)—24no—
162C2×#170—22no—
163C4⋊AGL(1,𝔽5)—22no—
164C4⋊AGL(1,𝔽5)C3×D414yes—
165C4×AGL(1,𝔽5)—24no—
166D5⋊C8—22no—
167D5⋊C8—22no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4C2×C3×C411yescharacteristic
170C5×C22⋊C4—21no—
171C22⋊Dic5—21no—
172#685—21no—
173#685—21no—
174C4⋊Dic5—21no—
175C4⋊Dic5C2×C3×C411yescharacteristic
176#683—21no—
177#683—21no—
178#683—21no—
179C2×C22⋊Q8—51no—
180C2×#117—51no—
181C2×#103—52no—
182C2×C22⋊C8—51no—
183#340—52no—
184C22×C3×C5—21no—
185C22×C3×C5—21no—
186C22×C3×C5—21no—
187C22×C3×C5C2×C22⋊C411yescharacteristic
188C2×C3×D5—21no—
189C2×C3×D5—21no—
190C2×C3×D5—21no—
191C2×C3×D5—42no—
192C2×C3×D5—42no—
193C2×C3×D5#12011yescharacteristic
194C2×C3×D5#12311yescharacteristic
195C2×C3×D5#13512yes—
196C2×C3×D5#13712yes—
197C3×AGL(1,𝔽5)—44no—
198C3×C4×C5—21no—
199C3×C4×C5—41no—
200C3×C4×C5—41no—
201C3×C4×C5C4×D412yes—
202C3×Dic5—21no—
203C3×Dic5—41no—
204C3×Dic5—41no—
205C3×Dic5#12112yes—
206C24×C3—51no—
207C2×C3×Q8—51no—
208C2×C3×Q8—101no—
209C2×C3×Q8—102no—
210C2×C3×Q8—102no—
211C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
212C22×C3×C4—51no—
213C22×C3×C4—51no—
214C22×C3×C4—51no—
215C22×C3×C4—51no—
216C22×C3×C4—51no—
217C22×C3×C4—102no—
218C22×C3×C4—102no—
219C22×C3×C4—101no—
220C22×C3×C4—101no—
221C22×C3×C4—102no—
222C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
223C2×C3×C8—102no—
224C2×C3×C8—102no—
225C2×C3×C8—102no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—54no—
228C3×C4⋊C4—51no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4—101no—
232C3×C4⋊C4—102no—
233C3×C4⋊C4—101no—
234C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
235C3×C22⋊C4—21no—
236C3×C22⋊C4—101no—
237C3×C22⋊C4—101no—
238C3×C22⋊C4—104no—
239C3×C22⋊C4—101no—
240C3×C42—104no—
241C23×C5C3×C22⋊C411yescharacteristic
242C22×D5—21no—
243C22×D5—21no—
244C22×D5—22no—
245C22×D5—22no—
246C22×D5C3×C22⋊C411yescharacteristic
247C22×D5C2×C3×D411yescharacteristic
248C2×AGL(1,𝔽5)—24no—
249C2×AGL(1,𝔽5)—42no—
250C2×AGL(1,𝔽5)—42no—
251C5×Q8—22no—
252C2×C4×C5—22no—
253C2×C4×C5—21no—
254C2×C4×C5—21no—
255C2×C4×C5C3×C22⋊C411yescharacteristic
256C2×C4×C5C22×C3×C411yescharacteristic
257C2×Dic5—22no—
258C2×Dic5—21no—
259C2×Dic5—21no—
260C2×Dic5C3×C22⋊C411yescharacteristic
261C2×Dic5C2×C3×D411yescharacteristic
262C4×D5—21no—
263C4×D5—21no—
264C4×D5—22no—
265C4×D5—22no—
266C4×D5—41no—
267C4×D5—41no—
268C4×D5—42no—
269C4×D5C2×C3×D412yes—
270C4×D5C3×#4112yes—
271Dic10—22no—
272Dic10—42no—
273#170—42no—
274C22×Q8—51no—
275C23×C4—51no—
276C22×C8—52no—
277C22⋊Q8—51no—
278C22⋊Q8—51no—
279C22⋊Q8—51no—
280C22⋊Q8—101no—
281C22⋊Q8—101no—
282C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
283#117—104no—
284C2×C4⋊C4—52no—
285C2×C4⋊C4—51no—
286C2×C4⋊C4—101no—
287C2×C22⋊C4—101no—
288C2×C22⋊C4—101no—
289C2×C42—101no—
290#103—104no—
291C22⋊C8—101no—
292C22⋊C8—101no—
293C2×C3×C5—41no—
294C2×C3×C5#38611yescharacteristic
295C2×C3×C5#41611yescharacteristic
296C2×C3×C5#41911yescharacteristic
297C3×D5—41no—
298C3×D5#45212yes—
299C3×D5#48512yes—
300C23×C3—51no—
301C23×C3—51no—
302C23×C3—101no—
303C23×C3—101no—
304C23×C3—102no—
305C23×C3—102no—
306C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
307C3×Q8—22no—
308C3×Q8—102no—
309C3×Q8—202no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—51no—
314C2×C3×C4—52no—
315C2×C3×C4—52no—
316C2×C3×C4—51no—
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—101no—
324C2×C3×C4—101no—
325C2×C3×C4—202no—
326C2×C3×C4—201no—
327C2×C3×C4—201no—
328C2×C3×C4—202no—
329C2×C3×C4—202no—
330C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
331C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
332C3×C8—202no—
333C22×C5—21no—
334C22×C5—21no—
335C22×C5—21no—
336C22×C5C2×C3×C22⋊C411yescharacteristic
337C2×D5—21no—
338C2×D5—21no—
339C2×D5—21no—
340C2×D5—42no—
341C2×D5—42no—
342C2×D5C3×#12011yescharacteristic
343C2×D5C3×#12311yescharacteristic
344C2×D5C3×#13512yes—
345C2×D5C3×#13712yes—
346AGL(1,𝔽5)—44no—
347C4×C5—21no—
348C4×C5—41no—
349C4×C5—41no—
350C4×C5C3×C4×D412yes—
351Dic5—21no—
352Dic5—41no—
353Dic5—41no—
354Dic5C3×#12112yes—
355C24—51no—
356C2×Q8—51no—
357C2×Q8—101no—
358C2×Q8—102no—
359C2×Q8—102no—
360C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
361C22×C4—51no—
362C22×C4—51no—
363C22×C4—51no—
364C22×C4—51no—
365C22×C4—51no—
366C22×C4—102no—
367C22×C4—102no—
368C22×C4—101no—
369C22×C4—101no—
370C22×C4—102no—
371C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
372C2×C8—102no—
373C2×C8—102no—
374C2×C8—102no—
375C4⋊C4—21no—
376C4⋊C4—54no—
377C4⋊C4—51no—
378C4⋊C4—101no—
379C4⋊C4—101no—
380C4⋊C4—101no—
381C4⋊C4—102no—
382C4⋊C4—101no—
383C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
384C22⋊C4—21no—
385C22⋊C4—101no—
386C22⋊C4—101no—
387C22⋊C4—104no—
388C22⋊C4—101no—
389C42—104no—
390C3×C5#187311yescharacteristic, complemented, Hall
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—52no—
395C22×C3—51no—
396C22×C3—51no—
397C22×C3—52no—
398C22×C3—101no—
399C22×C3—101no—
400C22×C3—101no—
401C22×C3—202no—
402C22×C3—202no—
403C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
404C3×C4—21no—
405C3×C4—41no—
406C3×C4—41no—
407C3×C4—52no—
408C3×C4—101no—
409C3×C4—204no—
410C3×C4—201no—
411C3×C4—201no—
412C3×C4D4×AGL(1,𝔽5)12yes—
413C2×C5—41no—
414C2×C5C3×#38611yescharacteristic
415C2×C5C3×#41611yescharacteristic
416C2×C5C3×#41911yescharacteristic
417D5—41no—
418D5C3×#45212yes—
419D5C3×#48512yes—
420C23—51no—
421C23—51no—
422C23—101no—
423C23—101no—
424C23—102no—
425C23—102no—
426C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
427Q8—22no—
428Q8—102no—
429Q8—202no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—51no—
434C2×C4—52no—
435C2×C4—52no—
436C2×C4—51no—
437C2×C4—102no—
438C2×C4—101no—
439C2×C4—101no—
440C2×C4—102no—
441C2×C4—102no—
442C2×C4—104no—
443C2×C4—101no—
444C2×C4—101no—
445C2×C4—201no—
446C2×C4—201no—
447C2×C4—202no—
448C2×C4—202no—
449C2×C4—202no—
450C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
451C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
452C8—202no—
453C2×C3—41no—
454C2×C3—52no—
455C2×C3—52no—
456C2×C3—201no—
457C2×C3#6577911yescharacteristic
458C2×C3#6578211yescharacteristic
459C2×C3#6579411yescharacteristic
460C5C3×#187311yescharacteristic, complemented, Sylow
461C22—21no—
462C22—21no—
463C22—21no—
464C22—51no—
465C22—51no—
466C22—52no—
467C22—52no—
468C22—101no—
469C22—101no—
470C22—101no—
471C22—202no—
472C22—202no—
473C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
474C4—21no—
475C4—41no—
476C4—41no—
477C4—52no—
478C4—101no—
479C4—204no—
480C4—201no—
481C4—201no—
482C4C3×D4×AGL(1,𝔽5)12yes—
483C3#1060452411yescharacteristic, direct product factor, Sylow
484C2—41no—
485C2—52no—
486C2—52no—
487C2—201no—
488C2C3×#6577911yescharacteristic
489C2C3×#6578211yescharacteristic
490C2C3×#6579411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C22⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C4×D5C23×C3
hypercenterC3×C22⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1873
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×C3×#4009C2semidirect
5. C3×#64874C2semidirect
6. #10604524C3direct (N × H)
14. C3×C5×C22⋊Q8C4semidirect
15. C3×#4092C4semidirect
16. C3×#4070C4semidirect
17. C3×#4043C4semidirect
29. C2×#4009C2×C3semidirect
31. #64874C2×C3semidirect
79. C5×C22⋊Q8C3×C4semidirect
80. #4092C3×C4semidirect
81. #4070C3×C4semidirect
82. #4043C3×C4semidirect
133. C3×C22⋊Q8AGL(1,𝔽5)semidirect
282. C22⋊Q8C3×AGL(1,𝔽5)semidirect
390. C3×C5#1873semidirect
460. C5C3×#1873semidirect
483. C3#10604524direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41302
C3×D5×C22⋊Q82
C2×C3×#40092
C3×D5⋊(C22⋊C8)2
C3×#648742
#106045243
C3×#18735