Finite Group Databasebeta

← C3×#10604527 #14300540

Subgroups — C3×#10604527 #14300540

Order 1920.

752 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#700C211yesmaximal, characteristic, complemented
2C3×#65961C211yesmaximal, characteristic
3C2×C3×#4010C212yesmaximal, complemented
4C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5C3×#64875C212yesmaximal, complemented
6#10604527C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C22×C3×#170—21no—
11C2×C3×C4×AGL(1,𝔽5)C2212yes—
12C2×C3×#700—21no—
13C2×C3×#700—24no—
14C2×C3×#700C2212yes—
15C3×C5×C22⋊Q8C411yescharacteristic, complemented
16C3×#4092C411yescharacteristic, complemented
17C3×#4076C411yescharacteristic, complemented
18C3×#4041C411yescharacteristic, complemented
19C3×#4040—21no—
20C3×#4039—21no—
21C3×#4038C2211yescharacteristic
22C3×#4028—21no—
23C3×#4027—21no—
24C3×#4010—24no—
25C3×(C2×AGL(1,𝔽5))⋊C4—21no—
26C3×#1871—51nomaximal, Hall
27C22×#700C2×C311yescharacteristic, complemented
28#65961C2×C311yescharacteristic
29C2×#4010C2×C312yescomplemented
30C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
31#64875C2×C312yescomplemented
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—22no—
34C2×C3×C5×Q8C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic
36C2×C3×C22⋊D5—21no—
37C22×C3×Dic5D411yescharacteristic
38C3×#713—22no—
39C3×#713—22no—
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C2×C3×D20C2×C411yescharacteristic
43C2×C3×C4×D5—22no—
44C2×C3×C4×D5—22no—
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5C2×C411yescharacteristic
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×C4×D5D411yescharacteristic
49C2×C3×C4×D5C2311yescharacteristic
50C3×C22⋊AGL(1,𝔽5)—24no—
51C2×C3×#170—24no—
52C2×C3×#170—22no—
53C3×C4×AGL(1,𝔽5)—22no—
54C3×C4×AGL(1,𝔽5)—22no—
55C3×#700—24no—
56C3×#700—22no—
57C3×#700D414yescomplemented
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4C2×C411yescharacteristic
60C3×C5×C22⋊C4—21no—
61C3×#685—21no—
62C3×#685—21no—
63C3×#685—21no—
64C3×#685—21no—
65C3×C4⋊Dic5C2×C411yescharacteristic
66C3×#683—21no—
67C3×C4×Dic5—21no—
68C22×C3×M16—51no—
69C3×#525—51no—
70C2×C3×#104—52no—
71C3×C4×C22⋊C4—51no—
72C3×#339—52no—
73C2×#713C22×C311yescharacteristic
74C22×C4×D5C22×C311yescharacteristic
75C2×C22⋊AGL(1,𝔽5)—21no—
76C22×#170—21no—
77C2×C4×AGL(1,𝔽5)C22×C312yes—
78C2×#700—21no—
79C2×#700—24no—
80C2×#700C22×C312yes—
81C5×C22⋊Q8C3×C411yescharacteristic, complemented
82#4092C3×C411yescharacteristic, complemented
83#4076C3×C411yescharacteristic, complemented
84#4041C3×C411yescharacteristic, complemented
85#4040—21no—
86#4039—21no—
87#4038C22×C311yescharacteristic
88#4028—21no—
89#4027—21no—
90#4010—24no—
91(C2×AGL(1,𝔽5))⋊C4—21no—
92#1871—51noSylow
93C23×C3×C5C22⋊C411yescharacteristic
94C22×C3×D5—22no—
95C22×C3×D5—22no—
96C22×C3×D5—21no—
97C22×C3×D5—41no—
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×C3×AGL(1,𝔽5)—24no—
101C2×C3×AGL(1,𝔽5)—42no—
102C2×C3×AGL(1,𝔽5)—42no—
103C3×C5×Q8—22no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5C22⋊C411yescharacteristic
108C2×C3×C4×C5C22×C411yescharacteristic
109C3×C22⋊D5—42no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5C22⋊C411yescharacteristic
114C2×C3×Dic5C2×D411yescharacteristic
115C3×D20—22no—
116C3×D20—42no—
117C3×D20—42no—
118C3×C4×D5—22no—
119C3×C4×D5—22no—
120C3×C4×D5—21no—
121C3×C4×D5—21no—
122C3×C4×D5—41no—
123C3×C4×D5—41no—
124C3×C4×D5C2×D412yes—
125C3×C4×D5#4112yes—
126C3×#170—24no—
127C2×C3×#41—51no—
128C23×C3×C4—51no—
129C2×C3×M16—52no—
130C2×C3×M16—104no—
131C2×C3×M16—101no—
132C22×C3×C8—101no—
133C3×#123—101no—
134C3×C22⋊Q8—51no—
135C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
136C3×#121—51no—
137C3×#121—51no—
138C3×#120—101no—
139C3×C4×D4—101no—
140C3×C4×D4—101no—
141C3×#117—51no—
142C2×C3×C22⋊C4—101no—
143C2×C3×C42—52no—
144C3×#104—104no—
145C3×#95—101no—
146C23×D5C3×D411yescharacteristic
147C22×AGL(1,𝔽5)—22no—
148C2×C5×Q8C2×C3×C411yescharacteristic
149C22×C4×C5C2×C3×C411yescharacteristic
150C2×C22⋊D5—21no—
151C22×Dic5C3×D411yescharacteristic
152#713—22no—
153#713—22no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20C2×C3×C411yescharacteristic
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5—21no—
160C2×C4×D5C2×C3×C411yescharacteristic
161C2×C4×D5C3×D411yescharacteristic
162C2×C4×D5C3×D411yescharacteristic
163C2×C4×D5C23×C311yescharacteristic
164C22⋊AGL(1,𝔽5)—24no—
165C2×#170—24no—
166C2×#170—22no—
167C4×AGL(1,𝔽5)—22no—
168C4×AGL(1,𝔽5)—22no—
169#700—24no—
170#700—22no—
171#700C3×D414yescomplemented
172C5×C4⋊C4—21no—
173C5×C4⋊C4C2×C3×C411yescharacteristic
174C5×C22⋊C4—21no—
175#685—21no—
176#685—21no—
177#685—21no—
178#685—21no—
179C4⋊Dic5C2×C3×C411yescharacteristic
180#683—21no—
181C4×Dic5—21no—
182C22×M16—51no—
183#525—51no—
184C2×#104—52no—
185C4×C22⋊C4—51no—
186#339—52no—
187C22×C3×C5—21no—
188C22×C3×C5—21no—
189C22×C3×C5#10412yescomplemented
190C22×C3×C5C2×C22⋊C411yescharacteristic
191C2×C3×D5—21no—
192C2×C3×D5—21no—
193C2×C3×D5—21no—
194C2×C3×D5—21no—
195C2×C3×D5—21no—
196C2×C3×D5—44no—
197C2×C3×D5—41no—
198C2×C3×D5—41no—
199C2×C3×D5—41no—
200C2×C3×D5#12112yes—
201C3×AGL(1,𝔽5)—44no—
202C3×C4×C5—21no—
203C3×C4×C5—41no—
204C3×C4×C5—41no—
205C3×C4×C5C4×D412yes—
206C3×Dic5—21no—
207C3×Dic5—41no—
208C3×Dic5#12011yescharacteristic
209C3×Dic5#12311yescharacteristic
210C24×C3—51no—
211C3×#41—102no—
212C3×#41—102no—
213C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
214C2×C3×D4—51no—
215C2×C3×D4—101no—
216C2×C3×D4—101no—
217C2×C3×D4—101no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4—51no—
223C22×C3×C4—102no—
224C22×C3×C4—102no—
225C22×C3×C4—101no—
226C22×C3×C4—102no—
227C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
228C3×M16—54no—
229C3×M16—102no—
230C3×M16—104no—
231C2×C3×C8—102no—
232C2×C3×C8—104no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—51no—
235C3×C4⋊C4—101no—
236C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
237C3×C22⋊C4—21no—
238C3×C22⋊C4—101no—
239C3×C22⋊C4—101no—
240C3×C22⋊C4—101no—
241C3×C22⋊C4—104no—
242C3×C22⋊C4—101no—
243C3×C42—101no—
244C3×C42—102no—
245C3×C42—102no—
246C23×C5C3×C22⋊C411yescharacteristic
247C22×D5—22no—
248C22×D5—22no—
249C22×D5—21no—
250C22×D5—41no—
251C22×D5C3×C22⋊C411yescharacteristic
252C22×D5C2×C3×D411yescharacteristic
253C2×AGL(1,𝔽5)—24no—
254C2×AGL(1,𝔽5)—42no—
255C2×AGL(1,𝔽5)—42no—
256C5×Q8—22no—
257C2×C4×C5—22no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5C3×C22⋊C411yescharacteristic
261C2×C4×C5C22×C3×C411yescharacteristic
262C22⋊D5—42no—
263C2×Dic5—21no—
264C2×Dic5—21no—
265C2×Dic5—21no—
266C2×Dic5C3×C22⋊C411yescharacteristic
267C2×Dic5C2×C3×D411yescharacteristic
268D20—22no—
269D20—42no—
270D20—42no—
271C4×D5—22no—
272C4×D5—22no—
273C4×D5—21no—
274C4×D5—21no—
275C4×D5—41no—
276C4×D5—41no—
277C4×D5C2×C3×D412yes—
278C4×D5C3×#4112yes—
279#170—24no—
280C2×#41—51no—
281C23×C4—51no—
282C2×M16—52no—
283C2×M16—104no—
284C2×M16—101no—
285C22×C8—101no—
286#123—101no—
287C22⋊Q8—51no—
288C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
289#121—51no—
290#121—51no—
291#120—101no—
292C4×D4—101no—
293C4×D4—101no—
294#117—51no—
295C2×C22⋊C4—101no—
296C2×C42—52no—
297#104—104no—
298#95—101no—
299C2×C3×C5—22no—
300C2×C3×C5#38611yescharacteristic
301C2×C3×C5C2×#10411yescharacteristic
302C2×C3×C5#42111yescharacteristic
303C3×D5—22no—
304C3×D5—41no—
305C3×D5—81no—
306C23×C3—51no—
307C23×C3—51no—
308C23×C3—102no—
309C23×C3—102no—
310C23×C3—101no—
311C23×C3—201no—
312C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
313C3×Q8—22no—
314C3×D4—102no—
315C3×D4—202no—
316C3×D4—202no—
317C3×D4—202no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—51no—
322C2×C3×C4—52no—
323C2×C3×C4—52no—
324C2×C3×C4—51no—
325C2×C3×C4—101no—
326C2×C3×C4—101no—
327C2×C3×C4—102no—
328C2×C3×C4—101no—
329C2×C3×C4—101no—
330C2×C3×C4—102no—
331C2×C3×C4—104no—
332C2×C3×C4—101no—
333C2×C3×C4—201no—
334C2×C3×C4—201no—
335C2×C3×C4—202no—
336C2×C3×C4—202no—
337C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
338C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
339C3×C8—104no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5C3×#10412yescomplemented
343C22×C5C2×C3×C22⋊C411yescharacteristic
344C2×D5—21no—
345C2×D5—21no—
346C2×D5—21no—
347C2×D5—21no—
348C2×D5—21no—
349C2×D5—44no—
350C2×D5—41no—
351C2×D5—41no—
352C2×D5—41no—
353C2×D5C3×#12112yes—
354AGL(1,𝔽5)—44no—
355C4×C5—21no—
356C4×C5—41no—
357C4×C5—41no—
358C4×C5C3×C4×D412yes—
359Dic5—21no—
360Dic5—41no—
361Dic5C3×#12011yescharacteristic
362Dic5C3×#12311yescharacteristic
363C24—51no—
364#41—102no—
365#41—102no—
366C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
367C2×D4—51no—
368C2×D4—101no—
369C2×D4—101no—
370C2×D4—101no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—102no—
377C22×C4—102no—
378C22×C4—101no—
379C22×C4—102no—
380C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
381M16—54no—
382M16—102no—
383M16—104no—
384C2×C8—102no—
385C2×C8—104no—
386C4⋊C4—21no—
387C4⋊C4—51no—
388C4⋊C4—101no—
389C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
390C22⋊C4—21no—
391C22⋊C4—101no—
392C22⋊C4—101no—
393C22⋊C4—101no—
394C22⋊C4—104no—
395C22⋊C4—101no—
396C42—101no—
397C42—102no—
398C42—102no—
399C3×C5#187111yescharacteristic, complemented, Hall
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—52no—
403C22×C3—101no—
404C22×C3—101no—
405C22×C3—101no—
406C22×C3—101no—
407C22×C3—101no—
408C22×C3—204no—
409C22×C3—201no—
410C22×C3—201no—
411C22×C3—201no—
412C22×C3#401012yescomplemented
413C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—51no—
418C3×C4—51no—
419C3×C4—101no—
420C3×C4—204no—
421C3×C4—201no—
422C3×C4D4×AGL(1,𝔽5)12yes—
423C2×C5—22no—
424C2×C5C3×#38611yescharacteristic
425C2×C5C2×C3×#10411yescharacteristic
426C2×C5C3×#42111yescharacteristic
427D5—22no—
428D5—41no—
429D5—81no—
430C23—51no—
431C23—51no—
432C23—102no—
433C23—102no—
434C23—101no—
435C23—201no—
436C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
437Q8—22no—
438D4—102no—
439D4—202no—
440D4—202no—
441D4—202no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—51no—
446C2×C4—52no—
447C2×C4—52no—
448C2×C4—51no—
449C2×C4—101no—
450C2×C4—101no—
451C2×C4—102no—
452C2×C4—101no—
453C2×C4—101no—
454C2×C4—102no—
455C2×C4—104no—
456C2×C4—101no—
457C2×C4—201no—
458C2×C4—201no—
459C2×C4—202no—
460C2×C4—202no—
461C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
462C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
463C8—104no—
464C2×C3—22no—
465C2×C3—102no—
466C2×C3—201no—
467C2×C3—401no—
468C2×C3#6577911yescharacteristic
469C2×C3C2×#401011yescharacteristic
470C2×C3#6579511yescharacteristic
471C5C3×#187111yescharacteristic, complemented, Sylow
472C22—21no—
473C22—21no—
474C22—52no—
475C22—101no—
476C22—101no—
477C22—101no—
478C22—101no—
479C22—101no—
480C22—204no—
481C22—201no—
482C22—201no—
483C22—201no—
484C22C3×#401012yescomplemented
485C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
486C4—21no—
487C4—41no—
488C4—41no—
489C4—51no—
490C4—51no—
491C4—101no—
492C4—204no—
493C4—201no—
494C4C3×D4×AGL(1,𝔽5)12yes—
495C3#1060452711yescharacteristic, direct product factor, Sylow
496C2—22no—
497C2—102no—
498C2—201no—
499C2—401no—
500C2C3×#6577911yescharacteristic
501C2C2×C3×#401011yescharacteristic
502C2C3×#6579511yescharacteristic
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 radicalC3×#65961C2
cosocleC2×C4×D5C23×C3
hypercenterC3×C22⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1871
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
1. C22×C3×#700C2semidirect
3. C2×C3×#4010C2semidirect
4. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
5. C3×#64875C2semidirect
6. #10604527C3direct (N × H)
15. C3×C5×C22⋊Q8C4semidirect
16. C3×#4092C4semidirect
17. C3×#4076C4semidirect
18. C3×#4041C4semidirect
27. C22×#700C2×C3semidirect
29. C2×#4010C2×C3semidirect
30. C4×C22⋊AGL(1,𝔽5)C2×C3semidirect
31. #64875C2×C3semidirect
57. C3×#700D4semidirect
81. C5×C22⋊Q8C3×C4semidirect
82. #4092C3×C4semidirect
83. #4076C3×C4semidirect
84. #4041C3×C4semidirect
135. C3×C22⋊Q8AGL(1,𝔽5)semidirect
171. #700C3×D4semidirect
189. C22×C3×C5#104semidirect
288. C22⋊Q8C3×AGL(1,𝔽5)semidirect
342. C22×C5C3×#104semidirect
399. C3×C5#1871semidirect
412. C22×C3#4010semidirect
471. C5C3×#1871semidirect
484. C22C3×#4010semidirect
495. C3#10604527direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7002
C3×#659612
C2×C3×#40102
C3×C4×C22⋊AGL(1,𝔽5)2
C3×#648752
#106045273
C3×#18715