Finite Group Databasebeta

← C3×#10604460 #14313270

Subgroups — C3×#10604460 #14313270

Order 1920.

532 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65926C211yesmaximal, characteristic
2C3×#65777C211yesmaximal, characteristic, complemented
3C3×#65771C211yesmaximal, characteristic, complemented
4C3×#64934C212yesmaximal, complemented
5C3×#64864C212yesmaximal, complemented
6#10604460C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C22⋊D5C411yescharacteristic, complemented
8C2×C3×#711C2211yescharacteristic
9C2×C3×#704—21no—
10C3×#4130—21no—
11C3×C5×#120C411yescharacteristic, complemented
12C3×#4099C2211yescharacteristic
13C3×#4086—21no—
14C3×#4085—21no—
15C3×#4083C411yescharacteristic, complemented
16C3×#4031—21no—
17C3×#4030—21no—
18C3×#4028C2211yescharacteristic
19C3×#4023C411yescharacteristic, complemented
20C3×#4013C2212yes—
21C3×#4011—21no—
22C3×#4011C2212yes—
23C3×#4005—21no—
24C3×#2122—51nomaximal, Hall
25#65926C2×C311yescharacteristic
26#65777C2×C311yescharacteristic, complemented
27#65771C2×C311yescharacteristic, complemented
28#64934C2×C312yescomplemented
29#64864C2×C312yescomplemented
30C24×C3×C5C2×C411yescharacteristic
31C23×C3×D5C2×C411yescharacteristic
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4C2×C411yescharacteristic
34C2×C3×C22⋊D5—21no—
35C2×C3×C22⋊D5—21no—
36C2×C3×C22⋊D5—21no—
37C2×C3×C22⋊D5—21no—
38C2×C3×C22⋊D5—41no—
39C2×C3×C22⋊D5—41no—
40C2×C3×C22⋊D5D411yescharacteristic
41C2×C3×C22⋊D5C2311yescharacteristic
42C22×C3×Dic5C2×C411yescharacteristic
43C22×C3×Dic5D411yescharacteristic
44C3×#711—41no—
45C3×#711—41no—
46C2×C3×D20—21no—
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×Dic10D411yescharacteristic
49C3×C22⋊AGL(1,𝔽5)—22no—
50C3×#704—22no—
51C3×#704—41no—
52C2×C3×#170—41no—
53C3×C4⋊AGL(1,𝔽5)—41no—
54C3×C4×AGL(1,𝔽5)—41no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4C2×C411yescharacteristic
57C3×C22⋊Dic5—21no—
58C3×C22⋊Dic5C2×C411yescharacteristic
59C3×#685—21no—
60C3×#685—21no—
61C3×#683—21no—
62C3×C4×Dic5—21no—
63C3×#535—51no—
64C3×#413—51no—
65C3×#410—51no—
66C3×AΣL(1,𝔽16)—52no—
67C3×AΣL(1,𝔽16)—52no—
68C22×C22⋊D5C3×C411yescharacteristic, complemented
69C2×#711C22×C311yescharacteristic
70C2×#704—21no—
71#4130—21no—
72C5×#120C3×C411yescharacteristic, complemented
73#4099C22×C311yescharacteristic
74#4086—21no—
75#4085—21no—
76#4083C3×C411yescharacteristic, complemented
77#4031—21no—
78#4030—21no—
79#4028C22×C311yescharacteristic
80#4023C3×C411yescharacteristic, complemented
81#4013C22×C312yes—
82#4011—21no—
83#4011C22×C312yes—
84#4005—21no—
85#2122—51noSylow
86C23×C3×C5—21no—
87C23×C3×C5—41no—
88C23×C3×C5—81no—
89C23×C3×C5C22⋊C411yescharacteristic
90C23×C3×C5C22×C411yescharacteristic
91C22×C3×D5—21no—
92C22×C3×D5—41no—
93C22×C3×D5—81no—
94C22×C3×D5C2×D411yescharacteristic
95C2×C3×AGL(1,𝔽5)—42no—
96C3×C5×D4—41no—
97C3×C5×D4—81no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C3×C22⋊D5—41no—
101C3×C22⋊D5—41no—
102C3×C22⋊D5—42no—
103C3×C22⋊D5—41no—
104C3×C22⋊D5—41no—
105C3×C22⋊D5—81no—
106C3×C22⋊D5—81no—
107C2×C3×Dic5—21no—
108C2×C3×Dic5—21no—
109C2×C3×Dic5—21no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5C22⋊C411yescharacteristic
113C2×C3×Dic5C22⋊C411yescharacteristic
114C2×C3×Dic5C2×D411yescharacteristic
115C3×D20—81no—
116C3×C4×D5—21no—
117C3×C4×D5—21no—
118C3×Dic10—41no—
119C3×#170—42no—
120C2×C3×#41—51no—
121C22×C3×D4—51no—
122C2×C3×M16—101no—
123C3×#127—101no—
124C3×#124—101no—
125C3×#121—101no—
126C3×#121—101no—
127C3×#120—51no—
128C3×#120—51no—
129C3×#120—51no—
130C3×#120AGL(1,𝔽5)11yescharacteristic, complemented
131C3×#117—51no—
132C3×#117—101no—
133C3×#101—101no—
134C3×#100—52no—
135C3×#99—52no—
136C3×#99—101no—
137C24×C5C2×C3×C411yescharacteristic
138C23×D5C2×C3×C411yescharacteristic
139C2×C5×D4—21no—
140C2×C5×D4C2×C3×C411yescharacteristic
141C2×C22⋊D5—21no—
142C2×C22⋊D5—21no—
143C2×C22⋊D5—21no—
144C2×C22⋊D5—21no—
145C2×C22⋊D5—41no—
146C2×C22⋊D5—41no—
147C2×C22⋊D5C3×D411yescharacteristic
148C2×C22⋊D5C23×C311yescharacteristic
149C22×Dic5C2×C3×C411yescharacteristic
150C22×Dic5C3×D411yescharacteristic
151#711—41no—
152#711—41no—
153C2×D20—21no—
154C2×C4×D5C3×D411yescharacteristic
155C2×Dic10C3×D411yescharacteristic
156C22⋊AGL(1,𝔽5)—22no—
157#704—22no—
158#704—41no—
159C2×#170—41no—
160C4⋊AGL(1,𝔽5)—41no—
161C4×AGL(1,𝔽5)—41no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4C2×C3×C411yescharacteristic
164C22⋊Dic5—21no—
165C22⋊Dic5C2×C3×C411yescharacteristic
166#685—21no—
167#685—21no—
168#683—21no—
169C4×Dic5—21no—
170#535—51no—
171#413—51no—
172#410—51no—
173AΣL(1,𝔽16)—52no—
174AΣL(1,𝔽16)—52no—
175C22×C3×C5—21no—
176C22×C3×C5—21no—
177C22×C3×C5—41no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5—41no—
182C22×C3×C5—82no—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C2×C3×D5—21no—
185C2×C3×D5—41no—
186C2×C3×D5—41no—
187C2×C3×D5—41no—
188C2×C3×D5—41no—
189C2×C3×D5—82no—
190C3×AGL(1,𝔽5)—82no—
191C3×C4×C5—21no—
192C3×C4×C5—41no—
193C3×Dic5—21no—
194C3×Dic5—21no—
195C3×Dic5—41no—
196C3×Dic5#9911yescharacteristic
197C3×Dic5#9911yescharacteristic
198C24×C3—51no—
199C24×C3C2×AGL(1,𝔽5)11yescharacteristic
200C3×#41—201no—
201C3×#41—201no—
202C2×C3×Q8—51no—
203C2×C3×D4—21no—
204C2×C3×D4—51no—
205C2×C3×D4—51no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C2×C3×D4—101no—
209C2×C3×D4—101no—
210C2×C3×D4—101no—
211C2×C3×D4—201no—
212C2×C3×D4—201no—
213C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
214C22×C3×C4—51no—
215C22×C3×C4—51no—
216C22×C3×C4—51no—
217C3×M16—102no—
218C3×M16—201no—
219C2×C3×C8—201no—
220C3×C4⋊C4—101no—
221C3×C4⋊C4—201no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—51no—
224C3×C22⋊C4—101no—
225C3×C22⋊C4—101no—
226C3×C22⋊C4—101no—
227C3×C22⋊C4—102no—
228C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
229C3×C42—101no—
230C3×C42—201no—
231C23×C5—21no—
232C23×C5—41no—
233C23×C5—81no—
234C23×C5C3×C22⋊C411yescharacteristic
235C23×C5C22×C3×C411yescharacteristic
236C22×D5—21no—
237C22×D5—41no—
238C22×D5—81no—
239C22×D5C2×C3×D411yescharacteristic
240C2×AGL(1,𝔽5)—42no—
241C5×D4—41no—
242C5×D4—81no—
243C2×C4×C5—21no—
244C2×C4×C5C3×C22⋊C411yescharacteristic
245C22⋊D5—41no—
246C22⋊D5—41no—
247C22⋊D5—42no—
248C22⋊D5—41no—
249C22⋊D5—41no—
250C22⋊D5—81no—
251C22⋊D5—81no—
252C2×Dic5—21no—
253C2×Dic5—21no—
254C2×Dic5—21no—
255C2×Dic5—21no—
256C2×Dic5—21no—
257C2×Dic5C3×C22⋊C411yescharacteristic
258C2×Dic5C3×C22⋊C411yescharacteristic
259C2×Dic5C2×C3×D411yescharacteristic
260D20—81no—
261C4×D5—21no—
262C4×D5—21no—
263Dic10—41no—
264#170—42no—
265C2×#41—51no—
266C22×D4—51no—
267C2×M16—101no—
268#127—101no—
269#124—101no—
270#121—101no—
271#121—101no—
272#120—51no—
273#120—51no—
274#120—51no—
275#120C3×AGL(1,𝔽5)11yescharacteristic, complemented
276#117—51no—
277#117—101no—
278#101—101no—
279#100—52no—
280#99—52no—
281#99—101no—
282C2×C3×C5—21no—
283C2×C3×C5—41no—
284C2×C3×C5—41no—
285C2×C3×C5—81no—
286C2×C3×C5C2×#9911yescharacteristic
287C3×D5—41no—
288C3×D5—81no—
289C23×C3—21no—
290C23×C3—41no—
291C23×C3—51no—
292C23×C3—81no—
293C23×C3—101no—
294C23×C3—201no—
295C23×C3—401no—
296C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
297C23×C3C22×AGL(1,𝔽5)11yescharacteristic
298C3×Q8—201no—
299C3×D4—41no—
300C3×D4—81no—
301C3×D4—201no—
302C3×D4—201no—
303C3×D4—201no—
304C3×D4—201no—
305C3×D4—202no—
306C3×D4—401no—
307C3×D4—401no—
308C3×D4—401no—
309C2×C3×C4—21no—
310C2×C3×C4—51no—
311C2×C3×C4—51no—
312C2×C3×C4—51no—
313C2×C3×C4—101no—
314C2×C3×C4—101no—
315C2×C3×C4—101no—
316C2×C3×C4—101no—
317C2×C3×C4—101no—
318C2×C3×C4—101no—
319C2×C3×C4—101no—
320C2×C3×C4—202no—
321C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
322C3×C8—202no—
323C22×C5—21no—
324C22×C5—21no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5—41no—
328C22×C5—41no—
329C22×C5—41no—
330C22×C5—82no—
331C22×C5C2×C3×C22⋊C411yescharacteristic
332C2×D5—21no—
333C2×D5—41no—
334C2×D5—41no—
335C2×D5—41no—
336C2×D5—41no—
337C2×D5—82no—
338AGL(1,𝔽5)—82no—
339C4×C5—21no—
340C4×C5—41no—
341Dic5—21no—
342Dic5—21no—
343Dic5—41no—
344Dic5C3×#9911yescharacteristic
345Dic5C3×#9911yescharacteristic
346C24—51no—
347C24C2×C3×AGL(1,𝔽5)11yescharacteristic
348#41—201no—
349#41—201no—
350C2×Q8—51no—
351C2×D4—21no—
352C2×D4—51no—
353C2×D4—51no—
354C2×D4—101no—
355C2×D4—101no—
356C2×D4—101no—
357C2×D4—101no—
358C2×D4—101no—
359C2×D4—201no—
360C2×D4—201no—
361C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
362C22×C4—51no—
363C22×C4—51no—
364C22×C4—51no—
365M16—102no—
366M16—201no—
367C2×C8—201no—
368C4⋊C4—101no—
369C4⋊C4—201no—
370C22⋊C4—21no—
371C22⋊C4—51no—
372C22⋊C4—101no—
373C22⋊C4—101no—
374C22⋊C4—101no—
375C22⋊C4—102no—
376C22⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
377C42—101no—
378C42—201no—
379C3×C5#212211yescharacteristic, complemented, Hall
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—82no—
388C22×C3—101no—
389C22×C3—201no—
390C22×C3—201no—
391C22×C3—201no—
392C22×C3—201no—
393C22×C3—402no—
394C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
395C3×C4—21no—
396C3×C4—41no—
397C3×C4—51no—
398C3×C4—51no—
399C3×C4—101no—
400C3×C4—101no—
401C3×C4—201no—
402C3×C4—402no—
403C2×C5—21no—
404C2×C5—41no—
405C2×C5—41no—
406C2×C5—81no—
407C2×C5C2×C3×#9911yescharacteristic
408D5—41no—
409D5—81no—
410C23—21no—
411C23—41no—
412C23—51no—
413C23—81no—
414C23—101no—
415C23—201no—
416C23—401no—
417C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
418C23C22×C3×AGL(1,𝔽5)11yescharacteristic
419Q8—201no—
420D4—41no—
421D4—81no—
422D4—201no—
423D4—201no—
424D4—201no—
425D4—201no—
426D4—202no—
427D4—401no—
428D4—401no—
429D4—401no—
430C2×C4—21no—
431C2×C4—51no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—101no—
435C2×C4—101no—
436C2×C4—101no—
437C2×C4—101no—
438C2×C4—101no—
439C2×C4—101no—
440C2×C4—101no—
441C2×C4—202no—
442C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
443C8—202no—
444C2×C3—21no—
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—81no—
448C2×C3—201no—
449C2×C3—401no—
450C2×C3D5⋊#9911yescharacteristic
451C5C3×#212211yescharacteristic, complemented, Sylow
452C22—21no—
453C22—21no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—82no—
460C22—101no—
461C22—201no—
462C22—201no—
463C22—201no—
464C22—201no—
465C22—402no—
466C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
467C4—21no—
468C4—41no—
469C4—51no—
470C4—51no—
471C4—101no—
472C4—101no—
473C4—201no—
474C4—402no—
475C3#1060446011yescharacteristic, direct product factor, Sylow
476C2—21no—
477C2—41no—
478C2—41no—
479C2—81no—
480C2—201no—
481C2—401no—
482C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC3×C5×#120C4
socleC2×C3×C5C2×#99
prime radicalC3×#65926C2
cosocleC2×C22⋊D5C23×C3
hypercenterC3×#120AGL(1,𝔽5)
hypocenterC5C3×#2122
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
2. C3×#65777C2semidirect
3. C3×#65771C2semidirect
4. C3×#64934C2semidirect
5. C3×#64864C2semidirect
6. #10604460C3direct (N × H)
7. C22×C3×C22⋊D5C4semidirect
11. C3×C5×#120C4semidirect
15. C3×#4083C4semidirect
19. C3×#4023C4semidirect
26. #65777C2×C3semidirect
27. #65771C2×C3semidirect
28. #64934C2×C3semidirect
29. #64864C2×C3semidirect
68. C22×C22⋊D5C3×C4semidirect
72. C5×#120C3×C4semidirect
76. #4083C3×C4semidirect
80. #4023C3×C4semidirect
130. C3×#120AGL(1,𝔽5)semidirect
275. #120C3×AGL(1,𝔽5)semidirect
379. C3×C5#2122semidirect
451. C5C3×#2122semidirect
475. C3#10604460direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659262
C3×#657772
C3×#657712
C3×#649342
C3×#648642
#106044603
C3×#21225