Finite Group Databasebeta

← C3×#10604502 #14283426

Subgroups — C3×#10604502 #14283426

Order 1920.

572 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65961C211yesmaximal, characteristic
2C3×#65765C211yesmaximal, characteristic, complemented
3C3×#65755C211yesmaximal, characteristic, complemented
4C3×#64926C212yesmaximal, complemented
5C3×#64868C212yesmaximal, complemented
6#10604502C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C2×C3×#704—21no—
11C2×C3×C4⋊AGL(1,𝔽5)—21no—
12C3×C5×C22⋊Q8C411yescharacteristic, complemented
13C3×#4092C2211yescharacteristic
14C3×#4076C411yescharacteristic, complemented
15C3×#4041C2211yescharacteristic
16C3×#4040—21no—
17C3×#4039—21no—
18C3×#4038C411yescharacteristic, complemented
19C3×#4028—21no—
20C3×#4027—21no—
21C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
22C3×#4003—21no—
23C3×#4003C2212yes—
24C3×#1513—51nomaximal, Hall
25#65961C2×C311yescharacteristic
26#65765C2×C311yescharacteristic, complemented
27#65755C2×C311yescharacteristic, complemented
28#64926C2×C312yescomplemented
29#64868C2×C312yescomplemented
30C23×C3×D5D411yescharacteristic
31C22×C3×AGL(1,𝔽5)—22no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C22×C3×C4×C5C2×C411yescharacteristic
34C2×C3×C22⋊D5—21no—
35C22×C3×Dic5D411yescharacteristic
36C3×#713—42no—
37C2×C3×D20—21no—
38C2×C3×D20—21no—
39C2×C3×D20C2×C411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5—42no—
42C2×C3×C4×D5C2×C411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5D411yescharacteristic
45C2×C3×C4×D5C2311yescharacteristic
46C3×C22⋊AGL(1,𝔽5)—42no—
47C3×#704—42no—
48C2×C3×#170—22no—
49C3×C4⋊AGL(1,𝔽5)—42no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4C2×C411yescharacteristic
52C3×C5×C22⋊C4—21no—
53C3×#685—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×#685—21no—
57C3×C4⋊Dic5C2×C411yescharacteristic
58C3×#683—21no—
59C3×C4×Dic5—21no—
60C3×#525—51no—
61C3×#407—51no—
62C3×#380—51no—
63C3×#328—52no—
64C3×#327—52no—
65C2×#713C3×C411yescharacteristic, complemented
66C22×C4×D5C22×C311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—21no—
68C2×#704—21no—
69C2×C4⋊AGL(1,𝔽5)—21no—
70C5×C22⋊Q8C3×C411yescharacteristic, complemented
71#4092C22×C311yescharacteristic
72#4076C3×C411yescharacteristic, complemented
73#4041C22×C311yescharacteristic
74#4040—21no—
75#4039—21no—
76#4038C3×C411yescharacteristic, complemented
77#4028—21no—
78#4027—21no—
79(C2×AGL(1,𝔽5))⋊C4C22×C312yes—
80#4003—21no—
81#4003C22×C312yes—
82#1513—51noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C3×D5—21no—
85C22×C3×D5—21no—
86C22×C3×D5—21no—
87C22×C3×D5—41no—
88C22×C3×D5—41no—
89C22×C3×D5C22⋊C411yescharacteristic
90C22×C3×D5C2×D411yescharacteristic
91C2×C3×AGL(1,𝔽5)—42no—
92C2×C3×AGL(1,𝔽5)—42no—
93C2×C3×AGL(1,𝔽5)—42no—
94C3×C5×Q8—41no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C22×C411yescharacteristic
100C3×C22⋊D5—42no—
101C2×C3×Dic5—22no—
102C2×C3×Dic5—21no—
103C2×C3×Dic5C22⋊C411yescharacteristic
104C2×C3×Dic5C2×D411yescharacteristic
105C3×D20—41no—
106C3×D20—42no—
107C3×D20—41no—
108C3×D20—41no—
109C3×C4×D5—22no—
110C3×C4×D5—22no—
111C3×C4×D5—41no—
112C3×C4×D5—41no—
113C3×C4×D5—42no—
114C3×#170—42no—
115C2×C3×#41—51no—
116C23×C3×C4—51no—
117C2×C3×M16—101no—
118C3×#123—101no—
119C3×C22⋊Q8—51no—
120C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
121C3×#121—51no—
122C3×#121—51no—
123C3×#120—101no—
124C3×C4×D4—101no—
125C3×C4×D4—101no—
126C3×#117—51no—
127C2×C3×C4⋊C4—101no—
128C2×C3×C22⋊C4—101no—
129C3×C22⋊C8—52no—
130C3×C22⋊C8—101no—
131C3×#95—52no—
132C23×D5C3×D411yescharacteristic
133C22×AGL(1,𝔽5)—22no—
134C2×C5×Q8C2×C3×C411yescharacteristic
135C22×C4×C5C2×C3×C411yescharacteristic
136C2×C22⋊D5—21no—
137C22×Dic5C3×D411yescharacteristic
138#713—42no—
139C2×D20—21no—
140C2×D20—21no—
141C2×D20C2×C3×C411yescharacteristic
142C2×C4×D5—21no—
143C2×C4×D5—42no—
144C2×C4×D5C2×C3×C411yescharacteristic
145C2×C4×D5C3×D411yescharacteristic
146C2×C4×D5C3×D411yescharacteristic
147C2×C4×D5C23×C311yescharacteristic
148C22⋊AGL(1,𝔽5)—42no—
149#704—42no—
150C2×#170—22no—
151C4⋊AGL(1,𝔽5)—42no—
152C5×C4⋊C4—21no—
153C5×C4⋊C4C2×C3×C411yescharacteristic
154C5×C22⋊C4—21no—
155#685—21no—
156#685—21no—
157#685—21no—
158#685—21no—
159C4⋊Dic5C2×C3×C411yescharacteristic
160#683—21no—
161C4×Dic5—21no—
162#525—51no—
163#407—51no—
164#380—51no—
165#328—52no—
166#327—52no—
167C22×C3×C5—21no—
168C22×C3×C5—21no—
169C22×C3×C5—21no—
170C22×C3×C5C2×C22⋊C411yescharacteristic
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—42no—
177C2×C3×D5—42no—
178C2×C3×D5—41no—
179C2×C3×D5—41no—
180C2×C3×D5—41no—
181C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
182C2×C3×D5#13711yescharacteristic
183C3×AGL(1,𝔽5)—82no—
184C3×C4×C5—21no—
185C3×C4×C5—21no—
186C3×C4×C5—41no—
187C3×C4×C5—41no—
188C3×Dic5—21no—
189C3×Dic5—41no—
190C3×Dic5#9912yes—
191C24×C3—51no—
192C3×#41—202no—
193C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
194C2×C3×D4—51no—
195C2×C3×D4—101no—
196C2×C3×D4—101no—
197C2×C3×D4—101no—
198C22×C3×C4—51no—
199C22×C3×C4—51no—
200C22×C3×C4—51no—
201C22×C3×C4—51no—
202C22×C3×C4—51no—
203C22×C3×C4—101no—
204C22×C3×C4—102no—
205C22×C3×C4—202no—
206C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
207C3×M16—202no—
208C2×C3×C8—102no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—51no—
211C3×C4⋊C4—101no—
212C3×C4⋊C4—202no—
213C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
214C3×C22⋊C4—21no—
215C3×C22⋊C4—101no—
216C3×C22⋊C4—101no—
217C3×C22⋊C4—101no—
218C3×C22⋊C4—101no—
219C3×C22⋊C4—202no—
220C3×C42—101no—
221C23×C5C3×C22⋊C411yescharacteristic
222C22×D5—21no—
223C22×D5—21no—
224C22×D5—21no—
225C22×D5—41no—
226C22×D5—41no—
227C22×D5C3×C22⋊C411yescharacteristic
228C22×D5C2×C3×D411yescharacteristic
229C2×AGL(1,𝔽5)—42no—
230C2×AGL(1,𝔽5)—42no—
231C2×AGL(1,𝔽5)—42no—
232C5×Q8—41no—
233C2×C4×C5—21no—
234C2×C4×C5—21no—
235C2×C4×C5—41no—
236C2×C4×C5C3×C22⋊C411yescharacteristic
237C2×C4×C5C22×C3×C411yescharacteristic
238C22⋊D5—42no—
239C2×Dic5—22no—
240C2×Dic5—21no—
241C2×Dic5C3×C22⋊C411yescharacteristic
242C2×Dic5C2×C3×D411yescharacteristic
243D20—41no—
244D20—42no—
245D20—41no—
246D20—41no—
247C4×D5—22no—
248C4×D5—22no—
249C4×D5—41no—
250C4×D5—41no—
251C4×D5—42no—
252#170—42no—
253C2×#41—51no—
254C23×C4—51no—
255C2×M16—101no—
256#123—101no—
257C22⋊Q8—51no—
258C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
259#121—51no—
260#121—51no—
261#120—101no—
262C4×D4—101no—
263C4×D4—101no—
264#117—51no—
265C2×C4⋊C4—101no—
266C2×C22⋊C4—101no—
267C22⋊C8—52no—
268C22⋊C8—101no—
269#95—52no—
270C2×C3×C5—41no—
271C2×C3×C5C2×#9911yescharacteristic
272C2×C3×C5#41711yescharacteristic
273C2×C3×C5#42111yescharacteristic
274C3×D5—21no—
275C3×D5—21no—
276C3×D5—41no—
277C3×D5—81no—
278C23×C3—51no—
279C23×C3—51no—
280C23×C3—101no—
281C23×C3—101no—
282C23×C3—101no—
283C23×C3—201no—
284C23×C3—201no—
285C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
286C3×Q8—41no—
287C3×D4—201no—
288C3×D4—201no—
289C3×D4—201no—
290C3×D4—202no—
291C3×D4—202no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—41no—
295C2×C3×C4—51no—
296C2×C3×C4—51no—
297C2×C3×C4—102no—
298C2×C3×C4—102no—
299C2×C3×C4—102no—
300C2×C3×C4—101no—
301C2×C3×C4—201no—
302C2×C3×C4—201no—
303C2×C3×C4—202no—
304C2×C3×C4—202no—
305C2×C3×C4—202no—
306C2×C3×C4—202no—
307C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
308C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
309C3×C8—202no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5C2×C3×C22⋊C411yescharacteristic
314C2×D5—21no—
315C2×D5—21no—
316C2×D5—21no—
317C2×D5—21no—
318C2×D5—21no—
319C2×D5—42no—
320C2×D5—42no—
321C2×D5—41no—
322C2×D5—41no—
323C2×D5—41no—
324C2×D5C3×AGL(1,ℤ/8)11yescharacteristic
325C2×D5C3×#13711yescharacteristic
326AGL(1,𝔽5)—82no—
327C4×C5—21no—
328C4×C5—21no—
329C4×C5—41no—
330C4×C5—41no—
331Dic5—21no—
332Dic5—41no—
333Dic5C3×#9912yes—
334C24—51no—
335#41—202no—
336C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
337C2×D4—51no—
338C2×D4—101no—
339C2×D4—101no—
340C2×D4—101no—
341C22×C4—51no—
342C22×C4—51no—
343C22×C4—51no—
344C22×C4—51no—
345C22×C4—51no—
346C22×C4—101no—
347C22×C4—102no—
348C22×C4—202no—
349C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
350M16—202no—
351C2×C8—102no—
352C4⋊C4—21no—
353C4⋊C4—51no—
354C4⋊C4—101no—
355C4⋊C4—202no—
356C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
357C22⋊C4—21no—
358C22⋊C4—101no—
359C22⋊C4—101no—
360C22⋊C4—101no—
361C22⋊C4—101no—
362C22⋊C4—202no—
363C42—101no—
364C3×C5#151311yescharacteristic, complemented, Hall
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—51no—
369C22×C3—51no—
370C22×C3—101no—
371C22×C3—101no—
372C22×C3—101no—
373C22×C3—101no—
374C22×C3—101no—
375C22×C3—202no—
376C22×C3—202no—
377C22×C3—201no—
378C22×C3—201no—
379C22×C3—201no—
380C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—52no—
386C3×C4—101no—
387C3×C4—201no—
388C3×C4—402no—
389C2×C5—41no—
390C2×C5C2×C3×#9911yescharacteristic
391C2×C5C3×#41711yescharacteristic
392C2×C5C3×#42111yescharacteristic
393D5—21no—
394D5—21no—
395D5—41no—
396D5—81no—
397C23—51no—
398C23—51no—
399C23—101no—
400C23—101no—
401C23—101no—
402C23—201no—
403C23—201no—
404C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
405Q8—41no—
406D4—201no—
407D4—201no—
408D4—201no—
409D4—202no—
410D4—202no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—51no—
415C2×C4—51no—
416C2×C4—102no—
417C2×C4—102no—
418C2×C4—102no—
419C2×C4—101no—
420C2×C4—201no—
421C2×C4—201no—
422C2×C4—202no—
423C2×C4—202no—
424C2×C4—202no—
425C2×C4—202no—
426C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
427C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
428C8—202no—
429C2×C3—41no—
430C2×C3—101no—
431C2×C3—101no—
432C2×C3—201no—
433C2×C3—401no—
434C2×C3D5⋊#9911yescharacteristic
435C2×C3#6578411yescharacteristic
436C2×C3#6579311yescharacteristic
437C5C3×#151311yescharacteristic, complemented, Sylow
438C22—21no—
439C22—21no—
440C22—21no—
441C22—51no—
442C22—51no—
443C22—101no—
444C22—101no—
445C22—101no—
446C22—101no—
447C22—101no—
448C22—202no—
449C22—202no—
450C22—201no—
451C22—201no—
452C22—201no—
453C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
454C4—21no—
455C4—21no—
456C4—41no—
457C4—41no—
458C4—52no—
459C4—101no—
460C4—201no—
461C4—402no—
462C3#1060450211yescharacteristic, direct product factor, Sylow
463C2—41no—
464C2—101no—
465C2—101no—
466C2—201no—
467C2—401no—
468C2C3×D5⋊#9911yescharacteristic
469C2C3×#6578411yescharacteristic
470C2C3×#6579311yescharacteristic
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×#1513
LCSG ▷ C2×C4×C5 ▷ C22×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
2. C3×#65765C2semidirect
3. C3×#65755C2semidirect
4. C3×#64926C2semidirect
5. C3×#64868C2semidirect
6. #10604502C3direct (N × H)
7. C2×C3×#713C4semidirect
12. C3×C5×C22⋊Q8C4semidirect
14. C3×#4076C4semidirect
18. C3×#4038C4semidirect
26. #65765C2×C3semidirect
27. #65755C2×C3semidirect
28. #64926C2×C3semidirect
29. #64868C2×C3semidirect
65. C2×#713C3×C4semidirect
70. C5×C22⋊Q8C3×C4semidirect
72. #4076C3×C4semidirect
76. #4038C3×C4semidirect
120. C3×C22⋊Q8AGL(1,𝔽5)semidirect
258. C22⋊Q8C3×AGL(1,𝔽5)semidirect
364. C3×C5#1513semidirect
437. C5C3×#1513semidirect
462. C3#10604502direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659612
C3×#657652
C3×#657552
C3×#649262
C3×#648682
#106045023
C3×#15135