Finite Group Databasebeta

← C3×#10604500 #14283162

Subgroups — C3×#10604500 #14283162

Order 1920.

588 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×C22⋊Q8C211yesmaximal, characteristic
2C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C3×#65755C211yesmaximal, characteristic
4C3×#64926C212yesmaximal, complemented
5C3×#64869C212yesmaximal, complemented
6#10604500C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C411yescharacteristic, 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×#4070C411yescharacteristic, complemented
15C3×#4043C2211yescharacteristic
16C3×#4042—21no—
17C3×D5×C4⋊C4—21no—
18C3×D5×C4⋊C4C411yescharacteristic, complemented
19C3×D5×C22⋊C4—21no—
20C3×#4024—21no—
21C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
22C3×#4003—21no—
23C3×#4003C2212yes—
24C3×#1508—51nomaximal, Hall
25D5×C22⋊Q8C2×C311yescharacteristic
26C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
27#65755C2×C311yescharacteristic
28#64926C2×C312yescomplemented
29#64869C2×C312yescomplemented
30C23×C3×D5D411yescharacteristic
31C22×C3×AGL(1,𝔽5)—22no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×C3×Dic5D411yescharacteristic
35C3×Q8×D5—42no—
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5—42no—
39C2×C3×C4×D5C2×C411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5C2311yescharacteristic
43C2×C3×Dic10—21no—
44C2×C3×Dic10C2×C411yescharacteristic
45C3×C22⋊AGL(1,𝔽5)—24no—
46C3×#704—42no—
47C2×C3×#170—22no—
48C3×C4×AGL(1,𝔽5)—42no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C3×C5×C22⋊C4—21no—
52C3×C22⋊Dic5—21no—
53C3×#685—21no—
54C3×#685—21no—
55C3×C4⋊Dic5—21no—
56C3×C4⋊Dic5C2×C411yescharacteristic
57C3×#683—21no—
58C3×#683—21no—
59C3×#683—21no—
60C2×C3×C22⋊Q8—51no—
61C3×#407—51no—
62C3×C4×C22⋊C4—51no—
63C3×#328—52no—
64C3×#328—52no—
65C2×Q8×D5C3×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#4070C3×C411yescharacteristic, complemented
73#4043C22×C311yescharacteristic
74#4042—21no—
75D5×C4⋊C4—21no—
76D5×C4⋊C4C3×C411yescharacteristic, complemented
77D5×C22⋊C4—21no—
78#4024—21no—
79(C2×AGL(1,𝔽5))⋊C4C22×C312yes—
80#4003—21no—
81#4003C22×C312yes—
82#1508—51noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C3×D5—22no—
85C22×C3×D5—22no—
86C22×C3×D5—41no—
87C22×C3×D5C22⋊C411yescharacteristic
88C22×C3×D5C2×D411yescharacteristic
89C2×C3×AGL(1,𝔽5)—24no—
90C2×C3×AGL(1,𝔽5)—42no—
91C2×C3×AGL(1,𝔽5)—42no—
92C3×C5×Q8—41no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5C22⋊C411yescharacteristic
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×C3×Dic5—22no—
99C2×C3×Dic5—21no—
100C2×C3×Dic5—21no—
101C2×C3×Dic5C22⋊C411yescharacteristic
102C2×C3×Dic5C2×D411yescharacteristic
103C3×C4×D5—22no—
104C3×C4×D5—22no—
105C3×C4×D5—41no—
106C3×C4×D5—41no—
107C3×C4×D5—42no—
108C3×C4×D5—42no—
109C3×Dic10—42no—
110C3×Dic10—41no—
111C3×#170—42no—
112C22×C3×Q8—51no—
113C23×C3×C4—51no—
114C2×C3×M16—101no—
115C3×C22⋊Q8—51no—
116C3×C22⋊Q8—51no—
117C3×C22⋊Q8—51no—
118C3×C22⋊Q8—101no—
119C3×C22⋊Q8—101no—
120C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
121C2×C3×C4⋊C4—51no—
122C2×C3×C4⋊C4—101no—
123C2×C3×C22⋊C4—101no—
124C2×C3×C22⋊C4—101no—
125C2×C3×C42—101no—
126C3×C22⋊C8—52no—
127C3×C22⋊C8—101no—
128C3×#95—52no—
129C23×D5C3×D411yescharacteristic
130C22×AGL(1,𝔽5)—22no—
131C2×C5×Q8C2×C3×C411yescharacteristic
132C22×C4×C5C2×C3×C411yescharacteristic
133C22×Dic5C3×D411yescharacteristic
134Q8×D5—42no—
135C2×C4×D5—21no—
136C2×C4×D5—21no—
137C2×C4×D5—42no—
138C2×C4×D5C2×C3×C411yescharacteristic
139C2×C4×D5C3×D411yescharacteristic
140C2×C4×D5C3×D411yescharacteristic
141C2×C4×D5C23×C311yescharacteristic
142C2×Dic10—21no—
143C2×Dic10C2×C3×C411yescharacteristic
144C22⋊AGL(1,𝔽5)—24no—
145#704—42no—
146C2×#170—22no—
147C4×AGL(1,𝔽5)—42no—
148C5×C4⋊C4—21no—
149C5×C4⋊C4C2×C3×C411yescharacteristic
150C5×C22⋊C4—21no—
151C22⋊Dic5—21no—
152#685—21no—
153#685—21no—
154C4⋊Dic5—21no—
155C4⋊Dic5C2×C3×C411yescharacteristic
156#683—21no—
157#683—21no—
158#683—21no—
159C2×C22⋊Q8—51no—
160#407—51no—
161C4×C22⋊C4—51no—
162#328—52no—
163#328—52no—
164C22×C3×C5—21no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5C2×C22⋊C411yescharacteristic
168C2×C3×D5—21no—
169C2×C3×D5—21no—
170C2×C3×D5—21no—
171C2×C3×D5—21no—
172C2×C3×D5—44no—
173C2×C3×D5#10412yes—
174C2×C3×D5#13712yes—
175C3×AGL(1,𝔽5)—44no—
176C3×C4×C5—21no—
177C3×C4×C5—21no—
178C3×C4×C5—41no—
179C3×C4×C5—41no—
180C3×Dic5—21no—
181C3×Dic5—41no—
182C3×Dic5—41no—
183C3×Dic5#9912yes—
184C24×C3—51no—
185C2×C3×Q8—51no—
186C2×C3×Q8—101no—
187C2×C3×Q8—202no—
188C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—51no—
192C22×C3×C4—51no—
193C22×C3×C4—51no—
194C22×C3×C4—101no—
195C22×C3×C4—101no—
196C22×C3×C4—102no—
197C22×C3×C4—202no—
198C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
199C3×M16—202no—
200C2×C3×C8—102no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—51no—
203C3×C4⋊C4—101no—
204C3×C4⋊C4—101no—
205C3×C4⋊C4—101no—
206C3×C4⋊C4—101no—
207C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
208C3×C22⋊C4—21no—
209C3×C22⋊C4—101no—
210C3×C22⋊C4—101no—
211C3×C22⋊C4—104no—
212C3×C22⋊C4—101no—
213C3×C42—202no—
214C23×C5C3×C22⋊C411yescharacteristic
215C22×D5—22no—
216C22×D5—22no—
217C22×D5—41no—
218C22×D5C3×C22⋊C411yescharacteristic
219C22×D5C2×C3×D411yescharacteristic
220C2×AGL(1,𝔽5)—24no—
221C2×AGL(1,𝔽5)—42no—
222C2×AGL(1,𝔽5)—42no—
223C5×Q8—41no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5—41no—
227C2×C4×C5C3×C22⋊C411yescharacteristic
228C2×C4×C5C22×C3×C411yescharacteristic
229C2×Dic5—22no—
230C2×Dic5—21no—
231C2×Dic5—21no—
232C2×Dic5C3×C22⋊C411yescharacteristic
233C2×Dic5C2×C3×D411yescharacteristic
234C4×D5—22no—
235C4×D5—22no—
236C4×D5—41no—
237C4×D5—41no—
238C4×D5—42no—
239C4×D5—42no—
240Dic10—42no—
241Dic10—41no—
242#170—42no—
243C22×Q8—51no—
244C23×C4—51no—
245C2×M16—101no—
246C22⋊Q8—51no—
247C22⋊Q8—51no—
248C22⋊Q8—51no—
249C22⋊Q8—101no—
250C22⋊Q8—101no—
251C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
252C2×C4⋊C4—51no—
253C2×C4⋊C4—101no—
254C2×C22⋊C4—101no—
255C2×C22⋊C4—101no—
256C2×C42—101no—
257C22⋊C8—52no—
258C22⋊C8—101no—
259#95—52no—
260C2×C3×C5—41no—
261C2×C3×C5C2×#9911yescharacteristic
262C2×C3×C5#41911yescharacteristic
263C2×C3×C5C2×#10411yescharacteristic
264C3×D5—22no—
265C3×D5—22no—
266C23×C3—51no—
267C23×C3—51no—
268C23×C3—102no—
269C23×C3—102no—
270C23×C3—201no—
271C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
272C3×Q8—41no—
273C3×Q8—201no—
274C3×Q8—202no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—41no—
278C2×C3×C4—51no—
279C2×C3×C4—51no—
280C2×C3×C4—102no—
281C2×C3×C4—102no—
282C2×C3×C4—102no—
283C2×C3×C4—104no—
284C2×C3×C4—101no—
285C2×C3×C4—101no—
286C2×C3×C4—201no—
287C2×C3×C4—201no—
288C2×C3×C4—202no—
289C2×C3×C4—202no—
290C2×C3×C4—202no—
291C2×C3×C4—202no—
292C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
293C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
294C3×C8—202no—
295C22×C5—21no—
296C22×C5—21no—
297C22×C5—21no—
298C22×C5C2×C3×C22⋊C411yescharacteristic
299C2×D5—21no—
300C2×D5—21no—
301C2×D5—21no—
302C2×D5—21no—
303C2×D5—44no—
304C2×D5C3×#10412yes—
305C2×D5C3×#13712yes—
306AGL(1,𝔽5)—44no—
307C4×C5—21no—
308C4×C5—21no—
309C4×C5—41no—
310C4×C5—41no—
311Dic5—21no—
312Dic5—41no—
313Dic5—41no—
314Dic5C3×#9912yes—
315C24—51no—
316C2×Q8—51no—
317C2×Q8—101no—
318C2×Q8—202no—
319C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—51no—
323C22×C4—51no—
324C22×C4—51no—
325C22×C4—101no—
326C22×C4—101no—
327C22×C4—102no—
328C22×C4—202no—
329C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
330M16—202no—
331C2×C8—102no—
332C4⋊C4—21no—
333C4⋊C4—51no—
334C4⋊C4—101no—
335C4⋊C4—101no—
336C4⋊C4—101no—
337C4⋊C4—101no—
338C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—101no—
341C22⋊C4—101no—
342C22⋊C4—104no—
343C22⋊C4—101no—
344C42—202no—
345C3×C5#150811yescharacteristic, complemented, Hall
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—52no—
350C22×C3—52no—
351C22×C3—101no—
352C22×C3—101no—
353C22×C3—101no—
354C22×C3—101no—
355C22×C3—204no—
356C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—52no—
362C3×C4—101no—
363C3×C4—204no—
364C3×C4—201no—
365C3×C4—201no—
366C2×C5—41no—
367C2×C5C2×C3×#9911yescharacteristic
368C2×C5C3×#41911yescharacteristic
369C2×C5C2×C3×#10411yescharacteristic
370D5—22no—
371D5—22no—
372C23—51no—
373C23—51no—
374C23—102no—
375C23—102no—
376C23—201no—
377C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
378Q8—41no—
379Q8—201no—
380Q8—202no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—51no—
385C2×C4—51no—
386C2×C4—102no—
387C2×C4—102no—
388C2×C4—102no—
389C2×C4—104no—
390C2×C4—101no—
391C2×C4—101no—
392C2×C4—201no—
393C2×C4—201no—
394C2×C4—202no—
395C2×C4—202no—
396C2×C4—202no—
397C2×C4—202no—
398C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
399C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
400C8—202no—
401C2×C3—41no—
402C2×C3—102no—
403C2×C3—102no—
404C2×C3D5⋊#9911yescharacteristic
405C2×C3#6578211yescharacteristic
406C2×C3D5⋊#10411yescharacteristic
407C5C3×#150811yescharacteristic, complemented, Sylow
408C22—21no—
409C22—21no—
410C22—21no—
411C22—52no—
412C22—52no—
413C22—101no—
414C22—101no—
415C22—101no—
416C22—101no—
417C22—204no—
418C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
419C4—21no—
420C4—21no—
421C4—41no—
422C4—41no—
423C4—52no—
424C4—101no—
425C4—204no—
426C4—201no—
427C4—201no—
428C3#1060450011yescharacteristic, direct product factor, Sylow
429C2—41no—
430C2—102no—
431C2—102no—
432C2C3×D5⋊#9911yescharacteristic
433C2C3×#6578211yescharacteristic
434C2C3×D5⋊#10411yescharacteristic
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×#1508
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
4. C3×#64926C2semidirect
5. C3×#64869C2semidirect
6. #10604500C3direct (N × H)
7. C2×C3×Q8×D5C4semidirect
12. C3×C5×C22⋊Q8C4semidirect
14. C3×#4070C4semidirect
18. C3×D5×C4⋊C4C4semidirect
28. #64926C2×C3semidirect
29. #64869C2×C3semidirect
65. C2×Q8×D5C3×C4semidirect
70. C5×C22⋊Q8C3×C4semidirect
72. #4070C3×C4semidirect
76. D5×C4⋊C4C3×C4semidirect
120. C3×C22⋊Q8AGL(1,𝔽5)semidirect
251. C22⋊Q8C3×AGL(1,𝔽5)semidirect
345. C3×C5#1508semidirect
407. C5C3×#1508semidirect
428. C3#10604500direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×C22⋊Q82
C3×C4×C22⋊AGL(1,𝔽5)2
C3×#657552
C3×#649262
C3×#648692
#106045003
C3×#15085