Finite Group Databasebeta

← C3×#10604498 #14282830

Subgroups — C3×#10604498 #14282830

Order 1920.

608 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65961C211yesmaximal, characteristic
2C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C2×C3×#4003C211yesmaximal, characteristic, complemented
4C3×#64926C212yesmaximal, complemented
5C3×#64868C212yesmaximal, complemented
6#10604498C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C22×C3×#170—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×#1501—51nomaximal, Hall
25#65961C2×C311yescharacteristic
26C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
27C2×#4003C2×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)—24no—
47C2×C3×#170—22no—
48C2×C3×#170—42no—
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—
61C2×C3×C22⋊C8—51no—
62C3×C4×C22⋊C4—51no—
63C3×#328—52no—
64C3×#327—52no—
65C2×#713C3×C411yescharacteristic, complemented
66C22×C4×D5C22×C311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—21no—
68C22×#170—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#1501—51noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C3×D5—22no—
85C22×C3×D5—22no—
86C22×C3×D5—21no—
87C22×C3×D5—41no—
88C22×C3×D5C22⋊C411yescharacteristic
89C22×C3×D5C2×D411yescharacteristic
90C2×C3×AGL(1,𝔽5)—24no—
91C2×C3×AGL(1,𝔽5)—42no—
92C2×C3×AGL(1,𝔽5)—42no—
93C3×C5×Q8—41no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22×C411yescharacteristic
99C3×C22⋊D5—42no—
100C2×C3×Dic5—22no—
101C2×C3×Dic5—21no—
102C2×C3×Dic5C22⋊C411yescharacteristic
103C2×C3×Dic5C2×D411yescharacteristic
104C3×D20—41no—
105C3×D20—42no—
106C3×D20—42no—
107C3×C4×D5—22no—
108C3×C4×D5—22no—
109C3×C4×D5—41no—
110C3×C4×D5—41no—
111C3×C4×D5—42no—
112C3×#170—42no—
113C2×C3×#41—51no—
114C23×C3×C4—51no—
115C22×C3×C8—101no—
116C3×#123—101no—
117C3×C22⋊Q8—51no—
118C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
119C3×#121—51no—
120C3×#121—51no—
121C3×#120—101no—
122C3×C4×D4—101no—
123C3×C4×D4—101no—
124C3×#117—51no—
125C2×C3×C22⋊C4—101no—
126C2×C3×C42—101no—
127C3×C22⋊C8—52no—
128C3×C22⋊C8—101no—
129C3×#95—52no—
130C23×D5C3×D411yescharacteristic
131C22×AGL(1,𝔽5)—22no—
132C2×C5×Q8C2×C3×C411yescharacteristic
133C22×C4×C5C2×C3×C411yescharacteristic
134C2×C22⋊D5—21no—
135C22×Dic5C3×D411yescharacteristic
136#713—42no—
137C2×D20—21no—
138C2×D20—21no—
139C2×D20C2×C3×C411yescharacteristic
140C2×C4×D5—21no—
141C2×C4×D5—42no—
142C2×C4×D5C2×C3×C411yescharacteristic
143C2×C4×D5C3×D411yescharacteristic
144C2×C4×D5C3×D411yescharacteristic
145C2×C4×D5C23×C311yescharacteristic
146C22⋊AGL(1,𝔽5)—24no—
147C2×#170—22no—
148C2×#170—42no—
149C4×AGL(1,𝔽5)—42no—
150C5×C4⋊C4—21no—
151C5×C4⋊C4C2×C3×C411yescharacteristic
152C5×C22⋊C4—21no—
153#685—21no—
154#685—21no—
155#685—21no—
156#685—21no—
157C4⋊Dic5C2×C3×C411yescharacteristic
158#683—21no—
159C4×Dic5—21no—
160#525—51no—
161C2×C22⋊C8—51no—
162C4×C22⋊C4—51no—
163#328—52no—
164#327—52no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5#10412yes—
168C22×C3×C5C2×C22⋊C411yescharacteristic
169C2×C3×D5—21no—
170C2×C3×D5—21no—
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—44no—
175C2×C3×D5—41no—
176C2×C3×D5—41no—
177C2×C3×D5—41no—
178C2×C3×D5#13512yes—
179C3×AGL(1,𝔽5)—44no—
180C3×C4×C5—21no—
181C3×C4×C5—21no—
182C3×C4×C5—41no—
183C3×C4×C5—41no—
184C3×Dic5—21no—
185C3×Dic5—41no—
186C3×Dic5#9912yes—
187C24×C3—51no—
188C3×#41—202no—
189C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
190C2×C3×D4—51no—
191C2×C3×D4—101no—
192C2×C3×D4—101no—
193C2×C3×D4—101no—
194C22×C3×C4—51no—
195C22×C3×C4—51no—
196C22×C3×C4—51no—
197C22×C3×C4—51no—
198C22×C3×C4—51no—
199C22×C3×C4—101no—
200C22×C3×C4—102no—
201C22×C3×C4—202no—
202C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
203C2×C3×C8—102no—
204C2×C3×C8—202no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—51no—
207C3×C4⋊C4—101no—
208C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
209C3×C22⋊C4—21no—
210C3×C22⋊C4—101no—
211C3×C22⋊C4—101no—
212C3×C22⋊C4—101no—
213C3×C22⋊C4—104no—
214C3×C22⋊C4—101no—
215C3×C42—101no—
216C3×C42—202no—
217C23×C5C3×C22⋊C411yescharacteristic
218C22×D5—22no—
219C22×D5—22no—
220C22×D5—21no—
221C22×D5—41no—
222C22×D5C3×C22⋊C411yescharacteristic
223C22×D5C2×C3×D411yescharacteristic
224C2×AGL(1,𝔽5)—24no—
225C2×AGL(1,𝔽5)—42no—
226C2×AGL(1,𝔽5)—42no—
227C5×Q8—41no—
228C2×C4×C5—21no—
229C2×C4×C5—21no—
230C2×C4×C5—41no—
231C2×C4×C5C3×C22⋊C411yescharacteristic
232C2×C4×C5C22×C3×C411yescharacteristic
233C22⋊D5—42no—
234C2×Dic5—22no—
235C2×Dic5—21no—
236C2×Dic5C3×C22⋊C411yescharacteristic
237C2×Dic5C2×C3×D411yescharacteristic
238D20—41no—
239D20—42no—
240D20—42no—
241C4×D5—22no—
242C4×D5—22no—
243C4×D5—41no—
244C4×D5—41no—
245C4×D5—42no—
246#170—42no—
247C2×#41—51no—
248C23×C4—51no—
249C22×C8—101no—
250#123—101no—
251C22⋊Q8—51no—
252C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
253#121—51no—
254#121—51no—
255#120—101no—
256C4×D4—101no—
257C4×D4—101no—
258#117—51no—
259C2×C22⋊C4—101no—
260C2×C42—101no—
261C22⋊C8—52no—
262C22⋊C8—101no—
263#95—52no—
264C2×C3×C5—22no—
265C2×C3×C5C2×#9911yescharacteristic
266C2×C3×C5#41611yescharacteristic
267C2×C3×C5C2×#10411yescharacteristic
268C3×D5—22no—
269C3×D5—41no—
270C3×D5—81no—
271C23×C3—51no—
272C23×C3—51no—
273C23×C3—102no—
274C23×C3—102no—
275C23×C3—101no—
276C23×C3—201no—
277C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
278C3×Q8—41no—
279C3×D4—201no—
280C3×D4—202no—
281C3×D4—202no—
282C3×D4—202no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—41no—
286C2×C3×C4—51no—
287C2×C3×C4—51no—
288C2×C3×C4—102no—
289C2×C3×C4—102no—
290C2×C3×C4—102no—
291C2×C3×C4—104no—
292C2×C3×C4—101no—
293C2×C3×C4—201no—
294C2×C3×C4—201no—
295C2×C3×C4—202no—
296C2×C3×C4—202no—
297C2×C3×C4—202no—
298C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
299C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
300C3×C8—202no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5C3×#10412yes—
304C22×C5C2×C3×C22⋊C411yescharacteristic
305C2×D5—21no—
306C2×D5—21no—
307C2×D5—21no—
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—44no—
311C2×D5—41no—
312C2×D5—41no—
313C2×D5—41no—
314C2×D5C3×#13512yes—
315AGL(1,𝔽5)—44no—
316C4×C5—21no—
317C4×C5—21no—
318C4×C5—41no—
319C4×C5—41no—
320Dic5—21no—
321Dic5—41no—
322Dic5C3×#9912yes—
323C24—51no—
324#41—202no—
325C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
326C2×D4—51no—
327C2×D4—101no—
328C2×D4—101no—
329C2×D4—101no—
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—51no—
333C22×C4—51no—
334C22×C4—51no—
335C22×C4—101no—
336C22×C4—102no—
337C22×C4—202no—
338C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
339C2×C8—102no—
340C2×C8—202no—
341C4⋊C4—21no—
342C4⋊C4—51no—
343C4⋊C4—101no—
344C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—101no—
347C22⋊C4—101no—
348C22⋊C4—101no—
349C22⋊C4—104no—
350C22⋊C4—101no—
351C42—101no—
352C42—202no—
353C3×C5#150111yescharacteristic, complemented, Hall
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—52no—
357C22×C3—101no—
358C22×C3—101no—
359C22×C3—101no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3—204no—
363C22×C3—201no—
364C22×C3—201no—
365C22×C3—201no—
366C22×C3#401012yes—
367C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—52no—
373C3×C4—101no—
374C3×C4—204no—
375C3×C4—201no—
376C2×C5—22no—
377C2×C5C2×C3×#9911yescharacteristic
378C2×C5C3×#41611yescharacteristic
379C2×C5C2×C3×#10411yescharacteristic
380D5—22no—
381D5—41no—
382D5—81no—
383C23—51no—
384C23—51no—
385C23—102no—
386C23—102no—
387C23—101no—
388C23—201no—
389C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
390Q8—41no—
391D4—201no—
392D4—202no—
393D4—202no—
394D4—202no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—51no—
399C2×C4—51no—
400C2×C4—102no—
401C2×C4—102no—
402C2×C4—102no—
403C2×C4—104no—
404C2×C4—101no—
405C2×C4—201no—
406C2×C4—201no—
407C2×C4—202no—
408C2×C4—202no—
409C2×C4—202no—
410C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
411C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
412C8—202no—
413C2×C3—22no—
414C2×C3—102no—
415C2×C3—201no—
416C2×C3—401no—
417C2×C3D5⋊#9911yescharacteristic
418C2×C3C2×#401011yescharacteristic
419C2×C3#6579411yescharacteristic
420C5C3×#150111yescharacteristic, complemented, Sylow
421C22—21no—
422C22—21no—
423C22—52no—
424C22—101no—
425C22—101no—
426C22—101no—
427C22—101no—
428C22—101no—
429C22—204no—
430C22—201no—
431C22—201no—
432C22—201no—
433C22C3×#401012yes—
434C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
435C4—21no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—52no—
440C4—101no—
441C4—204no—
442C4—201no—
443C3#1060449811yescharacteristic, direct product factor, Sylow
444C2—22no—
445C2—102no—
446C2—201no—
447C2—401no—
448C2C3×D5⋊#9911yescharacteristic
449C2C2×C3×#401011yescharacteristic
450C2C3×#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 radicalC3×#65961C2
cosocleC2×C4×D5C23×C3
hypercenterC3×C22⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1501
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×C4×C22⋊AGL(1,𝔽5)C2semidirect
3. C2×C3×#4003C2semidirect
4. C3×#64926C2semidirect
5. C3×#64868C2semidirect
6. #10604498C3direct (N × H)
7. C2×C3×#713C4semidirect
12. C3×C5×C22⋊Q8C4semidirect
14. C3×#4076C4semidirect
18. C3×#4038C4semidirect
26. C4×C22⋊AGL(1,𝔽5)C2×C3semidirect
27. C2×#4003C2×C3semidirect
28. #64926C2×C3semidirect
29. #64868C2×C3semidirect
65. C2×#713C3×C4semidirect
70. C5×C22⋊Q8C3×C4semidirect
72. #4076C3×C4semidirect
76. #4038C3×C4semidirect
118. C3×C22⋊Q8AGL(1,𝔽5)semidirect
252. C22⋊Q8C3×AGL(1,𝔽5)semidirect
353. C3×C5#1501semidirect
420. C5C3×#1501semidirect
443. C3#10604498direct (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×C4×C22⋊AGL(1,𝔽5)2
C2×C3×#40032
C3×#649262
C3×#648682
#106044983
C3×#15015