Finite Group Databasebeta

← C3×#10604450 #14295644

Subgroups — C3×#10604450 #14295644

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#713C211yesmaximal, characteristic
2C2×C3×#4010C212yesmaximal
3C3×#65686C211yesmaximal, characteristic, complemented
4C3×C4×#700C211yesmaximal, characteristic, complemented
5C3×#64875C212yesmaximal, complemented
6#10604450C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×C4×AGL(1,𝔽5)C2212yes—
9C2×C3×#700C2212yes—
10C3×C4×C5×Q8C411yescharacteristic, complemented
11C3×Q8×Dic5C411yescharacteristic, complemented
12C3×#4039—21no—
13C3×#4039C411yescharacteristic, complemented
14C3×#4038—21no—
15C3×#4038C2211yescharacteristic
16C3×C4×D20—21no—
17C3×C4×D20C411yescharacteristic, complemented
18C3×C42×D5—21no—
19C3×C42×D5C2211yescharacteristic
20C3×#4010C418yescomplemented
21C3×(C2×AGL(1,𝔽5))⋊C4—21no—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×#4002—21no—
24C3×C4×#170—21no—
25C3×#1759—51nomaximal, Hall
26C4×#713C2×C311yescharacteristic
27C2×#4010C2×C312yes—
28#65686C2×C311yescharacteristic, complemented
29C4×#700C2×C311yescharacteristic, complemented
30#64875C2×C312yescomplemented
31C22×C3×AGL(1,𝔽5)—22no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C3×#713—22no—
34C3×#713C2×C414yes—
35C2×C3×D20—21no—
36C2×C3×D20C2×C411yescharacteristic
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5—21no—
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5C2311yescharacteristic
43C2×C3×#170—22no—
44C3×C4×AGL(1,𝔽5)—22no—
45C3×C4×AGL(1,𝔽5)C2×C414yescomplemented
46C3×#700—22no—
47C3×#700C2×C414yescomplemented
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4C2×C411yescharacteristic
50C3×C42×C5—21no—
51C3×C42×C5C2×C411yescharacteristic
52C3×#685—21no—
53C3×#685—21no—
54C3×#685—21no—
55C3×C4⋊Dic5—21no—
56C3×C4⋊Dic5C2×C411yescharacteristic
57C3×C4×Dic5—21no—
58C3×C4×Dic5—21no—
59C3×C4×Dic5C2×C411yescharacteristic
60C3×C4×Dic5D411yescharacteristic
61C3×C4×Dic5D411yescharacteristic
62C3×C4×#41—51no—
63C2×C3×#104—52no—
64C3×C4×M16—51no—
65C3×#376—51no—
66C3×#339—52no—
67C2×#713C22×C311yescharacteristic
68C2×C4×AGL(1,𝔽5)C22×C312yes—
69C2×#700C22×C312yes—
70C4×C5×Q8C3×C411yescharacteristic, complemented
71Q8×Dic5C3×C411yescharacteristic, complemented
72#4039—21no—
73#4039C3×C411yescharacteristic, complemented
74#4038—21no—
75#4038C22×C311yescharacteristic
76C4×D20—21no—
77C4×D20C3×C411yescharacteristic, complemented
78C42×D5—21no—
79C42×D5C22×C311yescharacteristic
80#4010C3×C418yescomplemented
81(C2×AGL(1,𝔽5))⋊C4—21no—
82(C2×AGL(1,𝔽5))⋊C4—21no—
83#4002—21no—
84C4×#170—21no—
85#1759—51noSylow
86C22×C3×D5—21no—
87C22×C3×D5C2×D411yescharacteristic
88C2×C3×AGL(1,𝔽5)—24no—
89C2×C3×AGL(1,𝔽5)—42no—
90C2×C3×AGL(1,𝔽5)—42no—
91C3×C5×Q8—21no—
92C3×C5×Q8C4212yes—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5C22⋊C411yescharacteristic
96C2×C3×C4×C5C22⋊C411yescharacteristic
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×C3×Dic5—21no—
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C22⋊C411yescharacteristic
101C2×C3×Dic5C2×D411yescharacteristic
102C3×D20—24no—
103C3×D20—21no—
104C3×D20C4212yes—
105C3×C4×D5—24no—
106C3×C4×D5—41no—
107C3×C4×D5—41no—
108C3×C4×D5—42no—
109C3×C4×D5—42no—
110C3×C4×D5C22×C412yes—
111C3×C4×D5#4112yes—
112C3×#170—24no—
113C2×C3×#41—51no—
114C2×C3×M16—52no—
115C3×C4×Q8—51no—
116C3×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
117C3×C4×D4—51no—
118C3×C4×D4—51no—
119C3×C4×D4—101no—
120C3×C4×D4—101no—
121C3×#117—51no—
122C3×#117—101no—
123C2×C3×C42—51no—
124C2×C3×C42—52no—
125C2×C3×C42—101no—
126C3×#104—58no—
127C3×#97—101no—
128C3×C4×C8—101no—
129C3×#95—101no—
130C3×#95—101no—
131C22×AGL(1,𝔽5)—22no—
132C2×C5×Q8C2×C3×C411yescharacteristic
133#713—22no—
134#713C2×C3×C414yes—
135C2×D20—21no—
136C2×D20C2×C3×C411yescharacteristic
137C2×C4×D5—21no—
138C2×C4×D5—21no—
139C2×C4×D5—21no—
140C2×C4×D5C3×D411yescharacteristic
141C2×C4×D5C3×D411yescharacteristic
142C2×C4×D5C23×C311yescharacteristic
143C2×#170—22no—
144C4×AGL(1,𝔽5)—22no—
145C4×AGL(1,𝔽5)C2×C3×C414yescomplemented
146#700—22no—
147#700C2×C3×C414yescomplemented
148C5×C4⋊C4—21no—
149C5×C4⋊C4C2×C3×C411yescharacteristic
150C42×C5—21no—
151C42×C5C2×C3×C411yescharacteristic
152#685—21no—
153#685—21no—
154#685—21no—
155C4⋊Dic5—21no—
156C4⋊Dic5C2×C3×C411yescharacteristic
157C4×Dic5—21no—
158C4×Dic5—21no—
159C4×Dic5C2×C3×C411yescharacteristic
160C4×Dic5C3×D411yescharacteristic
161C4×Dic5C3×D411yescharacteristic
162C4×#41—51no—
163C2×#104—52no—
164C4×M16—51no—
165#376—51no—
166#339—52no—
167C22×C3×C5C2×C22⋊C411yescharacteristic
168C2×C3×D5—21no—
169C2×C3×D5—21no—
170C2×C3×D5—22no—
171C2×C3×D5—41no—
172C2×C3×D5—41no—
173C2×C3×D5C4×D412yes—
174C3×AGL(1,𝔽5)—44no—
175C3×C4×C5—22no—
176C3×C4×C5—21no—
177C3×C4×C5—21no—
178C3×C4×C5—41no—
179C3×C4×C5C2×C4211yescharacteristic
180C3×C4×C5#11711yescharacteristic
181C3×Dic5—21no—
182C3×Dic5—21no—
183C3×Dic5—41no—
184C3×Dic5C4×D412yes—
185C3×#41—54no—
186C3×#41—102no—
187C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
188C2×C3×D4—51no—
189C2×C3×D4—101no—
190C22×C3×C4—51no—
191C22×C3×C4—51no—
192C22×C3×C4—51no—
193C22×C3×C4—101no—
194C22×C3×C4—101no—
195C22×C3×C4—102no—
196C22×C3×C4—101no—
197C3×M16—54no—
198C3×M16—102no—
199C2×C3×C8—102no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—51no—
202C3×C4⋊C4—101no—
203C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
204C3×C22⋊C4—101no—
205C3×C22⋊C4—101no—
206C3×C22⋊C4—101no—
207C3×C42—21no—
208C3×C42—51no—
209C3×C42—51no—
210C3×C42—51no—
211C3×C42—54no—
212C3×C42—101no—
213C3×C42—101no—
214C3×C42—102no—
215C3×C42C2×AGL(1,𝔽5)11yescharacteristic
216C22×D5—21no—
217C22×D5C2×C3×D411yescharacteristic
218C2×AGL(1,𝔽5)—24no—
219C2×AGL(1,𝔽5)—42no—
220C2×AGL(1,𝔽5)—42no—
221C5×Q8—21no—
222C5×Q8C3×C4212yes—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5C3×C22⋊C411yescharacteristic
226C2×C4×C5C3×C22⋊C411yescharacteristic
227C2×C4×C5C22×C3×C411yescharacteristic
228C2×Dic5—21no—
229C2×Dic5C3×C22⋊C411yescharacteristic
230C2×Dic5C3×C22⋊C411yescharacteristic
231C2×Dic5C2×C3×D411yescharacteristic
232D20—24no—
233D20—21no—
234D20C3×C4212yes—
235C4×D5—24no—
236C4×D5—41no—
237C4×D5—41no—
238C4×D5—42no—
239C4×D5—42no—
240C4×D5C22×C3×C412yes—
241C4×D5C3×#4112yes—
242#170—24no—
243C2×#41—51no—
244C2×M16—52no—
245C4×Q8—51no—
246C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
247C4×D4—51no—
248C4×D4—51no—
249C4×D4—101no—
250C4×D4—101no—
251#117—51no—
252#117—101no—
253C2×C42—51no—
254C2×C42—52no—
255C2×C42—101no—
256#104—58no—
257#97—101no—
258C4×C8—101no—
259#95—101no—
260#95—101no—
261C2×C3×C5C4×C22⋊C411yescharacteristic
262C2×C3×C5#42111yescharacteristic
263C2×C3×C5#42111yescharacteristic
264C3×D5—22no—
265C3×D5—42no—
266C23×C3—51no—
267C23×C3—101no—
268C3×Q8—21no—
269C3×Q8C4×AGL(1,𝔽5)12yes—
270C3×D4—52no—
271C3×D4—104no—
272C3×D4—101no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—51no—
276C2×C3×C4—52no—
277C2×C3×C4—52no—
278C2×C3×C4—51no—
279C2×C3×C4—51no—
280C2×C3×C4—104no—
281C2×C3×C4—104no—
282C2×C3×C4—101no—
283C2×C3×C4—201no—
284C2×C3×C4—201no—
285C2×C3×C4—202no—
286C2×C3×C4—202no—
287C2×C3×C4—202no—
288C2×C3×C4—202no—
289C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
290C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
291C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
292C3×C8—104no—
293C22×C5C2×C3×C22⋊C411yescharacteristic
294C2×D5—21no—
295C2×D5—21no—
296C2×D5—22no—
297C2×D5—41no—
298C2×D5—41no—
299C2×D5C3×C4×D412yes—
300AGL(1,𝔽5)—44no—
301C4×C5—22no—
302C4×C5—21no—
303C4×C5—21no—
304C4×C5—41no—
305C4×C5C2×C3×C4211yescharacteristic
306C4×C5C3×#11711yescharacteristic
307Dic5—21no—
308Dic5—21no—
309Dic5—41no—
310Dic5C3×C4×D412yes—
311#41—54no—
312#41—102no—
313C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
314C2×D4—51no—
315C2×D4—101no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—101no—
320C22×C4—101no—
321C22×C4—102no—
322C22×C4—101no—
323M16—54no—
324M16—102no—
325C2×C8—102no—
326C4⋊C4—21no—
327C4⋊C4—51no—
328C4⋊C4—101no—
329C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
330C22⋊C4—101no—
331C22⋊C4—101no—
332C22⋊C4—101no—
333C42—21no—
334C42—51no—
335C42—51no—
336C42—51no—
337C42—54no—
338C42—101no—
339C42—101no—
340C42—102no—
341C42C2×C3×AGL(1,𝔽5)11yescharacteristic
342C3×C5#175911yescharacteristic, complemented, Hall
343C22×C3—52no—
344C22×C3—101no—
345C22×C3—101no—
346C22×C3—102no—
347C22×C3—201no—
348C22×C3—201no—
349C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
350C3×C4—21no—
351C3×C4—22no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—52no—
355C3×C4—101no—
356C3×C4—101no—
357C3×C4—204no—
358C3×C4—201no—
359C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
360C3×C4#413011yescharacteristic
361C2×C5C3×C4×C22⋊C411yescharacteristic
362C2×C5C3×#42111yescharacteristic
363C2×C5C3×#42111yescharacteristic
364D5—22no—
365D5—42no—
366C23—51no—
367C23—101no—
368Q8—21no—
369Q8C3×C4×AGL(1,𝔽5)12yes—
370D4—52no—
371D4—104no—
372D4—101no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—51no—
376C2×C4—52no—
377C2×C4—52no—
378C2×C4—51no—
379C2×C4—51no—
380C2×C4—104no—
381C2×C4—104no—
382C2×C4—101no—
383C2×C4—201no—
384C2×C4—201no—
385C2×C4—202no—
386C2×C4—202no—
387C2×C4—202no—
388C2×C4—202no—
389C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
390C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
391C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
392C8—104no—
393C2×C3—102no—
394C2×C3—202no—
395C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
396C2×C3#6578411yescharacteristic
397C2×C3#6579511yescharacteristic
398C5C3×#175911yescharacteristic, complemented, Sylow
399C22—52no—
400C22—101no—
401C22—101no—
402C22—102no—
403C22—201no—
404C22—201no—
405C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
406C4—22no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—52no—
411C4—101no—
412C4—101no—
413C4—204no—
414C4—201no—
415C4C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
416C4C3×#413011yescharacteristic
417C3#1060445011yescharacteristic, direct product factor, Sylow
418C2—102no—
419C2—202no—
420C2C3×C4×C22⋊AGL(1,𝔽5)11yescharacteristic
421C2C3×#6578411yescharacteristic
422C2C3×#6579511yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC4×C5C2×C3×C42
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#713C22
cosocleC2×C4×D5C23×C3
hypercenterC3×C4×Q8AGL(1,𝔽5)
hypocenterC5C3×#1759
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#65686C2semidirect
4. C3×C4×#700C2semidirect
5. C3×#64875C2semidirect
6. #10604450C3direct (N × H)
10. C3×C4×C5×Q8C4semidirect
11. C3×Q8×Dic5C4semidirect
13. C3×#4039C4semidirect
17. C3×C4×D20C4semidirect
20. C3×#4010C4semidirect
28. #65686C2×C3semidirect
29. C4×#700C2×C3semidirect
30. #64875C2×C3semidirect
45. C3×C4×AGL(1,𝔽5)C2×C4semidirect
47. C3×#700C2×C4semidirect
70. C4×C5×Q8C3×C4semidirect
71. Q8×Dic5C3×C4semidirect
73. #4039C3×C4semidirect
77. C4×D20C3×C4semidirect
80. #4010C3×C4semidirect
116. C3×C4×Q8AGL(1,𝔽5)semidirect
145. C4×AGL(1,𝔽5)C2×C3×C4semidirect
147. #700C2×C3×C4semidirect
246. C4×Q8C3×AGL(1,𝔽5)semidirect
342. C3×C5#1759semidirect
398. C5C3×#1759semidirect
417. C3#10604450direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#7132
C2×C3×#40102
C3×#656862
C3×C4×#7002
C3×#648752
#106044503
C3×#17595