Finite Group Databasebeta

← C3×#10600814 #14300320

Subgroups — C3×#10600814 #14300320

Order 1920.

748 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×#66132C211yesmaximal, characteristic
2C2×C3×C5×#99C211yesmaximal, characteristic
3C3×#65509C211yesmaximal, characteristic
4C3×#65239C212yesmaximal, complemented
5C3×#64747C212yesmaximal, complemented
6#10600814C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C2211yescharacteristic
8C23×C3×Dic5C2211yescharacteristic
9C2×C3×C5×C22⋊C4C2212yes—
10C2×C3×C22⋊Dic5—21no—
11C2×C3×C22⋊Dic5—21no—
12C2×C3×C22⋊Dic5C2211yescharacteristic
13C2×C3×C22⋊Dic5C2212yes—
14C3×D4×Dic5—22no—
15C3×D4×Dic5—22no—
16C3×#4072—21no—
17C2×C3×C4⋊Dic5—22no—
18C3×C5×#99C414yescomplemented
19C3×#3963—21no—
20C3×#3963—22no—
21C3×#1867—51nomaximal, Hall
22#66132C2×C311yescharacteristic
23C2×C5×#99C2×C311yescharacteristic
24#65509C2×C311yescharacteristic
25#65239C2×C312yescomplemented
26#64747C2×C312yescomplemented
27C24×C3×C5D411yescharacteristic
28C24×C3×C5C2311yescharacteristic
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4C2×C412yes—
32C2×C3×C5×D4Q812yes—
33C22×C3×C4×C5—22no—
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×Dic5—21no—
36C22×C3×Dic5—22no—
37C22×C3×Dic5—21no—
38C22×C3×Dic5—21no—
39C22×C3×Dic5—21no—
40C22×C3×Dic5—42no—
41C22×C3×Dic5D412yes—
42C22×C3×Dic5D412yes—
43C3×C5×C22⋊C4—22no—
44C3×C5×C22⋊C4C2×C414yescomplemented
45C3×C22⋊Dic5—22no—
46C3×C22⋊Dic5—22no—
47C3×C22⋊Dic5—22no—
48C3×C22⋊Dic5—22no—
49C3×C22⋊Dic5—42no—
50C3×C4⋊Dic5—21no—
51C3×C4⋊Dic5—21no—
52C3×C4⋊Dic5—44no—
53C3×C4×Dic5—22no—
54C3×#518—51no—
55C2×C3×#99D511yescharacteristic
56C3×#386—51no—
57C3×#385—52no—
58C3×#342—52no—
59C22×C5×D4C22×C311yescharacteristic
60C23×Dic5C22×C311yescharacteristic
61C2×C5×C22⋊C4C22×C312yes—
62C2×C22⋊Dic5—21no—
63C2×C22⋊Dic5—21no—
64C2×C22⋊Dic5C22×C311yescharacteristic
65C2×C22⋊Dic5C22×C312yes—
66D4×Dic5—22no—
67D4×Dic5—22no—
68#4072—21no—
69C2×C4⋊Dic5—22no—
70C5×#99C3×C414yescomplemented
71#3963—21no—
72#3963—22no—
73#1867—51noSylow
74C23×C3×C5—21no—
75C23×C3×C5—21no—
76C23×C3×C5—41no—
77C23×C3×C5—41no—
78C23×C3×C5—41no—
79C23×C3×C5—41no—
80C23×C3×C5C4⋊C412yescomplemented
81C23×C3×C5C22×C411yescharacteristic
82C23×C3×C5C2×D411yescharacteristic
83C23×C3×C5C2×D412yes—
84C23×C3×C5C2×Q811yescharacteristic
85C23×C3×C5#4112yes—
86C3×C5×D4—24no—
87C3×C5×D4—24no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—24no—
91C2×C3×C4×C5—42no—
92C2×C3×C4×C5—42no—
93C2×C3×C4×C5C4⋊C412yescomplemented
94C2×C3×Dic5—24no—
95C2×C3×Dic5—22no—
96C2×C3×Dic5—22no—
97C2×C3×Dic5—22no—
98C2×C3×Dic5—42no—
99C2×C3×Dic5—41no—
100C2×C3×Dic5—41no—
101C2×C3×Dic5—42no—
102C2×C3×Dic5—42no—
103C2×C3×Dic5—41no—
104C2×C3×Dic5—41no—
105C2×C3×Dic5—41no—
106C2×C3×Dic5—81no—
107C2×C3×Dic5—81no—
108C22×C3×D4C2×D511yescharacteristic
109C23×C3×C4—51no—
110C3×C4×D4—102no—
111C3×C4×D4—102no—
112C3×#117—101no—
113C2×C3×C4⋊C4—102no—
114C2×C3×C22⋊C4—52no—
115C2×C3×C22⋊C4—51no—
116C2×C3×C22⋊C4—101no—
117C2×C3×C22⋊C4—101no—
118C2×C3×C22⋊C4C2×D512yes—
119C3×#99Dic514yescomplemented
120C3×#95—102no—
121C3×#95—101no—
122C24×C5C3×D411yescharacteristic
123C24×C5C23×C311yescharacteristic
124C2×C5×D4—22no—
125C2×C5×D4—22no—
126C2×C5×D4C2×C3×C412yes—
127C2×C5×D4C3×Q812yes—
128C22×C4×C5—22no—
129C22×C4×C5C3×D411yescharacteristic
130C22×Dic5—21no—
131C22×Dic5—22no—
132C22×Dic5—21no—
133C22×Dic5—21no—
134C22×Dic5—21no—
135C22×Dic5—42no—
136C22×Dic5C3×D412yes—
137C22×Dic5C3×D412yes—
138C5×C22⋊C4—22no—
139C5×C22⋊C4C2×C3×C414yescomplemented
140C22⋊Dic5—22no—
141C22⋊Dic5—22no—
142C22⋊Dic5—22no—
143C22⋊Dic5—22no—
144C22⋊Dic5—42no—
145C4⋊Dic5—21no—
146C4⋊Dic5—21no—
147C4⋊Dic5—44no—
148C4×Dic5—22no—
149#518—51no—
150C2×#99C3×D511yescharacteristic
151#386—51no—
152#385—52no—
153#342—52no—
154C22×C3×C5—22no—
155C22×C3×C5—21no—
156C22×C3×C5—22no—
157C22×C3×C5—21no—
158C22×C3×C5—22no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5—42no—
162C22×C3×C5—42no—
163C22×C3×C5—41no—
164C22×C3×C5—41no—
165C22×C3×C5—81no—
166C22×C3×C5C2×C4⋊C411yescharacteristic
167C22×C3×C5C4×D412yes—
168C22×C3×C5#12011yescharacteristic
169C22×C3×C5C22⋊Q812yes—
170C22×C3×C5#12311yescharacteristic
171C3×C4×C5—22no—
172C3×C4×C5—44no—
173C3×Dic5—42no—
174C3×Dic5—42no—
175C3×Dic5—42no—
176C3×Dic5—81no—
177C24×C3D2011yescharacteristic
178C24×C3C22×D511yescharacteristic
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4Dic1012yes—
182C2×C3×D4C2×Dic512yes—
183C22×C3×C4—22no—
184C22×C3×C4—52no—
185C22×C3×C4—52no—
186C22×C3×C4—101no—
187C22×C3×C4—102no—
188C22×C3×C4—101no—
189C22×C3×C4—101no—
190C22×C3×C4—101no—
191C22×C3×C4—202no—
192C22×C3×C4D2011yescharacteristic
193C3×C4⋊C4—101no—
194C3×C4⋊C4—101no—
195C3×C4⋊C4—204no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—102no—
198C3×C22⋊C4—102no—
199C3×C22⋊C4—102no—
200C3×C22⋊C4—102no—
201C3×C22⋊C4—202no—
202C3×C22⋊C4C2×Dic514yescomplemented
203C3×C42—102no—
204C23×C5—21no—
205C23×C5—21no—
206C23×C5—41no—
207C23×C5—41no—
208C23×C5—41no—
209C23×C5—41no—
210C23×C5C3×C4⋊C412yescomplemented
211C23×C5C22×C3×C411yescharacteristic
212C23×C5C2×C3×D411yescharacteristic
213C23×C5C2×C3×D412yes—
214C23×C5C2×C3×Q811yescharacteristic
215C23×C5C3×#4112yes—
216C5×D4—24no—
217C5×D4—24no—
218C2×C4×C5—21no—
219C2×C4×C5—21no—
220C2×C4×C5—24no—
221C2×C4×C5—42no—
222C2×C4×C5—42no—
223C2×C4×C5C3×C4⋊C412yescomplemented
224C2×Dic5—24no—
225C2×Dic5—22no—
226C2×Dic5—22no—
227C2×Dic5—22no—
228C2×Dic5—42no—
229C2×Dic5—41no—
230C2×Dic5—41no—
231C2×Dic5—42no—
232C2×Dic5—42no—
233C2×Dic5—41no—
234C2×Dic5—41no—
235C2×Dic5—41no—
236C2×Dic5—81no—
237C2×Dic5—81no—
238C22×D4C2×C3×D511yescharacteristic
239C23×C4—51no—
240C4×D4—102no—
241C4×D4—102no—
242#117—101no—
243C2×C4⋊C4—102no—
244C2×C22⋊C4—52no—
245C2×C22⋊C4—51no—
246C2×C22⋊C4—101no—
247C2×C22⋊C4—101no—
248C2×C22⋊C4C2×C3×D512yes—
249#99C3×Dic514yescomplemented
250#95—102no—
251#95—101no—
252C2×C3×C5—21no—
253C2×C3×C5—22no—
254C2×C3×C5—21no—
255C2×C3×C5—22no—
256C2×C3×C5—42no—
257C2×C3×C5#38511yescharacteristic
258C2×C3×C5B(4,𝔽2)11yescharacteristic
259C2×C3×C5#45811yescharacteristic
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3C4⋊Dic512yescomplemented
267C23×C3C2×Dic1011yescharacteristic
268C23×C3C2×D2011yescharacteristic
269C23×C3D4×D512yes—
270C23×C3#71112yes—
271C23×C3C22×Dic511yescharacteristic
272C3×D4—24no—
273C3×D4—24no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—102no—
280C2×C3×C4—104no—
281C2×C3×C4—102no—
282C2×C3×C4—102no—
283C2×C3×C4—202no—
284C2×C3×C4—201no—
285C2×C3×C4—201no—
286C2×C3×C4—202no—
287C2×C3×C4—202no—
288C2×C3×C4—201no—
289C2×C3×C4—201no—
290C2×C3×C4—201no—
291C2×C3×C4—401no—
292C2×C3×C4—401no—
293C2×C3×C4C4⋊Dic512yescomplemented
294C22×C5—22no—
295C22×C5—21no—
296C22×C5—22no—
297C22×C5—21no—
298C22×C5—22no—
299C22×C5—41no—
300C22×C5—41no—
301C22×C5—42no—
302C22×C5—42no—
303C22×C5—41no—
304C22×C5—41no—
305C22×C5—81no—
306C22×C5C2×C3×C4⋊C411yescharacteristic
307C22×C5C3×C4×D412yes—
308C22×C5C3×#12011yescharacteristic
309C22×C5C3×C22⋊Q812yes—
310C22×C5C3×#12311yescharacteristic
311C4×C5—22no—
312C4×C5—44no—
313Dic5—42no—
314Dic5—42no—
315Dic5—42no—
316Dic5—81no—
317C24C3×D2011yescharacteristic
318C24C22×C3×D511yescharacteristic
319C2×D4—22no—
320C2×D4—22no—
321C2×D4C3×Dic1012yes—
322C2×D4C2×C3×Dic512yes—
323C22×C4—22no—
324C22×C4—52no—
325C22×C4—52no—
326C22×C4—101no—
327C22×C4—102no—
328C22×C4—101no—
329C22×C4—101no—
330C22×C4—101no—
331C22×C4—202no—
332C22×C4C3×D2011yescharacteristic
333C4⋊C4—101no—
334C4⋊C4—101no—
335C4⋊C4—204no—
336C22⋊C4—22no—
337C22⋊C4—102no—
338C22⋊C4—102no—
339C22⋊C4—102no—
340C22⋊C4—102no—
341C22⋊C4—202no—
342C22⋊C4C2×C3×Dic514yescomplemented
343C42—102no—
344C3×C5#186711yescharacteristic, complemented, Hall
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—41no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—81no—
357C22×C3#402412yes—
358C22×C3#402811yescharacteristic
359C22×C3#403211yescharacteristic
360C22×C3C2×C4⋊Dic511yescharacteristic
361C22×C3D4×Dic512yes—
362C3×C4—22no—
363C3×C4—44no—
364C3×C4—202no—
365C3×C4—202no—
366C3×C4—202no—
367C3×C4—401no—
368C2×C5—21no—
369C2×C5—22no—
370C2×C5—21no—
371C2×C5—22no—
372C2×C5—42no—
373C2×C5C3×#38511yescharacteristic
374C2×C5C3×B(4,𝔽2)11yescharacteristic
375C2×C5C3×#45811yescharacteristic
376C23—21no—
377C23—21no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23C3×C4⋊Dic512yescomplemented
383C23C2×C3×Dic1011yescharacteristic
384C23C2×C3×D2011yescharacteristic
385C23C3×D4×D512yes—
386C23C3×#71112yes—
387C23C22×C3×Dic511yescharacteristic
388D4—24no—
389D4—24no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—102no—
396C2×C4—104no—
397C2×C4—102no—
398C2×C4—102no—
399C2×C4—202no—
400C2×C4—201no—
401C2×C4—201no—
402C2×C4—202no—
403C2×C4—202no—
404C2×C4—201no—
405C2×C4—201no—
406C2×C4—201no—
407C2×C4—401no—
408C2×C4—401no—
409C2×C4C3×C4⋊Dic512yescomplemented
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—22no—
414C2×C3—42no—
415C2×C3#6503011yescharacteristic
416C2×C3#6503111yescharacteristic
417C2×C3#6523911yescharacteristic
418C5C3×#186711yescharacteristic, complemented, Sylow
419C22—22no—
420C22—21no—
421C22—22no—
422C22—21no—
423C22—22no—
424C22—41no—
425C22—42no—
426C22—41no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22—81no—
431C22C3×#402412yes—
432C22C3×#402811yescharacteristic
433C22C3×#403211yescharacteristic
434C22C2×C3×C4⋊Dic511yescharacteristic
435C22C3×D4×Dic512yes—
436C4—22no—
437C4—44no—
438C4—202no—
439C4—202no—
440C4—202no—
441C4—401no—
442C3#1060081411yescharacteristic, direct product factor, Sylow
443C2—22no—
444C2—21no—
445C2—21no—
446C2—22no—
447C2—42no—
448C2C3×#6503011yescharacteristic
449C2C3×#6503111yescharacteristic
450C2C3×#6523911yescharacteristic
subgroupG/H
centerC22×C3#4028
commutatorC23×C5C22×C3×C4
FrattiniC24C22×C3×D5
FittingC2×C3×C5×#99C2
socleC22×C3×C5#120
prime radicalC22×C3×C5×D4C22
cosocleC24×C5C23×C3
hypercenterC2×C3×#99D5
hypocenterC5C3×#1867
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×#99

Complements

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

NH ≅ G/Ntype
4. C3×#65239C2semidirect
5. C3×#64747C2semidirect
6. #10600814C3direct (N × H)
18. C3×C5×#99C4semidirect
25. #65239C2×C3semidirect
26. #64747C2×C3semidirect
44. C3×C5×C22⋊C4C2×C4semidirect
70. C5×#99C3×C4semidirect
80. C23×C3×C5C4⋊C4semidirect
93. C2×C3×C4×C5C4⋊C4semidirect
119. C3×#99Dic5semidirect
139. C5×C22⋊C4C2×C3×C4semidirect
202. C3×C22⋊C4C2×Dic5semidirect
210. C23×C5C3×C4⋊C4semidirect
223. C2×C4×C5C3×C4⋊C4semidirect
249. #99C3×Dic5semidirect
266. C23×C3C4⋊Dic5semidirect
293. C2×C3×C4C4⋊Dic5semidirect
342. C22⋊C4C2×C3×Dic5semidirect
344. C3×C5#1867semidirect
382. C23C3×C4⋊Dic5semidirect
409. C2×C4C3×C4⋊Dic5semidirect
418. C5C3×#1867semidirect
442. C3#10600814direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661322
C2×C3×C5×#992
C3×#655092
C3×#652392
C3×#647472
#106008143
C3×#18675