Finite Group Databasebeta

← C3×#10604320 #14300479

Subgroups — C3×#10604320 #14300479

Order 1920.

816 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#700C211yesmaximal, characteristic, complemented
2C2×C3×#4039C211yesmaximal, characteristic
3C2×C3×#4003C211yesmaximal, characteristic, complemented
4C2×C3×#4003C212yesmaximal, complemented
5C3×#64918C212yesmaximal, complemented
6#10604320C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×D20C411yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C22×C3×C4×D5C2211yescharacteristic
10C22×C3×#170C2212yes—
11C22×C3×#170C2212yes—
12C2×C3×#700—22no—
13C2×C3×#700—24no—
14C2×C3×C5×C4⋊C4C411yescharacteristic
15C2×C3×#685C411yescharacteristic
16C2×C3×#685C411yescharacteristic
17C2×C3×C4×Dic5C2211yescharacteristic
18C3×#4039—22no—
19C3×#4039—22no—
20C3×#4003—22no—
21C3×#4003—24no—
22C3×#1870—51nomaximal, Hall
23C22×#700C2×C311yescharacteristic, complemented
24C2×#4039C2×C311yescharacteristic
25C2×#4003C2×C311yescharacteristic, complemented
26C2×#4003C2×C312yescomplemented
27#64918C2×C312yescomplemented
28C23×C3×D5C2×C411yescharacteristic
29C23×C3×D5C2×C411yescharacteristic
30C22×C3×C4×C5C2×C411yescharacteristic
31C22×C3×C4×C5C2×C411yescharacteristic
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×Dic5C2×C411yescharacteristic
34C22×C3×Dic5C2311yescharacteristic
35C2×C3×D20—22no—
36C2×C3×D20—22no—
37C2×C3×D20—22no—
38C2×C3×C4×D5—22no—
39C2×C3×C4×D5—22no—
40C2×C3×C4×D5—22no—
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—22no—
44C2×C3×C4×D5D412yes—
45C2×C3×C4×D5D412yes—
46C2×C3×#170—22no—
47C2×C3×#170—22no—
48C2×C3×#170—22no—
49C2×C3×#170—22no—
50C2×C3×#170—22no—
51C2×C3×#170D414yescomplemented
52C3×#700—28no—
53C3×C5×C4⋊C4—22no—
54C3×#685—22no—
55C3×#685—22no—
56C3×C4×Dic5—21no—
57C3×C4×Dic5—21no—
58C22×C3×M16—51no—
59C2×C3×C4×D4—51no—
60C2×C3×C22⋊C8—52no—
61C2×C3×C22⋊C8—51no—
62C3×#336—52no—
63C22×D20C3×C411yescharacteristic
64C22×C4×D5C22×C311yescharacteristic
65C22×C4×D5C22×C311yescharacteristic
66C22×#170C22×C312yes—
67C22×#170C22×C312yes—
68C2×#700—22no—
69C2×#700—24no—
70C2×C5×C4⋊C4C3×C411yescharacteristic
71C2×#685C3×C411yescharacteristic
72C2×#685C3×C411yescharacteristic
73C2×C4×Dic5C22×C311yescharacteristic
74#4039—22no—
75#4039—22no—
76#4003—22no—
77#4003—24no—
78#1870—51noSylow
79C23×C3×C5C22×C411yescharacteristic
80C22×C3×D5—21no—
81C22×C3×D5—22no—
82C22×C3×D5—22no—
83C22×C3×D5—21no—
84C22×C3×D5—21no—
85C22×C3×D5—21no—
86C22×C3×D5—22no—
87C22×C3×D5—22no—
88C22×C3×D5C22⋊C412yes—
89C22×C3×D5M1612yescomplemented
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—22no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5C22⋊C412yes—
96C2×C3×C4×C5M1612yescomplemented
97C2×C3×Dic5—21no—
98C2×C3×Dic5—22no—
99C2×C3×Dic5C2×D412yes—
100C2×C3×Dic5C2×D411yescharacteristic
101C2×C3×Dic5C2×D411yescharacteristic
102C2×C3×Dic5#4112yes—
103C3×D20—28no—
104C3×C4×D5—22no—
105C3×C4×D5—24no—
106C3×C4×D5—22no—
107C3×C4×D5—41no—
108C3×C4×D5—42no—
109C3×C4×D5—41no—
110C3×#170—24no—
111C3×#170—42no—
112C22×C3×D4—51no—
113C23×C3×C4—51no—
114C23×C3×C4—51no—
115C2×C3×M16—104no—
116C2×C3×M16—102no—
117C22×C3×C8—52no—
118C22×C3×C8—52no—
119C3×C4×D4—102no—
120C3×C4×D4—102no—
121C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic
122C2×C3×C22⋊C4—51no—
123C2×C3×C22⋊C4—51no—
124C2×C3×C42—51no—
125C3×C22⋊C8—104no—
126C3×C22⋊C8—102no—
127C23×D5C2×C3×C411yescharacteristic
128C23×D5C2×C3×C411yescharacteristic
129C22×C4×C5C2×C3×C411yescharacteristic
130C22×C4×C5C2×C3×C411yescharacteristic
131C22×C4×C5C2×C3×C411yescharacteristic
132C22×Dic5C2×C3×C411yescharacteristic
133C22×Dic5C23×C311yescharacteristic
134C2×D20—22no—
135C2×D20—22no—
136C2×D20—22no—
137C2×C4×D5—22no—
138C2×C4×D5—22no—
139C2×C4×D5—22no—
140C2×C4×D5—21no—
141C2×C4×D5—21no—
142C2×C4×D5—22no—
143C2×C4×D5C3×D412yes—
144C2×C4×D5C3×D412yes—
145C2×#170—22no—
146C2×#170—22no—
147C2×#170—22no—
148C2×#170—22no—
149C2×#170—22no—
150C2×#170C3×D414yescomplemented
151#700—28no—
152C5×C4⋊C4—22no—
153#685—22no—
154#685—22no—
155C4×Dic5—21no—
156C4×Dic5—21no—
157C22×M16—51no—
158C2×C4×D4—51no—
159C2×C22⋊C8—52no—
160C2×C22⋊C8—51no—
161#336—52no—
162C22×C3×C5C2×C22⋊C411yescharacteristic
163C22×C3×C5C4×D412yes—
164C22×C3×C5C2×M1611yescharacteristic
165C22×C3×C5C2×M1611yescharacteristic
166C22×C3×C5#13112yes—
167C2×C3×D5—22no—
168C2×C3×D5—22no—
169C2×C3×D5—22no—
170C2×C3×D5—22no—
171C2×C3×D5—24no—
172C2×C3×D5—41no—
173C2×C3×D5—41no—
174C2×C3×D5—42no—
175C2×C3×D5—42no—
176C2×C3×D5—42no—
177C3×C4×C5—22no—
178C3×C4×C5—22no—
179C3×C4×C5—41no—
180C3×Dic5—41no—
181C3×Dic5#12011yescharacteristic
182C3×Dic5#12112yes—
183C3×Dic5#12311yescharacteristic
184C24×C3—51no—
185C24×C3—51no—
186C2×C3×D4—102no—
187C2×C3×D4—102no—
188C2×C3×D4—102no—
189C22×C3×C4—51no—
190C22×C3×C4—52no—
191C22×C3×C4—52no—
192C22×C3×C4—51no—
193C22×C3×C4—102no—
194C22×C3×C4—102no—
195C22×C3×C4—102no—
196C22×C3×C4—101no—
197C22×C3×C4—101no—
198C22×C3×C4—102no—
199C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
200C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
201C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
202C3×M16—108no—
203C2×C3×C8—54no—
204C2×C3×C8—102no—
205C2×C3×C8—102no—
206C2×C3×C8—102no—
207C2×C3×C8—102no—
208C2×C3×C8—102no—
209C3×C4⋊C4—22no—
210C3×C22⋊C4—102no—
211C3×C22⋊C4—102no—
212C3×C42—101no—
213C3×C42—101no—
214C23×C5C22×C3×C411yescharacteristic
215C22×D5—21no—
216C22×D5—22no—
217C22×D5—22no—
218C22×D5—21no—
219C22×D5—21no—
220C22×D5—21no—
221C22×D5—22no—
222C22×D5—22no—
223C22×D5C3×C22⋊C412yes—
224C22×D5C3×M1612yescomplemented
225C2×C4×C5—21no—
226C2×C4×C5—22no—
227C2×C4×C5—21no—
228C2×C4×C5—21no—
229C2×C4×C5—22no—
230C2×C4×C5C3×C22⋊C412yes—
231C2×C4×C5C3×M1612yescomplemented
232C2×Dic5—21no—
233C2×Dic5—22no—
234C2×Dic5C2×C3×D412yes—
235C2×Dic5C2×C3×D411yescharacteristic
236C2×Dic5C2×C3×D411yescharacteristic
237C2×Dic5C3×#4112yes—
238D20—28no—
239C4×D5—22no—
240C4×D5—24no—
241C4×D5—22no—
242C4×D5—41no—
243C4×D5—42no—
244C4×D5—41no—
245#170—24no—
246#170—42no—
247C22×D4—51no—
248C23×C4—51no—
249C23×C4—51no—
250C2×M16—104no—
251C2×M16—102no—
252C22×C8—52no—
253C22×C8—52no—
254C4×D4—102no—
255C4×D4—102no—
256C2×C4⋊C4C3×AGL(1,𝔽5)11yescharacteristic
257C2×C22⋊C4—51no—
258C2×C22⋊C4—51no—
259C2×C42—51no—
260C22⋊C8—104no—
261C22⋊C8—102no—
262C2×C3×C5#38611yescharacteristic
263C2×C3×C5#40711yescharacteristic
264C2×C3×C5#40811yescharacteristic
265C2×C3×C5#43512yes—
266C2×C3×C5#43612yes—
267C3×D5—42no—
268C3×D5—42no—
269C23×C3—52no—
270C23×C3—52no—
271C23×C3—101no—
272C23×C3—101no—
273C23×C3—101no—
274C23×C3—102no—
275C23×C3—102no—
276C23×C3—102no—
277C23×C3—102no—
278C23×C3—101no—
279C23×C3C22×AGL(1,𝔽5)11yescharacteristic
280C3×D4—108no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—52no—
287C2×C3×C4—52no—
288C2×C3×C4—51no—
289C2×C3×C4—51no—
290C2×C3×C4—102no—
291C2×C3×C4—104no—
292C2×C3×C4—102no—
293C2×C3×C4—102no—
294C2×C3×C4—101no—
295C2×C3×C4—201no—
296C2×C3×C4—202no—
297C2×C3×C4—201no—
298C2×C3×C4#70012yescomplemented
299C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
300C3×C8—104no—
301C3×C8—202no—
302C22×C5C2×C3×C22⋊C411yescharacteristic
303C22×C5C3×C4×D412yes—
304C22×C5C2×C3×M1611yescharacteristic
305C22×C5C2×C3×M1611yescharacteristic
306C22×C5C3×#13112yes—
307C2×D5—22no—
308C2×D5—22no—
309C2×D5—22no—
310C2×D5—22no—
311C2×D5—24no—
312C2×D5—41no—
313C2×D5—41no—
314C2×D5—42no—
315C2×D5—42no—
316C2×D5—42no—
317C4×C5—22no—
318C4×C5—22no—
319C4×C5—41no—
320Dic5—41no—
321Dic5C3×#12011yescharacteristic
322Dic5C3×#12112yes—
323Dic5C3×#12311yescharacteristic
324C24—51no—
325C24—51no—
326C2×D4—102no—
327C2×D4—102no—
328C2×D4—102no—
329C22×C4—51no—
330C22×C4—52no—
331C22×C4—52no—
332C22×C4—51no—
333C22×C4—102no—
334C22×C4—102no—
335C22×C4—102no—
336C22×C4—101no—
337C22×C4—101no—
338C22×C4—102no—
339C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
340C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
341C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
342M16—108no—
343C2×C8—54no—
344C2×C8—102no—
345C2×C8—102no—
346C2×C8—102no—
347C2×C8—102no—
348C2×C8—102no—
349C4⋊C4—22no—
350C22⋊C4—102no—
351C22⋊C4—102no—
352C42—101no—
353C42—101no—
354C3×C5#187011yescharacteristic, complemented, Hall
355C22×C3—104no—
356C22×C3—102no—
357C22×C3—102no—
358C22×C3—102no—
359C22×C3—102no—
360C22×C3—202no—
361C22×C3—202no—
362C22×C3—202no—
363C22×C3—201no—
364C22×C3—201no—
365C22×C3C2×#70011yescharacteristic
366C22×C3D5⋊M1611yescharacteristic
367C22×C3D4×AGL(1,𝔽5)12yes—
368C22×C3#413312yes—
369C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—41no—
373C3×C4—51no—
374C3×C4—52no—
375C3×C4—51no—
376C3×C4—201no—
377C2×C5C3×#38611yescharacteristic
378C2×C5C3×#40711yescharacteristic
379C2×C5C3×#40811yescharacteristic
380C2×C5C3×#43512yes—
381C2×C5C3×#43612yes—
382D5—42no—
383D5—42no—
384C23—52no—
385C23—52no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23—102no—
390C23—102no—
391C23—102no—
392C23—102no—
393C23—101no—
394C23C22×C3×AGL(1,𝔽5)11yescharacteristic
395D4—108no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—52no—
402C2×C4—52no—
403C2×C4—51no—
404C2×C4—51no—
405C2×C4—102no—
406C2×C4—104no—
407C2×C4—102no—
408C2×C4—102no—
409C2×C4—101no—
410C2×C4—201no—
411C2×C4—202no—
412C2×C4—201no—
413C2×C4C3×#70012yescomplemented
414C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
415C8—104no—
416C8—202no—
417C2×C3—202no—
418C2×C3—202no—
419C2×C3#6570412yes—
420C2×C3#6570512yes—
421C2×C3#6575611yescharacteristic
422C2×C3#6577911yescharacteristic
423C2×C3#6578711yescharacteristic
424C5C3×#187011yescharacteristic, complemented, Sylow
425C22—104no—
426C22—102no—
427C22—102no—
428C22—102no—
429C22—102no—
430C22—202no—
431C22—202no—
432C22—202no—
433C22—201no—
434C22—201no—
435C22C2×C3×#70011yescharacteristic
436C22C3×D5⋊M1611yescharacteristic
437C22C3×D4×AGL(1,𝔽5)12yes—
438C22C3×#413312yes—
439C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
440C4—22no—
441C4—22no—
442C4—41no—
443C4—51no—
444C4—52no—
445C4—51no—
446C4—201no—
447C3#1060432011yescharacteristic, direct product factor, Sylow
448C2—202no—
449C2—202no—
450C2C3×#6570412yes—
451C2C3×#6570512yes—
452C2C3×#6575611yescharacteristic
453C2C3×#6577911yescharacteristic
454C2C3×#6578711yescharacteristic
subgroupG/H
centerC23×C3C22×AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C4⋊C4C4
socleC23×C3×C5C22×C4
prime radicalC22×C3×D20C4
cosocleC22×Dic5C23×C3
hypercenterC2×C3×C4⋊C4AGL(1,𝔽5)
hypocenterC5C3×#1870
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#700C2semidirect
3. C2×C3×#4003C2semidirect
4. C2×C3×#4003C2semidirect
5. C3×#64918C2semidirect
6. #10604320C3direct (N × H)
23. C22×#700C2×C3semidirect
25. C2×#4003C2×C3semidirect
26. C2×#4003C2×C3semidirect
27. #64918C2×C3semidirect
51. C2×C3×#170D4semidirect
89. C22×C3×D5M16semidirect
96. C2×C3×C4×C5M16semidirect
150. C2×#170C3×D4semidirect
224. C22×D5C3×M16semidirect
231. C2×C4×C5C3×M16semidirect
298. C2×C3×C4#700semidirect
354. C3×C5#1870semidirect
413. C2×C4C3×#700semidirect
424. C5C3×#1870semidirect
447. C3#10604320direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7002
C2×C3×#40392
C2×C3×#40032
C3×#649182
#106043203
C3×#18705