Finite Group Databasebeta

← C3×#10604947 #14300477

Subgroups — C3×#10604947 #14300477

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×#704C211yesmaximal, characteristic, complemented
2C2×C3×D4×Dic5C211yesmaximal, characteristic
3C2×C3×C22⋊#170C211yesmaximal, characteristic, complemented
4C2×C3×C22⋊#170C212yesmaximal, complemented
5C3×#64918C212yesmaximal, complemented
6#10604947C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C411yescharacteristic
8C23×C3×Dic5C2211yescharacteristic
9C23×C3×Dic5C2211yescharacteristic
10C2×C3×#704—22no—
11C2×C3×#704—24no—
12C22×C3×#170C2212yes—
13C22×C3×#170C2212yes—
14C2×C3×C22⋊Dic5C411yescharacteristic
15C2×C3×C22⋊Dic5C411yescharacteristic
16C3×D4×Dic5—22no—
17C3×D4×Dic5—22no—
18C2×C3×C4⋊Dic5C411yescharacteristic
19C2×C3×C4×Dic5C2211yescharacteristic
20C3×C22⋊#170—22no—
21C3×C22⋊#170—24no—
22C3×#1870—51nomaximal, Hall
23C22×#704C2×C311yescharacteristic, complemented
24C2×D4×Dic5C2×C311yescharacteristic
25C2×C22⋊#170C2×C311yescharacteristic, complemented
26C2×C22⋊#170C2×C312yescomplemented
27#64918C2×C312yescomplemented
28C24×C3×C5C2×C411yescharacteristic
29C24×C3×C5C2×C411yescharacteristic
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—22no—
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×C3×Dic5—22no—
35C22×C3×Dic5—22no—
36C22×C3×Dic5—21no—
37C22×C3×Dic5—21no—
38C22×C3×Dic5—22no—
39C22×C3×Dic5—22no—
40C22×C3×Dic5C2×C411yescharacteristic
41C22×C3×Dic5C2×C411yescharacteristic
42C22×C3×Dic5C2×C411yescharacteristic
43C22×C3×Dic5D412yes—
44C22×C3×Dic5D412yes—
45C22×C3×Dic5C2311yescharacteristic
46C3×#704—28no—
47C2×C3×#170—22no—
48C2×C3×#170—22no—
49C2×C3×#170—22no—
50C2×C3×#170—22no—
51C2×C3×#170—22no—
52C2×C3×#170D414yescomplemented
53C3×C22⋊Dic5—22no—
54C3×C22⋊Dic5—22no—
55C3×C4⋊Dic5—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×C5×D4C3×C411yescharacteristic
64C23×Dic5C22×C311yescharacteristic
65C23×Dic5C22×C311yescharacteristic
66C2×#704—22no—
67C2×#704—24no—
68C22×#170C22×C312yes—
69C22×#170C22×C312yes—
70C2×C22⋊Dic5C3×C411yescharacteristic
71C2×C22⋊Dic5C3×C411yescharacteristic
72D4×Dic5—22no—
73D4×Dic5—22no—
74C2×C4⋊Dic5C3×C411yescharacteristic
75C2×C4×Dic5C22×C311yescharacteristic
76C22⋊#170—22no—
77C22⋊#170—24no—
78#1870—51noSylow
79C23×C3×C5—21no—
80C23×C3×C5—22no—
81C23×C3×C5—22no—
82C23×C3×C5—21no—
83C23×C3×C5—21no—
84C23×C3×C5—21no—
85C23×C3×C5—22no—
86C23×C3×C5—22no—
87C23×C3×C5C22⋊C412yes—
88C23×C3×C5M1612yescomplemented
89C23×C3×C5C22×C411yescharacteristic
90C3×C5×D4—28no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5M1612yescomplemented
94C2×C3×Dic5—22no—
95C2×C3×Dic5—22no—
96C2×C3×Dic5—24no—
97C2×C3×Dic5—22no—
98C2×C3×Dic5—21no—
99C2×C3×Dic5—21no—
100C2×C3×Dic5—22no—
101C2×C3×Dic5—22no—
102C2×C3×Dic5—41no—
103C2×C3×Dic5—42no—
104C2×C3×Dic5—41no—
105C2×C3×Dic5C22⋊C412yes—
106C2×C3×Dic5C2×D412yes—
107C2×C3×Dic5C2×D411yescharacteristic
108C2×C3×Dic5C2×D411yescharacteristic
109C2×C3×Dic5#4112yes—
110C3×#170—24no—
111C3×#170—42no—
112C22×C3×D4AGL(1,𝔽5)11yescharacteristic
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⋊C4—51no—
122C2×C3×C22⋊C4—51no—
123C2×C3×C22⋊C4—51no—
124C2×C3×C42—51no—
125C3×C22⋊C8—104no—
126C3×C22⋊C8—102no—
127C24×C5C2×C3×C411yescharacteristic
128C24×C5C2×C3×C411yescharacteristic
129C2×C5×D4—22no—
130C2×C5×D4—22no—
131C2×C5×D4—22no—
132C22×C4×C5C2×C3×C411yescharacteristic
133C22×Dic5—22no—
134C22×Dic5—22no—
135C22×Dic5—22no—
136C22×Dic5—21no—
137C22×Dic5—21no—
138C22×Dic5—22no—
139C22×Dic5C2×C3×C411yescharacteristic
140C22×Dic5C2×C3×C411yescharacteristic
141C22×Dic5C2×C3×C411yescharacteristic
142C22×Dic5C3×D412yes—
143C22×Dic5C3×D412yes—
144C22×Dic5C23×C311yescharacteristic
145#704—28no—
146C2×#170—22no—
147C2×#170—22no—
148C2×#170—22no—
149C2×#170—22no—
150C2×#170—22no—
151C2×#170C3×D414yescomplemented
152C22⋊Dic5—22no—
153C22⋊Dic5—22no—
154C4⋊Dic5—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×C5—22no—
163C22×C3×C5—22no—
164C22×C3×C5—22no—
165C22×C3×C5—22no—
166C22×C3×C5—24no—
167C22×C3×C5—41no—
168C22×C3×C5—41no—
169C22×C3×C5—42no—
170C22×C3×C5—42no—
171C22×C3×C5—42no—
172C22×C3×C5C2×C22⋊C411yescharacteristic
173C22×C3×C5C4×D412yes—
174C22×C3×C5C2×M1611yescharacteristic
175C22×C3×C5C2×M1611yescharacteristic
176C22×C3×C5#13112yes—
177C3×C4×C5—22no—
178C3×Dic5—22no—
179C3×Dic5—41no—
180C3×Dic5—41no—
181C3×Dic5#12011yescharacteristic
182C3×Dic5#12112yes—
183C3×Dic5#12311yescharacteristic
184C24×C3C2×AGL(1,𝔽5)11yescharacteristic
185C24×C3C2×AGL(1,𝔽5)11yescharacteristic
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C22×C3×C4—51no—
190C22×C3×C4—52no—
191C22×C3×C4—52no—
192C22×C3×C4—51no—
193C22×C3×C4—51no—
194C22×C3×C4—51no—
195C22×C3×C4—102no—
196C22×C3×C4—102no—
197C22×C3×C4—102no—
198C22×C3×C4—101no—
199C22×C3×C4—101no—
200C22×C3×C4—102no—
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—102no—
210C3×C22⋊C4—102no—
211C3×C22⋊C4—102no—
212C3×C42—101no—
213C3×C42—101no—
214C23×C5—21no—
215C23×C5—22no—
216C23×C5—22no—
217C23×C5—21no—
218C23×C5—21no—
219C23×C5—21no—
220C23×C5—22no—
221C23×C5—22no—
222C23×C5C3×C22⋊C412yes—
223C23×C5C3×M1612yescomplemented
224C23×C5C22×C3×C411yescharacteristic
225C5×D4—28no—
226C2×C4×C5—21no—
227C2×C4×C5—21no—
228C2×C4×C5C3×M1612yescomplemented
229C2×Dic5—22no—
230C2×Dic5—22no—
231C2×Dic5—24no—
232C2×Dic5—22no—
233C2×Dic5—21no—
234C2×Dic5—21no—
235C2×Dic5—22no—
236C2×Dic5—22no—
237C2×Dic5—41no—
238C2×Dic5—42no—
239C2×Dic5—41no—
240C2×Dic5C3×C22⋊C412yes—
241C2×Dic5C2×C3×D412yes—
242C2×Dic5C2×C3×D411yescharacteristic
243C2×Dic5C2×C3×D411yescharacteristic
244C2×Dic5C3×#4112yes—
245#170—24no—
246#170—42no—
247C22×D4C3×AGL(1,𝔽5)11yescharacteristic
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⋊C4—51no—
257C2×C22⋊C4—51no—
258C2×C22⋊C4—51no—
259C2×C42—51no—
260C22⋊C8—104no—
261C22⋊C8—102no—
262C2×C3×C5—42no—
263C2×C3×C5—42no—
264C2×C3×C5#38611yescharacteristic
265C2×C3×C5#40711yescharacteristic
266C2×C3×C5#40811yescharacteristic
267C2×C3×C5#43512yes—
268C2×C3×C5#43612yes—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3#70412yescomplemented
278C23×C3C22⋊AGL(1,𝔽5)12yes—
279C23×C3C22×AGL(1,𝔽5)11yescharacteristic
280C3×D4—28no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—52no—
284C2×C3×C4—52no—
285C2×C3×C4—51no—
286C2×C3×C4—51no—
287C2×C3×C4—52no—
288C2×C3×C4—102no—
289C2×C3×C4—104no—
290C2×C3×C4—102no—
291C2×C3×C4—102no—
292C2×C3×C4—102no—
293C2×C3×C4—102no—
294C2×C3×C4—101no—
295C2×C3×C4—101no—
296C2×C3×C4—201no—
297C2×C3×C4—202no—
298C2×C3×C4—201no—
299C2×C3×C4#70412yescomplemented
300C3×C8—104no—
301C3×C8—202no—
302C22×C5—22no—
303C22×C5—22no—
304C22×C5—22no—
305C22×C5—22no—
306C22×C5—24no—
307C22×C5—41no—
308C22×C5—41no—
309C22×C5—42no—
310C22×C5—42no—
311C22×C5—42no—
312C22×C5C2×C3×C22⋊C411yescharacteristic
313C22×C5C3×C4×D412yes—
314C22×C5C2×C3×M1611yescharacteristic
315C22×C5C2×C3×M1611yescharacteristic
316C22×C5C3×#13112yes—
317C4×C5—22no—
318Dic5—22no—
319Dic5—41no—
320Dic5—41no—
321Dic5C3×#12011yescharacteristic
322Dic5C3×#12112yes—
323Dic5C3×#12311yescharacteristic
324C24C2×C3×AGL(1,𝔽5)11yescharacteristic
325C24C2×C3×AGL(1,𝔽5)11yescharacteristic
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C22×C4—51no—
330C22×C4—52no—
331C22×C4—52no—
332C22×C4—51no—
333C22×C4—51no—
334C22×C4—51no—
335C22×C4—102no—
336C22×C4—102no—
337C22×C4—102no—
338C22×C4—101no—
339C22×C4—101no—
340C22×C4—102no—
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—102no—
350C22⋊C4—102no—
351C22⋊C4—102no—
352C42—101no—
353C42—101no—
354C3×C5#187011yescharacteristic, complemented, Hall
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—24no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3#413112yes—
366C22×C3D4×AGL(1,𝔽5)12yes—
367C22×C3C2×#70411yescharacteristic
368C22×C3C2×#70411yescharacteristic
369C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
370C3×C4—22no—
371C3×C4—51no—
372C3×C4—52no—
373C3×C4—51no—
374C3×C4—102no—
375C3×C4—201no—
376C3×C4—201no—
377C2×C5—42no—
378C2×C5—42no—
379C2×C5C3×#38611yescharacteristic
380C2×C5C3×#40711yescharacteristic
381C2×C5C3×#40811yescharacteristic
382C2×C5C3×#43512yes—
383C2×C5C3×#43612yes—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23C3×#70412yescomplemented
393C23C3×C22⋊AGL(1,𝔽5)12yes—
394C23C22×C3×AGL(1,𝔽5)11yescharacteristic
395D4—28no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—52no—
399C2×C4—52no—
400C2×C4—51no—
401C2×C4—51no—
402C2×C4—52no—
403C2×C4—102no—
404C2×C4—104no—
405C2×C4—102no—
406C2×C4—102no—
407C2×C4—102no—
408C2×C4—102no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—201no—
412C2×C4—202no—
413C2×C4—201no—
414C2×C4C3×#70412yescomplemented
415C8—104no—
416C8—202no—
417C2×C3—42no—
418C2×C3—42no—
419C2×C3#6577312yes—
420C2×C3#6577412yes—
421C2×C3#6577511yescharacteristic
422C2×C3#6577911yescharacteristic
423C2×C3#6579811yescharacteristic
424C5C3×#187011yescharacteristic, complemented, Sylow
425C22—24no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22C3×#413112yes—
436C22C3×D4×AGL(1,𝔽5)12yes—
437C22C2×C3×#70411yescharacteristic
438C22C2×C3×#70411yescharacteristic
439C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
440C4—22no—
441C4—51no—
442C4—52no—
443C4—51no—
444C4—102no—
445C4—201no—
446C4—201no—
447C3#1060494711yescharacteristic, direct product factor, Sylow
448C2—42no—
449C2—42no—
450C2C3×#6577312yes—
451C2C3×#6577412yes—
452C2C3×#6577511yescharacteristic
453C2C3×#6577911yescharacteristic
454C2C3×#6579811yescharacteristic
subgroupG/H
centerC23×C3C22×AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
Fitting, prime radicalC22×C3×C5×D4C4
socleC23×C3×C5C22×C4
cosocleC22×Dic5C23×C3
hypercenterC22×C3×D4AGL(1,𝔽5)
hypocenterC5C3×#1870
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#704C2semidirect
3. C2×C3×C22⋊#170C2semidirect
4. C2×C3×C22⋊#170C2semidirect
5. C3×#64918C2semidirect
6. #10604947C3direct (N × H)
23. C22×#704C2×C3semidirect
25. C2×C22⋊#170C2×C3semidirect
26. C2×C22⋊#170C2×C3semidirect
27. #64918C2×C3semidirect
52. C2×C3×#170D4semidirect
88. C23×C3×C5M16semidirect
93. C2×C3×C4×C5M16semidirect
151. C2×#170C3×D4semidirect
223. C23×C5C3×M16semidirect
228. C2×C4×C5C3×M16semidirect
277. C23×C3#704semidirect
299. C2×C3×C4#704semidirect
354. C3×C5#1870semidirect
392. C23C3×#704semidirect
414. C2×C4C3×#704semidirect
424. C5C3×#1870semidirect
447. C3#10604947direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7042
C2×C3×D4×Dic52
C2×C3×C22⋊#1702
C3×#649182
#106049473
C3×#18705