Finite Group Databasebeta

← C2×C3×#65756 #14411637

Subgroups — C2×C3×#65756 #14411637

Order 1920.

2016 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×D5C211yesmaximal, characteristic
2C22×C3×#700C212yesmaximal, complemented
3C2×C3×C22⋊#170C212yesmaximal, complemented
4C3×#65756C218yesmaximal, direct product factor
5C2×C3×#4003C212yesmaximal, complemented
6C2×#65756C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×D5C411yescharacteristic
8C23×C3×C4×C5C411yescharacteristic
9C23×C3×Dic5C2211yescharacteristic
10C22×C3×C4×D5—28no—
11C22×C3×C4×D5C411yescharacteristic
12C22×C3×C4×D5C411yescharacteristic
13C22×C3×C4×D5C414yes—
14C22×C3×C4×D5C2211yescharacteristic
15C22×C3×C4×D5C2214yes—
16C22×C3×C4×D5C2211yescharacteristic
17C22×C3×#170C2214yescomplemented
18C2×C3×#700—28no—
19C2×C3×#700C2218yescomplemented
20C3×C22⋊#170C2218yescomplemented
21C3×#4003C2218yescomplemented
22C2×C3×#407—51nomaximal, Hall
23C23×C4×D5C2×C311yescharacteristic
24C22×#700C2×C312yescomplemented
25C2×C22⋊#170C2×C312yescomplemented
26#65756C2×C318yesdirect product factor
27C2×#4003C2×C312yescomplemented
28C24×C3×C5C2×C411yescharacteristic
29C23×C3×D5—24no—
30C23×C3×D5—24no—
31C23×C3×D5—24no—
32C23×C3×D5C2×C414yes—
33C23×C3×D5C2×C411yescharacteristic
34C23×C3×D5C2×C411yescharacteristic
35C22×C3×C4×C5—24no—
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×C4×C5C2×C414yes—
38C22×C3×C4×C5C2×C411yescharacteristic
39C22×C3×Dic5—24no—
40C22×C3×Dic5C2×C411yescharacteristic
41C22×C3×Dic5C2×C412yes—
42C22×C3×Dic5C2312yes—
43C22×C3×Dic5C2311yescharacteristic
44C2×C3×C4×D5—24no—
45C2×C3×C4×D5—24no—
46C2×C3×C4×D5—24no—
47C2×C3×C4×D5—216no—
48C2×C3×C4×D5—216no—
49C2×C3×C4×D5C2×C414yes—
50C2×C3×C4×D5C2×C414yes—
51C2×C3×C4×D5C2×C412yes—
52C2×C3×C4×D5C2×C412yes—
53C2×C3×C4×D5D418yes—
54C2×C3×C4×D5C2312yes—
55C2×C3×C4×D5C2312yes—
56C2×C3×#170—28no—
57C2×C3×#170C2318yescomplemented
58C3×#700—216no—
59C24×C3×C4—51no—
60C22×C3×M16—52no—
61C3×#407—58no—
62C2×C3×C22⋊C8—52no—
63C2×C3×C22⋊C8—52no—
64C24×D5C3×C411yescharacteristic
65C23×C4×C5C3×C411yescharacteristic
66C23×Dic5C22×C311yescharacteristic
67C22×C4×D5—28no—
68C22×C4×D5C3×C411yescharacteristic
69C22×C4×D5C3×C411yescharacteristic
70C22×C4×D5C3×C414yes—
71C22×C4×D5C22×C311yescharacteristic
72C22×C4×D5C22×C314yes—
73C22×C4×D5C22×C311yescharacteristic
74C22×#170C22×C314yescomplemented
75C2×#700—28no—
76C2×#700C22×C318yescomplemented
77C22⋊#170C22×C318yescomplemented
78#4003C22×C318yescomplemented
79C2×#407—51noSylow
80C23×C3×C5—22no—
81C23×C3×C5—22no—
82C23×C3×C5M1614yescomplemented
83C23×C3×C5C22×C412yes—
84C23×C3×C5C22×C411yescharacteristic
85C22×C3×D5—232no—
86C22×C3×D5—22no—
87C22×C3×D5—22no—
88C22×C3×D5—22no—
89C22×C3×D5—22no—
90C22×C3×D5—28no—
91C22×C3×D5—28no—
92C22×C3×D5—28no—
93C22×C3×D5C22⋊C414yes—
94C22×C3×D5M1614yescomplemented
95C22×C3×D5C22×C412yes—
96C22×C3×D5C22×C412yes—
97C2×C3×C4×C5—22no—
98C2×C3×C4×C5—28no—
99C2×C3×C4×C5C22⋊C414yes—
100C2×C3×C4×C5C22×C412yes—
101C2×C3×C4×C5C22×C412yes—
102C2×C3×Dic5—22no—
103C2×C3×Dic5—24no—
104C2×C3×Dic5—24no—
105C2×C3×Dic5C22×C412yes—
106C2×C3×Dic5C22×C411yescharacteristic
107C2×C3×Dic5C2×D414yes—
108C2×C3×Dic5C2411yescharacteristic
109C3×C4×D5—28no—
110C3×C4×D5—28no—
111C3×C4×D5—24no—
112C3×C4×D5—24no—
113C3×C4×D5C22⋊C418yes—
114C3×C4×D5C2×D418yes—
115C3×#170—28no—
116C25×C3—51no—
117C23×C3×C4—51no—
118C23×C3×C4—51no—
119C23×C3×C4—54no—
120C23×C3×C4—51no—
121C23×C3×C4—51no—
122C23×C3×C4—51no—
123C23×C3×C4—54no—
124C23×C3×C4—108no—
125C23×C3×C4AGL(1,𝔽5)11yescharacteristic
126C2×C3×M16—58no—
127C2×C3×M16—108no—
128C22×C3×C8—54no—
129C3×C22⋊C8—58no—
130C3×C22⋊C8—58no—
131C24×C5C2×C3×C411yescharacteristic
132C23×D5—24no—
133C23×D5—24no—
134C23×D5—24no—
135C23×D5C2×C3×C414yes—
136C23×D5C2×C3×C411yescharacteristic
137C23×D5C2×C3×C411yescharacteristic
138C22×C4×C5—24no—
139C22×C4×C5C2×C3×C411yescharacteristic
140C22×C4×C5C2×C3×C414yes—
141C22×C4×C5C2×C3×C411yescharacteristic
142C22×Dic5—24no—
143C22×Dic5C2×C3×C411yescharacteristic
144C22×Dic5C2×C3×C412yes—
145C22×Dic5C23×C312yes—
146C22×Dic5C23×C311yescharacteristic
147C2×C4×D5—24no—
148C2×C4×D5—24no—
149C2×C4×D5—24no—
150C2×C4×D5—216no—
151C2×C4×D5—216no—
152C2×C4×D5C2×C3×C414yes—
153C2×C4×D5C2×C3×C414yes—
154C2×C4×D5C2×C3×C412yes—
155C2×C4×D5C2×C3×C412yes—
156C2×C4×D5C3×D418yes—
157C2×C4×D5C23×C312yes—
158C2×C4×D5C23×C312yes—
159C2×#170—28no—
160C2×#170C23×C318yescomplemented
161#700—216no—
162C24×C4—51no—
163C22×M16—52no—
164#407—58no—
165C2×C22⋊C8—52no—
166C2×C22⋊C8—52no—
167C22×C3×C5—22no—
168C22×C3×C5—22no—
169C22×C3×C5—28no—
170C22×C3×C5C2×C22⋊C412yes—
171C22×C3×C5C2×M1612yes—
172C22×C3×C5C2×M1612yes—
173C22×C3×C5C2×M1614yescomplemented
174C22×C3×C5C23×C411yescharacteristic
175C2×C3×D5—232no—
176C2×C3×D5—22no—
177C2×C3×D5—22no—
178C2×C3×D5—22no—
179C2×C3×D5—22no—
180C2×C3×D5—28no—
181C2×C3×D5—28no—
182C2×C3×D5C2×C22⋊C414yes—
183C2×C3×D5C2×M1614yescomplemented
184C3×C4×C5—22no—
185C3×C4×C5C2×C22⋊C414yes—
186C3×Dic5—22no—
187C3×Dic5C2×C22⋊C412yes—
188C3×Dic5C22×D412yes—
189C24×C3—51no—
190C24×C3—51no—
191C24×C3—54no—
192C24×C3—104no—
193C24×C3—104no—
194C24×C3—104no—
195C24×C3C2×AGL(1,𝔽5)11yescharacteristic
196C22×C3×C4—24no—
197C22×C3×C4—51no—
198C22×C3×C4—58no—
199C22×C3×C4—52no—
200C22×C3×C4—52no—
201C22×C3×C4—52no—
202C22×C3×C4—52no—
203C22×C3×C4—51no—
204C22×C3×C4—52no—
205C22×C3×C4—52no—
206C22×C3×C4—54no—
207C22×C3×C4—54no—
208C22×C3×C4—104no—
209C22×C3×C4—1016no—
210C22×C3×C4—1016no—
211C22×C3×C4—104no—
212C22×C3×C4—104no—
213C22×C3×C4—104no—
214C22×C3×C4C2×AGL(1,𝔽5)14yes—
215C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
216C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
217C3×M16—1016no—
218C2×C3×C8—58no—
219C2×C3×C8—108no—
220C23×C5—22no—
221C23×C5—22no—
222C23×C5C3×M1614yescomplemented
223C23×C5C22×C3×C412yes—
224C23×C5C22×C3×C411yescharacteristic
225C22×D5—232no—
226C22×D5—22no—
227C22×D5—22no—
228C22×D5—22no—
229C22×D5—22no—
230C22×D5—28no—
231C22×D5—28no—
232C22×D5—28no—
233C22×D5C3×C22⋊C414yes—
234C22×D5C3×M1614yescomplemented
235C22×D5C22×C3×C412yes—
236C22×D5C22×C3×C412yes—
237C2×C4×C5—22no—
238C2×C4×C5—28no—
239C2×C4×C5C3×C22⋊C414yes—
240C2×C4×C5C22×C3×C412yes—
241C2×C4×C5C22×C3×C412yes—
242C2×Dic5—22no—
243C2×Dic5—24no—
244C2×Dic5—24no—
245C2×Dic5C22×C3×C412yes—
246C2×Dic5C22×C3×C411yescharacteristic
247C2×Dic5C2×C3×D414yes—
248C2×Dic5C24×C311yescharacteristic
249C4×D5—28no—
250C4×D5—28no—
251C4×D5—24no—
252C4×D5—24no—
253C4×D5C3×C22⋊C418yes—
254C4×D5C2×C3×D418yes—
255#170—28no—
256C25—51no—
257C23×C4—51no—
258C23×C4—51no—
259C23×C4—54no—
260C23×C4—51no—
261C23×C4—51no—
262C23×C4—51no—
263C23×C4—54no—
264C23×C4—108no—
265C23×C4C3×AGL(1,𝔽5)11yescharacteristic
266C2×M16—58no—
267C2×M16—108no—
268C22×C8—54no—
269C22⋊C8—58no—
270C22⋊C8—58no—
271C2×C3×C5—24no—
272C2×C3×C5#40714yescomplemented
273C2×C3×C5C22×C22⋊C411yescharacteristic
274C2×C3×C5C22×M1611yescharacteristic
275C2×C3×C5C22×M1611yescharacteristic
276C3×D5—24no—
277C3×D5—24no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—54no—
281C23×C3—54no—
282C23×C3—52no—
283C23×C3—52no—
284C23×C3—108no—
285C23×C3—108no—
286C23×C3—1032no—
287C23×C3—102no—
288C23×C3—102no—
289C23×C3—108no—
290C23×C3—102no—
291C23×C3—102no—
292C23×C3#70014yescomplemented
293C23×C3C22×AGL(1,𝔽5)12yes—
294C23×C3C22×AGL(1,𝔽5)11yescharacteristic
295C2×C3×C4—22no—
296C2×C3×C4—28no—
297C2×C3×C4—54no—
298C2×C3×C4—51no—
299C2×C3×C4—51no—
300C2×C3×C4—58no—
301C2×C3×C4—58no—
302C2×C3×C4—52no—
303C2×C3×C4—104no—
304C2×C3×C4—104no—
305C2×C3×C4—108no—
306C2×C3×C4—104no—
307C2×C3×C4—104no—
308C2×C3×C4—102no—
309C2×C3×C4—108no—
310C2×C3×C4C22⋊AGL(1,𝔽5)14yes—
311C2×C3×C4C22×AGL(1,𝔽5)12yes—
312C2×C3×C4C22×AGL(1,𝔽5)12yes—
313C3×C8—108no—
314C22×C5—22no—
315C22×C5—28no—
316C22×C5—22no—
317C22×C5C2×C3×C22⋊C412yes—
318C22×C5C2×C3×M1612yes—
319C22×C5C2×C3×M1612yes—
320C22×C5C2×C3×M1614yescomplemented
321C22×C5C23×C3×C411yescharacteristic
322C2×D5—232no—
323C2×D5—22no—
324C2×D5—22no—
325C2×D5—22no—
326C2×D5—28no—
327C2×D5—28no—
328C2×D5—22no—
329C2×D5C2×C3×C22⋊C414yes—
330C2×D5C2×C3×M1614yescomplemented
331C4×C5—22no—
332C4×C5C2×C3×C22⋊C414yes—
333Dic5—22no—
334Dic5C2×C3×C22⋊C412yes—
335Dic5C22×C3×D412yes—
336C24—51no—
337C24—51no—
338C24—54no—
339C24—104no—
340C24—104no—
341C24—104no—
342C24C2×C3×AGL(1,𝔽5)11yescharacteristic
343C22×C4—24no—
344C22×C4—51no—
345C22×C4—58no—
346C22×C4—52no—
347C22×C4—52no—
348C22×C4—52no—
349C22×C4—52no—
350C22×C4—51no—
351C22×C4—52no—
352C22×C4—52no—
353C22×C4—54no—
354C22×C4—54no—
355C22×C4—104no—
356C22×C4—1016no—
357C22×C4—1016no—
358C22×C4—104no—
359C22×C4—104no—
360C22×C4—104no—
361C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
362C22×C4C2×C3×AGL(1,𝔽5)14yes—
363C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
364M16—1016no—
365C2×C8—58no—
366C2×C8—108no—
367C3×C5C2×#40711yescharacteristic, complemented, Hall
368C22×C3—22no—
369C22×C3—28no—
370C22×C3—22no—
371C22×C3—54no—
372C22×C3—54no—
373C22×C3—1032no—
374C22×C3—102no—
375C22×C3—102no—
376C22×C3—102no—
377C22×C3—108no—
378C22×C3—108no—
379C22×C3—102no—
380C22×C3C2×#70012yes—
381C22×C3C2×#70014yescomplemented
382C22×C3D5⋊M1612yes—
383C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
384C22×C3C23×AGL(1,𝔽5)11yescharacteristic
385C3×C4—22no—
386C3×C4—52no—
387C3×C4—52no—
388C3×C4—102no—
389C3×C4C2×C22⋊AGL(1,𝔽5)14yes—
390C2×C5—24no—
391C2×C5C3×#40714yescomplemented
392C2×C5C22×C3×C22⋊C411yescharacteristic
393C2×C5C22×C3×M1611yescharacteristic
394C2×C5C22×C3×M1611yescharacteristic
395D5—24no—
396D5—24no—
397C23—22no—
398C23—22no—
399C23—54no—
400C23—54no—
401C23—52no—
402C23—52no—
403C23—108no—
404C23—108no—
405C23—1032no—
406C23—102no—
407C23—102no—
408C23—108no—
409C23—102no—
410C23—102no—
411C23C3×#70014yescomplemented
412C23C22×C3×AGL(1,𝔽5)12yes—
413C23C22×C3×AGL(1,𝔽5)11yescharacteristic
414C2×C4—22no—
415C2×C4—28no—
416C2×C4—54no—
417C2×C4—51no—
418C2×C4—51no—
419C2×C4—58no—
420C2×C4—58no—
421C2×C4—52no—
422C2×C4—104no—
423C2×C4—104no—
424C2×C4—108no—
425C2×C4—104no—
426C2×C4—104no—
427C2×C4—102no—
428C2×C4—108no—
429C2×C4C3×C22⋊AGL(1,𝔽5)14yes—
430C2×C4C22×C3×AGL(1,𝔽5)12yes—
431C2×C4C22×C3×AGL(1,𝔽5)12yes—
432C8—108no—
433C2×C3—24no—
434C2×C3—104no—
435C2×C3—104no—
436C2×C3#6575614yesdirect product factor
437C2×C3C22×#70011yescharacteristic
438C2×C3C2×D5⋊M1611yescharacteristic
439C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
440C5C2×C3×#40711yescharacteristic, complemented, Sylow
441C22—28no—
442C22—22no—
443C22—22no—
444C22—54no—
445C22—54no—
446C22—1032no—
447C22—108no—
448C22—108no—
449C22—102no—
450C22—102no—
451C22—102no—
452C22—102no—
453C22C2×C3×#70012yes—
454C22C2×C3×#70014yescomplemented
455C22C3×D5⋊M1612yes—
456C22C2×C3×C22⋊AGL(1,𝔽5)12yes—
457C22C23×C3×AGL(1,𝔽5)11yescharacteristic
458C4—22no—
459C4—52no—
460C4—52no—
461C4—102no—
462C4C2×C3×C22⋊AGL(1,𝔽5)14yes—
463C3C2×#6575611yescharacteristic, direct product factor, Sylow
464C2—24no—
465C2—104no—
466C2—104no—
467C2C3×#6575614yesdirect product factor
468C2C22×C3×#70011yescharacteristic
469C2C2×C3×D5⋊M1611yescharacteristic
470C2C22×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC23×C3C22×AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC23×C3×C4×C5C4
socleC23×C3×C5C22×C4
prime radicalC24×C3×D5C4
cosocleC2×Dic5C24×C3
hypercenterC23×C3×C4AGL(1,𝔽5)
hypocenterC5C2×C3×#407
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#700C2semidirect
3. C2×C3×C22⋊#170C2semidirect
4. C3×#65756C2direct (N × H)
5. C2×C3×#4003C2semidirect
6. C2×#65756C3direct (N × H)
17. C22×C3×#170C22semidirect
19. C2×C3×#700C22semidirect
20. C3×C22⋊#170C22semidirect
21. C3×#4003C22semidirect
24. C22×#700C2×C3semidirect
25. C2×C22⋊#170C2×C3semidirect
26. #65756C2×C3direct (N × H)
27. C2×#4003C2×C3semidirect
57. C2×C3×#170C23semidirect
74. C22×#170C22×C3semidirect
76. C2×#700C22×C3semidirect
77. C22⋊#170C22×C3semidirect
78. #4003C22×C3semidirect
82. C23×C3×C5M16semidirect
94. C22×C3×D5M16semidirect
160. C2×#170C23×C3semidirect
173. C22×C3×C5C2×M16semidirect
183. C2×C3×D5C2×M16semidirect
222. C23×C5C3×M16semidirect
234. C22×D5C3×M16semidirect
272. C2×C3×C5#407semidirect
292. C23×C3#700semidirect
320. C22×C5C2×C3×M16semidirect
330. C2×D5C2×C3×M16semidirect
367. C3×C5C2×#407semidirect
381. C22×C3C2×#700semidirect
391. C2×C5C3×#407semidirect
411. C23C3×#700semidirect
436. C2×C3#65756direct (N × H)
440. C5C2×C3×#407semidirect
454. C22C2×C3×#700semidirect
463. C3C2×#65756direct (N × H)
467. C2C3×#65756direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×D52
C22×C3×#7002
C2×C3×C22⋊#1702
C3×#657562
C2×C3×#40032
C2×#657563
C2×C3×#4075