Finite Group Databasebeta

← C2×C3×#65910 #14482815

Subgroups — C2×C3×#65910 #14482815

Order 1920.

3240 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
1C22×C3×#713C211yesmaximal, characteristic, complemented
2C22×C3×Q8×D5C211yesmaximal, characteristic, complemented
3C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
4C3×#65910C2116yesmaximal, direct product factor
5C2×C3×#4043C213yesmaximal, complemented
6C2×C3×#4041C213yesmaximal, complemented
7C2×C3×#4021C213yesmaximal, complemented
8C2×C3×C4×D20C213yesmaximal, complemented
9C2×#65910C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C2×C3×#713—28no—
12C2×C3×#713C2218yescomplemented
13C2×C3×Q8×D5—28no—
14C2×C3×Q8×D5C2218yescomplemented
15C22×C3×D20C2213yes—
16C22×C3×D20C2213yes—
17C22×C3×C4×D5C2213yes—
18C22×C3×C4×D5C2213yes—
19C22×C3×Dic10C2213yes—
20C3×C4×C5×Q8C2218yescomplemented
21C2×C3×C5×C4⋊C4C2213yes—
22C2×C3×C42×C5C2213yes—
23C2×C3×#685C2216yes—
24C2×C3×#685C2211yescharacteristic
25C2×C3×#685C2213yes—
26C2×C3×C4⋊Dic5C2213yes—
27C3×#4043C22124yescomplemented
28C3×#4041C22124yescomplemented
29C3×#4021C22124yescomplemented
30C3×C4×D20C22124yescomplemented
31C2×C3×#548—51nomaximal, Hall
32C22×#713C2×C311yescharacteristic, complemented
33C22×Q8×D5C2×C311yescharacteristic, complemented
34C2×C4×C5×Q8C2×C311yescharacteristic, complemented
35#65910C2×C3116yesdirect product factor
36C2×#4043C2×C313yescomplemented
37C2×#4041C2×C313yescomplemented
38C2×#4021C2×C313yescomplemented
39C2×C4×D20C2×C313yescomplemented
40C23×C3×D5C2313yes—
41C23×C3×D5C2311yescharacteristic
42C2×C3×C5×Q8D418yescomplemented
43C2×C3×C5×Q8C2314yes—
44C22×C3×C4×C5C2313yes—
45C22×C3×C4×C5C2311yescharacteristic
46C22×C3×C4×C5C2313yes—
47C22×C3×Dic5C2313yes—
48C22×C3×Dic5C2311yescharacteristic
49C3×#713—232no—
50C3×Q8×D5—232no—
51C2×C3×D20—212no—
52C2×C3×D20—212no—
53C2×C3×D20C23112yes—
54C2×C3×D20C23112yes—
55C2×C3×C4×D5—212no—
56C2×C3×C4×D5—212no—
57C2×C3×C4×D5C23112yes—
58C2×C3×C4×D5C23112yes—
59C2×C3×Dic10—212no—
60C2×C3×Dic10C23112yes—
61C3×C5×C4⋊C4C23112yes—
62C3×C42×C5C23112yes—
63C3×#685C2314yes—
64C3×#685C23112yes—
65C3×#685C23124yes—
66C3×C4⋊Dic5C23112yes—
67C22×C3×#41—51no—
68C23×C3×Q8—51no—
69C3×#548—516no—
70C2×C3×#124—53no—
71C2×C3×C22⋊Q8—53no—
72C2×C3×#121—53no—
73C2×C3×C4×Q8D511yescharacteristic, complemented
74C2×C3×C4×D4—53no—
75C22×C5×Q8C22×C311yescharacteristic
76C2×#713—28no—
77C2×#713C22×C318yescomplemented
78C2×Q8×D5—28no—
79C2×Q8×D5C22×C318yescomplemented
80C22×D20C22×C313yes—
81C22×D20C22×C313yes—
82C22×C4×D5C22×C313yes—
83C22×C4×D5C22×C313yes—
84C22×Dic10C22×C313yes—
85C4×C5×Q8C22×C318yescomplemented
86C2×C5×C4⋊C4C22×C313yes—
87C2×C42×C5C22×C313yes—
88C2×#685C22×C311yescharacteristic
89C2×#685C22×C313yes—
90C2×#685C22×C316yes—
91C2×C4⋊Dic5C22×C313yes—
92#4043C22×C3124yescomplemented
93#4041C22×C3124yescomplemented
94#4021C22×C3124yescomplemented
95C4×D20C22×C3124yescomplemented
96C2×#548—51noSylow
97C23×C3×C5C2411yescharacteristic
98C22×C3×D5—22no—
99C22×C3×D5—22no—
100C22×C3×D5—26no—
101C22×C3×D5—26no—
102C22×C3×D5—26no—
103C22×C3×D5#4114yescomplemented
104C22×C3×D5C2416yes—
105C22×C3×D5C2412yes—
106C3×C5×Q8C2×D4116yescomplemented
107C2×C3×C4×C5—26no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5C2×D4112yes—
110C2×C3×C4×C5C2416yes—
111C2×C3×C4×C5C2412yes—
112C2×C3×C4×C5C2416yes—
113C2×C3×Dic5—26no—
114C2×C3×Dic5—22no—
115C2×C3×Dic5C2412yes—
116C2×C3×Dic5C2416yes—
117C3×D20—224no—
118C3×D20—224no—
119C3×C4×D5—224no—
120C3×C4×D5—224no—
121C3×Dic10—224no—
122C2×C3×#41—58no—
123C2×C3×#41—108no—
124C22×C3×Q8—58no—
125C22×C3×Q8—53no—
126C22×C3×Q8—108no—
127C22×C3×Q8C2×D511yescharacteristic
128C22×C3×D4—53no—
129C22×C3×D4—53no—
130C23×C3×C4—53no—
131C23×C3×C4—53no—
132C3×#124—524no—
133C3×C22⋊Q8—524no—
134C3×#121—524no—
135C3×C4×Q8C2×D518yescomplemented
136C3×C4×D4—524no—
137C2×C3×C4⋊C4—53no—
138C2×C3×C4⋊C4C2×D513yes—
139C2×C3×C22⋊C4—51no—
140C2×C3×C22⋊C4—53no—
141C2×C3×C22⋊C4—56no—
142C2×C3×C42C2×D513yes—
143C23×D5C23×C313yes—
144C23×D5C23×C311yescharacteristic
145C2×C5×Q8C3×D418yescomplemented
146C2×C5×Q8C23×C314yes—
147C22×C4×C5C23×C311yescharacteristic
148C22×C4×C5C23×C313yes—
149C22×C4×C5C23×C313yes—
150C22×Dic5C23×C313yes—
151C22×Dic5C23×C311yescharacteristic
152#713—232no—
153Q8×D5—232no—
154C2×D20—212no—
155C2×D20—212no—
156C2×D20C23×C3112yes—
157C2×D20C23×C3112yes—
158C2×C4×D5—212no—
159C2×C4×D5—212no—
160C2×C4×D5C23×C3112yes—
161C2×C4×D5C23×C3112yes—
162C2×Dic10—212no—
163C2×Dic10C23×C3112yes—
164C5×C4⋊C4C23×C3112yes—
165C42×C5C23×C3112yes—
166#685C23×C314yes—
167#685C23×C3112yes—
168#685C23×C3124yes—
169C4⋊Dic5C23×C3112yes—
170C22×#41—51no—
171C23×Q8—51no—
172#548—516no—
173C2×#124—53no—
174C2×C22⋊Q8—53no—
175C2×#121—53no—
176C2×C4×Q8C3×D511yescharacteristic, complemented
177C2×C4×D4—53no—
178C22×C3×C5C22×D412yes—
179C22×C3×C5C2×#4112yes—
180C22×C3×C52−1+412yes—
181C22×C3×C5C2511yescharacteristic
182C2×C3×D5—28no—
183C2×C3×D5—22no—
184C2×C3×D5—22no—
185C2×C3×D5—26no—
186C2×C3×D5—26no—
187C2×C3×D5—26no—
188C2×C3×D5—412no—
189C2×C3×D5C2×#4114yescomplemented
190C3×C4×C5—26no—
191C3×C4×C5—22no—
192C3×C4×C5C22×D4112yes—
193C3×Dic5—26no—
194C3×Dic5—22no—
195C24×C3—51no—
196C24×C3—53no—
197C3×#41—1032no—
198C2×C3×Q8—512no—
199C2×C3×Q8—1032no—
200C2×C3×Q8—1012no—
201C2×C3×Q8D2018yescomplemented
202C2×C3×Q8C22×D514yes—
203C2×C3×D4—512no—
204C2×C3×D4—512no—
205C2×C3×D4—1012no—
206C2×C3×D4—1012no—
207C22×C3×C4—53no—
208C22×C3×C4—51no—
209C22×C3×C4—512no—
210C22×C3×C4—512no—
211C22×C3×C4—1012no—
212C22×C3×C4—1012no—
213C22×C3×C4C22×D511yescharacteristic
214C22×C3×C4C22×D513yes—
215C22×C3×C4C22×D513yes—
216C3×C4⋊C4—512no—
217C3×C4⋊C4C22×D5112yes—
218C3×C22⋊C4—54no—
219C3×C22⋊C4—512no—
220C3×C22⋊C4—524no—
221C3×C42C22×D5112yes—
222C23×C5C24×C311yescharacteristic
223C22×D5—22no—
224C22×D5—22no—
225C22×D5—26no—
226C22×D5—26no—
227C22×D5—26no—
228C22×D5C3×#4114yescomplemented
229C22×D5C24×C316yes—
230C22×D5C24×C312yes—
231C5×Q8C2×C3×D4116yescomplemented
232C2×C4×C5—26no—
233C2×C4×C5—22no—
234C2×C4×C5C2×C3×D4112yes—
235C2×C4×C5C24×C316yes—
236C2×C4×C5C24×C312yes—
237C2×C4×C5C24×C316yes—
238C2×Dic5—26no—
239C2×Dic5—22no—
240C2×Dic5C24×C312yes—
241C2×Dic5C24×C316yes—
242D20—224no—
243D20—224no—
244C4×D5—224no—
245C4×D5—224no—
246Dic10—224no—
247C2×#41—58no—
248C2×#41—108no—
249C22×Q8—58no—
250C22×Q8—53no—
251C22×Q8—108no—
252C22×Q8C2×C3×D511yescharacteristic
253C22×D4—53no—
254C22×D4—53no—
255C23×C4—53no—
256C23×C4—53no—
257#124—524no—
258C22⋊Q8—524no—
259#121—524no—
260C4×Q8C2×C3×D518yescomplemented
261C4×D4—524no—
262C2×C4⋊C4—53no—
263C2×C4⋊C4C2×C3×D513yes—
264C2×C22⋊C4—51no—
265C2×C22⋊C4—53no—
266C2×C22⋊C4—56no—
267C2×C42C2×C3×D513yes—
268C2×C3×C5#54814yescomplemented
269C2×C3×C5C23×D411yescharacteristic
270C2×C3×C5C22×#4111yescharacteristic
271C2×C3×C5C2×2−1+411yescharacteristic
272C3×D5—24no—
273C3×D5—46no—
274C23×C3—54no—
275C23×C3—52no—
276C23×C3—56no—
277C23×C3—102no—
278C23×C3—102no—
279C23×C3—106no—
280C23×C3—106no—
281C23×C3—106no—
282C23×C3C23×D511yescharacteristic
283C3×Q8—1024no—
284C3×Q8C2×D20116yescomplemented
285C3×D4—1024no—
286C3×D4—1024no—
287C2×C3×C4—22no—
288C2×C3×C4—26no—
289C2×C3×C4—56no—
290C2×C3×C4—52no—
291C2×C3×C4—102no—
292C2×C3×C4—106no—
293C2×C3×C4—1024no—
294C2×C3×C4—1024no—
295C2×C3×C4C2×D20112yes—
296C2×C3×C4C23×D512yes—
297C2×C3×C4C23×D516yes—
298C2×C3×C4C23×D516yes—
299C22×C5C22×C3×D412yes—
300C22×C5C2×C3×#4112yes—
301C22×C5C3×2−1+412yes—
302C22×C5C25×C311yescharacteristic
303C2×D5—22no—
304C2×D5—22no—
305C2×D5—28no—
306C2×D5—26no—
307C2×D5—26no—
308C2×D5—26no—
309C2×D5—412no—
310C2×D5C2×C3×#4114yescomplemented
311C4×C5—26no—
312C4×C5—22no—
313C4×C5C22×C3×D4112yes—
314Dic5—26no—
315Dic5—22no—
316C24—51no—
317C24—53no—
318#41—1032no—
319C2×Q8—512no—
320C2×Q8—1032no—
321C2×Q8—1012no—
322C2×Q8C3×D2018yescomplemented
323C2×Q8C22×C3×D514yes—
324C2×D4—512no—
325C2×D4—512no—
326C2×D4—1012no—
327C2×D4—1012no—
328C22×C4—53no—
329C22×C4—51no—
330C22×C4—512no—
331C22×C4—512no—
332C22×C4—1012no—
333C22×C4—1012no—
334C22×C4C22×C3×D511yescharacteristic
335C22×C4C22×C3×D513yes—
336C22×C4C22×C3×D513yes—
337C4⋊C4—512no—
338C4⋊C4C22×C3×D5112yes—
339C22⋊C4—54no—
340C22⋊C4—512no—
341C22⋊C4—524no—
342C42C22×C3×D5112yes—
343C3×C5C2×#54811yescharacteristic, complemented, Hall
344C22×C3—54no—
345C22×C3—108no—
346C22×C3—102no—
347C22×C3—102no—
348C22×C3—106no—
349C22×C3—106no—
350C22×C3—106no—
351C22×C3—2012no—
352C22×C3C22×D2012yes—
353C22×C3#414712yes—
354C22×C3D5×#4112yes—
355C22×C3C24×D511yescharacteristic
356C3×C4—22no—
357C3×C4—26no—
358C3×C4—102no—
359C3×C4—106no—
360C3×C4C22×D20112yes—
361C2×C5C3×#54814yescomplemented
362C2×C5C23×C3×D411yescharacteristic
363C2×C5C22×C3×#4111yescharacteristic
364C2×C5C2×C3×2−1+411yescharacteristic
365D5—24no—
366D5—46no—
367C23—54no—
368C23—52no—
369C23—56no—
370C23—102no—
371C23—102no—
372C23—106no—
373C23—106no—
374C23—106no—
375C23C23×C3×D511yescharacteristic
376Q8—1024no—
377Q8C2×C3×D20116yescomplemented
378D4—1024no—
379D4—1024no—
380C2×C4—22no—
381C2×C4—26no—
382C2×C4—56no—
383C2×C4—52no—
384C2×C4—102no—
385C2×C4—106no—
386C2×C4—1024no—
387C2×C4—1024no—
388C2×C4C2×C3×D20112yes—
389C2×C4C23×C3×D512yes—
390C2×C4C23×C3×D516yes—
391C2×C4C23×C3×D516yes—
392C2×C3—104no—
393C2×C3—206no—
394C2×C3#6591014yesdirect product factor
395C2×C3C23×D2011yescharacteristic
396C2×C3C2×#414711yescharacteristic
397C2×C3C2×D5×#4111yescharacteristic
398C5C2×C3×#54811yescharacteristic, complemented, Sylow
399C22—54no—
400C22—102no—
401C22—102no—
402C22—108no—
403C22—106no—
404C22—106no—
405C22—106no—
406C22—2012no—
407C22C22×C3×D2012yes—
408C22C3×#414712yes—
409C22C3×D5×#4112yes—
410C22C24×C3×D511yescharacteristic
411C4—22no—
412C4—26no—
413C4—102no—
414C4—106no—
415C4C22×C3×D20112yes—
416C3C2×#6591011yescharacteristic, direct product factor, Sylow
417C2—104no—
418C2—206no—
419C2C3×#6591014yesdirect product factor
420C2C23×C3×D2011yescharacteristic
421C2C2×C3×#414711yescharacteristic
422C2C2×C3×D5×#4111yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C24
hypercenterC2×C3×C4×Q8D5
hypocenterC5C2×C3×#548
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×#713C2semidirect
2. C22×C3×Q8×D5C2semidirect
3. C2×C3×C4×C5×Q8C2semidirect
4. C3×#65910C2direct (N × H)
5. C2×C3×#4043C2semidirect
6. C2×C3×#4041C2semidirect
7. C2×C3×#4021C2semidirect
8. C2×C3×C4×D20C2semidirect
9. C2×#65910C3direct (N × H)
12. C2×C3×#713C22semidirect
14. C2×C3×Q8×D5C22semidirect
20. C3×C4×C5×Q8C22semidirect
27. C3×#4043C22semidirect
28. C3×#4041C22semidirect
29. C3×#4021C22semidirect
30. C3×C4×D20C22semidirect
32. C22×#713C2×C3semidirect
33. C22×Q8×D5C2×C3semidirect
34. C2×C4×C5×Q8C2×C3semidirect
35. #65910C2×C3direct (N × H)
36. C2×#4043C2×C3semidirect
37. C2×#4041C2×C3semidirect
38. C2×#4021C2×C3semidirect
39. C2×C4×D20C2×C3semidirect
42. C2×C3×C5×Q8D4semidirect
73. C2×C3×C4×Q8D5semidirect
77. C2×#713C22×C3semidirect
79. C2×Q8×D5C22×C3semidirect
85. C4×C5×Q8C22×C3semidirect
92. #4043C22×C3semidirect
93. #4041C22×C3semidirect
94. #4021C22×C3semidirect
95. C4×D20C22×C3semidirect
103. C22×C3×D5#41semidirect
106. C3×C5×Q8C2×D4semidirect
135. C3×C4×Q8C2×D5semidirect
145. C2×C5×Q8C3×D4semidirect
176. C2×C4×Q8C3×D5semidirect
189. C2×C3×D5C2×#41semidirect
201. C2×C3×Q8D20semidirect
228. C22×D5C3×#41semidirect
231. C5×Q8C2×C3×D4semidirect
260. C4×Q8C2×C3×D5semidirect
268. C2×C3×C5#548semidirect
284. C3×Q8C2×D20semidirect
310. C2×D5C2×C3×#41semidirect
322. C2×Q8C3×D20semidirect
343. C3×C5C2×#548semidirect
361. C2×C5C3×#548semidirect
377. Q8C2×C3×D20semidirect
394. C2×C3#65910direct (N × H)
398. C5C2×C3×#548semidirect
416. C3C2×#65910direct (N × H)
419. C2C3×#65910direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C22×C3×Q8×D52
C2×C3×C4×C5×Q82
C3×#659102
C2×C3×#40432
C2×C3×#40412
C2×C3×#40212
C2×C3×C4×D202
C2×#659103
C2×C3×#5485