Finite Group Databasebeta

← C2×C3×#66165 #14482800

Subgroups — C2×C3×#66165 #14482800

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×C5×#41C211yesmaximal, characteristic, complemented
2C22×C3×Q8×D5C211yesmaximal, characteristic, complemented
3C3×#66165C2116yesmaximal, direct product factor
4C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
5C2×C3×#4084C213yesmaximal, complemented
6C2×C3×#4082C213yesmaximal, complemented
7C2×C3×C4×C22⋊D5C213yesmaximal, complemented
8C2×C3×#4070C213yesmaximal, complemented
9C2×#66165C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×C5×#41—28no—
11C2×C3×C5×#41C2218yescomplemented
12C22×C3×C5×Q8C2211yescharacteristic
13C22×C3×C5×D4C2213yes—
14C23×C3×C4×C5C2213yes—
15C22×C3×C22⋊D5C2213yes—
16C2×C3×Q8×D5—28no—
17C2×C3×Q8×D5C2218yescomplemented
18C22×C3×C4×D5C2213yes—
19C22×C3×Dic10C2213yes—
20C2×C3×C22⋊Dic5C2216yes—
21C2×C3×C22⋊Dic5C2213yes—
22C3×Q8×Dic5C2218yescomplemented
23C3×#4084C22124yescomplemented
24C3×#4082C22124yescomplemented
25C3×C4×C22⋊D5C22124yescomplemented
26C2×C3×#685C2211yescharacteristic
27C2×C3×C4⋊Dic5C2213yes—
28C3×#4070C22124yescomplemented
29C2×C3×#683C2213yes—
30C2×C3×C4×Dic5C2213yes—
31C2×C3×#548—51nomaximal, Hall
32C22×C5×#41C2×C311yescharacteristic, complemented
33C22×Q8×D5C2×C311yescharacteristic, complemented
34#66165C2×C3116yesdirect product factor
35C2×Q8×Dic5C2×C311yescharacteristic, complemented
36C2×#4084C2×C313yescomplemented
37C2×#4082C2×C313yescomplemented
38C2×C4×C22⋊D5C2×C313yescomplemented
39C2×#4070C2×C313yescomplemented
40C24×C3×C5C2313yes—
41C23×C3×D5C2311yescharacteristic
42C3×C5×#41—232no—
43C2×C3×C5×Q8D418yescomplemented
44C2×C3×C5×Q8C2314yes—
45C2×C3×C5×D4—212no—
46C2×C3×C5×D4C23112yes—
47C22×C3×C4×C5—212no—
48C22×C3×C4×C5C2311yescharacteristic
49C22×C3×C4×C5C23112yes—
50C22×C3×C4×C5C2313yes—
51C2×C3×C22⋊D5—212no—
52C2×C3×C22⋊D5C23112yes—
53C22×C3×Dic5C2313yes—
54C22×C3×Dic5C2311yescharacteristic
55C22×C3×Dic5C2313yes—
56C3×Q8×D5—232no—
57C2×C3×C4×D5—212no—
58C2×C3×C4×D5C23112yes—
59C2×C3×Dic10—212no—
60C2×C3×Dic10C23112yes—
61C3×C22⋊Dic5C23112yes—
62C3×C22⋊Dic5C23124yes—
63C3×#685C2314yes—
64C3×C4⋊Dic5C23112yes—
65C3×#683C23112yes—
66C3×C4×Dic5C23112yes—
67C22×C3×#41D511yescharacteristic, complemented
68C23×C3×Q8—51no—
69C3×#548—516no—
70C2×C3×#124—53no—
71C2×C3×C22⋊Q8—53no—
72C2×C3×#121—53no—
73C2×C3×C4×Q8—51no—
74C2×C3×C4×D4—53no—
75C2×C5×#41—28no—
76C2×C5×#41C22×C318yescomplemented
77C22×C5×Q8C22×C311yescharacteristic
78C22×C5×D4C22×C313yes—
79C23×C4×C5C22×C313yes—
80C22×C22⋊D5C22×C313yes—
81C2×Q8×D5—28no—
82C2×Q8×D5C22×C318yescomplemented
83C22×C4×D5C22×C313yes—
84C22×Dic10C22×C313yes—
85C2×C22⋊Dic5C22×C316yes—
86C2×C22⋊Dic5C22×C313yes—
87Q8×Dic5C22×C318yescomplemented
88#4084C22×C3124yescomplemented
89#4082C22×C3124yescomplemented
90C4×C22⋊D5C22×C3124yescomplemented
91C2×#685C22×C311yescharacteristic
92C2×C4⋊Dic5C22×C313yes—
93#4070C22×C3124yescomplemented
94C2×#683C22×C313yes—
95C2×C4×Dic5C22×C313yes—
96C2×#548—51noSylow
97C23×C3×C5—26no—
98C23×C3×C5—26no—
99C23×C3×C5—26no—
100C23×C3×C5C2416yes—
101C23×C3×C5C2411yescharacteristic
102C22×C3×D5—22no—
103C22×C3×D5—22no—
104C22×C3×D5#4114yescomplemented
105C22×C3×D5C2412yes—
106C3×C5×Q8C2×D4116yescomplemented
107C3×C5×D4—224no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—224no—
110C2×C3×C4×C5C2×D4112yes—
111C2×C3×C4×C5C2412yes—
112C2×C3×C4×C5C2416yes—
113C3×C22⋊D5—224no—
114C2×C3×Dic5—26no—
115C2×C3×Dic5—26no—
116C2×C3×Dic5—22no—
117C2×C3×Dic5C2412yes—
118C2×C3×Dic5C2416yes—
119C2×C3×Dic5C2416yes—
120C3×C4×D5—224no—
121C3×Dic10—224no—
122C2×C3×#41—28no—
123C2×C3×#41C2×D518yescomplemented
124C22×C3×Q8—58no—
125C22×C3×Q8—53no—
126C22×C3×Q8—108no—
127C22×C3×Q8C2×D511yescharacteristic
128C22×C3×D4—53no—
129C22×C3×D4C2×D513yes—
130C23×C3×C4—53no—
131C23×C3×C4C2×D513yes—
132C3×#124—524no—
133C3×C22⋊Q8—524no—
134C3×#121—524no—
135C3×C4×Q8—58no—
136C3×C4×D4—524no—
137C2×C3×C4⋊C4—53no—
138C2×C3×C4⋊C4—53no—
139C2×C3×C22⋊C4—53no—
140C2×C3×C22⋊C4—56no—
141C2×C3×C22⋊C4—51no—
142C2×C3×C42—53no—
143C24×C5C23×C313yes—
144C23×D5C23×C311yescharacteristic
145C5×#41—232no—
146C2×C5×Q8C3×D418yescomplemented
147C2×C5×Q8C23×C314yes—
148C2×C5×D4—212no—
149C2×C5×D4C23×C3112yes—
150C22×C4×C5—212no—
151C22×C4×C5C23×C311yescharacteristic
152C22×C4×C5C23×C3112yes—
153C22×C4×C5C23×C313yes—
154C2×C22⋊D5—212no—
155C2×C22⋊D5C23×C3112yes—
156C22×Dic5C23×C313yes—
157C22×Dic5C23×C311yescharacteristic
158C22×Dic5C23×C313yes—
159Q8×D5—232no—
160C2×C4×D5—212no—
161C2×C4×D5C23×C3112yes—
162C2×Dic10—212no—
163C2×Dic10C23×C3112yes—
164C22⋊Dic5C23×C3124yes—
165C22⋊Dic5C23×C3112yes—
166#685C23×C314yes—
167C4⋊Dic5C23×C3112yes—
168#683C23×C3112yes—
169C4×Dic5C23×C3112yes—
170C22×#41C3×D511yescharacteristic, complemented
171C23×Q8—51no—
172#548—516no—
173C2×#124—53no—
174C2×C22⋊Q8—53no—
175C2×#121—53no—
176C2×C4×Q8—51no—
177C2×C4×D4—53no—
178C22×C3×C5—26no—
179C22×C3×C5—26no—
180C22×C3×C5—26no—
181C22×C3×C5—412no—
182C22×C3×C5C22×D412yes—
183C22×C3×C5C2×#4112yes—
184C22×C3×C52−1+412yes—
185C22×C3×C5C2511yescharacteristic
186C2×C3×D5—28no—
187C2×C3×D5—22no—
188C2×C3×D5—22no—
189C2×C3×D5C2×#4114yescomplemented
190C3×C4×C5—22no—
191C3×C4×C5C22×D4112yes—
192C3×Dic5—26no—
193C3×Dic5—26no—
194C3×Dic5—22no—
195C24×C3—51no—
196C24×C3C22×D513yes—
197C3×#41—232no—
198C2×C3×Q8—512no—
199C2×C3×Q8—1032no—
200C2×C3×Q8—1012no—
201C2×C3×Q8C22⋊D518yescomplemented
202C2×C3×Q8C22×D514yes—
203C2×C3×D4—212no—
204C2×C3×D4—512no—
205C2×C3×D4—1012no—
206C2×C3×D4C22×D5112yes—
207C22×C3×C4—212no—
208C22×C3×C4—51no—
209C22×C3×C4—53no—
210C22×C3×C4—53no—
211C22×C3×C4—512no—
212C22×C3×C4—1012no—
213C22×C3×C4C22×D5112yes—
214C22×C3×C4C22×D511yescharacteristic
215C22×C3×C4C22×D513yes—
216C3×C4⋊C4—512no—
217C3×C4⋊C4—512no—
218C3×C22⋊C4—512no—
219C3×C22⋊C4—524no—
220C3×C22⋊C4—54no—
221C3×C42—512no—
222C23×C5—26no—
223C23×C5—26no—
224C23×C5—26no—
225C23×C5C24×C316yes—
226C23×C5C24×C311yescharacteristic
227C22×D5—22no—
228C22×D5—22no—
229C22×D5C3×#4114yescomplemented
230C22×D5C24×C312yes—
231C5×Q8C2×C3×D4116yescomplemented
232C5×D4—224no—
233C2×C4×C5—22no—
234C2×C4×C5—224no—
235C2×C4×C5C2×C3×D4112yes—
236C2×C4×C5C24×C312yes—
237C2×C4×C5C24×C316yes—
238C22⋊D5—224no—
239C2×Dic5—26no—
240C2×Dic5—26no—
241C2×Dic5—22no—
242C2×Dic5C24×C312yes—
243C2×Dic5C24×C316yes—
244C2×Dic5C24×C316yes—
245C4×D5—224no—
246Dic10—224no—
247C2×#41—28no—
248C2×#41C2×C3×D518yescomplemented
249C22×Q8—58no—
250C22×Q8—53no—
251C22×Q8—108no—
252C22×Q8C2×C3×D511yescharacteristic
253C22×D4—53no—
254C22×D4C2×C3×D513yes—
255C23×C4—53no—
256C23×C4C2×C3×D513yes—
257#124—524no—
258C22⋊Q8—524no—
259#121—524no—
260C4×Q8—58no—
261C4×D4—524no—
262C2×C4⋊C4—53no—
263C2×C4⋊C4—53no—
264C2×C22⋊C4—53no—
265C2×C22⋊C4—56no—
266C2×C22⋊C4—51no—
267C2×C42—53no—
268C2×C3×C5—46no—
269C2×C3×C5#54814yescomplemented
270C2×C3×C5C23×D411yescharacteristic
271C2×C3×C5C22×#4111yescharacteristic
272C2×C3×C5C2×2−1+411yescharacteristic
273C3×D5—24no—
274C23×C3—26no—
275C23×C3—26no—
276C23×C3—26no—
277C23×C3—54no—
278C23×C3—52no—
279C23×C3—102no—
280C23×C3—102no—
281C23×C3C23×D516yes—
282C23×C3C23×D511yescharacteristic
283C3×Q8—1024no—
284C3×Q8C2×C22⋊D5116yescomplemented
285C3×D4—224no—
286C3×D4—1024no—
287C2×C3×C4—22no—
288C2×C3×C4—224no—
289C2×C3×C4—52no—
290C2×C3×C4—56no—
291C2×C3×C4—56no—
292C2×C3×C4—106no—
293C2×C3×C4—102no—
294C2×C3×C4—106no—
295C2×C3×C4—1024no—
296C2×C3×C4C2×C22⋊D5112yes—
297C2×C3×C4C23×D512yes—
298C2×C3×C4C23×D516yes—
299C22×C5—26no—
300C22×C5—26no—
301C22×C5—26no—
302C22×C5—412no—
303C22×C5C22×C3×D412yes—
304C22×C5C2×C3×#4112yes—
305C22×C5C3×2−1+412yes—
306C22×C5C25×C311yescharacteristic
307C2×D5—22no—
308C2×D5—22no—
309C2×D5—28no—
310C2×D5C2×C3×#4114yescomplemented
311C4×C5—22no—
312C4×C5C22×C3×D4112yes—
313Dic5—26no—
314Dic5—26no—
315Dic5—22no—
316C24—51no—
317C24C22×C3×D513yes—
318#41—232no—
319C2×Q8—512no—
320C2×Q8—1032no—
321C2×Q8—1012no—
322C2×Q8C3×C22⋊D518yescomplemented
323C2×Q8C22×C3×D514yes—
324C2×D4—212no—
325C2×D4—512no—
326C2×D4—1012no—
327C2×D4C22×C3×D5112yes—
328C22×C4—212no—
329C22×C4—51no—
330C22×C4—53no—
331C22×C4—53no—
332C22×C4—512no—
333C22×C4—1012no—
334C22×C4C22×C3×D511yescharacteristic
335C22×C4C22×C3×D5112yes—
336C22×C4C22×C3×D513yes—
337C4⋊C4—512no—
338C4⋊C4—512no—
339C22⋊C4—512no—
340C22⋊C4—524no—
341C22⋊C4—54no—
342C42—512no—
343C3×C5C2×#54811yescharacteristic, complemented, Hall
344C22×C3—26no—
345C22×C3—26no—
346C22×C3—26no—
347C22×C3—412no—
348C22×C3—54no—
349C22×C3—108no—
350C22×C3—102no—
351C22×C3—102no—
352C22×C3D5×#4112yes—
353C22×C3#415012yes—
354C22×C3C22×C22⋊D512yes—
355C22×C3C24×D511yescharacteristic
356C3×C4—22no—
357C3×C4—106no—
358C3×C4—102no—
359C3×C4—106no—
360C3×C4C22×C22⋊D5112yes—
361C2×C5—46no—
362C2×C5C3×#54814yescomplemented
363C2×C5C23×C3×D411yescharacteristic
364C2×C5C22×C3×#4111yescharacteristic
365C2×C5C2×C3×2−1+411yescharacteristic
366D5—24no—
367C23—26no—
368C23—26no—
369C23—26no—
370C23—54no—
371C23—52no—
372C23—102no—
373C23—102no—
374C23C23×C3×D516yes—
375C23C23×C3×D511yescharacteristic
376Q8—1024no—
377Q8C2×C3×C22⋊D5116yescomplemented
378D4—224no—
379D4—1024no—
380C2×C4—22no—
381C2×C4—224no—
382C2×C4—52no—
383C2×C4—56no—
384C2×C4—56no—
385C2×C4—106no—
386C2×C4—102no—
387C2×C4—106no—
388C2×C4—1024no—
389C2×C4C2×C3×C22⋊D5112yes—
390C2×C4C23×C3×D512yes—
391C2×C4C23×C3×D516yes—
392C2×C3—46no—
393C2×C3—104no—
394C2×C3#6616514yesdirect product factor
395C2×C3C2×D5×#4111yescharacteristic
396C2×C3C2×#415011yescharacteristic
397C2×C3C23×C22⋊D511yescharacteristic
398C5C2×C3×#54811yescharacteristic, complemented, Sylow
399C22—26no—
400C22—26no—
401C22—26no—
402C22—412no—
403C22—54no—
404C22—102no—
405C22—102no—
406C22—108no—
407C22C3×D5×#4112yes—
408C22C3×#415012yes—
409C22C22×C3×C22⋊D512yes—
410C22C24×C3×D511yescharacteristic
411C4—22no—
412C4—106no—
413C4—102no—
414C4—106no—
415C4C22×C3×C22⋊D5112yes—
416C3C2×#6616511yescharacteristic, direct product factor, Sylow
417C2—46no—
418C2—104no—
419C2C3×#6616514yesdirect product factor
420C2C2×C3×D5×#4111yescharacteristic
421C2C2×C3×#415011yescharacteristic
422C2C23×C3×C22⋊D511yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC22×C3×C5×#41C2
socleC23×C3×C5C24
hypercenterC22×C3×#41D5
hypocenterC5C2×C3×#548
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×#41

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×#41C2semidirect
2. C22×C3×Q8×D5C2semidirect
3. C3×#66165C2direct (N × H)
4. C2×C3×Q8×Dic5C2semidirect
5. C2×C3×#4084C2semidirect
6. C2×C3×#4082C2semidirect
7. C2×C3×C4×C22⋊D5C2semidirect
8. C2×C3×#4070C2semidirect
9. C2×#66165C3direct (N × H)
11. C2×C3×C5×#41C22semidirect
17. C2×C3×Q8×D5C22semidirect
22. C3×Q8×Dic5C22semidirect
23. C3×#4084C22semidirect
24. C3×#4082C22semidirect
25. C3×C4×C22⋊D5C22semidirect
28. C3×#4070C22semidirect
32. C22×C5×#41C2×C3semidirect
33. C22×Q8×D5C2×C3semidirect
34. #66165C2×C3direct (N × H)
35. C2×Q8×Dic5C2×C3semidirect
36. C2×#4084C2×C3semidirect
37. C2×#4082C2×C3semidirect
38. C2×C4×C22⋊D5C2×C3semidirect
39. C2×#4070C2×C3semidirect
43. C2×C3×C5×Q8D4semidirect
67. C22×C3×#41D5semidirect
76. C2×C5×#41C22×C3semidirect
82. C2×Q8×D5C22×C3semidirect
87. Q8×Dic5C22×C3semidirect
88. #4084C22×C3semidirect
89. #4082C22×C3semidirect
90. C4×C22⋊D5C22×C3semidirect
93. #4070C22×C3semidirect
104. C22×C3×D5#41semidirect
106. C3×C5×Q8C2×D4semidirect
123. C2×C3×#41C2×D5semidirect
146. C2×C5×Q8C3×D4semidirect
170. C22×#41C3×D5semidirect
189. C2×C3×D5C2×#41semidirect
201. C2×C3×Q8C22⋊D5semidirect
229. C22×D5C3×#41semidirect
231. C5×Q8C2×C3×D4semidirect
248. C2×#41C2×C3×D5semidirect
269. C2×C3×C5#548semidirect
284. C3×Q8C2×C22⋊D5semidirect
310. C2×D5C2×C3×#41semidirect
322. C2×Q8C3×C22⋊D5semidirect
343. C3×C5C2×#548semidirect
362. C2×C5C3×#548semidirect
377. Q8C2×C3×C22⋊D5semidirect
394. C2×C3#66165direct (N × H)
398. C5C2×C3×#548semidirect
416. C3C2×#66165direct (N × H)
419. C2C3×#66165direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×#412
C22×C3×Q8×D52
C3×#661652
C2×C3×Q8×Dic52
C2×C3×#40842
C2×C3×#40822
C2×C3×C4×C22⋊D52
C2×C3×#40702
C2×#661653
C2×C3×#5485