Finite Group Databasebeta

← C2×C3×#65177 #14431615

Subgroups — C2×C3×#65177 #14431615

Order 1920.

832 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4043C211yesmaximal, characteristic
2C2×C3×#4038C211yesmaximal, characteristic
3C2×C3×C5×#107C211yesmaximal, characteristic, complemented
4C2×C3×#3952C211yesmaximal, characteristic
5C2×C3×#3948C211yesmaximal, characteristic, complemented
6C2×C3×#3942C211yesmaximal, characteristic, complemented
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C3×#65177C218yesmaximal, direct product factor
9C2×#65177C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C4×D5C2211yescharacteristic
11C22×C3×Dic10C2211yescharacteristic
12C22×C3×C5×C8C2211yescharacteristic
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×#685—21no—
16C2×C3×#685—21no—
17C2×C3×C4⋊Dic5—21no—
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×C3×C4×Dic5—21no—
20C22×C3×#168C2211yescharacteristic
21C3×#4043—22no—
22C3×#4043C2214yes—
23C3×#4038—22no—
24C3×#4038C2214yes—
25C3×C5×#107C2214yescomplemented
26C3×#3952C2214yes—
27C3×#3948C2214yescomplemented
28C3×#3942C2214yescomplemented
29C3×#3940C2214yescomplemented
30C2×C3×#485—51nomaximal, Hall
31C2×#4043C2×C311yescharacteristic
32C2×#4038C2×C311yescharacteristic
33C2×C5×#107C2×C311yescharacteristic, complemented
34C2×#3952C2×C311yescharacteristic
35C2×#3948C2×C311yescharacteristic, complemented
36C2×#3942C2×C311yescharacteristic, complemented
37C2×#3940C2×C311yescharacteristic, complemented
38#65177C2×C318yesdirect product factor
39C23×C3×D5D411yescharacteristic
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×Dic5—21no—
44C22×C3×Dic5—21no—
45C22×C3×Dic5D411yescharacteristic
46C2×C3×C4×D5—22no—
47C2×C3×C4×D5—22no—
48C2×C3×C4×D5D412yes—
49C2×C3×C4×D5C2312yes—
50C2×C3×Dic10—22no—
51C2×C3×Dic10—22no—
52C2×C3×Dic10—21no—
53C2×C3×Dic10C2312yes—
54C2×C3×C5×C8—22no—
55C2×C3×C5×C8C2312yes—
56C3×C5×C4⋊C4—21no—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4C2312yes—
59C3×C5×C4⋊C4C2312yes—
60C3×#685—24no—
61C3×#685—24no—
62C3×C4⋊Dic5—21no—
63C3×C4⋊Dic5—24no—
64C3×C4⋊Dic5C2312yes—
65C3×C4×Dic5—24no—
66C2×C3×#168—22no—
67C2×C3×#168C2312yes—
68C2×C3×C22⋊Q8—51no—
69C2×C3×#117—51no—
70C3×#485—58no—
71C2×C3×#107D511yescharacteristic, complemented
72C2×C3×#106—51no—
73C2×C3×#103—51no—
74C2×C3×#103—51no—
75C2×C3×C22⋊C8—51no—
76C22×C4×D5C22×C311yescharacteristic
77C22×Dic10C22×C311yescharacteristic
78C22×C5×C8C22×C311yescharacteristic
79C2×C5×C4⋊C4C22×C311yescharacteristic
80C2×C5×C4⋊C4C22×C311yescharacteristic
81C2×#685—21no—
82C2×#685—21no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5C22×C311yescharacteristic
85C2×C4×Dic5—21no—
86C22×#168C22×C311yescharacteristic
87#4043—22no—
88#4043C22×C314yes—
89#4038—22no—
90#4038C22×C314yes—
91C5×#107C22×C314yescomplemented
92#3952C22×C314yes—
93#3948C22×C314yescomplemented
94#3942C22×C314yescomplemented
95#3940C22×C314yescomplemented
96C2×#485—51noSylow
97C23×C3×C5C2×D411yescharacteristic
98C22×C3×D5—22no—
99C22×C3×D5—22no—
100C22×C3×D5—22no—
101C22×C3×D5C2×D412yes—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—42no—
105C2×C3×C4×C5—42no—
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5#4112yes—
108C2×C3×C4×C5#4112yes—
109C2×C3×C4×C5C2411yescharacteristic
110C2×C3×Dic5—22no—
111C2×C3×Dic5—22no—
112C2×C3×Dic5—22no—
113C2×C3×Dic5—42no—
114C2×C3×Dic5—42no—
115C2×C3×Dic5C2×D412yes—
116C3×C4×D5—24no—
117C3×C4×D5—22no—
118C3×C4×D5—22no—
119C3×Dic10—22no—
120C3×Dic10—22no—
121C3×Dic10—42no—
122C3×C5×C8—22no—
123C3×#168—22no—
124C22×C3×Q8—51no—
125C23×C3×C4—51no—
126C22×C3×C8—51no—
127C22×C3×C8C2×D511yescharacteristic
128C3×C22⋊Q8—54no—
129C3×C22⋊Q8—102no—
130C3×#117—54no—
131C3×#117—102no—
132C2×C3×C4⋊C4—51no—
133C2×C3×C4⋊C4—101no—
134C2×C3×C4⋊C4C2×D511yescharacteristic
135C2×C3×C4⋊C4C2×D511yescharacteristic
136C2×C3×C22⋊C4—101no—
137C2×C3×C22⋊C4—101no—
138C2×C3×C42—101no—
139C3×#107C2×D514yescomplemented
140C3×#106—54no—
141C3×#103—54no—
142C3×#103—54no—
143C3×C22⋊C8—54no—
144C23×D5C3×D411yescharacteristic
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5C23×C311yescharacteristic
148C22×Dic5—21no—
149C22×Dic5—21no—
150C22×Dic5C3×D411yescharacteristic
151C2×C4×D5—22no—
152C2×C4×D5—22no—
153C2×C4×D5C3×D412yes—
154C2×C4×D5C23×C312yes—
155C2×Dic10—22no—
156C2×Dic10—22no—
157C2×Dic10—21no—
158C2×Dic10C23×C312yes—
159C2×C5×C8—22no—
160C2×C5×C8C23×C312yes—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4C23×C312yes—
164C5×C4⋊C4C23×C312yes—
165#685—24no—
166#685—24no—
167C4⋊Dic5—21no—
168C4⋊Dic5—24no—
169C4⋊Dic5C23×C312yes—
170C4×Dic5—24no—
171C2×#168—22no—
172C2×#168C23×C312yes—
173C2×C22⋊Q8—51no—
174C2×#117—51no—
175#485—58no—
176C2×#107C3×D511yescharacteristic, complemented
177C2×#106—51no—
178C2×#103—51no—
179C2×#103—51no—
180C2×C22⋊C8—51no—
181C22×C3×C5#12312yes—
182C22×C3×C5#13512yes—
183C22×C3×C5#13712yes—
184C22×C3×C5C22×D411yescharacteristic
185C2×C3×D5—22no—
186C2×C3×D5—22no—
187C2×C3×D5—22no—
188C2×C3×D5—44no—
189C3×C4×C5—42no—
190C3×C4×C5—42no—
191C3×C4×C5#12312yes—
192C3×C4×C5C2×#4111yescharacteristic
193C3×C4×C5C2×#4111yescharacteristic
194C3×Dic5—22no—
195C3×Dic5—42no—
196C3×Dic5—42no—
197C24×C3—51no—
198C2×C3×Q8—52no—
199C2×C3×Q8—101no—
200C2×C3×Q8—102no—
201C2×C3×Q8—102no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—51no—
205C22×C3×C4—52no—
206C22×C3×C4—52no—
207C22×C3×C4—102no—
208C22×C3×C4—102no—
209C22×C3×C4—101no—
210C22×C3×C4—101no—
211C22×C3×C4C22×D511yescharacteristic
212C2×C3×C8—22no—
213C2×C3×C8—52no—
214C2×C3×C8—102no—
215C2×C3×C8C22×D512yes—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—52no—
219C3×C4⋊C4—104no—
220C3×C4⋊C4—101no—
221C3×C4⋊C4C22×D512yes—
222C3×C4⋊C4C22×D512yes—
223C3×C22⋊C4—104no—
224C3×C22⋊C4—104no—
225C3×C42—104no—
226C23×C5C2×C3×D411yescharacteristic
227C22×D5—22no—
228C22×D5—22no—
229C22×D5—22no—
230C22×D5C2×C3×D412yes—
231C2×C4×C5—22no—
232C2×C4×C5—22no—
233C2×C4×C5—42no—
234C2×C4×C5—42no—
235C2×C4×C5C2×C3×D411yescharacteristic
236C2×C4×C5C3×#4112yes—
237C2×C4×C5C3×#4112yes—
238C2×C4×C5C24×C311yescharacteristic
239C2×Dic5—22no—
240C2×Dic5—22no—
241C2×Dic5—22no—
242C2×Dic5—42no—
243C2×Dic5—42no—
244C2×Dic5C2×C3×D412yes—
245C4×D5—24no—
246C4×D5—22no—
247C4×D5—22no—
248Dic10—22no—
249Dic10—22no—
250Dic10—42no—
251C5×C8—22no—
252#168—22no—
253C22×Q8—51no—
254C23×C4—51no—
255C22×C8—51no—
256C22×C8C2×C3×D511yescharacteristic
257C22⋊Q8—54no—
258C22⋊Q8—102no—
259#117—54no—
260#117—102no—
261C2×C4⋊C4—51no—
262C2×C4⋊C4—101no—
263C2×C4⋊C4C2×C3×D511yescharacteristic
264C2×C4⋊C4C2×C3×D511yescharacteristic
265C2×C22⋊C4—101no—
266C2×C22⋊C4—101no—
267C2×C42—101no—
268#107C2×C3×D514yescomplemented
269#106—54no—
270#103—54no—
271#103—54no—
272C22⋊C8—54no—
273C2×C3×C5#48514yescomplemented
274C2×C3×C5C2×#12311yescharacteristic
275C2×C3×C5C2×#13511yescharacteristic
276C2×C3×C5C2×#13711yescharacteristic
277C3×D5—42no—
278C23×C3—52no—
279C23×C3—102no—
280C23×C3—102no—
281C23×C3—102no—
282C23×C3D4×D511yescharacteristic
283C3×Q8—102no—
284C3×Q8—102no—
285C3×Q8—202no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—52no—
291C2×C3×C4—102no—
292C2×C3×C4—102no—
293C2×C3×C4—102no—
294C2×C3×C4—104no—
295C2×C3×C4—102no—
296C2×C3×C4—102no—
297C2×C3×C4—202no—
298C2×C3×C4—202no—
299C2×C3×C4#70912yes—
300C2×C3×C4D4×D511yescharacteristic
301C2×C3×C4#71312yes—
302C2×C3×C4C23×D511yescharacteristic
303C3×C8—22no—
304C3×C8—102no—
305C22×C5C3×#12312yes—
306C22×C5C3×#13512yes—
307C22×C5C3×#13712yes—
308C22×C5C22×C3×D411yescharacteristic
309C2×D5—22no—
310C2×D5—22no—
311C2×D5—22no—
312C2×D5—44no—
313C4×C5—42no—
314C4×C5—42no—
315C4×C5C3×#12312yes—
316C4×C5C2×C3×#4111yescharacteristic
317C4×C5C2×C3×#4111yescharacteristic
318Dic5—22no—
319Dic5—42no—
320Dic5—42no—
321C24—51no—
322C2×Q8—52no—
323C2×Q8—101no—
324C2×Q8—102no—
325C2×Q8—102no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—51no—
329C22×C4—52no—
330C22×C4—52no—
331C22×C4—102no—
332C22×C4—102no—
333C22×C4—101no—
334C22×C4—101no—
335C22×C4C22×C3×D511yescharacteristic
336C2×C8—22no—
337C2×C8—52no—
338C2×C8—102no—
339C2×C8C22×C3×D512yes—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—52no—
343C4⋊C4—104no—
344C4⋊C4—101no—
345C4⋊C4C22×C3×D512yes—
346C4⋊C4C22×C3×D512yes—
347C22⋊C4—104no—
348C22⋊C4—104no—
349C42—104no—
350C3×C5C2×#48511yescharacteristic, complemented, Hall
351C22×C3—102no—
352C22×C3—102no—
353C22×C3—102no—
354C22×C3—204no—
355C22×C3#404012yes—
356C22×C3#405812yes—
357C22×C3#406412yes—
358C22×C3C2×D4×D511yescharacteristic
359C3×C4—42no—
360C3×C4—42no—
361C3×C4—102no—
362C3×C4—202no—
363C3×C4—202no—
364C3×C4#404012yes—
365C3×C4C2×#70911yescharacteristic
366C3×C4C2×#71311yescharacteristic
367C2×C5C3×#48514yescomplemented
368C2×C5C2×C3×#12311yescharacteristic
369C2×C5C2×C3×#13511yescharacteristic
370C2×C5C2×C3×#13711yescharacteristic
371D5—42no—
372C23—52no—
373C23—102no—
374C23—102no—
375C23—102no—
376C23C3×D4×D511yescharacteristic
377Q8—102no—
378Q8—102no—
379Q8—202no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—52no—
385C2×C4—102no—
386C2×C4—102no—
387C2×C4—102no—
388C2×C4—104no—
389C2×C4—102no—
390C2×C4—102no—
391C2×C4—202no—
392C2×C4—202no—
393C2×C4C3×#70912yes—
394C2×C4C3×D4×D511yescharacteristic
395C2×C4C3×#71312yes—
396C2×C4C23×C3×D511yescharacteristic
397C8—22no—
398C8—102no—
399C2×C3—202no—
400C2×C3#6517714yesdirect product factor
401C2×C3C2×#404011yescharacteristic
402C2×C3C2×#405811yescharacteristic
403C2×C3C2×#406411yescharacteristic
404C5C2×C3×#48511yescharacteristic, complemented, Sylow
405C22—102no—
406C22—102no—
407C22—102no—
408C22—204no—
409C22C3×#404012yes—
410C22C3×#405812yes—
411C22C3×#406412yes—
412C22C2×C3×D4×D511yescharacteristic
413C4—42no—
414C4—42no—
415C4—102no—
416C4—202no—
417C4—202no—
418C4C3×#404012yes—
419C4C2×C3×#70911yescharacteristic
420C4C2×C3×#71311yescharacteristic
421C3C2×#6517711yescharacteristic, direct product factor, Sylow
422C2—202no—
423C2C3×#6517714yesdirect product factor
424C2C2×C3×#404011yescharacteristic
425C2C2×C3×#405811yescharacteristic
426C2C2×C3×#406411yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#107C2
socleC23×C3×C5C2×D4
prime radicalC23×C3×D5D4
hypercenterC2×C3×#107D5
hypocenterC5C2×C3×#485
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#107

Complements

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

NH ≅ G/Ntype
3. C2×C3×C5×#107C2semidirect
5. C2×C3×#3948C2semidirect
6. C2×C3×#3942C2semidirect
7. C2×C3×#3940C2semidirect
8. C3×#65177C2direct (N × H)
9. C2×#65177C3direct (N × H)
25. C3×C5×#107C22semidirect
27. C3×#3948C22semidirect
28. C3×#3942C22semidirect
29. C3×#3940C22semidirect
33. C2×C5×#107C2×C3semidirect
35. C2×#3948C2×C3semidirect
36. C2×#3942C2×C3semidirect
37. C2×#3940C2×C3semidirect
38. #65177C2×C3direct (N × H)
71. C2×C3×#107D5semidirect
91. C5×#107C22×C3semidirect
93. #3948C22×C3semidirect
94. #3942C22×C3semidirect
95. #3940C22×C3semidirect
139. C3×#107C2×D5semidirect
176. C2×#107C3×D5semidirect
268. #107C2×C3×D5semidirect
273. C2×C3×C5#485semidirect
350. C3×C5C2×#485semidirect
367. C2×C5C3×#485semidirect
400. C2×C3#65177direct (N × H)
404. C5C2×C3×#485semidirect
421. C3C2×#65177direct (N × H)
423. C2C3×#65177direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40432
C2×C3×#40382
C2×C3×C5×#1072
C2×C3×#39522
C2×C3×#39482
C2×C3×#39422
C2×C3×#39402
C3×#651772
C2×#651773
C2×C3×#4855