Finite Group Databasebeta

← C2×C3×#65180 #14429938

Subgroups — C2×C3×#65180 #14429938

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4039C211yesmaximal, characteristic
2C2×C3×#4036C211yesmaximal, characteristic, complemented
3C2×C3×C5×#107C211yesmaximal, characteristic, complemented
4C2×C3×#3953C211yesmaximal, characteristic
5C2×C3×#3946C211yesmaximal, characteristic, complemented
6C2×C3×#3941C211yesmaximal, characteristic
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C3×#65180C218yesmaximal, direct product factor
9C2×#65180C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×D20C2211yescharacteristic
11C22×C3×C4×D5—21no—
12C22×C3×C5×C8C2211yescharacteristic
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×#685—21no—
16C2×C3×C4⋊Dic5—21no—
17C2×C3×C4⋊Dic5C2211yescharacteristic
18C2×C3×#683—21no—
19C2×C3×C4×Dic5C2211yescharacteristic
20C22×C3×#168C2211yescharacteristic
21C3×#4039—22no—
22C3×#4039C2214yes—
23C3×#4036—22no—
24C3×#4036C2214yescomplemented
25C3×C5×#107C2214yescomplemented
26C3×#3953C2214yes—
27C3×#3946C2214yescomplemented
28C3×#3941C2214yes—
29C3×#3940C2214yescomplemented
30C2×C3×#478—51nomaximal, Hall
31C2×#4039C2×C311yescharacteristic
32C2×#4036C2×C311yescharacteristic, complemented
33C2×C5×#107C2×C311yescharacteristic, complemented
34C2×#3953C2×C311yescharacteristic
35C2×#3946C2×C311yescharacteristic, complemented
36C2×#3941C2×C311yescharacteristic
37C2×#3940C2×C311yescharacteristic, complemented
38#65180C2×C318yesdirect product factor
39C23×C3×D5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×Dic5—21no—
44C22×C3×Dic5D411yescharacteristic
45C22×C3×Dic5D411yescharacteristic
46C2×C3×D20—21no—
47C2×C3×D20—22no—
48C2×C3×D20Q814yes—
49C2×C3×D20C2312yes—
50C2×C3×C4×D5—24no—
51C2×C3×C4×D5—44no—
52C2×C3×C5×C8—22no—
53C2×C3×C5×C8C2312yes—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4—21no—
56C3×C5×C4⋊C4C2312yes—
57C3×C5×C4⋊C4C2312yes—
58C3×#685—24no—
59C3×C4⋊Dic5—21no—
60C3×C4⋊Dic5—24no—
61C3×C4⋊Dic5C2312yes—
62C3×#683—24no—
63C3×C4×Dic5D412yes—
64C3×C4×Dic5C2312yes—
65C2×C3×#168—22no—
66C2×C3×#168C2312yes—
67C2×C3×#125—51no—
68C2×C3×C4×D4—51no—
69C3×#478—58no—
70C2×C3×#107D511yescharacteristic, complemented
71C2×C3×#106—51no—
72C2×C3×C4⋊C8—51no—
73C2×C3×#102—51no—
74C2×C3×#102—51no—
75C22×D20C22×C311yescharacteristic
76C22×C4×D5—21no—
77C22×C5×C8C22×C311yescharacteristic
78C2×C5×C4⋊C4C22×C311yescharacteristic
79C2×C5×C4⋊C4C22×C311yescharacteristic
80C2×#685—21no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5C22×C311yescharacteristic
83C2×#683—21no—
84C2×C4×Dic5C22×C311yescharacteristic
85C22×#168C22×C311yescharacteristic
86#4039—22no—
87#4039C22×C314yes—
88#4036—22no—
89#4036C22×C314yescomplemented
90C5×#107C22×C314yescomplemented
91#3953C22×C314yes—
92#3946C22×C314yescomplemented
93#3941C22×C314yes—
94#3940C22×C314yescomplemented
95C2×#478—51noSylow
96C23×C3×C5C2×D411yescharacteristic
97C22×C3×D5—24no—
98C22×C3×D5—22no—
99C22×C3×D5—42no—
100C22×C3×D5—42no—
101C2×C3×C4×C5—22no—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—42no—
104C2×C3×C4×C5—42no—
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×Q812yes—
107C2×C3×C4×C5#4112yes—
108C2×C3×C4×C5C2411yescharacteristic
109C2×C3×Dic5—22no—
110C2×C3×Dic5—22no—
111C2×C3×Dic5—22no—
112C2×C3×Dic5—42no—
113C2×C3×Dic5C2×D412yes—
114C2×C3×Dic5C2×D412yes—
115C3×D20—24no—
116C3×D20—22no—
117C3×D20C2×Q814yes—
118C3×C4×D5—48no—
119C3×C5×C8—22no—
120C3×#168—22no—
121C22×C3×D4—51no—
122C23×C3×C4—101no—
123C22×C3×C8—51no—
124C22×C3×C8C2×D511yescharacteristic
125C3×#125—54no—
126C3×#125—102no—
127C3×C4×D4—54no—
128C3×C4×D4—102no—
129C2×C3×C4⋊C4—51no—
130C2×C3×C4⋊C4—101no—
131C2×C3×C4⋊C4—101no—
132C2×C3×C4⋊C4C2×D511yescharacteristic
133C2×C3×C4⋊C4C2×D511yescharacteristic
134C2×C3×C22⋊C4—101no—
135C2×C3×C42—51no—
136C3×#107C2×D514yescomplemented
137C3×#106—54no—
138C3×C4⋊C8—54no—
139C3×#102—54no—
140C3×#102—54no—
141C23×D5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5C23×C311yescharacteristic
145C22×Dic5—21no—
146C22×Dic5C3×D411yescharacteristic
147C22×Dic5C3×D411yescharacteristic
148C2×D20—21no—
149C2×D20—22no—
150C2×D20C3×Q814yes—
151C2×D20C23×C312yes—
152C2×C4×D5—24no—
153C2×C4×D5—44no—
154C2×C5×C8—22no—
155C2×C5×C8C23×C312yes—
156C5×C4⋊C4—21no—
157C5×C4⋊C4—21no—
158C5×C4⋊C4C23×C312yes—
159C5×C4⋊C4C23×C312yes—
160#685—24no—
161C4⋊Dic5—21no—
162C4⋊Dic5—24no—
163C4⋊Dic5C23×C312yes—
164#683—24no—
165C4×Dic5C3×D412yes—
166C4×Dic5C23×C312yes—
167C2×#168—22no—
168C2×#168C23×C312yes—
169C2×#125—51no—
170C2×C4×D4—51no—
171#478—58no—
172C2×#107C3×D511yescharacteristic, complemented
173C2×#106—51no—
174C2×C4⋊C8—51no—
175C2×#102—51no—
176C2×#102—51no—
177C22×C3×C5C22⋊Q812yes—
178C22×C3×C5#13512yes—
179C22×C3×C5AGL(1,ℤ/8)12yes—
180C22×C3×C5C22×D411yescharacteristic
181C2×C3×D5—24no—
182C2×C3×D5—42no—
183C2×C3×D5—42no—
184C2×C3×D5—48no—
185C3×C4×C5—42no—
186C3×C4×C5—42no—
187C3×C4×C5C22⋊Q812yes—
188C3×C4×C5C22×Q811yescharacteristic
189C3×C4×C5C2×#4111yescharacteristic
190C3×Dic5—22no—
191C3×Dic5—22no—
192C3×Dic5—42no—
193C24×C3—101no—
194C2×C3×D4—54no—
195C2×C3×D4—52no—
196C2×C3×D4—102no—
197C2×C3×D4—101no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—51no—
201C22×C3×C4—51no—
202C22×C3×C4—104no—
203C22×C3×C4—101no—
204C22×C3×C4—204no—
205C22×C3×C4C22×D511yescharacteristic
206C2×C3×C8—22no—
207C2×C3×C8—52no—
208C2×C3×C8—102no—
209C2×C3×C8C22×D512yes—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—52no—
213C3×C4⋊C4—104no—
214C3×C4⋊C4—104no—
215C3×C4⋊C4—101no—
216C3×C4⋊C4C22×D512yes—
217C3×C4⋊C4C22×D512yes—
218C3×C22⋊C4—104no—
219C3×C42—52no—
220C3×C42—52no—
221C23×C5C2×C3×D411yescharacteristic
222C22×D5—24no—
223C22×D5—22no—
224C22×D5—42no—
225C22×D5—42no—
226C2×C4×C5—22no—
227C2×C4×C5—22no—
228C2×C4×C5—42no—
229C2×C4×C5—42no—
230C2×C4×C5C2×C3×D411yescharacteristic
231C2×C4×C5C2×C3×Q812yes—
232C2×C4×C5C3×#4112yes—
233C2×C4×C5C24×C311yescharacteristic
234C2×Dic5—22no—
235C2×Dic5—22no—
236C2×Dic5—22no—
237C2×Dic5—42no—
238C2×Dic5C2×C3×D412yes—
239C2×Dic5C2×C3×D412yes—
240D20—24no—
241D20—22no—
242D20C2×C3×Q814yes—
243C4×D5—48no—
244C5×C8—22no—
245#168—22no—
246C22×D4—51no—
247C23×C4—101no—
248C22×C8—51no—
249C22×C8C2×C3×D511yescharacteristic
250#125—54no—
251#125—102no—
252C4×D4—54no—
253C4×D4—102no—
254C2×C4⋊C4—51no—
255C2×C4⋊C4—101no—
256C2×C4⋊C4—101no—
257C2×C4⋊C4C2×C3×D511yescharacteristic
258C2×C4⋊C4C2×C3×D511yescharacteristic
259C2×C22⋊C4—101no—
260C2×C42—51no—
261#107C2×C3×D514yescomplemented
262#106—54no—
263C4⋊C8—54no—
264#102—54no—
265#102—54no—
266C2×C3×C5#47814yescomplemented
267C2×C3×C5C2×C22⋊Q811yescharacteristic
268C2×C3×C5C2×#13511yescharacteristic
269C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
270C3×D5—44no—
271C23×C3—102no—
272C23×C3—104no—
273C23×C3—202no—
274C23×C3—202no—
275C23×C3D4×D511yescharacteristic
276C3×D4—54no—
277C3×D4—102no—
278C3×D4—104no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—52no—
284C2×C3×C4—52no—
285C2×C3×C4—102no—
286C2×C3×C4—102no—
287C2×C3×C4—102no—
288C2×C3×C4—208no—
289C2×C3×C4—202no—
290C2×C3×C4#70912yes—
291C2×C3×C4D4×D511yescharacteristic
292C2×C3×C4Q8×D512yes—
293C2×C3×C4C23×D511yescharacteristic
294C3×C8—22no—
295C3×C8—102no—
296C22×C5C3×C22⋊Q812yes—
297C22×C5C3×#13512yes—
298C22×C5C3×AGL(1,ℤ/8)12yes—
299C22×C5C22×C3×D411yescharacteristic
300C2×D5—24no—
301C2×D5—42no—
302C2×D5—42no—
303C2×D5—48no—
304C4×C5—42no—
305C4×C5—42no—
306C4×C5C3×C22⋊Q812yes—
307C4×C5C22×C3×Q811yescharacteristic
308C4×C5C2×C3×#4111yescharacteristic
309Dic5—22no—
310Dic5—22no—
311Dic5—42no—
312C24—101no—
313C2×D4—54no—
314C2×D4—52no—
315C2×D4—102no—
316C2×D4—101no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—104no—
322C22×C4—101no—
323C22×C4—204no—
324C22×C4C22×C3×D511yescharacteristic
325C2×C8—22no—
326C2×C8—52no—
327C2×C8—102no—
328C2×C8C22×C3×D512yes—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—52no—
332C4⋊C4—104no—
333C4⋊C4—104no—
334C4⋊C4—101no—
335C4⋊C4C22×C3×D512yes—
336C4⋊C4C22×C3×D512yes—
337C22⋊C4—104no—
338C42—52no—
339C42—52no—
340C3×C5C2×#47811yescharacteristic, complemented, Hall
341C22×C3—104no—
342C22×C3—202no—
343C22×C3—202no—
344C22×C3—208no—
345C22×C3#404212yes—
346C22×C3#405712yes—
347C22×C3#406512yes—
348C22×C3C2×D4×D511yescharacteristic
349C3×C4—42no—
350C3×C4—42no—
351C3×C4—102no—
352C3×C4—102no—
353C3×C4—202no—
354C3×C4#404212yes—
355C3×C4C2×#70911yescharacteristic
356C3×C4C2×Q8×D511yescharacteristic
357C2×C5C3×#47814yescomplemented
358C2×C5C2×C3×C22⋊Q811yescharacteristic
359C2×C5C2×C3×#13511yescharacteristic
360C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
361D5—44no—
362C23—102no—
363C23—104no—
364C23—202no—
365C23—202no—
366C23C3×D4×D511yescharacteristic
367D4—54no—
368D4—102no—
369D4—104no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—52no—
375C2×C4—52no—
376C2×C4—102no—
377C2×C4—102no—
378C2×C4—102no—
379C2×C4—208no—
380C2×C4—202no—
381C2×C4C3×#70912yes—
382C2×C4C3×D4×D511yescharacteristic
383C2×C4C3×Q8×D512yes—
384C2×C4C23×C3×D511yescharacteristic
385C8—22no—
386C8—102no—
387C2×C3—204no—
388C2×C3#6518014yesdirect product factor
389C2×C3C2×#404211yescharacteristic
390C2×C3C2×#405711yescharacteristic
391C2×C3C2×#406511yescharacteristic
392C5C2×C3×#47811yescharacteristic, complemented, Sylow
393C22—104no—
394C22—202no—
395C22—208no—
396C22—202no—
397C22C3×#404212yes—
398C22C3×#405712yes—
399C22C3×#406512yes—
400C22C2×C3×D4×D511yescharacteristic
401C4—42no—
402C4—42no—
403C4—102no—
404C4—102no—
405C4—202no—
406C4C3×#404212yes—
407C4C2×C3×#70911yescharacteristic
408C4C2×C3×Q8×D511yescharacteristic
409C3C2×#6518011yescharacteristic, direct product factor, Sylow
410C2—204no—
411C2C3×#6518014yesdirect product factor
412C2C2×C3×#404211yescharacteristic
413C2C2×C3×#405711yescharacteristic
414C2C2×C3×#406511yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#107C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
hypercenterC2×C3×#107D5
hypocenterC5C2×C3×#478
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
2. C2×C3×#4036C2semidirect
3. C2×C3×C5×#107C2semidirect
5. C2×C3×#3946C2semidirect
7. C2×C3×#3940C2semidirect
8. C3×#65180C2direct (N × H)
9. C2×#65180C3direct (N × H)
24. C3×#4036C22semidirect
25. C3×C5×#107C22semidirect
27. C3×#3946C22semidirect
29. C3×#3940C22semidirect
32. C2×#4036C2×C3semidirect
33. C2×C5×#107C2×C3semidirect
35. C2×#3946C2×C3semidirect
37. C2×#3940C2×C3semidirect
38. #65180C2×C3direct (N × H)
70. C2×C3×#107D5semidirect
89. #4036C22×C3semidirect
90. C5×#107C22×C3semidirect
92. #3946C22×C3semidirect
94. #3940C22×C3semidirect
136. C3×#107C2×D5semidirect
172. C2×#107C3×D5semidirect
261. #107C2×C3×D5semidirect
266. C2×C3×C5#478semidirect
340. C3×C5C2×#478semidirect
357. C2×C5C3×#478semidirect
388. C2×C3#65180direct (N × H)
392. C5C2×C3×#478semidirect
409. C3C2×#65180direct (N × H)
411. C2C3×#65180direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40392
C2×C3×#40362
C2×C3×C5×#1072
C2×C3×#39532
C2×C3×#39462
C2×C3×#39412
C2×C3×#39402
C3×#651802
C2×#651803
C2×C3×#4785