Finite Group Databasebeta

← C3×D4×C4⋊Dic5 #14327011

Subgroups — C3×D4×C4⋊Dic5 #14327011

Order 1920.

1552 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
1C2×C3×C4×C5×D4C211yesmaximal, characteristic
2C2×C3×D4×Dic5C212yesmaximal
3C22×C3×C4⋊Dic5C212yesmaximal, complemented
4C3×#65501C212yesmaximal, complemented
5C3×#65274C212yesmaximal, complemented
6C3×#65239C214yesmaximal, complemented
7C3×#65224C211yesmaximal, characteristic, complemented
8C3×C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
9D4×C4⋊Dic5C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C23×C3×C4×C5C2212yes—
12C23×C3×Dic5C2214yes—
13C3×C4×C5×D4C418yescomplemented
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C22⋊C4C2212yes—
16C2×C3×C42×C5C2211yescharacteristic
17C2×C3×C22⋊Dic5C2214yes—
18C3×D4×Dic5—28no—
19C2×C3×C4⋊Dic5—28no—
20C2×C3×C4⋊Dic5C2212yes—
21C2×C3×C4⋊Dic5C2214yes—
22C2×C3×C4⋊Dic5C2212yes—
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×C4⋊Dic5C2212yes—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2212yes—
27C3×#3963C2212yes—
28C3×#3963C2214yes—
29C3×D4×C4⋊C4—51nomaximal, Hall
30C2×C4×C5×D4C2×C311yescharacteristic
31C2×D4×Dic5C2×C312yes—
32C22×C4⋊Dic5C2×C312yescomplemented
33#65501C2×C312yescomplemented
34#65274C2×C312yescomplemented
35#65239C2×C314yescomplemented
36#65224C2×C311yescharacteristic, complemented
37C4×C4⋊Dic5C2×C311yescharacteristic, complemented
38C24×C3×C5C2312yes—
39C2×C3×C5×D4C2×C414yes—
40C2×C3×C5×D4D414yes—
41C2×C3×C5×D4Q814yes—
42C22×C3×C4×C5—24no—
43C22×C3×C4×C5—24no—
44C22×C3×C4×C5C2×C418yescomplemented
45C22×C3×C4×C5C2311yescharacteristic
46C22×C3×C4×C5C2312yes—
47C22×C3×C4×C5C2311yescharacteristic
48C22×C3×C4×C5C2311yescharacteristic
49C22×C3×Dic5—28no—
50C22×C3×Dic5—28no—
51C22×C3×Dic5—44no—
52C22×C3×Dic5C2312yes—
53C22×C3×Dic5C2314yes—
54C22×C3×Dic5C2312yes—
55C3×C5×C4⋊C4C2×C414yescomplemented
56C3×C5×C22⋊C4C2×C418yescomplemented
57C3×C42×C5C2×C414yescomplemented
58C3×C22⋊Dic5—28no—
59C3×C4⋊Dic5—24no—
60C3×C4⋊Dic5—24no—
61C3×C4⋊Dic5—22no—
62C3×C4⋊Dic5—28no—
63C3×C4⋊Dic5—42no—
64C3×C4⋊Dic5D414yesdirect product factor
65C3×C4×Dic5—24no—
66C2×C3×C4×D4—52no—
67C2×C3×C4×D4D511yescharacteristic
68C22×C3×C4⋊C4—52no—
69C3×#389—52no—
70C3×#385—54no—
71C3×#384—51no—
72C3×#380—52no—
73C3×C4×C4⋊C4—51no—
74C22×C5×D4C22×C311yescharacteristic
75C23×C4×C5C22×C312yes—
76C23×Dic5C22×C314yes—
77C4×C5×D4C3×C418yescomplemented
78C2×C5×C4⋊C4C22×C311yescharacteristic
79C2×C5×C22⋊C4C22×C312yes—
80C2×C42×C5C22×C311yescharacteristic
81C2×C22⋊Dic5C22×C314yes—
82D4×Dic5—28no—
83C2×C4⋊Dic5—28no—
84C2×C4⋊Dic5C22×C312yes—
85C2×C4⋊Dic5C22×C314yes—
86C2×C4⋊Dic5C22×C312yes—
87C2×C4⋊Dic5C22×C311yescharacteristic
88C2×C4⋊Dic5C22×C312yes—
89C2×C4⋊Dic5C22×C311yescharacteristic
90C2×C4×Dic5C22×C312yes—
91#3963C22×C312yes—
92#3963C22×C314yes—
93D4×C4⋊C4—51noSylow
94C23×C3×C5—22no—
95C23×C3×C5—22no—
96C23×C3×C5—22no—
97C23×C3×C5—22no—
98C23×C3×C5C22×C414yes—
99C23×C3×C5C2×D414yes—
100C23×C3×C5C2×Q814yes—
101C23×C3×C5C2411yescharacteristic
102C3×C5×D4C4⋊C4116yescomplemented
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—216no—
108C2×C3×C4×C5C22×C412yes—
109C2×C3×C4×C5C22×C414yes—
110C2×C3×C4×C5C22×C412yes—
111C2×C3×C4×C5C22×C412yes—
112C2×C3×C4×C5C2×D412yes—
113C2×C3×C4×C5C2×D412yes—
114C2×C3×C4×C5C2×Q812yes—
115C2×C3×C4×C5#4112yes—
116C2×C3×Dic5—22no—
117C2×C3×Dic5—22no—
118C2×C3×Dic5—22no—
119C2×C3×Dic5—24no—
120C2×C3×Dic5—24no—
121C2×C3×Dic5—22no—
122C2×C3×Dic5—24no—
123C2×C3×Dic5—22no—
124C2×C3×Dic5—416no—
125C2×C3×Dic5C2×D414yes—
126C22×C3×D4C2×D511yescharacteristic
127C23×C3×C4—54no—
128C23×C3×C4C2×D512yes—
129C3×C4×D4—108no—
130C3×C4×D4Dic518yescomplemented
131C2×C3×C4⋊C4—51no—
132C2×C3×C4⋊C4—51no—
133C2×C3×C4⋊C4—52no—
134C2×C3×C4⋊C4—52no—
135C2×C3×C4⋊C4—54no—
136C2×C3×C4⋊C4—52no—
137C2×C3×C4⋊C4—108no—
138C2×C3×C4⋊C4C2×D511yescharacteristic
139C2×C3×C22⋊C4—54no—
140C2×C3×C22⋊C4C2×D512yes—
141C2×C3×C42—52no—
142C2×C3×C42C2×D511yescharacteristic
143C3×#95—52no—
144C3×#95—54no—
145C24×C5C23×C312yes—
146C2×C5×D4C2×C3×C414yes—
147C2×C5×D4C3×D414yes—
148C2×C5×D4C3×Q814yes—
149C22×C4×C5—24no—
150C22×C4×C5—24no—
151C22×C4×C5C2×C3×C418yescomplemented
152C22×C4×C5C23×C311yescharacteristic
153C22×C4×C5C23×C312yes—
154C22×C4×C5C23×C311yescharacteristic
155C22×C4×C5C23×C311yescharacteristic
156C22×Dic5—28no—
157C22×Dic5—28no—
158C22×Dic5—44no—
159C22×Dic5C23×C312yes—
160C22×Dic5C23×C314yes—
161C22×Dic5C23×C312yes—
162C5×C4⋊C4C2×C3×C414yescomplemented
163C5×C22⋊C4C2×C3×C418yescomplemented
164C42×C5C2×C3×C414yescomplemented
165C22⋊Dic5—28no—
166C4⋊Dic5—24no—
167C4⋊Dic5—24no—
168C4⋊Dic5—22no—
169C4⋊Dic5—28no—
170C4⋊Dic5—42no—
171C4⋊Dic5C3×D414yesdirect product factor
172C4×Dic5—24no—
173C2×C4×D4—52no—
174C2×C4×D4C3×D511yescharacteristic
175C22×C4⋊C4—52no—
176#389—52no—
177#385—54no—
178#384—51no—
179#380—52no—
180C4×C4⋊C4—51no—
181C22×C3×C5—24no—
182C22×C3×C5—24no—
183C22×C3×C5—24no—
184C22×C3×C5—24no—
185C22×C3×C5—24no—
186C22×C3×C5—24no—
187C22×C3×C5C2×C4⋊C418yescomplemented
188C22×C3×C5C23×C411yescharacteristic
189C22×C3×C5C22×D411yescharacteristic
190C22×C3×C5C22×D411yescharacteristic
191C22×C3×C5C22×Q811yescharacteristic
192C22×C3×C5C2×#4111yescharacteristic
193C22×C3×C5H(5,𝔽2)11yescharacteristic
194C22×C3×C52−1+411yescharacteristic
195C3×C4×C5—24no—
196C3×C4×C5—22no—
197C3×C4×C5C2×C4⋊C414yescomplemented
198C3×C4×C5C4×D414yescomplemented
199C3×Dic5—24no—
200C3×Dic5—44no—
201C3×Dic5—42no—
202C24×C3C22×D512yes—
203C2×C3×D4Dic1014yes—
204C2×C3×D4D2014yes—
205C2×C3×D4C2×Dic514yes—
206C22×C3×C4—24no—
207C22×C3×C4—24no—
208C22×C3×C4—52no—
209C22×C3×C4—52no—
210C22×C3×C4—54no—
211C22×C3×C4—108no—
212C22×C3×C4—108no—
213C22×C3×C4—204no—
214C22×C3×C4C2×Dic518yescomplemented
215C22×C3×C4C22×D511yescharacteristic
216C22×C3×C4C22×D512yes—
217C22×C3×C4C22×D511yescharacteristic
218C22×C3×C4C22×D511yescharacteristic
219C3×C4⋊C4—54no—
220C3×C4⋊C4—108no—
221C3×C4⋊C4—104no—
222C3×C4⋊C4—104no—
223C3×C4⋊C4—102no—
224C3×C4⋊C4—202no—
225C3×C4⋊C4C2×Dic514yescomplemented
226C3×C22⋊C4—108no—
227C3×C22⋊C4C2×Dic518yescomplemented
228C3×C42—104no—
229C3×C42C2×Dic514yescomplemented
230C23×C5—22no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—22no—
234C23×C5C22×C3×C414yes—
235C23×C5C2×C3×D414yes—
236C23×C5C2×C3×Q814yes—
237C23×C5C24×C311yescharacteristic
238C5×D4C3×C4⋊C4116yescomplemented
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—216no—
244C2×C4×C5C22×C3×C412yes—
245C2×C4×C5C22×C3×C414yes—
246C2×C4×C5C22×C3×C412yes—
247C2×C4×C5C22×C3×C412yes—
248C2×C4×C5C2×C3×D412yes—
249C2×C4×C5C2×C3×D412yes—
250C2×C4×C5C2×C3×Q812yes—
251C2×C4×C5C3×#4112yes—
252C2×Dic5—22no—
253C2×Dic5—22no—
254C2×Dic5—22no—
255C2×Dic5—24no—
256C2×Dic5—24no—
257C2×Dic5—22no—
258C2×Dic5—24no—
259C2×Dic5—22no—
260C2×Dic5—416no—
261C2×Dic5C2×C3×D414yes—
262C22×D4C2×C3×D511yescharacteristic
263C23×C4—54no—
264C23×C4C2×C3×D512yes—
265C4×D4—108no—
266C4×D4C3×Dic518yescomplemented
267C2×C4⋊C4—51no—
268C2×C4⋊C4—52no—
269C2×C4⋊C4—54no—
270C2×C4⋊C4—52no—
271C2×C4⋊C4—51no—
272C2×C4⋊C4—52no—
273C2×C4⋊C4—108no—
274C2×C4⋊C4C2×C3×D511yescharacteristic
275C2×C22⋊C4—54no—
276C2×C22⋊C4C2×C3×D512yes—
277C2×C42—52no—
278C2×C42C2×C3×D511yescharacteristic
279#95—52no—
280#95—54no—
281C2×C3×C5—28no—
282C2×C3×C5C22×C4⋊C411yescharacteristic
283C2×C3×C5C2×C4×D411yescharacteristic
284C2×C3×C5#52011yescharacteristic
285C2×C3×C5D4211yescharacteristic
286C2×C3×C5#54711yescharacteristic
287C2×C3×C5D4×Q811yescharacteristic
288C2×C3×C5#55411yescharacteristic
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3C2×Dic1014yes—
294C23×C3C2×D2014yes—
295C23×C3C22×Dic514yes—
296C23×C3C23×D511yescharacteristic
297C3×D4C4⋊Dic5116yesdirect product factor
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—216no—
303C2×C3×C4—54no—
304C2×C3×C4—102no—
305C2×C3×C4—102no—
306C2×C3×C4—102no—
307C2×C3×C4—104no—
308C2×C3×C4—104no—
309C2×C3×C4—102no—
310C2×C3×C4—104no—
311C2×C3×C4—102no—
312C2×C3×C4—2016no—
313C2×C3×C4C2×Dic1012yes—
314C2×C3×C4C2×D2012yes—
315C2×C3×C4D4×D512yes—
316C2×C3×C4#71112yes—
317C2×C3×C4C22×Dic512yes—
318C2×C3×C4C22×Dic514yes—
319C2×C3×C4C22×Dic512yes—
320C2×C3×C4C22×Dic512yes—
321C22×C5—24no—
322C22×C5—24no—
323C22×C5—24no—
324C22×C5—24no—
325C22×C5—24no—
326C22×C5—24no—
327C22×C5C2×C3×C4⋊C418yescomplemented
328C22×C5C23×C3×C411yescharacteristic
329C22×C5C22×C3×D411yescharacteristic
330C22×C5C22×C3×D411yescharacteristic
331C22×C5C22×C3×Q811yescharacteristic
332C22×C5C2×C3×#4111yescharacteristic
333C22×C5C3×H(5,𝔽2)11yescharacteristic
334C22×C5C3×2−1+411yescharacteristic
335C4×C5—24no—
336C4×C5—22no—
337C4×C5C2×C3×C4⋊C414yescomplemented
338C4×C5C3×C4×D414yescomplemented
339Dic5—24no—
340Dic5—44no—
341Dic5—42no—
342C24C22×C3×D512yes—
343C2×D4C3×Dic1014yes—
344C2×D4C3×D2014yes—
345C2×D4C2×C3×Dic514yes—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—52no—
349C22×C4—52no—
350C22×C4—54no—
351C22×C4—108no—
352C22×C4—108no—
353C22×C4—204no—
354C22×C4C2×C3×Dic518yescomplemented
355C22×C4C22×C3×D511yescharacteristic
356C22×C4C22×C3×D512yes—
357C22×C4C22×C3×D511yescharacteristic
358C22×C4C22×C3×D511yescharacteristic
359C4⋊C4—54no—
360C4⋊C4—108no—
361C4⋊C4—104no—
362C4⋊C4—104no—
363C4⋊C4—102no—
364C4⋊C4—202no—
365C4⋊C4C2×C3×Dic514yescomplemented
366C22⋊C4—108no—
367C22⋊C4C2×C3×Dic518yescomplemented
368C42—104no—
369C42C2×C3×Dic514yescomplemented
370C3×C5D4×C4⋊C411yescharacteristic, complemented, Hall
371C22×C3—24no—
372C22×C3—24no—
373C22×C3—24no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3C2×C4⋊Dic518yescomplemented
378C22×C3C22×Dic1011yescharacteristic
379C22×C3C22×D2011yescharacteristic
380C22×C3C2×D4×D511yescharacteristic
381C22×C3C2×#71111yescharacteristic
382C22×C3#414911yescharacteristic
383C22×C3#415011yescharacteristic
384C22×C3C23×Dic511yescharacteristic
385C3×C4—24no—
386C3×C4—22no—
387C3×C4—104no—
388C3×C4—204no—
389C3×C4—202no—
390C3×C4C2×C4⋊Dic514yescomplemented
391C3×C4D4×Dic514yescomplemented
392C2×C5—28no—
393C2×C5C22×C3×C4⋊C411yescharacteristic
394C2×C5C2×C3×C4×D411yescharacteristic
395C2×C5C3×#52011yescharacteristic
396C2×C5C3×D4211yescharacteristic
397C2×C5C3×#54711yescharacteristic
398C2×C5C3×D4×Q811yescharacteristic
399C2×C5C3×#55411yescharacteristic
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23C2×C3×Dic1014yes—
405C23C2×C3×D2014yes—
406C23C22×C3×Dic514yes—
407C23C23×C3×D511yescharacteristic
408D4C3×C4⋊Dic5116yesdirect product factor
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—216no—
414C2×C4—54no—
415C2×C4—102no—
416C2×C4—102no—
417C2×C4—102no—
418C2×C4—104no—
419C2×C4—104no—
420C2×C4—102no—
421C2×C4—104no—
422C2×C4—102no—
423C2×C4—2016no—
424C2×C4C2×C3×Dic1012yes—
425C2×C4C2×C3×D2012yes—
426C2×C4C3×D4×D512yes—
427C2×C4C3×#71112yes—
428C2×C4C22×C3×Dic512yes—
429C2×C4C22×C3×Dic514yes—
430C2×C4C22×C3×Dic512yes—
431C2×C4C22×C3×Dic512yes—
432C2×C3—28no—
433C2×C3D4×Dic1011yescharacteristic
434C2×C3#6587711yescharacteristic
435C2×C3D4×D2011yescharacteristic
436C2×C3#6588911yescharacteristic
437C2×C3C22×C4⋊Dic511yescharacteristic
438C2×C3C2×D4×Dic511yescharacteristic
439C2×C3#6616111yescharacteristic
440C5C3×D4×C4⋊C411yescharacteristic, complemented, Sylow
441C22—24no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—24no—
447C22C2×C3×C4⋊Dic518yescomplemented
448C22C22×C3×Dic1011yescharacteristic
449C22C22×C3×D2011yescharacteristic
450C22C2×C3×D4×D511yescharacteristic
451C22C2×C3×#71111yescharacteristic
452C22C3×#414911yescharacteristic
453C22C3×#415011yescharacteristic
454C22C23×C3×Dic511yescharacteristic
455C4—24no—
456C4—22no—
457C4—104no—
458C4—204no—
459C4—202no—
460C4C2×C3×C4⋊Dic514yescomplemented
461C4C3×D4×Dic514yescomplemented
462C3D4×C4⋊Dic511yescharacteristic, direct product factor, Sylow
463C2—28no—
464C2C3×D4×Dic1011yescharacteristic
465C2C3×#6587711yescharacteristic
466C2C3×D4×D2011yescharacteristic
467C2C3×#6588911yescharacteristic
468C2C22×C3×C4⋊Dic511yescharacteristic
469C2C2×C3×D4×Dic511yescharacteristic
470C2C3×#6616111yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC2×C3×C4×C5×D4C2
socleC23×C3×C5C24
prime radicalC22×C3×C5×D4C22
cosocleC23×C5C24×C3
hypercenterC2×C3×C4×D4D5
hypocenterC5C3×D4×C4⋊C4
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C22×C3×C4⋊Dic5C2semidirect
4. C3×#65501C2semidirect
5. C3×#65274C2semidirect
6. C3×#65239C2semidirect
7. C3×#65224C2semidirect
8. C3×C4×C4⋊Dic5C2semidirect
9. D4×C4⋊Dic5C3direct (N × H)
13. C3×C4×C5×D4C4semidirect
32. C22×C4⋊Dic5C2×C3semidirect
33. #65501C2×C3semidirect
34. #65274C2×C3semidirect
35. #65239C2×C3semidirect
36. #65224C2×C3semidirect
37. C4×C4⋊Dic5C2×C3semidirect
44. C22×C3×C4×C5C2×C4semidirect
55. C3×C5×C4⋊C4C2×C4semidirect
56. C3×C5×C22⋊C4C2×C4semidirect
57. C3×C42×C5C2×C4semidirect
64. C3×C4⋊Dic5D4direct (N × H)
77. C4×C5×D4C3×C4semidirect
102. C3×C5×D4C4⋊C4semidirect
130. C3×C4×D4Dic5semidirect
151. C22×C4×C5C2×C3×C4semidirect
162. C5×C4⋊C4C2×C3×C4semidirect
163. C5×C22⋊C4C2×C3×C4semidirect
164. C42×C5C2×C3×C4semidirect
171. C4⋊Dic5C3×D4direct (N × H)
187. C22×C3×C5C2×C4⋊C4semidirect
197. C3×C4×C5C2×C4⋊C4semidirect
198. C3×C4×C5C4×D4semidirect
214. C22×C3×C4C2×Dic5semidirect
225. C3×C4⋊C4C2×Dic5semidirect
227. C3×C22⋊C4C2×Dic5semidirect
229. C3×C42C2×Dic5semidirect
238. C5×D4C3×C4⋊C4semidirect
266. C4×D4C3×Dic5semidirect
297. C3×D4C4⋊Dic5direct (N × H)
327. C22×C5C2×C3×C4⋊C4semidirect
337. C4×C5C2×C3×C4⋊C4semidirect
338. C4×C5C3×C4×D4semidirect
354. C22×C4C2×C3×Dic5semidirect
365. C4⋊C4C2×C3×Dic5semidirect
367. C22⋊C4C2×C3×Dic5semidirect
369. C42C2×C3×Dic5semidirect
370. C3×C5D4×C4⋊C4semidirect
377. C22×C3C2×C4⋊Dic5semidirect
390. C3×C4C2×C4⋊Dic5semidirect
391. C3×C4D4×Dic5semidirect
408. D4C3×C4⋊Dic5direct (N × H)
440. C5C3×D4×C4⋊C4semidirect
447. C22C2×C3×C4⋊Dic5semidirect
460. C4C2×C3×C4⋊Dic5semidirect
461. C4C3×D4×Dic5semidirect
462. C3D4×C4⋊Dic5direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×D42
C2×C3×D4×Dic52
C22×C3×C4⋊Dic52
C3×#655012
C3×#652742
C3×#652392
C3×#652242
C3×C4×C4⋊Dic52
D4×C4⋊Dic53
C3×D4×C4⋊C45