Finite Group Databasebeta

← #14294738

Subgroups — #14294738

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91923C212yesmaximal
2C2×#91393C211yesmaximal, characteristic
3#11746788C211yesmaximal, characteristic
4C3×#64917C211yesmaximal, characteristic
5#11746573C212yesmaximal
6#10604165—31nomaximal, Hall
7C22×D5×Dic3C2211yescharacteristic
8C2×#6763C2212yes—
9C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
10C2×Dic3⋊AGL(1,𝔽5)—22no—
11#91923—24no—
12#91923C418yescomplemented
13C2×Dic3×AGL(1,𝔽5)—22no—
14C2×#6712C411yescharacteristic, complemented
15C2×C4×C5×Dic3C411yescharacteristic, complemented
16C2×C3×#685C2211yescharacteristic
17C2×#6669C411yescharacteristic, complemented
18C2×#6665C2211yescharacteristic
19C2×Dic3×Dic5C411yescharacteristic, complemented
20#91393—22no—
21#91393—22no—
22#68951—51nomaximal, Hall
23C2×#4130—32no—
24C2×#4018—31no—
25#64917—31no—
26#64917—32no—
27#64917D311yescharacteristic
28C23×C3×D5C2311yescharacteristic
29C22×ΓL(1,𝔽16)—22no—
30C22×C3×AGL(1,𝔽5)—22no—
31C22×C3×C4×C5C2×C411yescharacteristic
32C22×Dic15C2×C411yescharacteristic
33C22×Dic15C2×C411yescharacteristic
34C22×C5×Dic3C2×C411yescharacteristic
35C22×C5×Dic3C2×C411yescharacteristic
36C22×C3×Dic5C2×C411yescharacteristic
37C2×D5×Dic3C2×C414yes—
38C2×D5×Dic3D412yes—
39C2×D5×Dic3D412yes—
40C2×D5×Dic3D412yes—
41C2×D5×Dic3Q812yes—
42#6763—22no—
43#6763C2×C414yes—
44C3×C22⋊AGL(1,𝔽5)—22no—
45C3×C22⋊AGL(1,𝔽5)C2×C414yes—
46Dic3⋊AGL(1,𝔽5)—28no—
47Dic3×AGL(1,𝔽5)—28no—
48#6712—22no—
49C4×C5×Dic3—21no—
50C4×C5×Dic3—21no—
51C3×#685—22no—
52#6669—22no—
53#6665—22no—
54Dic3×Dic5—21no—
55Dic3×Dic5—21no—
56C2×#877—52no—
57C2×#877—51no—
58C3×#342—51no—
59#4527—52no—
60#4526—51no—
61C22×C4×D5—31no—
62C2×C22⋊AGL(1,𝔽5)—32no—
63C2×C22⋊AGL(1,𝔽5)C2×D312yes—
64#4130—38no—
65#4130—64no—
66C2×C4⋊AGL(1,𝔽5)—62no—
67C2×C4×AGL(1,𝔽5)—62no—
68C2×C42×C5—31no—
69C2×#685—31no—
70C2×#685C2×D311yescharacteristic
71C2×#683—31no—
72C2×#683—31no—
73C2×C4×Dic5—31no—
74#4018—62no—
75#4018—62no—
76#1740—151noSylow
77C23×C3×C5C22×C411yescharacteristic
78C22×C3×D5—41no—
79C22×C3×D5—41no—
80C22×C3×D5C22×C412yes—
81C22×C3×D5C2×D412yes—
82C22×C3×D5C2×D411yescharacteristic
83C22×C3×D5C2×Q811yescharacteristic
84C2×ΓL(1,𝔽16)—24no—
85C2×ΓL(1,𝔽16)—42no—
86C2×ΓL(1,𝔽16)—42no—
87C2×C3×AGL(1,𝔽5)—24no—
88C2×C3×AGL(1,𝔽5)—42no—
89C2×C3×AGL(1,𝔽5)—42no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—41no—
92C2×Dic15—21no—
93C2×Dic15—41no—
94C2×Dic15C4212yes—
95C2×Dic15C22⋊C412yes—
96C2×Dic15C4⋊C412yes—
97C2×C5×Dic3—21no—
98C2×C5×Dic3—41no—
99C2×C5×Dic3C4212yes—
100C2×C5×Dic3C22⋊C411yescharacteristic
101C2×C5×Dic3C22⋊C411yescharacteristic
102C2×C5×Dic3C4⋊C412yes—
103C2×C3×Dic5—21no—
104C2×C3×Dic5—41no—
105D5×Dic3C22⋊C414yes—
106D5×Dic3C22⋊C414yes—
107D5×Dic3C4⋊C414yes—
108D5×Dic3C4⋊C414yes—
109C23×Dic3—51no—
110C2×C3×C22⋊C4—52no—
111C2×C3×C22⋊C4—51no—
112C2×#217—52no—
113C2×#217—51no—
114C2×Dic3⋊C4—51no—
115C2×Dic3⋊C4—51no—
116C2×Dic3⋊C4—102no—
117C2×C4×Dic3—51no—
118C2×C4×Dic3—102no—
119C2×C4×Dic3AGL(1,𝔽5)11yescharacteristic, complemented
120#877—58no—
121#877—102no—
122#877—102no—
123#877—104no—
124C23×D5C22×D311yescharacteristic
125C22×AGL(1,𝔽5)—22no—
126C22×AGL(1,𝔽5)—62no—
127C22×C4×C5—31no—
128C22×C4×C5—31no—
129C22×C4×C5C4×D311yescharacteristic
130C22×Dic5—31no—
131C22×Dic5—31no—
132C22×Dic5C4×D311yescharacteristic
133C2×C4×D5—34no—
134C2×C4×D5—32no—
135C2×C4×D5—32no—
136C2×C4×D5—32no—
137C2×C4×D5—32no—
138C22⋊AGL(1,𝔽5)—22no—
139C22⋊AGL(1,𝔽5)—34no—
140C22⋊AGL(1,𝔽5)—62no—
141C22⋊AGL(1,𝔽5)C4×D314yes—
142C4⋊AGL(1,𝔽5)—68no—
143C4×AGL(1,𝔽5)—68no—
144C42×C5—61no—
145C42×C5—61no—
146#685—22no—
147#685—62no—
148#683—62no—
149#683—62no—
150C4×Dic5—61no—
151C4×Dic5—61no—
152C2×#117—151no—
153C2×#117—152no—
154#342—51no—
155#342—151no—
156#342—152no—
157C22×C3×C5—21no—
158C22×C3×C5—21no—
159C22×C3×C5C2×C4211yescharacteristic
160C22×C3×C5C2×C22⋊C411yescharacteristic
161C22×C3×C5C2×C4⋊C411yescharacteristic
162C2×C3×D5—22no—
163C2×C3×D5—22no—
164C2×C3×D5—42no—
165C2×C3×D5—41no—
166C2×C3×D5—41no—
167C2×C3×D5C2×C22⋊C412yes—
168C2×C3×D5C2×C4⋊C412yes—
169ΓL(1,𝔽16)—44no—
170C3×AGL(1,𝔽5)—44no—
171C3×C4×C5—41no—
172Dic15—41no—
173Dic15#9514yes—
174C3×Dic5—41no—
175C5×Dic3—41no—
176C5×Dic3#9512yes—
177C5×Dic3#9512yes—
178C24×C3—51no—
179C22×C3×C4—51no—
180C22×C3×C4—102no—
181C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
182C22×Dic3—51no—
183C22×Dic3—54no—
184C22×Dic3—52no—
185C22×Dic3—52no—
186C22×Dic3—52no—
187C22×Dic3—52no—
188C22×Dic3—51no—
189C22×Dic3—102no—
190C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
191C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
192C3×C22⋊C4—54no—
193C3×C22⋊C4—102no—
194C3×C22⋊C4—102no—
195#217—54no—
196#217—102no—
197#217—102no—
198Dic3⋊C4—102no—
199Dic3⋊C4—102no—
200Dic3⋊C4—108no—
201C4×Dic3—21no—
202C4×Dic3—21no—
203C4×Dic3—108no—
204C4×Dic3—101no—
205C4×Dic3—101no—
206C23×C5C2×C4×D311yescharacteristic
207C22×D5—41no—
208C22×D5—41no—
209C22×D5C2×C4×D312yes—
210C22×D5D3×D412yes—
211C22×D5D3×D411yescharacteristic
212C22×D5D3×Q811yescharacteristic
213C2×AGL(1,𝔽5)—24no—
214C2×AGL(1,𝔽5)—42no—
215C2×AGL(1,𝔽5)—42no—
216C2×AGL(1,𝔽5)—64no—
217C2×AGL(1,𝔽5)—122no—
218C2×AGL(1,𝔽5)—122no—
219C2×C4×C5—21no—
220C2×C4×C5—32no—
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—32no—
224C2×C4×C5—41no—
225C2×C4×C5—61no—
226C2×C4×C5—121no—
227C2×Dic5—21no—
228C2×Dic5—32no—
229C2×Dic5—32no—
230C2×Dic5—32no—
231C2×Dic5—41no—
232C2×Dic5—61no—
233C2×Dic5—121no—
234C4×D5—34no—
235C4×D5—34no—
236C4×D5—34no—
237C4×D5—34no—
238C23×C4—151no—
239#117—158no—
240#117—302no—
241#117—302no—
242#117—304no—
243C2×C4⋊C4—151no—
244C2×C4⋊C4—151no—
245C2×C4⋊C4—302no—
246C2×C22⋊C4—52no—
247C2×C22⋊C4—51no—
248C2×C22⋊C4—152no—
249C2×C22⋊C4—151no—
250C2×C42—31no—
251C2×C42—151no—
252C2×C42—302no—
253C2×C3×C5—21no—
254C2×C3×C5—21no—
255C2×C3×C5C2×#9511yescharacteristic
256C2×C3×C5#41011yescharacteristic
257C2×C3×C5#41011yescharacteristic
258C3×D5—22no—
259C3×D5—22no—
260C23×C3—52no—
261C23×C3—52no—
262C23×C3—51no—
263C23×C3—51no—
264C23×C3—201no—
265C23×C3—201no—
266C23×C3C22×AGL(1,𝔽5)11yescharacteristic
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—104no—
270C2×C3×C4—101no—
271C2×C3×C4—202no—
272C2×C3×C4—202no—
273C2×C3×C4—201no—
274C2×Dic3—21no—
275C2×Dic3—41no—
276C2×Dic3—52no—
277C2×Dic3—52no—
278C2×Dic3—52no—
279C2×Dic3—54no—
280C2×Dic3—54no—
281C2×Dic3—54no—
282C2×Dic3—54no—
283C2×Dic3—104no—
284C2×Dic3—101no—
285C2×Dic3—202no—
286C2×Dic3—202no—
287C2×Dic3—201no—
288C2×Dic3C4×AGL(1,𝔽5)12yes—
289C2×Dic3C4⋊AGL(1,𝔽5)12yes—
290C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
291C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
292C22×C5—21no—
293C22×C5—21no—
294C22×C5C42×D311yescharacteristic
295C22×C5D3×C22⋊C411yescharacteristic
296C22×C5D3×C4⋊C411yescharacteristic
297C2×D5—22no—
298C2×D5—22no—
299C2×D5—42no—
300C2×D5—41no—
301C2×D5—41no—
302C2×D5D3×C22⋊C412yes—
303C2×D5D3×C4⋊C412yes—
304AGL(1,𝔽5)—44no—
305AGL(1,𝔽5)—124no—
306C4×C5—32no—
307C4×C5—32no—
308C4×C5—41no—
309C4×C5—121no—
310Dic5—34no—
311Dic5—41no—
312Dic5—121no—
313C24—51no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—51no—
317C22×C4—102no—
318C22×C4—152no—
319C22×C4—152no—
320C22×C4—154no—
321C22×C4—152no—
322C22×C4—152no—
323C22×C4—151no—
324C22×C4—151no—
325C22×C4—302no—
326C22×C4D3×AGL(1,𝔽5)11yescharacteristic
327C4⋊C4—302no—
328C4⋊C4—302no—
329C4⋊C4—308no—
330C22⋊C4—54no—
331C22⋊C4—102no—
332C22⋊C4—102no—
333C22⋊C4—154no—
334C22⋊C4—302no—
335C22⋊C4—302no—
336C42—61no—
337C42—61no—
338C42—301no—
339C42—301no—
340C42—308no—
341C3×C5#174011yescharacteristic, complemented, Hall
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—52no—
345C22×C3—52no—
346C22×C3—102no—
347C22×C3—102no—
348C22×C3—202no—
349C22×C3—201no—
350C22×C3—201no—
351C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
352C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
353C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
354C3×C4—41no—
355C3×C4—204no—
356C3×C4—201no—
357Dic3—41no—
358Dic3—54no—
359Dic3—204no—
360Dic3—201no—
361Dic3(C2×AGL(1,𝔽5))⋊C412yes—
362Dic3(C2×AGL(1,𝔽5))⋊C412yes—
363C2×C5—21no—
364C2×C5—21no—
365C2×C5D3×#9511yescharacteristic
366C2×C5#474011yescharacteristic
367C2×C5#474011yescharacteristic
368D5—22no—
369D5—22no—
370C23—52no—
371C23—52no—
372C23—51no—
373C23—51no—
374C23—201no—
375C23—201no—
376C23C2×D3×AGL(1,𝔽5)11yescharacteristic
377C2×C4—21no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—41no—
383C2×C4—61no—
384C2×C4—104no—
385C2×C4—101no—
386C2×C4—121no—
387C2×C4—154no—
388C2×C4—154no—
389C2×C4—154no—
390C2×C4—154no—
391C2×C4—152no—
392C2×C4—152no—
393C2×C4—152no—
394C2×C4—202no—
395C2×C4—202no—
396C2×C4—201no—
397C2×C4—304no—
398C2×C4—301no—
399C2×C4—602no—
400C2×C4—602no—
401C2×C4—601no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—102no—
405C2×C3—102no—
406C2×C3#6574511yescharacteristic
407C2×C3#6574511yescharacteristic
408C2×C3C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
409C5#6895111yescharacteristic, complemented, Sylow
410C22—21no—
411C22—21no—
412C22—52no—
413C22—52no—
414C22—102no—
415C22—102no—
416C22—202no—
417C22—201no—
418C22—201no—
419C22C4×D3×AGL(1,𝔽5)11yescharacteristic
420C22D3×C4⋊AGL(1,𝔽5)11yescharacteristic
421C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
422C4—32no—
423C4—32no—
424C4—41no—
425C4—121no—
426C4—154no—
427C4—204no—
428C4—201no—
429C4—604no—
430C4—601no—
431C3#1060416511yescharacteristic, complemented, Sylow
432C2—21no—
433C2—21no—
434C2—102no—
435C2—102no—
436C2#1175186211yescharacteristic
437C2D3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
438C2#1175203011yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC22×C3×C5C2×C42
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical, cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1740
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
12. #91923C4semidirect
14. C2×#6712C4semidirect
15. C2×C4×C5×Dic3C4semidirect
17. C2×#6669C4semidirect
19. C2×Dic3×Dic5C4semidirect
119. C2×C4×Dic3AGL(1,𝔽5)semidirect
341. C3×C5#1740semidirect
409. C5#68951semidirect
431. C3#10604165semidirect

Maximal subgroups

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

subgroupindex
C2×#919232
C2×#913932
#117467882
C3×#649172
#117465732
#106041653
#689515