Finite Group Databasebeta

← #14297826

Subgroups — #14297826

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#713C211yesmaximal, characteristic
2C2×#91239C212yesmaximal
3C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
4#11752341C211yesmaximal, characteristic, complemented
5#11746746C212yesmaximal, complemented
6#10604456—31nomaximal, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C4⋊ΓL(1,𝔽16)—21no—
9C2×C4×ΓL(1,𝔽16)C2212yes—
10C2×#6758C2212yes—
11C3×C4×C5×Q8C411yescharacteristic, complemented
12C3×Q8×Dic5C411yescharacteristic, complemented
13C3×#4039—21no—
14C3×#4039C411yescharacteristic, complemented
15C3×#4038—21no—
16C3×#4038C2211yescharacteristic
17C3×C4×D20—21no—
18C3×C4×D20C411yescharacteristic, complemented
19C3×C42×D5—21no—
20C3×C42×D5C2211yescharacteristic
21#91239—24no—
22#91235—21no—
23#91233—21no—
24#91233—21no—
25#71646—51nomaximal, Hall
26C4×#713D311yescharacteristic
27C2×#4010—32no—
28C4×C4⋊AGL(1,𝔽5)—31no—
29#65682—31no—
30#64875—32no—
31C22×ΓL(1,𝔽16)—22no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C3×#713—21no—
34C3×#713—21no—
35C3×#713D412yes—
36C3×#713Q812yes—
37C2×C3×D20—21no—
38C2×C3×D20C2×C411yescharacteristic
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3⋊#170—22no—
46C4⋊ΓL(1,𝔽16)—24no—
47C4×ΓL(1,𝔽16)—22no—
48C4×ΓL(1,𝔽16)—22no—
49#6758—22no—
50#6758—22no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4C2×C411yescharacteristic
53C3×C42×C5—21no—
54C3×C42×C5C2×C411yescharacteristic
55C3×#685—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×C4⋊Dic5—21no—
59C3×C4⋊Dic5C2×C411yescharacteristic
60C3×C4×Dic5—21no—
61C3×C4×Dic5—21no—
62C3×C4×Dic5C2×C411yescharacteristic
63C3×C4×Dic5D411yescharacteristic
64C3×C4×Dic5D411yescharacteristic
65C3×C4×#41—51no—
66C2×#837—52no—
67C4×C4⋊Dic3—51no—
68#4924—51no—
69#4532—52no—
70C2×#713C2×D311yescharacteristic
71C2×C4⋊AGL(1,𝔽5)—61no—
72C2×C4×AGL(1,𝔽5)—32no—
73C2×#700—32no—
74C4×C5×Q8Dic311yescharacteristic, complemented
75Q8×Dic5Dic311yescharacteristic, complemented
76#4039—21no—
77#4039Dic311yescharacteristic, complemented
78#4038—21no—
79#4038C2×D311yescharacteristic
80C4×D20—21no—
81C4×D20Dic311yescharacteristic, complemented
82C42×D5—21no—
83C42×D5C2×D311yescharacteristic
84#4010—64no—
85(C2×AGL(1,𝔽5))⋊C4—61no—
86#4004—61no—
87#4004—61no—
88#1806—151noSylow
89C22×C3×D5—21no—
90C22×C3×D5C2×D411yescharacteristic
91C2×ΓL(1,𝔽16)—24no—
92C2×ΓL(1,𝔽16)—42no—
93C2×ΓL(1,𝔽16)—42no—
94C3×C5×Q8—21no—
95C3×C5×Q8C4⋊C412yescomplemented
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C2×C3×C4×C5C22×C411yescharacteristic
101C2×C3×Dic5—21no—
102C2×C3×Dic5C22⋊C411yescharacteristic
103C2×C3×Dic5C22⋊C411yescharacteristic
104C2×C3×Dic5C2×D411yescharacteristic
105C3×D20—22no—
106C3×D20—22no—
107C3×D20—21no—
108C3×D20C4⋊C412yescomplemented
109C3×C4×D5—22no—
110C3×C4×D5—22no—
111C3×C4×D5—22no—
112C3×C4×D5—41no—
113C3×C4×D5—42no—
114C3×C4×D5—42no—
115C3×C4×D5C2×D411yescharacteristic
116C3×C4×D5C2×Q811yescharacteristic
117C3×C4×D5#4111yescharacteristic
118C3×C4×D5#4111yescharacteristic
119C3⋊#170—42no—
120C2×C3×#41—51no—
121C3×C4×Q8—51no—
122C3×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
123C3×C4×D4—51no—
124C3×C4×D4—51no—
125C3×C4×D4—101no—
126C3×C4×D4—101no—
127C3×#117—51no—
128C3×#117—101no—
129C2×C3×C42—51no—
130C2×C3×C42—101no—
131C2×C4⋊Dic3—101no—
132C2×C4×Dic3—52no—
133C2×#208—52no—
134#837—104no—
135C3⋊#95—101no—
136C4⋊(C3⋊C8)—101no—
137C4⋊(C3⋊C8)—101no—
138C22×AGL(1,𝔽5)—62no—
139C2×C5×Q8C2×Dic311yescharacteristic
140#713—21no—
141#713—21no—
142#713Dic612yes—
143#713D1212yes—
144C2×D20—21no—
145C2×D20C2×Dic311yescharacteristic
146C2×C4×D5—21no—
147C2×C4×D5—21no—
148C2×C4×D5—21no—
149C2×C4×D5#6711yescharacteristic
150C2×C4×D5#6711yescharacteristic
151C2×C4×D5C22×D311yescharacteristic
152C2×#170—62no—
153C4⋊AGL(1,𝔽5)—64no—
154C4×AGL(1,𝔽5)—62no—
155C4×AGL(1,𝔽5)—62no—
156#700—62no—
157#700—62no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4C2×Dic311yescharacteristic
160C42×C5—21no—
161C42×C5C2×Dic311yescharacteristic
162#685—21no—
163#685—21no—
164#685—21no—
165C4⋊Dic5—21no—
166C4⋊Dic5C2×Dic311yescharacteristic
167C4×Dic5—21no—
168C4×Dic5—21no—
169C4×Dic5C2×Dic311yescharacteristic
170C4×Dic5#6711yescharacteristic
171C4×Dic5#6711yescharacteristic
172C4×#41—51no—
173#423—151no—
174C2×#104—152no—
175C4×C4⋊C4—151no—
176#339—152no—
177C22×C3×C5C2×C22⋊C411yescharacteristic
178C2×C3×D5—21no—
179C2×C3×D5—21no—
180C2×C3×D5—22no—
181C2×C3×D5—41no—
182C2×C3×D5—41no—
183C2×C3×D5#12111yescharacteristic
184C2×C3×D5C22⋊Q811yescharacteristic
185ΓL(1,𝔽16)—44no—
186C3×C4×C5—22no—
187C3×C4×C5—21no—
188C3×C4×C5—21no—
189C3×C4×C5—41no—
190C3×C4×C5C2×C4⋊C411yescharacteristic
191C3×C4×C5#11711yescharacteristic
192C3×Dic5—21no—
193C3×Dic5—41no—
194C3×Dic5#10412yescomplemented
195C3×Dic5#12111yescharacteristic
196C3×Dic5C22⋊Q811yescharacteristic
197C3×#41—52no—
198C3×#41—52no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
202C2×C3×D4—51no—
203C2×C3×D4—101no—
204C22×C3×C4—51no—
205C22×C3×C4—51no—
206C22×C3×C4—51no—
207C22×C3×C4—101no—
208C22×C3×C4—101no—
209C22×C3×C4—101no—
210C22×Dic3—102no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—51no—
213C3×C4⋊C4—101no—
214C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
215C3×C22⋊C4—101no—
216C3×C22⋊C4—101no—
217C3×C22⋊C4—101no—
218C3×C42—21no—
219C3×C42—51no—
220C3×C42—51no—
221C3×C42—51no—
222C3×C42—101no—
223C3×C42—101no—
224C3×C42C2×AGL(1,𝔽5)11yescharacteristic
225C4⋊Dic3—104no—
226C4×Dic3—102no—
227C4×Dic3—102no—
228#208—102no—
229#208—102no—
230C2×C3⋊C8—102no—
231C22×D5—21no—
232C22×D5C2×#6711yescharacteristic
233C2×AGL(1,𝔽5)—64no—
234C2×AGL(1,𝔽5)—122no—
235C2×AGL(1,𝔽5)—122no—
236C5×Q8—21no—
237C5×Q8C4⋊Dic312yescomplemented
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5#21711yescharacteristic
241C2×C4×C5#21711yescharacteristic
242C2×C4×C5C22×Dic311yescharacteristic
243C2×Dic5—21no—
244C2×Dic5#21711yescharacteristic
245C2×Dic5#21711yescharacteristic
246C2×Dic5C2×#6711yescharacteristic
247D20—22no—
248D20—22no—
249D20—21no—
250D20C4⋊Dic312yescomplemented
251C4×D5—22no—
252C4×D5—22no—
253C4×D5—22no—
254C4×D5—41no—
255C4×D5—42no—
256C4×D5—42no—
257C4×D5C2×Dic611yescharacteristic
258C4×D5C2×D1211yescharacteristic
259C4×D5#23511yescharacteristic
260C4×D5#23511yescharacteristic
261#170—122no—
262C2×#41—51no—
263C2×M16—152no—
264C4×Q8—51no—
265C4×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
266C4×D4—51no—
267C4×D4—51no—
268C4×D4—101no—
269C4×D4—101no—
270#117—51no—
271#117—101no—
272C2×C4⋊C4—301no—
273C2×C42—51no—
274C2×C42—101no—
275C2×C42—152no—
276C4⋊C8—301no—
277C4⋊C8—301no—
278#104—304no—
279#95—301no—
280C2×C3×C5#38011yescharacteristic
281C2×C3×C5C2×#10411yescharacteristic
282C2×C3×C5#42111yescharacteristic
283C3×D5—21no—
284C3×D5—21no—
285C3×D5—42no—
286C23×C3—51no—
287C23×C3—101no—
288C3×Q8—21no—
289C3×Q8C4⋊AGL(1,𝔽5)12yescomplemented
290C3×D4—52no—
291C3×D4—102no—
292C3×D4—102no—
293C3×D4—101no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—51no—
297C2×C3×C4—51no—
298C2×C3×C4—51no—
299C2×C3×C4—51no—
300C2×C3×C4—51no—
301C2×C3×C4—51no—
302C2×C3×C4—51no—
303C2×C3×C4—102no—
304C2×C3×C4—102no—
305C2×C3×C4—102no—
306C2×C3×C4—101no—
307C2×C3×C4—201no—
308C2×C3×C4—202no—
309C2×C3×C4—202no—
310C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
311C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
312C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
313C2×Dic3—104no—
314C2×Dic3—202no—
315C2×Dic3—202no—
316C3⋊C8—202no—
317C22×C5C2×#21711yescharacteristic
318C2×D5—21no—
319C2×D5—21no—
320C2×D5—22no—
321C2×D5—41no—
322C2×D5—41no—
323C2×D5#92411yescharacteristic
324C2×D5#93011yescharacteristic
325AGL(1,𝔽5)—124no—
326C4×C5—22no—
327C4×C5—21no—
328C4×C5—21no—
329C4×C5—41no—
330C4×C5C2×C4⋊Dic311yescharacteristic
331C4×C5#92611yescharacteristic
332Dic5—21no—
333Dic5—41no—
334Dic5#83712yescomplemented
335Dic5#92411yescharacteristic
336Dic5#93011yescharacteristic
337#41—52no—
338#41—52no—
339#41—101no—
340#41—101no—
341C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
342C2×D4—51no—
343C2×D4—101no—
344C22×C4—51no—
345C22×C4—51no—
346C22×C4—51no—
347C22×C4—101no—
348C22×C4—101no—
349C22×C4—101no—
350C22×C4—302no—
351M16—302no—
352M16—302no—
353C2×C8—302no—
354C4⋊C4—21no—
355C4⋊C4—51no—
356C4⋊C4—101no—
357C4⋊C4—304no—
358C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
359C22⋊C4—101no—
360C22⋊C4—101no—
361C22⋊C4—101no—
362C42—21no—
363C42—51no—
364C42—51no—
365C42—51no—
366C42—101no—
367C42—101no—
368C42—302no—
369C42—302no—
370C42C2×ΓL(1,𝔽16)11yescharacteristic
371C3×C5#180611yescharacteristic, complemented, Hall
372C22×C3—51no—
373C22×C3—51no—
374C22×C3—101no—
375C22×C3—101no—
376C22×C3—102no—
377C22×C3—201no—
378C22×C3—201no—
379C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
380C3×C4—21no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—52no—
385C3×C4—51no—
386C3×C4—51no—
387C3×C4—101no—
388C3×C4—201no—
389C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
390C3×C4#413011yescharacteristic
391Dic3—204no—
392C2×C5#521211yescharacteristic
393C2×C5C2×#83711yescharacteristic
394C2×C5#523611yescharacteristic
395D5—21no—
396D5—21no—
397D5—42no—
398C23—51no—
399C23—101no—
400Q8—21no—
401Q8C4⋊ΓL(1,𝔽16)12yescomplemented
402D4—52no—
403D4—102no—
404D4—102no—
405D4—101no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—51no—
409C2×C4—51no—
410C2×C4—51no—
411C2×C4—51no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—102no—
416C2×C4—102no—
417C2×C4—102no—
418C2×C4—101no—
419C2×C4—201no—
420C2×C4—202no—
421C2×C4—202no—
422C2×C4—304no—
423C2×C4—602no—
424C2×C4—602no—
425C2×C4#676311yescharacteristic
426C2×C4#676311yescharacteristic
427C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
428C8—602no—
429C2×C3—101no—
430C2×C3—101no—
431C2×C3—202no—
432C2×C3#6576511yescharacteristic
433C2×C3#6578411yescharacteristic
434C2×C3D5⋊#10411yescharacteristic
435C5#7164611yescharacteristic, complemented, Sylow
436C22—51no—
437C22—51no—
438C22—101no—
439C22—101no—
440C22—102no—
441C22—201no—
442C22—201no—
443C22C2×#676311yescharacteristic
444C4—21no—
445C4—22no—
446C4—21no—
447C4—41no—
448C4—52no—
449C4—51no—
450C4—51no—
451C4—101no—
452C4—201no—
453C4—604no—
454C4C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
455C4#9198911yescharacteristic
456C3#1060445611yescharacteristic, complemented, Sylow
457C2—101no—
458C2—101no—
459C2—202no—
460C2#1175242411yescharacteristic
461C2#1175244311yescharacteristic
462C2D5⋊#83711yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#713C22
cosocleC2×C3×C4×D5C23
hypercenterC4×Q8ΓL(1,𝔽16)
hypocenterC3×C5#1806
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. C4×C4⋊ΓL(1,𝔽16)C2semidirect
4. #11752341C2semidirect
5. #11746746C2semidirect
11. C3×C4×C5×Q8C4semidirect
12. C3×Q8×Dic5C4semidirect
14. C3×#4039C4semidirect
18. C3×C4×D20C4semidirect
74. C4×C5×Q8Dic3semidirect
75. Q8×Dic5Dic3semidirect
77. #4039Dic3semidirect
81. C4×D20Dic3semidirect
95. C3×C5×Q8C4⋊C4semidirect
108. C3×D20C4⋊C4semidirect
122. C3×C4×Q8AGL(1,𝔽5)semidirect
194. C3×Dic5#104semidirect
237. C5×Q8C4⋊Dic3semidirect
250. D20C4⋊Dic3semidirect
265. C4×Q8ΓL(1,𝔽16)semidirect
289. C3×Q8C4⋊AGL(1,𝔽5)semidirect
334. Dic5#837semidirect
371. C3×C5#1806semidirect
401. Q8C4⋊ΓL(1,𝔽16)semidirect
435. C5#71646semidirect
456. C3#10604456semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#7132
C2×#912392
C4×C4⋊ΓL(1,𝔽16)2
#117523412
#117467462
#106044563
#716465