Finite Group Databasebeta

← #14322790

Subgroups — #14322790

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#872C211yesmaximal, characteristic
2#11752348C211yesmaximal, characteristic, complemented
3C3×#65688C211yesmaximal, characteristic, complemented
4#11751855C212yesmaximal
5Dic3×C4⋊AGL(1,𝔽5)C212yesmaximal, complemented
6#11751836C214yesmaximal
7#11751835C214yesmaximal, complemented
8#10617130—31nomaximal, Hall
9C2×C4×D3×D5C2211yescharacteristic
10C2×C4⋊ΓL(1,𝔽16)C2212yes—
11C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
12C2×C3×C4⋊AGL(1,𝔽5)C2214yes—
13C2×(C2×D3)⋊AGL(1,𝔽5)C2214yes—
14C2×Dic3×AGL(1,𝔽5)C2214yes—
15#91761C411yescharacteristic, complemented
16C5×#872C411yescharacteristic, complemented
17C3×C42×D5C2211yescharacteristic
18D5×(C2×D3)⋊C4C2212yes—
19#91402C412yescomplemented
20#91398C412yescomplemented
21D5×Dic3⋊C4C2212yes—
22C4×D5×Dic3C2211yescharacteristic
23#91386C411yescharacteristic, complemented
24#91371C411yescharacteristic, complemented
25#91235C2214yes—
26(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
27(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
28#74358—51nomaximal, Hall
29D5×#117—31no—
30#65765—32no—
31#65711—34no—
32#65700—34no—
33#65689—31no—
34#65688D311yescharacteristic, complemented
35C4×C4⋊AGL(1,𝔽5)—32no—
36C22×D3×D5C2311yescharacteristic
37C22×ΓL(1,𝔽16)C2314yes—
38C22×C3×AGL(1,𝔽5)C2314yes—
39C2×C4×D15C2×C411yescharacteristic
40C2×C4×C5×D3C2×C411yescharacteristic
41C2×C3×C4×D5C2312yes—
42C2×C3×C4×D5C2311yescharacteristic
43C2×#1200C2×C411yescharacteristic
44C2×D3×Dic5C2×C411yescharacteristic
45C2×D5×Dic3C2311yescharacteristic
46C2×D5×Dic3C2312yes—
47C4×D3×D5—22no—
48C4×D3×D5—22no—
49C4⋊ΓL(1,𝔽16)—24no—
50C3×C4⋊AGL(1,𝔽5)—24no—
51C3×C4⋊AGL(1,𝔽5)—44no—
52(C2×D3)⋊AGL(1,𝔽5)—44no—
53Dic3×AGL(1,𝔽5)—44no—
54C3×C42×C5C2×C411yescharacteristic
55#6714C2×C412yes—
56#6712C2×C412yes—
57C4×Dic15C2×C411yescharacteristic
58C5×(C2×D3)⋊C4C2×C412yes—
59C5×Dic3⋊C4C2×C412yes—
60C4×C5×Dic3C2×C411yescharacteristic
61C3×C4×Dic5C2×C412yes—
62C3×C4×Dic5C2×C411yescharacteristic
63Dic3⋊Dic5C2×C412yes—
64#6669C2×C412yes—
65#6667C2×C412yes—
66(C2×D3)⋊Dic5C2×C412yes—
67Dic3×Dic5C2×C412yes—
68C2×#872—51no—
69C3×#384—51no—
70#4990—54no—
71#4987—52no—
72#4978—54no—
73Dic3×C4⋊C4—52no—
74#4936—51no—
75C22×C4×D5—31no—
76C2×C22⋊AGL(1,𝔽5)—34no—
77C2×C4⋊AGL(1,𝔽5)—32no—
78C2×C4⋊AGL(1,𝔽5)C2×D312yes—
79C2×C4⋊AGL(1,𝔽5)C2×D314yes—
80C2×C4×AGL(1,𝔽5)—34no—
81C5×#117—31no—
82#4072—31no—
83#4038—32no—
84D5×C4⋊C4—32no—
85D5×C22⋊C4—32no—
86#4023—32no—
87#4018—31no—
88#4018—31no—
89C42×D5—31no—
90C42×D5C2×D311yescharacteristic
91(C2×AGL(1,𝔽5))⋊C4—34no—
92(C2×AGL(1,𝔽5))⋊C4—34no—
93(C2×AGL(1,𝔽5))⋊C4—34no—
94#2314—151noSylow
95C22×D15C22×C411yescharacteristic
96C22×C5×D3C22×C411yescharacteristic
97C22×C3×D5C2411yescharacteristic
98C2×D3×D5—21no—
99C2×D3×D5—21no—
100C2×D3×D5—22no—
101C2×D3×D5—22no—
102C2×ΓL(1,𝔽16)—24no—
103C2×ΓL(1,𝔽16)—24no—
104C2×ΓL(1,𝔽16)—24no—
105C2×C3×AGL(1,𝔽5)—24no—
106C2×C3×AGL(1,𝔽5)—24no—
107C2×C3×AGL(1,𝔽5)—24no—
108C2×C3×C4×C5C22×C412yes—
109C2×C3×C4×C5C22×C411yescharacteristic
110C2×Dic15C22×C412yes—
111C2×Dic15C22×C411yescharacteristic
112C4×D15—22no—
113C2×C5×Dic3C22×C412yes—
114C2×C5×Dic3C22×C411yescharacteristic
115C4×C5×D3—22no—
116C2×C3×Dic5C22×C412yes—
117C2×C3×Dic5C22×C411yescharacteristic
118C3×C4×D5—22no—
119C3×C4×D5—22no—
120C3×C4×D5#4112yes—
121C3×C4×D5#4112yes—
122#1200—22no—
123D3×Dic5—22no—
124D5×Dic3—21no—
125D5×Dic3—21no—
126D5×Dic3—22no—
127D5×Dic3—22no—
128C22×C4×D3—51no—
129C2×C3×C4⋊C4—52no—
130C2×C3×C4⋊C4—54no—
131C2×C3×C42—51no—
132C2×(C2×D3)⋊C4—52no—
133C2×(C2×D3)⋊C4—54no—
134C2×C4⋊Dic3—52no—
135C2×Dic3⋊C4—52no—
136C2×C4×Dic3—51no—
137C2×C4×Dic3—54no—
138#872—51no—
139#872—51no—
140#872—52no—
141#872—52no—
142#872—51no—
143#872AGL(1,𝔽5)11yescharacteristic, complemented
144C3⋊#95—54no—
145C3⋊#95—54no—
146C3⋊#95—54no—
147C23×D5—31no—
148C22×AGL(1,𝔽5)—34no—
149C22×AGL(1,𝔽5)C22×D314yes—
150C22×C4×C5—31no—
151C22×Dic5—31no—
152C2×C4×D5—32no—
153C2×C4×D5—31no—
154C2×C4×D5—31no—
155C2×C4×D5—31no—
156C2×C4×D5—62no—
157C2×C4×D5—62no—
158C2×C4×D5C22×D312yes—
159C2×C4×D5C22×D311yescharacteristic
160C22⋊AGL(1,𝔽5)—124no—
161C4⋊AGL(1,𝔽5)—24no—
162C4⋊AGL(1,𝔽5)—44no—
163C4⋊AGL(1,𝔽5)—64no—
164C4×AGL(1,𝔽5)—124no—
165C5×C4⋊C4—32no—
166C5×C22⋊C4—32no—
167C42×C5—31no—
168C42×C5C4×D311yescharacteristic
169C22⋊Dic5—32no—
170#685—32no—
171#685—32no—
172C4⋊Dic5—32no—
173#683—32no—
174#683—32no—
175C4×Dic5—31no—
176C4×Dic5—32no—
177C4×Dic5C4×D311yescharacteristic
178C4×Dic5C4×D312yes—
179C2×#117—151no—
180#388—154no—
181#387—154no—
182#384—51no—
183#382—151no—
184#380—152no—
185C4×C4⋊C4—152no—
186C22×C3×C5C23×C411yescharacteristic
187C2×D15—21no—
188C2×D15—22no—
189C2×C5×D3—21no—
190C2×C5×D3—22no—
191C2×C3×D5C2×#4111yescharacteristic
192C2×C3×D5C2×#4111yescharacteristic
193C2×C3×D5H(5,𝔽2)12yes—
194C2×C3×D52−1+412yes—
195D3×D5—41no—
196D3×D5—41no—
197D3×D5—42no—
198ΓL(1,𝔽16)—44no—
199C3×AGL(1,𝔽5)—44no—
200C3×C4×C5—22no—
201C3×C4×C5#11712yes—
202Dic15—21no—
203Dic15—22no—
204C3×Dic5—22no—
205C3×Dic5#11712yes—
206C5×Dic3—21no—
207C5×Dic3—22no—
208C23×D3—51no—
209C22×C3×C4—51no—
210C22×C3×C4—52no—
211C22×C3×C4—54no—
212C22×Dic3—51no—
213C22×Dic3—52no—
214C22×Dic3—54no—
215C2×C4×D3—51no—
216C2×C4×D3—51no—
217C2×C4×D3—51no—
218C2×C4×D3—102no—
219C2×C4×D3—102no—
220C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
221C3×C4⋊C4—104no—
222C3×C4⋊C4—204no—
223C3×C42—52no—
224C3×C42—51no—
225C3×C42C2×AGL(1,𝔽5)11yescharacteristic
226(C2×D3)⋊C4—52no—
227(C2×D3)⋊C4—52no—
228(C2×D3)⋊C4—52no—
229(C2×D3)⋊C4—204no—
230(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
231C4⋊Dic3—104no—
232Dic3⋊C4—52no—
233Dic3⋊C4—52no—
234Dic3⋊C4—52no—
235Dic3⋊C4C2×AGL(1,𝔽5)12yes—
236C4×Dic3—51no—
237C4×Dic3—52no—
238C4×Dic3—204no—
239C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
240C23×C5—31no—
241C22×D5—31no—
242C22×D5—61no—
243C22×D5—62no—
244C22×D5—61no—
245C22×D5—62no—
246C22×D5C23×D311yescharacteristic
247C2×AGL(1,𝔽5)—24no—
248C2×AGL(1,𝔽5)—24no—
249C2×AGL(1,𝔽5)—24no—
250C2×AGL(1,𝔽5)—64no—
251C2×AGL(1,𝔽5)—64no—
252C2×AGL(1,𝔽5)—64no—
253C2×C4×C5—32no—
254C2×C4×C5—31no—
255C2×C4×C5—62no—
256C2×C4×C5C2×C4×D312yes—
257C2×C4×C5C2×C4×D311yescharacteristic
258C2×Dic5—32no—
259C2×Dic5—31no—
260C2×Dic5—62no—
261C2×Dic5C2×C4×D312yes—
262C2×Dic5C2×C4×D311yescharacteristic
263C4×D5—22no—
264C4×D5—22no—
265C4×D5—61no—
266C4×D5—62no—
267C4×D5—61no—
268C4×D5—62no—
269C4×D5—62no—
270C4×D5—62no—
271C4×D5#23712yes—
272C4×D5#23912yes—
273C23×C4—151no—
274#117—31no—
275#117—151no—
276#117—152no—
277#117—152no—
278#117—151no—
279#117—151no—
280C2×C4⋊C4—52no—
281C2×C4⋊C4—54no—
282C2×C4⋊C4—152no—
283C2×C4⋊C4—152no—
284C2×C22⋊C4—152no—
285C2×C22⋊C4—154no—
286C2×C42—51no—
287C2×C42—151no—
288C2×C42—154no—
289#95—154no—
290#95—154no—
291#95—154no—
292C2×C3×C5C2×#11711yescharacteristic
293C2×C3×C5#52012yes—
294D15—41no—
295C3×D5#54011yescharacteristic
296C3×D5#54111yescharacteristic
297C3×D5#54212yes—
298C5×D3—41no—
299C23×C3—51no—
300C22×D3—51no—
301C22×D3—102no—
302C22×D3—101no—
303C22×D3—101no—
304C22×D3—102no—
305C22×D3C22×AGL(1,𝔽5)11yescharacteristic
306C2×C3×C4—51no—
307C2×C3×C4—52no—
308C2×C3×C4—52no—
309C2×C3×C4—52no—
310C2×C3×C4—102no—
311C2×C3×C4—102no—
312C2×C3×C4—104no—
313C2×C3×C4—104no—
314C2×C3×C4—104no—
315C2×C3×C4C22×AGL(1,𝔽5)12yes—
316C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
317C2×Dic3—52no—
318C2×Dic3—51no—
319C2×Dic3—102no—
320C2×Dic3—101no—
321C2×Dic3—101no—
322C2×Dic3—102no—
323C2×Dic3—104no—
324C2×Dic3—104no—
325C2×Dic3—104no—
326C2×Dic3C22×AGL(1,𝔽5)12yes—
327C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
328C4×D3—22no—
329C4×D3—102no—
330C4×D3—102no—
331C4×D3—102no—
332C22×C5—61no—
333C22×C5—62no—
334C22×C5C22×C4×D311yescharacteristic
335C2×D5—61no—
336C2×D5—62no—
337C2×D5—121no—
338C2×D5—121no—
339C2×D5—122no—
340C2×D5C2×#23711yescharacteristic
341C2×D5#100412yes—
342C2×D5C2×#23911yescharacteristic
343C2×D5#100712yes—
344AGL(1,𝔽5)—44no—
345AGL(1,𝔽5)—124no—
346C4×C5—22no—
347C4×C5—61no—
348C4×C5—62no—
349C4×C5#89212yes—
350Dic5—22no—
351Dic5—61no—
352Dic5—62no—
353Dic5#89212yes—
354C24—151no—
355C22×C4—31no—
356C22×C4—51no—
357C22×C4—52no—
358C22×C4—54no—
359C22×C4—151no—
360C22×C4—151no—
361C22×C4—152no—
362C22×C4—151no—
363C22×C4—151no—
364C22×C4—154no—
365C22×C4—302no—
366C22×C4—302no—
367C4⋊C4—32no—
368C4⋊C4—104no—
369C4⋊C4—152no—
370C4⋊C4—152no—
371C4⋊C4—152no—
372C4⋊C4—204no—
373C4⋊C4—304no—
374C22⋊C4—32no—
375C22⋊C4—152no—
376C22⋊C4—152no—
377C22⋊C4—152no—
378C22⋊C4—604no—
379C42—31no—
380C42—52no—
381C42—51no—
382C42—152no—
383C42—151no—
384C42—604no—
385C42D3×AGL(1,𝔽5)11yescharacteristic
386C3×C5#231411yescharacteristic, complemented, Hall
387C22×C3—51no—
388C22×C3—51no—
389C22×C3—52no—
390C22×C3—52no—
391C22×C3C23×AGL(1,𝔽5)11yescharacteristic
392C2×D3—21no—
393C2×D3—22no—
394C2×D3—101no—
395C2×D3—102no—
396C2×D3—202no—
397C2×D3—201no—
398C2×D3—201no—
399C3×C4—22no—
400C3×C4—52no—
401C3×C4—102no—
402C3×C4—204no—
403C3×C4#413012yes—
404Dic3—21no—
405Dic3—22no—
406Dic3—101no—
407Dic3—102no—
408Dic3—204no—
409C2×C5—121no—
410C2×C5C2×#89211yescharacteristic
411C2×C5#550412yes—
412D5—121no—
413D5#571911yescharacteristic
414D5#573111yescharacteristic
415D5#573312yes—
416C23—31no—
417C23—51no—
418C23—151no—
419C23—302no—
420C23—301no—
421C23—301no—
422C23—302no—
423C2×C4—32no—
424C2×C4—31no—
425C2×C4—51no—
426C2×C4—52no—
427C2×C4—52no—
428C2×C4—52no—
429C2×C4—62no—
430C2×C4—102no—
431C2×C4—102no—
432C2×C4—104no—
433C2×C4—104no—
434C2×C4—104no—
435C2×C4—152no—
436C2×C4—151no—
437C2×C4—302no—
438C2×C4—302no—
439C2×C4—302no—
440C2×C4—301no—
441C2×C4—301no—
442C2×C4—302no—
443C2×C4—302no—
444C2×C4—304no—
445C2×C4—304no—
446C2×C4—304no—
447C2×C4C2×D3×AGL(1,𝔽5)12yes—
448C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
449C2×C3—52no—
450C2×C3—51no—
451C2×C3—51no—
452C2×C3C2×#413011yescharacteristic
453C2×C3#6626612yes—
454D3—41no—
455D3—201no—
456C5#7435811yescharacteristic, complemented, Sylow
457C22—51no—
458C22—51no—
459C22—52no—
460C22—52no—
461C22—61no—
462C22—62no—
463C22—301no—
464C22—302no—
465C22—601no—
466C22—602no—
467C22—601no—
468C22C22×D3×AGL(1,𝔽5)11yescharacteristic
469C4—22no—
470C4—52no—
471C4—61no—
472C4—62no—
473C4—102no—
474C4—204no—
475C4—301no—
476C4—302no—
477C4—604no—
478C4#9190712yes—
479C3#1061713011yescharacteristic, complemented, Sylow
480C2—51no—
481C2—52no—
482C2—51no—
483C2—121no—
484C2—601no—
485C2C2×#9190711yescharacteristic
486C2#1175683612yes—
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C42×C5C2×C4
prime radicalC22×D3×D5C23
cosocleC22×C3×D5C24
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#2314
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11752348C2semidirect
3. C3×#65688C2semidirect
5. Dic3×C4⋊AGL(1,𝔽5)C2semidirect
7. #11751835C2semidirect
15. #91761C4semidirect
16. C5×#872C4semidirect
19. #91402C4semidirect
20. #91398C4semidirect
23. #91386C4semidirect
24. #91371C4semidirect
34. #65688D3semidirect
143. #872AGL(1,𝔽5)semidirect
386. C3×C5#2314semidirect
456. C5#74358semidirect
479. C3#10617130semidirect

Maximal subgroups

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

subgroupindex
D5×#8722
#117523482
C3×#656882
#117518552
Dic3×C4⋊AGL(1,𝔽5)2
#117518362
#117518352
#106171303
#743585