Finite Group Databasebeta

← #14283184

Subgroups — #14283184

Order 1920.

588 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#4043C211yesmaximal, characteristic
2#11751848C211yesmaximal, characteristic
3#11751802C211yesmaximal, characteristic
4#11746740C211yesmaximal, characteristic, complemented
5C3×#64869C211yesmaximal, characteristic, complemented
6#11746503C211yesmaximal, characteristic, complemented
7#11746501C211yesmaximal, characteristic, complemented
8#10604259—31nomaximal, Hall
9C2×C4×D3×D5C2211yescharacteristic
10C2×D3×Dic10C411yescharacteristic, complemented
11C2×#6740—21no—
12C2×(C2×D3)⋊AGL(1,𝔽5)—21no—
13C2×Dic3×AGL(1,𝔽5)—21no—
14#91786C411yescharacteristic, complemented
15C5×D3×C4⋊C4C411yescharacteristic, complemented
16C3×#4043C2211yescharacteristic
17D3×#685—21no—
18#91429—21no—
19D3×C4⋊Dic5—21no—
20#91422—21no—
21#91390C411yescharacteristic, complemented
22#91360C2211yescharacteristic
23#91235C2211yescharacteristic
24#91232C2211yescharacteristic
25C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
26C3×#4003C2211yescharacteristic
27(C2×D3)⋊#170—21no—
28#68335—51nomaximal, Hall
29C2×#4043—31no—
30C4×C22⋊AGL(1,𝔽5)—31no—
31#65756—31no—
32#64869—31no—
33#64869—31no—
34#64869—31no—
35#64869D311yescharacteristic, complemented
36C22×D3×D5D411yescharacteristic
37C22×ΓL(1,𝔽16)—21no—
38C22×C3×AGL(1,𝔽5)—21no—
39C2×C4×D15C2×C411yescharacteristic
40C2×Dic30C2×C411yescharacteristic
41C2×C4×C5×D3—21no—
42C2×C4×C5×D3C2×C411yescharacteristic
43C2×C3×C4×D5C2311yescharacteristic
44C2×C3×Dic10C2×C411yescharacteristic
45C2×#1204—21no—
46C2×#1200D411yescharacteristic
47C2×D3×Dic5—21no—
48C2×D3×Dic5D411yescharacteristic
49C2×D5×Dic3D411yescharacteristic
50C4×D3×D5—42no—
51D3×Dic10—42no—
52C2×C3⋊#170—21no—
53C2×C3×#170—21no—
54#6740—42no—
55(C2×D3)⋊AGL(1,𝔽5)—24no—
56Dic3×AGL(1,𝔽5)—42no—
57C3×C5×C4⋊C4C2×C411yescharacteristic
58#6714—21no—
59C4⋊Dic15—21no—
60C5×C4⋊Dic3C2×C411yescharacteristic
61C5×Dic3⋊C4—21no—
62C3×#685—21no—
63C3×C4⋊Dic5—21no—
64Dic3⋊Dic5—21no—
65Dic3⋊Dic5—21no—
66#6667—21no—
67#6665—21no—
68D3×C22⋊Q8—51no—
69#4726—51no—
70#4666—51no—
71C3×#328—51no—
72#4538—51no—
73#4467—51no—
74#4452—51no—
75C22×C4×D5—31no—
76C22×Dic10—31no—
77C2×C22⋊AGL(1,𝔽5)—61no—
78C2×C4×AGL(1,𝔽5)—61no—
79C2×#700—61no—
80C2×C5×C4⋊C4—31no—
81C2×#685—61no—
82C2×C4⋊Dic5—61no—
83#4043—31no—
84#4043—31no—
85#4043—31no—
86#4043—61no—
87#4043—61no—
88#4043C2×D311yescharacteristic
89C22⋊#170—61no—
90(C2×AGL(1,𝔽5))⋊C4—31no—
91(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
92#4003—31no—
93#4003C2×D311yescharacteristic
94#1508—151noSylow
95C22×D15C22⋊C411yescharacteristic
96C22×C5×D3C22⋊C411yescharacteristic
97C22×C3×D5C2×D411yescharacteristic
98C2×D3×D5—22no—
99C2×D3×D5—21no—
100C2×D3×D5—21no—
101C2×D3×D5—41no—
102C2×ΓL(1,𝔽16)—22no—
103C2×ΓL(1,𝔽16)—41no—
104C2×ΓL(1,𝔽16)—41no—
105C2×C3×AGL(1,𝔽5)—22no—
106C2×C3×AGL(1,𝔽5)—41no—
107C2×C3×AGL(1,𝔽5)—41no—
108C2×C3×C4×C5—21no—
109C2×C3×C4×C5C22×C411yescharacteristic
110C2×Dic15—21no—
111C2×Dic15C22⋊C411yescharacteristic
112C4×D15—41no—
113Dic30—41no—
114C2×C5×Dic3—21no—
115C2×C5×Dic3C22⋊C411yescharacteristic
116C4×C5×D3—22no—
117C4×C5×D3—42no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5C2×D411yescharacteristic
120C3×C4×D5—41no—
121C3×Dic10—41no—
122#1204—42no—
123#1200—22no—
124D3×Dic5—41no—
125D3×Dic5—42no—
126D5×Dic3—22no—
127C3⋊#170—41no—
128C3×#170—41no—
129C2×D3×Q8—51no—
130C22×C4×D3—51no—
131C3×C22⋊Q8—51no—
132#946—51no—
133C2×(C2×D3)⋊C4—101no—
134#924—51no—
135C2×C4×Dic3—101no—
136C2×ΓL(1,𝔽25)—101no—
137#897—51no—
138#896—101no—
139D3×C4⋊C4—101no—
140D3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
141D3×C22⋊C4—101no—
142#878—101no—
143C3×C22⋊C8—51no—
144C3×#95—51no—
145C3⋊#95—51no—
146#830—51no—
147(C2×D3)⋊C8—101no—
148C23×D5—31no—
149C22×AGL(1,𝔽5)—21no—
150C22×AGL(1,𝔽5)—61no—
151C22×C4×C5—31no—
152C22×C4×C5—61no—
153C22×Dic5—31no—
154C22×Dic5—61no—
155C2×C4×D5—31no—
156C2×C4×D5—31no—
157C2×C4×D5—31no—
158C2×C4×D5—122no—
159C2×C4×D5C22×D311yescharacteristic
160C2×Dic10—31no—
161C2×Dic10—61no—
162C2×Dic10—122no—
163C2×Dic10C4×D311yescharacteristic
164C22⋊AGL(1,𝔽5)—64no—
165C2×#170—21no—
166C2×#170—61no—
167C4×AGL(1,𝔽5)—122no—
168#700—122no—
169C5×C4⋊C4—31no—
170C5×C4⋊C4—61no—
171C5×C4⋊C4C4×D311yescharacteristic
172#685—21no—
173#685—61no—
174#685—61no—
175#685—61no—
176C4⋊Dic5—21no—
177C4⋊Dic5—61no—
178C4⋊Dic5—61no—
179C4⋊Dic5—61no—
180C2×C22⋊Q8—151no—
181#407—151no—
182C4×C22⋊C4—151no—
183#328—51no—
184#328—151no—
185#328—151no—
186#328—151no—
187C22×C3×C5C2×C22⋊C411yescharacteristic
188C2×D15—21no—
189C2×D15—21no—
190C2×D15—21no—
191C2×C5×D3—21no—
192C2×C5×D3—21no—
193C2×C5×D3#10412yes—
194C2×C3×D5—21no—
195C2×C3×D5—21no—
196C2×C3×D5#13711yescharacteristic
197C2×C3×D5#13711yescharacteristic
198D3×D5—42no—
199D3×D5—42no—
200ΓL(1,𝔽16)—42no—
201C3×AGL(1,𝔽5)—42no—
202C3×C4×C5—21no—
203C3×C4×C5—41no—
204Dic15—21no—
205Dic15—41no—
206C3×Dic5—21no—
207C3×Dic5—41no—
208C5×Dic3—41no—
209C5×Dic3#9912yes—
210C23×D3—51no—
211C2×C3×Q8—51no—
212C22×C3×C4—51no—
213C22×C3×C4—101no—
214C22×Dic3—51no—
215C22×Dic3—101no—
216D3×Q8—202no—
217C2×C4×D3—21no—
218C2×C4×D3—51no—
219C2×C4×D3—51no—
220C2×C4×D3—51no—
221C2×C4×D3—101no—
222C2×C4×D3—202no—
223C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
224C2×Dic6—51no—
225C2×Dic6—101no—
226C2×C3×C8—101no—
227C3×C4⋊C4—101no—
228C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
229C3×C22⋊C4—101no—
230#217—101no—
231(C2×D3)⋊C4—101no—
232(C2×D3)⋊C4—101no—
233(C2×D3)⋊C4—104no—
234C4⋊Dic3—101no—
235C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
236Dic3⋊C4—21no—
237Dic3⋊C4—101no—
238Dic3⋊C4—101no—
239C4×Dic3—202no—
240C2×C3⋊C8—101no—
241ΓL(1,𝔽25)—202no—
242C23×C5—31no—
243C22×D5—31no—
244C22×D5—62no—
245C22×D5—61no—
246C22×D5—61no—
247C22×D5—121no—
248C22×D5D3×D411yescharacteristic
249C2×AGL(1,𝔽5)—22no—
250C2×AGL(1,𝔽5)—41no—
251C2×AGL(1,𝔽5)—41no—
252C2×AGL(1,𝔽5)—62no—
253C2×AGL(1,𝔽5)—121no—
254C2×AGL(1,𝔽5)—121no—
255C2×C4×C5—21no—
256C2×C4×C5—31no—
257C2×C4×C5—62no—
258C2×C4×C5—61no—
259C2×C4×C5—122no—
260C2×C4×C5C2×C4×D311yescharacteristic
261C2×Dic5—21no—
262C2×Dic5—31no—
263C2×Dic5—61no—
264C2×Dic5—121no—
265C2×Dic5—122no—
266C2×Dic5D3×D411yescharacteristic
267C4×D5—41no—
268C4×D5—62no—
269C4×D5—62no—
270C4×D5—121no—
271Dic10—41no—
272Dic10—122no—
273Dic10—121no—
274#170—41no—
275#170—121no—
276C22×Q8—151no—
277C23×C4—151no—
278C2×M16—301no—
279C22⋊Q8—51no—
280C22⋊Q8—151no—
281C22⋊Q8—151no—
282C22⋊Q8—151no—
283C22⋊Q8—301no—
284C22⋊Q8—301no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—301no—
287C2×C22⋊C4—301no—
288C2×C22⋊C4—301no—
289C2×C42—301no—
290C22⋊C8—51no—
291C22⋊C8—151no—
292C22⋊C8—301no—
293#95—51no—
294#95—151no—
295C2×C3×C5C2×#9911yescharacteristic
296C2×C3×C5#41911yescharacteristic
297C2×C3×C5C2×#10411yescharacteristic
298D15—41no—
299C3×D5—21no—
300C3×D5—21no—
301C5×D3—22no—
302C23×C3—51no—
303C22×D3—51no—
304C22×D3—101no—
305C22×D3—101no—
306C22×D3—102no—
307C22×D3—201no—
308C22×D3C22⋊AGL(1,𝔽5)11yescharacteristic
309C3×Q8—201no—
310C2×C3×C4—21no—
311C2×C3×C4—51no—
312C2×C3×C4—102no—
313C2×C3×C4—101no—
314C2×C3×C4—201no—
315C2×C3×C4—201no—
316C2×C3×C4—201no—
317C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
318C2×Dic3—21no—
319C2×Dic3—51no—
320C2×Dic3—102no—
321C2×Dic3—102no—
322C2×Dic3—101no—
323C2×Dic3—201no—
324C2×Dic3—201no—
325C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
326C4×D3—22no—
327C4×D3—42no—
328C4×D3—102no—
329C4×D3—201no—
330C4×D3—201no—
331C4×D3—202no—
332Dic6—202no—
333Dic6—201no—
334C3×C8—201no—
335C3⋊C8—201no—
336C22×C5—32no—
337C22×C5—61no—
338C22×C5—61no—
339C22×C5D3×C22⋊C411yescharacteristic
340C2×D5—21no—
341C2×D5—21no—
342C2×D5—61no—
343C2×D5—61no—
344C2×D5—61no—
345C2×D5—122no—
346C2×D5—122no—
347C2×D5#91511yescharacteristic
348C2×D5#91811yescharacteristic
349AGL(1,𝔽5)—42no—
350AGL(1,𝔽5)—122no—
351C4×C5—21no—
352C4×C5—32no—
353C4×C5—41no—
354C4×C5—121no—
355Dic5—21no—
356Dic5—41no—
357Dic5—61no—
358Dic5—121no—
359C24—151no—
360C2×Q8—51no—
361C2×Q8—151no—
362C2×Q8—301no—
363C2×Q8—602no—
364C22×C4—31no—
365C22×C4—51no—
366C22×C4—61no—
367C22×C4—101no—
368C22×C4—151no—
369C22×C4—151no—
370C22×C4—151no—
371C22×C4—151no—
372C22×C4—301no—
373C22×C4—301no—
374C22×C4—602no—
375M16—602no—
376C2×C8—101no—
377C2×C8—301no—
378C4⋊C4—31no—
379C4⋊C4—61no—
380C4⋊C4—101no—
381C4⋊C4—301no—
382C4⋊C4—301no—
383C4⋊C4—301no—
384C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
385C22⋊C4—101no—
386C22⋊C4—301no—
387C22⋊C4—301no—
388C22⋊C4—301no—
389C22⋊C4—304no—
390C42—602no—
391C3×C5#150811yescharacteristic, complemented, Hall
392C22×C3—51no—
393C22×C3—51no—
394C22×C3—101no—
395C22×C3—101no—
396C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
397C2×D3—21no—
398C2×D3—21no—
399C2×D3—101no—
400C2×D3—101no—
401C2×D3—101no—
402C2×D3—202no—
403C2×D3—202no—
404C2×D3#400812yes—
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—101no—
408C3×C4—202no—
409C3×C4—201no—
410Dic3—41no—
411Dic3—101no—
412Dic3—202no—
413Dic3—201no—
414Dic3#399912yes—
415C2×C5—62no—
416C2×C5D3×#9911yescharacteristic
417C2×C5#480211yescharacteristic
418C2×C5D3×#10411yescharacteristic
419D5—21no—
420D5—21no—
421D5—121no—
422C23—31no—
423C23—51no—
424C23—151no—
425C23—302no—
426C23—301no—
427C23—301no—
428C23—601no—
429Q8—201no—
430Q8—601no—
431Q8—602no—
432C2×C4—21no—
433C2×C4—31no—
434C2×C4—51no—
435C2×C4—62no—
436C2×C4—61no—
437C2×C4—102no—
438C2×C4—101no—
439C2×C4—122no—
440C2×C4—151no—
441C2×C4—201no—
442C2×C4—201no—
443C2×C4—201no—
444C2×C4—302no—
445C2×C4—302no—
446C2×C4—302no—
447C2×C4—301no—
448C2×C4—601no—
449C2×C4—601no—
450C2×C4—602no—
451C2×C4—601no—
452C2×C4—601no—
453C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
454C8—201no—
455C8—601no—
456C2×C3—101no—
457C2×C3—101no—
458C2×C3C2×#399911yescharacteristic
459C2×C3C2×#400811yescharacteristic
460C2×C3#6578211yescharacteristic
461D3—22no—
462D3—201no—
463C5#6833511yescharacteristic, complemented, Sylow
464C22—32no—
465C22—51no—
466C22—51no—
467C22—61no—
468C22—61no—
469C22—101no—
470C22—101no—
471C22—301no—
472C22—301no—
473C22—301no—
474C22—602no—
475C22—602no—
476C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
477C4—21no—
478C4—32no—
479C4—41no—
480C4—101no—
481C4—121no—
482C4—202no—
483C4—201no—
484C4—301no—
485C4—602no—
486C4—601no—
487C3#1060425911yescharacteristic, complemented, Sylow
488C2—62no—
489C2—101no—
490C2—101no—
491C2—601no—
492C2D3×#399911yescharacteristic
493C2D3×#400811yescharacteristic
494C2#1175196211yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×D3×D5D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1508
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11746740C2semidirect
5. C3×#64869C2semidirect
6. #11746503C2semidirect
7. #11746501C2semidirect
10. C2×D3×Dic10C4semidirect
14. #91786C4semidirect
15. C5×D3×C4⋊C4C4semidirect
21. #91390C4semidirect
35. #64869D3semidirect
140. D3×C4⋊C4AGL(1,𝔽5)semidirect
391. C3×C5#1508semidirect
463. C5#68335semidirect
487. C3#10604259semidirect

Maximal subgroups

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

subgroupindex
D3×#40432
#117518482
#117518022
#117467402
C3×#648692
#117465032
#117465012
#106042593
#683355