Finite Group Databasebeta

← #14283020

Subgroups — #14283020

Order 1920.

668 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 86 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
1#11755276C211yesmaximal, characteristic
2D3×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic, complemented
3#11751919C211yesmaximal, characteristic, complemented
4#11746797C211yesmaximal, characteristic, complemented
5C3×#64926C211yesmaximal, characteristic, complemented
6#11746498C211yesmaximal, characteristic, complemented
7#11746496C211yesmaximal, characteristic, complemented
8#10604990—31nomaximal, Hall
9C22×D3×AGL(1,𝔽5)—21no—
10C2×C5×#239C411yescharacteristic, complemented
11C2×C4×D3×D5C2211yescharacteristic
12C2×#6739—21no—
13#91835C411yescharacteristic, complemented
14C3×#4092C2211yescharacteristic
15#91475—21no—
16#91466C411yescharacteristic, complemented
17#91449C2211yescharacteristic
18#91446—21no—
19#91441—21no—
20#91423—21no—
21#91338C411yescharacteristic, complemented
22#91235C2211yescharacteristic
23#91232C2211yescharacteristic
24C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
25C3×#4003C2211yescharacteristic
26#91171—21no—
27(C2×AGL(1,𝔽5))⋊Dic3—21no—
28#68334—51nomaximal, Hall
29#66162—31no—
30C2×(C2×AGL(1,𝔽5))⋊C4—31no—
31#65755—31no—
32#64926—31no—
33#64926D311yescharacteristic, complemented
34#64920—31no—
35#64920—31no—
36C22×D3×D5D411yescharacteristic
37C22×ΓL(1,𝔽16)—21no—
38C22×C3×AGL(1,𝔽5)—21no—
39C2×D3×AGL(1,𝔽5)—24no—
40C2×D3×AGL(1,𝔽5)—42no—
41C2×D3×AGL(1,𝔽5)—42no—
42C2×C3×C5×Q8C2×C411yescharacteristic, complemented
43C2×C4×D15C2×C411yescharacteristic
44C5×#239—42no—
45C2×C5×D12—21no—
46C2×C5×D12C2×C411yescharacteristic, complemented
47C2×C4×C5×D3—21no—
48C2×C4×C5×D3C2×C411yescharacteristic
49C2×C3×C4×D5C2311yescharacteristic
50C2×#1203—21no—
51C2×#1201—21no—
52C2×#1200D411yescharacteristic
53C2×D3×Dic5D411yescharacteristic
54C2×D5×Dic3D411yescharacteristic
55C4×D3×D5—42no—
56C2×C3⋊#170—21no—
57C2×C3×#170—21no—
58#6739—42no—
59#6714—21no—
60C4⋊Dic15C2×C411yescharacteristic, complemented
61#6712—21no—
62C3×#685—21no—
63C3×C4⋊Dic5C2×C411yescharacteristic, complemented
64C3×#683—21no—
65(C2×D3)⋊Dic5—21no—
66(C2×D3)⋊Dic5—21no—
67(C2×D3)⋊Dic5—21no—
68Dic3×Dic5—21no—
69#5627—51no—
70#4726—51no—
71D3×#95—51no—
72C3×#328—51no—
73#4538—51no—
74#4470—51no—
75#4451—51no—
76C23×AGL(1,𝔽5)—61no—
77C2×C5×#41—31no—
78C22×C4×D5—31no—
79C2×#704—61no—
80#4092—31no—
81#4092C2×D311yescharacteristic
82#4084—31no—
83#4084—31no—
84#4083—61no—
85#4081—61no—
86C4×C22⋊D5—61no—
87C4×C22⋊D5—61no—
88#4072—31no—
89(C2×AGL(1,𝔽5))⋊C4—31no—
90(C2×AGL(1,𝔽5))⋊C4—61no—
91(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
92#4003—31no—
93#4003—61no—
94#4003C2×D311yescharacteristic
95#1504—151noSylow
96C22×D15C22⋊C411yescharacteristic
97C22×C5×D3—21no—
98C22×C5×D3C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×D3×D5—22no—
101C2×D3×D5—22no—
102C2×D3×D5—21no—
103C2×D3×D5—21no—
104C2×ΓL(1,𝔽16)—22no—
105C2×ΓL(1,𝔽16)—41no—
106C2×ΓL(1,𝔽16)—41no—
107C2×C3×AGL(1,𝔽5)—22no—
108C2×C3×AGL(1,𝔽5)—41no—
109C2×C3×AGL(1,𝔽5)—41no—
110D3×AGL(1,𝔽5)—48no—
111C3×C5×Q8—41no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5C22×C411yescharacteristic
114C2×Dic15—21no—
115C2×Dic15C22⋊C411yescharacteristic
116C4×D15—41no—
117C2×C5×Dic3C22⋊C411yescharacteristic
118C5×D12—41no—
119C5×D12—42no—
120C4×C5×D3—22no—
121C4×C5×D3—42no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5C2×D411yescharacteristic
124C3×C4×D5—41no—
125#1203—42no—
126#1201—41no—
127#1201—41no—
128#1200—41no—
129D3×Dic5—22no—
130D5×Dic3—41no—
131C3⋊#170—41no—
132C3×#170—41no—
133C2×#239AGL(1,𝔽5)11yescharacteristic, complemented
134C22×C4×D3—51no—
135C22×C4×D3—101no—
136C3×C22⋊Q8—51no—
137#946—51no—
138#930—51no—
139C2×ΓL(1,𝔽25)—101no—
140#895—51no—
141#894—101no—
142#893—101no—
143#892—51no—
144#882—101no—
145#67⋊C4—101no—
146C3×C22⋊C8—51no—
147C3×#95—51no—
148C3⋊#95—51no—
149C3⋊#95—101no—
150#830—51no—
151(C2×D3)⋊C8—101no—
152C23×D5—31no—
153C22×AGL(1,𝔽5)—21no—
154C22×AGL(1,𝔽5)—64no—
155C22×AGL(1,𝔽5)—61no—
156C22×AGL(1,𝔽5)—122no—
157C22×AGL(1,𝔽5)—122no—
158C5×#41—122no—
159C2×C5×Q8C4×D311yescharacteristic, complemented
160C2×C5×D4—31no—
161C2×C5×D4—61no—
162C22×C4×C5—31no—
163C22×C4×C5—61no—
164C2×C22⋊D5—61no—
165C2×C22⋊D5—61no—
166C22×Dic5—31no—
167C2×C4×D5—31no—
168C2×C4×D5—31no—
169C2×C4×D5—31no—
170C2×C4×D5—122no—
171C2×C4×D5C22×D311yescharacteristic
172#704—122no—
173C2×#170—21no—
174C2×#170—61no—
175C22⋊Dic5—61no—
176C22⋊Dic5—61no—
177C22⋊Dic5—61no—
178#685—21no—
179#685—61no—
180C4⋊Dic5—31no—
181C4⋊Dic5C4×D311yescharacteristic, complemented
182#683—21no—
183#683—61no—
184C4×Dic5—61no—
185#525—151no—
186#407—151no—
187C2×#95—151no—
188#328—51no—
189#328—151no—
190#327—151no—
191#327—151no—
192C22×C3×C5C2×C22⋊C411yescharacteristic
193C2×D15—21no—
194C2×D15—21no—
195C2×D15#10412yes—
196C2×C5×D3—21no—
197C2×C5×D3—21no—
198C2×C5×D3—41no—
199C2×C5×D3—41no—
200C2×C5×D3—41no—
201C2×C5×D3#9912yes—
202C2×C3×D5—21no—
203C2×C3×D5—21no—
204C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
205C2×C3×D5#13711yescharacteristic
206D3×D5—24no—
207D3×D5—24no—
208ΓL(1,𝔽16)—42no—
209C3×AGL(1,𝔽5)—42no—
210C3×C4×C5—21no—
211C3×C4×C5—41no—
212Dic15—21no—
213Dic15—41no—
214C3×Dic5—21no—
215C3×Dic5—41no—
216C5×Dic3—21no—
217C23×D3—51no—
218C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
219C22×C3×C4—51no—
220C22×C3×C4—101no—
221C2×#67—101no—
222C22×Dic3—51no—
223C22×Dic3—101no—
224#239—42no—
225C2×D12—21no—
226C2×D12—101no—
227C2×D12C2×AGL(1,𝔽5)11yescharacteristic, complemented
228C2×C4×D3—21no—
229C2×C4×D3—51no—
230C2×C4×D3—51no—
231C2×C4×D3—51no—
232C2×C4×D3—104no—
233C2×C4×D3—202no—
234C2×C4×D3—202no—
235C2×C4×D3—202no—
236C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
237C2×C3×C8—101no—
238C3×C4⋊C4—51no—
239C3×C4⋊C4—101no—
240C3×C22⋊C4—101no—
241(C2×D3)⋊C4—101no—
242(C2×D3)⋊C4—101no—
243(C2×D3)⋊C4—101no—
244(C2×D3)⋊C4—101no—
245C4⋊Dic3—51no—
246Dic3⋊C4—101no—
247C4×Dic3—101no—
248C2×C3⋊C8—101no—
249ΓL(1,𝔽25)—202no—
250C23×C5—31no—
251C23×C5—61no—
252C22×D5—31no—
253C22×D5—62no—
254C22×D5—62no—
255C22×D5—61no—
256C22×D5—61no—
257C22×D5D3×D411yescharacteristic
258C2×AGL(1,𝔽5)—22no—
259C2×AGL(1,𝔽5)—41no—
260C2×AGL(1,𝔽5)—41no—
261C2×AGL(1,𝔽5)—62no—
262C2×AGL(1,𝔽5)—128no—
263C2×AGL(1,𝔽5)—121no—
264C2×AGL(1,𝔽5)—121no—
265C5×Q8—41no—
266C5×D4—121no—
267C5×D4—122no—
268C2×C4×C5—21no—
269C2×C4×C5—31no—
270C2×C4×C5—62no—
271C2×C4×C5—122no—
272C2×C4×C5C2×C4×D311yescharacteristic
273C22⋊D5—122no—
274C22⋊D5—121no—
275C22⋊D5—121no—
276C2×Dic5—21no—
277C2×Dic5—31no—
278C2×Dic5—62no—
279C2×Dic5—61no—
280C2×Dic5D3×D411yescharacteristic
281C4×D5—41no—
282C4×D5—121no—
283C4×D5—121no—
284C4×D5—121no—
285#170—41no—
286#170—121no—
287C2×#41—31no—
288C23×C4—151no—
289C23×C4—301no—
290C2×M16—301no—
291#123—301no—
292C22⋊Q8—51no—
293C22⋊Q8—151no—
294#121—151no—
295#121—151no—
296#120—301no—
297C4×D4—301no—
298C4×D4—301no—
299#117—151no—
300C22⋊C8—51no—
301C22⋊C8—151no—
302C22⋊C8—301no—
303#95—51no—
304#95—151no—
305#95—301no—
306C2×C3×C5C2×#9911yescharacteristic
307C2×C3×C5#41711yescharacteristic
308C2×C3×C5C2×#10411yescharacteristic
309D15—22no—
310C3×D5—21no—
311C3×D5—21no—
312C5×D3—22no—
313C5×D3—81no—
314C23×C3—51no—
315C22×D3—21no—
316C22×D3—51no—
317C22×D3—102no—
318C22×D3—101no—
319C22×D3—101no—
320C22×D3—102no—
321C22×D3C22⋊AGL(1,𝔽5)11yescharacteristic
322C3×Q8—41no—
323C2×C3×C4—21no—
324C2×C3×C4—51no—
325C2×C3×C4—102no—
326C2×C3×C4—101no—
327C2×C3×C4—201no—
328C2×C3×C4—201no—
329C2×C3×C4—201no—
330C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
331#67—201no—
332#67—201no—
333C2×Dic3—51no—
334C2×Dic3—102no—
335C2×Dic3—101no—
336C2×Dic3—201no—
337C2×Dic3—201no—
338C2×Dic3—201no—
339C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
340D12—41no—
341D12—42no—
342D12—202no—
343C4×D3—22no—
344C4×D3—42no—
345C4×D3—102no—
346C4×D3—201no—
347C4×D3—201no—
348C4×D3—208no—
349C3×C8—201no—
350C3⋊C8—201no—
351C22×C5—32no—
352C22×C5—61no—
353C22×C5—61no—
354C22×C5—121no—
355C22×C5—121no—
356C22×C5—121no—
357C22×C5D3×C22⋊C411yescharacteristic
358C2×D5—21no—
359C2×D5—21no—
360C2×D5—32no—
361C2×D5—64no—
362C2×D5—64no—
363C2×D5—61no—
364C2×D5—61no—
365C2×D5#91411yescharacteristic
366C2×D5#91811yescharacteristic
367AGL(1,𝔽5)—42no—
368AGL(1,𝔽5)—122no—
369C4×C5—21no—
370C4×C5—41no—
371C4×C5—61no—
372Dic5—21no—
373Dic5—41no—
374Dic5—61no—
375Dic5—121no—
376C24—151no—
377#41—122no—
378C2×Q8D3×AGL(1,𝔽5)11yescharacteristic, complemented
379C2×D4—31no—
380C2×D4—61no—
381C2×D4—301no—
382C2×D4—301no—
383C22×C4—31no—
384C22×C4—51no—
385C22×C4—61no—
386C22×C4—101no—
387C22×C4—151no—
388C22×C4—151no—
389C22×C4—151no—
390C22×C4—151no—
391C22×C4—304no—
392C22×C4—301no—
393C22×C4—602no—
394C22×C4—602no—
395C22×C4—602no—
396M16—602no—
397C2×C8—101no—
398C2×C8—301no—
399C4⋊C4—51no—
400C4⋊C4—101no—
401C4⋊C4—151no—
402C4⋊C4—301no—
403C22⋊C4—101no—
404C22⋊C4—301no—
405C22⋊C4—301no—
406C22⋊C4—301no—
407C22⋊C4—301no—
408C42—301no—
409C3×C5#150411yescharacteristic, complemented, Hall
410C22×C3—51no—
411C22×C3—51no—
412C22×C3—101no—
413C22×C3—101no—
414C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
415C2×D3—21no—
416C2×D3—21no—
417C2×D3—41no—
418C2×D3—41no—
419C2×D3—41no—
420C2×D3—52no—
421C2×D3—104no—
422C2×D3—104no—
423C2×D3—101no—
424C2×D3—101no—
425C2×D3#401112yes—
426C3×C4—21no—
427C3×C4—41no—
428C3×C4—101no—
429C3×C4—202no—
430C3×C4—201no—
431Dic3—21no—
432Dic3—101no—
433Dic3—202no—
434Dic3—201no—
435C2×C5—62no—
436C2×C5—241no—
437C2×C5D3×#9911yescharacteristic
438C2×C5#480311yescharacteristic
439C2×C5D3×#10411yescharacteristic
440D5—21no—
441D5—21no—
442D5—62no—
443C23—31no—
444C23—51no—
445C23—61no—
446C23—151no—
447C23—302no—
448C23—301no—
449C23—301no—
450C23—302no—
451Q8—41no—
452D4—121no—
453D4—122no—
454D4—602no—
455D4—601no—
456D4—601no—
457C2×C4—21no—
458C2×C4—31no—
459C2×C4—51no—
460C2×C4—62no—
461C2×C4—102no—
462C2×C4—101no—
463C2×C4—122no—
464C2×C4—151no—
465C2×C4—201no—
466C2×C4—201no—
467C2×C4—201no—
468C2×C4—302no—
469C2×C4—302no—
470C2×C4—301no—
471C2×C4—601no—
472C2×C4—601no—
473C2×C4—601no—
474C2×C4—608no—
475C2×C4—601no—
476C2×C4—601no—
477C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
478C8—201no—
479C8—601no—
480C2×C3—101no—
481C2×C3—101no—
482C2×C3D5⋊#10411yescharacteristic
483C2×C3#6579311yescharacteristic
484C2×C3C2×#401111yescharacteristic
485D3—22no—
486D3—81no—
487D3—102no—
488C5#6833411yescharacteristic, complemented, Sylow
489C22—32no—
490C22—51no—
491C22—51no—
492C22—61no—
493C22—61no—
494C22—101no—
495C22—101no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—152no—
500C22—304no—
501C22—304no—
502C22—301no—
503C22—301no—
504C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
505C4—21no—
506C4—41no—
507C4—61no—
508C4—101no—
509C4—202no—
510C4—201no—
511C4—301no—
512C4—602no—
513C4—601no—
514C3#1060499011yescharacteristic, complemented, Sylow
515C2—62no—
516C2—101no—
517C2—101no—
518C2—241no—
519C2—302no—
520C2#1175195911yescharacteristic
521C2#1175196111yescharacteristic
522C2D3×#401111yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical#11755276C2
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1504
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. D3×(C2×AGL(1,𝔽5))⋊C4C2semidirect
3. #11751919C2semidirect
4. #11746797C2semidirect
5. C3×#64926C2semidirect
6. #11746498C2semidirect
7. #11746496C2semidirect
10. C2×C5×#239C4semidirect
13. #91835C4semidirect
16. #91466C4semidirect
21. #91338C4semidirect
33. #64926D3semidirect
42. C2×C3×C5×Q8C2×C4semidirect
46. C2×C5×D12C2×C4semidirect
60. C4⋊Dic15C2×C4semidirect
63. C3×C4⋊Dic5C2×C4semidirect
133. C2×#239AGL(1,𝔽5)semidirect
159. C2×C5×Q8C4×D3semidirect
181. C4⋊Dic5C4×D3semidirect
218. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
227. C2×D12C2×AGL(1,𝔽5)semidirect
378. C2×Q8D3×AGL(1,𝔽5)semidirect
409. C3×C5#1504semidirect
488. C5#68334semidirect
514. C3#10604990semidirect

Maximal subgroups

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

subgroupindex
#117552762
D3×(C2×AGL(1,𝔽5))⋊C42
#117519192
#117467972
C3×#649262
#117464982
#117464962
#106049903
#683345