Finite Group Databasebeta

← #14300578

Subgroups — #14300578

Order 1920.

752 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C2×C3×D5⋊M16C211yesmaximal, characteristic, complemented
2#11754526C211yesmaximal, characteristic
3C4×#6763C211yesmaximal, characteristic, complemented
4C2×#91155C212yesmaximal, complemented
5#11746524C212yesmaximal, complemented
6#10604496—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6772C2211yescharacteristic
9C2×#6763—21no—
10C2×C4×ΓL(1,𝔽16)C2212yes—
11C2×C3×#704—21no—
12C3×D5⋊M16—24no—
13C2×C3×#700C2212yes—
14C2×C3×D5⋊C8—21no—
15#91813C411yescharacteristic, complemented
16C5×#930C411yescharacteristic, complemented
17#91563—21no—
18Dic5×#67—21no—
19#91449C411yescharacteristic, complemented
20#91447—21no—
21#91440—21no—
22#91360C411yescharacteristic, complemented
23#91358C2211yescharacteristic
24#91235—21no—
25#91155—24no—
26#71984—51nomaximal, Hall
27C2×D5⋊M16D311yescharacteristic, complemented
28#65940—31no—
29C2×#4008—32no—
30C4×C22⋊AGL(1,𝔽5)—31no—
31#64875—32no—
32C23×C3×D5D411yescharacteristic
33C22×ΓL(1,𝔽16)—22no—
34C22×C3×C4×C5C2×C411yescharacteristic
35C2×Dic30C2×C411yescharacteristic
36C2×C5×#67—21no—
37C2×C5×D12C2×C411yescharacteristic
38C22×C3×Dic5D411yescharacteristic
39C2×C3×C4×D5—22no—
40C2×C3×C4×D5—22no—
41C2×C3×C4×D5C2×C411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×#1201—21no—
46C2×#1201—21no—
47C2×D3×Dic5—21no—
48#6772—22no—
49#6772—22no—
50#6763—24no—
51C4×ΓL(1,𝔽16)—22no—
52C4×ΓL(1,𝔽16)—22no—
53C3×#704—24no—
54C2×C3×#170—22no—
55C3×#700—22no—
56C3×#700D414yescomplemented
57C3×D5⋊C8—24no—
58#6719—21no—
59C4⋊Dic15C2×C411yescharacteristic
60#6712—21no—
61C5×(C2×D3)⋊C4—21no—
62C5×C4⋊Dic3C2×C411yescharacteristic
63Dic3⋊Dic5—21no—
64(C2×D3)⋊Dic5—21no—
65#6665—21no—
66#6665—21no—
67Dic3×Dic5—21no—
68C22×C3×M16—51no—
69#5791—51no—
70C4×#217—51no—
71C2×#825—52no—
72#4555—52no—
73C2×#711—31no—
74C22×C4×D5C2×D311yescharacteristic
75C2×C22⋊AGL(1,𝔽5)—61no—
76C2×#704—21no—
77C2×C4×AGL(1,𝔽5)—32no—
78D5⋊M16—24no—
79C2×#700C2×D312yes—
80C2×D5⋊C8—21no—
81C5×#121—31no—
82#4084—31no—
83#4083—61no—
84D4×Dic5—61no—
85#4070—31no—
86#4043—31no—
87#4038—31no—
88#4029—61no—
89#4027—61no—
90#4008—64no—
91(C2×AGL(1,𝔽5))⋊C4—61no—
92#1871—151noSylow
93C23×C3×C5C22⋊C411yescharacteristic
94C22×C5×D3—21no—
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5—41no—
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×ΓL(1,𝔽16)—24no—
101C2×ΓL(1,𝔽16)—42no—
102C2×ΓL(1,𝔽16)—42no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5C22⋊C411yescharacteristic
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×Dic15—21no—
107C2×Dic15—21no—
108Dic30—22no—
109C5×#67—42no—
110C2×C5×Dic3—21no—
111C5×D12—22no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5C22⋊C411yescharacteristic
115C2×C3×Dic5C2×D411yescharacteristic
116C3×C4×D5—22no—
117C3×C4×D5—22no—
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5C2×D412yes—
121C3×C4×D5#4112yes—
122#1201—42no—
123#1201—42no—
124D3×Dic5—41no—
125D3×Dic5—41no—
126C3×#170—24no—
127C23×C3×C4—51no—
128C2×#235—51no—
129C2×C3×M16—52no—
130C2×C3×M16—104no—
131C2×C3×M16—101no—
132C22×C3×C8—101no—
133#953—101no—
134C2×#217—101no—
135#930—51no—
136#930AGL(1,𝔽5)11yescharacteristic, complemented
137#929—101no—
138C4×#67—101no—
139C4×#67—101no—
140#926—51no—
141#924—51no—
142#924—51no—
143C2×C4×Dic3—52no—
144C3⋊#95—101no—
145#825—104no—
146C23×D5D1211yescharacteristic
147C22×AGL(1,𝔽5)—62no—
148C2×C5×D4—31no—
149C2×C5×D4—61no—
150C22×C4×C5C4×D311yescharacteristic
151C2×C22⋊D5—61no—
152C2×C22⋊D5—61no—
153C22×Dic5—61no—
154C22×Dic5#6711yescharacteristic
155#711—62no—
156#711—62no—
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5C4×D311yescharacteristic
160C2×C4×D5D1211yescharacteristic
161C2×C4×D5#6711yescharacteristic
162C2×C4×D5C22×D311yescharacteristic
163C2×Dic10—31no—
164C22⋊AGL(1,𝔽5)—64no—
165#704—24no—
166C2×#170—22no—
167C4×AGL(1,𝔽5)—62no—
168C4×AGL(1,𝔽5)—62no—
169#700—22no—
170#700#6714yescomplemented
171D5⋊C8—24no—
172C5×C4⋊C4—31no—
173C5×C22⋊C4—61no—
174C22⋊Dic5—61no—
175C22⋊Dic5—61no—
176#685—61no—
177#685—61no—
178C4⋊Dic5—31no—
179C4⋊Dic5—61no—
180#683—61no—
181C4×Dic5—61no—
182C22×M16—51no—
183#525—151no—
184C2×#104—152no—
185C4×C22⋊C4—151no—
186#339—152no—
187C22×C3×C5—21no—
188C22×C3×C5—21no—
189C22×C3×C5—21no—
190C22×C3×C5C2×C22⋊C411yescharacteristic
191C2×C5×D3—41no—
192C2×C5×D3—41no—
193C2×C5×D3—41no—
194C2×C3×D5—21no—
195C2×C3×D5—21no—
196C2×C3×D5—21no—
197C2×C3×D5—21no—
198C2×C3×D5—44no—
199C2×C3×D5#10412yescomplemented
200C2×C3×D5#12112yes—
201ΓL(1,𝔽16)—44no—
202C3×C4×C5—21no—
203C3×C4×C5C4×D412yes—
204Dic15—41no—
205Dic15—41no—
206C3×Dic5—21no—
207C3×Dic5#12011yescharacteristic
208C3×Dic5#12311yescharacteristic
209C5×Dic3—41no—
210C24×C3—51no—
211C22×C3×C4—51no—
212C22×C3×C4—51no—
213C22×C3×C4—51no—
214C22×C3×C4—51no—
215C22×C3×C4—51no—
216C22×C3×C4—102no—
217C22×C3×C4—102no—
218C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
219C2×#67—21no—
220C2×#67—101no—
221C2×#67—101no—
222C22×Dic3—102no—
223#235—102no—
224#235—102no—
225C2×D12C2×AGL(1,𝔽5)11yescharacteristic
226C2×C4×D3—101no—
227C2×Dic6—51no—
228C3×M16—54no—
229C3×M16—102no—
230C3×M16—104no—
231C2×C3×C8—102no—
232C2×C3×C8—104no—
233#217—101no—
234#217—101no—
235#217—104no—
236#217—101no—
237(C2×D3)⋊C4—21no—
238(C2×D3)⋊C4—101no—
239C4⋊Dic3—51no—
240C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
241Dic3⋊C4—101no—
242Dic3⋊C4—101no—
243C4×Dic3—101no—
244C4×Dic3—102no—
245C4×Dic3—102no—
246C23×C5—61no—
247C23×C5(C2×D3)⋊C411yescharacteristic
248C22×D5—22no—
249C22×D5—22no—
250C22×D5—41no—
251C22×D5(C2×D3)⋊C411yescharacteristic
252C22×D5C2×D1211yescharacteristic
253C2×AGL(1,𝔽5)—64no—
254C2×AGL(1,𝔽5)—122no—
255C2×AGL(1,𝔽5)—122no—
256C5×D4—62no—
257C5×D4—122no—
258C2×C4×C5—22no—
259C2×C4×C5—61no—
260C2×C4×C5(C2×D3)⋊C411yescharacteristic
261C2×C4×C5C2×C4×D311yescharacteristic
262C22⋊D5—122no—
263C22⋊D5—122no—
264C2×Dic5—21no—
265C2×Dic5—21no—
266C2×Dic5—61no—
267C2×Dic5—61no—
268C2×Dic5—121no—
269C2×Dic5—121no—
270C2×Dic5(C2×D3)⋊C411yescharacteristic
271C2×Dic5C2×#6711yescharacteristic
272C4×D5—22no—
273C4×D5—22no—
274C4×D5—21no—
275C4×D5—21no—
276C4×D5#23512yes—
277C4×D5C2×#6712yes—
278Dic10—62no—
279#170—24no—
280C2×#41—151no—
281C23×C4—51no—
282C2×M16—52no—
283C2×M16—104no—
284C2×M16—101no—
285C22×C8—101no—
286#123—301no—
287C22⋊Q8—151no—
288C22⋊Q8—151no—
289#121—31no—
290#121—151no—
291#120—301no—
292C4×D4—301no—
293C4×D4—301no—
294#117—151no—
295C2×C22⋊C4—301no—
296C2×C42—152no—
297#104—304no—
298#95—301no—
299C2×C3×C5—41no—
300C2×C3×C5#38611yescharacteristic
301C2×C3×C5C2×#10411yescharacteristic
302C2×C3×C5#42111yescharacteristic
303C3×D5—22no—
304C3×D5—22no—
305C5×D3—81no—
306C23×C3—51no—
307C23×C3—51no—
308C23×C3—102no—
309C23×C3—102no—
310C23×C3—201no—
311C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
312C22×D3—21no—
313C2×C3×C4—22no—
314C2×C3×C4—51no—
315C2×C3×C4—52no—
316C2×C3×C4—52no—
317C2×C3×C4—51no—
318C2×C3×C4—101no—
319C2×C3×C4—101no—
320C2×C3×C4—102no—
321C2×C3×C4—101no—
322C2×C3×C4—101no—
323C2×C3×C4—102no—
324C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
325C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
326#67—42no—
327#67—202no—
328#67—202no—
329C2×Dic3—21no—
330C2×Dic3—104no—
331C2×Dic3—101no—
332C2×Dic3—101no—
333C2×Dic3—202no—
334C2×Dic3—202no—
335D12—22no—
336C4×D3—201no—
337C4×D3—201no—
338Dic6—102no—
339C3×C8—104no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5—21no—
343C22×C5—121no—
344C22×C5—121no—
345C22×C5—121no—
346C22×C5C2×(C2×D3)⋊C411yescharacteristic
347C2×D5—21no—
348C2×D5—21no—
349C2×D5—21no—
350C2×D5—21no—
351C2×D5—44no—
352C2×D5#82512yescomplemented
353C2×D5#93012yes—
354AGL(1,𝔽5)—124no—
355C4×C5—21no—
356C4×C5—121no—
357C4×C5C4×#6712yes—
358Dic5—21no—
359Dic5—121no—
360Dic5—121no—
361Dic5#92911yescharacteristic
362Dic5#95311yescharacteristic
363C24—51no—
364#41—302no—
365#41—302no—
366C2×Q8—151no—
367C2×D4—31no—
368C2×D4—61no—
369C2×D4—301no—
370C2×D4—301no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—102no—
377C22×C4—102no—
378C22×C4—301no—
379C22×C4—302no—
380C22×C4D3×AGL(1,𝔽5)11yescharacteristic
381M16—54no—
382M16—102no—
383M16—104no—
384C2×C8—102no—
385C2×C8—104no—
386C4⋊C4—31no—
387C4⋊C4—151no—
388C4⋊C4—301no—
389C4⋊C4—301no—
390C22⋊C4—61no—
391C22⋊C4—301no—
392C22⋊C4—301no—
393C22⋊C4—301no—
394C22⋊C4—304no—
395C22⋊C4—301no—
396C42—301no—
397C42—302no—
398C42—302no—
399C3×C5#187111yescharacteristic, complemented, Hall
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—52no—
404C22×C3—52no—
405C22×C3—101no—
406C22×C3—101no—
407C22×C3—101no—
408C22×C3—101no—
409C22×C3—204no—
410C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C3×C4—21no—
415C3×C4—51no—
416C3×C4—51no—
417C3×C4—101no—
418C3×C4D4×AGL(1,𝔽5)12yes—
419Dic3—41no—
420Dic3—204no—
421Dic3—201no—
422Dic3—201no—
423C2×C5—41no—
424C2×C5—241no—
425C2×C5C2×#82511yescharacteristic
426C2×C5#513911yescharacteristic
427C2×C5#521311yescharacteristic
428D5—22no—
429D5—22no—
430C23—51no—
431C23—51no—
432C23—61no—
433C23—102no—
434C23—102no—
435C23—201no—
436C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
437Q8—302no—
438D4—62no—
439D4—122no—
440D4—602no—
441D4—602no—
442C2×C4—22no—
443C2×C4—51no—
444C2×C4—52no—
445C2×C4—52no—
446C2×C4—51no—
447C2×C4—61no—
448C2×C4—101no—
449C2×C4—101no—
450C2×C4—102no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—102no—
454C2×C4—304no—
455C2×C4—301no—
456C2×C4—301no—
457C2×C4—601no—
458C2×C4—601no—
459C2×C4—602no—
460C2×C4—602no—
461C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
462C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
463C8—104no—
464C2×C3—41no—
465C2×C3—102no—
466C2×C3—102no—
467C2×C3#6576911yescharacteristic
468C2×C3#6577911yescharacteristic
469C2×C3D5⋊#10411yescharacteristic
470D3—81no—
471C5#7198411yescharacteristic, complemented, Sylow
472C22—21no—
473C22—21no—
474C22—21no—
475C22—52no—
476C22—52no—
477C22—101no—
478C22—101no—
479C22—101no—
480C22—101no—
481C22—121no—
482C22—121no—
483C22—121no—
484C22—204no—
485C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
486C4—21no—
487C4—51no—
488C4—51no—
489C4—101no—
490C4—121no—
491C4—604no—
492C4—601no—
493C4—601no—
494C4AGL(1,𝔽5)×#6712yes—
495C3#1060449611yescharacteristic, complemented, Sylow
496C2—41no—
497C2—102no—
498C2—102no—
499C2—241no—
500C2D5⋊#82511yescharacteristic
501C2#1175171811yescharacteristic
502C2#1175202211yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical#11754526C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1871
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D5⋊M16C2semidirect
3. C4×#6763C2semidirect
4. C2×#91155C2semidirect
5. #11746524C2semidirect
15. #91813C4semidirect
16. C5×#930C4semidirect
19. #91449C4semidirect
22. #91360C4semidirect
27. C2×D5⋊M16D3semidirect
56. C3×#700D4semidirect
136. #930AGL(1,𝔽5)semidirect
170. #700#67semidirect
199. C2×C3×D5#104semidirect
352. C2×D5#825semidirect
399. C3×C5#1871semidirect
471. C5#71984semidirect
495. C3#10604496semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D5⋊M162
#117545262
C4×#67632
C2×#911552
#117465242
#106044963
#719845