Finite Group Databasebeta

← #14302027

Subgroups — #14302027

Order 1920.

716 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 118 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×#4130C211yesmaximal, characteristic, complemented
2#11754527C211yesmaximal, characteristic
3#11752415C211yesmaximal, characteristic, complemented
4C2×#91158C212yesmaximal, complemented
5#11746526C212yesmaximal, complemented
6#10604522—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6773C2211yescharacteristic
9C2×#6758C2212yes—
10C2×C3×C22⋊AGL(1,𝔽5)—21no—
11C3×#4130—24no—
12C2×C3×C4⋊AGL(1,𝔽5)—21no—
13C2×C3×C4×AGL(1,𝔽5)C2212yes—
14#91819C411yescharacteristic, complemented
15C5×#924C411yescharacteristic, complemented
16#91564—21no—
17#91552—21no—
18#91450C411yescharacteristic, complemented
19#91444—21no—
20#91359C411yescharacteristic, complemented
21#91358C2211yescharacteristic
22#91348—21no—
23#91241—21no—
24#91232—21no—
25#91158—24no—
26#71591—51nomaximal, Hall
27C2×#4130D311yescharacteristic, complemented
28#65961—31no—
29C2×#4010—32no—
30#65756—31no—
31#64875—32no—
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—22no—
34C22×C3×C4×C5C2×C411yescharacteristic
35C2×#1220—21no—
36C2×D60C2×C411yescharacteristic
37C2×C5×Dic6C2×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×#1202—21no—
46C2×#1202—21no—
47C2×#1200—21no—
48#6773—22no—
49#6773—22no—
50C2×C3⋊#170—22no—
51#6758—22no—
52#6758—22no—
53C3×C22⋊AGL(1,𝔽5)—24no—
54C3×C4⋊AGL(1,𝔽5)—24no—
55C3×C4×AGL(1,𝔽5)—22no—
56C3×C4×AGL(1,𝔽5)D414yescomplemented
57#6714—21no—
58C4⋊Dic15C2×C411yescharacteristic
59C5×#217—21no—
60C5×C4⋊Dic3C2×C411yescharacteristic
61C5×Dic3⋊C4—21no—
62#6669—21no—
63#6667—21no—
64#6665—21no—
65#6665—21no—
66Dic3×Dic5—21no—
67C2×C3×#117—51no—
68#5791—51no—
69#5207—51no—
70C2×#805—52no—
71#4523—52no—
72C2×#713—31no—
73C22×C4×D5C2×D311yescharacteristic
74C2×C22⋊AGL(1,𝔽5)—21no—
75#4130—24no—
76C2×C4⋊AGL(1,𝔽5)—21no—
77C2×C4×AGL(1,𝔽5)C2×D312yes—
78C2×#700—32no—
79C5×C22⋊Q8—31no—
80#4092—31no—
81#4076—31no—
82#4041—31no—
83#4040—61no—
84#4039—61no—
85#4038—31no—
86#4028—61no—
87#4027—61no—
88C22⋊#170—61no—
89#4010—64no—
90#4003—61no—
91#1897—151noSylow
92C23×C3×C5C22⋊C411yescharacteristic
93C22×D15—21no—
94C22×C3×D5—21no—
95C22×C3×D5—21no—
96C22×C3×D5—41no—
97C22×C3×D5—41no—
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×C3×AGL(1,𝔽5)—24no—
101C2×C3×AGL(1,𝔽5)—42no—
102C2×C3×AGL(1,𝔽5)—42no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5C22⋊C411yescharacteristic
105C2×C3×C4×C5C22×C411yescharacteristic
106#1220—42no—
107C2×Dic15—21no—
108D60—22no—
109C2×C5×Dic3—21no—
110C2×C5×Dic3—21no—
111C5×Dic6—22no—
112C2×C3×Dic5—22no—
113C2×C3×Dic5C22⋊C411yescharacteristic
114C2×C3×Dic5C2×D411yescharacteristic
115C3×C4×D5—21no—
116C3×C4×D5—21no—
117C3×C4×D5—22no—
118C3×C4×D5—22no—
119C3×C4×D5C2×D412yes—
120C3×C4×D5#4112yes—
121#1202—42no—
122#1202—41no—
123#1202—41no—
124#1200—42no—
125C3⋊#170—42no—
126C23×C3×C4—51no—
127C2×#235—51no—
128C3×#117—104no—
129C2×C3×C4⋊C4—101no—
130C2×C3×C22⋊C4—101no—
131C2×C3×C42—52no—
132#953—101no—
133#930—51no—
134#930—51no—
135#929—101no—
136C4×#67—101no—
137C4×#67—101no—
138#926—51no—
139#924—51no—
140#924AGL(1,𝔽5)11yescharacteristic, complemented
141C2×#208—52no—
142#830—101no—
143#830—101no—
144#805—104no—
145C23×D5#6711yescharacteristic
146C22×AGL(1,𝔽5)—22no—
147C2×C5×Q8—31no—
148C22×C4×C5C4×D311yescharacteristic
149C2×C22⋊D5—61no—
150C22×Dic5D1211yescharacteristic
151#713—62no—
152#713—62no—
153C2×D20—31no—
154C2×D20—61no—
155C2×D20—61no—
156C2×C4×D5—22no—
157C2×C4×D5—22no—
158C2×C4×D5—61no—
159C2×C4×D5C4×D311yescharacteristic
160C2×C4×D5D1211yescharacteristic
161C2×C4×D5#6711yescharacteristic
162C2×C4×D5C22×D311yescharacteristic
163C22⋊AGL(1,𝔽5)—24no—
164C2×#170—62no—
165C4⋊AGL(1,𝔽5)—24no—
166C4×AGL(1,𝔽5)—22no—
167C4×AGL(1,𝔽5)#6714yescomplemented
168#700—62no—
169#700—62no—
170C5×C4⋊C4—31no—
171C5×C4⋊C4—61no—
172C5×C22⋊C4—61no—
173#685—61no—
174#685—61no—
175#685—61no—
176#685—61no—
177C4⋊Dic5—31no—
178#683—61no—
179C4×Dic5—61no—
180#525—151no—
181C2×#117—51no—
182C2×#104—152no—
183#407—151no—
184#339—152no—
185C22×C3×C5—21no—
186C22×C3×C5—21no—
187C22×C3×C5—21no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C2×D15—41no—
190C2×D15—41no—
191C2×D15—41no—
192C2×C3×D5—21no—
193C2×C3×D5—21no—
194C2×C3×D5—21no—
195C2×C3×D5—21no—
196C2×C3×D5—21no—
197C2×C3×D5—42no—
198C2×C3×D5—42no—
199C2×C3×D5#12011yescharacteristic
200C2×C3×D5#12311yescharacteristic
201C3×AGL(1,𝔽5)—44no—
202C3×C4×C5—21no—
203C3×C4×C5C4×D412yes—
204Dic15—41no—
205C3×Dic5—21no—
206C3×Dic5#12112yes—
207C5×Dic3—41no—
208C5×Dic3—41no—
209C24×C3—51no—
210C22×C3×C4—51no—
211C22×C3×C4—51no—
212C22×C3×C4—51no—
213C22×C3×C4—51no—
214C22×C3×C4—51no—
215C22×C3×C4—102no—
216C22×C3×C4—102no—
217C22×C3×C4—102no—
218C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
219C2×#67—101no—
220C2×#67—101no—
221C2×#67—101no—
222#235—102no—
223#235—102no—
224C2×D12—51no—
225C2×C4×D3—101no—
226C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
227C3×C4⋊C4—104no—
228C3×C22⋊C4—104no—
229C3×C42—54no—
230C3×C42—102no—
231#217—21no—
232#217—101no—
233#217—101no—
234(C2×D3)⋊C4—101no—
235(C2×D3)⋊C4—101no—
236C4⋊Dic3—51no—
237C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
238Dic3⋊C4—21no—
239Dic3⋊C4—101no—
240C4×Dic3—101no—
241#208—102no—
242#208—102no—
243C2×C3⋊C8—102no—
244C23×C5(C2×D3)⋊C411yescharacteristic
245C22×D5—21no—
246C22×D5—21no—
247C22×D5—41no—
248C22×D5—41no—
249C22×D5—61no—
250C22×D5(C2×D3)⋊C411yescharacteristic
251C22×D5C2×#6711yescharacteristic
252C2×AGL(1,𝔽5)—24no—
253C2×AGL(1,𝔽5)—42no—
254C2×AGL(1,𝔽5)—42no—
255C5×Q8—62no—
256C2×C4×C5—22no—
257C2×C4×C5—61no—
258C2×C4×C5—61no—
259C2×C4×C5(C2×D3)⋊C411yescharacteristic
260C2×C4×C5C2×C4×D311yescharacteristic
261C22⋊D5—122no—
262C2×Dic5—22no—
263C2×Dic5—61no—
264C2×Dic5(C2×D3)⋊C411yescharacteristic
265C2×Dic5C2×D1211yescharacteristic
266D20—62no—
267D20—122no—
268D20—121no—
269D20—121no—
270C4×D5—21no—
271C4×D5—21no—
272C4×D5—22no—
273C4×D5—22no—
274C4×D5—122no—
275C4×D5#23512yes—
276C4×D5C2×#6712yes—
277#170—122no—
278C2×#41—151no—
279C23×C4—51no—
280C2×M16—152no—
281#123—301no—
282C22⋊Q8—31no—
283C22⋊Q8—151no—
284#121—151no—
285#121—151no—
286#120—301no—
287C4×D4—301no—
288C4×D4—301no—
289#117—104no—
290#117—151no—
291C2×C4⋊C4—101no—
292C2×C22⋊C4—101no—
293C2×C42—52no—
294#104—304no—
295C22⋊C8—301no—
296C22⋊C8—301no—
297C2×C3×C5—41no—
298C2×C3×C5#38611yescharacteristic
299C2×C3×C5#42111yescharacteristic
300C2×C3×C5#42111yescharacteristic
301D15—81no—
302C3×D5—21no—
303C3×D5—21no—
304C3×D5—41no—
305C23×C3—51no—
306C23×C3—51no—
307C23×C3—101no—
308C23×C3—101no—
309C23×C3—201no—
310C23×C3—201no—
311C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
312C22×D3—101no—
313C2×C3×C4—22no—
314C2×C3×C4—51no—
315C2×C3×C4—52no—
316C2×C3×C4—52no—
317C2×C3×C4—51no—
318C2×C3×C4—102no—
319C2×C3×C4—101no—
320C2×C3×C4—101no—
321C2×C3×C4—102no—
322C2×C3×C4—102no—
323C2×C3×C4—104no—
324C2×C3×C4—202no—
325C2×C3×C4—202no—
326C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
327C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
328#67—201no—
329#67—201no—
330#67—202no—
331#67—202no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—101no—
335D12—102no—
336C4×D3—202no—
337Dic6—22no—
338C3⋊C8—202no—
339C22×C5—21no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5C2×(C2×D3)⋊C411yescharacteristic
343C2×D5—21no—
344C2×D5—21no—
345C2×D5—21no—
346C2×D5—21no—
347C2×D5—21no—
348C2×D5—42no—
349C2×D5—42no—
350C2×D5—121no—
351C2×D5—121no—
352C2×D5—121no—
353C2×D5#92911yescharacteristic
354C2×D5#95311yescharacteristic
355AGL(1,𝔽5)—44no—
356C4×C5—21no—
357C4×C5—121no—
358C4×C5—121no—
359C4×C5C4×#6712yes—
360Dic5—21no—
361Dic5—121no—
362Dic5#93012yes—
363C24—51no—
364#41—302no—
365#41—302no—
366C2×Q8—31no—
367C2×D4—151no—
368C2×D4—301no—
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—102no—
379C22×C4—301no—
380C22×C4D3×AGL(1,𝔽5)11yescharacteristic
381M16—302no—
382M16—302no—
383C2×C8—302no—
384C4⋊C4—31no—
385C4⋊C4—61no—
386C4⋊C4—104no—
387C4⋊C4—151no—
388C4⋊C4—301no—
389C22⋊C4—61no—
390C22⋊C4—104no—
391C22⋊C4—301no—
392C22⋊C4—301no—
393C22⋊C4—301no—
394C22⋊C4—301no—
395C42—54no—
396C42—102no—
397C42—301no—
398C3×C5#189711yescharacteristic, complemented, Hall
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—51no—
403C22×C3—51no—
404C22×C3—101no—
405C22×C3—101no—
406C22×C3—101no—
407C22×C3—101no—
408C22×C3—101no—
409C22×C3—202no—
410C22×C3—202no—
411C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
412C2×D3—201no—
413C2×D3—201no—
414C2×D3—201no—
415C3×C4—21no—
416C3×C4—52no—
417C3×C4—101no—
418C3×C4—204no—
419C3×C4D4×AGL(1,𝔽5)12yes—
420Dic3—41no—
421Dic3—41no—
422Dic3—201no—
423C2×C5—41no—
424C2×C5#500511yescharacteristic
425C2×C5#500511yescharacteristic
426C2×C5#521311yescharacteristic
427D5—21no—
428D5—21no—
429D5—41no—
430D5—241no—
431C23—51no—
432C23—51no—
433C23—101no—
434C23—101no—
435C23—201no—
436C23—201no—
437C23—301no—
438C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
439Q8—62no—
440D4—302no—
441D4—601no—
442D4—601no—
443D4—602no—
444D4—602no—
445C2×C4—22no—
446C2×C4—51no—
447C2×C4—52no—
448C2×C4—52no—
449C2×C4—51no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—102no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—104no—
458C2×C4—202no—
459C2×C4—202no—
460C2×C4—301no—
461C2×C4—602no—
462C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
463C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
464C8—602no—
465C2×C3—41no—
466C2×C3—101no—
467C2×C3—101no—
468C2×C3—201no—
469C2×C3#6577911yescharacteristic
470C2×C3#6578411yescharacteristic
471C2×C3#6579511yescharacteristic
472D3—401no—
473C5#7159111yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—21no—
477C22—51no—
478C22—51no—
479C22—101no—
480C22—101no—
481C22—101no—
482C22—101no—
483C22—101no—
484C22—202no—
485C22—202no—
486C22—601no—
487C22—601no—
488C22—601no—
489C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
490C4—21no—
491C4—52no—
492C4—101no—
493C4—121no—
494C4—121no—
495C4—204no—
496C4—601no—
497C4AGL(1,𝔽5)×#6712yes—
498C3#1060452211yescharacteristic, complemented, Sylow
499C2—41no—
500C2—101no—
501C2—101no—
502C2—201no—
503C2—1201no—
504C2#1175172011yescharacteristic
505C2#1175172411yescharacteristic
506C2#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#11754527C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1897
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×#4130C2semidirect
3. #11752415C2semidirect
4. C2×#91158C2semidirect
5. #11746526C2semidirect
14. #91819C4semidirect
15. C5×#924C4semidirect
18. #91450C4semidirect
20. #91359C4semidirect
27. C2×#4130D3semidirect
56. C3×C4×AGL(1,𝔽5)D4semidirect
140. #924AGL(1,𝔽5)semidirect
167. C4×AGL(1,𝔽5)#67semidirect
398. C3×C5#1897semidirect
473. C5#71591semidirect
498. C3#10604522semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41302
#117545272
#117524152
C2×#911582
#117465262
#106045223
#715915