Finite Group Databasebeta

← #14296882

Subgroups — #14296882

Order 1920.

1224 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22×#6771C211yesmaximal, characteristic
2#11752423C211yesmaximal, characteristic, complemented
3C3×#65764C211yesmaximal, characteristic, complemented
4C2×#91173C212yesmaximal, complemented
5C2×#91172C212yesmaximal, complemented
6#10604891—31nomaximal, Hall
7C22×C4×D15C411yescharacteristic, complemented
8C22×C4×C5×D3C411yescharacteristic, complemented
9C22×C3×C4×D5C2211yescharacteristic
10C22×#1204C411yescharacteristic, complemented
11C22×#1203C411yescharacteristic, complemented
12C22×#1202C2211yescharacteristic
13C22×#1201C2211yescharacteristic
14C2×#6771—22no—
15C2×#6771—22no—
16C2×#6771—24no—
17C2×#6771—24no—
18C2×#6763C2212yes—
19C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
20#91235—22no—
21C3×(C2×AGL(1,𝔽5))⋊C4—22no—
22#91173—24no—
23#91172—24no—
24#69537—51nomaximal, Hall
25C22×#709—31no—
26C2×#4011—32no—
27#65764—31no—
28#65764D311yescharacteristic, complemented
29C2×#3999—32no—
30C23×D15C2×C411yescharacteristic
31C23×C5×D3C2×C411yescharacteristic
32C23×C3×D5C2311yescharacteristic
33C22×ΓL(1,𝔽16)—22no—
34C22×C3×AGL(1,𝔽5)—22no—
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×Dic15C2×C411yescharacteristic
37C2×C4×D15—22no—
38C2×C4×D15—22no—
39C2×C4×D15—22no—
40C22×C5×Dic3C2×C411yescharacteristic
41C2×C4×C5×D3—22no—
42C2×C4×C5×D3—22no—
43C2×C4×C5×D3—22no—
44C22×C3×Dic5C2×C411yescharacteristic
45C2×C3×C4×D5—22no—
46C2×C3×C4×D5—22no—
47C2×C3×C4×D5D412yes—
48C2×C3×C4×D5D412yes—
49C2×#1204—22no—
50C2×#1204—22no—
51C2×#1204—22no—
52C2×#1203—22no—
53C2×#1203—22no—
54C2×#1203—22no—
55C2×#1202—21no—
56C2×#1202—22no—
57C2×#1202—21no—
58C2×#1202D414yes—
59C2×#1201—21no—
60C2×#1201—22no—
61C2×#1201—21no—
62C2×#1201D414yes—
63#6771—216no—
64#6771—216no—
65#6763—22no—
66#6763—22no—
67C3×C22⋊AGL(1,𝔽5)—22no—
68C3×C22⋊AGL(1,𝔽5)—22no—
69C22×#235—51no—
70C3×#381—51no—
71#5211—51no—
72C2×#806—52no—
73C2×#806—52no—
74C23×C4×C5—31no—
75C22×C22⋊D5—31no—
76C22×C22⋊D5—31no—
77C2×#709—64no—
78C2×#709—64no—
79C2×#709—62no—
80C2×#709—62no—
81C22×D20—31no—
82C22×C4×D5—31no—
83C22×C4×D5C2×D311yescharacteristic
84C22×Dic10—31no—
85C2×C22⋊AGL(1,𝔽5)—32no—
86C2×C22⋊AGL(1,𝔽5)C2×D312yes—
87#4011—64no—
88(C2×AGL(1,𝔽5))⋊C4—22no—
89(C2×AGL(1,𝔽5))⋊C4—62no—
90#3999—64no—
91#1782—151noSylow
92C23×C3×C5C22×C411yescharacteristic
93C22×D15—21no—
94C22×D15—21no—
95C22×D15—42no—
96C22×D15C22⋊C412yes—
97C22×C5×D3—21no—
98C22×C5×D3—21no—
99C22×C5×D3—42no—
100C22×C5×D3C22⋊C412yes—
101C22×C3×D5—42no—
102C22×C3×D5C2×D411yescharacteristic
103C22×C3×D5C2×D411yescharacteristic
104C22×C3×D5C2×D412yes—
105C22×C3×D5C2×D412yes—
106C2×ΓL(1,𝔽16)—44no—
107C2×ΓL(1,𝔽16)—42no—
108C2×C3×AGL(1,𝔽5)—44no—
109C2×C3×AGL(1,𝔽5)—42no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5C22⋊C412yes—
113C2×Dic15—21no—
114C2×Dic15—21no—
115C2×Dic15C22⋊C412yes—
116C4×D15—28no—
117C2×C5×Dic3—21no—
118C2×C5×Dic3—21no—
119C2×C5×Dic3C22⋊C412yes—
120C4×C5×D3—28no—
121C2×C3×Dic5—21no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5C22⋊C412yes—
124C3×C4×D5—24no—
125C3×C4×D5—24no—
126#1204—28no—
127#1203—28no—
128#1202—24no—
129#1202—24no—
130#1201—24no—
131#1201—24no—
132C23×C3×C4—51no—
133C22×#67—51no—
134C22×#67—51no—
135C2×#235—104no—
136C2×#235—104no—
137C2×#235—102no—
138C2×#235—102no—
139C22×D12—51no—
140C22×C4×D3—51no—
141C22×C4×D3AGL(1,𝔽5)11yescharacteristic, complemented
142C22×Dic6—51no—
143C2×C3×C22⋊C4—52no—
144C2×#217—52no—
145C3×#95—102no—
146C3⋊#95—102no—
147#806—104no—
148#806—104no—
149C24×C5—31no—
150C23×D5—31no—
151C23×D5C22×D311yescharacteristic
152C22×AGL(1,𝔽5)—22no—
153C22×AGL(1,𝔽5)—62no—
154C22×C4×C5—31no—
155C22×C4×C5—62no—
156C22×C4×C5—62no—
157C22×C4×C5—62no—
158C22×C4×C5C4×D311yescharacteristic
159C2×C22⋊D5—34no—
160C2×C22⋊D5—62no—
161C2×C22⋊D5—62no—
162C2×C22⋊D5—61no—
163C2×C22⋊D5—62no—
164C2×C22⋊D5—61no—
165C2×C22⋊D5—62no—
166C22×Dic5—31no—
167C22×Dic5C4×D311yescharacteristic
168#709—616no—
169#709—616no—
170C2×D20—34no—
171C2×D20—61no—
172C2×D20—62no—
173C2×D20—61no—
174C2×C4×D5—22no—
175C2×C4×D5—22no—
176C2×C4×D5—62no—
177C2×C4×D5—62no—
178C2×C4×D5—62no—
179C2×C4×D5D1212yes—
180C2×C4×D5#6712yes—
181C2×Dic10—62no—
182C2×Dic10—62no—
183C2×Dic10—62no—
184C22⋊AGL(1,𝔽5)—22no—
185C22⋊AGL(1,𝔽5)—22no—
186C22⋊AGL(1,𝔽5)—62no—
187C22⋊AGL(1,𝔽5)—62no—
188C22×#41—151no—
189C2×#99—152no—
190C2×#99—152no—
191#381—51no—
192#381—151no—
193C22×C3×C5—22no—
194C22×C3×C5C2×C22⋊C411yescharacteristic
195C22×C3×C5C2×C22⋊C411yescharacteristic
196C22×C3×C5C2×C22⋊C411yescharacteristic
197C2×D15—22no—
198C2×D15—42no—
199C2×D15—42no—
200C2×D15—42no—
201C2×C5×D3—22no—
202C2×C5×D3—42no—
203C2×C5×D3—42no—
204C2×C5×D3—42no—
205C2×C3×D5—22no—
206C2×C3×D5—22no—
207C2×C3×D5—42no—
208C2×C3×D5—42no—
209C2×C3×D5#12012yes—
210C2×C3×D5#12012yes—
211ΓL(1,𝔽16)—82no—
212C3×AGL(1,𝔽5)—82no—
213C3×C4×C5—22no—
214Dic15—22no—
215C3×Dic5—22no—
216C5×Dic3—22no—
217C24×C3—51no—
218C23×D3—51no—
219C23×D3C2×AGL(1,𝔽5)11yescharacteristic
220C22×C3×C4—51no—
221C22×C3×C4—52no—
222C22×C3×C4—52no—
223C22×C3×C4—102no—
224C22×C3×C4—102no—
225C22×C3×C4—102no—
226C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
227C2×#67—54no—
228C2×#67—54no—
229C2×#67—101no—
230C2×#67—101no—
231C2×#67—102no—
232C2×#67—101no—
233C2×#67—101no—
234C2×#67—102no—
235C22×Dic3—51no—
236C22×Dic3—102no—
237C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
238#235—1016no—
239#235—1016no—
240C2×D12—102no—
241C2×D12—102no—
242C2×D12—102no—
243C2×C4×D3—22no—
244C2×C4×D3—22no—
245C2×C4×D3—22no—
246C2×C4×D3—102no—
247C2×C4×D3—102no—
248C2×C4×D3—102no—
249C2×Dic6—102no—
250C2×Dic6—102no—
251C2×Dic6—102no—
252C3×C22⋊C4—102no—
253C3×C22⋊C4—102no—
254#217—102no—
255#217—102no—
256C23×C5—32no—
257C23×C5—61no—
258C23×C5—61no—
259C23×C5—122no—
260C23×C5C2×C4×D311yescharacteristic
261C22×D5—32no—
262C22×D5—42no—
263C22×D5—61no—
264C22×D5—61no—
265C22×D5—122no—
266C22×D5C2×D1211yescharacteristic
267C22×D5D3×D412yes—
268C22×D5D3×D412yes—
269C22×D5C2×#6711yescharacteristic
270C2×AGL(1,𝔽5)—44no—
271C2×AGL(1,𝔽5)—42no—
272C2×AGL(1,𝔽5)—124no—
273C2×AGL(1,𝔽5)—122no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—32no—
277C2×C4×C5—68no—
278C2×C4×C5—61no—
279C2×C4×C5—61no—
280C2×C4×C5(C2×D3)⋊C412yes—
281C22⋊D5—68no—
282C22⋊D5—64no—
283C22⋊D5—64no—
284C2×Dic5—21no—
285C2×Dic5—21no—
286C2×Dic5—32no—
287C2×Dic5—61no—
288C2×Dic5—61no—
289C2×Dic5(C2×D3)⋊C412yes—
290D20—64no—
291D20—64no—
292C4×D5—24no—
293C4×D5—24no—
294C4×D5—68no—
295Dic10—68no—
296C2×#41—302no—
297C2×#41—302no—
298C2×#41—304no—
299C2×#41—304no—
300C22×Q8—151no—
301C22×D4—151no—
302C22×D4—151no—
303C22×D4—151no—
304C23×C4—31no—
305C23×C4—51no—
306C23×C4—151no—
307C2×C22⋊C4—52no—
308C2×C22⋊C4—152no—
309#99—304no—
310#99—304no—
311#95—102no—
312#95—302no—
313C2×C3×C5—21no—
314C2×C3×C5—21no—
315C2×C3×C5#37911yescharacteristic
316C2×C3×C5#41012yes—
317D15—42no—
318C3×D5—22no—
319C3×D5—22no—
320C5×D3—42no—
321C23×C3—51no—
322C23×C3—51no—
323C23×C3—52no—
324C23×C3—52no—
325C23×C3—202no—
326C23×C3C22×AGL(1,𝔽5)11yescharacteristic
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—42no—
330C22×D3—52no—
331C22×D3—101no—
332C22×D3—101no—
333C22×D3—202no—
334C22×D3C22⋊AGL(1,𝔽5)12yes—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—52no—
338C2×C3×C4—104no—
339C2×C3×C4—104no—
340C2×C3×C4—101no—
341C2×C3×C4—101no—
342C2×C3×C4—202no—
343C2×C3×C4—204no—
344C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
345#67—104no—
346#67—104no—
347#67—104no—
348#67—104no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—52no—
352C2×Dic3—101no—
353C2×Dic3—101no—
354C2×Dic3—202no—
355C2×Dic3—204no—
356C2×Dic3C22⋊AGL(1,𝔽5)12yes—
357D12—108no—
358C4×D3—28no—
359C4×D3—108no—
360Dic6—108no—
361C22×C5—22no—
362C22×C5—62no—
363C22×C5—122no—
364C22×C5—122no—
365C22×C5—122no—
366C22×C5D3×C22⋊C411yescharacteristic
367C22×C5D3×C22⋊C411yescharacteristic
368C22×C5C2×(C2×D3)⋊C411yescharacteristic
369C2×D5—22no—
370C2×D5—22no—
371C2×D5—42no—
372C2×D5—42no—
373C2×D5—62no—
374C2×D5—122no—
375C2×D5—122no—
376C2×D5—122no—
377C2×D5#88212yes—
378C2×D5#93712yes—
379AGL(1,𝔽5)—82no—
380AGL(1,𝔽5)—242no—
381C4×C5—22no—
382C4×C5—62no—
383Dic5—22no—
384Dic5—62no—
385C24—31no—
386C24—51no—
387C24—151no—
388#41—3016no—
389#41—3016no—
390C2×Q8—302no—
391C2×Q8—302no—
392C2×Q8—302no—
393C2×D4—154no—
394C2×D4—154no—
395C2×D4—302no—
396C2×D4—301no—
397C2×D4—301no—
398C2×D4—302no—
399C2×D4—302no—
400C2×D4—302no—
401C2×D4—301no—
402C2×D4—301no—
403C2×D4—302no—
404C22×C4—31no—
405C22×C4—51no—
406C22×C4—52no—
407C22×C4—52no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—102no—
414C22×C4—151no—
415C22×C4—302no—
416C22×C4—302no—
417C22×C4—302no—
418C22×C4—302no—
419C22×C4D3×AGL(1,𝔽5)11yescharacteristic
420C22⋊C4—102no—
421C22⋊C4—102no—
422C22⋊C4—302no—
423C22⋊C4—302no—
424C3×C5#178211yescharacteristic, complemented, Hall
425C22×C3—22no—
426C22×C3—52no—
427C22×C3—52no—
428C22×C3—102no—
429C22×C3—102no—
430C22×C3—202no—
431C22×C3—202no—
432C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
433C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
434C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
435C2×D3—22no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—102no—
440C2×D3—202no—
441C2×D3—202no—
442C2×D3—202no—
443C3×C4—22no—
444C3×C4—102no—
445C3×C4—402no—
446Dic3—22no—
447Dic3—102no—
448Dic3—402no—
449C2×C5—21no—
450C2×C5—21no—
451C2×C5—122no—
452C2×C5#495511yescharacteristic
453C2×C5#500612yes—
454D5—22no—
455D5—22no—
456D5—122no—
457C23—32no—
458C23—51no—
459C23—51no—
460C23—52no—
461C23—52no—
462C23—61no—
463C23—61no—
464C23—122no—
465C23—152no—
466C23—202no—
467C23—301no—
468C23—301no—
469C23—602no—
470C23C2×D3×AGL(1,𝔽5)11yescharacteristic
471Q8—308no—
472D4—304no—
473D4—304no—
474D4—308no—
475D4—304no—
476D4—304no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—32no—
480C2×C4—52no—
481C2×C4—68no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—104no—
485C2×C4—104no—
486C2×C4—101no—
487C2×C4—101no—
488C2×C4—152no—
489C2×C4—202no—
490C2×C4—204no—
491C2×C4—308no—
492C2×C4—301no—
493C2×C4—301no—
494C2×C4—602no—
495C2×C4—604no—
496C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—102no—
500C2×C3—102no—
501C2×C3#6574512yes—
502C2×C3#6580011yescharacteristic
503D3—42no—
504D3—202no—
505C5#6953711yescharacteristic, complemented, Sylow
506C22—22no—
507C22—52no—
508C22—52no—
509C22—62no—
510C22—102no—
511C22—102no—
512C22—122no—
513C22—122no—
514C22—122no—
515C22—202no—
516C22—202no—
517C22—302no—
518C22—602no—
519C22—602no—
520C22—602no—
521C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
522C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
523C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
524C4—22no—
525C4—62no—
526C4—102no—
527C4—302no—
528C4—402no—
529C4—1202no—
530C3#1060489111yescharacteristic, complemented, Sylow
531C2—21no—
532C2—21no—
533C2—102no—
534C2—102no—
535C2—122no—
536C2—602no—
537C2#1175181512yes—
538C2#1175202511yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#6771C2
cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1782
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752423C2semidirect
3. C3×#65764C2semidirect
4. C2×#91173C2semidirect
5. C2×#91172C2semidirect
7. C22×C4×D15C4semidirect
8. C22×C4×C5×D3C4semidirect
10. C22×#1204C4semidirect
11. C22×#1203C4semidirect
28. #65764D3semidirect
141. C22×C4×D3AGL(1,𝔽5)semidirect
424. C3×C5#1782semidirect
505. C5#69537semidirect
530. C3#10604891semidirect

Maximal subgroups

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

subgroupindex
C22×#67712
#117524232
C3×#657642
C2×#911732
C2×#911722
#106048913
#695375