Finite Group Databasebeta

← #14301137

Subgroups — #14301137

Order 1920.

748 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6706C211yesmaximal, characteristic
2C2×#91340C211yesmaximal, characteristic, complemented
3#11748789C211yesmaximal, characteristic
4C2×#90974C211yesmaximal, characteristic, complemented
5C2×#90968C211yesmaximal, characteristic
6C5×#4550C211yesmaximal, characteristic, complemented
7C3×#64771C211yesmaximal, characteristic, complemented
8#10599479—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×Dic30C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C2×#6706—22no—
13C2×#6706—22no—
14C2×#6706C414yescomplemented
15C2×C4×C5×Dic3C2211yescharacteristic
16C22×C5×C3⋊C8C2211yescharacteristic
17C2×C3×C4⋊Dic5C2211yescharacteristic
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×#6669—21no—
20C2×#6667—21no—
21C2×#6667—21no—
22C2×Dic3×Dic5—21no—
23#91340—22no—
24#91340—22no—
25#90974—22no—
26#90968—22no—
27#68909—51nomaximal, Hall
28C22×#677—31no—
29C2×C4×Dic10—31no—
30C2×#3953—31no—
31C2×#3948—31no—
32#65228—31no—
33C5×#336—31no—
34#64771D311yescharacteristic, complemented
35C23×D15—21no—
36C22×C3×C4×C5C2311yescharacteristic
37C22×Dic15—21no—
38C2×D60—21no—
39C2×D60—21no—
40C2×D60—21no—
41C2×D60—21no—
42C2×D60—21no—
43C2×D60C2×C412yes—
44C2×Dic30—21no—
45C2×Dic30—21no—
46C2×Dic30—21no—
47C2×Dic30C2×C412yes—
48C2×Dic30D412yes—
49C2×Dic30D412yes—
50C22×C5×Dic3D411yescharacteristic
51C22×C5×Dic3D411yescharacteristic
52C22×C3×Dic5—21no—
53C22×C3×Dic5—21no—
54C2×C3×C5×C8—21no—
55C2×C3×C5×C8—21no—
56C2×C3×C5×C8C2×C412yes—
57#6706—28no—
58C4×C5×Dic3—21no—
59C4×C5×Dic3—21no—
60C2×C5×C3⋊C8—21no—
61C2×C5×C3⋊C8—21no—
62C2×C5×C3⋊C8—21no—
63C3×C4⋊Dic5—21no—
64C3×C4⋊Dic5—21no—
65C3×C4⋊Dic5—21no—
66C3×C4⋊Dic5D412yes—
67#6669—24no—
68#6667—42no—
69#6667—42no—
70Dic3×Dic5—24no—
71C22×#204—51no—
72C2×#887—51no—
73#4988—51no—
74C2×#810—51no—
75C2×#809—51no—
76C3×#340—51no—
77#4550D511yescharacteristic, complemented
78C22×D20—31no—
79C22×Dic10—31no—
80C22×C5×C8—31no—
81C22×C5×C8C2×D311yescharacteristic
82C2×C42×C5—31no—
83C2×#685—61no—
84C2×#685—61no—
85C2×C4⋊Dic5C2×D311yescharacteristic
86C2×C4⋊Dic5C2×D311yescharacteristic
87C2×#683—61no—
88C2×C4×Dic5—61no—
89C2×#677—34no—
90C2×#677—62no—
91C2×#677—62no—
92C4×Dic10—62no—
93C4×Dic10—62no—
94#3953—62no—
95#3948—62no—
96#1880—151noSylow
97C23×C3×C5C2×D411yescharacteristic
98C22×D15—22no—
99C22×D15—41no—
100C22×D15—41no—
101C22×D15—41no—
102C22×D15—41no—
103C22×D15—41no—
104C22×D15—41no—
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×Dic15—22no—
112C2×Dic15—22no—
113C2×Dic15—22no—
114D60—22no—
115D60—22no—
116D60—41no—
117D60—41no—
118Dic30—22no—
119Dic30—22no—
120Dic30—22no—
121Dic30C22⋊C414yescomplemented
122C2×C5×Dic3—21no—
123C2×C5×Dic3—21no—
124C2×C5×Dic3—21no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—21no—
127C2×C5×Dic3SD1612yescomplemented
128C2×C3×Dic5—22no—
129C2×C3×Dic5—41no—
130C2×C3×Dic5—41no—
131C2×C3×Dic5—41no—
132C2×C3×Dic5—41no—
133C2×C3×Dic5—41no—
134C3×C5×C8—22no—
135C5×C3⋊C8—41no—
136C22×D12—51no—
137C22×Dic6—51no—
138C22×C3×C8C2×D511yescharacteristic
139C2×C3×C4⋊C4—51no—
140C2×C3×C4⋊C4—51no—
141C2×(C2×D3)⋊C4—101no—
142C2×(C2×D3)⋊C4—101no—
143C2×Dic3⋊C4—101no—
144C2×C4×Dic3—101no—
145C2×C4×Dic3C2×D511yescharacteristic
146C22×C3⋊C8C2×D511yescharacteristic
147C2×#204—54no—
148C2×#204—102no—
149C2×#204—102no—
150#887—102no—
151#887—102no—
152#810—102no—
153#809—102no—
154C23×D5—61no—
155C22×C4×C5—31no—
156C22×C4×C5—31no—
157C22×C4×C5C22×D311yescharacteristic
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—61no—
161C2×D20—32no—
162C2×D20—61no—
163C2×D20—61no—
164C2×D20—61no—
165C2×D20—61no—
166C2×D20—61no—
167C2×Dic10—32no—
168C2×Dic10—32no—
169C2×Dic10—32no—
170C2×Dic10—61no—
171C2×Dic10—61no—
172C2×Dic10—61no—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—61no—
176C2×C5×C8—61no—
177C2×C5×C8—61no—
178C2×C5×C8C4×D312yes—
179C42×C5—61no—
180C42×C5—61no—
181#685—122no—
182#685—122no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5D1212yes—
187#683—64no—
188C4×Dic5—64no—
189#677—68no—
190C22×SD16—151no—
191C2×C4×Q8—151no—
192C2×#103—151no—
193C2×#102—151no—
194#390—151no—
195#340—51no—
196#336—31no—
197C22×C3×C5C4×D411yescharacteristic
198C22×C3×C5#12011yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5C2×SD1611yescharacteristic
201C22×C3×C5#13511yescharacteristic
202C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
203C22×C3×C5#13711yescharacteristic
204C2×D15—42no—
205C2×D15—42no—
206C2×D15—42no—
207C2×D15—81no—
208C2×D15—81no—
209C2×D15—81no—
210C2×D15—81no—
211C3×C4×C5C2×C22⋊C411yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5#12111yescharacteristic
214C3×C4×C5#12311yescharacteristic
215Dic15—24no—
216C3×Dic5—42no—
217C3×Dic5—81no—
218C5×Dic3—22no—
219C5×Dic3—41no—
220C23×D3—101no—
221C22×C3×C4—101no—
222C22×C3×C4—101no—
223C22×C3×C4C22×D511yescharacteristic
224C22×Dic3—101no—
225C22×Dic3D2011yescharacteristic
226C22×Dic3D2011yescharacteristic
227C2×D12—52no—
228C2×D12—101no—
229C2×D12—101no—
230C2×D12—101no—
231C2×D12—101no—
232C2×D12—101no—
233C2×Dic6—52no—
234C2×Dic6—52no—
235C2×Dic6—52no—
236C2×Dic6—101no—
237C2×Dic6—101no—
238C2×Dic6—101no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8C4×D512yes—
242C3×C4⋊C4—52no—
243C3×C4⋊C4—101no—
244C3×C4⋊C4—101no—
245C3×C4⋊C4—101no—
246(C2×D3)⋊C4—202no—
247(C2×D3)⋊C4—202no—
248Dic3⋊C4—104no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—104no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255#204—108no—
256C23×C5D3×D411yescharacteristic
257C22×D5—62no—
258C22×D5—121no—
259C22×D5—121no—
260C22×D5—121no—
261C22×D5—121no—
262C22×D5—121no—
263C22×D5—121no—
264C2×C4×C5—32no—
265C2×C4×C5—61no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5C2×C4×D311yescharacteristic
271C2×C4×C5C2×D1211yescharacteristic
272C2×C4×C5#23511yescharacteristic
273C2×C4×C5D3×D411yescharacteristic
274C2×C4×C5D3×D411yescharacteristic
275C2×C4×C5#23911yescharacteristic
276C2×Dic5—22no—
277C2×Dic5—41no—
278C2×Dic5—41no—
279C2×Dic5—41no—
280C2×Dic5—41no—
281C2×Dic5—41no—
282C2×Dic5—62no—
283C2×Dic5—62no—
284C2×Dic5—62no—
285D20—62no—
286D20—62no—
287D20—121no—
288D20—121no—
289Dic10—34no—
290Dic10—62no—
291Dic10—62no—
292Dic10—62no—
293C5×C8—22no—
294C5×C8—121no—
295C22×Q8—151no—
296C22×D4—151no—
297C2×SD16—154no—
298C2×SD16—302no—
299C2×SD16—302no—
300C22×C8—31no—
301C22×C8D3×D511yescharacteristic
302C4×Q8—302no—
303C4×Q8—302no—
304C2×C4⋊C4—51no—
305C2×C4⋊C4—51no—
306C2×C4⋊C4—301no—
307C2×C22⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—31no—
310C2×C42—301no—
311#103—302no—
312#102—302no—
313C2×C3×C5#38611yescharacteristic
314C2×C3×C5C4×SD1611yescharacteristic
315C2×C3×C5#44011yescharacteristic
316C2×C3×C5#44811yescharacteristic
317C2×C3×C5#44911yescharacteristic
318C2×C3×C5#46011yescharacteristic
319C2×C3×C5#46411yescharacteristic
320D15—82no—
321C23×C3C2×D2011yescharacteristic
322C22×D3—102no—
323C22×D3—201no—
324C22×D3—201no—
325C22×D3—201no—
326C22×D3—201no—
327C22×D3—201no—
328C22×D3—201no—
329C2×C3×C4—102no—
330C2×C3×C4—201no—
331C2×C3×C4—201no—
332C2×C3×C4—201no—
333C2×C3×C4—201no—
334C2×C3×C4—201no—
335C2×C3×C4C2×C4×D511yescharacteristic
336C2×C3×C4#70911yescharacteristic
337C2×C3×C4D4×D511yescharacteristic
338C2×C3×C4D4×D511yescharacteristic
339C2×C3×C4D4×D511yescharacteristic
340C2×C3×C4#71311yescharacteristic
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—102no—
347C2×Dic3—102no—
348C2×Dic3—102no—
349C2×Dic3#67712yescomplemented
350D12—102no—
351D12—102no—
352D12—201no—
353D12—201no—
354Dic6—54no—
355Dic6—102no—
356Dic6—102no—
357Dic6—102no—
358C3×C8—22no—
359C3⋊C8—41no—
360C22×C5#88211yescharacteristic
361C22×C5#88411yescharacteristic
362C22×C5#89311yescharacteristic
363C22×C5#91111yescharacteristic
364C22×C5D3×SD1611yescharacteristic
365C22×C5#91611yescharacteristic
366C22×C5#91811yescharacteristic
367C2×D5—122no—
368C2×D5—122no—
369C2×D5—122no—
370C2×D5—241no—
371C2×D5—241no—
372C2×D5—241no—
373C2×D5—241no—
374C4×C5—62no—
375C4×C5—121no—
376C4×C5C4×D1211yescharacteristic
377C4×C5D3×C22⋊C411yescharacteristic
378C4×C5#89411yescharacteristic
379C4×C5#89511yescharacteristic
380Dic5—42no—
381Dic5—64no—
382Dic5—81no—
383C24—301no—
384C2×Q8—152no—
385C2×Q8—152no—
386C2×Q8—152no—
387C2×Q8—301no—
388C2×Q8—301no—
389C2×Q8—301no—
390C2×D4—152no—
391C2×D4—301no—
392C2×D4—301no—
393C2×D4—301no—
394C2×D4—301no—
395C2×D4—301no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—101no—
399C22×C4—101no—
400C22×C4—301no—
401C22×C4C2×D3×D511yescharacteristic
402SD16—308no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8#120012yes—
409C4⋊C4—52no—
410C4⋊C4—101no—
411C4⋊C4—101no—
412C4⋊C4—101no—
413C4⋊C4—304no—
414C22⋊C4—602no—
415C22⋊C4—602no—
416C42—61no—
417C42—61no—
418C42—304no—
419C3×C5#188011yescharacteristic, complemented, Hall
420C22×C3C4×D2011yescharacteristic
421C22×C3#402811yescharacteristic
422C22×C3#404111yescharacteristic
423C22×C3C2×#67711yescharacteristic
424C22×C3#405011yescharacteristic
425C22×C3#405411yescharacteristic
426C22×C3#405511yescharacteristic
427C2×D3—202no—
428C2×D3—202no—
429C2×D3—202no—
430C2×D3—401no—
431C2×D3—401no—
432C2×D3—401no—
433C2×D3—401no—
434C3×C4—202no—
435C3×C4—401no—
436C3×C4D5×C22⋊C411yescharacteristic
437C3×C4#403011yescharacteristic
438C3×C4#403911yescharacteristic
439C3×C4#404011yescharacteristic
440Dic3—22no—
441Dic3—41no—
442Dic3—104no—
443C2×C5#466511yescharacteristic
444C2×C5#476211yescharacteristic
445C2×C5#478111yescharacteristic
446C2×C5#479811yescharacteristic
447C2×C5#480611yescharacteristic
448C2×C5#485211yescharacteristic
449C2×C5#488911yescharacteristic
450D5—242no—
451C23—302no—
452C23—601no—
453C23—601no—
454C23—601no—
455C23—601no—
456C23—601no—
457C23—601no—
458C23D3×D2011yescharacteristic
459Q8—154no—
460Q8—302no—
461Q8—302no—
462Q8—302no—
463D4—302no—
464D4—302no—
465D4—601no—
466D4—601no—
467C2×C4—32no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—102no—
474C2×C4—201no—
475C2×C4—201no—
476C2×C4—201no—
477C2×C4—201no—
478C2×C4—201no—
479C2×C4—302no—
480C2×C4—302no—
481C2×C4—302no—
482C2×C4#676911yescharacteristic
483C2×C4#677311yescharacteristic
484C2×C4D5×D1211yescharacteristic
485C2×C4C2×#120011yescharacteristic
486C2×C4#679011yescharacteristic
487C2×C4#679011yescharacteristic
488C8—22no—
489C8—121no—
490C2×C3#6495411yescharacteristic
491C2×C3C4×#67711yescharacteristic
492C2×C3#6499911yescharacteristic
493C2×C3#6502311yescharacteristic
494C2×C3#6502711yescharacteristic
495C2×C3#6513411yescharacteristic
496C2×C3#6513711yescharacteristic
497D3—402no—
498C5#6890911yescharacteristic, complemented, Sylow
499C22—602no—
500C22—602no—
501C22—602no—
502C22—1201no—
503C22—1201no—
504C22—1201no—
505C22—1201no—
506C22D3×#67711yescharacteristic
507C22#9125411yescharacteristic
508C22#9126311yescharacteristic
509C22#9127211yescharacteristic
510C22#9142811yescharacteristic
511C22#9145911yescharacteristic
512C22#9147611yescharacteristic
513C4—62no—
514C4—121no—
515C4—202no—
516C4—304no—
517C4—401no—
518C4#9141811yescharacteristic
519C4#9142011yescharacteristic
520C4#9143311yescharacteristic
521C4#9156111yescharacteristic
522C3#1059947911yescharacteristic, complemented, Sylow
523C2—1202no—
524C2#1174778911yescharacteristic
525C2#1174780011yescharacteristic
526C2#1174787211yescharacteristic
527C2#1174800411yescharacteristic
528C2#1174813211yescharacteristic
529C2#1174816111yescharacteristic
530C2#1174890411yescharacteristic
subgroupG/H
centerC23D3×D20
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×C8C22
socleC23×C3×C5C2×D4
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenterC22×C8D3×D5
hypocenterC3×C5#1880
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C2×#91340C2semidirect
4. C2×#90974C2semidirect
6. C5×#4550C2semidirect
7. C3×#64771C2semidirect
14. C2×#6706C4semidirect
34. #64771D3semidirect
77. #4550D5semidirect
121. Dic30C22⋊C4semidirect
127. C2×C5×Dic3SD16semidirect
349. C2×Dic3#677semidirect
419. C3×C5#1880semidirect
498. C5#68909semidirect
522. C3#10599479semidirect

Maximal subgroups

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

subgroupindex
C22×#67062
C2×#913402
#117487892
C2×#909742
C2×#909682
C5×#45502
C3×#647712
#105994793
#689095