Finite Group Databasebeta

← #14283172

Subgroups — #14283172

Order 1920.

588 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 96 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
1D15×C22⋊Q8C211yesmaximal, characteristic
2#11748670C211yesmaximal, characteristic
3#11748509C211yesmaximal, characteristic
4C5×#4538C211yesmaximal, characteristic, complemented
5C3×#64758C211yesmaximal, characteristic, complemented
6#11745965C211yesmaximal, characteristic, complemented
7#11745964C211yesmaximal, characteristic, complemented
8#10599128—31nomaximal, Hall
9C2×Q8×D15C411yescharacteristic, complemented
10C22×C4×D15C2211yescharacteristic
11C3×C5×C22⋊Q8C2211yescharacteristic
12#91835C411yescharacteristic, complemented
13#91813C2211yescharacteristic
14#91786C411yescharacteristic, complemented
15#91785—21no—
16C4⋊C4×D15—21no—
17C4⋊C4×D15C411yescharacteristic, complemented
18C22⋊C4×D15—21no—
19#91767—21no—
20C2×#6667—21no—
21C2×Dic3×Dic5—21no—
22C2×#6651—21no—
23C5×C3⋊#95C2211yescharacteristic
24C5×#830C2211yescharacteristic
25C3×#3963C2211yescharacteristic
26C3×C22⋊#168C2211yescharacteristic
27#90957—21no—
28#68335—51nomaximal, Hall
29D5×C22⋊Q8—31no—
30#65015—31no—
31#64955—31no—
32C5×#328—31no—
33#64758D311yescharacteristic, complemented
34#64687—31no—
35#64672—31no—
36C23×D15D411yescharacteristic
37C2×C3×C5×Q8C2×C411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C22×Dic15D411yescharacteristic
40Q8×D15—42no—
41C2×C4×D15—21no—
42C2×C4×D15—21no—
43C2×C4×D15—42no—
44C2×C4×D15C2×C411yescharacteristic
45C2×C4×D15C2×C411yescharacteristic
46C2×C4×D15D411yescharacteristic
47C2×C4×D15D411yescharacteristic
48C2×Dic30—21no—
49C2×Dic30C2×C411yescharacteristic
50C22×C5×Dic3—21no—
51C22×C3×Dic5—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4C2×C411yescharacteristic
54C3×C5×C22⋊C4—21no—
55#6719—21no—
56#6714—21no—
57#6714—21no—
58C4⋊Dic15—21no—
59C4⋊Dic15C2×C411yescharacteristic
60#6712—21no—
61#6712—21no—
62#6712—21no—
63C2×C5×C3⋊C8—21no—
64C2×C3×#168—21no—
65#6667—24no—
66Dic3×Dic5—42no—
67#6651—42no—
68D3×C22⋊Q8—51no—
69#4726—51no—
70#4666—51no—
71C3×#328—51no—
72#4538D511yescharacteristic, complemented
73#4467—51no—
74#4452—51no—
75C2×Q8×D5—31no—
76C22×C4×D5—31no—
77C5×C22⋊Q8C2×D311yescharacteristic
78#4092—31no—
79C2×#685—61no—
80#4070—31no—
81C2×C4×Dic5—61no—
82C2×#676—61no—
83#4043—31no—
84#4042—61no—
85D5×C4⋊C4—31no—
86D5×C4⋊C4—61no—
87D5×C22⋊C4—61no—
88#4024—61no—
89C5×C22⋊C8—31no—
90C5×#95—31no—
91#3963C2×D311yescharacteristic
92C22⋊#168C2×D311yescharacteristic
93#3952—61no—
94#1508—151noSylow
95C23×C3×C5C2×D411yescharacteristic
96C22×D15—21no—
97C22×D15—21no—
98C22×D15—22no—
99C22×D15—41no—
100C22×D15C22⋊C411yescharacteristic
101C22×D15C22⋊C411yescharacteristic
102C3×C5×Q8—41no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5C22×C411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×Dic15—22no—
109C2×Dic15—21no—
110C2×Dic15—21no—
111C2×Dic15C22⋊C411yescharacteristic
112C2×Dic15C22⋊C411yescharacteristic
113C4×D15—22no—
114C4×D15—22no—
115C4×D15—41no—
116C4×D15—41no—
117C4×D15—42no—
118C4×D15—42no—
119Dic30—42no—
120Dic30—41no—
121C2×C5×Dic3—22no—
122C2×C5×Dic3—41no—
123C2×C5×Dic3—41no—
124C2×C3×Dic5—22no—
125C2×C3×Dic5—41no—
126C2×C3×Dic5—41no—
127C3×#168—41no—
128C5×C3⋊C8—41no—
129C2×D3×Q8—51no—
130C22×C4×D3—51no—
131C3×C22⋊Q8C2×D511yescharacteristic
132#946—51no—
133C2×(C2×D3)⋊C4—101no—
134#924—51no—
135C2×C4×Dic3—101no—
136C2×ΓL(1,𝔽25)—101no—
137#897—51no—
138#896—101no—
139D3×C4⋊C4—51no—
140D3×C4⋊C4—101no—
141D3×C22⋊C4—101no—
142#878—101no—
143C3×C22⋊C8—51no—
144C3×#95—51no—
145C3⋊#95C2×D511yescharacteristic
146#830C2×D511yescharacteristic
147(C2×D3)⋊C8—101no—
148C23×D5—31no—
149C2×C5×Q8C4×D311yescharacteristic
150C22×C4×C5—61no—
151C22×C4×C5C22×D311yescharacteristic
152C22×Dic5—21no—
153C22×Dic5—31no—
154Q8×D5—122no—
155C2×C4×D5—31no—
156C2×C4×D5—31no—
157C2×C4×D5—31no—
158C2×C4×D5—31no—
159C2×C4×D5—61no—
160C2×C4×D5—61no—
161C2×C4×D5—122no—
162C2×Dic10—31no—
163C2×Dic10—61no—
164C2×C5×C8—61no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4C4×D311yescharacteristic
167C5×C22⋊C4—21no—
168C22⋊Dic5—61no—
169#685—64no—
170#685—61no—
171#685—61no—
172C4⋊Dic5—31no—
173C4⋊Dic5—61no—
174#683—61no—
175#683—61no—
176#683—61no—
177C4×Dic5—122no—
178C2×#168—21no—
179#676—122no—
180C2×C22⋊Q8—151no—
181#407—151no—
182C4×C22⋊C4—151no—
183#328—31no—
184#328—51no—
185#328—151no—
186#328—151no—
187C22×C3×C5—21no—
188C22×C3×C5—21no—
189C22×C3×C5C2×C22⋊C411yescharacteristic
190C22×C3×C5#13711yescharacteristic
191C22×C3×C5#13711yescharacteristic
192C2×D15—21no—
193C2×D15—21no—
194C2×D15—21no—
195C2×D15—21no—
196C2×D15—21no—
197C2×D15—42no—
198C2×D15—42no—
199C2×D15#10412yes—
200C3×C4×C5—21no—
201C3×C4×C5—21no—
202C3×C4×C5—41no—
203C3×C4×C5—41no—
204Dic15—21no—
205Dic15—41no—
206Dic15—41no—
207Dic15#9912yes—
208C3×Dic5—42no—
209C5×Dic3—42no—
210C23×D3—51no—
211C2×C3×Q8C4×D511yescharacteristic
212C22×C3×C4—101no—
213C22×C3×C4C22×D511yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—51no—
216D3×Q8—202no—
217C2×C4×D3—51no—
218C2×C4×D3—51no—
219C2×C4×D3—51no—
220C2×C4×D3—51no—
221C2×C4×D3—101no—
222C2×C4×D3—101no—
223C2×C4×D3—202no—
224C2×Dic6—51no—
225C2×Dic6—101no—
226C2×C3×C8—101no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C4×D511yescharacteristic
229C3×C22⋊C4—21no—
230#217—101no—
231(C2×D3)⋊C4—104no—
232(C2×D3)⋊C4—101no—
233(C2×D3)⋊C4—101no—
234C4⋊Dic3—51no—
235C4⋊Dic3—101no—
236Dic3⋊C4—101no—
237Dic3⋊C4—101no—
238Dic3⋊C4—101no—
239C4×Dic3—202no—
240C2×C3⋊C8—21no—
241ΓL(1,𝔽25)—202no—
242C23×C5D3×D411yescharacteristic
243C22×D5—31no—
244C22×D5—31no—
245C22×D5—62no—
246C22×D5—61no—
247C22×D5—61no—
248C22×D5—121no—
249C5×Q8—41no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—41no—
253C2×C4×C5—62no—
254C2×C4×C5—121no—
255C2×C4×C5—121no—
256C2×C4×C5C2×C4×D311yescharacteristic
257C2×C4×C5D3×D411yescharacteristic
258C2×Dic5—22no—
259C2×Dic5—31no—
260C2×Dic5—31no—
261C2×Dic5—41no—
262C2×Dic5—41no—
263C2×Dic5—62no—
264C2×Dic5—61no—
265C2×Dic5—61no—
266C4×D5—62no—
267C4×D5—62no—
268C4×D5—121no—
269C4×D5—121no—
270C4×D5—122no—
271C4×D5—122no—
272Dic10—122no—
273Dic10—121no—
274C5×C8—121no—
275#168—41no—
276C22×Q8—151no—
277C23×C4—151no—
278C2×M16—301no—
279C22⋊Q8—151no—
280C22⋊Q8—151no—
281C22⋊Q8—151no—
282C22⋊Q8—301no—
283C22⋊Q8—301no—
284C22⋊Q8D3×D511yescharacteristic
285C2×C4⋊C4—151no—
286C2×C4⋊C4—301no—
287C2×C22⋊C4—301no—
288C2×C22⋊C4—301no—
289C2×C42—301no—
290C22⋊C8—31no—
291C22⋊C8—51no—
292C22⋊C8—301no—
293#95—31no—
294#95—51no—
295C2×C3×C5—21no—
296C2×C3×C5—21no—
297C2×C3×C5C2×#9911yescharacteristic
298C2×C3×C5#41911yescharacteristic
299C2×C3×C5C2×#10411yescharacteristic
300D15—22no—
301D15—41no—
302C23×C3D4×D511yescharacteristic
303C22×D3—51no—
304C22×D3—51no—
305C22×D3—102no—
306C22×D3—101no—
307C22×D3—101no—
308C22×D3—201no—
309C3×Q8—41no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—41no—
313C2×C3×C4—102no—
314C2×C3×C4—201no—
315C2×C3×C4—201no—
316C2×C3×C4C2×C4×D511yescharacteristic
317C2×C3×C4D4×D511yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—51no—
322C2×Dic3—51no—
323C2×Dic3—102no—
324C2×Dic3—101no—
325C2×Dic3—101no—
326C4×D3—102no—
327C4×D3—102no—
328C4×D3—201no—
329C4×D3—201no—
330C4×D3—202no—
331C4×D3—202no—
332Dic6—202no—
333Dic6—201no—
334C3×C8—201no—
335C3⋊C8—41no—
336C22×C5—21no—
337C22×C5—21no—
338C22×C5D3×C22⋊C411yescharacteristic
339C22×C5#91511yescharacteristic
340C22×C5#91811yescharacteristic
341C2×D5—32no—
342C2×D5—61no—
343C2×D5—61no—
344C2×D5—61no—
345C2×D5—61no—
346C2×D5—61no—
347C2×D5—122no—
348C2×D5—122no—
349C4×C5—21no—
350C4×C5—21no—
351C4×C5—41no—
352C4×C5—41no—
353C4×C5—122no—
354Dic5—32no—
355Dic5—42no—
356Dic5—61no—
357Dic5—121no—
358Dic5—121no—
359C24—151no—
360C2×Q8—151no—
361C2×Q8—301no—
362C2×Q8—602no—
363C2×Q8#120011yescharacteristic
364C22×C4—61no—
365C22×C4—101no—
366C22×C4—151no—
367C22×C4—151no—
368C22×C4—151no—
369C22×C4—151no—
370C22×C4—151no—
371C22×C4—301no—
372C22×C4—301no—
373C22×C4—602no—
374C22×C4C2×D3×D511yescharacteristic
375M16—602no—
376C2×C8—61no—
377C2×C8—101no—
378C4⋊C4—21no—
379C4⋊C4—151no—
380C4⋊C4—301no—
381C4⋊C4—301no—
382C4⋊C4—301no—
383C4⋊C4—301no—
384C4⋊C4#120011yescharacteristic
385C22⋊C4—21no—
386C22⋊C4—301no—
387C22⋊C4—301no—
388C22⋊C4—304no—
389C22⋊C4—301no—
390C42—602no—
391C3×C5#150811yescharacteristic, complemented, Hall
392C22×C3—21no—
393C22×C3—21no—
394C22×C3D5×C22⋊C411yescharacteristic
395C22×C3#406111yescharacteristic
396C22×C3#406411yescharacteristic
397C2×D3—52no—
398C2×D3—101no—
399C2×D3—101no—
400C2×D3—101no—
401C2×D3—101no—
402C2×D3—101no—
403C2×D3—202no—
404C2×D3—202no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—202no—
410Dic3—42no—
411Dic3—52no—
412Dic3—101no—
413Dic3—201no—
414Dic3—201no—
415C2×C5—21no—
416C2×C5—21no—
417C2×C5D3×#9911yescharacteristic
418C2×C5#480211yescharacteristic
419C2×C5D3×#10411yescharacteristic
420D5—62no—
421D5—121no—
422C23—151no—
423C23—151no—
424C23—302no—
425C23—301no—
426C23—301no—
427C23—601no—
428C23#679011yescharacteristic
429Q8—41no—
430Q8—601no—
431Q8—602no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—62no—
436C2×C4—102no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—151no—
440C2×C4—151no—
441C2×C4—201no—
442C2×C4—201no—
443C2×C4—302no—
444C2×C4—302no—
445C2×C4—302no—
446C2×C4—301no—
447C2×C4—301no—
448C2×C4—601no—
449C2×C4—601no—
450C2×C4—602no—
451C2×C4—602no—
452C2×C4C2×#120011yescharacteristic
453C2×C4#679011yescharacteristic
454C8—121no—
455C8—201no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3D5×#9911yescharacteristic
459C2×C3#6509111yescharacteristic
460C2×C3D5×#10411yescharacteristic
461D3—102no—
462D3—201no—
463C5#6833511yescharacteristic, complemented, Sylow
464C22—21no—
465C22—21no—
466C22—152no—
467C22—301no—
468C22—301no—
469C22—301no—
470C22—301no—
471C22—301no—
472C22—602no—
473C22—602no—
474C22#9152211yescharacteristic
475C22#9152411yescharacteristic
476C22#9156111yescharacteristic
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—122no—
482C4—152no—
483C4—202no—
484C4—301no—
485C4—601no—
486C4—601no—
487C3#1059912811yescharacteristic, complemented, Sylow
488C2—21no—
489C2—21no—
490C2—302no—
491C2—601no—
492C2#1174917111yescharacteristic
493C2#1174921811yescharacteristic
494C2#1174943011yescharacteristic
subgroupG/H
centerC22#91561
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×C22⋊Q8C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×D15D4
cosocleC22×C3×C4×C5C23
hypercenterC22⋊Q8D3×D5
hypocenterC3×C5#1508
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. C5×#4538C2semidirect
5. C3×#64758C2semidirect
6. #11745965C2semidirect
7. #11745964C2semidirect
9. C2×Q8×D15C4semidirect
12. #91835C4semidirect
14. #91786C4semidirect
17. C4⋊C4×D15C4semidirect
33. #64758D3semidirect
72. #4538D5semidirect
391. C3×C5#1508semidirect
463. C5#68335semidirect
487. C3#10599128semidirect

Maximal subgroups

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

subgroupindex
D15×C22⋊Q82
#117486702
#117485092
C5×#45382
C3×#647582
#117459652
#117459642
#105991283
#683355