Finite Group Databasebeta

← #14328028

Subgroups — #14328028

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×#67C211yesmaximal, characteristic
2#11751312C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊Dic5C211yesmaximal, characteristic, complemented
4#11749482C212yesmaximal
5#11749479C212yesmaximal, complemented
6#11748777C211yesmaximal, characteristic, complemented
7#11748747C212yesmaximal, complemented
8#11748719C211yesmaximal, characteristic, complemented
9#11748667C211yesmaximal, characteristic, complemented
10Dic5×Dic3⋊C4C212yesmaximal, complemented
11#11748600C211yesmaximal, characteristic, complemented
12#10614069—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×#67C2211yescharacteristic
15C22×C4×C5×D3C2211yescharacteristic
16C2×#6719C2212yes—
17C2×C5×#217C2211yescharacteristic
18C4×C5×#67C418yescomplemented
19C2×C5×(C2×D3)⋊C4C2211yescharacteristic
20C2×C5×Dic3⋊C4C2211yescharacteristic
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×C22⋊Dic5C2212yes—
23C2×C3×C4×Dic5C2212yes—
24C2×Dic3⋊Dic5C2212yes—
25C2×Dic3⋊Dic5C2212yes—
26C2×(C2×D3)⋊Dic5C2212yes—
27C2×(C2×D3)⋊Dic5C2212yes—
28C2×Dic3×Dic5C2212yes—
29C2×Dic3×Dic5C2212yes—
30#91115C2212yes—
31#91115C2211yescharacteristic
32#91115C2211yescharacteristic
33C3×#3963C2211yescharacteristic
34C3×#3963C2211yescharacteristic
35#90994C2212yes—
36#90994C2212yes—
37#73463—51nomaximal, Hall
38C2×C4×C5×D4—31no—
39#65510—32no—
40#65500—31no—
41#65500—31no—
42C4×C22⋊Dic5—31no—
43C4×C22⋊Dic5D311yescharacteristic, complemented
44C4⋊C4×Dic5—32no—
45#65240—32no—
46#65240—32no—
47#65225—31no—
48C4×C4⋊Dic5—31no—
49C24×C3×C5C2311yescharacteristic
50C23×C5×D3C2311yescharacteristic
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2312yes—
57C2×C5×#67—22no—
58C2×C5×#67—22no—
59C2×C5×#67C2×C414yes—
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C2×C4×C5×D3—42no—
64C2×C4×C5×D3C2×C414yes—
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2312yes—
67#6719—42no—
68C5×#217C2×C414yes—
69C5×(C2×D3)⋊C4C2×C414yes—
70C5×Dic3⋊C4C2×C414yes—
71C4×C5×Dic3C2×C414yes—
72C3×C22⋊Dic5—24no—
73C3×C4×Dic5—42no—
74Dic3⋊Dic5—24no—
75Dic3⋊Dic5—24no—
76(C2×D3)⋊Dic5—24no—
77(C2×D3)⋊Dic5—42no—
78Dic3×Dic5—24no—
79Dic3×Dic5—42no—
80C2×C4×#67D511yescharacteristic
81C3×C4×C22⋊C4—51no—
82#5211—51no—
83#4957—52no—
84#4945—52no—
85#4940—52no—
86C4×Dic3⋊C4—52no—
87#4666—51no—
88#4664—51no—
89#4650—51no—
90#4649—51no—
91C22×C5×D4—31no—
92C23×C4×C5—31no—
93C23×C4×C5C2×D311yescharacteristic
94C4×C5×D4—38no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C22⋊C4—31no—
97C2×C5×C22⋊C4—31no—
98C2×C42×C5—31no—
99C2×C22⋊Dic5—32no—
100C2×C22⋊Dic5—32no—
101C2×C22⋊Dic5—32no—
102C2×C22⋊Dic5C2×D312yes—
103C2×C4⋊Dic5—32no—
104C2×C4⋊Dic5—32no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5—32no—
107C2×C4×Dic5C2×D312yes—
108#3963—31no—
109#3963—31no—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963C2×D311yescharacteristic
114#3963C2×D311yescharacteristic
115#2357—151noSylow
116C23×C3×C5—22no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5—22no—
120C23×C3×C5C22×C412yes—
121C23×C3×C5C2411yescharacteristic
122C22×C5×D3—22no—
123C22×C5×D3—21no—
124C22×C5×D3—21no—
125C22×C5×D3—22no—
126C22×C5×D3C22×C412yes—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—44no—
130C2×C3×C4×C5C22×C412yes—
131C2×C3×C4×C5C22×C412yes—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15—22no—
138C5×#67—28no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3C22×C412yes—
142C2×C5×Dic3C22×C412yes—
143C2×C5×Dic3C22×C412yes—
144C2×C5×Dic3#4112yes—
145C2×C5×Dic3#4112yes—
146C4×C5×D3—44no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5#4114yes—
153C23×C3×C4C2×D511yescharacteristic
154C22×#67C2×D511yescharacteristic
155C22×C4×D3C2×D511yescharacteristic
156C2×C3×C22⋊C4—52no—
157C2×C3×C42—52no—
158C2×#217—52no—
159C2×#217C2×D511yescharacteristic
160C4×#67Dic518yescomplemented
161C2×(C2×D3)⋊C4—52no—
162C2×(C2×D3)⋊C4—52no—
163C2×(C2×D3)⋊C4C2×D511yescharacteristic
164C2×Dic3⋊C4—52no—
165C2×Dic3⋊C4—52no—
166C2×Dic3⋊C4C2×D511yescharacteristic
167C2×C4×Dic3—52no—
168C2×C4×Dic3—52no—
169C2×C4×Dic3C2×D511yescharacteristic
170C3×#95—51no—
171C3×#95—51no—
172C3⋊#95—51no—
173C3⋊#95—51no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C24×C5—31no—
178C24×C5C22×D311yescharacteristic
179C2×C5×D4—34no—
180C2×C5×D4—62no—
181C2×C5×D4—62no—
182C22×C4×C5—31no—
183C22×C4×C5—31no—
184C22×C4×C5—31no—
185C22×C4×C5—34no—
186C22×C4×C5—42no—
187C22×C4×C5—122no—
188C22×C4×C5C4×D314yes—
189C22×C4×C5C22×D311yescharacteristic
190C22×C4×C5C22×D311yescharacteristic
191C22×Dic5—32no—
192C22×Dic5—32no—
193C22×Dic5C22×D312yes—
194C22×Dic5C22×D312yes—
195C5×C4⋊C4—34no—
196C5×C22⋊C4—34no—
197C5×C22⋊C4—34no—
198C42×C5—34no—
199C22⋊Dic5—24no—
200C22⋊Dic5—64no—
201C22⋊Dic5—122no—
202C22⋊Dic5—122no—
203C4⋊Dic5—64no—
204C4⋊Dic5—64no—
205C4×Dic5—42no—
206C4×Dic5—64no—
207C4×Dic5—122no—
208C2×C4×D4—31no—
209#390—152no—
210#388—152no—
211#388—152no—
212#382—151no—
213#381—151no—
214#381—151no—
215C4×C4⋊C4—151no—
216C4×C4⋊C4—152no—
217C4×C22⋊C4—51no—
218C4×C22⋊C4—151no—
219C22×C3×C5—22no—
220C22×C3×C5—22no—
221C22×C3×C5—22no—
222C22×C3×C5—42no—
223C22×C3×C5—42no—
224C22×C3×C5C23×C411yescharacteristic
225C22×C3×C5C2×#4111yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4112yes—
228C22×C3×C5H(5,𝔽2)12yes—
229C2×C5×D3—22no—
230C2×C5×D3—22no—
231C2×C5×D3—22no—
232C2×C5×D3—42no—
233C2×C5×D3—42no—
234C3×C4×C5—22no—
235C3×C4×C5—22no—
236Dic15—42no—
237Dic15—42no—
238C3×Dic5—24no—
239C3×Dic5—42no—
240C5×Dic3—22no—
241C5×Dic3—22no—
242C5×Dic3#11714yes—
243C24×C3C22×D511yescharacteristic
244C23×D3C22×D511yescharacteristic
245C22×C3×C4—42no—
246C22×C3×C4—52no—
247C22×C3×C4—52no—
248C22×C3×C4C2×Dic514yes—
249C22×C3×C4C22×D511yescharacteristic
250C22×C3×C4C22×D511yescharacteristic
251C2×#67—22no—
252C2×#67—22no—
253C2×#67C2×Dic514yes—
254C22×Dic3—52no—
255C22×Dic3—52no—
256C22×Dic3C22×D511yescharacteristic
257C22×Dic3C22×D511yescharacteristic
258C22×Dic3C22×D511yescharacteristic
259C2×C4×D3—42no—
260C2×C4×D3C2×Dic514yes—
261C3×C22⋊C4—104no—
262C3×C42—202no—
263#217—202no—
264#217C2×Dic514yes—
265(C2×D3)⋊C4—104no—
266(C2×D3)⋊C4—202no—
267(C2×D3)⋊C4C2×Dic514yes—
268Dic3⋊C4—104no—
269Dic3⋊C4—104no—
270Dic3⋊C4C2×Dic514yes—
271C4×Dic3—104no—
272C4×Dic3—202no—
273C4×Dic3C2×Dic514yes—
274C23×C5—22no—
275C23×C5—22no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5—32no—
279C23×C5—62no—
280C23×C5—62no—
281C23×C5—61no—
282C23×C5—61no—
283C23×C5C2×C4×D312yes—
284C23×C5C23×D311yescharacteristic
285C5×D4—68no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—32no—
289C2×C4×C5—32no—
290C2×C4×C5—32no—
291C2×C4×C5—32no—
292C2×C4×C5—32no—
293C2×C4×C5—44no—
294C2×C4×C5—62no—
295C2×C4×C5—62no—
296C2×C4×C5—124no—
297C2×C4×C5C2×C4×D312yes—
298C2×C4×C5C2×C4×D312yes—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5—62no—
307C2×Dic5—62no—
308C2×Dic5—62no—
309C2×Dic5—62no—
310C2×Dic5#23514yes—
311C22×D4—31no—
312C23×C4—31no—
313C23×C4D3×D511yescharacteristic
314C4×D4—38no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—152no—
317C2×C4⋊C4—152no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—31no—
320C2×C22⋊C4—52no—
321C2×C22⋊C4—152no—
322C2×C22⋊C4—152no—
323C2×C22⋊C4—152no—
324C2×C42—31no—
325C2×C42—52no—
326C2×C42—152no—
327C2×C42—152no—
328#95—51no—
329#95—51no—
330#95—152no—
331#95—152no—
332#95—151no—
333#95—151no—
334#95—152no—
335C2×C3×C5—42no—
336C2×C3×C5C2×#11711yescharacteristic
337C2×C3×C5C4×#4111yescharacteristic
338C2×C3×C5#51811yescharacteristic
339C2×C3×C5#55312yes—
340C2×C3×C5#55912yes—
341C5×D3—42no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3C22×Dic512yes—
347C23×C3C23×D511yescharacteristic
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—22no—
351C22×D3—22no—
352C22×D3C22×Dic512yes—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—44no—
356C2×C3×C4—54no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4—102no—
360C2×C3×C4—102no—
361C2×C3×C4—102no—
362C2×C3×C4C22×Dic512yes—
363C2×C3×C4C22×Dic512yes—
364#67—28no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—102no—
368C2×Dic3—102no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3—102no—
372C2×Dic3—102no—
373C2×Dic3#70912yes—
374C2×Dic3#70912yes—
375C2×Dic3C22×Dic512yes—
376C2×Dic3C22×Dic512yes—
377C2×Dic3C22×Dic512yes—
378C4×D3—44no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—42no—
383C22×C5—42no—
384C22×C5—62no—
385C22×C5—62no—
386C22×C5—62no—
387C22×C5—122no—
388C22×C5—122no—
389C22×C5C22×C4×D311yescharacteristic
390C22×C5C2×#23512yes—
391C22×C5#100412yes—
392C22×C5D3×#4111yescharacteristic
393C22×C5D3×#4111yescharacteristic
394C4×C5—22no—
395C4×C5—22no—
396C4×C5—34no—
397C4×C5—62no—
398C4×C5—62no—
399Dic5—24no—
400Dic5—42no—
401Dic5—122no—
402Dic5—122no—
403C24—31no—
404C24C2×D3×D511yescharacteristic
405C2×D4—34no—
406C2×D4—62no—
407C2×D4—62no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—34no—
412C22×C4—42no—
413C22×C4—52no—
414C22×C4—52no—
415C22×C4—122no—
416C22×C4—152no—
417C22×C4—152no—
418C22×C4D3×Dic514yes—
419C22×C4C2×D3×D511yescharacteristic
420C22×C4C2×D3×D511yescharacteristic
421C4⋊C4—34no—
422C4⋊C4—304no—
423C4⋊C4—304no—
424C22⋊C4—34no—
425C22⋊C4—34no—
426C22⋊C4—104no—
427C22⋊C4—304no—
428C22⋊C4—602no—
429C22⋊C4—602no—
430C42—34no—
431C42—202no—
432C42—304no—
433C42—602no—
434C3×C5#235711yescharacteristic, complemented, Hall
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3C2×#70911yescharacteristic
441C22×C3C2×#70911yescharacteristic
442C22×C3#414412yes—
443C22×C3D5×#4112yes—
444C22×C3C23×Dic511yescharacteristic
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—22no—
448C2×D3—42no—
449C2×D3—42no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—104no—
453C3×C4—202no—
454Dic3—22no—
455Dic3—22no—
456Dic3—202no—
457Dic3—202no—
458Dic3#407214yes—
459C2×C5—42no—
460C2×C5—122no—
461C2×C5C4×#23511yescharacteristic
462C2×C5#548611yescharacteristic
463C2×C5D3×#11711yescharacteristic
464C2×C5#554012yes—
465C2×C5#555912yes—
466C23—22no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—32no—
471C23—62no—
472C23—61no—
473C23—61no—
474C23—62no—
475C23C2×D3×Dic512yes—
476C23C22×D3×D511yescharacteristic
477D4—68no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—44no—
486C2×C4—54no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4—102no—
492C2×C4—102no—
493C2×C4—102no—
494C2×C4—124no—
495C2×C4—302no—
496C2×C4—302no—
497C2×C4—302no—
498C2×C4—302no—
499C2×C4—302no—
500C2×C4—302no—
501C2×C4C2×D3×Dic512yes—
502C2×C4C2×D3×Dic512yes—
503C2×C3—42no—
504C2×C3#6589212yes—
505C2×C3#6589312yes—
506C2×C3C2×#407211yescharacteristic
507C2×C3#6613211yescharacteristic
508C2×C3#41×Dic511yescharacteristic
509D3—42no—
510C5#7346311yescharacteristic, complemented, Sylow
511C22—22no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—122no—
520C22—122no—
521C22D5×#23512yes—
522C22D3×#70911yescharacteristic
523C22D3×#70911yescharacteristic
524C22C22×D3×Dic511yescharacteristic
525C22#9204912yes—
526C4—22no—
527C4—22no—
528C4—34no—
529C4—62no—
530C4—62no—
531C4—104no—
532C4—202no—
533C4—602no—
534C4—602no—
535C3#1061406911yescharacteristic, complemented, Sylow
536C2—42no—
537C2—122no—
538C2#1175440412yes—
539C2Dic5×#23511yescharacteristic
540C2D3×#407211yescharacteristic
541C2#1175457312yes—
542C2#1175541011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C5×#67C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2357
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. #11751312C2semidirect
3. C3×C4×C22⋊Dic5C2semidirect
5. #11749479C2semidirect
6. #11748777C2semidirect
7. #11748747C2semidirect
8. #11748719C2semidirect
9. #11748667C2semidirect
10. Dic5×Dic3⋊C4C2semidirect
11. #11748600C2semidirect
18. C4×C5×#67C4semidirect
43. C4×C22⋊Dic5D3semidirect
160. C4×#67Dic5semidirect
434. C3×C5#2357semidirect
510. C5#73463semidirect
535. C3#10614069semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×#672
#117513122
C3×C4×C22⋊Dic52
#117494822
#117494792
#117487772
#117487472
#117487192
#117486672
Dic5×Dic3⋊C42
#117486002
#106140693
#734635