Finite Group Databasebeta

← #14326830

Subgroups — #14326830

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91386C211yesmaximal, characteristic
2#11750767C211yesmaximal, characteristic
3C5×#4648C211yesmaximal, characteristic, complemented
4C3×#64937C211yesmaximal, characteristic, complemented
5#11748863C211yesmaximal, characteristic
6C4×#6669C212yesmaximal, complemented
7#11748804C212yesmaximal, complemented
8#11748730C212yesmaximal
9#11748726C212yesmaximal
10#11748600C211yesmaximal, characteristic, complemented
11#11748580C211yesmaximal, characteristic, complemented
12#10605123—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C2×#6714C2212yes—
15C2×C4×Dic15C2212yes—
16C2×C4×C5×Dic3C2212yes—
17C2×C4×C5×Dic3C2211yescharacteristic
18C2×C3×C4×Dic5C2212yes—
19C2×C3×C4×Dic5C2211yescharacteristic
20C2×Dic3⋊Dic5C2212yes—
21C2×#6669C2212yes—
22C2×#6669C2211yescharacteristic
23C2×#6669C2211yescharacteristic
24C2×#6668C2212yes—
25C2×#6667C2212yes—
26C2×#6667C2211yescharacteristic
27C2×#6667C2211yescharacteristic
28#91386C418yescomplemented
29C3×C5×#95C2211yescharacteristic
30#91115C2211yescharacteristic
31C5×C3⋊#95C2212yes—
32C5×C3⋊#95C2211yescharacteristic
33C3×#3963C2212yes—
34C3×#3963C2211yescharacteristic
35#90994C2212yes—
36#90994C2212yes—
37#73392—51nomaximal, Hall
38C2×#4018—31no—
39C5×#376—31no—
40#65229—32no—
41#65225—31no—
42C4×#683—32no—
43#64959—32no—
44#64956—31no—
45#64955—31no—
46#64939—31no—
47#64937D311yescharacteristic, complemented
48#64936—32no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×Dic15C2312yes—
53C22×Dic15C2311yescharacteristic
54C2×C4×D15—24no—
55C2×C4×D15C2×C414yes—
56C22×C5×Dic3C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2311yescharacteristic
59C22×C3×Dic5C2312yes—
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2311yescharacteristic
62#6714—24no—
63C4×Dic15—24no—
64C4×C5×Dic3—24no—
65C4×C5×Dic3C2×C414yes—
66C3×C4×Dic5—24no—
67C3×C4×Dic5C2×C414yes—
68Dic3⋊Dic5—42no—
69#6669—24no—
70#6669C2×C414yes—
71#6669C2×C414yes—
72#6668—42no—
73#6667—42no—
74#6667C2×C414yes—
75#6667C2×C414yes—
76C2×#872—51no—
77C3×#376—51no—
78#4940—52no—
79#4936—51no—
80C4×Dic3⋊C4—52no—
81#4670—52no—
82#4667—51no—
83#4666—51no—
84#4650—51no—
85#4648D511yescharacteristic, complemented
86#4647—52no—
87C22×C4×D5—31no—
88C2×C42×C5—31no—
89C2×C42×C5—32no—
90C2×#685—31no—
91C2×#685—31no—
92C2×#685—32no—
93C2×#685—32no—
94C2×C4⋊Dic5—32no—
95C2×#683—31no—
96C2×#683—31no—
97C2×#683—32no—
98C2×#683—32no—
99C2×C4×Dic5—32no—
100C2×C4×Dic5C2×D312yes—
101C2×C4×Dic5C2×D311yescharacteristic
102#4018—38no—
103C5×#95—32no—
104C5×#95—31no—
105C5×#95C2×D311yescharacteristic
106#3963—31no—
107#3963—32no—
108#3963—32no—
109#3963C2×D312yes—
110#3963C2×D311yescharacteristic
111#2346—151noSylow
112C23×C3×C5C2411yescharacteristic
113C22×D15—22no—
114C22×D15—22no—
115C22×D15—21no—
116C22×D15—21no—
117C22×D15C22×C412yes—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5C22×C412yes—
122C2×C3×C4×C5#4114yes—
123C2×Dic15—22no—
124C2×Dic15—22no—
125C2×Dic15—22no—
126C2×Dic15C22×C412yes—
127C2×Dic15#4114yes—
128C4×D15—28no—
129C2×C5×Dic3—22no—
130C2×C5×Dic3—21no—
131C2×C5×Dic3—21no—
132C2×C5×Dic3—22no—
133C2×C5×Dic3—22no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3C22×C412yes—
136C2×C5×Dic3C22×C412yes—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—21no—
139C2×C3×Dic5—21no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5C22×C412yes—
144C2×C3×Dic5C22×C412yes—
145C22×C4×D3—51no—
146C2×C3×C42—51no—
147C2×C3×C42—52no—
148C2×(C2×D3)⋊C4—51no—
149C2×(C2×D3)⋊C4—51no—
150C2×(C2×D3)⋊C4—52no—
151C2×(C2×D3)⋊C4—52no—
152C2×C4⋊Dic3—52no—
153C2×Dic3⋊C4—51no—
154C2×Dic3⋊C4—51no—
155C2×Dic3⋊C4—52no—
156C2×Dic3⋊C4—52no—
157C2×C4×Dic3—52no—
158C2×C4×Dic3C2×D512yes—
159C2×C4×Dic3C2×D511yescharacteristic
160#872—58no—
161C3×#95—52no—
162C3×#95—51no—
163C3×#95C2×D511yescharacteristic
164C3⋊#95—51no—
165C3⋊#95—52no—
166C3⋊#95—52no—
167C3⋊#95C2×D512yes—
168C3⋊#95C2×D511yescharacteristic
169C23×D5—31no—
170C22×C4×C5—31no—
171C22×C4×C5—31no—
172C22×C4×C5—32no—
173C22×C4×C5C22×D311yescharacteristic
174C22×C4×C5C22×D312yes—
175C22×Dic5—31no—
176C22×Dic5—32no—
177C22×Dic5C22×D312yes—
178C22×Dic5C22×D311yescharacteristic
179C22×Dic5C22×D311yescharacteristic
180C2×C4×D5—34no—
181C2×C4×D5—64no—
182C42×C5—34no—
183C42×C5—64no—
184#685—34no—
185#685—34no—
186#685—64no—
187#685—122no—
188C4⋊Dic5—122no—
189#683—34no—
190#683—34no—
191#683—64no—
192#683—122no—
193C4×Dic5—24no—
194C4×Dic5—64no—
195C4×Dic5C4×D314yes—
196C2×#117—151no—
197#389—152no—
198#388—152no—
199#388—152no—
200#382—151no—
201#382—151no—
202#380—151no—
203C4×C4⋊C4—152no—
204C4×C22⋊C4—151no—
205#376—31no—
206#376—51no—
207C22×C3×C5C23×C411yescharacteristic
208C22×C3×C5C2×#4112yes—
209C22×C3×C5C2×#4112yes—
210C22×C3×C5H(5,𝔽2)11yescharacteristic
211C22×C3×C52−1+411yescharacteristic
212C2×D15—22no—
213C2×D15—22no—
214C2×D15—22no—
215C2×D15—42no—
216C2×D15—42no—
217C3×C4×C5—22no—
218C3×C4×C5—24no—
219Dic15—42no—
220Dic15#11714yes—
221C3×Dic5—22no—
222C3×Dic5—22no—
223C3×Dic5—42no—
224C5×Dic3—22no—
225C5×Dic3—22no—
226C5×Dic3—42no—
227C23×D3—51no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—52no—
231C22×C3×C4C22×D511yescharacteristic
232C22×C3×C4C22×D512yes—
233C22×Dic3—51no—
234C22×Dic3—52no—
235C22×Dic3C22×D512yes—
236C22×Dic3C22×D511yescharacteristic
237C22×Dic3C22×D511yescharacteristic
238C2×C4×D3—54no—
239C2×C4×D3—104no—
240C3×C42—54no—
241C3×C42—104no—
242(C2×D3)⋊C4—54no—
243(C2×D3)⋊C4—54no—
244(C2×D3)⋊C4—104no—
245(C2×D3)⋊C4—202no—
246C4⋊Dic3—202no—
247Dic3⋊C4—54no—
248Dic3⋊C4—54no—
249Dic3⋊C4—104no—
250Dic3⋊C4—202no—
251C4×Dic3—24no—
252C4×Dic3—104no—
253C4×Dic3C4×D514yes—
254C23×C5C23×D311yescharacteristic
255C22×D5—32no—
256C22×D5—62no—
257C22×D5—62no—
258C22×D5—61no—
259C22×D5—61no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—32no—
264C2×C4×C5—32no—
265C2×C4×C5—62no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—62no—
269C2×C4×C5—62no—
270C2×C4×C5—62no—
271C2×C4×C5C2×C4×D312yes—
272C2×C4×C5#23514yes—
273C2×Dic5—22no—
274C2×Dic5—21no—
275C2×Dic5—21no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—32no—
280C2×Dic5—34no—
281C2×Dic5—62no—
282C2×Dic5—62no—
283C2×Dic5—62no—
284C2×Dic5C2×C4×D312yes—
285C2×Dic5C2×C4×D312yes—
286C4×D5—68no—
287C23×C4—151no—
288#117—158no—
289C2×C4⋊C4—151no—
290C2×C4⋊C4—151no—
291C2×C4⋊C4—152no—
292C2×C4⋊C4—152no—
293C2×C4⋊C4—152no—
294C2×C22⋊C4—151no—
295C2×C22⋊C4—151no—
296C2×C22⋊C4—152no—
297C2×C22⋊C4—152no—
298C2×C42—31no—
299C2×C42—32no—
300C2×C42—51no—
301C2×C42—52no—
302C2×C42—152no—
303#95—31no—
304#95—32no—
305#95—51no—
306#95—52no—
307#95—152no—
308#95—152no—
309#95—151no—
310#95D3×D511yescharacteristic
311C2×C3×C5C2×#11711yescharacteristic
312C2×C3×C5C4×#4111yescharacteristic
313C2×C3×C5#52011yescharacteristic
314C2×C3×C5#55212yes—
315C2×C3×C5#55512yes—
316D15—42no—
317C23×C3C23×D511yescharacteristic
318C22×D3—52no—
319C22×D3—102no—
320C22×D3—102no—
321C22×D3—101no—
322C22×D3—101no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—52no—
327C2×C3×C4—52no—
328C2×C3×C4—102no—
329C2×C3×C4—101no—
330C2×C3×C4—101no—
331C2×C3×C4—102no—
332C2×C3×C4—102no—
333C2×C3×C4—102no—
334C2×C3×C4C2×C4×D512yes—
335C2×C3×C4#70914yes—
336C2×Dic3—22no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—52no—
343C2×Dic3—54no—
344C2×Dic3—102no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347C2×Dic3C2×C4×D512yes—
348C2×Dic3C2×C4×D512yes—
349C4×D3—108no—
350C22×C5C22×C4×D311yescharacteristic
351C22×C5C2×#23512yes—
352C22×C5D3×#4112yes—
353C22×C5#100911yescharacteristic
354C22×C5#101011yescharacteristic
355C2×D5—62no—
356C2×D5—62no—
357C2×D5—62no—
358C2×D5—122no—
359C2×D5—122no—
360C4×C5—22no—
361C4×C5—24no—
362C4×C5—62no—
363C4×C5—62no—
364C4×C5—122no—
365Dic5—22no—
366Dic5—22no—
367Dic5—34no—
368Dic5—42no—
369Dic5—122no—
370C24—151no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—32no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—52no—
377C22×C4—151no—
378C22×C4—154no—
379C22×C4—152no—
380C22×C4—304no—
381C22×C4C2×D3×D512yes—
382C22×C4C2×D3×D511yescharacteristic
383C4⋊C4—154no—
384C4⋊C4—154no—
385C4⋊C4—304no—
386C4⋊C4—602no—
387C4⋊C4—602no—
388C22⋊C4—154no—
389C22⋊C4—154no—
390C22⋊C4—304no—
391C22⋊C4—602no—
392C42—34no—
393C42—54no—
394C42—64no—
395C42—104no—
396C42—304no—
397C3×C5#234611yescharacteristic, complemented, Hall
398C22×C3C22×C4×D511yescharacteristic
399C22×C3C2×#70912yes—
400C22×C3D5×#4112yes—
401C22×C3#414911yescharacteristic
402C22×C3#415011yescharacteristic
403C2×D3—102no—
404C2×D3—102no—
405C2×D3—102no—
406C2×D3—202no—
407C2×D3—202no—
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—102no—
411C3×C4—102no—
412C3×C4—202no—
413Dic3—22no—
414Dic3—22no—
415Dic3—42no—
416Dic3—54no—
417Dic3—202no—
418C2×C5C4×#23511yescharacteristic
419C2×C5D3×#11711yescharacteristic
420C2×C5#552311yescharacteristic
421C2×C5#553112yes—
422C2×C5#553212yes—
423D5—122no—
424C23—152no—
425C23—302no—
426C23—301no—
427C23—302no—
428C23—301no—
429C23C22×D3×D511yescharacteristic
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—52no—
436C2×C4—52no—
437C2×C4—62no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—102no—
444C2×C4—101no—
445C2×C4—101no—
446C2×C4—102no—
447C2×C4—102no—
448C2×C4—102no—
449C2×C4—152no—
450C2×C4—154no—
451C2×C4—308no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4#677114yes—
456C2×C4C4×D3×D512yes—
457C2×C3C4×#70911yescharacteristic
458C2×C3D5×#11711yescharacteristic
459C2×C3#6585711yescharacteristic
460C2×C3#6586512yes—
461C2×C3#6586612yes—
462D3—202no—
463C5#7339211yescharacteristic, complemented, Sylow
464C22—302no—
465C22—302no—
466C22—302no—
467C22—602no—
468C22—602no—
469C22C2×#677112yes—
470C22C2×C4×D3×D511yescharacteristic
471C22#9202011yescharacteristic
472C22#9202412yes—
473C22#9203511yescharacteristic
474C4—24no—
475C4—22no—
476C4—62no—
477C4—62no—
478C4—102no—
479C4—102no—
480C4—122no—
481C4—154no—
482C4—202no—
483C4—602no—
484C3#1060512311yescharacteristic, complemented, Sylow
485C2—602no—
486C2C4×#677111yescharacteristic
487C2#1175281412yes—
488C2#1175282511yescharacteristic
489C2#1175367011yescharacteristic
490C2#1175379512yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenter#95D3×D5
hypocenterC3×C5#2346
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. C5×#4648C2semidirect
4. C3×#64937C2semidirect
6. C4×#6669C2semidirect
7. #11748804C2semidirect
10. #11748600C2semidirect
11. #11748580C2semidirect
28. #91386C4semidirect
47. #64937D3semidirect
85. #4648D5semidirect
397. C3×C5#2346semidirect
463. C5#73392semidirect
484. C3#10605123semidirect

Maximal subgroups

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

subgroupindex
C2×#913862
#117507672
C5×#46482
C3×#649372
#117488632
C4×#66692
#117488042
#117487302
#117487262
#117486002
#117485802
#106051233
#733925