Finite Group Databasebeta

← #14325830

Subgroups — #14325830

Order 1920.

1080 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4021C211yesmaximal, characteristic
2#11751038C211yesmaximal, characteristic, complemented
3C4×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11748899C212yesmaximal, complemented
5#11748896C212yesmaximal
6Dic3×#685C212yesmaximal, complemented
7#11748771C212yesmaximal, complemented
8#11748770C212yesmaximal, complemented
9#11748672C212yesmaximal, complemented
10#10605363—31nomaximal, Hall
11C22×C3×D20C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C22×D5×Dic3C2212yes—
14C2×C3×C42×C5C2211yescharacteristic
15C2×C5×C4⋊Dic3C2212yes—
16C2×C4×C5×Dic3C2211yescharacteristic
17C2×C4×C5×Dic3C2211yescharacteristic
18C2×C3×#685C2212yes—
19C2×C3×#685C2212yes—
20C3×#4021C418yescomplemented
21C2×#6668C2212yes—
22C2×#6665C2212yes—
23C2×#6665C2212yes—
24C2×#6665C2212yes—
25C2×Dic3×Dic5C2212yes—
26#91115C2212yes—
27#91115C2212yes—
28#91115C2211yescharacteristic
29#91115C2211yescharacteristic
30C5×C3⋊#95C2212yes—
31#90994C2212yes—
32#90994C2212yes—
33#82954—51nomaximal, Hall
34C2×#4021D311yescharacteristic
35C4×C5×C4⋊C4—31no—
36#65279—32no—
37#65278—32no—
38#65229—32no—
39C4×#685—32no—
40#65226—31no—
41#64954—32no—
42#64935—32no—
43C23×C3×D5C2312yes—
44C22×C3×C4×C5C2311yescharacteristic
45C22×C3×C4×C5C2311yescharacteristic
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15C2312yes—
48C22×Dic15C2312yes—
49C22×C5×Dic3C2312yes—
50C22×C5×Dic3C2312yes—
51C22×C3×Dic5C2312yes—
52C2×C3×D20—42no—
53C2×C3×D20C2×C414yes—
54C2×C3×Dic10—42no—
55C2×C3×Dic10C2×C414yescomplemented
56C2×D5×Dic3—24no—
57C2×D5×Dic3—42no—
58C2×D5×Dic3—42no—
59C3×C42×C5C2×C414yescomplemented
60C5×C4⋊Dic3—24no—
61C4×C5×Dic3—41no—
62C4×C5×Dic3—41no—
63C3×#685C2×C418yescomplemented
64C3×#685C2×C418yescomplemented
65#6668—42no—
66#6665—24no—
67#6665—42no—
68#6665—42no—
69Dic3×Dic5—42no—
70C2×C3×#124—51no—
71#5229—52no—
72#5220—52no—
73#4951—52no—
74#4951—52no—
75#4950—52no—
76Dic3×C22⋊C4—52no—
77#4937—51no—
78C4×C4⋊Dic3D511yescharacteristic, complemented
79C22×D20C2×D311yescharacteristic
80C22×C4×D5—32no—
81C22×Dic10C2×D311yescharacteristic
82C2×C5×C4⋊C4—32no—
83C2×C42×C5—31no—
84C2×C42×C5—31no—
85C2×C42×C5C2×D311yescharacteristic
86C2×#685—32no—
87C2×#685—32no—
88C2×#685—32no—
89C2×#685C2×D312yes—
90C2×#685C2×D312yes—
91C2×#683—32no—
92C2×C4×Dic5—32no—
93#4021Dic318yescomplemented
94C5×#95—32no—
95#3963—32no—
96#3963—32no—
97#3963—31no—
98#3963—31no—
99#3963—32no—
100#3963—32no—
101#2337—151noSylow
102C23×C3×C5C2411yescharacteristic
103C22×C3×D5—22no—
104C22×C3×D5—22no—
105C22×C3×D5—22no—
106C22×C3×D5—22no—
107C22×C3×D5—22no—
108C22×C3×D5—22no—
109C22×C3×D5C22×C414yes—
110C2×C3×C4×C5—22no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5C22×C412yes—
115C2×C3×C4×C5C22×C412yes—
116C2×C3×C4×C5C22×C412yes—
117C2×Dic15—22no—
118C2×Dic15—22no—
119C2×Dic15—22no—
120C2×Dic15—22no—
121C2×Dic15—22no—
122C2×Dic15—22no—
123C2×C5×Dic3—22no—
124C2×C5×Dic3—22no—
125C2×C5×Dic3—22no—
126C2×C5×Dic3—22no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3#4114yes—
129C2×C3×Dic5—22no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5C22×C414yes—
132C3×D20—44no—
133C3×Dic10—44no—
134D5×Dic3—48no—
135C22×C3×Q8—51no—
136C22×C3×D4—51no—
137C23×Dic3—52no—
138C3×#124—58no—
139C2×C3×C22⋊C4—52no—
140C2×C3×C22⋊C4—52no—
141C2×C3×C42C2×D511yescharacteristic
142C2×#217—52no—
143C2×#217—52no—
144C2×#217—52no—
145C2×C4⋊Dic3—52no—
146C2×C4⋊Dic3C2×D512yes—
147C2×C4×Dic3—52no—
148C2×C4×Dic3C2×D511yescharacteristic
149C2×C4×Dic3C2×D511yescharacteristic
150C3⋊#95—52no—
151C3⋊#95—51no—
152C3⋊#95—51no—
153C3⋊#95—52no—
154C3⋊#95—52no—
155C3⋊#95—52no—
156C3⋊#95C2×D512yes—
157C23×D5C22×D312yes—
158C22×C4×C5—32no—
159C22×C4×C5—32no—
160C22×C4×C5C22×D311yescharacteristic
161C22×C4×C5C22×D311yescharacteristic
162C22×C4×C5C22×D311yescharacteristic
163C22×Dic5—32no—
164C22×Dic5—32no—
165C22×Dic5C22×D312yes—
166C2×D20—42no—
167C2×D20C2×Dic314yes—
168C2×C4×D5—64no—
169C2×C4×D5—122no—
170C2×C4×D5—122no—
171C2×Dic10—42no—
172C2×Dic10C2×Dic314yescomplemented
173C5×C4⋊C4—64no—
174C42×C5—121no—
175C42×C5—121no—
176C42×C5C2×Dic314yescomplemented
177#685—64no—
178#685—122no—
179#685—122no—
180#685C2×Dic318yescomplemented
181#685C2×Dic318yescomplemented
182#683—122no—
183C4×Dic5—122no—
184C2×#124—51no—
185#391—152no—
186#388—152no—
187#388—152no—
188#386—152no—
189#385—152no—
190#383—151no—
191C4×C4⋊C4—31no—
192C4×C22⋊C4—152no—
193C22×C3×C5C23×C411yescharacteristic
194C22×C3×C5C2×#4112yes—
195C22×C3×C5H(5,𝔽2)11yescharacteristic
196C22×C3×C5H(5,𝔽2)12yes—
197C22×C3×C52−1+411yescharacteristic
198C2×C3×D5—24no—
199C2×C3×D5—24no—
200C2×C3×D5—24no—
201C2×C3×D5—42no—
202C2×C3×D5—42no—
203C2×C3×D5—42no—
204C2×C3×D5—42no—
205C3×C4×C5—22no—
206C3×C4×C5—22no—
207C3×C4×C5—22no—
208Dic15—42no—
209Dic15—42no—
210C3×Dic5—24no—
211C5×Dic3—24no—
212C5×Dic3—42no—
213C24×C3—52no—
214C2×C3×Q8—54no—
215C2×C3×Q8—202no—
216C2×C3×D4—54no—
217C2×C3×D4—202no—
218C22×C3×C4—52no—
219C22×C3×C4C22×D511yescharacteristic
220C22×C3×C4C22×D511yescharacteristic
221C22×C3×C4C22×D511yescharacteristic
222C22×Dic3—52no—
223C22×Dic3—52no—
224C22×Dic3—104no—
225C22×Dic3—202no—
226C22×Dic3—202no—
227C22×Dic3C22×D512yes—
228C22×Dic3C22×D512yes—
229C3×C22⋊C4—58no—
230C3×C22⋊C4—58no—
231C3×C42C4×D514yescomplemented
232#217—104no—
233#217—202no—
234#217—202no—
235C4⋊Dic3—24no—
236C4⋊Dic3—202no—
237C4×Dic3—41no—
238C4×Dic3—41no—
239C4×Dic3—202no—
240C23×C5C23×D311yescharacteristic
241C22×D5—22no—
242C22×D5—22no—
243C22×D5—22no—
244C22×D5—22no—
245C22×D5—22no—
246C22×D5—22no—
247C22×D5C22×Dic314yes—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—34no—
253C2×C4×C5—62no—
254C2×C4×C5—62no—
255C2×C4×C5—62no—
256C2×C4×C5—62no—
257C2×C4×C5—62no—
258C2×C4×C5C22×Dic312yes—
259C2×C4×C5C22×Dic312yes—
260C2×C4×C5C22×Dic312yes—
261C2×Dic5—22no—
262C2×Dic5—22no—
263C2×Dic5—62no—
264C2×Dic5—62no—
265C2×Dic5—62no—
266C2×Dic5—62no—
267C2×Dic5—62no—
268C2×Dic5—62no—
269C2×Dic5C22×Dic314yes—
270D20—44no—
271C4×D5—128no—
272Dic10—44no—
273C22×Q8—51no—
274C22×D4—51no—
275C23×C4—152no—
276#124—58no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—152no—
279C2×C22⋊C4—52no—
280C2×C22⋊C4—52no—
281C2×C22⋊C4—152no—
282C2×C22⋊C4—152no—
283C2×C22⋊C4—152no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—152no—
287C2×C42D3×D511yescharacteristic
288#95—32no—
289#95—152no—
290#95—152no—
291#95—152no—
292#95—151no—
293#95—152no—
294#95—151no—
295C2×C3×C5C4×#4111yescharacteristic
296C2×C3×C5#51811yescharacteristic
297C2×C3×C5#52011yescharacteristic
298C2×C3×C5#53812yes—
299C2×C3×C5#54212yes—
300C3×D5—44no—
301C23×C3—54no—
302C23×C3—102no—
303C23×C3—102no—
304C23×C3—102no—
305C23×C3—102no—
306C23×C3—102no—
307C23×C3—102no—
308C23×C3C23×D511yescharacteristic
309C3×Q8—204no—
310C3×D4—204no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—54no—
316C2×C3×C4—102no—
317C2×C3×C4—102no—
318C2×C3×C4C2×C4×D512yes—
319C2×C3×C4C2×C4×D512yes—
320C2×C3×C4C2×C4×D512yes—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—102no—
327C2×Dic3—102no—
328C2×Dic3—102no—
329C2×Dic3—102no—
330C2×Dic3—102no—
331C2×Dic3—102no—
332C2×Dic3—208no—
333C2×Dic3#70914yes—
334C22×C5#100412yes—
335C22×C5D3×#4112yes—
336C22×C5#100911yescharacteristic
337C22×C5#101011yescharacteristic
338C22×C5C23×Dic311yescharacteristic
339C2×D5—24no—
340C2×D5—24no—
341C2×D5—24no—
342C2×D5—42no—
343C2×D5—42no—
344C2×D5—42no—
345C2×D5—42no—
346C4×C5—22no—
347C4×C5—22no—
348C4×C5—22no—
349C4×C5—64no—
350C4×C5—122no—
351Dic5—24no—
352Dic5—122no—
353Dic5—122no—
354C24—52no—
355C2×Q8—54no—
356C2×Q8—202no—
357C2×D4—54no—
358C2×D4—202no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—52no—
362C22×C4—152no—
363C22×C4—152no—
364C22×C4—304no—
365C22×C4—602no—
366C22×C4—602no—
367C22×C4C2×D3×D511yescharacteristic
368C22×C4C2×D3×D511yescharacteristic
369C22×C4C2×D3×D511yescharacteristic
370C4⋊C4—64no—
371C4⋊C4—602no—
372C22⋊C4—58no—
373C22⋊C4—58no—
374C22⋊C4—304no—
375C22⋊C4—602no—
376C22⋊C4—602no—
377C42—121no—
378C42—121no—
379C42—602no—
380C42D5×Dic314yescomplemented
381C3×C5#233711yescharacteristic, complemented, Hall
382C22×C3—104no—
383C22×C3—104no—
384C22×C3—104no—
385C22×C3—202no—
386C22×C3—202no—
387C22×C3—202no—
388C22×C3—202no—
389C22×C3C22×C4×D511yescharacteristic
390C22×C3C2×#70912yes—
391C22×C3#414411yescharacteristic
392C22×C3#414711yescharacteristic
393C22×C3#414912yes—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—104no—
398Dic3—24no—
399Dic3—42no—
400Dic3—202no—
401Dic3—202no—
402C2×C5#566912yes—
403C2×C5#567812yes—
404C2×C5#579811yescharacteristic
405C2×C5Dic3×#4111yescharacteristic
406C2×C5#582711yescharacteristic
407D5—44no—
408C23—54no—
409C23—102no—
410C23—102no—
411C23—102no—
412C23—102no—
413C23—102no—
414C23—102no—
415C23C22×D3×D511yescharacteristic
416Q8—204no—
417D4—204no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—34no—
423C2×C4—54no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—102no—
430C2×C4—102no—
431C2×C4—302no—
432C2×C4—302no—
433C2×C4—302no—
434C2×C4—302no—
435C2×C4—302no—
436C2×C4—302no—
437C2×C4—608no—
438C2×C4C2×D5×Dic312yes—
439C2×C4C2×D5×Dic312yes—
440C2×C4C2×D5×Dic312yes—
441C2×C3—204no—
442C2×C3C4×#70911yescharacteristic
443C2×C3#6583811yescharacteristic
444C2×C3#6585511yescharacteristic
445C2×C3#6589412yes—
446C2×C3#6591612yes—
447C5#8295411yescharacteristic, complemented, Sylow
448C22—104no—
449C22—104no—
450C22—104no—
451C22—202no—
452C22—202no—
453C22—202no—
454C22—202no—
455C22#9200011yescharacteristic
456C22D3×#70912yes—
457C22#9201712yes—
458C22#9201811yescharacteristic
459C22C22×D5×Dic311yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—64no—
464C4—104no—
465C4—122no—
466C4—602no—
467C4—602no—
468C3#1060536311yescharacteristic, complemented, Sylow
469C2—204no—
470C2#1175259812yes—
471C2#1175276312yes—
472C2Dic3×#70911yescharacteristic
473C2#1175448811yescharacteristic
474C2#1175449911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×D20C22
hypercenterC2×C42D3×D5
hypocenterC3×C5#2337
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. #11751038C2semidirect
3. C4×C5×C4⋊Dic3C2semidirect
4. #11748899C2semidirect
6. Dic3×#685C2semidirect
7. #11748771C2semidirect
8. #11748770C2semidirect
9. #11748672C2semidirect
20. C3×#4021C4semidirect
55. C2×C3×Dic10C2×C4semidirect
59. C3×C42×C5C2×C4semidirect
63. C3×#685C2×C4semidirect
64. C3×#685C2×C4semidirect
78. C4×C4⋊Dic3D5semidirect
93. #4021Dic3semidirect
172. C2×Dic10C2×Dic3semidirect
176. C42×C5C2×Dic3semidirect
180. #685C2×Dic3semidirect
181. #685C2×Dic3semidirect
231. C3×C42C4×D5semidirect
380. C42D5×Dic3semidirect
381. C3×C5#2337semidirect
447. C5#82954semidirect
468. C3#10605363semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40212
#117510382
C4×C5×C4⋊Dic32
#117488992
#117488962
Dic3×#6852
#117487712
#117487702
#117486722
#106053633
#829545