Finite Group Databasebeta

← #14307501

Subgroups — #14307501

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4070C211yesmaximal, characteristic
2#11751313C211yesmaximal, characteristic
3C4×C5×#217C211yesmaximal, characteristic
4#11746019C214yesmaximal
5#10600250—31nomaximal, Hall
6C23×C3×C4×C5C2211yescharacteristic
7C22×C3×Dic10C411yescharacteristic, complemented
8C2×#6719C2212yes—
9C2×C4⋊Dic15—21no—
10C2×C5×#217C2212yes—
11C2×C4×C5×Dic3—21no—
12C2×C3×C22⋊Dic5C2212yes—
13C2×C3×C4⋊Dic5C411yescharacteristic, complemented
14C3×#4070—21no—
15C3×#4070—21no—
16C3×#4070—21no—
17C3×#4070—21no—
18C2×C3×#683C412yescomplemented
19#91115—21no—
20C5×C3⋊#95—21no—
21#71701—51nomaximal, Hall
22C2×#4070D311yescharacteristic
23C4×C5×C22⋊C4—31no—
24#65501—31no—
25#64747—34no—
26C24×C3×C5C2311yescharacteristic
27C22×C3×C4×C5—42no—
28C22×C3×C4×C5C2×C411yescharacteristic
29C22×C3×C4×C5C2×C411yescharacteristic
30C22×C3×C4×C5D411yescharacteristic
31C22×C3×C4×C5D411yescharacteristic
32C22×C3×C4×C5D411yescharacteristic
33C22×C3×C4×C5D411yescharacteristic
34C22×Dic15—22no—
35C22×C5×Dic3—22no—
36C22×C3×Dic5C2×C412yes—
37C22×C3×Dic5C2×C412yes—
38C2×C3×Dic10—21no—
39C2×C3×Dic10—21no—
40C2×C3×Dic10—42no—
41#6719—24no—
42C4⋊Dic15—42no—
43C5×#217—22no—
44C5×#217Q814yescomplemented
45C4×C5×Dic3—42no—
46C3×C22⋊Dic5—22no—
47C3×C22⋊Dic5—22no—
48C3×C4⋊Dic5—21no—
49C3×C4⋊Dic5—21no—
50C3×#683—22no—
51C3×#683—22no—
52C2×C3×C22⋊Q8—51no—
53#5212—51no—
54C4×#217D511yescharacteristic
55#4526—54no—
56C23×C4×C5C2×D311yescharacteristic
57C22×Dic10Dic311yescharacteristic, complemented
58C2×C5×C22⋊C4—32no—
59C2×C42×C5—61no—
60C2×C22⋊Dic5—32no—
61C2×C22⋊Dic5C2×D312yes—
62C2×C4⋊Dic5—61no—
63C2×C4⋊Dic5Dic311yescharacteristic, complemented
64#4070—21no—
65#4070—21no—
66#4070—21no—
67#4070—21no—
68C2×#683Dic312yescomplemented
69C5×#95—61no—
70#3963—61no—
71#1996—151noSylow
72C23×C3×C5—41no—
73C23×C3×C5—41no—
74C23×C3×C5C22×C411yescharacteristic
75C23×C3×C5C2×D411yescharacteristic
76C23×C3×C5C2×D411yescharacteristic
77C23×C3×C5C2×Q812yes—
78C23×C3×C5#4112yes—
79C2×C3×C4×C5—22no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5C22⋊C411yescharacteristic
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×Dic15—42no—
90C2×Dic15—42no—
91C2×Dic15—42no—
92C2×C5×Dic3—24no—
93C2×C5×Dic3—42no—
94C2×C5×Dic3—42no—
95C2×C3×Dic5—22no—
96C2×C3×Dic5—22no—
97C2×C3×Dic5—42no—
98C2×C3×Dic5—42no—
99C3×Dic10—42no—
100C3×Dic10—42no—
101C22×C3×Q8—51no—
102C23×C3×C4C2×D511yescharacteristic
103C3×C22⋊Q8—101no—
104C3×C22⋊Q8—101no—
105C3×C22⋊Q8—101no—
106C3×C22⋊Q8—101no—
107C2×C3×C4⋊C4—52no—
108C2×C3×C4⋊C4—51no—
109C2×C3×C22⋊C4—52no—
110C2×#217—52no—
111C2×#217C2×D512yes—
112C2×C4⋊Dic3—101no—
113C2×C4×Dic3—21no—
114C3⋊#95—21no—
115C3⋊#95—101no—
116C24×C5C22×D311yescharacteristic
117C22×C4×C5—42no—
118C22×C4×C5—62no—
119C22×C4×C5C2×Dic311yescharacteristic
120C22×C4×C5C2×Dic311yescharacteristic
121C22×C4×C5#6711yescharacteristic
122C22×C4×C5#6711yescharacteristic
123C22×C4×C5#6711yescharacteristic
124C22×C4×C5#6711yescharacteristic
125C22×Dic5—62no—
126C22×Dic5C2×Dic312yes—
127C22×Dic5C2×Dic312yes—
128C2×Dic10—21no—
129C2×Dic10—21no—
130C2×Dic10—42no—
131C5×C22⋊C4—34no—
132C5×C22⋊C4—62no—
133C42×C5—122no—
134C22⋊Dic5—22no—
135C22⋊Dic5—22no—
136C22⋊Dic5—64no—
137C4⋊Dic5—21no—
138C4⋊Dic5—21no—
139C4⋊Dic5—122no—
140#683—22no—
141#683—22no—
142C2×C22⋊Q8—51no—
143#380—151no—
144C4×C22⋊C4—31no—
145#342—154no—
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5—21no—
152C22×C3×C5—42no—
153C22×C3×C5—42no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C22×C3×C5C4×Q812yes—
156C22×C3×C5C22⋊Q812yes—
157C22×C3×C5#12411yescharacteristic
158C22×C3×C5C4⋊Q811yescharacteristic
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162C3×C4×C5#9912yes—
163Dic15—82no—
164C3×Dic5—42no—
165C3×Dic5—42no—
166C5×Dic3—44no—
167C24×C3C22×D511yescharacteristic
168C2×C3×Q8—101no—
169C2×C3×Q8—101no—
170C2×C3×Q8—202no—
171C22×C3×C4—42no—
172C22×C3×C4—52no—
173C22×C3×C4—52no—
174C22×C3×C4C4×D511yescharacteristic
175C22×C3×C4C4×D511yescharacteristic
176C22×C3×C4D2011yescharacteristic
177C22×C3×C4D2011yescharacteristic
178C22×C3×C4C22⋊D511yescharacteristic
179C22×C3×C4C22⋊D511yescharacteristic
180C22×Dic3—22no—
181C22×Dic3—102no—
182C3×C4⋊C4—102no—
183C3×C4⋊C4—102no—
184C3×C4⋊C4—101no—
185C3×C4⋊C4—101no—
186C3×C22⋊C4—102no—
187C3×C22⋊C4—102no—
188#217—22no—
189#217—104no—
190#217Dic1014yescomplemented
191C4⋊Dic3—202no—
192C4×Dic3—42no—
193C23×C5—41no—
194C23×C5—41no—
195C23×C5D3×Q812yes—
196C23×C5#23912yes—
197C23×C5C22×Dic311yescharacteristic
198C23×C5C2×#6711yescharacteristic
199C23×C5C2×#6711yescharacteristic
200C2×C4×C5—22no—
201C2×C4×C5—22no—
202C2×C4×C5—22no—
203C2×C4×C5—41no—
204C2×C4×C5—41no—
205C2×C4×C5—41no—
206C2×C4×C5—64no—
207C2×C4×C5—122no—
208C2×C4×C5—122no—
209C2×C4×C5#21711yescharacteristic
210C2×C4×C5#21711yescharacteristic
211C2×C4×C5#21711yescharacteristic
212C2×C4×C5#21711yescharacteristic
213C2×Dic5—22no—
214C2×Dic5—22no—
215C2×Dic5—42no—
216C2×Dic5—42no—
217C2×Dic5—122no—
218C2×Dic5—122no—
219C2×Dic5—122no—
220Dic10—42no—
221Dic10—42no—
222C22×Q8—51no—
223C23×C4D3×D511yescharacteristic
224C22⋊Q8—101no—
225C22⋊Q8—101no—
226C22⋊Q8—101no—
227C22⋊Q8—101no—
228C2×C4⋊C4—52no—
229C2×C4⋊C4—51no—
230C2×C4⋊C4—301no—
231C2×C22⋊C4—32no—
232C2×C22⋊C4—52no—
233C2×C22⋊C4—152no—
234C2×C42—61no—
235#95—61no—
236#95—301no—
237C2×C3×C5—22no—
238C2×C3×C5—21no—
239C2×C3×C5—22no—
240C2×C3×C5—21no—
241C2×C3×C5#39111yescharacteristic
242C2×C3×C5C2×#9911yescharacteristic
243C2×C3×C5#41011yescharacteristic
244C23×C3—41no—
245C23×C3—41no—
246C23×C3C2×Dic1012yes—
247C23×C3C2×C4×D511yescharacteristic
248C23×C3C2×D2011yescharacteristic
249C23×C3#70912yes—
250C23×C3C2×C22⋊D511yescharacteristic
251C3×Q8—202no—
252C3×Q8—202no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—102no—
260C2×C3×C4—102no—
261C2×C3×C4—202no—
262C2×C3×C4—202no—
263C2×C3×C4#68511yescharacteristic
264C2×C3×C4#68511yescharacteristic
265C2×C3×C4#68511yescharacteristic
266C2×C3×C4#68511yescharacteristic
267C2×Dic3—24no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—202no—
271C2×Dic3—202no—
272C2×Dic3—202no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5—21no—
276C22×C5—21no—
277C22×C5—21no—
278C22×C5—21no—
279C22×C5—42no—
280C22×C5—42no—
281C22×C5#94411yescharacteristic
282C22×C5Q8×Dic312yes—
283C22×C5#94612yes—
284C22×C5#94711yescharacteristic
285C22×C5C2×#21711yescharacteristic
286C4×C5—21no—
287C4×C5—21no—
288C4×C5—21no—
289C4×C5—124no—
290C4×C5#83412yes—
291Dic5—42no—
292Dic5—42no—
293Dic5—242no—
294C24C2×D3×D511yescharacteristic
295C2×Q8—101no—
296C2×Q8—101no—
297C2×Q8—202no—
298C22×C4—42no—
299C22×C4—52no—
300C22×C4—52no—
301C22×C4—62no—
302C22×C4—302no—
303C22×C4D5×Dic311yescharacteristic
304C22×C4D5×Dic311yescharacteristic
305C22×C4#120111yescharacteristic
306C22×C4#120111yescharacteristic
307C22×C4#120211yescharacteristic
308C22×C4#120211yescharacteristic
309C4⋊C4—102no—
310C4⋊C4—102no—
311C4⋊C4—101no—
312C4⋊C4—101no—
313C4⋊C4—602no—
314C22⋊C4—34no—
315C22⋊C4—62no—
316C22⋊C4—102no—
317C22⋊C4—102no—
318C22⋊C4—304no—
319C42—122no—
320C3×C5#199611yescharacteristic, complemented, Hall
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3C4×Dic1012yes—
330C22×C3#401511yescharacteristic
331C22×C3#402111yescharacteristic
332C22×C3#407012yes—
333C22×C3C2×#68511yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—202no—
338C3×C4—202no—
339C3×C4#393812yes—
340Dic3—44no—
341Dic3—402no—
342C2×C5—21no—
343C2×C5—22no—
344C2×C5—21no—
345C2×C5—22no—
346C2×C5C2×#83411yescharacteristic
347C2×C5#522911yescharacteristic
348C2×C5#524011yescharacteristic
349C23—41no—
350C23—41no—
351C23D3×Dic1012yes—
352C23#676912yes—
353C23C2×D5×Dic311yescharacteristic
354C23C2×#120111yescharacteristic
355C23C2×#120211yescharacteristic
356Q8—202no—
357Q8—202no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—64no—
365C2×C4—102no—
366C2×C4—102no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—202no—
370C2×C4—202no—
371C2×C4—602no—
372C2×C4—602no—
373C2×C4—602no—
374C2×C4#666511yescharacteristic
375C2×C4#666511yescharacteristic
376C2×C4#666511yescharacteristic
377C2×C4#666511yescharacteristic
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—21no—
382C2×C3#6521811yescharacteristic
383C2×C3C2×#393811yescharacteristic
384C2×C3#6529511yescharacteristic
385C5#7170111yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22Dic3×Dic1012yes—
395C22#9136912yes—
396C22#9137411yescharacteristic
397C22#9138911yescharacteristic
398C22C2×#666511yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—124no—
403C4—202no—
404C4—202no—
405C4—1202no—
406C4#9099512yes—
407C3#1060025011yescharacteristic, complemented, Sylow
408C2—22no—
409C2—21no—
410C2—22no—
411C2—21no—
412C2#1174860911yescharacteristic
413C2#1174862411yescharacteristic
414C2C2×#9099511yescharacteristic
subgroupG/H
centerC22C2×#6665
commutatorC23×C3×C5C22×C4
FrattiniC24C2×D3×D5
FittingC23×C3×C4×C5C22
socleC22×C3×C5C2×C22⋊C4
prime radical, cosocleC24×C3×C5C23
hypercenterC23×C4D3×D5
hypocenterC3×C5#1996
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
7. C22×C3×Dic10C4semidirect
13. C2×C3×C4⋊Dic5C4semidirect
18. C2×C3×#683C4semidirect
44. C5×#217Q8semidirect
57. C22×Dic10Dic3semidirect
63. C2×C4⋊Dic5Dic3semidirect
68. C2×#683Dic3semidirect
190. #217Dic10semidirect
320. C3×C5#1996semidirect
385. C5#71701semidirect
407. C3#10600250semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40702
#117513132
C4×C5×#2172
#117460192
#106002503
#717015