Finite Group Databasebeta

← #14301603

Subgroups — #14301603

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91764C211yesmaximal, characteristic
2#11754213C211yesmaximal, characteristic, complemented
3#11748396C211yesmaximal, characteristic
4#11748392C211yesmaximal, characteristic
5C2×#90963C211yesmaximal, characteristic, complemented
6C5×#4523C211yesmaximal, characteristic, complemented
7C3×#64743C211yesmaximal, characteristic, complemented
8#10599559—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×Dic30C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12C2×#6714—22no—
13#91764—22no—
14#91764C414yescomplemented
15C5×#926C2211yescharacteristic
16C2×C5×#208C2211yescharacteristic
17C3×#4072C2211yescharacteristic
18C2×C3×#681C2211yescharacteristic
19#91532—22no—
20#91340—22no—
21#91333—22no—
22#90969—22no—
23#90968—22no—
24#90963—22no—
25#68766—51nomaximal, Hall
26#65850—31no—
27C2×#4021—31no—
28C2×#3956—31no—
29#65420—31no—
30#65290—31no—
31C5×#339—31no—
32#64743D311yescharacteristic, complemented
33C23×D15—21no—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C22×Dic15D412yes—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60—21no—
40C2×D60—21no—
41C2×D60—21no—
42C2×D60C2×C412yes—
43C2×Dic30—21no—
44C2×Dic30—21no—
45C2×Dic30—21no—
46C2×Dic30—21no—
47C2×Dic30C2×C412yes—
48C2×Dic30D412yes—
49C3×C42×C5—21no—
50C3×C42×C5C2×C412yes—
51#6714—28no—
52C5×#217—21no—
53C5×C4⋊Dic3D412yes—
54C4×C5×Dic3—21no—
55C5×#208—21no—
56C5×#208—21no—
57C2×C5×C3⋊C8—21no—
58C3×C22⋊Dic5—21no—
59C3×C4⋊Dic5D412yes—
60C3×C4×Dic5—21no—
61C3×#681—21no—
62C3×#681—21no—
63C2×C3×#168—21no—
64#6669—24no—
65Dic3×Dic5—24no—
66#5516—51no—
67C2×#875—51no—
68C2×#824—51no—
69#5131—51no—
70#5001—51no—
71C3×#339—51no—
72#4523D511yescharacteristic, complemented
73C22×D20—31no—
74C22×Dic10—31no—
75C2×C5×M16—31no—
76C5×#117—31no—
77C2×C42×C5C2×D311yescharacteristic
78C2×#685—62no—
79#4072C2×D311yescharacteristic
80C2×#681C2×D311yescharacteristic
81#4033—62no—
82#4023—62no—
83#4021—34no—
84#4021—62no—
85C4×Dic10—62no—
86#3956—62no—
87#3953—62no—
88#3941—62no—
89#1888—151noSylow
90C23×C3×C5C2×D411yescharacteristic
91C22×D15—22no—
92C22×D15—44no—
93C22×D15—41no—
94C22×D15—41no—
95C2×C3×C4×C5—22no—
96C2×C3×C4×C5—42no—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102C2×C3×C4×C5#4111yescharacteristic
103C2×Dic15—22no—
104C2×Dic15—22no—
105C2×Dic15C22⋊C414yes—
106D60—22no—
107D60—22no—
108D60—41no—
109D60—41no—
110Dic30—22no—
111Dic30—22no—
112Dic30—22no—
113Dic30—22no—
114C2×C5×Dic3—22no—
115C2×C3×Dic5—22no—
116C3×#168—41no—
117C5×C3⋊C8—41no—
118C22×D12—51no—
119C22×Dic6—51no—
120C2×C3×M16—51no—
121C3×#117—51no—
122C2×C3×C42C2×D511yescharacteristic
123C2×(C2×D3)⋊C4—102no—
124#926C2×D511yescharacteristic
125C2×#208C2×D511yescharacteristic
126#887—102no—
127#877—102no—
128#875—54no—
129#875—102no—
130C4×Dic6—102no—
131#824—102no—
132#821—102no—
133#809—102no—
134C23×D5—61no—
135C22×C4×C5—21no—
136C22×C4×C5C22×D311yescharacteristic
137C22×Dic5—32no—
138C2×D20—32no—
139C2×D20—61no—
140C2×D20—61no—
141C2×D20—61no—
142C2×D20—61no—
143C2×D20—61no—
144C2×Dic10—32no—
145C2×Dic10—32no—
146C2×Dic10—61no—
147C2×Dic10—61no—
148C2×Dic10—61no—
149C2×Dic10—61no—
150C5×M16—61no—
151C5×M16—61no—
152C2×C5×C8—61no—
153C5×C4⋊C4—32no—
154C5×C22⋊C4—61no—
155C42×C5—21no—
156C42×C5—61no—
157C42×C5C4×D312yes—
158C22⋊Dic5—21no—
159#685—68no—
160C4⋊Dic5D1212yes—
161#683—64no—
162C4×Dic5—21no—
163C4×Dic5—64no—
164#681—21no—
165#681—21no—
166C2×#168—21no—
167C2×#124—151no—
168#519—151no—
169#418—151no—
170#418—151no—
171C2×#101—151no—
172#339—31no—
173#339—51no—
174C22×C3×C5—22no—
175C22×C3×C5C2×C22⋊C411yescharacteristic
176C22×C3×C5#12011yescharacteristic
177C22×C3×C5#12311yescharacteristic
178C2×D15—44no—
179C2×D15—42no—
180C2×D15—82no—
181C2×D15—82no—
182C3×C4×C5—42no—
183C3×C4×C5C4×D411yescharacteristic
184C3×C4×C5C4×D411yescharacteristic
185C3×C4×C5#12111yescharacteristic
186C3×C4×C5#12111yescharacteristic
187Dic15—24no—
188C3×Dic5—42no—
189C5×Dic3—42no—
190C23×D3—101no—
191C22×C3×C4—21no—
192C22×C3×C4C22×D511yescharacteristic
193C22×Dic3—52no—
194C2×D12—52no—
195C2×D12—101no—
196C2×D12—101no—
197C2×D12—101no—
198C2×D12—101no—
199C2×D12—101no—
200C2×Dic6—52no—
201C2×Dic6—52no—
202C2×Dic6—101no—
203C2×Dic6—101no—
204C2×Dic6—101no—
205C2×Dic6—101no—
206C3×M16—101no—
207C3×M16—101no—
208C2×C3×C8—101no—
209C3×C4⋊C4—52no—
210C3×C22⋊C4—101no—
211C3×C42—21no—
212C3×C42—101no—
213C3×C42C4×D512yes—
214#217—21no—
215(C2×D3)⋊C4—108no—
216C4⋊Dic3D2012yes—
217Dic3⋊C4—104no—
218C4×Dic3—21no—
219C4×Dic3—104no—
220#208—21no—
221#208—21no—
222C2×C3⋊C8—21no—
223C23×C5D3×D411yescharacteristic
224C22×D5—62no—
225C22×D5—124no—
226C22×D5—121no—
227C22×D5—121no—
228C2×C4×C5—22no—
229C2×C4×C5—42no—
230C2×C4×C5—62no—
231C2×C4×C5C2×C4×D311yescharacteristic
232C2×C4×C5C2×D1211yescharacteristic
233C2×C4×C5#23511yescharacteristic
234C2×C4×C5D3×D411yescharacteristic
235C2×C4×C5D3×D411yescharacteristic
236C2×C4×C5#23911yescharacteristic
237C2×Dic5—22no—
238C2×Dic5—34no—
239C2×Dic5—62no—
240C2×Dic5—62no—
241D20—62no—
242D20—62no—
243D20—121no—
244D20—121no—
245Dic10—62no—
246Dic10—62no—
247Dic10—62no—
248Dic10—62no—
249C5×C8—121no—
250#168—41no—
251C22×Q8—151no—
252C22×D4—151no—
253C2×M16—31no—
254C2×M16—51no—
255#124—154no—
256#124—302no—
257C4×Q8—302no—
258C4×Q8—302no—
259#117—31no—
260#117—51no—
261#117—302no—
262C2×C22⋊C4—302no—
263C2×C42D3×D511yescharacteristic
264#102—302no—
265#102—302no—
266#101—302no—
267C2×C3×C5—21no—
268C2×C3×C5—21no—
269C2×C3×C5#38611yescharacteristic
270C2×C3×C5#45412yes—
271D15—82no—
272C23×C3D4×D511yescharacteristic
273C22×D3—102no—
274C22×D3—204no—
275C22×D3—201no—
276C22×D3—201no—
277C2×C3×C4—22no—
278C2×C3×C4—42no—
279C2×C3×C4—102no—
280C2×C3×C4C2×C4×D511yescharacteristic
281C2×C3×C4C2×D2011yescharacteristic
282C2×C3×C4#70911yescharacteristic
283C2×C3×C4D4×D511yescharacteristic
284C2×C3×C4D4×D511yescharacteristic
285C2×C3×C4#71311yescharacteristic
286C2×Dic3—22no—
287C2×Dic3—54no—
288C2×Dic3—102no—
289C2×Dic3—102no—
290D12—102no—
291D12—102no—
292D12—201no—
293D12—201no—
294Dic6—102no—
295Dic6—102no—
296Dic6—102no—
297Dic6—102no—
298C3×C8—201no—
299C3⋊C8—41no—
300C22×C5—22no—
301C22×C5D3×C22⋊C411yescharacteristic
302C22×C5#88211yescharacteristic
303C22×C5#89411yescharacteristic
304C2×D5—124no—
305C2×D5—122no—
306C2×D5—242no—
307C2×D5—242no—
308C4×C5—42no—
309C4×C5—122no—
310C4×C5C4×D1211yescharacteristic
311C4×C5#88411yescharacteristic
312C4×C5#89311yescharacteristic
313C4×C5#89511yescharacteristic
314Dic5—42no—
315Dic5—64no—
316C24—301no—
317C2×Q8—152no—
318C2×Q8—152no—
319C2×Q8—301no—
320C2×Q8—301no—
321C2×Q8—301no—
322C2×Q8—301no—
323C2×D4—152no—
324C2×D4—301no—
325C2×D4—301no—
326C2×D4—301no—
327C2×D4—301no—
328C2×D4—301no—
329C22×C4—21no—
330C22×C4—152no—
331C22×C4C2×D3×D511yescharacteristic
332M16—61no—
333M16—61no—
334M16—101no—
335M16—101no—
336C2×C8—61no—
337C2×C8—101no—
338C4⋊C4—32no—
339C4⋊C4—52no—
340C4⋊C4—304no—
341C22⋊C4—61no—
342C22⋊C4—101no—
343C22⋊C4—308no—
344C42—21no—
345C42—61no—
346C42—101no—
347C42—304no—
348C42#120012yes—
349C3×C5#188811yescharacteristic, complemented, Hall
350C22×C3—22no—
351C22×C3D5×C22⋊C411yescharacteristic
352C22×C3#402811yescharacteristic
353C22×C3#404011yescharacteristic
354C2×D3—204no—
355C2×D3—202no—
356C2×D3—402no—
357C2×D3—402no—
358C3×C4—42no—
359C3×C4—202no—
360C3×C4C4×D2011yescharacteristic
361C3×C4#403011yescharacteristic
362C3×C4#403911yescharacteristic
363C3×C4#404111yescharacteristic
364Dic3—42no—
365Dic3—104no—
366C2×C5—21no—
367C2×C5—21no—
368C2×C5#466511yescharacteristic
369C2×C5#482712yes—
370D5—242no—
371C23—302no—
372C23—604no—
373C23—601no—
374C23—601no—
375C23#679011yescharacteristic
376Q8—302no—
377Q8—302no—
378Q8—302no—
379Q8—302no—
380D4—302no—
381D4—302no—
382D4—601no—
383D4—601no—
384C2×C4—22no—
385C2×C4—42no—
386C2×C4—62no—
387C2×C4—102no—
388C2×C4—154no—
389C2×C4—302no—
390C2×C4—302no—
391C2×C4#676911yescharacteristic
392C2×C4#677311yescharacteristic
393C2×C4D5×D1211yescharacteristic
394C2×C4D3×D2011yescharacteristic
395C2×C4C2×#120011yescharacteristic
396C2×C4#679011yescharacteristic
397C8—121no—
398C8—201no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3#6495411yescharacteristic
402C2×C3#6511612yes—
403D3—402no—
404C5#6876611yescharacteristic, complemented, Sylow
405C22—22no—
406C22—604no—
407C22—602no—
408C22—1202no—
409C22—1202no—
410C22#9143311yescharacteristic
411C22#9147611yescharacteristic
412C22#9156111yescharacteristic
413C4—42no—
414C4—122no—
415C4—202no—
416C4—304no—
417C4#9141811yescharacteristic
418C4#9142011yescharacteristic
419C4#9142811yescharacteristic
420C4#9145911yescharacteristic
421C3#1059955911yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—1202no—
425C2#1174712512yes—
426C2#1174890411yescharacteristic
subgroupG/H
centerC22#91476
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5#120
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#1888
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. #11754213C2semidirect
5. C2×#90963C2semidirect
6. C5×#4523C2semidirect
7. C3×#64743C2semidirect
14. #91764C4semidirect
32. #64743D3semidirect
72. #4523D5semidirect
349. C3×C5#1888semidirect
404. C5#68766semidirect
421. C3#10599559semidirect

Maximal subgroups

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

subgroupindex
C2×#917642
#117542132
#117483962
#117483922
C2×#909632
C5×#45232
C3×#647432
#105995593
#687665