Finite Group Databasebeta

← #14301590

Subgroups — #14301590

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
1C5×#5516C211yesmaximal, characteristic, complemented
2C2×#91375C211yesmaximal, characteristic
3#11751102C211yesmaximal, characteristic
4#11748396C211yesmaximal, characteristic
5C2×#90962C211yesmaximal, characteristic, complemented
6C3×#64752C211yesmaximal, characteristic, complemented
7#11745977C211yesmaximal, characteristic, complemented
8#10602454—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×C5×Dic6C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C5×#887—22no—
15C5×#877—22no—
16C4×C5×Dic6—22no—
17C2×C3×C4×Dic5C2211yescharacteristic
18C2×C3×#681C2211yescharacteristic
19C2×#6667—22no—
20#91375—22no—
21#91375C414yescomplemented
22#91093—22no—
23#90969—22no—
24#90962—22no—
25#68766—51nomaximal, Hall
26C5×#519—31no—
27C2×#4093—31no—
28C2×#3968—31no—
29#65290—31no—
30#65290—31no—
31#64752—31no—
32#64752D311yescharacteristic, complemented
33C23×D15—21no—
34C22×C3×C4×C5C2311yescharacteristic
35C2×D60—21no—
36C2×D60—21no—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60—21no—
40C2×D60C2×C412yes—
41C22×C5×Dic3D412yes—
42C2×C5×Dic6—21no—
43C2×C5×Dic6—21no—
44C2×C5×Dic6—21no—
45C2×C5×Dic6—21no—
46C2×C5×Dic6C2×C412yes—
47C2×C5×Dic6D412yes—
48C22×C3×Dic5—21no—
49C3×C5×C4⋊C4D412yes—
50C3×C5×C22⋊C4—21no—
51C3×C42×C5—21no—
52#6710—21no—
53#6710—21no—
54C2×C3⋊#168—21no—
55C5×#217—21no—
56C5×C4⋊Dic3D412yes—
57C5×Dic3⋊C4—24no—
58C4×C5×Dic3—24no—
59C4×C5×Dic3—21no—
60C3×C4×Dic5—21no—
61C3×C4×Dic5C2×C412yes—
62C3×#681—21no—
63C3×#681—21no—
64C2×C3×#168—21no—
65#6667—28no—
66#5516D511yescharacteristic, complemented
67C2×#875—51no—
68C2×#824—51no—
69#5131—51no—
70#5001—51no—
71C3×#339—51no—
72#4523—51no—
73C22×C5×Q8—31no—
74C22×D20—31no—
75C4×C5×Q8—62no—
76C4×C5×Q8—62no—
77C5×#117—31no—
78C5×#117—62no—
79C5×#117C2×D311yescharacteristic
80#4093—34no—
81#4093—62no—
82C2×#685—62no—
83C2×C4×Dic5C2×D311yescharacteristic
84C2×#681—31no—
85C2×#681C2×D311yescharacteristic
86#3968—62no—
87#3941—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×C5C22×C411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5#4111yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102D60—22no—
103D60—22no—
104D60—41no—
105D60—41no—
106C2×C5×Dic3—22no—
107C2×C5×Dic3—22no—
108C2×C5×Dic3—22no—
109C2×C5×Dic3C22⋊C414yes—
110C5×Dic6—22no—
111C5×Dic6—22no—
112C5×Dic6—22no—
113C5×Dic6—22no—
114C2×C3×Dic5—22no—
115C2×C3×Dic5—42no—
116C3⋊#168—41no—
117C3×#168—41no—
118C22×D12—51no—
119C22×Dic6C2×D511yescharacteristic
120C2×C3×M16—51no—
121C3×#117C2×D511yescharacteristic
122C2×C3×C42—51no—
123C2×(C2×D3)⋊C4—102no—
124#926C2×D511yescharacteristic
125C2×#208—51no—
126#887—22no—
127#877—22no—
128#875—54no—
129#875—102no—
130C4×Dic6—22no—
131#824—102no—
132#821—102no—
133#809—102no—
134C23×D5—61no—
135C2×C5×Q8—32no—
136C2×C5×Q8—32no—
137C2×C5×Q8—61no—
138C2×C5×Q8—61no—
139C2×C5×Q8—61no—
140C2×C5×Q8—61no—
141C22×C4×C5—32no—
142C22×C4×C5C22×D311yescharacteristic
143C22×Dic5—21no—
144C2×D20—32no—
145C2×D20—61no—
146C2×D20—61no—
147C2×D20—61no—
148C2×D20—61no—
149C2×D20—61no—
150C5×C4⋊C4—32no—
151C5×C4⋊C4—64no—
152C5×C4⋊C4D1212yes—
153C5×C22⋊C4—21no—
154C5×C22⋊C4—61no—
155C42×C5—21no—
156C42×C5—64no—
157C42×C5—61no—
158#685—68no—
159C4×Dic5—21no—
160C4×Dic5C4×D312yes—
161#681—21no—
162#681—21no—
163#681—61no—
164#681—61no—
165C2×#168—21no—
166C2×#168—61no—
167C2×#124—151no—
168#519—31no—
169#418—151no—
170#418—151no—
171C2×#101—151no—
172#339—51no—
173#339—151no—
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
187C3×Dic5—42no—
188C5×Dic3—24no—
189C5×Dic3—42no—
190C23×D3—101no—
191C22×C3×C4—101no—
192C22×C3×C4C22×D511yescharacteristic
193C22×Dic3C22⋊D512yes—
194C2×D12—52no—
195C2×D12—101no—
196C2×D12—101no—
197C2×D12—101no—
198C2×D12—101no—
199C2×D12—101no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6C4×D512yes—
205C2×Dic6D2012yes—
206C3×M16—101no—
207C3×M16—101no—
208C2×C3×C8—101no—
209C3×C4⋊C4C22⋊D512yes—
210C3×C22⋊C4—21no—
211C3×C42—21no—
212C3×C42—52no—
213C3×C42—101no—
214#217—21no—
215(C2×D3)⋊C4—108no—
216C4⋊Dic3C22⋊D512yes—
217Dic3⋊C4—24no—
218C4×Dic3—24no—
219C4×Dic3—21no—
220#208—101no—
221#208—101no—
222C2×C3⋊C8—101no—
223C23×C5D3×D411yescharacteristic
224C22×D5—62no—
225C22×D5—124no—
226C22×D5—121no—
227C22×D5—121no—
228C5×Q8—62no—
229C5×Q8—62no—
230C5×Q8—62no—
231C5×Q8—62no—
232C2×C4×C5—22no—
233C2×C4×C5—34no—
234C2×C4×C5—62no—
235C2×C4×C5—62no—
236C2×C4×C5—62no—
237C2×C4×C5C2×C4×D311yescharacteristic
238C2×C4×C5C2×D1211yescharacteristic
239C2×C4×C5#23511yescharacteristic
240C2×C4×C5D3×D411yescharacteristic
241C2×C4×C5D3×D411yescharacteristic
242C2×C4×C5#23911yescharacteristic
243C2×Dic5—22no—
244C2×Dic5—42no—
245D20—62no—
246D20—62no—
247D20—121no—
248D20—121no—
249#168—41no—
250#168—121no—
251C22×Q8—31no—
252C22×D4—151no—
253C2×M16—51no—
254C2×M16—151no—
255#124—154no—
256#124—302no—
257C4×Q8—62no—
258C4×Q8—62no—
259#117—31no—
260#117—62no—
261#117D3×D511yescharacteristic
262C2×C22⋊C4—302no—
263C2×C42—51no—
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×C3C2×C22⋊D511yescharacteristic
273C22×D3—102no—
274C22×D3—204no—
275C22×D3—201no—
276C22×D3—201no—
277C2×C3×C4—22no—
278C2×C3×C4—102no—
279C2×C3×C4—202no—
280C2×C3×C4C2×C4×D511yescharacteristic
281C2×C3×C4C2×D2011yescharacteristic
282C2×C3×C4#70911yescharacteristic
283C2×C3×C4#70911yescharacteristic
284C2×C3×C4C2×C22⋊D511yescharacteristic
285C2×C3×C4C2×C22⋊D511yescharacteristic
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3#68514yes—
290D12—102no—
291D12—102no—
292D12—201no—
293D12—201no—
294Dic6—22no—
295Dic6—22no—
296Dic6—22no—
297Dic6—22no—
298C3×C8—201no—
299C3⋊C8—201no—
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—64no—
310C4×C5—122no—
311C4×C5C4×D1211yescharacteristic
312C4×C5#88411yescharacteristic
313C4×C5#89311yescharacteristic
314C4×C5#89511yescharacteristic
315Dic5—42no—
316C24—301no—
317C2×Q8—32no—
318C2×Q8—32no—
319C2×Q8—61no—
320C2×Q8—61no—
321C2×Q8—61no—
322C2×Q8—61no—
323C2×D4—152no—
324C2×D4—301no—
325C2×D4—301no—
326C2×D4—301no—
327C2×D4—301no—
328C2×D4—301no—
329C22×C4—32no—
330C22×C4—101no—
331C22×C4C2×D3×D511yescharacteristic
332M16—101no—
333M16—101no—
334M16—301no—
335M16—301no—
336C2×C8—101no—
337C2×C8—301no—
338C4⋊C4—32no—
339C4⋊C4—64no—
340C4⋊C4#120312yes—
341C22⋊C4—21no—
342C22⋊C4—61no—
343C22⋊C4—308no—
344C42—21no—
345C42—52no—
346C42—64no—
347C42—61no—
348C42—101no—
349C3×C5#188811yescharacteristic, complemented, Hall
350C22×C3—22no—
351C22×C3C2×#68511yescharacteristic
352C22×C3#407511yescharacteristic
353C22×C3#409911yescharacteristic
354C2×D3—204no—
355C2×D3—202no—
356C2×D3—402no—
357C2×D3—402no—
358C3×C4—42no—
359C3×C4—202no—
360C3×C4C4×C22⋊D511yescharacteristic
361C3×C4C4×C22⋊D511yescharacteristic
362C3×C4#407611yescharacteristic
363C3×C4#407611yescharacteristic
364Dic3—24no—
365Dic3—42no—
366C2×C5—21no—
367C2×C5—21no—
368C2×C5#466511yescharacteristic
369C2×C5#482712yes—
370D5—242no—
371C23—302no—
372C23—604no—
373C23—601no—
374C23—601no—
375C23D3×C22⋊D511yescharacteristic
376Q8—62no—
377Q8—62no—
378Q8—62no—
379Q8—62no—
380D4—302no—
381D4—302no—
382D4—601no—
383D4—601no—
384C2×C4—22no—
385C2×C4—34no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—102no—
390C2×C4—202no—
391C2×C4#676911yescharacteristic
392C2×C4#677111yescharacteristic
393C2×C4C4×D3×D511yescharacteristic
394C2×C4D3×D2011yescharacteristic
395C2×C4C2×#120311yescharacteristic
396C2×C4D3×C22⋊D511yescharacteristic
397C8—201no—
398C8—601no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3#6550211yescharacteristic
402C2×C3#6553412yes—
403D3—402no—
404C5#6876611yescharacteristic, complemented, Sylow
405C22—22no—
406C22—604no—
407C22—602no—
408C22—1202no—
409C22—1202no—
410C22#9135111yescharacteristic
411C22D3×#68511yescharacteristic
412C22#9157011yescharacteristic
413C4—42no—
414C4—64no—
415C4—122no—
416C4—202no—
417C4C4×#120311yescharacteristic
418C4#9144611yescharacteristic
419C4#9145411yescharacteristic
420C4#9146011yescharacteristic
421C3#1060245411yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—1202no—
425C2#1174891411yescharacteristic
426C2#1174945912yes—
subgroupG/H
centerC22#91570
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5#120
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1888
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5516C2semidirect
5. C2×#90962C2semidirect
6. C3×#64752C2semidirect
7. #11745977C2semidirect
21. #91375C4semidirect
32. #64752D3semidirect
66. #5516D5semidirect
349. C3×C5#1888semidirect
404. C5#68766semidirect
421. C3#10602454semidirect

Maximal subgroups

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

subgroupindex
C5×#55162
C2×#913752
#117511022
#117483962
C2×#909622
C3×#647522
#117459772
#106024543
#687665