Finite Group Databasebeta

← #14302323

Subgroups — #14302323

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91797C211yesmaximal, characteristic
2#11756504C211yesmaximal, characteristic, complemented
3C2×#91107C211yesmaximal, characteristic, complemented
4#11751106C211yesmaximal, characteristic, complemented
5#11751102C211yesmaximal, characteristic, complemented
6C3×C5×#344C211yesmaximal, characteristic, complemented
7#11746381C211yesmaximal, characteristic, complemented
8#10599831—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×D60C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C5×M16C2211yescharacteristic
13C3×C5×#117C2211yescharacteristic
14C2×#6710C2211yescharacteristic
15C2×#6710C2211yescharacteristic
16#91797—22no—
17#91797—22no—
18#91796—22no—
19M16×D15—22no—
20C2×D120C412yescomplemented
21C2×#6706C412yescomplemented
22#91790—22no—
23#91107—22no—
24#91093—22no—
25#91089—22no—
26#69038—51nomaximal, Hall
27C2×#4054—31no—
28#66079—31no—
29C2×#3955—31no—
30#65294—31no—
31#65290—31no—
32C5×#344D311yescharacteristic, complemented
33#64780—31no—
34C23×D15—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C2×#1220—21no—
37#6817—21no—
38#6817—21no—
39#6817D412yes—
40#6817D412yes—
41C2×D60—21no—
42C2×D60—21no—
43C2×D60—21no—
44C2×D60—21no—
45C2×D60—21no—
46C2×D60C2×C412yes—
47C2×D60C2×C411yescharacteristic
48C2×C4×D15—21no—
49C2×Dic30C2×C411yescharacteristic
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C3×C5×M16—21no—
53C3×C5×M16D412yes—
54C2×C3×C5×C8—21no—
55C2×C3×C5×C8C2×C412yes—
56C3×C5×C4⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C42×C5—21no—
59#6710—21no—
60#6710—21no—
61#6710—21no—
62#6710—21no—
63C2×C3⋊#168—21no—
64C2×C3⋊#168D412yes—
65D120—24no—
66#6706—24no—
67#6705—24no—
68C8×D15—24no—
69C2×#908—51no—
70#5745—51no—
71C2×#823—51no—
72#5005—51no—
73#5001—51no—
74C3×#344D511yescharacteristic, complemented
75C3⋊#338—51no—
76C2×#709—31no—
77C22×D20—31no—
78C2×C5×M16C2×D311yescharacteristic
79C2×C5×M16C2×D311yescharacteristic
80C5×#117C2×D311yescharacteristic
81C2×#681—31no—
82C2×#681—31no—
83#4054—62no—
84#4054—62no—
85#4053—62no—
86D5×M16—62no—
87C2×D40—32no—
88C2×#677—32no—
89#4047—62no—
90#3955—62no—
91#3941—62no—
92#3937—62no—
93#1903—151noSylow
94C23×C3×C5#4111yescharacteristic
95C22×D15—21no—
96C22×D15—22no—
97C22×D15—41no—
98C22×D15—41no—
99C22×D15—82no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5C22×C411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107#1220—22no—
108#1220—22no—
109C2×Dic15—21no—
110D60—22no—
111D60—21no—
112D60—22no—
113D60—41no—
114D60—41no—
115D60C22⋊C412yes—
116C4×D15—22no—
117C4×D15—22no—
118Dic30—21no—
119Dic30C22⋊C412yes—
120C3×C5×C8—22no—
121C3×C5×C8—22no—
122C3⋊#168—22no—
123C3⋊#168—41no—
124C2×#235—51no—
125C22×D12—51no—
126C2×C3×M16C2×D511yescharacteristic
127C2×C3×M16C2×D511yescharacteristic
128C3×#117C2×D511yescharacteristic
129C2×#208—51no—
130C2×#208—51no—
131#908—102no—
132#908—102no—
133#907—102no—
134D3×M16—102no—
135C2×D24—52no—
136C2×#204—52no—
137#901—102no—
138#823—102no—
139#809—102no—
140#805—102no—
141C23×D5—61no—
142C22×C4×C5C22×D311yescharacteristic
143C2×C22⋊D5—61no—
144#709—32no—
145#709—32no—
146#709—61no—
147#709—61no—
148C2×D20—31no—
149C2×D20—32no—
150C2×D20—61no—
151C2×D20—61no—
152C2×D20—61no—
153C2×D20—61no—
154C2×D20—61no—
155C2×C4×D5—61no—
156C2×Dic10—31no—
157C5×M16—21no—
158C5×M16—21no—
159C5×M16—21no—
160C5×M16D1212yes—
161C2×C5×C8—21no—
162C2×C5×C8C4×D312yes—
163C5×C4⋊C4—21no—
164C5×C22⋊C4—21no—
165C42×C5—21no—
166#681—61no—
167#681—61no—
168#681—61no—
169#681—61no—
170C2×#168—32no—
171C2×#168—61no—
172D40—64no—
173#677—64no—
174#676—64no—
175C8×D5—64no—
176C2×AGL(1,ℤ/8)—151no—
177#568—151no—
178#421—151no—
179#418—151no—
180C2×#100—151no—
181#344D1511yescharacteristic, complemented
182#338—151no—
183C22×C3×C5—41no—
184C22×C3×C5C4×D411yescharacteristic
185C22×C3×C5#12111yescharacteristic
186C22×C3×C5#12311yescharacteristic
187C2×D15—22no—
188C2×D15—42no—
189C2×D15—81no—
190C2×D15—82no—
191C2×D15—82no—
192C2×D15—82no—
193C3×C4×C5—81no—
194C3×C4×C5C2×C22⋊C411yescharacteristic
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5#12011yescharacteristic
197C3×C4×C5#12111yescharacteristic
198Dic15—22no—
199C23×D3—101no—
200C22×C3×C4C22×D511yescharacteristic
201C2×#67—101no—
202#235—52no—
203#235—52no—
204#235—101no—
205#235—101no—
206C2×D12—51no—
207C2×D12—52no—
208C2×D12—101no—
209C2×D12—101no—
210C2×D12—101no—
211C2×D12—101no—
212C2×D12—101no—
213C2×C4×D3—101no—
214C2×Dic6—51no—
215C3×M16—21no—
216C3×M16—21no—
217C3×M16—21no—
218C3×M16D2012yes—
219C2×C3×C8—21no—
220C2×C3×C8C4×D512yes—
221C3×C4⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C42—21no—
224#208—101no—
225#208—101no—
226#208—101no—
227#208—101no—
228C2×C3⋊C8—52no—
229C2×C3⋊C8—101no—
230D24—104no—
231#204—104no—
232ΓL(1,𝔽25)—104no—
233D3×C8—104no—
234C23×C5#23511yescharacteristic
235C22×D5—61no—
236C22×D5—62no—
237C22×D5—121no—
238C22×D5—121no—
239C22×D5—242no—
240C2×C4×C5—41no—
241C2×C4×C5C2×C4×D311yescharacteristic
242C2×C4×C5C2×D1211yescharacteristic
243C2×C4×C5D3×D411yescharacteristic
244C2×C4×C5D3×D411yescharacteristic
245C2×C4×C5D3×D411yescharacteristic
246C2×C4×C5#23911yescharacteristic
247C22⋊D5—62no—
248C22⋊D5—62no—
249C2×Dic5—61no—
250D20—32no—
251D20—62no—
252D20—61no—
253D20—62no—
254D20—121no—
255D20—121no—
256C4×D5—62no—
257C4×D5—62no—
258Dic10—32no—
259Dic10—61no—
260C5×C8—22no—
261C5×C8—22no—
262#168—62no—
263#168—121no—
264C2×#41—151no—
265C22×D4—151no—
266AGL(1,ℤ/8)—302no—
267AGL(1,ℤ/8)—302no—
268C2×SD16—152no—
269C2×D8—152no—
270#131—302no—
271#131—302no—
272C2×M16—151no—
273C2×M16—151no—
274C2×M16—302no—
275C2×M16C2×D1511yescharacteristic
276C2×M16C2×D1511yescharacteristic
277#117C2×D1511yescharacteristic
278#104—302no—
279#102—302no—
280#100—302no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5—21no—
284C2×C3×C5#38611yescharacteristic
285D15—42no—
286D15—82no—
287C23×C3#70911yescharacteristic
288C22×D3—101no—
289C22×D3—102no—
290C22×D3—201no—
291C22×D3—201no—
292C22×D3—402no—
293C2×C3×C4—41no—
294C2×C3×C4C2×C4×D511yescharacteristic
295C2×C3×C4C2×D2011yescharacteristic
296C2×C3×C4D4×D511yescharacteristic
297C2×C3×C4D4×D511yescharacteristic
298C2×C3×C4D4×D511yescharacteristic
299C2×C3×C4#71311yescharacteristic
300#67—102no—
301#67—102no—
302C2×Dic3—101no—
303D12—52no—
304D12—102no—
305D12—101no—
306D12—102no—
307D12—201no—
308D12—201no—
309C4×D3—102no—
310C4×D3—102no—
311Dic6—52no—
312Dic6—101no—
313C3×C8—22no—
314C3×C8—22no—
315C3⋊C8—102no—
316C3⋊C8—201no—
317C22×C5—41no—
318C22×C5C4×D1211yescharacteristic
319C22×C5#88411yescharacteristic
320C22×C5#89411yescharacteristic
321C2×D5—62no—
322C2×D5—122no—
323C2×D5—241no—
324C2×D5—242no—
325C2×D5—242no—
326C2×D5—242no—
327C4×C5—81no—
328C4×C5D3×C22⋊C411yescharacteristic
329C4×C5#88211yescharacteristic
330C4×C5#89311yescharacteristic
331C4×C5#89511yescharacteristic
332Dic5—62no—
333C24—301no—
334#41—152no—
335#41—152no—
336#41—301no—
337#41—301no—
338C2×Q8—151no—
339C2×D4—151no—
340C2×D4—152no—
341C2×D4—301no—
342C2×D4—301no—
343C2×D4—301no—
344C2×D4—301no—
345C2×D4—301no—
346C2×D4—301no—
347C22×C4—301no—
348C22×C4C22×D1511yescharacteristic
349SD16—304no—
350D8—304no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—304no—
355M16—301no—
356M16—301no—
357M16—301no—
358M16—301no—
359M16D6012yes—
360C2×C8—21no—
361C2×C8—152no—
362C2×C8—304no—
363C2×C8—301no—
364C2×C8C4×D1512yes—
365C4⋊C4—21no—
366C22⋊C4—21no—
367C42—21no—
368C3×C5#190311yescharacteristic, complemented, Hall
369C22×C3—41no—
370C22×C3C4×D2011yescharacteristic
371C22×C3#403011yescharacteristic
372C22×C3#404011yescharacteristic
373C2×D3—102no—
374C2×D3—202no—
375C2×D3—401no—
376C2×D3—402no—
377C2×D3—402no—
378C2×D3—402no—
379C3×C4—81no—
380C3×C4D5×C22⋊C411yescharacteristic
381C3×C4#402811yescharacteristic
382C3×C4#403911yescharacteristic
383C3×C4#404111yescharacteristic
384Dic3—102no—
385C2×C5—21no—
386C2×C5—21no—
387C2×C5—21no—
388C2×C5#466511yescharacteristic
389D5—122no—
390D5—242no—
391C23—301no—
392C23—302no—
393C23—601no—
394C23—601no—
395C23—1202no—
396C23#681711yescharacteristic
397Q8—152no—
398Q8—301no—
399D4—152no—
400D4—302no—
401D4—302no—
402D4—302no—
403D4—301no—
404D4—302no—
405D4—601no—
406D4—601no—
407C2×C4—41no—
408C2×C4—301no—
409C2×C4—302no—
410C2×C4—302no—
411C2×C4C2×C4×D1511yescharacteristic
412C2×C4C2×D6011yescharacteristic
413C2×C4D4×D1511yescharacteristic
414C2×C4D4×D1511yescharacteristic
415C2×C4D4×D1511yescharacteristic
416C2×C4#682111yescharacteristic
417C8—22no—
418C8—22no—
419C8—302no—
420C8—601no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3#6495411yescharacteristic
425D3—202no—
426D3—402no—
427C5#6903811yescharacteristic, complemented, Sylow
428C22—41no—
429C22—302no—
430C22—602no—
431C22—1201no—
432C22—1202no—
433C22—1202no—
434C22—1202no—
435C22C4×D6011yescharacteristic
436C22#9177311yescharacteristic
437C22#9178311yescharacteristic
438C4—81no—
439C4—302no—
440C4C22⋊C4×D1511yescharacteristic
441C4#9177111yescharacteristic
442C4#9178211yescharacteristic
443C4#9178411yescharacteristic
444C3#1059983111yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—21no—
448C2—602no—
449C2—1202no—
450C2#1175076611yescharacteristic
subgroupG/H
centerC2#11750766
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#344C2
socleC2×C3×C5#386
prime radicalC2×#91797C2
cosocleC22×C3×C4×C5C23
hypercenter#344D15
hypocenterC3×C5#1903
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #11756504C2semidirect
3. C2×#91107C2semidirect
4. #11751106C2semidirect
5. #11751102C2semidirect
6. C3×C5×#344C2semidirect
7. #11746381C2semidirect
20. C2×D120C4semidirect
21. C2×#6706C4semidirect
32. C5×#344D3semidirect
74. C3×#344D5semidirect
181. #344D15semidirect
368. C3×C5#1903semidirect
427. C5#69038semidirect
444. C3#10599831semidirect

Maximal subgroups

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

subgroupindex
C2×#917972
#117565042
C2×#911072
#117511062
#117511022
C3×C5×#3442
#117463812
#105998313
#690385