Finite Group Databasebeta

← #14302320

Subgroups — #14302320

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×#91838C211yesmaximal, characteristic
2#11756578C211yesmaximal, characteristic, complemented
3C3×C5×#421C211yesmaximal, characteristic, complemented
4C2×#91107C211yesmaximal, characteristic, complemented
5#11751102C211yesmaximal, characteristic, complemented
6#11746381C211yesmaximal, characteristic, complemented
7#11746354C211yesmaximal, characteristic, complemented
8#10601193—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×D60C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13#91838—22no—
14#91838—22no—
15#91837—22no—
16#91837—22no—
17C2×#6717C412yescomplemented
18C2×#6715C412yescomplemented
19C2×#6710—22no—
20C2×#6710C2211yescharacteristic
21C2×#6710C2211yescharacteristic
22C2×#6710C2211yescharacteristic
23C3×C5×#104—22no—
24#91107—22no—
25#91093—22no—
26#70400—51nomaximal, Hall
27C2×#4095—31no—
28#66153—31no—
29C5×#421D311yescharacteristic, complemented
30C2×#3955—31no—
31#65290—31no—
32#64780—31no—
33#64753—31no—
34C23×D15—21no—
35C3×C5×#41—21no—
36C3×C5×#41—21no—
37C3×C5×#41D412yes—
38C3×C5×#41D412yes—
39C2×C3×C5×Q8C2×C411yescharacteristic
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4C2×C411yescharacteristic
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C2×D60—21no—
45C2×D60—21no—
46C2×D60—21no—
47C2×D60—21no—
48C2×D60—21no—
49C2×D60C2×C412yes—
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C2×C3×C5×C8—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C42×C5—21no—
56#6717—24no—
57#6715—24no—
58#6710—24no—
59#6710—21no—
60#6710—21no—
61#6710—21no—
62#6710—21no—
63#6710—21no—
64#6710D412yes—
65C2×C3⋊#168—24no—
66C2×C3⋊#168—21no—
67C2×C3⋊#168C2×C412yes—
68C2×C3⋊#168D412yes—
69C2×#949—51no—
70#5819—51no—
71C3×#421D511yescharacteristic, complemented
72C2×#823—51no—
73#5001—51no—
74C3⋊#338—51no—
75#4533—51no—
76C2×C5×#41C2×D311yescharacteristic
77C22×D20—31no—
78C2×C5×M16C2×D311yescharacteristic
79C5×#117C2×D311yescharacteristic
80#4095—62no—
81#4095—62no—
82#4094—62no—
83#4094—62no—
84C2×#688—32no—
85C2×#686—32no—
86C2×#681—31no—
87C2×#681—31no—
88C2×#681—31no—
89C2×#681—62no—
90C5×#104—22no—
91#3955—62no—
92#3941—62no—
93#1903—151noSylow
94C23×C3×C5—21no—
95C23×C3×C5#4111yescharacteristic
96C22×D15—22no—
97C22×D15—41no—
98C22×D15—41no—
99C22×D15—82no—
100C3×C5×Q8—21no—
101C3×C5×Q8C22⋊C412yes—
102C3×C5×D4—22no—
103C3×C5×D4—22no—
104C3×C5×D4—21no—
105C3×C5×D4C22⋊C412yes—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5#4111yescharacteristic
116D60—22no—
117D60—22no—
118D60—41no—
119D60—41no—
120C3×C5×C8—41no—
121C3⋊#168—22no—
122C3⋊#168—22no—
123C3⋊#168—22no—
124C2×C3×#41C2×D511yescharacteristic
125C22×D12—51no—
126C2×C3×M16C2×D511yescharacteristic
127C3×#117C2×D511yescharacteristic
128#949—102no—
129#949—102no—
130#948—102no—
131#948—102no—
132C2×#215—52no—
133C2×#213—52no—
134C2×#208—51no—
135C2×#208—51no—
136C2×#208—51no—
137C2×#208—102no—
138C3×#104—22no—
139#823—102no—
140#809—102no—
141C23×D5—61no—
142C5×#41—21no—
143C5×#41—21no—
144C5×#41D1212yes—
145C5×#41#6712yes—
146C2×C5×Q8C4×D311yescharacteristic
147C2×C5×D4—21no—
148C2×C5×D4C4×D311yescharacteristic
149C22×C4×C5—21no—
150C22×C4×C5C22×D311yescharacteristic
151C2×D20—32no—
152C2×D20—61no—
153C2×D20—61no—
154C2×D20—61no—
155C2×D20—61no—
156C2×D20—61no—
157C5×M16—21no—
158C5×M16—21no—
159C2×C5×C8—21no—
160C5×C4⋊C4—21no—
161C5×C22⋊C4—21no—
162C42×C5—21no—
163#688—64no—
164#686—64no—
165#681—32no—
166#681—64no—
167#681—61no—
168#681—61no—
169#681—61no—
170#681—61no—
171#681—61no—
172C2×#168—32no—
173C2×#168—32no—
174C2×#168—64no—
175C2×#168—61no—
176C2×AGL(1,ℤ/8)—151no—
177#568—151no—
178#421D1511yescharacteristic, complemented
179#418—151no—
180C2×#100—151no—
181#344—151no—
182#338—151no—
183C22×C3×C5—22no—
184C22×C3×C5—41no—
185C22×C3×C5—81no—
186C22×C3×C5C4×D411yescharacteristic
187C22×C3×C5#12111yescharacteristic
188C22×C3×C5#12311yescharacteristic
189C2×D15—42no—
190C2×D15—82no—
191C2×D15—82no—
192C2×D15—82no—
193C3×C4×C5—22no—
194C3×C4×C5—81no—
195C3×C4×C5C2×C22⋊C411yescharacteristic
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12111yescharacteristic
199C23×D3—101no—
200C3×#41—21no—
201C3×#41—21no—
202C3×#41D2012yes—
203C3×#41C22⋊D512yes—
204C2×C3×Q8C4×D511yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4C4×D511yescharacteristic
207C22×C3×C4—21no—
208C22×C3×C4C22×D511yescharacteristic
209C2×D12—52no—
210C2×D12—101no—
211C2×D12—101no—
212C2×D12—101no—
213C2×D12—101no—
214C2×D12—101no—
215C3×M16—21no—
216C3×M16—21no—
217C2×C3×C8—21no—
218C3×C4⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C42—21no—
221#215—104no—
222#213—104no—
223#208—52no—
224#208—104no—
225#208—101no—
226#208—101no—
227#208—101no—
228#208—101no—
229#208—101no—
230C2×C3⋊C8—52no—
231C2×C3⋊C8—52no—
232C2×C3⋊C8—104no—
233C2×C3⋊C8—101no—
234C23×C5—21no—
235C23×C5#23711yescharacteristic
236C22×D5—62no—
237C22×D5—121no—
238C22×D5—121no—
239C22×D5—242no—
240C5×Q8—21no—
241C5×Q8(C2×D3)⋊C412yes—
242C5×D4—22no—
243C5×D4—22no—
244C5×D4—21no—
245C5×D4(C2×D3)⋊C412yes—
246C2×C4×C5—21no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—41no—
250C2×C4×C5C2×C4×D311yescharacteristic
251C2×C4×C5C2×D1211yescharacteristic
252C2×C4×C5D3×D411yescharacteristic
253C2×C4×C5D3×D411yescharacteristic
254C2×C4×C5#23711yescharacteristic
255C2×C4×C5C2×#6711yescharacteristic
256D20—62no—
257D20—62no—
258D20—121no—
259D20—121no—
260C5×C8—41no—
261#168—62no—
262#168—62no—
263#168—62no—
264C2×#41C2×D1511yescharacteristic
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—151no—
275C2×M16—302no—
276C2×M16C2×D1511yescharacteristic
277#117C2×D1511yescharacteristic
278#104—22no—
279#102—302no—
280#100—302no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5—21no—
284C2×C3×C5—42no—
285C2×C3×C5#38611yescharacteristic
286D15—82no—
287C23×C3—21no—
288C23×C3#71111yescharacteristic
289C22×D3—102no—
290C22×D3—201no—
291C22×D3—201no—
292C22×D3—402no—
293C3×Q8—21no—
294C3×Q8#68512yes—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—21no—
298C3×D4#68512yes—
299C2×C3×C4—21no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—41no—
303C2×C3×C4C2×C4×D511yescharacteristic
304C2×C3×C4C2×D2011yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4D4×D511yescharacteristic
307C2×C3×C4#71111yescharacteristic
308C2×C3×C4C2×C22⋊D511yescharacteristic
309D12—102no—
310D12—102no—
311D12—201no—
312D12—201no—
313C3×C8—41no—
314C3⋊C8—102no—
315C3⋊C8—102no—
316C3⋊C8—102no—
317C22×C5—22no—
318C22×C5—41no—
319C22×C5—81no—
320C22×C5#67⋊C411yescharacteristic
321C22×C5#88611yescharacteristic
322C22×C5#93911yescharacteristic
323C2×D5—122no—
324C2×D5—242no—
325C2×D5—242no—
326C2×D5—242no—
327C4×C5—22no—
328C4×C5—81no—
329C4×C5#67⋊C411yescharacteristic
330C4×C5#88211yescharacteristic
331C4×C5C2×(C2×D3)⋊C411yescharacteristic
332C4×C5#93911yescharacteristic
333C24—301no—
334#41—21no—
335#41—21no—
336#41D6012yes—
337#41#122012yes—
338C2×Q8C4×D1511yescharacteristic
339C2×D4—21no—
340C2×D4—152no—
341C2×D4—301no—
342C2×D4—301no—
343C2×D4—301no—
344C2×D4—301no—
345C2×D4—301no—
346C2×D4C4×D1511yescharacteristic
347C22×C4—21no—
348C22×C4C22×D1511yescharacteristic
349SD16—304no—
350D8—304no—
351M16—21no—
352M16—21no—
353M16—152no—
354M16—304no—
355M16—301no—
356M16—301no—
357M16—301no—
358M16—301no—
359M16—301no—
360C2×C8—21no—
361C2×C8—152no—
362C2×C8—152no—
363C2×C8—304no—
364C2×C8—301no—
365C4⋊C4—21no—
366C22⋊C4—21no—
367C42—21no—
368C3×C5#190311yescharacteristic, complemented, Hall
369C22×C3—22no—
370C22×C3—41no—
371C22×C3—81no—
372C22×C3#402711yescharacteristic
373C22×C3#403211yescharacteristic
374C22×C3#408511yescharacteristic
375C2×D3—202no—
376C2×D3—402no—
377C2×D3—402no—
378C2×D3—402no—
379C3×C4—22no—
380C3×C4—81no—
381C3×C4#402711yescharacteristic
382C3×C4#402811yescharacteristic
383C3×C4C2×#68511yescharacteristic
384C3×C4#408511yescharacteristic
385C2×C5—21no—
386C2×C5—21no—
387C2×C5—21no—
388C2×C5—42no—
389C2×C5#495811yescharacteristic
390D5—242no—
391C23—21no—
392C23—302no—
393C23—601no—
394C23—601no—
395C23—1202no—
396C23#681911yescharacteristic
397Q8—21no—
398Q8#671412yes—
399D4—22no—
400D4—22no—
401D4—21no—
402D4—302no—
403D4—302no—
404D4—601no—
405D4—601no—
406D4#671412yes—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4C2×C4×D1511yescharacteristic
412C2×C4C2×D6011yescharacteristic
413C2×C4D4×D1511yescharacteristic
414C2×C4D4×D1511yescharacteristic
415C2×C4#681911yescharacteristic
416C2×C4C2×#122011yescharacteristic
417C8—41no—
418C8—302no—
419C8—302no—
420C8—302no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—42no—
425C2×C3#6524711yescharacteristic
426D3—402no—
427C5#7040011yescharacteristic, complemented, Sylow
428C22—22no—
429C22—41no—
430C22—81no—
431C22—602no—
432C22—1202no—
433C22—1202no—
434C22—1202no—
435C22#9177011yescharacteristic
436C22#9177511yescharacteristic
437C22#9182811yescharacteristic
438C4—22no—
439C4—81no—
440C4#9177011yescharacteristic
441C4#9177111yescharacteristic
442C4C2×#671411yescharacteristic
443C4#9182811yescharacteristic
444C3#1060119311yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—21no—
448C2—42no—
449C2—1202no—
450C2#1175105911yescharacteristic
subgroupG/H
centerC2#11751059
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#421C2
socleC2×C3×C5#386
prime radicalC2×#91838C2
cosocleC22×C3×C4×C5C23
hypercenter#421D15
hypocenterC3×C5#1903
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #11756578C2semidirect
3. C3×C5×#421C2semidirect
4. C2×#91107C2semidirect
5. #11751102C2semidirect
6. #11746381C2semidirect
7. #11746354C2semidirect
17. C2×#6717C4semidirect
18. C2×#6715C4semidirect
29. C5×#421D3semidirect
71. C3×#421D5semidirect
178. #421D15semidirect
368. C3×C5#1903semidirect
427. C5#70400semidirect
444. C3#10601193semidirect

Maximal subgroups

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

subgroupindex
C2×#918382
#117565782
C3×C5×#4212
C2×#911072
#117511022
#117463812
#117463542
#106011933
#704005