Finite Group Databasebeta

← #14302775

Subgroups — #14302775

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756575C211yesmaximal, characteristic
2#11756504C211yesmaximal, characteristic
3C3×C5×#419C211yesmaximal, characteristic, complemented
4#11751326C211yesmaximal, characteristic, complemented
5#11751107C211yesmaximal, characteristic
6#11746378C211yesmaximal, characteristic, complemented
7#11746352C211yesmaximal, characteristic, complemented
8#10601143—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13#91836C412yescomplemented
14#91835—22no—
15#91833C412yescomplemented
16#91818—22no—
17#91815C2211yescharacteristic
18#91815C2211yescharacteristic
19C2×#6710C2211yescharacteristic
20#91796—22no—
21M16×D15—22no—
22#91790—22no—
23C3×C5×#103—22no—
24#91119—22no—
25#91090—22no—
26#70350—51nomaximal, Hall
27#66150—31no—
28#66079—31no—
29C5×#419D311yescharacteristic, complemented
30#65514—31no—
31#65295—31no—
32#64777—31no—
33#64751—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8C2×C412yes—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5C2311yescharacteristic
42C2×#1220D412yes—
43#6817—21no—
44#6817—21no—
45#6817—21no—
46#6817—21no—
47C2×D60C2×C411yescharacteristic
48C2×C4×D15D412yes—
49C2×Dic30C2×C411yescharacteristic
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C2×C3×C5×C8D412yes—
53C3×C5×C4⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C42×C5—21no—
56#6719—21no—
57#6719—21no—
58#6714—24no—
59C4⋊Dic15—21no—
60C4⋊Dic15—21no—
61#6712—24no—
62C4×Dic15—21no—
63C4×Dic15C2×C412yes—
64#6710—21no—
65#6710—21no—
66C2×C3⋊#168D412yes—
67#6705—24no—
68C8×D15—24no—
69#5816—51no—
70#5745—51no—
71C3×#419D511yescharacteristic, complemented
72#5225—51no—
73#5006—51no—
74#4557—51no—
75#4531—51no—
76C22×C5×Q8C2×D311yescharacteristic
77C2×#709—31no—
78C2×C5×M16C2×D311yescharacteristic
79C5×#117C2×D311yescharacteristic
80#4093—32no—
81#4092—62no—
82#4090—32no—
83#4075—62no—
84#4072—31no—
85#4072—31no—
86C2×#681—31no—
87#4053—62no—
88D5×M16—62no—
89#4047—62no—
90C5×#103—22no—
91#3967—62no—
92#3938—62no—
93#1909—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15C22⋊C412yes—
96C3×C5×Q8—22no—
97C3×C5×Q8—22no—
98C3×C5×Q8—41no—
99C3×C5×Q8—41no—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110#1220—22no—
111#1220—22no—
112C2×Dic15—22no—
113C2×Dic15—41no—
114C2×Dic15C22⋊C412yes—
115D60—21no—
116D60—21no—
117C4×D15—22no—
118C4×D15—22no—
119Dic30—21no—
120Dic30—21no—
121C3×C5×C8—22no—
122C3⋊#168—22no—
123C22×C3×Q8C2×D511yescharacteristic
124C2×#235—51no—
125C2×C3×M16C2×D511yescharacteristic
126C3×#117C2×D511yescharacteristic
127#947—52no—
128#946—102no—
129#944—52no—
130#929—102no—
131#926—51no—
132#926—51no—
133C2×#208—51no—
134#907—102no—
135D3×M16—102no—
136#901—102no—
137C3×#103—22no—
138#835—102no—
139#806—102no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8C4×D312yes—
146C22×C4×C5—21no—
147C22×C4×C5C22×D311yescharacteristic
148C2×C22⋊D5—32no—
149#709—61no—
150#709—61no—
151#709—61no—
152#709—61no—
153C2×D20—31no—
154C2×C4×D5—32no—
155C2×Dic10—31no—
156C5×M16—21no—
157C5×M16—21no—
158C2×C5×C8#6712yes—
159C5×C4⋊C4—21no—
160C5×C22⋊C4—21no—
161C42×C5—21no—
162C22⋊Dic5—61no—
163C22⋊Dic5—61no—
164#685—64no—
165C4⋊Dic5—61no—
166C4⋊Dic5—61no—
167#683—64no—
168C4×Dic5—32no—
169C4×Dic5—61no—
170#681—61no—
171#681—61no—
172C2×#168—32no—
173#676—64no—
174C8×D5—64no—
175#568—151no—
176#536—151no—
177#419—151no—
178#419D1511yescharacteristic, complemented
179#410—151no—
180H(3,ℤ/4)—151no—
181H(3,ℤ/4)—151no—
182C22×C3×C5—41no—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C22×C3×C5#12111yescharacteristic
185C22×C3×C5#12111yescharacteristic
186C2×D15—22no—
187C2×D15—42no—
188C3×C4×C5—42no—
189C3×C4×C5—81no—
190C3×C4×C5C4×D411yescharacteristic
191C3×C4×C5C4×D411yescharacteristic
192C3×C4×C5#12011yescharacteristic
193C3×C4×C5#12311yescharacteristic
194Dic15—22no—
195Dic15—42no—
196Dic15—81no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C4×D512yes—
203C22×C3×C4—21no—
204C22×C3×C4C22×D511yescharacteristic
205C2×#67—52no—
206#235—101no—
207#235—101no—
208#235—101no—
209#235—101no—
210C2×D12—51no—
211C2×C4×D3—52no—
212C2×Dic6—51no—
213C3×M16—21no—
214C3×M16—21no—
215C2×C3×C8C22⋊D512yes—
216C3×C4⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C42—21no—
219#217—101no—
220#217—101no—
221(C2×D3)⋊C4—104no—
222C4⋊Dic3—101no—
223C4⋊Dic3—101no—
224Dic3⋊C4—104no—
225C4×Dic3—52no—
226C4×Dic3—101no—
227#208—101no—
228#208—101no—
229C2×C3⋊C8—52no—
230ΓL(1,𝔽25)—104no—
231D3×C8—104no—
232C23×C5D3×D411yescharacteristic
233C22×D5—32no—
234C5×Q8—22no—
235C5×Q8—22no—
236C5×Q8—41no—
237C5×Q8—41no—
238C2×C4×C5—22no—
239C2×C4×C5—41no—
240C2×C4×C5—41no—
241C2×C4×C5—41no—
242C2×C4×C5C2×C4×D311yescharacteristic
243C2×C4×C5#23511yescharacteristic
244C2×C4×C5D3×D411yescharacteristic
245C2×C4×C5D3×D411yescharacteristic
246C2×C4×C5#23711yescharacteristic
247C2×C4×C5C2×#6711yescharacteristic
248C22⋊D5—62no—
249C22⋊D5—62no—
250C2×Dic5—32no—
251C2×Dic5—62no—
252C2×Dic5—121no—
253D20—61no—
254D20—61no—
255C4×D5—62no—
256C4×D5—62no—
257Dic10—61no—
258Dic10—61no—
259C5×C8—22no—
260#168—62no—
261C2×#41—151no—
262C22×Q8C2×D1511yescharacteristic
263#131—302no—
264#131—302no—
265C2×M16—151no—
266C2×M16—302no—
267C2×M16C2×D1511yescharacteristic
268C4⋊Q8—152no—
269#124—152no—
270#123—302no—
271C22⋊Q8—302no—
272#117—151no—
273#117—151no—
274#117C2×D1511yescharacteristic
275#103—22no—
276#103—302no—
277#99—302no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38611yescharacteristic
282D15—42no—
283C23×C3D4×D511yescharacteristic
284C22×D3—52no—
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—41no—
288C3×Q8—41no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4C2×C4×D511yescharacteristic
294C2×C3×C4#70911yescharacteristic
295C2×C3×C4D4×D511yescharacteristic
296C2×C3×C4D4×D511yescharacteristic
297C2×C3×C4#71111yescharacteristic
298C2×C3×C4C2×C22⋊D511yescharacteristic
299#67—102no—
300#67—102no—
301C2×Dic3—52no—
302C2×Dic3—102no—
303C2×Dic3—201no—
304D12—101no—
305D12—101no—
306C4×D3—102no—
307C4×D3—102no—
308Dic6—101no—
309Dic6—101no—
310C3×C8—22no—
311C3⋊C8—102no—
312C22×C5—41no—
313C22×C5D3×C22⋊C411yescharacteristic
314C22×C5#88411yescharacteristic
315C22×C5#93811yescharacteristic
316C2×D5—62no—
317C2×D5—122no—
318C4×C5—42no—
319C4×C5—81no—
320C4×C5#67⋊C411yescharacteristic
321C4×C5#88311yescharacteristic
322C4×C5C4×#6711yescharacteristic
323C4×C5#93711yescharacteristic
324Dic5—62no—
325Dic5—122no—
326Dic5—241no—
327#41—301no—
328#41—301no—
329#41—301no—
330#41—301no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—151no—
337C2×Q8C4×D1512yes—
338C2×D4—152no—
339C2×D4—151no—
340C22×C4—21no—
341C22×C4—152no—
342C22×C4C22×D1511yescharacteristic
343M16—21no—
344M16—21no—
345M16—304no—
346M16—301no—
347M16—301no—
348C2×C8—152no—
349C2×C8—304no—
350C2×C8#122012yes—
351C4⋊C4—21no—
352C4⋊C4—304no—
353C4⋊C4—301no—
354C4⋊C4—301no—
355C22⋊C4—21no—
356C22⋊C4—304no—
357C22⋊C4—301no—
358C22⋊C4—301no—
359C42—21no—
360C42—152no—
361C42—301no—
362C3×C5#190911yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3D5×C22⋊C411yescharacteristic
365C22×C3#403011yescharacteristic
366C22×C3#408411yescharacteristic
367C2×D3—102no—
368C2×D3—202no—
369C3×C4—42no—
370C3×C4—81no—
371C3×C4#402711yescharacteristic
372C3×C4#402911yescharacteristic
373C3×C4C4×C22⋊D511yescharacteristic
374C3×C4#408311yescharacteristic
375Dic3—102no—
376Dic3—202no—
377Dic3—401no—
378C2×C5—21no—
379C2×C5—21no—
380C2×C5—21no—
381C2×C5#495611yescharacteristic
382D5—122no—
383C23—152no—
384C23D4×D1511yescharacteristic
385Q8—22no—
386Q8—22no—
387Q8—41no—
388Q8—41no—
389Q8—301no—
390Q8—301no—
391D4—302no—
392D4—302no—
393D4—301no—
394D4—301no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—152no—
400C2×C4—302no—
401C2×C4—302no—
402C2×C4—302no—
403C2×C4—601no—
404C2×C4C2×C4×D1511yescharacteristic
405C2×C4#681711yescharacteristic
406C2×C4D4×D1511yescharacteristic
407C2×C4D4×D1511yescharacteristic
408C2×C4#681911yescharacteristic
409C2×C4C2×#122011yescharacteristic
410C8—22no—
411C8—302no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3#6524511yescharacteristic
416D3—202no—
417C5#7035011yescharacteristic, complemented, Sylow
418C22—41no—
419C22—302no—
420C22—602no—
421C22C22⋊C4×D1511yescharacteristic
422C22#9177311yescharacteristic
423C22#9182711yescharacteristic
424C4—42no—
425C4—81no—
426C4—302no—
427C4—602no—
428C4—1201no—
429C4#9177011yescharacteristic
430C4#9177211yescharacteristic
431C4C4×#122011yescharacteristic
432C4#9182611yescharacteristic
433C3#1060114311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—21no—
437C2—602no—
438C2#1175105711yescharacteristic
subgroupG/H
centerC2#11751057
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#419C2
socleC2×C3×C5#386
prime radicalC2×#6817C22
cosocleC22×C3×C4×C5C23
hypercenter#419D15
hypocenterC3×C5#1909
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. C3×C5×#419C2semidirect
4. #11751326C2semidirect
6. #11746378C2semidirect
7. #11746352C2semidirect
13. #91836C4semidirect
15. #91833C4semidirect
29. C5×#419D3semidirect
71. C3×#419D5semidirect
178. #419D15semidirect
362. C3×C5#1909semidirect
417. C5#70350semidirect
433. C3#10601143semidirect

Maximal subgroups

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

subgroupindex
#117565752
#117565042
C3×C5×#4192
#117513262
#117511072
#117463782
#117463522
#106011433
#703505