Finite Group Databasebeta

← #14302810

Subgroups — #14302810

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
1C3×#66150C211yesmaximal, characteristic
2#11754120C211yesmaximal, characteristic
3#11751326C211yesmaximal, characteristic, complemented
4C5×#5225C211yesmaximal, characteristic, complemented
5#11748624C211yesmaximal, characteristic
6#11745970C211yesmaximal, characteristic, complemented
7#11745969C211yesmaximal, characteristic, complemented
8#10601143—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4093C412yescomplemented
16C3×#4092—22no—
17C3×#4090C412yescomplemented
18C3×#4075—22no—
19C3×#4072C2211yescharacteristic
20#91284—22no—
21#91283—22no—
22D5×#208—22no—
23#91119—22no—
24C5×#835—22no—
25#90995—22no—
26#71865—51nomaximal, Hall
27#66150D311yescharacteristic
28#66079—31no—
29C5×#419—31no—
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×C3×C22⋊D5D412yes—
43C3×#709—21no—
44C3×#709—21no—
45C3×#709—21no—
46C3×#709—21no—
47C2×C3×D20C2×C411yescharacteristic
48C2×C3×C4×D5D412yes—
49C2×C3×Dic10C2×C411yescharacteristic
50#6719—21no—
51C4⋊Dic15—21no—
52C4×Dic15—21no—
53#6710—21no—
54#6710—21no—
55C2×C3⋊#168D412yes—
56C5×#217—21no—
57C5×C4⋊Dic3—21no—
58C4×C5×Dic3—21no—
59C5×#208—21no—
60C5×#208—21no—
61C2×C5×C3⋊C8D412yes—
62C3×C22⋊Dic5—21no—
63C3×#685—24no—
64C3×C4⋊Dic5—21no—
65C3×#683—24no—
66C3×C4×Dic5C2×C412yes—
67#6649—24no—
68D5×C3⋊C8—24no—
69C3×#536—51no—
70#5819—51no—
71#5240—51no—
72#5225—51no—
73#5225D511yescharacteristic, complemented
74#4531—51no—
75#4531—51no—
76C22×C5×Q8C2×D311yescharacteristic
77C2×#709C2×D311yescharacteristic
78C2×C5×M16—31no—
79C5×#117—31no—
80#4093Dic312yescomplemented
81#4092—22no—
82#4090Dic312yescomplemented
83#4075—22no—
84#4072—31no—
85#4072C2×D311yescharacteristic
86C2×#681—31no—
87#4053—62no—
88D5×M16—62no—
89#4047—62no—
90C5×#103—62no—
91#3967—62no—
92#3938—62no—
93#1909—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5C22⋊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×C5C22×C411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×Dic15—41no—
110C2×C5×Dic3—41no—
111C3×C22⋊D5—22no—
112C3×C22⋊D5—22no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5C22⋊C412yes—
115C3×D20—21no—
116C3×D20—21no—
117C3×C4×D5—22no—
118C3×C4×D5—22no—
119C3×Dic10—21no—
120C3×Dic10—21no—
121C3⋊#168—22no—
122C5×C3⋊C8—22no—
123C2×C3×#41—51no—
124C22×C3×Q8C2×D511yescharacteristic
125C3×C4⋊Q8—52no—
126C3×#124—52no—
127C3×#123—102no—
128C3×C22⋊Q8—102no—
129C3×#117—51no—
130#948—102no—
131#948—102no—
132#926—51no—
133#926C2×D511yescharacteristic
134C2×#208—51no—
135C2×#208—102no—
136C2×#208C2×D511yescharacteristic
137#835—22no—
138#835—102no—
139#834—102no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8C2×Dic312yes—
146C22×C4×C5—21no—
147C22×C4×C5C22×D311yescharacteristic
148C2×C22⋊D5#6712yes—
149#709—21no—
150#709—21no—
151#709—21no—
152#709—21no—
153C2×D20C2×Dic311yescharacteristic
154C2×C4×D5#6712yes—
155C2×Dic10C2×Dic311yescharacteristic
156C5×M16—61no—
157C5×M16—61no—
158C2×C5×C8—32no—
159C5×C4⋊C4—61no—
160C5×C22⋊C4—61no—
161C42×C5—61no—
162C22⋊Dic5—21no—
163C22⋊Dic5—61no—
164#685—24no—
165C4⋊Dic5—21no—
166C4⋊Dic5—61no—
167#683—24no—
168C4×Dic5—61no—
169C4×Dic5C2×Dic312yes—
170#681—61no—
171#681—61no—
172C2×#168—32no—
173#676—64no—
174C8×D5—64no—
175#568—151no—
176#536—51no—
177#419—31no—
178#419—151no—
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×C3×D5—22no—
187C2×C3×D5—42no—
188C3×C4×C5—42no—
189C3×C4×C5C4×D411yescharacteristic
190C3×C4×C5C4×D411yescharacteristic
191C3×C4×C5#12011yescharacteristic
192C3×C4×C5#12311yescharacteristic
193Dic15—81no—
194C3×Dic5—22no—
195C3×Dic5—42no—
196C5×Dic3—81no—
197C3×#41—101no—
198C3×#41—101no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—51no—
207C2×C3×Q8C4×D512yes—
208C2×C3×D4—52no—
209C2×C3×D4—51no—
210C22×C3×C4—21no—
211C22×C3×C4—52no—
212C22×C3×C4C22×D511yescharacteristic
213C3×C4⋊C4—104no—
214C3×C4⋊C4—101no—
215C3×C22⋊C4—104no—
216C3×C22⋊C4—101no—
217C3×C42—52no—
218#217—21no—
219#217—101no—
220C4⋊Dic3—21no—
221C4⋊Dic3—101no—
222C4×Dic3—21no—
223C4×Dic3—101no—
224#208—21no—
225#208—21no—
226#208—104no—
227#208—101no—
228#208—101no—
229C2×C3⋊C8—52no—
230C2×C3⋊C8—104no—
231C2×C3⋊C8C22⋊D512yes—
232C23×C5C2×#6711yescharacteristic
233C22×D5#21712yes—
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—121no—
242C2×C4×C5D3×D411yescharacteristic
243C2×C4×C5D3×D411yescharacteristic
244C2×C4×C5#23711yescharacteristic
245C2×C4×C5#23711yescharacteristic
246C2×C4×C5C22×Dic311yescharacteristic
247C2×C4×C5C2×#6711yescharacteristic
248C22⋊D5—22no—
249C22⋊D5—22no—
250C2×Dic5—22no—
251C2×Dic5—121no—
252C2×Dic5#21712yes—
253D20—21no—
254D20—21no—
255C4×D5—22no—
256C4×D5—22no—
257Dic10—21no—
258Dic10—21no—
259C5×C8—62no—
260#168—62no—
261C2×#41—51no—
262C22×Q8D3×D511yescharacteristic
263#131—302no—
264#131—302no—
265C2×M16—31no—
266C2×M16—151no—
267C2×M16—302no—
268C4⋊Q8—52no—
269#124—52no—
270#123—102no—
271C22⋊Q8—102no—
272#117—31no—
273#117—51no—
274#117—151no—
275#103—62no—
276#103—302no—
277#99—302no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38611yescharacteristic
282C3×D5—42no—
283C23×C3—52no—
284C23×C3D4×D511yescharacteristic
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—41no—
288C3×Q8—41no—
289C3×Q8—101no—
290C3×Q8—101no—
291C3×D4—102no—
292C3×D4—102no—
293C3×D4—101no—
294C3×D4—101no—
295C2×C3×C4—22no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—52no—
299C2×C3×C4—102no—
300C2×C3×C4—102no—
301C2×C3×C4—102no—
302C2×C3×C4C2×C4×D511yescharacteristic
303C2×C3×C4#70911yescharacteristic
304C2×C3×C4D4×D511yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4#71111yescharacteristic
307C2×C3×C4C2×C22⋊D511yescharacteristic
308C2×Dic3—41no—
309C2×Dic3—201no—
310C3⋊C8—22no—
311C3⋊C8—102no—
312C22×C5—41no—
313C22×C5#93911yescharacteristic
314C22×C5#93911yescharacteristic
315C22×C5C2×#21711yescharacteristic
316C2×D5—22no—
317C2×D5—42no—
318C4×C5—42no—
319C4×C5—241no—
320C4×C5D4×Dic311yescharacteristic
321C4×C5D4×Dic311yescharacteristic
322C4×C5#93511yescharacteristic
323C4×C5#93711yescharacteristic
324Dic5—22no—
325Dic5—42no—
326Dic5—241no—
327#41—101no—
328#41—101no—
329#41—101no—
330#41—101no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—51no—
337C2×Q8D5×Dic312yes—
338C2×D4—52no—
339C2×D4—51no—
340C22×C4—21no—
341C22×C4—52no—
342C22×C4C2×D3×D511yescharacteristic
343M16—61no—
344M16—61no—
345M16—304no—
346M16—301no—
347M16—301no—
348C2×C8—32no—
349C2×C8—152no—
350C2×C8—304no—
351C4⋊C4—61no—
352C4⋊C4—104no—
353C4⋊C4—101no—
354C4⋊C4—301no—
355C22⋊C4—61no—
356C22⋊C4—104no—
357C22⋊C4—101no—
358C22⋊C4—301no—
359C42—52no—
360C42—61no—
361C42—301no—
362C3×C5#190911yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3—102no—
365C22×C3—202no—
366C22×C3D5×C22⋊C411yescharacteristic
367C22×C3#403011yescharacteristic
368C22×C3#408411yescharacteristic
369C3×C4—42no—
370C3×C4—102no—
371C3×C4—202no—
372C3×C4#402711yescharacteristic
373C3×C4#402911yescharacteristic
374C3×C4C4×C22⋊D511yescharacteristic
375C3×C4#408311yescharacteristic
376Dic3—81no—
377Dic3—401no—
378C2×C5—21no—
379C2×C5—21no—
380C2×C5—21no—
381C2×C5#522011yescharacteristic
382D5—42no—
383C23—52no—
384C23D5×#6711yescharacteristic
385Q8—22no—
386Q8—22no—
387Q8—41no—
388Q8—41no—
389Q8—101no—
390Q8—101no—
391D4—102no—
392D4—102no—
393D4—101no—
394D4—101no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—52no—
399C2×C4—102no—
400C2×C4—102no—
401C2×C4—102no—
402C2×C4—121no—
403C2×C4—601no—
404C2×C4C2×D5×Dic311yescharacteristic
405C2×C4#677911yescharacteristic
406C2×C4#678011yescharacteristic
407C2×C4D5×#6711yescharacteristic
408C2×C4D3×C22⋊D511yescharacteristic
409C2×C4#679011yescharacteristic
410C8—62no—
411C8—302no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—202no—
416C2×C3#6524511yescharacteristic
417C5#7186511yescharacteristic, complemented, Sylow
418C22—41no—
419C22—102no—
420C22—202no—
421C22D5×#21711yescharacteristic
422C22#9154911yescharacteristic
423C22#9155011yescharacteristic
424C4—42no—
425C4—102no—
426C4—202no—
427C4—241no—
428C4—1201no—
429C4#9154811yescharacteristic
430C4Dic3×C22⋊D511yescharacteristic
431C4#9155911yescharacteristic
432C4#9157711yescharacteristic
433C3#1060114311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—21no—
437C2—202no—
438C2#1174951811yescharacteristic
subgroupG/H
centerC2#11749518
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×Q8C22
socleC2×C3×C5#386
prime radicalC2×C3×#709C22
cosocleC22×C3×C4×C5C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5#1909
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #11751326C2semidirect
4. C5×#5225C2semidirect
6. #11745970C2semidirect
7. #11745969C2semidirect
15. C3×#4093C4semidirect
17. C3×#4090C4semidirect
73. #5225D5semidirect
80. #4093Dic3semidirect
82. #4090Dic3semidirect
362. C3×C5#1909semidirect
417. C5#71865semidirect
433. C3#10601143semidirect

Maximal subgroups

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

subgroupindex
C3×#661502
#117541202
#117513262
C5×#52252
#117486242
#117459702
#117459692
#106011433
#718655