Finite Group Databasebeta

← #14302815

Subgroups — #14302815

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
1C5×#5526C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic
3#11748625C211yesmaximal, characteristic
4#11748409C211yesmaximal, characteristic, complemented
5#11748407C211yesmaximal, characteristic, complemented
6#11746352C211yesmaximal, characteristic, complemented
7C3×#64751C211yesmaximal, characteristic, complemented
8#10602658—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C5×Dic6C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C5×#897—22no—
15C5×#886—22no—
16C5×#875C412yescomplemented
17C5×#869C412yescomplemented
18C3×#4072C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20#91288—22no—
21D3×#681—22no—
22#91286—22no—
23#90996—22no—
24#90973—22no—
25#90972—22no—
26#68712—51nomaximal, Hall
27C5×#536—31no—
28#66153—31no—
29#65529—31no—
30#65514—31no—
31#65514—31no—
32#64751—31no—
33#64751D311yescharacteristic, complemented
34C22×C3×C4×C5C2311yescharacteristic
35C2×C5×#67D412yes—
36C22×C5×Dic3—21no—
37C5×#235—21no—
38C5×#235—21no—
39C5×#235—21no—
40C5×#235—21no—
41C2×C5×D12C2×C411yescharacteristic
42C2×C4×C5×D3D412yes—
43C2×C5×Dic6—21no—
44C2×C5×Dic6—21no—
45C2×C5×Dic6—21no—
46C2×C5×Dic6—21no—
47C2×C5×Dic6—21no—
48C2×C5×Dic6C2×C412yes—
49C2×C5×Dic6C2×C411yescharacteristic
50C3×C5×C4⋊C4—21no—
51C3×C5×C22⋊C4—21no—
52C3×C42×C5C2×C412yes—
53#6719—21no—
54C4⋊Dic15—21no—
55C4×Dic15—21no—
56#6710—21no—
57#6710—21no—
58C2×C3⋊#168D412yes—
59C5×(C2×D3)⋊C4—24no—
60C5×C4⋊Dic3—24no—
61C3×C22⋊Dic5—21no—
62C3×C4⋊Dic5—21no—
63C3×C4×Dic5—21no—
64C3×#681—21no—
65C3×#681—21no—
66C2×C3×#168D412yes—
67#6650—24no—
68D3×#168—24no—
69#5745—51no—
70#5526D511yescharacteristic
71#5120—51no—
72#5006—51no—
73#5004—51no—
74C3×H(3,ℤ/4)—51no—
75#4531—51no—
76C2×C5×#41—31no—
77C22×C5×Q8—31no—
78C5×C4⋊Q8—32no—
79C5×#124—32no—
80C5×#123—62no—
81C5×C22⋊Q8—62no—
82C5×#117C2×D311yescharacteristic
83#4094—62no—
84#4094—62no—
85#4072—31no—
86#4072C2×D311yescharacteristic
87C2×#681—31no—
88C2×#681—62no—
89C2×#681C2×D311yescharacteristic
90#3967—62no—
91#3967—62no—
92#3966—62no—
93#1909—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3C22⋊C412yes—
96C2×C3×C4×C5—22no—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102C2×C3×C4×C5#4111yescharacteristic
103C2×Dic15—41no—
104C5×#67—22no—
105C5×#67—22no—
106C2×C5×Dic3—22no—
107C2×C5×Dic3—41no—
108C2×C5×Dic3—41no—
109C2×C5×Dic3C22⋊C412yes—
110C5×D12—21no—
111C5×D12—21no—
112C4×C5×D3—22no—
113C4×C5×D3—22no—
114C5×Dic6—21no—
115C5×Dic6—21no—
116C5×Dic6—22no—
117C5×Dic6—22no—
118C5×Dic6—41no—
119C5×Dic6—41no—
120C2×C3×Dic5—41no—
121C3⋊#168—22no—
122C3×#168—22no—
123C2×#235C2×D511yescharacteristic
124C22×Dic6C2×D511yescharacteristic
125C2×C3×M16—51no—
126C3×#117—51no—
127C3×#117C2×D511yescharacteristic
128#926—51no—
129C2×#208—51no—
130#907—102no—
131D3×M16—102no—
132#901—102no—
133#897—22no—
134#886—22no—
135#875Dic512yescomplemented
136#869Dic512yescomplemented
137#816—102no—
138#810—102no—
139#806—102no—
140C5×#41—61no—
141C5×#41—61no—
142C5×#41—61no—
143C5×#41—61no—
144C2×C5×Q8—31no—
145C2×C5×Q8—32no—
146C2×C5×Q8—61no—
147C2×C5×Q8—61no—
148C2×C5×Q8—61no—
149C2×C5×Q8—61no—
150C2×C5×Q8—61no—
151C2×C5×D4—32no—
152C2×C5×D4—31no—
153C22×C4×C5—32no—
154C22×C4×C5—61no—
155C22×C4×C5C22×D311yescharacteristic
156C5×C4⋊C4—21no—
157C5×C4⋊C4—64no—
158C5×C22⋊C4—21no—
159C5×C22⋊C4—64no—
160C42×C5C4×D312yes—
161C22⋊Dic5—21no—
162C22⋊Dic5—61no—
163C4⋊Dic5—21no—
164C4⋊Dic5—61no—
165C4×Dic5—21no—
166C4×Dic5—61no—
167#681—21no—
168#681—21no—
169#681—64no—
170#681—61no—
171#681—61no—
172C2×#168—32no—
173C2×#168—64no—
174C2×#168D1212yes—
175#568—151no—
176#536—31no—
177#419—151no—
178#419—151no—
179#410—151no—
180H(3,ℤ/4)—51no—
181H(3,ℤ/4)—151no—
182C22×C3×C5—41no—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C22×C3×C5#12111yescharacteristic
185C22×C3×C5#12111yescharacteristic
186C2×C5×D3—22no—
187C2×C5×D3—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—81no—
195C5×Dic3—22no—
196C5×Dic3—42no—
197C22×C3×C4C22×D511yescharacteristic
198C2×#67C22⋊D512yes—
199C22×Dic3—21no—
200#235—21no—
201#235—21no—
202#235—21no—
203#235—21no—
204C2×D12C2×Dic511yescharacteristic
205C2×C4×D3C22⋊D512yes—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6C2×Dic512yes—
212C2×Dic6C2×Dic511yescharacteristic
213C3×M16—101no—
214C3×M16—101no—
215C2×C3×C8—52no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—101no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—101no—
220C3×C42—101no—
221C3×C42C2×Dic512yes—
222#217—101no—
223(C2×D3)⋊C4—24no—
224C4⋊Dic3—24no—
225C4⋊Dic3—101no—
226C4×Dic3—101no—
227#208—101no—
228#208—101no—
229C2×C3⋊C8—52no—
230ΓL(1,𝔽25)—104no—
231D3×C8—104no—
232C23×C5—32no—
233C23×C5D3×D411yescharacteristic
234C5×Q8—61no—
235C5×Q8—61no—
236C5×Q8—62no—
237C5×Q8—62no—
238C5×Q8—121no—
239C5×Q8—121no—
240C5×D4—62no—
241C5×D4—62no—
242C5×D4—61no—
243C5×D4—61no—
244C2×C4×C5—22no—
245C2×C4×C5—32no—
246C2×C4×C5—62no—
247C2×C4×C5—62no—
248C2×C4×C5—62no—
249C2×C4×C5—121no—
250C2×C4×C5—121no—
251C2×C4×C5C2×C4×D311yescharacteristic
252C2×C4×C5C2×D1211yescharacteristic
253C2×C4×C5#23511yescharacteristic
254C2×C4×C5D3×D411yescharacteristic
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5#23911yescharacteristic
257C2×Dic5—41no—
258C2×Dic5—121no—
259#168—22no—
260#168—62no—
261C2×#41—31no—
262C22×Q8—31no—
263#131—302no—
264#131—302no—
265C2×M16—51no—
266C2×M16—151no—
267C2×M16—302no—
268C4⋊Q8—32no—
269#124—32no—
270#123—62no—
271C22⋊Q8—62no—
272#117—51no—
273#117—151no—
274#117D3×D511yescharacteristic
275#103—302no—
276#103—302no—
277#99—302no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38611yescharacteristic
282C5×D3—42no—
283C23×C3C2×C22⋊D511yescharacteristic
284C22×D3C22⋊Dic512yes—
285C2×C3×C4—22no—
286C2×C3×C4—201no—
287C2×C3×C4D4×D511yescharacteristic
288C2×C3×C4D4×D511yescharacteristic
289C2×C3×C4#71111yescharacteristic
290C2×C3×C4#71111yescharacteristic
291C2×C3×C4C22×Dic511yescharacteristic
292C2×C3×C4C2×C22⋊D511yescharacteristic
293#67—22no—
294#67—22no—
295C2×Dic3—22no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—201no—
299C2×Dic3C22⋊Dic512yes—
300D12—21no—
301D12—21no—
302C4×D3—22no—
303C4×D3—22no—
304Dic6—21no—
305Dic6—21no—
306Dic6—22no—
307Dic6—22no—
308Dic6—41no—
309Dic6—41no—
310C3×C8—102no—
311C3⋊C8—102no—
312C22×C5—41no—
313C22×C5—62no—
314C22×C5—122no—
315C22×C5D3×C22⋊C411yescharacteristic
316C22×C5#88411yescharacteristic
317C22×C5#89511yescharacteristic
318C4×C5—42no—
319C4×C5—62no—
320C4×C5—122no—
321C4×C5C4×D1211yescharacteristic
322C4×C5#88211yescharacteristic
323C4×C5#89311yescharacteristic
324C4×C5#89411yescharacteristic
325Dic5—81no—
326Dic5—241no—
327#41—61no—
328#41—61no—
329#41—61no—
330#41—61no—
331C2×Q8—31no—
332C2×Q8—32no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×D4—32no—
339C2×D4—31no—
340C22×C4—32no—
341C22×C4—61no—
342C22×C4C2×D3×D511yescharacteristic
343M16—101no—
344M16—101no—
345M16—304no—
346M16—301no—
347M16—301no—
348C2×C8—52no—
349C2×C8—152no—
350C2×C8—304no—
351C4⋊C4—21no—
352C4⋊C4—64no—
353C4⋊C4—101no—
354C4⋊C4—301no—
355C22⋊C4—21no—
356C22⋊C4—64no—
357C22⋊C4—101no—
358C22⋊C4—301no—
359C42—101no—
360C42—301no—
361C42D3×Dic512yes—
362C3×C5#190911yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3#408511yescharacteristic
365C22×C3#408511yescharacteristic
366C22×C3C2×C22⋊Dic511yescharacteristic
367C2×D3—22no—
368C2×D3—42no—
369C3×C4—42no—
370C3×C4—401no—
371C3×C4D4×Dic511yescharacteristic
372C3×C4D4×Dic511yescharacteristic
373C3×C4#408111yescharacteristic
374C3×C4#408311yescharacteristic
375Dic3—22no—
376Dic3—42no—
377Dic3—401no—
378C2×C5—21no—
379C2×C5—21no—
380C2×C5—21no—
381C2×C5—122no—
382C2×C5#466511yescharacteristic
383C23—32no—
384C23D3×C22⋊D511yescharacteristic
385Q8—61no—
386Q8—61no—
387Q8—62no—
388Q8—62no—
389Q8—121no—
390Q8—121no—
391D4—62no—
392D4—62no—
393D4—61no—
394D4—61no—
395C2×C4—22no—
396C2×C4—32no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—201no—
403C2×C4—601no—
404C2×C4#676811yescharacteristic
405C2×C4#677211yescharacteristic
406C2×C4D5×D1211yescharacteristic
407C2×C4#677711yescharacteristic
408C2×C4C2×D3×Dic511yescharacteristic
409C2×C4D3×C22⋊D511yescharacteristic
410C8—102no—
411C8—302no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3#6550911yescharacteristic
416D3—42no—
417C5#6871211yescharacteristic, complemented, Sylow
418C22—41no—
419C22—62no—
420C22—122no—
421C22#9141611yescharacteristic
422C22#9145311yescharacteristic
423C22D3×C22⋊Dic511yescharacteristic
424C4—42no—
425C4—62no—
426C4—122no—
427C4—401no—
428C4—1201no—
429C4Dic5×D1211yescharacteristic
430C4#9142311yescharacteristic
431C4#9143511yescharacteristic
432C4#9147511yescharacteristic
433C3#1060265811yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—21no—
437C2—122no—
438C2#1174890111yescharacteristic
subgroupG/H
centerC2#11748901
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radicalC2×C5×#235C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1909
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #11748409C2semidirect
5. #11748407C2semidirect
6. #11746352C2semidirect
7. C3×#64751C2semidirect
16. C5×#875C4semidirect
17. C5×#869C4semidirect
33. #64751D3semidirect
135. #875Dic5semidirect
136. #869Dic5semidirect
362. C3×C5#1909semidirect
417. C5#68712semidirect
433. C3#10602658semidirect

Maximal subgroups

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

subgroupindex
C5×#55262
#117541252
#117486252
#117484092
#117484072
#117463522
C3×#647512
#106026583
#687125