Finite Group Databasebeta

← #14302866

Subgroups — #14302866

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5525C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3#11748625C211yesmaximal, characteristic, complemented
4#11748389C211yesmaximal, characteristic, complemented
5#11748386C211yesmaximal, characteristic, complemented
6#11746352C211yesmaximal, characteristic, complemented
7C3×#64751C211yesmaximal, characteristic, complemented
8#10602647—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C5×D12C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C5×#895—22no—
15C5×#882—22no—
16C5×#875C412yescomplemented
17C5×#874C412yescomplemented
18C3×#4072C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20#91288—22no—
21D3×#681—22no—
22#91286—22no—
23#90996—22no—
24#90967—22no—
25#90966—22no—
26#68713—51nomaximal, Hall
27C5×#535—31no—
28#66153—31no—
29#65529—31no—
30#65504—31no—
31#65504—31no—
32#64751—31no—
33#64751D311yescharacteristic, complemented
34C23×C5×D3—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C2×C5×#67D412yes—
37C5×#235—21no—
38C5×#235—21no—
39C5×#235—21no—
40C5×#235—21no—
41C2×C5×D12—21no—
42C2×C5×D12—21no—
43C2×C5×D12—21no—
44C2×C5×D12—21no—
45C2×C5×D12—21no—
46C2×C5×D12—24no—
47C2×C5×D12C2×C412yes—
48C2×C5×D12C2×C411yescharacteristic
49C2×C4×C5×D3D412yes—
50C2×C5×Dic6C2×C411yescharacteristic
51C3×C5×C4⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C42×C5C2×C412yes—
54#6719—21no—
55C4⋊Dic15—21no—
56C4×Dic15—21no—
57#6710—21no—
58#6710—21no—
59C2×C3⋊#168D412yescomplemented
60C5×(C2×D3)⋊C4—24no—
61C3×C22⋊Dic5—21no—
62C3×C4⋊Dic5—21no—
63C3×C4×Dic5—21no—
64C3×#681—21no—
65C3×#681—21no—
66C2×C3×#168D412yescomplemented
67#6650—24no—
68D3×#168—24no—
69#5745—51no—
70#5525D511yescharacteristic
71#5131—51no—
72#5006—51no—
73#5001—51no—
74C3×H(3,ℤ/4)—51no—
75#4531—51no—
76C2×C5×#41—31no—
77C22×C5×D4—31no—
78C5×#127—32no—
79C5×#124—32no—
80C5×#121—62no—
81C5×#120—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#3966—62no—
91#3964—62no—
92#3964—62no—
93#1910—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3—22no—
96C22×C5×D3—41no—
97C22×C5×D3—41no—
98C22×C5×D3—82no—
99C22×C5×D3C22⋊C412yes—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5C22×C411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×Dic15—41no—
108C5×#67—22no—
109C5×#67—22no—
110C2×C5×Dic3C22⋊C412yes—
111C5×D12—21no—
112C5×D12—21no—
113C5×D12—22no—
114C5×D12—22no—
115C5×D12—41no—
116C5×D12—41no—
117C5×D12—84no—
118C4×C5×D3—22no—
119C4×C5×D3—22no—
120C5×Dic6—21no—
121C5×Dic6—21no—
122C2×C3×Dic5—41no—
123C3⋊#168—22no—
124C3×#168—22no—
125C2×#235C2×D511yescharacteristic
126C22×D12C2×D511yescharacteristic
127C2×C3×M16—51no—
128C3×#117—51no—
129C3×#117C2×D511yescharacteristic
130#926—51no—
131C2×#208—51no—
132#907—102no—
133D3×M16—102no—
134#901—102no—
135#895—22no—
136#882—22no—
137#875Dic512yescomplemented
138#874Dic512yescomplemented
139#821—102no—
140#809—102no—
141#806—102no—
142C24×C5—61no—
143C5×#41—61no—
144C5×#41—61no—
145C5×#41—61no—
146C5×#41—61no—
147C2×C5×Q8—31no—
148C2×C5×D4—32no—
149C2×C5×D4—31no—
150C2×C5×D4—32no—
151C2×C5×D4—64no—
152C2×C5×D4—61no—
153C2×C5×D4—61no—
154C2×C5×D4—61no—
155C2×C5×D4—61no—
156C2×C5×D4—61no—
157C22×C4×C5—32no—
158C22×C4×C5C22×D311yescharacteristic
159C5×C4⋊C4—21no—
160C5×C22⋊C4—21no—
161C5×C22⋊C4—64no—
162C42×C5C4×D312yes—
163C22⋊Dic5—21no—
164C22⋊Dic5—61no—
165C4⋊Dic5—21no—
166C4⋊Dic5—61no—
167C4×Dic5—21no—
168C4×Dic5—61no—
169#681—21no—
170#681—21no—
171#681—64no—
172#681—61no—
173#681—61no—
174C2×#168—32no—
175C2×#168—64no—
176C2×#168D1212yescomplemented
177#568—151no—
178#535—31no—
179#418—151no—
180#418—151no—
181#410—151no—
182H(3,ℤ/4)—51no—
183H(3,ℤ/4)—151no—
184C22×C3×C5—41no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C22×C3×C5#12111yescharacteristic
187C22×C3×C5#12111yescharacteristic
188C2×C5×D3—22no—
189C2×C5×D3—42no—
190C2×C5×D3—44no—
191C2×C5×D3—42no—
192C2×C5×D3—82no—
193C2×C5×D3—82no—
194C3×C4×C5—42no—
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12311yescharacteristic
199Dic15—81no—
200C3×Dic5—81no—
201C5×Dic3—22no—
202C23×D3—21no—
203C22×C3×C4C22×D511yescharacteristic
204C2×#67C22⋊D512yes—
205#235—21no—
206#235—21no—
207#235—21no—
208#235—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—24no—
215C2×D12C2×Dic512yes—
216C2×D12C2×Dic511yescharacteristic
217C2×C4×D3C22⋊D512yes—
218C2×Dic6C2×Dic511yescharacteristic
219C3×M16—101no—
220C3×M16—101no—
221C2×C3×C8—52no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—101no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—101no—
226C3×C42—101no—
227C3×C42C2×Dic512yes—
228#217—101no—
229(C2×D3)⋊C4—24no—
230C4⋊Dic3—101no—
231C4×Dic3—101no—
232#208—101no—
233#208—101no—
234C2×C3⋊C8—52no—
235ΓL(1,𝔽25)—104no—
236D3×C8—104no—
237C23×C5—32no—
238C23×C5—62no—
239C23×C5—121no—
240C23×C5—121no—
241C23×C5—242no—
242C23×C5D3×D411yescharacteristic
243C5×Q8—61no—
244C5×Q8—61no—
245C5×D4—62no—
246C5×D4—62no—
247C5×D4—61no—
248C5×D4—62no—
249C5×D4—61no—
250C5×D4—62no—
251C5×D4—121no—
252C5×D4—121no—
253C5×D4—244no—
254C2×C4×C5—22no—
255C2×C4×C5—32no—
256C2×C4×C5—62no—
257C2×C4×C5—62no—
258C2×C4×C5C2×C4×D311yescharacteristic
259C2×C4×C5C2×D1211yescharacteristic
260C2×C4×C5#23511yescharacteristic
261C2×C4×C5D3×D411yescharacteristic
262C2×C4×C5D3×D411yescharacteristic
263C2×C4×C5#23911yescharacteristic
264C2×Dic5—41no—
265C2×Dic5—121no—
266#168—22no—
267#168—62no—
268C2×#41—31no—
269C22×D4—31no—
270#131—302no—
271#131—302no—
272C2×M16—51no—
273C2×M16—151no—
274C2×M16—302no—
275#127—32no—
276#124—32no—
277#121—62no—
278#120—62no—
279#117—51no—
280#117—151no—
281#117D3×D511yescharacteristic
282#102—302no—
283#102—302no—
284#99—302no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5#38611yescharacteristic
289C5×D3—42no—
290C5×D3—82no—
291C23×C3C2×C22⋊D511yescharacteristic
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—82no—
296C22×D3C22⋊Dic512yes—
297C2×C3×C4—22no—
298C2×C3×C4—201no—
299C2×C3×C4D4×D511yescharacteristic
300C2×C3×C4D4×D511yescharacteristic
301C2×C3×C4#71111yescharacteristic
302C2×C3×C4#71111yescharacteristic
303C2×C3×C4C22×Dic511yescharacteristic
304C2×C3×C4C2×C22⋊D511yescharacteristic
305#67—22no—
306#67—22no—
307C2×Dic3—201no—
308C2×Dic3C22⋊Dic512yes—
309D12—21no—
310D12—21no—
311D12—22no—
312D12—22no—
313D12—41no—
314D12—41no—
315D12—84no—
316C4×D3—22no—
317C4×D3—22no—
318Dic6—21no—
319Dic6—21no—
320C3×C8—102no—
321C3⋊C8—102no—
322C22×C5—41no—
323C22×C5—62no—
324C22×C5—122no—
325C22×C5—124no—
326C22×C5—122no—
327C22×C5—242no—
328C22×C5—242no—
329C22×C5D3×C22⋊C411yescharacteristic
330C22×C5#88411yescharacteristic
331C22×C5#89511yescharacteristic
332C4×C5—42no—
333C4×C5—62no—
334C4×C5C4×D1211yescharacteristic
335C4×C5#88211yescharacteristic
336C4×C5#89311yescharacteristic
337C4×C5#89411yescharacteristic
338Dic5—81no—
339Dic5—241no—
340C24—61no—
341#41—61no—
342#41—61no—
343#41—61no—
344#41—61no—
345C2×Q8—31no—
346C2×D4—32no—
347C2×D4—31no—
348C2×D4—32no—
349C2×D4—64no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C22×C4—32no—
356C22×C4C2×D3×D511yescharacteristic
357M16—101no—
358M16—101no—
359M16—304no—
360M16—301no—
361M16—301no—
362C2×C8—52no—
363C2×C8—152no—
364C2×C8—304no—
365C4⋊C4—21no—
366C4⋊C4—101no—
367C4⋊C4—301no—
368C22⋊C4—21no—
369C22⋊C4—64no—
370C22⋊C4—101no—
371C22⋊C4—301no—
372C42—101no—
373C42—301no—
374C42D3×Dic512yes—
375C3×C5#191011yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3#408511yescharacteristic
378C22×C3#408511yescharacteristic
379C22×C3C2×C22⋊Dic511yescharacteristic
380C2×D3—22no—
381C2×D3—42no—
382C2×D3—44no—
383C2×D3—42no—
384C2×D3—82no—
385C2×D3—82no—
386C3×C4—42no—
387C3×C4—401no—
388C3×C4D4×Dic511yescharacteristic
389C3×C4D4×Dic511yescharacteristic
390C3×C4#408111yescharacteristic
391C3×C4#408311yescharacteristic
392Dic3—22no—
393Dic3—401no—
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5—122no—
398C2×C5—242no—
399C2×C5#466511yescharacteristic
400C23—32no—
401C23—62no—
402C23—121no—
403C23—121no—
404C23—242no—
405C23D3×C22⋊D511yescharacteristic
406Q8—61no—
407Q8—61no—
408D4—62no—
409D4—62no—
410D4—61no—
411D4—62no—
412D4—61no—
413D4—62no—
414D4—121no—
415D4—121no—
416D4—244no—
417C2×C4—22no—
418C2×C4—32no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—201no—
422C2×C4—601no—
423C2×C4#676811yescharacteristic
424C2×C4#677211yescharacteristic
425C2×C4D5×D1211yescharacteristic
426C2×C4#677711yescharacteristic
427C2×C4C2×D3×Dic511yescharacteristic
428C2×C4D3×C22⋊D511yescharacteristic
429C8—102no—
430C8—302no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3#6550911yescharacteristic
435D3—42no—
436D3—82no—
437C5#6871311yescharacteristic, complemented, Sylow
438C22—41no—
439C22—62no—
440C22—122no—
441C22—124no—
442C22—122no—
443C22—242no—
444C22—242no—
445C22#9141611yescharacteristic
446C22#9145311yescharacteristic
447C22D3×C22⋊Dic511yescharacteristic
448C4—42no—
449C4—62no—
450C4—401no—
451C4—1201no—
452C4Dic5×D1211yescharacteristic
453C4#9142311yescharacteristic
454C4#9143511yescharacteristic
455C4#9147511yescharacteristic
456C3#1060264711yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—122no—
461C2—242no—
462C2#1174890111yescharacteristic
subgroupG/H
centerC2#11748901
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radicalC5×#5525C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1910
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
2. #11754125C2semidirect
3. #11748625C2semidirect
4. #11748389C2semidirect
5. #11748386C2semidirect
6. #11746352C2semidirect
7. C3×#64751C2semidirect
16. C5×#875C4semidirect
17. C5×#874C4semidirect
33. #64751D3semidirect
59. C2×C3⋊#168D4semidirect
66. C2×C3×#168D4semidirect
137. #875Dic5semidirect
138. #874Dic5semidirect
176. C2×#168D12semidirect
375. C3×C5#1910semidirect
437. C5#68713semidirect
456. C3#10602647semidirect

Maximal subgroups

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

subgroupindex
C5×#55252
#117541252
#117486252
#117483892
#117483862
#117463522
C3×#647512
#106026473
#687135