Finite Group Databasebeta

← #14301196

Subgroups — #14301196

Order 1920.

620 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754145C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3#11751211C211yesmaximal, characteristic, complemented
4C5×#5113C211yesmaximal, characteristic, complemented
5#11748428C211yesmaximal, characteristic, complemented
6C3×#64772C211yesmaximal, characteristic, complemented
7#11745989C211yesmaximal, characteristic, complemented
8#10601086—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C3×#681C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91322C412yescomplemented
17#91321—21no—
18#91305C412yescomplemented
19#91304—21no—
20#91303—22no—
21#91288—22no—
22D3×#681—22no—
23#91286—22no—
24#91104—22no—
25C5×#821—21no—
26C5×#816—21no—
27#90980—22no—
28#68947—51nomaximal, Hall
29#66156—31no—
30#66153—31no—
31C5×#416—31no—
32#65399—31no—
33#65294—31no—
34#64772D311yescharacteristic, complemented
35#64753—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220—21no—
38#6817—21no—
39#6817—21no—
40#6817—21no—
41#6817—21no—
42C2×D60C2×C411yescharacteristic
43C2×C4×D15—21no—
44C2×Dic30C2×C411yescharacteristic
45C2×C5×#67—21no—
46C5×#235—21no—
47C5×#235—21no—
48C5×#235C2×C412yes—
49C5×#235D412yes—
50C2×C5×D12D411yescharacteristic
51C2×C4×C5×D3—21no—
52C2×C5×Dic6D411yescharacteristic
53C2×C3×C5×C8—21no—
54C2×C3×C5×C8—21no—
55C2×C3×C5×C8—21no—
56#6710—21no—
57#6710—21no—
58C2×C3⋊#168D412yes—
59C5×#217—21no—
60C5×C4⋊Dic3—21no—
61C4×C5×Dic3—21no—
62C3×#681—21no—
63C3×#681—21no—
64C3×#681C2×C412yes—
65C3×#681D412yes—
66C2×C3×#168—21no—
67C2×C3×#168—21no—
68#6663—22no—
69#6660—22no—
70#6659—22no—
71#6654—22no—
72#6650—24no—
73D3×#168—24no—
74#5748—51no—
75#5745—51no—
76#5139—51no—
77#5113D511yescharacteristic, complemented
78#5110—51no—
79C3×#341—51no—
80#4559—51no—
81C2×C5×#41—31no—
82C2×#709—31no—
83C22×C5×C8C2×D311yescharacteristic
84C5×#117—31no—
85#4097—32no—
86#4096—62no—
87#4095—32no—
88#4094—62no—
89#4094—62no—
90#4088—61no—
91#4078—61no—
92C2×#681—31no—
93C2×#681—62no—
94C2×#681C2×D311yescharacteristic
95C2×#681C2×D311yescharacteristic
96C5×#103—61no—
97C5×#102—61no—
98#3952—62no—
99#3937—62no—
100#1881—151noSylow
101C23×C3×C5#4111yescharacteristic
102C22×D15—21no—
103C22×C5×D3—21no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110#1220—41no—
111#1220—41no—
112C2×Dic15—21no—
113D60—21no—
114D60—21no—
115C4×D15—41no—
116C4×D15—41no—
117Dic30—21no—
118Dic30—21no—
119C5×#67—22no—
120C5×#67—22no—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—41no—
123C5×D12—21no—
124C5×D12C22⋊C412yes—
125C4×C5×D3—22no—
126C4×C5×D3—22no—
127C5×Dic6—21no—
128C5×Dic6C22⋊C412yes—
129C3×C5×C8—41no—
130C3⋊#168—22no—
131C3×#168—22no—
132C3×#168—22no—
133C2×#235—51no—
134C2×#235C2×D511yescharacteristic
135C2×C3×M16—51no—
136C2×C3×M16—51no—
137C22×C3×C8C2×D511yescharacteristic
138#926C2×D511yescharacteristic
139C2×#208—51no—
140#909—52no—
141#909—101no—
142#908—52no—
143#908—101no—
144#907—102no—
145D3×M16—102no—
146#904—102no—
147#901—102no—
148#825—102no—
149#821—21no—
150(C2×D3)⋊C8—102no—
151#816—21no—
152C5×#41—32no—
153C5×#41—32no—
154C5×#41—61no—
155C5×#41—61no—
156C2×C5×Q8—31no—
157C2×C5×D4—31no—
158C2×C5×D4—61no—
159C22×C4×C5—61no—
160C22×C4×C5C22×D311yescharacteristic
161C2×C22⋊D5—61no—
162#709—61no—
163#709—61no—
164#709—61no—
165#709—61no—
166C2×D20—31no—
167C2×C4×D5—61no—
168C2×Dic10—31no—
169C2×C5×C8—21no—
170C2×C5×C8—21no—
171C2×C5×C8—21no—
172C5×C4⋊C4—61no—
173C5×C22⋊C4—61no—
174C42×C5—61no—
175#689—62no—
176#688—62no—
177#687—62no—
178#686—62no—
179#681—21no—
180#681—21no—
181#681—64no—
182#681—61no—
183#681—61no—
184#681C4×D312yes—
185#681D1212yes—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—32no—
189C2×#168—64no—
190#575—151no—
191#568—151no—
192#421—151no—
193#416—31no—
194#408—151no—
195#344—151no—
196#341—51no—
197C22×C3×C5—41no—
198C22×C3×C5C4×D411yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5#12311yescharacteristic
201C2×D15—42no—
202C2×D15—41no—
203C2×C5×D3—22no—
204C2×C5×D3—81no—
205C3×C4×C5C2×C22⋊C411yescharacteristic
206C3×C4×C5C4×D411yescharacteristic
207C3×C4×C5#12011yescharacteristic
208C3×C4×C5#12111yescharacteristic
209Dic15—41no—
210C5×Dic3—22no—
211C5×Dic3—81no—
212C22×C3×C4C22×D511yescharacteristic
213C2×#67—21no—
214C2×#67—101no—
215#235—21no—
216#235—21no—
217#235—101no—
218#235—101no—
219#235—101no—
220#235—101no—
221#235C4×D512yes—
222#235D2012yes—
223C2×D12—51no—
224C2×D12C22⋊D511yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—101no—
227C2×Dic6—51no—
228C2×Dic6C22⋊D511yescharacteristic
229C3×M16—52no—
230C3×M16—52no—
231C3×M16—101no—
232C3×M16—101no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—101no—
237C2×C3×C8—101no—
238#217—21no—
239C4⋊Dic3—21no—
240C4×Dic3—21no—
241#208—101no—
242#208—101no—
243C2×C3⋊C8—52no—
244Dic12—102no—
245D24—102no—
246#204—102no—
247#204—102no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5—61no—
251C23×C5#23911yescharacteristic
252C22×D5—61no—
253C5×Q8—32no—
254C5×Q8—61no—
255C5×D4—32no—
256C5×D4—62no—
257C5×D4—62no—
258C5×D4—61no—
259C2×C4×C5—61no—
260C2×C4×C5—62no—
261C2×C4×C5—62no—
262C2×C4×C5—121no—
263C2×C4×C5C2×C4×D311yescharacteristic
264C2×C4×C5C2×D1211yescharacteristic
265C2×C4×C5#23511yescharacteristic
266C2×C4×C5D3×D411yescharacteristic
267C2×C4×C5D3×D411yescharacteristic
268C2×C4×C5D3×D411yescharacteristic
269C22⋊D5—121no—
270C22⋊D5—121no—
271C2×Dic5—61no—
272D20—61no—
273D20—61no—
274C4×D5—121no—
275C4×D5—121no—
276Dic10—61no—
277Dic10—61no—
278C5×C8—41no—
279#168—22no—
280#168—22no—
281#168—62no—
282C2×#41—31no—
283C2×#41—151no—
284#137—152no—
285#137—301no—
286AGL(1,ℤ/8)—152no—
287AGL(1,ℤ/8)—301no—
288#135—302no—
289#131—302no—
290#131—302no—
291C2×M16—51no—
292C2×M16—51no—
293C2×M16—151no—
294C2×M16—302no—
295C22×C8D3×D511yescharacteristic
296#117—31no—
297#104—302no—
298#103—61no—
299#102—61no—
300C22⋊C8—302no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5#38611yescharacteristic
305D15—81no—
306C5×D3—42no—
307C23×C3#71111yescharacteristic
308C22×D3—21no—
309C22×D3—101no—
310C2×C3×C4C2×C4×D511yescharacteristic
311C2×C3×C4C2×D2011yescharacteristic
312C2×C3×C4D4×D511yescharacteristic
313C2×C3×C4D4×D511yescharacteristic
314C2×C3×C4#71111yescharacteristic
315C2×C3×C4C2×C22⋊D511yescharacteristic
316#67—22no—
317#67—22no—
318#67—201no—
319#67—201no—
320C2×Dic3—21no—
321C2×Dic3—41no—
322C2×Dic3—101no—
323D12—21no—
324D12—101no—
325D12—101no—
326D12#68512yes—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—201no—
330C4×D3—201no—
331Dic6—21no—
332Dic6—101no—
333Dic6—101no—
334Dic6#68512yes—
335C3×C8—41no—
336C3×C8—102no—
337C3×C8—102no—
338C3⋊C8—102no—
339C22×C5—41no—
340C22×C5—62no—
341C22×C5—241no—
342C22×C5#89311yescharacteristic
343C22×C5#89411yescharacteristic
344C22×C5#89511yescharacteristic
345C2×D5—122no—
346C2×D5—121no—
347C4×C5—62no—
348C4×C5—241no—
349C4×C5C4×D1211yescharacteristic
350C4×C5D3×C22⋊C411yescharacteristic
351C4×C5#88211yescharacteristic
352C4×C5#88411yescharacteristic
353Dic5—121no—
354#41—32no—
355#41—32no—
356#41—61no—
357#41—61no—
358#41—301no—
359#41—301no—
360#41—301no—
361#41—301no—
362C2×Q8—31no—
363C2×Q8—151no—
364C2×D4—31no—
365C2×D4—61no—
366C2×D4—151no—
367C2×D4—301no—
368C22×C4—61no—
369C22×C4—301no—
370C22×C4C2×D3×D511yescharacteristic
371Q16—302no—
372SD16—302no—
373SD16—302no—
374D8—302no—
375M16—52no—
376M16—52no—
377M16—101no—
378M16—101no—
379M16—304no—
380M16—301no—
381M16—301no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—101no—
386C2×C8—101no—
387C2×C8—152no—
388C2×C8—304no—
389C4⋊C4—61no—
390C22⋊C4—61no—
391C42—61no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3#402711yescharacteristic
395C22×C3#403211yescharacteristic
396C22×C3#408511yescharacteristic
397C2×D3—22no—
398C2×D3—81no—
399C2×D3—202no—
400C2×D3—201no—
401C3×C4#402711yescharacteristic
402C3×C4#402811yescharacteristic
403C3×C4C2×#68511yescharacteristic
404C3×C4#408511yescharacteristic
405Dic3—22no—
406Dic3—81no—
407Dic3—201no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5—122no—
412C2×C5#466511yescharacteristic
413D5—241no—
414C23—61no—
415C23—301no—
416C23#676811yescharacteristic
417Q8—32no—
418Q8—61no—
419Q8—301no—
420Q8—301no—
421D4—32no—
422D4—62no—
423D4—62no—
424D4—61no—
425D4—301no—
426D4—301no—
427D4—601no—
428D4—601no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—121no—
433C2×C4—301no—
434C2×C4—601no—
435C2×C4—601no—
436C2×C4C4×D3×D511yescharacteristic
437C2×C4D5×D1211yescharacteristic
438C2×C4D3×D2011yescharacteristic
439C2×C4#678211yescharacteristic
440C2×C4D3×C22⋊D511yescharacteristic
441C2×C4#679011yescharacteristic
442C8—41no—
443C8—102no—
444C8—102no—
445C8—302no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3#6524711yescharacteristic
450D3—42no—
451D3—401no—
452C5#6894711yescharacteristic, complemented, Sylow
453C22—41no—
454C22—62no—
455C22—241no—
456C22—602no—
457C22—601no—
458C22#9140611yescharacteristic
459C22#9141611yescharacteristic
460C22#9142711yescharacteristic
461C4—62no—
462C4—241no—
463C4—601no—
464C4#9140511yescharacteristic
465C4D3×#68511yescharacteristic
466C4#9147611yescharacteristic
467C4#9155811yescharacteristic
468C3#1060108611yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—122no—
473C2—1201no—
474C2#1174888311yescharacteristic
subgroupG/H
centerC4#91476
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×C8C22
socleC2×C3×C5#386
prime radical#11754145C2
cosocleC22×C3×C4×C5C23
hypercenterC22×C8D3×D5
hypocenterC3×C5#1881
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. #11754125C2semidirect
3. #11751211C2semidirect
4. C5×#5113C2semidirect
5. #11748428C2semidirect
6. C3×#64772C2semidirect
7. #11745989C2semidirect
16. #91322C4semidirect
18. #91305C4semidirect
34. #64772D3semidirect
77. #5113D5semidirect
392. C3×C5#1881semidirect
452. C5#68947semidirect
468. C3#10601086semidirect

Maximal subgroups

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

subgroupindex
#117541452
#117541252
#117512112
C5×#51132
#117484282
C3×#647722
#117459892
#106010863
#689475