Finite Group Databasebeta

← #14301197

Subgroups — #14301197

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#11754138C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3C5×#5113C211yesmaximal, characteristic, complemented
4C3×#65399C211yesmaximal, characteristic, complemented
5#11748421C211yesmaximal, characteristic, complemented
6#11746373C211yesmaximal, characteristic, complemented
7#11745988C211yesmaximal, characteristic, complemented
8#10601086—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C5×#926C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91309C412yescomplemented
17#91307—21no—
18#91299—21no—
19#91298C412yescomplemented
20#91297—22no—
21#91288—22no—
22D3×#681—22no—
23#91286—22no—
24C5×#821—21no—
25C5×#816—21no—
26C3×#3952—22no—
27#90977—22no—
28#70012—51nomaximal, Hall
29#66156—31no—
30#66153—31no—
31C5×#416—31no—
32#65399D311yescharacteristic, complemented
33#65294—31no—
34#64772—31no—
35#64753—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C2×C5×#67—21no—
38C5×#235—21no—
39C5×#235—21no—
40C5×#235C2×C412yes—
41C5×#235D412yes—
42C2×C5×D12D411yescharacteristic
43C2×C4×C5×D3—21no—
44C2×C5×Dic6D411yescharacteristic
45C2×C3×C22⋊D5—21no—
46C3×#709—21no—
47C3×#709—21no—
48C3×#709—21no—
49C3×#709—21no—
50C2×C3×D20C2×C411yescharacteristic
51C2×C3×C4×D5—21no—
52C2×C3×Dic10C2×C411yescharacteristic
53C2×C3×C5×C8—21no—
54C2×C3×C5×C8—21no—
55C2×C3×C5×C8—21no—
56#6710—21no—
57#6710—21no—
58#6710C2×C412yes—
59#6710D412yes—
60C2×C3⋊#168—21no—
61C2×C3⋊#168—21no—
62C5×#217—21no—
63C5×C4⋊Dic3—21no—
64C4×C5×Dic3—21no—
65C3×#681—21no—
66C3×#681—21no—
67C2×C3×#168D412yes—
68#6661—22no—
69#6656—22no—
70#6655—22no—
71#6652—22no—
72#6650—24no—
73D3×#168—24no—
74#5822—51no—
75#5745—51no—
76C3×#408—51no—
77#5236—51no—
78#5113D511yescharacteristic, complemented
79#4556—51no—
80#4552—51no—
81C2×C5×#41—31no—
82C2×#709C2×D311yescharacteristic
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—31no—
94C2×#681—62no—
95C2×#681C2×D311yescharacteristic
96C5×#103—61no—
97C5×#102—61no—
98#3952—22no—
99#3937—62no—
100#1881—151noSylow
101C23×C3×C5#4111yescharacteristic
102C22×C5×D3—21no—
103C22×C3×D5—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
110C5×#67—22no—
111C5×#67—22no—
112C2×C5×Dic3—21no—
113C2×C5×Dic3—41no—
114C5×D12—21no—
115C5×D12C22⋊C412yes—
116C4×C5×D3—22no—
117C4×C5×D3—22no—
118C5×Dic6—21no—
119C5×Dic6C22⋊C412yes—
120C3×C22⋊D5—41no—
121C3×C22⋊D5—41no—
122C2×C3×Dic5—21no—
123C3×D20—21no—
124C3×D20—21no—
125C3×C4×D5—41no—
126C3×C4×D5—41no—
127C3×Dic10—21no—
128C3×Dic10—21no—
129C3×C5×C8—41no—
130C3⋊#168—22no—
131C3⋊#168—22no—
132C3×#168—22no—
133C2×C3×#41—51no—
134C2×#235C2×D511yescharacteristic
135C2×C3×M16—51no—
136C22×C3×C8C2×D511yescharacteristic
137#951—101no—
138#950—102no—
139#949—101no—
140#942—52no—
141#932—52no—
142#926C2×D511yescharacteristic
143C2×#208—51no—
144C2×#208—51no—
145#907—102no—
146D3×M16—102no—
147#901—102no—
148C3×C22⋊C8—102no—
149#837—102no—
150#821—21no—
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—21no—
162#709—21no—
163#709—21no—
164#709—21no—
165#709—21no—
166C2×D20C4×D311yescharacteristic
167C2×C4×D5—21no—
168C2×Dic10C4×D311yescharacteristic
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—32no—
182#681—32no—
183#681—64no—
184#681—61no—
185#681—61no—
186C2×#168—64no—
187C2×#168—61no—
188C2×#168—61no—
189C2×#168#6712yes—
190#575—151no—
191#568—151no—
192#421—151no—
193#416—31no—
194#408—51no—
195#344—151no—
196#341—151no—
197C22×C3×C5—41no—
198C22×C3×C5C4×D411yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5#12311yescharacteristic
201C2×C5×D3—22no—
202C2×C5×D3—81no—
203C2×C3×D5—42no—
204C2×C3×D5—41no—
205C3×C4×C5C2×C22⋊C411yescharacteristic
206C3×C4×C5C4×D411yescharacteristic
207C3×C4×C5#12011yescharacteristic
208C3×C4×C5#12111yescharacteristic
209C3×Dic5—41no—
210C5×Dic3—22no—
211C5×Dic3—81no—
212C3×#41—101no—
213C3×#41—101no—
214C3×#41—101no—
215C3×#41—101no—
216C2×C3×Q8—51no—
217C2×C3×D4—51no—
218C2×C3×D4—101no—
219C22×C3×C4—101no—
220C22×C3×C4C22×D511yescharacteristic
221C2×#67—21no—
222#235—21no—
223#235—21no—
224#235C4×D512yes—
225#235D2012yes—
226C2×D12C22⋊D511yescharacteristic
227C2×C4×D3—21no—
228C2×Dic6C22⋊D511yescharacteristic
229C3×M16—101no—
230C3×M16—101no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—52no—
235#217—21no—
236#216—102no—
237#215—102no—
238#214—102no—
239#213—102no—
240C4⋊Dic3—21no—
241C4×Dic3—21no—
242#208—52no—
243#208—52no—
244#208—101no—
245#208—101no—
246C2×C3⋊C8—101no—
247C2×C3⋊C8—101no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5—61no—
251C23×C5#23511yescharacteristic
252C22×D5—21no—
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×C5D3×D411yescharacteristic
265C2×C4×C5D3×D411yescharacteristic
266C2×C4×C5D3×D411yescharacteristic
267C2×C4×C5#23711yescharacteristic
268C2×C4×C5C2×#6711yescharacteristic
269C22⋊D5—41no—
270C22⋊D5—41no—
271C2×Dic5—21no—
272D20—21no—
273D20—21no—
274C4×D5—41no—
275C4×D5—41no—
276Dic10—21no—
277Dic10—21no—
278C5×C8—41no—
279#168—22no—
280#168—62no—
281#168—62no—
282C2×#41—31no—
283C2×#41—51no—
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—151no—
293C2×M16—151no—
294C2×M16—302no—
295C22×C8D3×D511yescharacteristic
296#117—31no—
297#104—302no—
298#103—61no—
299#102—61no—
300C22⋊C8—102no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5#38611yescharacteristic
305C3×D5—81no—
306C5×D3—42no—
307C23×C3—101no—
308C23×C3#71111yescharacteristic
309C22×D3—21no—
310C3×Q8—101no—
311C3×Q8—101no—
312C3×D4—101no—
313C3×D4—101no—
314C3×D4—201no—
315C3×D4—201no—
316C2×C3×C4—101no—
317C2×C3×C4—201no—
318C2×C3×C4—201no—
319C2×C3×C4C2×C4×D511yescharacteristic
320C2×C3×C4C2×D2011yescharacteristic
321C2×C3×C4D4×D511yescharacteristic
322C2×C3×C4D4×D511yescharacteristic
323C2×C3×C4#71111yescharacteristic
324C2×C3×C4C2×C22⋊D511yescharacteristic
325#67—22no—
326#67—22no—
327C2×Dic3—21no—
328C2×Dic3—41no—
329D12—21no—
330D12#68512yes—
331C4×D3—22no—
332C4×D3—22no—
333Dic6—21no—
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#88311yescharacteristic
343C22×C5#88411yescharacteristic
344C22×C5C4×#6711yescharacteristic
345C2×D5—42no—
346C2×D5—41no—
347C4×C5—62no—
348C4×C5—241no—
349C4×C5D3×C22⋊C411yescharacteristic
350C4×C5#67⋊C411yescharacteristic
351C4×C5#93711yescharacteristic
352C4×C5#93811yescharacteristic
353Dic5—41no—
354#41—32no—
355#41—32no—
356#41—61no—
357#41—61no—
358#41—101no—
359#41—101no—
360#41—101no—
361#41—101no—
362C2×Q8—31no—
363C2×Q8—51no—
364C2×D4—31no—
365C2×D4—51no—
366C2×D4—61no—
367C2×D4—101no—
368C22×C4—61no—
369C22×C4—101no—
370C22×C4C2×D3×D511yescharacteristic
371Q16—302no—
372SD16—302no—
373SD16—302no—
374D8—302no—
375M16—101no—
376M16—101no—
377M16—152no—
378M16—152no—
379M16—304no—
380M16—301no—
381M16—301no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—52no—
386C2×C8—304no—
387C2×C8—301no—
388C2×C8—301no—
389C4⋊C4—61no—
390C22⋊C4—61no—
391C42—61no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3—202no—
395C22×C3—201no—
396C22×C3#402711yescharacteristic
397C22×C3#403211yescharacteristic
398C22×C3#408511yescharacteristic
399C2×D3—22no—
400C2×D3—81no—
401C3×C4—201no—
402C3×C4#402711yescharacteristic
403C3×C4#402811yescharacteristic
404C3×C4C2×#68511yescharacteristic
405C3×C4#408511yescharacteristic
406Dic3—22no—
407Dic3—81no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5—122no—
412C2×C5#495611yescharacteristic
413D5—81no—
414C23—61no—
415C23—101no—
416C23#677211yescharacteristic
417Q8—32no—
418Q8—61no—
419Q8—101no—
420Q8—101no—
421D4—32no—
422D4—62no—
423D4—62no—
424D4—61no—
425D4—101no—
426D4—101no—
427D4—201no—
428D4—201no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—101no—
433C2×C4—121no—
434C2×C4—201no—
435C2×C4—201no—
436C2×C4C4×D3×D511yescharacteristic
437C2×C4D3×D2011yescharacteristic
438C2×C4#677711yescharacteristic
439C2×C4#678011yescharacteristic
440C2×C4D5×#6711yescharacteristic
441C2×C4D3×C22⋊D511yescharacteristic
442C8—41no—
443C8—102no—
444C8—302no—
445C8—302no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—401no—
450C2×C3#6524711yescharacteristic
451D3—42no—
452C5#7001211yescharacteristic, complemented, Sylow
453C22—41no—
454C22—62no—
455C22—202no—
456C22—201no—
457C22—241no—
458C22#9144011yescharacteristic
459C22#9145311yescharacteristic
460C22#9145711yescharacteristic
461C4—62no—
462C4—201no—
463C4—241no—
464C4#9144211yescharacteristic
465C4D3×#68511yescharacteristic
466C4#9147411yescharacteristic
467C4#9156511yescharacteristic
468C3#1060108611yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—122no—
473C2—401no—
474C2#1174891111yescharacteristic
subgroupG/H
centerC4#91474
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×C8C22
socleC2×C3×C5#386
prime radical#11754138C2
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. C5×#5113C2semidirect
4. C3×#65399C2semidirect
5. #11748421C2semidirect
6. #11746373C2semidirect
7. #11745988C2semidirect
16. #91309C4semidirect
19. #91298C4semidirect
32. #65399D3semidirect
78. #5113D5semidirect
392. C3×C5#1881semidirect
452. C5#70012semidirect
468. C3#10601086semidirect

Maximal subgroups

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

subgroupindex
#117541382
#117541252
C5×#51132
C3×#653992
#117484212
#117463732
#117459882
#106010863
#700125