Finite Group Databasebeta

← #14301195

Subgroups — #14301195

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#11754154C211yesmaximal, characteristic
2#11754130C211yesmaximal, characteristic, complemented
3#11751214C211yesmaximal, characteristic, complemented
4C3×#65399C211yesmaximal, characteristic, complemented
5#11748420C211yesmaximal, characteristic, complemented
6C5×#4552C211yesmaximal, characteristic, complemented
7#11745991C211yesmaximal, characteristic, complemented
8#10599740—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×C5×#208C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91315C412yescomplemented
17#91313—21no—
18#91312C412yescomplemented
19#91311—22no—
20#91301—21no—
21#91292—22no—
22#91291—22no—
23#91290—22no—
24#91105—21no—
25#91100—21no—
26C3×#3952—22no—
27#90976—22no—
28#70012—51nomaximal, Hall
29#66082—31no—
30#66079—31no—
31#65428—31no—
32#65402—31no—
33#65399D311yescharacteristic, complemented
34C5×#341—31no—
35#64779—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220—21no—
38#6817—21no—
39#6817—21no—
40#6817C2×C412yes—
41#6817D412yes—
42C2×D60D411yescharacteristic
43C2×C4×D15—21no—
44C2×Dic30D411yescharacteristic
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#6719—21no—
57C4⋊Dic15—21no—
58C4×Dic15—21no—
59C5×#208—21no—
60C5×#208—21no—
61C5×#208C2×C412yes—
62C5×#208D412yes—
63C2×C5×C3⋊C8—21no—
64C2×C5×C3⋊C8—21no—
65C3×#681—21no—
66C3×#681—21no—
67C2×C3×#168D412yes—
68#6662—22no—
69#6658—22no—
70#6657—22no—
71#6653—22no—
72#6651—24no—
73#6648—24no—
74#5822—51no—
75#5745—51no—
76C3×#408—51no—
77#5236—51no—
78#5113—51no—
79#4556—51no—
80#4552D511yescharacteristic, complemented
81C2×#709—31no—
82C2×#709C2×D311yescharacteristic
83C2×C5×M16—31no—
84C2×C5×M16—31no—
85C22×C5×C8C2×D311yescharacteristic
86#4072—31no—
87C2×#681C2×D311yescharacteristic
88#4055—32no—
89#4055—61no—
90#4054—32no—
91#4054—61no—
92#4053—62no—
93D5×M16—62no—
94#4050—62no—
95#4047—62no—
96#3957—62no—
97#3953—61no—
98#3952—22no—
99#3948—61no—
100#1881—151noSylow
101C23×C3×C5#4111yescharacteristic
102C22×D15—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
110#1220—22no—
111#1220—22no—
112C2×Dic15—21no—
113C2×Dic15—41no—
114D60—21no—
115D60C22⋊C412yes—
116C4×D15—22no—
117C4×D15—22no—
118Dic30—21no—
119Dic30C22⋊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—
131C5×C3⋊C8—22no—
132C5×C3⋊C8—22no—
133C2×C3×#41—51no—
134C2×#235—51no—
135C2×C3×M16—51no—
136C22×C3×C8C2×D511yescharacteristic
137#951—101no—
138#950—102no—
139#949—101no—
140#942—52no—
141#932—52no—
142#926—51no—
143C2×#208C2×D511yescharacteristic
144C2×#208C2×D511yescharacteristic
145#907—102no—
146D3×M16—102no—
147#901—102no—
148C3×C22⋊C8—102no—
149#837—102no—
150#821—101no—
151#816—101no—
152C22×C4×C5C22×D311yescharacteristic
153C2×C22⋊D5—21no—
154C2×C22⋊D5—61no—
155#709—21no—
156#709—21no—
157#709—21no—
158#709—21no—
159#709—32no—
160#709—32no—
161#709—61no—
162#709—61no—
163C2×D20—31no—
164C2×D20C4×D311yescharacteristic
165C2×C4×D5—21no—
166C2×C4×D5—61no—
167C2×Dic10—31no—
168C2×Dic10C4×D311yescharacteristic
169C5×M16—32no—
170C5×M16—32no—
171C5×M16—61no—
172C5×M16—61no—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—61no—
177C2×C5×C8—61no—
178C22⋊Dic5—61no—
179C4⋊Dic5—61no—
180C4×Dic5—61no—
181#681—21no—
182#681—21no—
183C2×#168#6712yes—
184Dic20—62no—
185D40—62no—
186#677—62no—
187#677—62no—
188#676—64no—
189C8×D5—64no—
190#575—151no—
191#568—151no—
192#421—151no—
193#416—151no—
194#408—51no—
195#344—151no—
196#341—31no—
197C22×C3×C5—41no—
198C22×C3×C5C4×D411yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5#12311yescharacteristic
201C2×D15—22no—
202C2×D15—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
209Dic15—22no—
210Dic15—81no—
211C3×Dic5—41no—
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—101no—
222#235—52no—
223#235—52no—
224#235—101no—
225#235—101no—
226C2×D12—51no—
227C2×C4×D3—101no—
228C2×Dic6—51no—
229C3×M16—101no—
230C3×M16—101no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—52no—
235#217—101no—
236#216—102no—
237#215—102no—
238#214—102no—
239#213—102no—
240C4⋊Dic3—101no—
241C4×Dic3—101no—
242#208—21no—
243#208—21no—
244#208C4×D512yes—
245#208D2012yes—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5#23511yescharacteristic
251C22×D5—21no—
252C22×D5—61no—
253C2×C4×C5C2×C4×D311yescharacteristic
254C2×C4×C5D3×D411yescharacteristic
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5D3×D411yescharacteristic
257C2×C4×C5#23711yescharacteristic
258C2×C4×C5C2×#6711yescharacteristic
259C22⋊D5—41no—
260C22⋊D5—41no—
261C22⋊D5—62no—
262C22⋊D5—62no—
263C2×Dic5—21no—
264C2×Dic5—61no—
265C2×Dic5—121no—
266D20—21no—
267D20—21no—
268D20—32no—
269D20—61no—
270C4×D5—41no—
271C4×D5—41no—
272C4×D5—62no—
273C4×D5—62no—
274Dic10—21no—
275Dic10—21no—
276Dic10—32no—
277Dic10—61no—
278C5×C8—41no—
279C5×C8—62no—
280C5×C8—62no—
281#168—22no—
282C2×#41—51no—
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—31no—
292C2×M16—31no—
293C2×M16—51no—
294C2×M16—302no—
295C22×C8D3×D511yescharacteristic
296#117—151no—
297#104—302no—
298#103—301no—
299#102—301no—
300C22⋊C8—102no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5#38611yescharacteristic
305D15—42no—
306C3×D5—81no—
307C23×C3—101no—
308C23×C3#71311yescharacteristic
309C22×D3—101no—
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×C4#70911yescharacteristic
322C2×C3×C4D4×D511yescharacteristic
323C2×C3×C4D4×D511yescharacteristic
324C2×C3×C4D4×D511yescharacteristic
325#67—102no—
326#67—102no—
327C2×Dic3—101no—
328C2×Dic3—201no—
329D12—52no—
330D12—101no—
331C4×D3—102no—
332C4×D3—102no—
333Dic6—52no—
334Dic6—101no—
335C3×C8—41no—
336C3×C8—102no—
337C3⋊C8—22no—
338C3⋊C8—22no—
339C22×C5—41no—
340C22×C5#88311yescharacteristic
341C22×C5#88411yescharacteristic
342C22×C5C4×#6711yescharacteristic
343C2×D5—42no—
344C2×D5—41no—
345C2×D5—62no—
346C2×D5—241no—
347C4×C5D3×C22⋊C411yescharacteristic
348C4×C5#67⋊C411yescharacteristic
349C4×C5#93711yescharacteristic
350C4×C5#93811yescharacteristic
351Dic5—41no—
352Dic5—62no—
353Dic5—241no—
354#41—101no—
355#41—101no—
356#41—101no—
357#41—101no—
358#41—152no—
359#41—152no—
360#41—301no—
361#41—301no—
362C2×Q8—51no—
363C2×Q8—151no—
364C2×D4—51no—
365C2×D4—101no—
366C2×D4—151no—
367C2×D4—301no—
368C22×C4—101no—
369C22×C4—301no—
370C22×C4C2×D3×D511yescharacteristic
371Q16—302no—
372SD16—302no—
373SD16—302no—
374D8—302no—
375M16—32no—
376M16—32no—
377M16—61no—
378M16—61no—
379M16—101no—
380M16—101no—
381M16—304no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—52no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—304no—
389C4⋊C4—301no—
390C22⋊C4—301no—
391C42—301no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3—202no—
395C22×C3—201no—
396C22×C3#403911yescharacteristic
397C22×C3#404011yescharacteristic
398C22×C3#404111yescharacteristic
399C2×D3—102no—
400C2×D3—401no—
401C3×C4—201no—
402C3×C4C4×D2011yescharacteristic
403C3×C4D5×C22⋊C411yescharacteristic
404C3×C4#402811yescharacteristic
405C3×C4#403011yescharacteristic
406Dic3—102no—
407Dic3—401no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5#495611yescharacteristic
412D5—81no—
413D5—122no—
414C23—101no—
415C23—301no—
416C23#677311yescharacteristic
417Q8—101no—
418Q8—101no—
419Q8—152no—
420Q8—301no—
421D4—101no—
422D4—101no—
423D4—152no—
424D4—201no—
425D4—201no—
426D4—302no—
427D4—302no—
428D4—301no—
429C2×C4—101no—
430C2×C4—201no—
431C2×C4—201no—
432C2×C4—301no—
433C2×C4—302no—
434C2×C4—302no—
435C2×C4—601no—
436C2×C4C4×D3×D511yescharacteristic
437C2×C4D3×D2011yescharacteristic
438C2×C4#677711yescharacteristic
439C2×C4#677911yescharacteristic
440C2×C4D5×#6711yescharacteristic
441C2×C4#679011yescharacteristic
442C8—41no—
443C8—62no—
444C8—62no—
445C8—102no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—401no—
450C2×C3#6495411yescharacteristic
451D3—202no—
452C5#7001211yescharacteristic, complemented, Sylow
453C22—41no—
454C22—202no—
455C22—201no—
456C22—302no—
457C22—1201no—
458C22#9135611yescharacteristic
459C22#9144411yescharacteristic
460C22#9145911yescharacteristic
461C4—201no—
462C4—302no—
463C4—1201no—
464C4#9139511yescharacteristic
465C4#9147311yescharacteristic
466C4#9147411yescharacteristic
467C4#9153311yescharacteristic
468C3#1059974011yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—401no—
473C2—602no—
474C2#1174870011yescharacteristic
subgroupG/H
centerC4#91474
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×C8C22
socleC2×C3×C5#386
prime radical#11754154C2
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. #11754130C2semidirect
3. #11751214C2semidirect
4. C3×#65399C2semidirect
5. #11748420C2semidirect
6. C5×#4552C2semidirect
7. #11745991C2semidirect
16. #91315C4semidirect
18. #91312C4semidirect
33. #65399D3semidirect
80. #4552D5semidirect
392. C3×C5#1881semidirect
452. C5#70012semidirect
468. C3#10599740semidirect

Maximal subgroups

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

subgroupindex
#117541542
#117541302
#117512142
C3×#653992
#117484202
C5×#45522
#117459912
#105997403
#700125