Finite Group Databasebeta

← #14301201

Subgroups — #14301201

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#11756504C211yesmaximal, characteristic, complemented
2#11754154C211yesmaximal, characteristic
3C3×#65418C211yesmaximal, characteristic, complemented
4#11748421C211yesmaximal, characteristic, complemented
5#11748393C211yesmaximal, characteristic, complemented
6C5×#4556C211yesmaximal, characteristic, complemented
7#11745984C211yesmaximal, characteristic, complemented
8#10599828—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13#91796—22no—
14M16×D15—22no—
15#91790—22no—
16C5×#926C2211yescharacteristic
17C2×C5×#208C2211yescharacteristic
18C22×C3×#168C2211yescharacteristic
19#91315C412yescomplemented
20#91313—21no—
21#91312C412yescomplemented
22#91311—22no—
23#91301—21no—
24C3×#3952—22no—
25#90977—22no—
26#90975—21no—
27#90969—21no—
28#70012—51nomaximal, Hall
29#66082—31no—
30#66079—31no—
31#65418D311yescharacteristic, complemented
32#65294—31no—
33#65291—31no—
34C5×#344—31no—
35#64778—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
53C3×C5×M16—21no—
54C3×C5×M16—21no—
55C2×C3×C5×C8D412yes—
56#6710—21no—
57#6710D412yes—
58C2×C3⋊#168—21no—
59#6705—24no—
60C8×D15—24no—
61C5×#217—21no—
62C5×C4⋊Dic3—21no—
63C4×C5×Dic3—21no—
64C5×#208—21no—
65C5×#208C2×C412yes—
66C2×C5×C3⋊C8—21no—
67C2×C3×#168—21no—
68C2×C3×#168—21no—
69C2×C3×#168—21no—
70#6662—22no—
71#6658—22no—
72#6657—22no—
73#6653—22no—
74#5822—51no—
75#5745—51no—
76C3×#408—51no—
77#5236—51no—
78#5113—51no—
79#4556D511yescharacteristic, complemented
80#4552—51no—
81C2×#709—31no—
82C2×#709C2×D311yescharacteristic
83C2×C5×M16—31no—
84C2×C5×M16C2×D311yescharacteristic
85C5×#117—31no—
86C2×#681—31no—
87C22×#168C2×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#3952—22no—
97#3942—61no—
98#3941—61no—
99#3937—62no—
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—
113D60—21no—
114D60C22⋊C412yes—
115C4×D15—22no—
116C4×D15—22no—
117Dic30—21no—
118Dic30C22⋊C412yes—
119C2×C5×Dic3—41no—
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—22no—
130C3⋊#168—22no—
131C3×#168—41no—
132C5×C3⋊C8—22no—
133C2×C3×#41—51no—
134C2×#235—51no—
135C2×C3×M16C2×D511yescharacteristic
136C22×C3×C8—51no—
137#951—101no—
138#950—102no—
139#949—101no—
140#942—52no—
141#932—52no—
142#926C2×D511yescharacteristic
143C2×#208—51no—
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—21no—
170C5×M16—21no—
171C5×M16—32no—
172C5×M16—61no—
173C2×C5×C8—61no—
174C2×C5×C8#6712yes—
175C5×C4⋊C4—61no—
176C5×C22⋊C4—61no—
177C42×C5—61no—
178#681—32no—
179#681—61no—
180C2×#168—21no—
181C2×#168—21no—
182C2×#168—21no—
183C2×#168—61no—
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—31no—
196#341—151no—
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—
210C3×Dic5—41no—
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—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—21no—
230C3×M16—21no—
231C2×C3×C8—101no—
232C2×C3×C8—101no—
233C2×C3×C8—101no—
234C2×C3×C8D2012yes—
235#217—21no—
236#216—102no—
237#215—102no—
238#214—102no—
239#213—102no—
240C4⋊Dic3—21no—
241C4×Dic3—21no—
242#208—21no—
243#208—52no—
244#208—101no—
245#208C4×D512yes—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—101no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5#23511yescharacteristic
251C22×D5—21no—
252C22×D5—61no—
253C2×C4×C5—121no—
254C2×C4×C5C2×C4×D311yescharacteristic
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5D3×D411yescharacteristic
257C2×C4×C5D3×D411yescharacteristic
258C2×C4×C5#23711yescharacteristic
259C2×C4×C5C2×#6711yescharacteristic
260C22⋊D5—41no—
261C22⋊D5—41no—
262C22⋊D5—62no—
263C22⋊D5—62no—
264C2×Dic5—21no—
265C2×Dic5—61no—
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—22no—
279C5×C8—62no—
280#168—41no—
281#168—62no—
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—151no—
293C2×M16—302no—
294C2×M16D3×D511yescharacteristic
295C22×C8—51no—
296#117—31no—
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#70911yescharacteristic
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×C4D4×D511yescharacteristic
322C2×C3×C4D4×D511yescharacteristic
323C2×C3×C4D4×D511yescharacteristic
324C2×C3×C4#71311yescharacteristic
325#67—102no—
326#67—102no—
327C2×Dic3—41no—
328C2×Dic3—101no—
329D12—52no—
330D12—101no—
331C4×D3—102no—
332C4×D3—102no—
333Dic6—52no—
334Dic6—101no—
335C3×C8—22no—
336C3×C8—201no—
337C3⋊C8—22no—
338C3⋊C8—102no—
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×C5—241no—
348C4×C5D3×C22⋊C411yescharacteristic
349C4×C5#67⋊C411yescharacteristic
350C4×C5#93711yescharacteristic
351C4×C5#93811yescharacteristic
352Dic5—41no—
353Dic5—62no—
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—21no—
376M16—21no—
377M16—32no—
378M16—61no—
379M16—152no—
380M16—304no—
381M16—301no—
382C2×C8—61no—
383C2×C8—101no—
384C2×C8—101no—
385C2×C8—101no—
386C2×C8—304no—
387C2×C8—301no—
388C2×C8#120212yes—
389C4⋊C4—61no—
390C22⋊C4—61no—
391C42—61no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3—202no—
395C22×C3—201no—
396C22×C3C4×D2011yescharacteristic
397C22×C3#403011yescharacteristic
398C22×C3#404011yescharacteristic
399C2×D3—102no—
400C2×D3—401no—
401C3×C4—201no—
402C3×C4D5×C22⋊C411yescharacteristic
403C3×C4#402811yescharacteristic
404C3×C4#403911yescharacteristic
405C3×C4#404111yescharacteristic
406Dic3—81no—
407Dic3—102no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5#495611yescharacteristic
412D5—81no—
413D5—122no—
414C23—101no—
415C23—301no—
416C23#677111yescharacteristic
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—121no—
431C2×C4—201no—
432C2×C4—201no—
433C2×C4—301no—
434C2×C4—302no—
435C2×C4—302no—
436C2×C4#676611yescharacteristic
437C2×C4C4×D3×D511yescharacteristic
438C2×C4#677711yescharacteristic
439C2×C4C2×#120211yescharacteristic
440C2×C4#679011yescharacteristic
441C2×C4#679011yescharacteristic
442C8—22no—
443C8—62no—
444C8—201no—
445C8—302no—
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#9135411yescharacteristic
459C22C4×#120211yescharacteristic
460C22#9146111yescharacteristic
461C4—201no—
462C4—241no—
463C4—302no—
464C4#9139611yescharacteristic
465C4#9146511yescharacteristic
466C4#9147311yescharacteristic
467C4#9157211yescharacteristic
468C3#1059982811yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—401no—
473C2—602no—
474C2#1174891711yescharacteristic
subgroupG/H
centerC4#91572
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11754154C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1881
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. #11756504C2semidirect
3. C3×#65418C2semidirect
4. #11748421C2semidirect
5. #11748393C2semidirect
6. C5×#4556C2semidirect
7. #11745984C2semidirect
19. #91315C4semidirect
21. #91312C4semidirect
31. #65418D3semidirect
79. #4556D5semidirect
392. C3×C5#1881semidirect
452. C5#70012semidirect
468. C3#10599828semidirect

Maximal subgroups

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

subgroupindex
#117565042
#117541542
C3×#654182
#117484212
#117483932
C5×#45562
#117459842
#105998283
#700125