Finite Group Databasebeta

← #14302862

Subgroups — #14302862

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
1#11756564C211yesmaximal, characteristic
2#11754130C211yesmaximal, characteristic, complemented
3C5×#5215C211yesmaximal, characteristic, complemented
4C3×#65504C211yesmaximal, characteristic, complemented
5#11748626C211yesmaximal, characteristic, complemented
6#11745971C211yesmaximal, characteristic, complemented
7#11745970C211yesmaximal, characteristic, complemented
8#10601123—31nomaximal, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11#91842—22no—
12#91829C412yescomplemented
13#91827—22no—
14#91825C412yescomplemented
15#91815C2211yescharacteristic
16C5×#926C2211yescharacteristic
17C2×C5×#208C2211yescharacteristic
18C3×#4072C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20#91292—22no—
21#91291—22no—
22#91290—22no—
23C5×#832—22no—
24C3×#3964—22no—
25#90997—22no—
26#70330—51nomaximal, Hall
27#66139—31no—
28#66079—31no—
29C5×#418—31no—
30#65504D311yescharacteristic, complemented
31#65295—31no—
32#64777—31no—
33#64751—31no—
34C24×C3×C5—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4C2×C412yes—
41C22×C3×C4×C5C2311yescharacteristic
42C2×#1220—24no—
43C2×#1220D412yes—
44#6817—21no—
45#6817—21no—
46#6817—21no—
47#6817—21no—
48C2×D60C2×C411yescharacteristic
49C2×C4×D15D412yes—
50C2×Dic30C2×C411yescharacteristic
51#6719—21no—
52#6719—24no—
53C4⋊Dic15—21no—
54C4×Dic15C2×C412yes—
55C5×#217—21no—
56C5×C4⋊Dic3—21no—
57C4×C5×Dic3—21no—
58C5×#208—21no—
59C5×#208—21no—
60C2×C5×C3⋊C8D412yescomplemented
61C3×C22⋊Dic5—21no—
62C3×C4⋊Dic5—21no—
63C3×C4×Dic5—21no—
64C3×#681—21no—
65C3×#681—21no—
66C2×C3×#168D412yescomplemented
67#6651—24no—
68#6648—24no—
69#5805—51no—
70#5745—51no—
71C3×#418—51no—
72#5215D511yescharacteristic, complemented
73#5006—51no—
74#4557—51no—
75#4531—51no—
76C22×C5×D4C2×D311yescharacteristic
77C2×#709—31no—
78C2×C5×M16—31no—
79C5×#117—31no—
80#4099—62no—
81#4086—32no—
82#4084—62no—
83#4082—32no—
84#4072—31no—
85#4072C2×D311yescharacteristic
86C2×#681C2×D311yescharacteristic
87#4053—62no—
88D5×M16—62no—
89#4047—62no—
90C5×#102—62no—
91#3964—22no—
92#3938—62no—
93#1910—151noSylow
94C23×C3×C5—22no—
95C23×C3×C5—41no—
96C23×C3×C5—41no—
97C23×C3×C5—82no—
98C23×C3×C5C2×D411yescharacteristic
99C22×D15C22⋊C412yes—
100C3×C5×D4—22no—
101C3×C5×D4—22no—
102C3×C5×D4—41no—
103C3×C5×D4—41no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110#1220—22no—
111#1220—22no—
112#1220—84no—
113C2×Dic15—22no—
114C2×Dic15C22⋊C412yes—
115D60—21no—
116D60—21no—
117C4×D15—22no—
118C4×D15—22no—
119Dic30—21no—
120Dic30—21no—
121C2×C5×Dic3—41no—
122C2×C3×Dic5—41no—
123C3×#168—22no—
124C5×C3⋊C8—22no—
125C22×C3×D4C2×D511yescharacteristic
126C2×#235—51no—
127C2×C3×M16—51no—
128C3×#117—51no—
129#953—102no—
130#940—52no—
131#938—102no—
132#936—52no—
133#926—51no—
134#926C2×D511yescharacteristic
135C2×#208C2×D511yescharacteristic
136#907—102no—
137D3×M16—102no—
138#901—102no—
139C3×#102—102no—
140#832—22no—
141#806—102no—
142C24×C5—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4C4×D312yes—
149C22×C4×C5C22×D311yescharacteristic
150C2×C22⋊D5—32no—
151C2×C22⋊D5—64no—
152#709—61no—
153#709—61no—
154#709—61no—
155#709—61no—
156C2×D20—31no—
157C2×C4×D5—32no—
158C2×Dic10—31no—
159C5×M16—61no—
160C5×M16—61no—
161C2×C5×C8—32no—
162C5×C4⋊C4—61no—
163C5×C22⋊C4—61no—
164C42×C5—61no—
165C22⋊Dic5—21no—
166C22⋊Dic5—64no—
167C22⋊Dic5—61no—
168C4⋊Dic5—21no—
169C4⋊Dic5—61no—
170C4×Dic5—21no—
171C4×Dic5—32no—
172#681—21no—
173#681—21no—
174C2×#168#6712yescomplemented
175#676—64no—
176C8×D5—64no—
177#568—151no—
178#535—151no—
179#418—31no—
180#418—51no—
181#410—151no—
182H(3,ℤ/4)—151no—
183H(3,ℤ/4)—151no—
184C22×C3×C5—41no—
185C22×C3×C5—44no—
186C22×C3×C5—42no—
187C22×C3×C5—82no—
188C22×C3×C5—82no—
189C22×C3×C5C2×C22⋊C411yescharacteristic
190C22×C3×C5#12111yescharacteristic
191C22×C3×C5#12111yescharacteristic
192C2×D15—22no—
193C2×D15—42no—
194C3×C4×C5C4×D411yescharacteristic
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5#12011yescharacteristic
197C3×C4×C5#12311yescharacteristic
198Dic15—22no—
199Dic15—42no—
200C3×Dic5—81no—
201C5×Dic3—81no—
202C24×C3—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4C4×D512yes—
209C22×C3×C4C22×D511yescharacteristic
210C2×#67—52no—
211C2×#67—104no—
212#235—101no—
213#235—101no—
214#235—101no—
215#235—101no—
216C2×D12—51no—
217C2×C4×D3—52no—
218C2×Dic6—51no—
219C3×M16—101no—
220C3×M16—101no—
221C2×C3×C8—52no—
222C3×C4⋊C4—101no—
223C3×C22⋊C4—101no—
224C3×C42—101no—
225#217—21no—
226#217—104no—
227#217—101no—
228C4⋊Dic3—21no—
229C4⋊Dic3—101no—
230C4×Dic3—21no—
231C4×Dic3—52no—
232#208—21no—
233#208—21no—
234C2×C3⋊C8C22⋊D512yescomplemented
235ΓL(1,𝔽25)—104no—
236D3×C8—104no—
237C23×C5—22no—
238C23×C5—41no—
239C23×C5—41no—
240C23×C5—82no—
241C23×C5D3×D411yescharacteristic
242C22×D5—32no—
243C5×D4—22no—
244C5×D4—22no—
245C5×D4—41no—
246C5×D4—41no—
247C2×C4×C5—121no—
248C2×C4×C5C2×C4×D311yescharacteristic
249C2×C4×C5#23511yescharacteristic
250C2×C4×C5D3×D411yescharacteristic
251C2×C4×C5D3×D411yescharacteristic
252C2×C4×C5#23711yescharacteristic
253C2×C4×C5C2×#6711yescharacteristic
254C22⋊D5—62no—
255C22⋊D5—62no—
256C22⋊D5—244no—
257C2×Dic5—32no—
258C2×Dic5—41no—
259C2×Dic5—62no—
260D20—61no—
261D20—61no—
262C4×D5—62no—
263C4×D5—62no—
264Dic10—61no—
265Dic10—61no—
266C5×C8—62no—
267#168—22no—
268C2×#41—151no—
269C22×D4D3×D511yescharacteristic
270#131—302no—
271#131—302no—
272C2×M16—31no—
273C2×M16—51no—
274C2×M16—302no—
275#127—152no—
276#124—152no—
277#121—302no—
278#120—302no—
279#117—31no—
280#117—51no—
281#117—151no—
282#102—62no—
283#102—102no—
284#99—302no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5—82no—
289C2×C3×C5#38611yescharacteristic
290D15—42no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—82no—
295C23×C3D4×D511yescharacteristic
296C22×D3—52no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—41no—
300C3×D4—41no—
301C2×C3×C4—201no—
302C2×C3×C4C2×C4×D511yescharacteristic
303C2×C3×C4#70911yescharacteristic
304C2×C3×C4D4×D511yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4#71111yescharacteristic
307C2×C3×C4C2×C22⋊D511yescharacteristic
308#67—102no—
309#67—102no—
310#67—404no—
311C2×Dic3—41no—
312C2×Dic3—52no—
313C2×Dic3—102no—
314D12—101no—
315D12—101no—
316C4×D3—102no—
317C4×D3—102no—
318Dic6—101no—
319Dic6—101no—
320C3×C8—102no—
321C3⋊C8—22no—
322C22×C5—41no—
323C22×C5—44no—
324C22×C5—42no—
325C22×C5—82no—
326C22×C5—82no—
327C22×C5D3×C22⋊C411yescharacteristic
328C22×C5#88411yescharacteristic
329C22×C5#93811yescharacteristic
330C2×D5—62no—
331C2×D5—122no—
332C4×C5—241no—
333C4×C5#67⋊C411yescharacteristic
334C4×C5#88311yescharacteristic
335C4×C5C4×#6711yescharacteristic
336C4×C5#93711yescharacteristic
337Dic5—62no—
338Dic5—81no—
339Dic5—122no—
340C24—21no—
341#41—301no—
342#41—301no—
343#41—301no—
344#41—301no—
345C2×Q8—151no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—152no—
352C2×D4—151no—
353C2×D4—304no—
354C2×D4#120012yes—
355C22×C4—152no—
356C22×C4C2×D3×D511yescharacteristic
357M16—61no—
358M16—61no—
359M16—101no—
360M16—101no—
361M16—304no—
362C2×C8—32no—
363C2×C8—52no—
364C2×C8—304no—
365C4⋊C4—61no—
366C4⋊C4—101no—
367C4⋊C4—301no—
368C22⋊C4—61no—
369C22⋊C4—101no—
370C22⋊C4—304no—
371C22⋊C4—301no—
372C42—61no—
373C42—101no—
374C42—152no—
375C3×C5#191011yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3—44no—
378C22×C3—42no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3D5×C22⋊C411yescharacteristic
382C22×C3#403011yescharacteristic
383C22×C3#408411yescharacteristic
384C2×D3—102no—
385C2×D3—202no—
386C3×C4—401no—
387C3×C4#402711yescharacteristic
388C3×C4#402911yescharacteristic
389C3×C4C4×C22⋊D511yescharacteristic
390C3×C4#408311yescharacteristic
391Dic3—81no—
392Dic3—102no—
393Dic3—202no—
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5—82no—
398C2×C5#495611yescharacteristic
399D5—122no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23—82no—
404C23—152no—
405C23#679011yescharacteristic
406Q8—301no—
407Q8—301no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412D4—302no—
413D4—302no—
414D4—301no—
415D4—301no—
416D4—1204no—
417C2×C4—121no—
418C2×C4—152no—
419C2×C4—201no—
420C2×C4—302no—
421C2×C4—302no—
422C2×C4—302no—
423C2×C4#677911yescharacteristic
424C2×C4#678211yescharacteristic
425C2×C4C2×#120011yescharacteristic
426C2×C4D5×#6711yescharacteristic
427C2×C4D3×C22⋊D511yescharacteristic
428C2×C4#679011yescharacteristic
429C8—62no—
430C8—102no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—82no—
435C2×C3#6524511yescharacteristic
436D3—202no—
437C5#7033011yescharacteristic, complemented, Sylow
438C22—41no—
439C22—44no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22—302no—
444C22—602no—
445C22#9153311yescharacteristic
446C22#9155711yescharacteristic
447C22#9156111yescharacteristic
448C4—241no—
449C4—302no—
450C4—401no—
451C4—602no—
452C4#9155211yescharacteristic
453C4#9155611yescharacteristic
454C4#9156211yescharacteristic
455C4#9157711yescharacteristic
456C3#1060112311yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—82no—
461C2—602no—
462C2#1174952011yescharacteristic
subgroupG/H
centerC2#11749520
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×D4C22
socleC2×C3×C5#386
prime radical#11756564C2
cosocleC22×C3×C4×C5C23
hypercenterC22×D4D3×D5
hypocenterC3×C5#1910
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #11754130C2semidirect
3. C5×#5215C2semidirect
4. C3×#65504C2semidirect
5. #11748626C2semidirect
6. #11745971C2semidirect
7. #11745970C2semidirect
12. #91829C4semidirect
14. #91825C4semidirect
30. #65504D3semidirect
60. C2×C5×C3⋊C8D4semidirect
66. C2×C3×#168D4semidirect
72. #5215D5semidirect
174. C2×#168#67semidirect
234. C2×C3⋊C8C22⋊D5semidirect
375. C3×C5#1910semidirect
437. C5#70330semidirect
456. C3#10601123semidirect

Maximal subgroups

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

subgroupindex
#117565642
#117541302
C5×#52152
C3×#655042
#117486262
#117459712
#117459702
#106011233
#703305