Finite Group Databasebeta

← #14302853

Subgroups — #14302853

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#11756504C211yesmaximal, characteristic, complemented
2#11754602C211yesmaximal, characteristic
3C3×#65420C211yesmaximal, characteristic, complemented
4#11748626C211yesmaximal, characteristic, complemented
5#11748366C211yesmaximal, characteristic, complemented
6C5×#4557C211yesmaximal, characteristic, complemented
7#11745969C211yesmaximal, characteristic, complemented
8#10599506—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13#91796—22no—
14M16×D15—22no—
15#91790—22no—
16C5×#926C2211yescharacteristic
17C5×#926C2211yescharacteristic
18C3×#4072C2211yescharacteristic
19#91564—22no—
20#91465—22no—
21#91458C412yescomplemented
22#91364C412yescomplemented
23C3×#3953—22no—
24#90997—22no—
25#90964—22no—
26#70330—51nomaximal, Hall
27#66079—31no—
28#65859—31no—
29#65420D311yescharacteristic, complemented
30#65295—31no—
31#65290—31no—
32C5×H(3,ℤ/4)—31no—
33#64751—31no—
34C23×C3×D5—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C2×#1220D412yes—
37#6817—21no—
38#6817—21no—
39#6817—21no—
40#6817—21no—
41C2×D60C2×C411yescharacteristic
42C2×C4×D15D412yes—
43C2×Dic30C2×C411yescharacteristic
44C2×C3×D20—21no—
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20—21no—
48C2×C3×D20—21no—
49C2×C3×D20C2×C412yes—
50C2×#1202—24no—
51C3×C5×M16—21no—
52C3×C5×M16—21no—
53C2×C3×C5×C8D412yescomplemented
54#6710—21no—
55#6710—21no—
56C2×C3⋊#168D412yescomplemented
57#6705—24no—
58C8×D15—24no—
59C5×#217—21no—
60C5×#217—21no—
61C5×C4⋊Dic3—21no—
62C5×C4⋊Dic3—21no—
63C4×C5×Dic3—21no—
64C4×C5×Dic3C2×C412yes—
65C3×C22⋊Dic5—21no—
66C3×C4⋊Dic5—21no—
67C3×C4×Dic5—21no—
68#6665—24no—
69#5805—51no—
70#5745—51no—
71C3×#418—51no—
72#5215—51no—
73#5006—51no—
74#4557D511yescharacteristic, complemented
75#4531—51no—
76C2×#709—31no—
77C22×D20C2×D311yescharacteristic
78C2×C5×M16C2×D311yescharacteristic
79C5×#117—31no—
80C5×#117—31no—
81#4072C2×D311yescharacteristic
82C2×#681—31no—
83#4053—62no—
84D5×M16—62no—
85#4047—62no—
86#4041—62no—
87#4028—62no—
88#4021—32no—
89#4020—32no—
90#3953—22no—
91#3941—62no—
92#3938—62no—
93#1910—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15C22⋊C412yes—
96C22×C3×D5—22no—
97C22×C3×D5—41no—
98C22×C3×D5—41no—
99C22×C3×D5—82no—
100C2×C3×C4×C5C22×C411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106#1220—22no—
107#1220—22no—
108C2×Dic15C22⋊C412yes—
109D60—21no—
110D60—21no—
111C4×D15—22no—
112C4×D15—22no—
113Dic30—21no—
114Dic30—21no—
115C2×C5×Dic3—22no—
116C2×C5×Dic3—41no—
117C2×C3×Dic5—41no—
118C3×D20—22no—
119C3×D20—22no—
120C3×D20—41no—
121C3×D20—41no—
122#1202—84no—
123C3×C5×C8—22no—
124C3⋊#168—22no—
125C22×C3×D4—51no—
126C2×#235—51no—
127C2×C3×M16C2×D511yescharacteristic
128C3×#117—51no—
129#953—102no—
130#940—52no—
131#938—102no—
132#936—52no—
133#926C2×D511yescharacteristic
134#926C2×D511yescharacteristic
135C2×#208—51no—
136#907—102no—
137D3×M16—102no—
138#901—102no—
139C3×#102—102no—
140#832—102no—
141#806—102no—
142C23×D5—21no—
143C22×C4×C5C22×D311yescharacteristic
144C2×C22⋊D5—32no—
145#709—61no—
146#709—61no—
147#709—61no—
148#709—61no—
149C2×D20—21no—
150C2×D20—21no—
151C2×D20—21no—
152C2×D20—21no—
153C2×D20—21no—
154C2×D20—31no—
155C2×D20—64no—
156C2×D20C4×D312yes—
157C2×C4×D5—32no—
158C2×Dic10—31no—
159C5×M16—21no—
160C5×M16—21no—
161C2×C5×C8#6712yescomplemented
162C5×C4⋊C4—61no—
163C5×C4⋊C4—61no—
164C5×C22⋊C4—61no—
165C5×C22⋊C4—61no—
166C42×C5—32no—
167C42×C5—61no—
168C22⋊Dic5—21no—
169#685—64no—
170C4⋊Dic5—21no—
171C4×Dic5—21no—
172#681—61no—
173#681—61no—
174C2×#168—32no—
175#676—64no—
176C8×D5—64no—
177#568—151no—
178#535—151no—
179#418—51no—
180#418—151no—
181#410—151no—
182H(3,ℤ/4)—31no—
183H(3,ℤ/4)—151no—
184C22×C3×C5—41no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C22×C3×C5#12111yescharacteristic
187C22×C3×C5#12111yescharacteristic
188C2×D15—22no—
189C2×D15—42no—
190C2×C3×D5—44no—
191C2×C3×D5—42no—
192C2×C3×D5—82no—
193C2×C3×D5—82no—
194C3×C4×C5C4×D411yescharacteristic
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5#12011yescharacteristic
197C3×C4×C5#12311yescharacteristic
198Dic15—22no—
199C3×Dic5—81no—
200C5×Dic3—42no—
201C5×Dic3—81no—
202C24×C3—101no—
203C2×C3×D4—52no—
204C2×C3×D4—101no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C2×C3×D4—101no—
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—21no—
220C3×M16—21no—
221C2×C3×C8D2012yescomplemented
222C3×C4⋊C4—101no—
223C3×C22⋊C4—101no—
224C3×C42—101no—
225#217—21no—
226#217—21no—
227#217—104no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4×Dic3—21no—
231C4×Dic3C4×D512yes—
232#208—101no—
233#208—101no—
234C2×C3⋊C8—52no—
235ΓL(1,𝔽25)—104no—
236D3×C8—104no—
237C23×C5D3×D411yescharacteristic
238C22×D5—22no—
239C22×D5—32no—
240C22×D5—41no—
241C22×D5—41no—
242C22×D5—82no—
243C2×C4×C5—62no—
244C2×C4×C5—121no—
245C2×C4×C5C2×C4×D311yescharacteristic
246C2×C4×C5#23511yescharacteristic
247C2×C4×C5D3×D411yescharacteristic
248C2×C4×C5D3×D411yescharacteristic
249C2×C4×C5#23711yescharacteristic
250C2×C4×C5C2×#6711yescharacteristic
251C22⋊D5—62no—
252C22⋊D5—62no—
253C2×Dic5—32no—
254C2×Dic5—41no—
255D20—22no—
256D20—22no—
257D20—41no—
258D20—41no—
259D20—61no—
260D20—61no—
261D20—244no—
262C4×D5—62no—
263C4×D5—62no—
264Dic10—61no—
265Dic10—61no—
266C5×C8—22no—
267#168—62no—
268C2×#41—151no—
269C22×D4—51no—
270#131—302no—
271#131—302no—
272C2×M16—151no—
273C2×M16—302no—
274C2×M16D3×D511yescharacteristic
275#127—152no—
276#124—152no—
277#121—302no—
278#120—302no—
279#117—31no—
280#117—31no—
281#117—51no—
282#102—102no—
283#102—302no—
284#99—302no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5#38611yescharacteristic
289D15—42no—
290C3×D5—82no—
291C23×C3—102no—
292C23×C3—201no—
293C23×C3—201no—
294C23×C3—402no—
295C23×C3D4×D511yescharacteristic
296C22×D3—52no—
297C3×D4—102no—
298C3×D4—102no—
299C3×D4—201no—
300C3×D4—201no—
301C2×C3×C4—201no—
302C2×C3×C4C2×C4×D511yescharacteristic
303C2×C3×C4C2×D2011yescharacteristic
304C2×C3×C4#70911yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4D4×D511yescharacteristic
307C2×C3×C4#71311yescharacteristic
308#67—102no—
309#67—102no—
310#67—404no—
311C2×Dic3—22no—
312C2×Dic3—41no—
313C2×Dic3—52no—
314D12—101no—
315D12—101no—
316C4×D3—102no—
317C4×D3—102no—
318Dic6—101no—
319Dic6—101no—
320C3×C8—22no—
321C3⋊C8—102no—
322C22×C5—41no—
323C22×C5D3×C22⋊C411yescharacteristic
324C22×C5#88411yescharacteristic
325C22×C5#93811yescharacteristic
326C2×D5—44no—
327C2×D5—42no—
328C2×D5—62no—
329C2×D5—82no—
330C2×D5—82no—
331C2×D5—122no—
332C4×C5—122no—
333C4×C5—241no—
334C4×C5#67⋊C411yescharacteristic
335C4×C5#88311yescharacteristic
336C4×C5C4×#6711yescharacteristic
337C4×C5#93711yescharacteristic
338Dic5—62no—
339Dic5—81no—
340C24—101no—
341#41—301no—
342#41—301no—
343#41—301no—
344#41—301no—
345C2×Q8—151no—
346C2×D4—52no—
347C2×D4—101no—
348C2×D4—101no—
349C2×D4—101no—
350C2×D4—101no—
351C2×D4—101no—
352C2×D4—152no—
353C2×D4—151no—
354C2×D4—304no—
355C22×C4—152no—
356C22×C4C2×D3×D511yescharacteristic
357M16—21no—
358M16—21no—
359M16—304no—
360M16—301no—
361M16—301no—
362C2×C8—152no—
363C2×C8—304no—
364C2×C8#120212yescomplemented
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—101no—
368C22⋊C4—61no—
369C22⋊C4—61no—
370C22⋊C4—101no—
371C22⋊C4—304no—
372C42—32no—
373C42—61no—
374C42—101no—
375C3×C5#191011yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3—204no—
378C22×C3—202no—
379C22×C3—402no—
380C22×C3—402no—
381C22×C3D5×C22⋊C411yescharacteristic
382C22×C3#403011yescharacteristic
383C22×C3#404111yescharacteristic
384C2×D3—102no—
385C2×D3—202no—
386C3×C4—401no—
387C3×C4C4×D2011yescharacteristic
388C3×C4#402811yescharacteristic
389C3×C4#403911yescharacteristic
390C3×C4#404011yescharacteristic
391Dic3—42no—
392Dic3—81no—
393Dic3—102no—
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5#495611yescharacteristic
398D5—82no—
399D5—122no—
400C23—102no—
401C23—152no—
402C23—201no—
403C23—201no—
404C23—402no—
405C23#677711yescharacteristic
406Q8—301no—
407Q8—301no—
408D4—102no—
409D4—102no—
410D4—201no—
411D4—201no—
412D4—302no—
413D4—302no—
414D4—301no—
415D4—301no—
416D4—1204no—
417C2×C4—62no—
418C2×C4—121no—
419C2×C4—152no—
420C2×C4—201no—
421C2×C4—302no—
422C2×C4—302no—
423C2×C4#676611yescharacteristic
424C2×C4#677111yescharacteristic
425C2×C4C4×D3×D511yescharacteristic
426C2×C4C2×#120211yescharacteristic
427C2×C4#679011yescharacteristic
428C2×C4#679011yescharacteristic
429C8—22no—
430C8—302no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—402no—
435C2×C3#6495411yescharacteristic
436D3—202no—
437C5#7033011yescharacteristic, complemented, Sylow
438C22—41no—
439C22—204no—
440C22—202no—
441C22—302no—
442C22—402no—
443C22—402no—
444C22—602no—
445C22#9146111yescharacteristic
446C22#9146511yescharacteristic
447C22#9147311yescharacteristic
448C4—122no—
449C4—241no—
450C4—302no—
451C4—401no—
452C4#9135411yescharacteristic
453C4#9139611yescharacteristic
454C4C4×#120211yescharacteristic
455C4#9157211yescharacteristic
456C3#1059950611yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—402no—
461C2—602no—
462C2#1174891711yescharacteristic
subgroupG/H
centerC2#11748917
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11754602C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1910
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. #11756504C2semidirect
3. C3×#65420C2semidirect
4. #11748626C2semidirect
5. #11748366C2semidirect
6. C5×#4557C2semidirect
7. #11745969C2semidirect
21. #91458C4semidirect
22. #91364C4semidirect
29. #65420D3semidirect
53. C2×C3×C5×C8D4semidirect
56. C2×C3⋊#168D4semidirect
74. #4557D5semidirect
161. C2×C5×C8#67semidirect
221. C2×C3×C8D20semidirect
364. C2×C8#1202semidirect
375. C3×C5#1910semidirect
437. C5#70330semidirect
456. C3#10599506semidirect

Maximal subgroups

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

subgroupindex
#117565042
#117546022
C3×#654202
#117486262
#117483662
C5×#45572
#117459692
#105995063
#703305