Finite Group Databasebeta

← #14303047

Subgroups — #14303047

Order 1920.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754600C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3C5×#5132C211yesmaximal, characteristic, complemented
4C2×C3×#3955C211yesmaximal, characteristic, complemented
5C2×#90958C211yesmaximal, characteristic, complemented
6#11746377C211yesmaximal, characteristic, complemented
7#11745990C211yesmaximal, characteristic, complemented
8#10600930—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91563C414yes—
17#91463—21no—
18#91456—22no—
19#91368—21no—
20#91288—22no—
21D3×#681—22no—
22#91286—22no—
23C5×#824—21no—
24C5×#823—21no—
25C3×#3955—22no—
26#90958—22no—
27#70089—51nomaximal, Hall
28#66164—31no—
29#66153—31no—
30C5×#413—31no—
31C2×#3955—31no—
32C2×#3955D311yescharacteristic, complemented
33#64776—31no—
34#64776—31no—
35C23×C3×D5C2×C412yes—
36C22×C3×C4×C5C2311yescharacteristic
37C2×C5×#67C2×C412yes—
38C5×#235—21no—
39C5×#235—21no—
40C5×#235—21no—
41C5×#235—21no—
42C2×C5×D12D411yescharacteristic
43C2×C4×C5×D3D412yes—
44C2×C5×Dic6D411yescharacteristic
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20—21no—
48C2×C3×D20—21no—
49C2×C3×D20—21no—
50C2×C3×D20—21no—
51C2×#1201—24no—
52C3×C5×M16—21no—
53C3×C5×M16—21no—
54C2×C3×C5×C8—21no—
55#6719C2×C412yes—
56C4⋊Dic15—21no—
57C4×Dic15—21no—
58#6710—21no—
59#6710D412yescomplemented
60C2×C3⋊#168—21no—
61C5×#208—21no—
62C5×#208—21no—
63C2×C5×C3⋊C8—21no—
64C3×#681—21no—
65C3×#681D412yescomplemented
66C2×C3×#168—21no—
67#6665—24no—
68#6650—24no—
69D3×#168—24no—
70#5805—51no—
71#5745—51no—
72C2×C3×#100—51no—
73C2×#833—51no—
74#5132D511yescharacteristic, complemented
75#4559—51no—
76#4556—51no—
77C2×C5×#41—31no—
78C22×D20C2×D311yescharacteristic
79C2×C5×M16—31no—
80C2×C5×M16C2×D311yescharacteristic
81#4094—62no—
82#4094—62no—
83#4093—61no—
84#4086—61no—
85#4083—34no—
86#4076—62no—
87#4072—31no—
88C2×#681—31no—
89C2×#681—62no—
90C2×#681C2×D311yescharacteristic
91C5×#101—61no—
92C5×#100—61no—
93#3955—22no—
94#3955—62no—
95#1914—151noSylow
96C23×C3×C5C22×C411yescharacteristic
97C22×C5×D3C22⋊C412yes—
98C22×C3×D5—22no—
99C22×C3×D5—22no—
100C22×C3×D5—22no—
101C22×C3×D5—82no—
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×Dic15—22no—
109C5×#67—22no—
110C5×#67—22no—
111C2×C5×Dic3C22⋊C412yes—
112C5×D12—21no—
113C5×D12—21no—
114C4×C5×D3—22no—
115C4×C5×D3—22no—
116C5×Dic6—21no—
117C5×Dic6—21no—
118C3×D20—41no—
119C3×D20—41no—
120C3×D20—41no—
121C3×D20—41no—
122#1201—84no—
123C3×C5×C8—41no—
124C3⋊#168—22no—
125C3×#168—22no—
126C5×C3⋊C8—41no—
127C22×C3×D4—51no—
128C2×#235C2×D511yescharacteristic
129C2×C3×M16—51no—
130C2×C3×M16C2×D511yescharacteristic
131#953—54no—
132#940—101no—
133#938—102no—
134#936—101no—
135#926—51no—
136C2×#208—51no—
137C2×#208C2×D511yescharacteristic
138#907—102no—
139D3×M16—102no—
140#901—102no—
141C3×#100—102no—
142#833—102no—
143#824—21no—
144#823—21no—
145C23×D5C4×D312yes—
146C5×#41—61no—
147C5×#41—61no—
148C5×#41—61no—
149C5×#41—61no—
150C2×C5×Q8—31no—
151C2×C5×D4—32no—
152C2×C5×D4—31no—
153C22×C4×C5—32no—
154C22×C4×C5C22×D311yescharacteristic
155C2×C22⋊D5—64no—
156C2×D20—21no—
157C2×D20—21no—
158C2×D20—21no—
159C2×D20—21no—
160C2×D20—21no—
161C2×D20—21no—
162C5×M16—21no—
163C5×M16—21no—
164C5×M16—61no—
165C5×M16—61no—
166C2×C5×C8—21no—
167C2×C5×C8—61no—
168C22⋊Dic5—32no—
169#685—64no—
170C4⋊Dic5—61no—
171C4×Dic5—61no—
172#681—21no—
173#681—32no—
174#681—64no—
175#681—61no—
176#681#6712yescomplemented
177C2×#168—21no—
178C2×#168—64no—
179C2×#168—61no—
180#568—151no—
181#535—151no—
182#413—31no—
183C2×#100—51no—
184C2×#100—151no—
185#344—151no—
186#344—151no—
187C22×C3×C5—41no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C22×C3×C5C4×D411yescharacteristic
190C22×C3×C5C4×D411yescharacteristic
191C2×C5×D3—22no—
192C2×C5×D3—42no—
193C2×C3×D5—44no—
194C2×C3×D5—42no—
195C2×C3×D5—82no—
196C2×C3×D5—82no—
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12111yescharacteristic
199C3×C4×C5#12111yescharacteristic
200C3×C4×C5#12311yescharacteristic
201Dic15—42no—
202C5×Dic3—22no—
203C24×C3—52no—
204C2×C3×D4—101no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C2×C3×D4—101no—
209C2×C3×D4—101no—
210C22×C3×C4C22×D511yescharacteristic
211C2×#67—104no—
212C2×#67C4×D512yes—
213#235—21no—
214#235—21no—
215#235—21no—
216#235—21no—
217C2×D12C22⋊D511yescharacteristic
218C2×C4×D3D2012yes—
219C2×Dic6C22⋊D511yescharacteristic
220C3×M16—21no—
221C3×M16—21no—
222C3×M16—52no—
223C3×M16—101no—
224C2×C3×C8—21no—
225C2×C3×C8—101no—
226#217—52no—
227#217—104no—
228C4⋊Dic3—101no—
229C4×Dic3—101no—
230#208—21no—
231#208—21no—
232#208—52no—
233#208—101no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—101no—
236ΓL(1,𝔽25)—104no—
237D3×C8—104no—
238C23×C5—32no—
239C23×C5C2×C4×D311yescharacteristic
240C22×D5—22no—
241C22×D5—22no—
242C22×D5—22no—
243C22×D5—82no—
244C5×Q8—61no—
245C5×Q8—61no—
246C5×D4—62no—
247C5×D4—62no—
248C5×D4—61no—
249C5×D4—61no—
250C2×C4×C5—32no—
251C2×C4×C5—62no—
252C2×C4×C5—62no—
253C2×C4×C5#23511yescharacteristic
254C2×C4×C5D3×D411yescharacteristic
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5D3×D411yescharacteristic
257C2×C4×C5#23711yescharacteristic
258C2×C4×C5C2×#6711yescharacteristic
259C22⋊D5—244no—
260C2×Dic5—62no—
261D20—41no—
262D20—41no—
263D20—41no—
264D20—41no—
265C5×C8—41no—
266C5×C8—121no—
267#168—22no—
268#168—62no—
269C2×#41—31no—
270C22×D4—51no—
271#131—302no—
272#131—302no—
273C2×M16—31no—
274C2×M16—51no—
275C2×M16—151no—
276C2×M16—302no—
277C2×M16D3×D511yescharacteristic
278#127—301no—
279#124—301no—
280#121—302no—
281#120—154no—
282#117—151no—
283#101—61no—
284#100—61no—
285#100—102no—
286#100—302no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5#38611yescharacteristic
291C3×D5—82no—
292C5×D3—42no—
293C23×C3—102no—
294C23×C3—102no—
295C23×C3—102no—
296C23×C3—402no—
297C23×C3C2×C4×D511yescharacteristic
298C22×D3#68512yes—
299C3×D4—201no—
300C3×D4—201no—
301C3×D4—201no—
302C3×D4—201no—
303C2×C3×C4C2×D2011yescharacteristic
304C2×C3×C4D4×D511yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4#71111yescharacteristic
307C2×C3×C4#71111yescharacteristic
308C2×C3×C4C2×C22⋊D511yescharacteristic
309#67—22no—
310#67—22no—
311#67—404no—
312C2×Dic3—102no—
313C2×Dic3#68512yes—
314D12—21no—
315D12—21no—
316C4×D3—22no—
317C4×D3—22no—
318Dic6—21no—
319Dic6—21no—
320C3×C8—41no—
321C3×C8—102no—
322C3⋊C8—41no—
323C3⋊C8—102no—
324C22×C5—41no—
325C22×C5—62no—
326C22×C5—122no—
327C22×C5D3×C22⋊C411yescharacteristic
328C22×C5#67⋊C411yescharacteristic
329C22×C5C4×#6711yescharacteristic
330C2×D5—44no—
331C2×D5—42no—
332C2×D5—82no—
333C2×D5—82no—
334C4×C5—62no—
335C4×C5#88311yescharacteristic
336C4×C5#88411yescharacteristic
337C4×C5#93711yescharacteristic
338C4×C5#93811yescharacteristic
339Dic5—122no—
340C24—52no—
341#41—61no—
342#41—61no—
343#41—61no—
344#41—61no—
345C2×Q8—31no—
346C2×D4—32no—
347C2×D4—31no—
348C2×D4—101no—
349C2×D4—101no—
350C2×D4—101no—
351C2×D4—101no—
352C2×D4—101no—
353C2×D4—101no—
354C2×D4—304no—
355C22×C4—32no—
356C22×C4C2×D3×D511yescharacteristic
357M16—21no—
358M16—21no—
359M16—52no—
360M16—61no—
361M16—61no—
362M16—101no—
363M16—152no—
364M16—304no—
365M16—301no—
366C2×C8—21no—
367C2×C8—61no—
368C2×C8—101no—
369C2×C8—304no—
370C2×C8—301no—
371C4⋊C4—301no—
372C22⋊C4—152no—
373C22⋊C4—304no—
374C42—301no—
375C3×C5#191411yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3—204no—
378C22×C3—202no—
379C22×C3—402no—
380C22×C3—402no—
381C22×C3#402711yescharacteristic
382C22×C3#402711yescharacteristic
383C22×C3C2×#68511yescharacteristic
384C2×D3—22no—
385C2×D3—42no—
386C3×C4#402811yescharacteristic
387C3×C4#403211yescharacteristic
388C3×C4#408511yescharacteristic
389C3×C4#408511yescharacteristic
390Dic3—22no—
391Dic3—202no—
392C2×C5—21no—
393C2×C5—21no—
394C2×C5—21no—
395C2×C5—122no—
396C2×C5#495611yescharacteristic
397D5—82no—
398C23—32no—
399C23—102no—
400C23—102no—
401C23—102no—
402C23—402no—
403C23C4×D3×D511yescharacteristic
404Q8—61no—
405Q8—61no—
406D4—62no—
407D4—62no—
408D4—61no—
409D4—61no—
410D4—201no—
411D4—201no—
412D4—201no—
413D4—201no—
414D4—1204no—
415C2×C4—32no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—302no—
419C2×C4#677211yescharacteristic
420C2×C4D3×D2011yescharacteristic
421C2×C4#677711yescharacteristic
422C2×C4#678011yescharacteristic
423C2×C4D5×#6711yescharacteristic
424C2×C4D3×C22⋊D511yescharacteristic
425C8—41no—
426C8—102no—
427C8—121no—
428C8—302no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—402no—
433C2×C3#6524711yescharacteristic
434D3—42no—
435C5#7008911yescharacteristic, complemented, Sylow
436C22—41no—
437C22—62no—
438C22—122no—
439C22—204no—
440C22—202no—
441C22—402no—
442C22—402no—
443C22#9144011yescharacteristic
444C22#9144211yescharacteristic
445C22D3×#68511yescharacteristic
446C4—62no—
447C4—602no—
448C4#9145311yescharacteristic
449C4#9145711yescharacteristic
450C4#9147411yescharacteristic
451C4#9156511yescharacteristic
452C3#1060093011yescharacteristic, complemented, Sylow
453C2—21no—
454C2—21no—
455C2—21no—
456C2—122no—
457C2—402no—
458C2#1174891111yescharacteristic
subgroupG/H
centerC2#11748911
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11754600C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1914
LCSG ▷ C23×C3×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
2. #11754125C2semidirect
3. C5×#5132C2semidirect
4. C2×C3×#3955C2semidirect
5. C2×#90958C2semidirect
6. #11746377C2semidirect
7. #11745990C2semidirect
32. C2×#3955D3semidirect
59. #6710D4semidirect
65. C3×#681D4semidirect
74. #5132D5semidirect
176. #681#67semidirect
375. C3×C5#1914semidirect
435. C5#70089semidirect
452. C3#10600930semidirect

Maximal subgroups

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

subgroupindex
#117546002
#117541252
C5×#51322
C2×C3×#39552
C2×#909582
#117463772
#117459902
#106009303
#700895