Finite Group Databasebeta

← #14303030

Subgroups — #14303030

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#11756564C211yesmaximal, characteristic
2#11756504C211yesmaximal, characteristic, complemented
3C2×C3×C5×#100C211yesmaximal, characteristic, complemented
4C2×#91117C211yesmaximal, characteristic, complemented
5#11751233C211yesmaximal, characteristic, complemented
6#11746380C211yesmaximal, characteristic, complemented
7#11746377C211yesmaximal, characteristic, complemented
8#10600882—31nomaximal, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C5×M16C2211yescharacteristic
13#91842C414yes—
14#91829—21no—
15#91827—22no—
16#91825—21no—
17#91815C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C2×#6710C2211yescharacteristic
20#91796—22no—
21M16×D15—22no—
22#91790—22no—
23C3×C5×#100—22no—
24#91117—22no—
25#91108—21no—
26#91107—21no—
27#70089—51nomaximal, Hall
28#66139—31no—
29#66079—31no—
30C2×C5×#100D311yescharacteristic, complemented
31C2×#3965—31no—
32#65421—31no—
33#64779—31no—
34#64776—31no—
35C24×C3×C5C2×C412yes—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C2×#1220—24no—
44C2×#1220C2×C412yes—
45#6817—21no—
46#6817—21no—
47#6817—21no—
48#6817—21no—
49C2×D60D411yescharacteristic
50C2×C4×D15D412yes—
51C2×Dic30D411yescharacteristic
52C3×C5×M16—21no—
53C3×C5×M16—21no—
54C3×C5×M16—21no—
55C3×C5×M16D412yescomplemented
56C2×C3×C5×C8—21no—
57C2×C3×C5×C8—21no—
58#6719—24no—
59#6719C2×C412yes—
60C4⋊Dic15—21no—
61C4×Dic15—21no—
62#6710—21no—
63#6710—21no—
64#6710—21no—
65#6710D412yescomplemented
66C2×C3⋊#168—21no—
67C2×C3⋊#168—21no—
68#6705—24no—
69C8×D15—24no—
70#5805—51no—
71#5745—51no—
72C2×C3×#100D511yescharacteristic, complemented
73C2×#833—51no—
74#5132—51no—
75#4559—51no—
76#4556—51no—
77C22×C5×D4C2×D311yescharacteristic
78C2×#709—31no—
79C2×C5×M16C2×D311yescharacteristic
80C2×C5×M16C2×D311yescharacteristic
81#4099—34no—
82#4086—61no—
83#4084—62no—
84#4082—61no—
85#4072—31no—
86C2×#681—31no—
87C2×#681—31no—
88#4053—62no—
89D5×M16—62no—
90#4047—62no—
91C5×#100—22no—
92#3965—62no—
93#3956—61no—
94#3955—61no—
95#1914—151noSylow
96C23×C3×C5—22no—
97C23×C3×C5—22no—
98C23×C3×C5—22no—
99C23×C3×C5—82no—
100C23×C3×C5C22×C411yescharacteristic
101C22×D15C22⋊C412yes—
102C3×C5×D4—41no—
103C3×C5×D4—41no—
104C3×C5×D4—41no—
105C3×C5×D4—41no—
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×C3×C4×C5#4111yescharacteristic
112#1220—22no—
113#1220—22no—
114#1220—84no—
115C2×Dic15—22no—
116C2×Dic15C22⋊C412yes—
117D60—21no—
118D60—21no—
119C4×D15—22no—
120C4×D15—22no—
121Dic30—21no—
122Dic30—21no—
123C3×C5×C8—22no—
124C3×C5×C8—41no—
125C3⋊#168—22no—
126C3⋊#168—41no—
127C22×C3×D4C2×D511yescharacteristic
128C2×#235—51no—
129C2×C3×M16C2×D511yescharacteristic
130C2×C3×M16C2×D511yescharacteristic
131#953—54no—
132#940—101no—
133#938—102no—
134#936—101no—
135#926—51no—
136C2×#208—51no—
137C2×#208—51no—
138#907—102no—
139D3×M16—102no—
140#901—102no—
141C3×#100—22no—
142#833—102no—
143#824—101no—
144#823—101no—
145C24×C5C4×D312yes—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4—21no—
150C2×C5×D4—21no—
151C2×C5×D4—21no—
152C22×C4×C5C22×D311yescharacteristic
153C2×C22⋊D5—32no—
154C2×C22⋊D5—64no—
155#709—61no—
156#709—61no—
157#709—61no—
158#709—61no—
159C2×D20—31no—
160C2×C4×D5—32no—
161C2×Dic10—31no—
162C5×M16—21no—
163C5×M16—21no—
164C5×M16—21no—
165C5×M16#6712yescomplemented
166C2×C5×C8—21no—
167C2×C5×C8—21no—
168C22⋊Dic5—32no—
169C22⋊Dic5—64no—
170C4⋊Dic5—61no—
171C4×Dic5—61no—
172#681—32no—
173#681—61no—
174#681—61no—
175#681—61no—
176C2×#168—61no—
177C2×#168—61no—
178#676—64no—
179C8×D5—64no—
180#568—151no—
181#535—151no—
182#413—151no—
183C2×#100—151no—
184C2×#100D1511yescharacteristic, complemented
185#344—151no—
186#344—151no—
187C22×C3×C5—41no—
188C22×C3×C5—44no—
189C22×C3×C5—42no—
190C22×C3×C5—82no—
191C22×C3×C5—82no—
192C22×C3×C5C2×C22⋊C411yescharacteristic
193C22×C3×C5C4×D411yescharacteristic
194C22×C3×C5C4×D411yescharacteristic
195C2×D15—22no—
196C2×D15—42no—
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12111yescharacteristic
199C3×C4×C5#12111yescharacteristic
200C3×C4×C5#12311yescharacteristic
201Dic15—22no—
202Dic15—42no—
203C24×C3C4×D512yes—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C22×C3×C4C22×D511yescharacteristic
211C2×#67—52no—
212C2×#67—104no—
213#235—101no—
214#235—101no—
215#235—101no—
216#235—101no—
217C2×D12—51no—
218C2×C4×D3—52no—
219C2×Dic6—51no—
220C3×M16—21no—
221C3×M16—21no—
222C3×M16—21no—
223C3×M16C22⋊D512yescomplemented
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226#217—52no—
227#217—104no—
228C4⋊Dic3—101no—
229C4×Dic3—101no—
230#208—52no—
231#208—101no—
232#208—101no—
233#208—101no—
234C2×C3⋊C8—101no—
235C2×C3⋊C8—101no—
236ΓL(1,𝔽25)—104no—
237D3×C8—104no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—82no—
242C23×C5C2×C4×D311yescharacteristic
243C22×D5—32no—
244C5×D4—41no—
245C5×D4—41no—
246C5×D4—41no—
247C5×D4—41no—
248C2×C4×C5#23511yescharacteristic
249C2×C4×C5D3×D411yescharacteristic
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—62no—
259D20—61no—
260D20—61no—
261C4×D5—62no—
262C4×D5—62no—
263Dic10—61no—
264Dic10—61no—
265C5×C8—22no—
266C5×C8—41no—
267#168—62no—
268#168—121no—
269C2×#41—151no—
270C22×D4C2×D1511yescharacteristic
271#131—302no—
272#131—302no—
273C2×M16—151no—
274C2×M16—151no—
275C2×M16—302no—
276C2×M16C2×D1511yescharacteristic
277C2×M16C2×D1511yescharacteristic
278#127—301no—
279#124—301no—
280#121—302no—
281#120—154no—
282#117—151no—
283#101—301no—
284#100—22no—
285#100—301no—
286#100—302no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5—82no—
291C2×C3×C5#38611yescharacteristic
292D15—42no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—82no—
297C23×C3C2×C4×D511yescharacteristic
298C22×D3—52no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4—41no—
303C2×C3×C4#70911yescharacteristic
304C2×C3×C4D4×D511yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4D4×D511yescharacteristic
307C2×C3×C4#71111yescharacteristic
308C2×C3×C4C2×C22⋊D511yescharacteristic
309#67—102no—
310#67—102no—
311#67—404no—
312C2×Dic3—52no—
313C2×Dic3—102no—
314D12—101no—
315D12—101no—
316C4×D3—102no—
317C4×D3—102no—
318Dic6—101no—
319Dic6—101no—
320C3×C8—22no—
321C3×C8—41no—
322C3⋊C8—102no—
323C3⋊C8—201no—
324C22×C5—41no—
325C22×C5—44no—
326C22×C5—42no—
327C22×C5—82no—
328C22×C5—82no—
329C22×C5D3×C22⋊C411yescharacteristic
330C22×C5#67⋊C411yescharacteristic
331C22×C5C4×#6711yescharacteristic
332C2×D5—62no—
333C2×D5—122no—
334C4×C5#88311yescharacteristic
335C4×C5#88411yescharacteristic
336C4×C5#93711yescharacteristic
337C4×C5#93811yescharacteristic
338Dic5—62no—
339Dic5—122no—
340C24C4×D1512yes—
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—21no—
352C2×D4—152no—
353C2×D4—151no—
354C2×D4—304no—
355C22×C4—152no—
356C22×C4C22×D1511yescharacteristic
357M16—21no—
358M16—21no—
359M16—21no—
360M16—152no—
361M16—304no—
362M16—301no—
363M16—301no—
364M16—301no—
365M16#122012yescomplemented
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—304no—
369C2×C8—301no—
370C2×C8—301no—
371C4⋊C4—301no—
372C22⋊C4—152no—
373C22⋊C4—304no—
374C42—301no—
375C3×C5#191411yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3—44no—
378C22×C3—42no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3D5×C22⋊C411yescharacteristic
382C22×C3#402711yescharacteristic
383C22×C3C4×C22⋊D511yescharacteristic
384C2×D3—102no—
385C2×D3—202no—
386C3×C4#402911yescharacteristic
387C3×C4#403011yescharacteristic
388C3×C4#408311yescharacteristic
389C3×C4#408411yescharacteristic
390Dic3—102no—
391Dic3—202no—
392C2×C5—21no—
393C2×C5—21no—
394C2×C5—21no—
395C2×C5—82no—
396C2×C5#495611yescharacteristic
397D5—122no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—82no—
402C23—152no—
403C23C2×C4×D1511yescharacteristic
404Q8—301no—
405Q8—301no—
406D4—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—302no—
411D4—302no—
412D4—301no—
413D4—301no—
414D4—1204no—
415C2×C4—152no—
416C2×C4—302no—
417C2×C4—302no—
418C2×C4—302no—
419C2×C4#681711yescharacteristic
420C2×C4D4×D1511yescharacteristic
421C2×C4D4×D1511yescharacteristic
422C2×C4D4×D1511yescharacteristic
423C2×C4#681911yescharacteristic
424C2×C4C2×#122011yescharacteristic
425C8—22no—
426C8—41no—
427C8—302no—
428C8—601no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3#6524511yescharacteristic
434D3—202no—
435C5#7008911yescharacteristic, complemented, Sylow
436C22—41no—
437C22—44no—
438C22—42no—
439C22—82no—
440C22—82no—
441C22—302no—
442C22—602no—
443C22C22⋊C4×D1511yescharacteristic
444C22#9177011yescharacteristic
445C22C4×#122011yescharacteristic
446C4—302no—
447C4—602no—
448C4#9177211yescharacteristic
449C4#9177311yescharacteristic
450C4#9182611yescharacteristic
451C4#9182711yescharacteristic
452C3#1060088211yescharacteristic, complemented, Sylow
453C2—21no—
454C2—21no—
455C2—21no—
456C2—82no—
457C2—602no—
458C2#1175105711yescharacteristic
subgroupG/H
centerC2#11751057
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C22×D15
FittingC2×C3×C5×#100C2
socleC2×C3×C5#386
prime radical#11756564C2
cosocleC22×C3×C4×C5C23
hypercenterC2×#100D15
hypocenterC3×C5#1914
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #11756504C2semidirect
3. C2×C3×C5×#100C2semidirect
4. C2×#91117C2semidirect
5. #11751233C2semidirect
6. #11746380C2semidirect
7. #11746377C2semidirect
30. C2×C5×#100D3semidirect
55. C3×C5×M16D4semidirect
65. #6710D4semidirect
72. C2×C3×#100D5semidirect
165. C5×M16#67semidirect
184. C2×#100D15semidirect
223. C3×M16C22⋊D5semidirect
365. M16#1220semidirect
375. C3×C5#1914semidirect
435. C5#70089semidirect
452. C3#10600882semidirect

Maximal subgroups

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

subgroupindex
#117565642
#117565042
C2×C3×C5×#1002
C2×#911172
#117512332
#117463802
#117463772
#106008823
#700895