Finite Group Databasebeta

← #14299604

Subgroups — #14299604

Order 1920.

648 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#11754488C211yesmaximal, characteristic
2C3×#65228C211yesmaximal, characteristic
3#11748470C211yesmaximal, characteristic, complemented
4#11748453C211yesmaximal, characteristic, complemented
5C2×#90958C211yesmaximal, characteristic
6#11746344C211yesmaximal, characteristic, complemented
7C5×#4523C211yesmaximal, characteristic, complemented
8#10599551—31nomaximal, Hall
9C22×C3×D20C2211yescharacteristic
10C2×C3×C42×C5C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×#685—22no—
16C2×C3×C4⋊Dic5C2211yescharacteristic
17D5×#217—22no—
18#91434—21no—
19#91434—21no—
20Dic3×D20—21no—
21Dic3×D20—21no—
22#90990—21no—
23#90989—21no—
24#90987—21no—
25#90986—21no—
26#90958C414yescomplemented
27#69642—51nomaximal, Hall
28#65855—31no—
29C2×#3955—31no—
30#65410—31no—
31#65284—31no—
32#65228D311yescharacteristic
33C5×#339—31no—
34#64743—31no—
35C23×C3×D5D412yes—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C22×C3×Dic5—21no—
39C2×C3×D20—21no—
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C2×C3×D20—21no—
43C2×C3×D20C2×C412yes—
44C2×C3×D20Q812yes—
45C2×D5×Dic3—22no—
46C2×D5×Dic3—22no—
47C3×C42×C5—21no—
48C3×C42×C5—21no—
49#6719—21no—
50C4⋊Dic15D411yescharacteristic
51C4⋊Dic15D411yescharacteristic
52C4×Dic15—21no—
53#6710—21no—
54#6710C2×C412yescomplemented
55C2×C3⋊#168—21no—
56C5×#217—21no—
57C5×C4⋊Dic3D411yescharacteristic
58C5×C4⋊Dic3D411yescharacteristic
59C4×C5×Dic3—21no—
60C5×#208—21no—
61C5×#208C2×C412yescomplemented
62C2×C5×C3⋊C8—21no—
63C3×#685—44no—
64C3×C4⋊Dic5—21no—
65C3×C4⋊Dic5—21no—
66#6665—22no—
67#6665—22no—
68#5798—51no—
69C3×#390—51no—
70C2×#833—51no—
71#4964—51no—
72#4964—51no—
73#4523—51no—
74#4523D511yescharacteristic, complemented
75C22×D20C2×D311yescharacteristic
76C2×C5×M16—31no—
77C5×#117—31no—
78C2×C42×C5C2×D311yescharacteristic
79C2×#685—22no—
80#4072—31no—
81C2×C4⋊Dic5C2×D311yescharacteristic
82C2×#681—31no—
83#4039—61no—
84#4039—61no—
85D5×C22⋊C4—62no—
86C4×D20—61no—
87C4×D20—61no—
88#3955—34no—
89#3950—61no—
90#3949—61no—
91#3940—61no—
92#3939—61no—
93#1856—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5—42no—
98C22×C3×D5—42no—
99C22×C3×D5C4⋊C414yes—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5C22×C411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×Q811yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×Dic15—21no—
110C2×Dic15—21no—
111C2×C5×Dic3—21no—
112C2×C5×Dic3—21no—
113C2×C3×Dic5—42no—
114C2×C3×Dic5—41no—
115C3×D20—22no—
116C3×D20—22no—
117C3×D20—22no—
118C3×D20—22no—
119D5×Dic3—82no—
120D5×Dic3—82no—
121C3⋊#168—22no—
122C5×C3⋊C8—22no—
123C22×C3×D4—51no—
124C2×C3×C4⋊C4—51no—
125C2×C3×C22⋊C4—102no—
126C2×C3×C42C2×D511yescharacteristic
127C2×#217—102no—
128D4×Dic3—101no—
129D4×Dic3—101no—
130D4×Dic3—101no—
131D4×Dic3—101no—
132#926—51no—
133#926C2×D511yescharacteristic
134C2×#208—51no—
135C2×#208C2×D511yescharacteristic
136#833—54no—
137#808—101no—
138#808—101no—
139#807—101no—
140#807—101no—
141C23×D5#6712yes—
142C22×C4×C5—21no—
143C22×C4×C5C22×D311yescharacteristic
144C22×Dic5—21no—
145C2×D20—21no—
146C2×D20—21no—
147C2×D20—21no—
148C2×D20—21no—
149C2×D20Dic612yes—
150C2×D20C4×D312yes—
151C2×C4×D5—62no—
152C2×C4×D5—62no—
153C5×M16—32no—
154C5×M16—61no—
155C2×C5×C8—61no—
156C5×C4⋊C4—31no—
157C5×C4⋊C4—31no—
158C5×C22⋊C4—61no—
159C42×C5—21no—
160C42×C5—21no—
161C42×C5—61no—
162C22⋊Dic5—61no—
163#685—44no—
164#685—62no—
165#685—62no—
166C4⋊Dic5—21no—
167C4⋊Dic5—21no—
168C4⋊Dic5—31no—
169C4⋊Dic5—31no—
170C4×Dic5—61no—
171#681—32no—
172#681—61no—
173C2×#168—61no—
174#518—151no—
175#428—151no—
176#428—151no—
177C2×#100—151no—
178#390—51no—
179#339—31no—
180#339—151no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5C2×C4⋊C411yescharacteristic
184C22×C3×C5#12011yescharacteristic
185C22×C3×C5#12311yescharacteristic
186C2×C3×D5—24no—
187C2×C3×D5—42no—
188C2×C3×D5—42no—
189C2×C3×D5—44no—
190C2×C3×D5—82no—
191C3×C4×C5—42no—
192C3×C4×C5C4×D411yescharacteristic
193C3×C4×C5C4×D411yescharacteristic
194C3×C4×C5C22⋊Q811yescharacteristic
195C3×C4×C5C22⋊Q811yescharacteristic
196Dic15—41no—
197Dic15—41no—
198C3×Dic5—81no—
199C5×Dic3—41no—
200C5×Dic3—41no—
201C24×C3—52no—
202C2×C3×D4—52no—
203C2×C3×D4—52no—
204C2×C3×D4—101no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C22×C3×C4—21no—
209C22×C3×C4—101no—
210C22×C3×C4C22×D511yescharacteristic
211C22×Dic3—102no—
212C22×Dic3—102no—
213C3×C4⋊C4—101no—
214C3×C4⋊C4—101no—
215C3×C22⋊C4—204no—
216C3×C42—21no—
217C3×C42—21no—
218#217—21no—
219#217—102no—
220#217—102no—
221#217—101no—
222C4⋊Dic3—51no—
223C4⋊Dic3—51no—
224C4⋊Dic3D2011yescharacteristic
225C4⋊Dic3D2011yescharacteristic
226C4×Dic3—21no—
227C4×Dic3—101no—
228#208—21no—
229#208—52no—
230#208—101no—
231#208C4×D512yescomplemented
232C2×C3⋊C8—21no—
233C2×C3⋊C8—101no—
234C23×C5C2×#6711yescharacteristic
235C22×D5—22no—
236C22×D5—22no—
237C22×D5—42no—
238C22×D5—42no—
239C22×D5Dic3⋊C414yes—
240C2×C4×C5—22no—
241C2×C4×C5—41no—
242C2×C4×C5—41no—
243C2×C4×C5—61no—
244C2×C4×C5—61no—
245C2×C4×C5C2×Dic611yescharacteristic
246C2×C4×C5C2×C4×D311yescharacteristic
247C2×C4×C5D3×D411yescharacteristic
248C2×C4×C5D3×D411yescharacteristic
249C2×C4×C5#23711yescharacteristic
250C2×C4×C5#23711yescharacteristic
251C2×Dic5—42no—
252C2×Dic5—41no—
253C2×Dic5—61no—
254C2×Dic5—61no—
255D20—22no—
256D20—22no—
257D20—22no—
258D20—22no—
259C4×D5—242no—
260C4×D5—242no—
261C5×C8—62no—
262#168—62no—
263C22×D4—51no—
264C2×M16—31no—
265C2×M16—151no—
266C4×D4—301no—
267C4×D4—301no—
268C4×D4—301no—
269C4×D4—301no—
270#117—31no—
271#117—151no—
272C2×C4⋊C4—51no—
273C2×C22⋊C4—102no—
274C2×C22⋊C4—302no—
275C2×C42D3×D511yescharacteristic
276#107—301no—
277#107—301no—
278#106—301no—
279#106—301no—
280#100—154no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5#38511yescharacteristic
284C2×C3×C5#45311yescharacteristic
285C2×C3×C5#45511yescharacteristic
286C3×D5—44no—
287C23×C3—54no—
288C23×C3—102no—
289C23×C3—102no—
290C23×C3—202no—
291C23×C3—202no—
292C23×C3D4×D511yescharacteristic
293C3×D4—102no—
294C3×D4—102no—
295C3×D4—102no—
296C3×D4—102no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—202no—
301C2×C3×C4—201no—
302C2×C3×C4C2×C4×D511yescharacteristic
303C2×C3×C4C2×D2011yescharacteristic
304C2×C3×C4#70911yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4#71111yescharacteristic
307C2×C3×C4Q8×D511yescharacteristic
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—101no—
311C2×Dic3—101no—
312C2×Dic3—402no—
313C2×Dic3—402no—
314C3⋊C8—22no—
315C3⋊C8—102no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5C2×Dic3⋊C411yescharacteristic
319C22×C5#93511yescharacteristic
320C22×C5#93711yescharacteristic
321C2×D5—24no—
322C2×D5—42no—
323C2×D5—42no—
324C2×D5—44no—
325C2×D5—82no—
326C4×C5—42no—
327C4×C5—121no—
328C4×C5—121no—
329C4×C5#87811yescharacteristic
330C4×C5#87811yescharacteristic
331C4×C5#67⋊C411yescharacteristic
332C4×C5#67⋊C411yescharacteristic
333Dic5—81no—
334Dic5—121no—
335Dic5—121no—
336C24—52no—
337C2×D4—52no—
338C2×D4—52no—
339C2×D4—101no—
340C2×D4—101no—
341C2×D4—101no—
342C2×D4—101no—
343C22×C4—21no—
344C22×C4—101no—
345C22×C4—302no—
346C22×C4—302no—
347C22×C4C2×D3×D511yescharacteristic
348M16—32no—
349M16—61no—
350M16—152no—
351M16—301no—
352C2×C8—61no—
353C2×C8—301no—
354C4⋊C4—31no—
355C4⋊C4—31no—
356C4⋊C4—101no—
357C4⋊C4—101no—
358C4⋊C4—151no—
359C4⋊C4—151no—
360C22⋊C4—61no—
361C22⋊C4—204no—
362C22⋊C4—302no—
363C22⋊C4—302no—
364C22⋊C4—301no—
365C42—21no—
366C42—21no—
367C42—61no—
368C42—301no—
369C3×C5#185611yescharacteristic, complemented, Hall
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—104no—
373C22×C3—204no—
374C22×C3—202no—
375C22×C3—202no—
376C22×C3—402no—
377C22×C3#402811yescharacteristic
378C22×C3#402911yescharacteristic
379C22×C3D5×C4⋊C411yescharacteristic
380C3×C4—42no—
381C3×C4—401no—
382C3×C4C4×D2011yescharacteristic
383C3×C4#402711yescharacteristic
384C3×C4#404211yescharacteristic
385C3×C4#404311yescharacteristic
386Dic3—41no—
387Dic3—41no—
388Dic3—201no—
389Dic3—201no—
390C2×C5—21no—
391C2×C5—21no—
392C2×C5#494611yescharacteristic
393C2×C5#506111yescharacteristic
394C2×C5#507711yescharacteristic
395D5—44no—
396C23—54no—
397C23—102no—
398C23—102no—
399C23—202no—
400C23—202no—
401C23D5×#6711yescharacteristic
402D4—102no—
403D4—102no—
404D4—102no—
405D4—102no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—202no—
412C2×C4—201no—
413C2×C4—301no—
414C2×C4—301no—
415C2×C4—1202no—
416C2×C4—1202no—
417C2×C4D5×Dic611yescharacteristic
418C2×C4C4×D3×D511yescharacteristic
419C2×C4D3×D2011yescharacteristic
420C2×C4#677711yescharacteristic
421C2×C4#677911yescharacteristic
422C2×C4#678011yescharacteristic
423C8—62no—
424C8—302no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—204no—
428C2×C3#6495711yescharacteristic
429C2×C3#6511111yescharacteristic
430C2×C3#6511411yescharacteristic
431C5#6964211yescharacteristic, complemented, Sylow
432C22—21no—
433C22—21no—
434C22—104no—
435C22—204no—
436C22—202no—
437C22—202no—
438C22—402no—
439C22D5×Dic3⋊C411yescharacteristic
440C22#9147411yescharacteristic
441C22#9154811yescharacteristic
442C4—42no—
443C4—121no—
444C4—121no—
445C4—401no—
446C4—601no—
447C4—601no—
448C4#9139511yescharacteristic
449C4#9142211yescharacteristic
450C4#9143011yescharacteristic
451C4#9144211yescharacteristic
452C3#1059955111yescharacteristic, complemented, Sylow
453C2—21no—
454C2—21no—
455C2—204no—
456C2#1174711411yescharacteristic
457C2#1174711511yescharacteristic
458C2#1174889011yescharacteristic
subgroupG/H
centerC22#91474
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5#120
prime radicalC22×C3×D20C22
cosocleC22×C3×C4×C5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#1856
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. #11748470C2semidirect
4. #11748453C2semidirect
6. #11746344C2semidirect
7. C5×#4523C2semidirect
26. #90958C4semidirect
54. #6710C2×C4semidirect
61. C5×#208C2×C4semidirect
74. #4523D5semidirect
231. #208C4×D5semidirect
369. C3×C5#1856semidirect
431. C5#69642semidirect
452. C3#10599551semidirect

Maximal subgroups

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

subgroupindex
#117544882
C3×#652282
#117484702
#117484532
C2×#909582
#117463442
C5×#45232
#105995513
#696425