Finite Group Databasebeta

← #14299606

Subgroups — #14299606

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#11754503C211yesmaximal, characteristic
2C5×#4939C211yesmaximal, characteristic
3#11748472C211yesmaximal, characteristic, complemented
4#11748459C211yesmaximal, characteristic, complemented
5C2×#90959C211yesmaximal, characteristic
6#11746344C211yesmaximal, characteristic, complemented
7C3×#64743C211yesmaximal, characteristic, complemented
8#10600435—31nomaximal, Hall
9C22×C5×D12C2211yescharacteristic
10C2×C3×C42×C5C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C5×(C2×D3)⋊C4—22no—
14C2×C5×C4⋊Dic3C2211yescharacteristic
15C3×#4072C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17D3×C22⋊Dic5—22no—
18#91435—21no—
19#91435—21no—
20Dic5×D12—21no—
21Dic5×D12—21no—
22#90989—21no—
23#90988—21no—
24#90986—21no—
25#90985—21no—
26#90959C414yescomplemented
27#68758—51nomaximal, Hall
28#66132—31no—
29C5×#390—31no—
30C2×#3965—31no—
31#65253—31no—
32#65253—31no—
33#64743—31no—
34#64743D311yescharacteristic, complemented
35C23×C5×D3D412yes—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C22×C5×Dic3—21no—
39C2×C5×D12—21no—
40C2×C5×D12—21no—
41C2×C5×D12—21no—
42C2×C5×D12—21no—
43C2×C5×D12C2×C412yes—
44C2×C5×D12Q812yes—
45C2×D3×Dic5—22no—
46C2×D3×Dic5—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×(C2×D3)⋊C4—44no—
57C5×C4⋊Dic3—21no—
58C5×C4⋊Dic3—21no—
59C3×C22⋊Dic5—21no—
60C3×C4⋊Dic5D411yescharacteristic
61C3×C4⋊Dic5D411yescharacteristic
62C3×C4×Dic5—21no—
63C3×#681—21no—
64C3×#681C2×C412yescomplemented
65C2×C3×#168—21no—
66(C2×D3)⋊Dic5—22no—
67(C2×D3)⋊Dic5—22no—
68#5521—51no—
69C2×#823—51no—
70#5121—51no—
71#4995—51no—
72#4939D511yescharacteristic
73C3×#339—51no—
74#4523—51no—
75C22×C5×D4—31no—
76C2×C5×C4⋊C4—31no—
77C2×C5×C22⋊C4—62no—
78C2×C42×C5C2×D311yescharacteristic
79C2×C22⋊Dic5—62no—
80D4×Dic5—61no—
81D4×Dic5—61no—
82D4×Dic5—61no—
83D4×Dic5—61no—
84#4072—31no—
85#4072C2×D311yescharacteristic
86C2×#681—31no—
87C2×#681C2×D311yescharacteristic
88#3965—34no—
89#3940—61no—
90#3940—61no—
91#3939—61no—
92#3939—61no—
93#1856—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3—22no—
96C22×C5×D3—22no—
97C22×C5×D3—42no—
98C22×C5×D3—42no—
99C22×C5×D3C4⋊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—42no—
112C2×C5×Dic3—41no—
113C5×D12—22no—
114C5×D12—22no—
115C5×D12—22no—
116C5×D12—22no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5—21no—
119D3×Dic5—82no—
120D3×Dic5—82no—
121C3⋊#168—22no—
122C3×#168—22no—
123C22×D12C2×D511yescharacteristic
124C2×C3×M16—51no—
125C3×#117—51no—
126C2×C3×C42C2×D511yescharacteristic
127C2×(C2×D3)⋊C4—22no—
128#926—51no—
129C2×C4⋊Dic3C2×D511yescharacteristic
130C2×#208—51no—
131#893—101no—
132#893—101no—
133D3×C22⋊C4—102no—
134C4×D12—101no—
135C4×D12—101no—
136#823—54no—
137#818—101no—
138#817—101no—
139#808—101no—
140#807—101no—
141C24×C5—32no—
142C2×C5×D4—32no—
143C2×C5×D4—32no—
144C2×C5×D4—61no—
145C2×C5×D4—61no—
146C2×C5×D4—61no—
147C2×C5×D4—61no—
148C22×C4×C5—21no—
149C22×C4×C5—61no—
150C22×C4×C5C22×D311yescharacteristic
151C22×Dic5—62no—
152C22×Dic5—62no—
153C5×C4⋊C4—61no—
154C5×C4⋊C4—61no—
155C5×C22⋊C4—124no—
156C42×C5—21no—
157C42×C5—21no—
158C22⋊Dic5—21no—
159C22⋊Dic5—62no—
160C22⋊Dic5—62no—
161C22⋊Dic5—61no—
162C4⋊Dic5—31no—
163C4⋊Dic5—31no—
164C4⋊Dic5D1211yescharacteristic
165C4⋊Dic5D1211yescharacteristic
166C4×Dic5—21no—
167C4×Dic5—61no—
168#681—21no—
169#681—32no—
170#681—61no—
171#681C4×D312yescomplemented
172C2×#168—21no—
173C2×#168—61no—
174#518—151no—
175#428—151no—
176#428—151no—
177C2×#100—151no—
178#390—31no—
179#339—51no—
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×C5×D3—24no—
187C2×C5×D3—42no—
188C2×C5×D3—42no—
189C2×C5×D3—44no—
190C2×C5×D3—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—41no—
199C3×Dic5—41no—
200C5×Dic3—81no—
201C23×D3C22⋊D512yes—
202C22×C3×C4—21no—
203C22×C3×C4C22×D511yescharacteristic
204C22×Dic3—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12Dic1012yes—
210C2×D12C4×D512yes—
211C2×C4×D3—102no—
212C2×C4×D3—102no—
213C3×M16—52no—
214C3×M16—101no—
215C2×C3×C8—101no—
216C3×C4⋊C4—51no—
217C3×C4⋊C4—51no—
218C3×C22⋊C4—101no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—101no—
222#217—101no—
223(C2×D3)⋊C4—44no—
224(C2×D3)⋊C4—102no—
225(C2×D3)⋊C4—102no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—51no—
229C4⋊Dic3—51no—
230C4×Dic3—101no—
231#208—52no—
232#208—101no—
233C2×C3⋊C8—101no—
234C23×C5—34no—
235C23×C5—62no—
236C23×C5—62no—
237C23×C5—122no—
238C23×C5—122no—
239C23×C5D3×D411yescharacteristic
240C5×D4—62no—
241C5×D4—62no—
242C5×D4—62no—
243C5×D4—62no—
244C2×C4×C5—22no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—122no—
248C2×C4×C5—121no—
249C2×C4×C5C2×C4×D311yescharacteristic
250C2×C4×C5C2×D1211yescharacteristic
251C2×C4×C5#23511yescharacteristic
252C2×C4×C5D3×D411yescharacteristic
253C2×C4×C5#23711yescharacteristic
254C2×C4×C5D3×Q811yescharacteristic
255C2×Dic5—21no—
256C2×Dic5—21no—
257C2×Dic5—61no—
258C2×Dic5—61no—
259C2×Dic5—242no—
260C2×Dic5—242no—
261#168—22no—
262#168—62no—
263C22×D4—31no—
264C2×M16—51no—
265C2×M16—151no—
266C4×D4—301no—
267C4×D4—301no—
268C4×D4—301no—
269C4×D4—301no—
270#117—51no—
271#117—151no—
272C2×C4⋊C4—31no—
273C2×C22⋊C4—62no—
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
286C5×D3—44no—
287C23×C3C2×C22⋊D511yescharacteristic
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3#68314yes—
293C2×C3×C4—22no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—101no—
297C2×C3×C4—101no—
298C2×C3×C4C2×Dic1011yescharacteristic
299C2×C3×C4C2×C4×D511yescharacteristic
300C2×C3×C4D4×D511yescharacteristic
301C2×C3×C4D4×D511yescharacteristic
302C2×C3×C4#71111yescharacteristic
303C2×C3×C4#71111yescharacteristic
304C2×Dic3—42no—
305C2×Dic3—41no—
306C2×Dic3—101no—
307C2×Dic3—101no—
308D12—22no—
309D12—22no—
310D12—22no—
311D12—22no—
312C4×D3—402no—
313C4×D3—402no—
314C3×C8—102no—
315C3⋊C8—102no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—64no—
319C22×C5—124no—
320C22×C5—122no—
321C22×C5—122no—
322C22×C5—242no—
323C22×C5#88211yescharacteristic
324C22×C5#88311yescharacteristic
325C22×C5D3×C4⋊C411yescharacteristic
326C4×C5—42no—
327C4×C5—241no—
328C4×C5C4×D1211yescharacteristic
329C4×C5#67⋊C411yescharacteristic
330C4×C5#89611yescharacteristic
331C4×C5#89711yescharacteristic
332Dic5—41no—
333Dic5—41no—
334Dic5—121no—
335Dic5—121no—
336C24—32no—
337C2×D4—32no—
338C2×D4—32no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—61no—
342C2×D4—61no—
343C22×C4—21no—
344C22×C4—61no—
345C22×C4—302no—
346C22×C4—302no—
347C22×C4C2×D3×D511yescharacteristic
348M16—52no—
349M16—101no—
350M16—152no—
351M16—301no—
352C2×C8—101no—
353C2×C8—301no—
354C4⋊C4—51no—
355C4⋊C4—51no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4—151no—
359C4⋊C4—151no—
360C22⋊C4—101no—
361C22⋊C4—124no—
362C22⋊C4—302no—
363C22⋊C4—302no—
364C22⋊C4—301no—
365C42—21no—
366C42—21no—
367C42—101no—
368C42—301no—
369C3×C5#185611yescharacteristic, complemented, Hall
370C22×C3—21no—
371C22×C3—21no—
372C22×C3C2×#68311yescharacteristic
373C22×C3#408111yescharacteristic
374C22×C3#408311yescharacteristic
375C2×D3—24no—
376C2×D3—42no—
377C2×D3—42no—
378C2×D3—44no—
379C2×D3—82no—
380C3×C4—42no—
381C3×C4—201no—
382C3×C4—201no—
383C3×C4#402411yescharacteristic
384C3×C4#402411yescharacteristic
385C3×C4#402711yescharacteristic
386C3×C4#402711yescharacteristic
387Dic3—81no—
388Dic3—201no—
389Dic3—201no—
390C2×C5—21no—
391C2×C5—21no—
392C2×C5—124no—
393C2×C5#466811yescharacteristic
394C2×C5#482211yescharacteristic
395C2×C5#482511yescharacteristic
396C23—34no—
397C23—62no—
398C23—62no—
399C23—122no—
400C23—122no—
401C23D3×C22⋊D511yescharacteristic
402D4—62no—
403D4—62no—
404D4—62no—
405D4—62no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—301no—
414C2×C4—301no—
415C2×C4—1202no—
416C2×C4—1202no—
417C2×C4D3×Dic1011yescharacteristic
418C2×C4C4×D3×D511yescharacteristic
419C2×C4D5×D1211yescharacteristic
420C2×C4#677711yescharacteristic
421C2×C4#678011yescharacteristic
422C2×C4#678211yescharacteristic
423C8—102no—
424C8—302no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3#6523511yescharacteristic
428C2×C3#6535011yescharacteristic
429C2×C3#6536611yescharacteristic
430D3—44no—
431C5#6875811yescharacteristic, complemented, Sylow
432C22—21no—
433C22—21no—
434C22—64no—
435C22—124no—
436C22—122no—
437C22—122no—
438C22—242no—
439C22D3×#68311yescharacteristic
440C22#9147511yescharacteristic
441C22#9155511yescharacteristic
442C4—42no—
443C4—201no—
444C4—201no—
445C4—241no—
446C4—601no—
447C4—601no—
448C4#9140511yescharacteristic
449C4#9141711yescharacteristic
450C4#9143611yescharacteristic
451C4#9144211yescharacteristic
452C3#1060043511yescharacteristic, complemented, Sylow
453C2—21no—
454C2—21no—
455C2—124no—
456C2#1174711611yescharacteristic
457C2#1174711711yescharacteristic
458C2#1174889311yescharacteristic
subgroupG/H
centerC22#91475
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5#120
prime radicalC22×C5×D12C22
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. #11748472C2semidirect
4. #11748459C2semidirect
6. #11746344C2semidirect
7. C3×#64743C2semidirect
26. #90959C4semidirect
34. #64743D3semidirect
54. #6710C2×C4semidirect
64. C3×#681C2×C4semidirect
171. #681C4×D3semidirect
369. C3×C5#1856semidirect
431. C5#68758semidirect
452. C3#10600435semidirect

Maximal subgroups

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

subgroupindex
#117545032
C5×#49392
#117484722
#117484592
C2×#909592
#117463442
C3×#647432
#106004353
#687585