Finite Group Databasebeta

← #14308916

Subgroups — #14308916

Order 1920.

564 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65853C211yesmaximal, characteristic, complemented
2C5×#5001C211yesmaximal, characteristic
3#11748797C211yesmaximal, characteristic
4#11748386C211yesmaximal, characteristic
5#11746353C211yesmaximal, characteristic, complemented
6#11745985C211yesmaximal, characteristic, complemented
7#11745973C211yesmaximal, characteristic, complemented
8#10601106—31nomaximal, Hall
9C22×C5×D12C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C2×C4×Dic15C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C5×#208C2211yescharacteristic
14C3×#4072C2211yescharacteristic
15C2×C3×C4⋊Dic5C2211yescharacteristic
16C3×#4036—21no—
17C3×#4034—21no—
18C3×#4024—22no—
19C3×#4016—21no—
20C3×#4015—21no—
21C2×(C2×D3)⋊Dic5—22no—
22C5×#809—21no—
23C5×#809—21no—
24#90966—21no—
25#90966—21no—
26#71903—51nomaximal, Hall
27#65853D311yescharacteristic, complemented
28C5×#418—31no—
29#65510—31no—
30#65504—31no—
31#64775—31no—
32#64774—31no—
33#64752—31no—
34C23×C5×D3—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C22×Dic15—21no—
37C2×C5×D12—21no—
38C2×C5×D12—21no—
39C2×C5×D12—21no—
40C2×C5×D12—21no—
41C2×C5×D12—21no—
42C2×C5×D12Q812yescomplemented
43C22×C3×Dic5D412yes—
44C2×C3×Dic10—22no—
45C3×C5×C4⋊C4D411yescharacteristic
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C5×C22⋊C4—21no—
48C3×C42×C5—21no—
49C4×Dic15—21no—
50C4×Dic15—21no—
51#6710—21no—
52#6710—21no—
53C2×C3⋊#168—21no—
54C5×#208—21no—
55C5×#208—21no—
56C2×C5×C3⋊C8—21no—
57C3×C22⋊Dic5—21no—
58C3×C4⋊Dic5—22no—
59C3×C4⋊Dic5—21no—
60C3×C4⋊Dic5—21no—
61C3×C4⋊Dic5D411yescharacteristic
62C3×C4⋊Dic5D411yescharacteristic
63C3×#683—22no—
64C3×#683—22no—
65C3×C4×Dic5—21no—
66(C2×D3)⋊Dic5—44no—
67C3×#544—51no—
68#5001—51no—
69#5001D511yescharacteristic
70#4988—51no—
71#4532—51no—
72#4532—51no—
73#4530—51no—
74C22×C5×D4—31no—
75C2×C5×M16—31no—
76C5×#117C2×D311yescharacteristic
77C2×C22⋊Dic5—62no—
78#4072C2×D311yescharacteristic
79C2×C4⋊Dic5C2×D311yescharacteristic
80C2×C4×Dic5—31no—
81C2×#681—31no—
82#4036—21no—
83#4034—21no—
84#4024—22no—
85#4016—21no—
86#4015—21no—
87C5×#102—61no—
88C5×#102—61no—
89#3964—61no—
90#3964—61no—
91#2027—151noSylow
92C23×C3×C5C2×D411yescharacteristic
93C22×C5×D3—22no—
94C22×C5×D3—42no—
95C22×C5×D3—42no—
96C22×C5×D3—41no—
97C22×C5×D3—41no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×Q811yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×Dic15—22no—
107C2×Dic15—41no—
108C2×Dic15—41no—
109C5×D12—22no—
110C5×D12—22no—
111C5×D12—41no—
112C5×D12—41no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—21no—
115C2×C3×Dic5—22no—
116C2×C3×Dic5—42no—
117C3×Dic10—82no—
118C3⋊#168—41no—
119C5×C3⋊C8—41no—
120C22×D12C2×D511yescharacteristic
121C3×C4⋊Q8—101no—
122C3×C4⋊Q8—101no—
123C3×#125—101no—
124C3×#125—101no—
125C3×C22⋊Q8—102no—
126C3×#117—51no—
127C3×#117C2×D511yescharacteristic
128C2×C3×C4⋊C4—51no—
129C2×(C2×D3)⋊C4—102no—
130C2×C4×Dic3—51no—
131C2×#208—51no—
132C2×#208C2×D511yescharacteristic
133#809—21no—
134#809—21no—
135#809—101no—
136#809—101no—
137C24×C5—61no—
138C2×C5×D4—32no—
139C2×C5×D4—61no—
140C2×C5×D4—61no—
141C2×C5×D4—61no—
142C2×C5×D4—61no—
143C2×C5×D4—61no—
144C22×C4×C5C22×D311yescharacteristic
145C22×Dic5—61no—
146C22×Dic5#6712yes—
147C2×Dic10—22no—
148C5×M16—61no—
149C5×M16—61no—
150C2×C5×C8—61no—
151C5×C4⋊C4#6711yescharacteristic
152C5×C4⋊C4#6711yescharacteristic
153C5×C22⋊C4—21no—
154C42×C5—21no—
155C22⋊Dic5—21no—
156C22⋊Dic5—124no—
157C4⋊Dic5—22no—
158C4⋊Dic5—21no—
159C4⋊Dic5—21no—
160C4⋊Dic5#6711yescharacteristic
161C4⋊Dic5#6711yescharacteristic
162#683—22no—
163#683—22no—
164C4×Dic5—21no—
165C4×Dic5—61no—
166C4×Dic5—61no—
167#681—61no—
168#681—61no—
169C2×#168—61no—
170#544—51no—
171#418—31no—
172#418—151no—
173#390—151no—
174#343—151no—
175#339—151no—
176#339—151no—
177C22×C3×C5—21no—
178C22×C3×C5—21no—
179C22×C3×C5#12011yescharacteristic
180C22×C3×C5C22⋊Q811yescharacteristic
181C22×C3×C5#12411yescharacteristic
182C2×C5×D3—44no—
183C2×C5×D3—42no—
184C2×C5×D3—82no—
185C2×C5×D3—81no—
186C2×C5×D3—81no—
187C3×C4×C5—41no—
188C3×C4×C5—41no—
189C3×C4×C5C22⋊Q811yescharacteristic
190C3×C4×C5C22⋊Q811yescharacteristic
191C3×C4×C5#12411yescharacteristic
192C3×C4×C5#12411yescharacteristic
193Dic15—42no—
194C3×Dic5—41no—
195C3×Dic5—42no—
196C3×Dic5—41no—
197C23×D3—21no—
198C2×C3×Q8—102no—
199C22×C3×C4—52no—
200C22×C3×C4C22×D511yescharacteristic
201C22×Dic3—101no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12Dic1012yescomplemented
208C3×C4⋊C4—51no—
209C3×C4⋊C4—51no—
210C3×C4⋊C4—102no—
211C3×C4⋊C4—102no—
212C3×C4⋊C4—102no—
213C3×C4⋊C4—101no—
214C3×C4⋊C4—101no—
215C3×C4⋊C4C22⋊D511yescharacteristic
216C3×C4⋊C4C22⋊D511yescharacteristic
217C3×C22⋊C4—21no—
218C3×C22⋊C4—101no—
219C3×C42—21no—
220C3×C42—101no—
221(C2×D3)⋊C4—204no—
222C4×Dic3—101no—
223C4×Dic3—101no—
224#208—21no—
225#208—21no—
226#208—101no—
227#208—101no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—101no—
230C23×C5—62no—
231C23×C5—122no—
232C23×C5—122no—
233C23×C5—121no—
234C23×C5—121no—
235C23×C5C2×#6711yescharacteristic
236C5×D4—62no—
237C5×D4—62no—
238C5×D4—121no—
239C5×D4—121no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5D3×Q811yescharacteristic
243C2×C4×C5#23911yescharacteristic
244C2×C4×C5#23911yescharacteristic
245C2×C4×C5#23911yescharacteristic
246C2×C4×C5C2×#6711yescharacteristic
247C2×C4×C5C2×#6711yescharacteristic
248C2×Dic5—21no—
249C2×Dic5—21no—
250C2×Dic5—22no—
251C2×Dic5—42no—
252C2×Dic5—62no—
253C2×Dic5—121no—
254C2×Dic5—121no—
255Dic10—82no—
256C5×C8—121no—
257#168—121no—
258C22×D4—31no—
259C2×M16—31no—
260C2×M16—151no—
261C4⋊Q8—101no—
262C4⋊Q8—101no—
263#125—101no—
264#125—101no—
265C22⋊Q8—102no—
266#117—51no—
267#117D3×D511yescharacteristic
268C2×C4⋊C4—51no—
269C2×C22⋊C4—302no—
270C2×C42—151no—
271#102—61no—
272#102—61no—
273#102—301no—
274#102—301no—
275C2×C3×C5—21no—
276C2×C3×C5—21no—
277C2×C3×C5#39311yescharacteristic
278C2×C3×C5#45311yescharacteristic
279C2×C3×C5#45411yescharacteristic
280C5×D3—82no—
281C23×C3D4×D511yescharacteristic
282C22×D3—22no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—41no—
286C22×D3—41no—
287C3×Q8—402no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—101no—
291C2×C3×C4—101no—
292C2×C3×C4—102no—
293C2×C3×C4—202no—
294C2×C3×C4C2×Dic1011yescharacteristic
295C2×C3×C4#70911yescharacteristic
296C2×C3×C4D4×D511yescharacteristic
297C2×C3×C4#71111yescharacteristic
298C2×C3×C4#71111yescharacteristic
299C2×C3×C4C2×C22⋊D511yescharacteristic
300C2×Dic3—102no—
301C2×Dic3—201no—
302C2×Dic3—201no—
303D12—22no—
304D12—22no—
305D12—41no—
306D12—41no—
307C3⋊C8—41no—
308C3⋊C8—201no—
309C22×C5—21no—
310C22×C5—21no—
311C22×C5—124no—
312C22×C5—122no—
313C22×C5—242no—
314C22×C5—241no—
315C22×C5—241no—
316C22×C5#94611yescharacteristic
317C22×C5#94711yescharacteristic
318C22×C5#95311yescharacteristic
319C4×C5—41no—
320C4×C5—41no—
321C4×C5#94611yescharacteristic
322C4×C5#94611yescharacteristic
323C4×C5#94711yescharacteristic
324C4×C5#94711yescharacteristic
325Dic5—41no—
326Dic5—42no—
327Dic5—41no—
328Dic5—122no—
329C24—61no—
330C2×Q8—102no—
331C2×D4—32no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C22×C4—52no—
338C22×C4—301no—
339C22×C4C2×D3×D511yescharacteristic
340M16—61no—
341M16—61no—
342M16—301no—
343M16—301no—
344C2×C8—61no—
345C2×C8—301no—
346C4⋊C4—51no—
347C4⋊C4—51no—
348C4⋊C4—102no—
349C4⋊C4—102no—
350C4⋊C4—102no—
351C4⋊C4—101no—
352C4⋊C4—101no—
353C4⋊C4#120111yescharacteristic
354C4⋊C4#120111yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—101no—
357C22⋊C4—604no—
358C42—21no—
359C42—101no—
360C42—301no—
361C42—301no—
362C3×C5#202711yescharacteristic, complemented, Hall
363C22×C3—21no—
364C22×C3—21no—
365C22×C3#402411yescharacteristic
366C22×C3#403111yescharacteristic
367C22×C3#408311yescharacteristic
368C2×D3—44no—
369C2×D3—42no—
370C2×D3—82no—
371C2×D3—81no—
372C2×D3—81no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—202no—
376C3×C4—201no—
377C3×C4—201no—
378C3×C4#402411yescharacteristic
379C3×C4#403111yescharacteristic
380C3×C4#407011yescharacteristic
381C3×C4#408211yescharacteristic
382Dic3—202no—
383C2×C5—21no—
384C2×C5—21no—
385C2×C5—242no—
386C2×C5#523111yescharacteristic
387C2×C5#524111yescharacteristic
388C2×C5#524511yescharacteristic
389C23—62no—
390C23—122no—
391C23—122no—
392C23—121no—
393C23—121no—
394C23D5×#6711yescharacteristic
395Q8—402no—
396D4—62no—
397D4—62no—
398D4—121no—
399D4—121no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—101no—
403C2×C4—101no—
404C2×C4—102no—
405C2×C4—202no—
406C2×C4—302no—
407C2×C4—601no—
408C2×C4—601no—
409C2×C4D3×Dic1011yescharacteristic
410C2×C4#676811yescharacteristic
411C2×C4#676811yescharacteristic
412C2×C4#676911yescharacteristic
413C2×C4C2×#120111yescharacteristic
414C2×C4D5×#6711yescharacteristic
415C8—121no—
416C8—601no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3#6523611yescharacteristic
420C2×C3#6548711yescharacteristic
421C2×C3#6549011yescharacteristic
422D3—82no—
423C5#7190311yescharacteristic, complemented, Sylow
424C22—21no—
425C22—21no—
426C22—124no—
427C22—122no—
428C22—242no—
429C22—241no—
430C22—241no—
431C22#9132611yescharacteristic
432C22#9141011yescharacteristic
433C22#9156311yescharacteristic
434C4—41no—
435C4—41no—
436C4—202no—
437C4—201no—
438C4—201no—
439C4—602no—
440C4#9132611yescharacteristic
441C4#9136911yescharacteristic
442C4#9137011yescharacteristic
443C4#9141011yescharacteristic
444C3#1060110611yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—242no—
448C2#1174874611yescharacteristic
449C2#1174941711yescharacteristic
450C2#1174945611yescharacteristic
subgroupG/H
centerC22#91563
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5#120
prime radicalC22×C5×D12C22
hypercenter#117D3×D5
hypocenterC3×C5#2027
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C3×#65853C2semidirect
5. #11746353C2semidirect
6. #11745985C2semidirect
7. #11745973C2semidirect
27. #65853D3semidirect
42. C2×C5×D12Q8semidirect
207. C2×D12Dic10semidirect
362. C3×C5#2027semidirect
423. C5#71903semidirect
444. C3#10601106semidirect

Maximal subgroups

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

subgroupindex
C3×#658532
C5×#50012
#117487972
#117483862
#117463532
#117459852
#117459732
#106011063
#719035