Finite Group Databasebeta

← #14301450

Subgroups — #14301450

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754515C211yesmaximal, characteristic
2C2×#91374C211yesmaximal, characteristic
3C2×#91107C211yesmaximal, characteristic
4C3×#65409C211yesmaximal, characteristic, complemented
5#11748399C211yesmaximal, characteristic, complemented
6C5×#4555C211yesmaximal, characteristic, complemented
7#11745974C211yesmaximal, characteristic, complemented
8#10599552—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×C3×Dic10C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C4×C5×Dic3C2211yescharacteristic
15C3×#4072C2211yescharacteristic
16#91561—22no—
17C2×#6667—22no—
18#91428—22no—
19#91418—22no—
20#91374—22no—
21#91374C414yescomplemented
22#91107—22no—
23C3×#3948—22no—
24#90970—22no—
25#70343—51nomaximal, Hall
26#65855—31no—
27C2×#4021—31no—
28C2×#3955—31no—
29#65409D311yescharacteristic, complemented
30#65293—31no—
31C5×#339—31no—
32#64743—31no—
33C23×D15D412yes—
34C22×C3×C4×C5C2311yescharacteristic
35C2×D60—21no—
36C2×D60—21no—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60C2×C412yes—
40C2×D60D412yes—
41C22×C5×Dic3—21no—
42C22×C3×Dic5—21no—
43C2×C3×Dic10—21no—
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—21no—
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C412yescomplemented
49C2×#1200—24no—
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C2×C3×C5×C8—21no—
53#6710—21no—
54#6710—21no—
55C2×C3⋊#168—21no—
56C5×#217—21no—
57C5×C4⋊Dic3D412yes—
58C4×C5×Dic3—21no—
59C4×C5×Dic3—21no—
60C4×C5×Dic3C2×C412yescomplemented
61C3×C22⋊Dic5—21no—
62C3×C4⋊Dic5D412yes—
63C3×C4×Dic5—21no—
64#6667—24no—
65#6667—28no—
66C2×#947—51no—
67#5521—51no—
68C3×#419—51no—
69#5225—51no—
70C2×#823—51no—
71#4555D511yescharacteristic, complemented
72#4532—51no—
73C22×D20—31no—
74C22×Dic10C2×D311yescharacteristic
75C2×C5×M16C2×D311yescharacteristic
76C5×#117—31no—
77C2×C42×C5—31no—
78C2×#685—62no—
79#4072C2×D311yescharacteristic
80C2×#681—31no—
81#4039—62no—
82D5×C22⋊C4—62no—
83#4021—34no—
84#4021—62no—
85C4×D20—62no—
86#3955—62no—
87#3948—22no—
88#3942—62no—
89#1885—151noSylow
90C23×C3×C5C2×D411yescharacteristic
91C22×D15—22no—
92C22×D15—22no—
93C22×D15—44no—
94C22×D15C22⋊C414yes—
95C2×C3×C4×C5C22×C411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5#4111yescharacteristic
100C2×C3×C4×C5#4111yescharacteristic
101D60—22no—
102D60—22no—
103D60—22no—
104D60—22no—
105C2×C5×Dic3—22no—
106C2×C5×Dic3—22no—
107C2×C5×Dic3—42no—
108C2×C3×Dic5—22no—
109C2×C3×Dic5—22no—
110C2×C3×Dic5—41no—
111C2×C3×Dic5—41no—
112C3×Dic10—22no—
113C3×Dic10—22no—
114C3×Dic10—41no—
115C3×Dic10—41no—
116#1200—84no—
117C3×C5×C8—41no—
118C3⋊#168—41no—
119C22×C3×Q8—51no—
120C22×D12—51no—
121C2×C3×M16C2×D511yescharacteristic
122C3×#117—51no—
123#947—54no—
124#947—102no—
125C2×(C2×D3)⋊C4—102no—
126#926C2×D511yescharacteristic
127C2×C4×Dic3C2×D511yescharacteristic
128C2×#208—51no—
129#893—102no—
130D3×C22⋊C4—102no—
131C4×D12—102no—
132C3×#103—102no—
133#835—102no—
134#823—102no—
135C23×D5—32no—
136C22×C4×C5—61no—
137C22×C4×C5C22×D311yescharacteristic
138C22×Dic5—21no—
139C2×D20—32no—
140C2×D20—32no—
141C2×D20—61no—
142C2×D20—61no—
143C2×D20—61no—
144C2×D20—61no—
145C2×C4×D5—64no—
146C2×Dic10—21no—
147C2×Dic10—21no—
148C2×Dic10—21no—
149C2×Dic10—21no—
150C2×Dic10—21no—
151C2×Dic10C4×D312yescomplemented
152C5×M16—21no—
153C5×M16—21no—
154C2×C5×C8—21no—
155C5×C4⋊C4—32no—
156C5×C22⋊C4—61no—
157C42×C5—32no—
158C42×C5—61no—
159C42×C5—61no—
160C22⋊Dic5—21no—
161#685—68no—
162#685—64no—
163C4⋊Dic5#6712yes—
164C4×Dic5—21no—
165#681—61no—
166#681—61no—
167C2×#168—61no—
168C2×#124—151no—
169#518—151no—
170#419—51no—
171#419—151no—
172C2×#100—151no—
173#339—31no—
174#339—151no—
175C22×C3×C5—22no—
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C22×C3×C5#12011yescharacteristic
178C22×C3×C5#12311yescharacteristic
179C2×D15—24no—
180C2×D15—44no—
181C2×D15—44no—
182C2×D15—82no—
183C3×C4×C5C4×D411yescharacteristic
184C3×C4×C5C4×D411yescharacteristic
185C3×C4×C5#12111yescharacteristic
186C3×C4×C5#12111yescharacteristic
187C3×Dic5—42no—
188C3×Dic5—42no—
189C5×Dic3—42no—
190C5×Dic3—42no—
191C23×D3—52no—
192C2×C3×Q8—52no—
193C2×C3×Q8—101no—
194C2×C3×Q8—101no—
195C2×C3×Q8—101no—
196C2×C3×Q8—101no—
197C2×C3×Q8—101no—
198C22×C3×C4—101no—
199C22×C3×C4C22×D511yescharacteristic
200C22×Dic3—21no—
201C2×D12—52no—
202C2×D12—52no—
203C2×D12—101no—
204C2×D12—101no—
205C2×D12—101no—
206C2×D12—101no—
207C2×C4×D3—104no—
208C3×M16—21no—
209C3×M16—21no—
210C2×C3×C8—21no—
211C3×C4⋊C4—52no—
212C3×C22⋊C4—101no—
213C3×C42—101no—
214#217—21no—
215(C2×D3)⋊C4—108no—
216(C2×D3)⋊C4—104no—
217C4⋊Dic3D2012yes—
218C4×Dic3—21no—
219C4×Dic3—21no—
220C4×Dic3C4×D512yescomplemented
221#208—101no—
222#208—101no—
223C2×C3⋊C8—101no—
224C23×C5D3×D411yescharacteristic
225C22×D5—34no—
226C22×D5—62no—
227C22×D5—62no—
228C22×D5—124no—
229C2×C4×C5—62no—
230C2×C4×C5—62no—
231C2×C4×C5—122no—
232C2×C4×C5C2×C4×D311yescharacteristic
233C2×C4×C5#23511yescharacteristic
234C2×C4×C5D3×D411yescharacteristic
235C2×C4×C5D3×D411yescharacteristic
236C2×C4×C5#23711yescharacteristic
237C2×C4×C5C2×#6711yescharacteristic
238C2×Dic5—22no—
239C2×Dic5—22no—
240C2×Dic5—41no—
241C2×Dic5—41no—
242D20—62no—
243D20—62no—
244D20—62no—
245D20—62no—
246C4×D5—244no—
247Dic10—22no—
248Dic10—22no—
249Dic10—41no—
250Dic10—41no—
251C5×C8—41no—
252#168—121no—
253C22×Q8—51no—
254C22×D4—151no—
255C2×M16—151no—
256C2×M16D3×D511yescharacteristic
257#124—154no—
258#124—302no—
259C4×D4—302no—
260C4×D4—302no—
261#117—31no—
262#117—51no—
263C2×C22⋊C4—302no—
264C2×C22⋊C4—302no—
265C2×C42—31no—
266#103—102no—
267#103—302no—
268#100—302no—
269C2×C3×C5—21no—
270C2×C3×C5—21no—
271C2×C3×C5#38611yescharacteristic
272C2×C3×C5#45512yes—
273D15—44no—
274C23×C3D4×D511yescharacteristic
275C22×D3—54no—
276C22×D3—102no—
277C22×D3—102no—
278C22×D3—204no—
279C3×Q8—102no—
280C3×Q8—102no—
281C3×Q8—201no—
282C3×Q8—201no—
283C2×C3×C4—102no—
284C2×C3×C4—102no—
285C2×C3×C4—201no—
286C2×C3×C4—201no—
287C2×C3×C4C2×C4×D511yescharacteristic
288C2×C3×C4C2×D2011yescharacteristic
289C2×C3×C4#70911yescharacteristic
290C2×C3×C4D4×D511yescharacteristic
291C2×C3×C4D4×D511yescharacteristic
292C2×C3×C4#71311yescharacteristic
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—42no—
296D12—102no—
297D12—102no—
298D12—102no—
299D12—102no—
300C4×D3—404no—
301C3×C8—41no—
302C3⋊C8—201no—
303C22×C5—22no—
304C22×C5D3×C22⋊C411yescharacteristic
305C22×C5#88311yescharacteristic
306C22×C5#93711yescharacteristic
307C2×D5—64no—
308C2×D5—124no—
309C2×D5—124no—
310C2×D5—242no—
311C4×C5—122no—
312C4×C5—122no—
313C4×C5#67⋊C411yescharacteristic
314C4×C5#88411yescharacteristic
315C4×C5C4×#6711yescharacteristic
316C4×C5#93811yescharacteristic
317Dic5—42no—
318Dic5—42no—
319C24—152no—
320C2×Q8—52no—
321C2×Q8—101no—
322C2×Q8—101no—
323C2×Q8—101no—
324C2×Q8—101no—
325C2×Q8—101no—
326C2×D4—152no—
327C2×D4—152no—
328C2×D4—301no—
329C2×D4—301no—
330C2×D4—301no—
331C2×D4—301no—
332C22×C4—61no—
333C22×C4—101no—
334C22×C4—304no—
335C22×C4C2×D3×D511yescharacteristic
336M16—21no—
337M16—21no—
338M16—301no—
339M16—301no—
340C2×C8—21no—
341C2×C8—301no—
342C4⋊C4—32no—
343C4⋊C4—52no—
344C22⋊C4—61no—
345C22⋊C4—101no—
346C22⋊C4—308no—
347C22⋊C4—304no—
348C42—32no—
349C42—61no—
350C42—61no—
351C42—101no—
352C3×C5#188511yescharacteristic, complemented, Hall
353C22×C3—22no—
354C22×C3D5×C22⋊C411yescharacteristic
355C22×C3#402811yescharacteristic
356C22×C3#404011yescharacteristic
357C2×D3—104no—
358C2×D3—204no—
359C2×D3—204no—
360C2×D3—402no—
361C3×C4—202no—
362C3×C4—202no—
363C3×C4C4×D2011yescharacteristic
364C3×C4#403011yescharacteristic
365C3×C4#403911yescharacteristic
366C3×C4#404111yescharacteristic
367Dic3—42no—
368Dic3—42no—
369C2×C5—21no—
370C2×C5—21no—
371C2×C5#495611yescharacteristic
372C2×C5#520212yes—
373D5—124no—
374C23—154no—
375C23—302no—
376C23—302no—
377C23—604no—
378C23#677711yescharacteristic
379Q8—102no—
380Q8—102no—
381Q8—201no—
382Q8—201no—
383D4—302no—
384D4—302no—
385D4—302no—
386D4—302no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—102no—
390C2×C4—102no—
391C2×C4—122no—
392C2×C4—201no—
393C2×C4—201no—
394C2×C4—1204no—
395C2×C4#677311yescharacteristic
396C2×C4C4×D3×D511yescharacteristic
397C2×C4D3×D2011yescharacteristic
398C2×C4#677911yescharacteristic
399C2×C4D5×#6711yescharacteristic
400C2×C4#679011yescharacteristic
401C8—41no—
402C8—601no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3#6495411yescharacteristic
406C2×C3#6511412yes—
407D3—204no—
408C5#7034311yescharacteristic, complemented, Sylow
409C22—22no—
410C22—304no—
411C22—604no—
412C22—604no—
413C22—1202no—
414C22#9135611yescharacteristic
415C22#9147311yescharacteristic
416C22#9147411yescharacteristic
417C4—122no—
418C4—122no—
419C4—202no—
420C4—202no—
421C4#9139511yescharacteristic
422C4#9144411yescharacteristic
423C4#9145911yescharacteristic
424C4#9153311yescharacteristic
425C3#1059955211yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—604no—
429C2#1174824012yes—
430C2#1174870011yescharacteristic
subgroupG/H
centerC22#91474
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5#120
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1885
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. C3×#65409C2semidirect
5. #11748399C2semidirect
6. C5×#4555C2semidirect
7. #11745974C2semidirect
21. #91374C4semidirect
29. #65409D3semidirect
48. C2×C3×Dic10C2×C4semidirect
60. C4×C5×Dic3C2×C4semidirect
71. #4555D5semidirect
151. C2×Dic10C4×D3semidirect
220. C4×Dic3C4×D5semidirect
352. C3×C5#1885semidirect
408. C5#70343semidirect
425. C3#10599552semidirect

Maximal subgroups

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

subgroupindex
#117545152
C2×#913742
C2×#911072
C3×#654092
#117483992
C5×#45552
#117459742
#105995523
#703435