Finite Group Databasebeta

← #14301592

Subgroups — #14301592

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65850C211yesmaximal, characteristic, complemented
2C2×#91374C211yesmaximal, characteristic
3#11751102C211yesmaximal, characteristic
4#11748392C211yesmaximal, characteristic
5C2×#90961C211yesmaximal, characteristic, complemented
6C5×#4532C211yesmaximal, characteristic, complemented
7#11745974C211yesmaximal, characteristic, complemented
8#10599559—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×C3×Dic10C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4072C2211yescharacteristic
16C3×#4033—22no—
17C3×#4023—22no—
18C3×C4×Dic10—22no—
19C2×#6667—22no—
20#91374—22no—
21#91374C414yescomplemented
22#91093—22no—
23#90968—22no—
24#90961—22no—
25#71661—51nomaximal, Hall
26#65850D311yescharacteristic, complemented
27C2×#4021—31no—
28C2×#3956—31no—
29#65420—31no—
30#65290—31no—
31C5×#339—31no—
32#64743—31no—
33C23×D15—21no—
34C22×C3×C4×C5C2311yescharacteristic
35C2×D60—21no—
36C2×D60—21no—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60—21no—
40C2×D60C2×C412yes—
41C22×C5×Dic3—21no—
42C22×C3×Dic5D412yes—
43C2×C3×Dic10—21no—
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—21no—
47C2×C3×Dic10C2×C412yes—
48C2×C3×Dic10D412yes—
49C3×C5×C4⋊C4D412yes—
50C3×C5×C22⋊C4—21no—
51C3×C42×C5—21no—
52#6710—21no—
53#6710—21no—
54C2×C3⋊#168—21no—
55C4×C5×Dic3—21no—
56C4×C5×Dic3C2×C412yes—
57C5×#208—21no—
58C5×#208—21no—
59C2×C5×C3⋊C8—21no—
60C3×C22⋊Dic5—21no—
61C3×C4⋊Dic5D412yes—
62C3×#683—24no—
63C3×C4×Dic5—24no—
64C3×C4×Dic5—21no—
65#6667—28no—
66C3×#519—51no—
67C2×#947—51no—
68C2×#836—51no—
69#5001—51no—
70#5001—51no—
71#4532—51no—
72#4532D511yescharacteristic, complemented
73C22×D20—31no—
74C22×Dic10C2×D311yescharacteristic
75C2×C5×M16—31no—
76C5×#117C2×D311yescharacteristic
77C2×C42×C5—31no—
78C2×#685—62no—
79#4072C2×D311yescharacteristic
80C2×#681—31no—
81#4033—22no—
82#4023—22no—
83#4021—34no—
84#4021—62no—
85C4×Dic10—22no—
86#3956—62no—
87#3953—62no—
88#3941—62no—
89#1888—151noSylow
90C23×C3×C5C2×D411yescharacteristic
91C22×D15—22no—
92C22×D15—44no—
93C22×D15—41no—
94C22×D15—41no—
95C2×C3×C4×C5—22no—
96C2×C3×C4×C5C22×C411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5#4111yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102D60—22no—
103D60—22no—
104D60—41no—
105D60—41no—
106C2×C5×Dic3—22no—
107C2×C5×Dic3—42no—
108C2×C3×Dic5—22no—
109C2×C3×Dic5—22no—
110C2×C3×Dic5—22no—
111C2×C3×Dic5C22⋊C414yes—
112C3×Dic10—22no—
113C3×Dic10—22no—
114C3×Dic10—22no—
115C3×Dic10—22no—
116C3⋊#168—41no—
117C5×C3⋊C8—41no—
118C22×C3×Q8—51no—
119C22×D12—51no—
120C3×C4×Q8—102no—
121C3×C4×Q8—102no—
122C3×#117—51no—
123C3×#117—102no—
124C3×#117C2×D511yescharacteristic
125#947—54no—
126#947—102no—
127C2×(C2×D3)⋊C4—102no—
128C2×C4×Dic3C2×D511yescharacteristic
129C2×#208—51no—
130C2×#208C2×D511yescharacteristic
131#836—102no—
132#809—102no—
133#809—102no—
134C23×D5—61no—
135C22×C4×C5—61no—
136C22×C4×C5C22×D311yescharacteristic
137C22×Dic5#6712yes—
138C2×D20—32no—
139C2×D20—61no—
140C2×D20—61no—
141C2×D20—61no—
142C2×D20—61no—
143C2×D20—61no—
144C2×Dic10—21no—
145C2×Dic10—21no—
146C2×Dic10—21no—
147C2×Dic10—21no—
148C2×Dic10C4×D312yes—
149C2×Dic10D1212yes—
150C5×M16—61no—
151C5×M16—61no—
152C2×C5×C8—61no—
153C5×C4⋊C4#6712yes—
154C5×C22⋊C4—21no—
155C42×C5—21no—
156C42×C5—32no—
157C42×C5—61no—
158C22⋊Dic5—21no—
159#685—68no—
160C4⋊Dic5#6712yes—
161#683—24no—
162C4×Dic5—24no—
163C4×Dic5—21no—
164#681—61no—
165#681—61no—
166C2×#168—61no—
167C2×#124—151no—
168#519—51no—
169#418—151no—
170#418—151no—
171C2×#101—151no—
172#339—31no—
173#339—151no—
174C22×C3×C5—22no—
175C22×C3×C5C2×C22⋊C411yescharacteristic
176C22×C3×C5#12011yescharacteristic
177C22×C3×C5#12311yescharacteristic
178C2×D15—44no—
179C2×D15—42no—
180C2×D15—82no—
181C2×D15—82no—
182C3×C4×C5—42no—
183C3×C4×C5C4×D411yescharacteristic
184C3×C4×C5C4×D411yescharacteristic
185C3×C4×C5#12111yescharacteristic
186C3×C4×C5#12111yescharacteristic
187C3×Dic5—24no—
188C3×Dic5—42no—
189C5×Dic3—42no—
190C23×D3—101no—
191C2×C3×Q8—52no—
192C2×C3×Q8—52no—
193C2×C3×Q8—101no—
194C2×C3×Q8—101no—
195C2×C3×Q8—101no—
196C2×C3×Q8—101no—
197C22×C3×C4—52no—
198C22×C3×C4C22×D511yescharacteristic
199C22×Dic3—21no—
200C2×D12—52no—
201C2×D12—101no—
202C2×D12—101no—
203C2×D12—101no—
204C2×D12—101no—
205C2×D12—101no—
206C3×C4⋊C4—52no—
207C3×C4⋊C4—104no—
208C3×C4⋊C4D2012yes—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—101no—
211C3×C42—21no—
212C3×C42—104no—
213C3×C42—101no—
214(C2×D3)⋊C4—108no—
215C4×Dic3—21no—
216C4×Dic3C4×D512yes—
217#208—21no—
218#208—21no—
219#208—101no—
220#208—101no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—101no—
223C23×C5C2×#6711yescharacteristic
224C22×D5—62no—
225C22×D5—124no—
226C22×D5—121no—
227C22×D5—121no—
228C2×C4×C5—22no—
229C2×C4×C5—62no—
230C2×C4×C5—122no—
231C2×C4×C5C2×C4×D311yescharacteristic
232C2×C4×C5C2×D1211yescharacteristic
233C2×C4×C5#23511yescharacteristic
234C2×C4×C5#23511yescharacteristic
235C2×C4×C5C2×#6711yescharacteristic
236C2×C4×C5C2×#6711yescharacteristic
237C2×Dic5—22no—
238C2×Dic5—22no—
239C2×Dic5—22no—
240C2×Dic5(C2×D3)⋊C414yes—
241D20—62no—
242D20—62no—
243D20—121no—
244D20—121no—
245Dic10—22no—
246Dic10—22no—
247Dic10—22no—
248Dic10—22no—
249C5×C8—121no—
250#168—121no—
251C22×Q8—51no—
252C22×D4—151no—
253C2×M16—31no—
254C2×M16—151no—
255#124—154no—
256#124—302no—
257C4×Q8—102no—
258C4×Q8—102no—
259#117—51no—
260#117—102no—
261#117D3×D511yescharacteristic
262C2×C22⋊C4—302no—
263C2×C42—31no—
264#102—302no—
265#102—302no—
266#101—302no—
267C2×C3×C5—21no—
268C2×C3×C5—21no—
269C2×C3×C5#38611yescharacteristic
270C2×C3×C5#45412yes—
271D15—82no—
272C23×C3D4×D511yescharacteristic
273C22×D3—102no—
274C22×D3—204no—
275C22×D3—201no—
276C22×D3—201no—
277C3×Q8—102no—
278C3×Q8—102no—
279C3×Q8—102no—
280C3×Q8—102no—
281C2×C3×C4—22no—
282C2×C3×C4—54no—
283C2×C3×C4—102no—
284C2×C3×C4—102no—
285C2×C3×C4—102no—
286C2×C3×C4C2×C4×D511yescharacteristic
287C2×C3×C4C2×D2011yescharacteristic
288C2×C3×C4#70911yescharacteristic
289C2×C3×C4D4×D511yescharacteristic
290C2×C3×C4D4×D511yescharacteristic
291C2×C3×C4#71311yescharacteristic
292C2×Dic3—22no—
293C2×Dic3—42no—
294D12—102no—
295D12—102no—
296D12—201no—
297D12—201no—
298C3⋊C8—41no—
299C3⋊C8—201no—
300C22×C5—22no—
301C22×C5C2×(C2×D3)⋊C411yescharacteristic
302C22×C5#92911yescharacteristic
303C22×C5#95311yescharacteristic
304C2×D5—124no—
305C2×D5—122no—
306C2×D5—242no—
307C2×D5—242no—
308C4×C5—42no—
309C4×C5—122no—
310C4×C5C4×#6711yescharacteristic
311C4×C5C4×#6711yescharacteristic
312C4×C5#93011yescharacteristic
313C4×C5#93011yescharacteristic
314Dic5—24no—
315Dic5—42no—
316C24—301no—
317C2×Q8—52no—
318C2×Q8—52no—
319C2×Q8—101no—
320C2×Q8—101no—
321C2×Q8—101no—
322C2×Q8—101no—
323C2×D4—152no—
324C2×D4—301no—
325C2×D4—301no—
326C2×D4—301no—
327C2×D4—301no—
328C2×D4—301no—
329C22×C4—52no—
330C22×C4—61no—
331C22×C4C2×D3×D511yescharacteristic
332M16—61no—
333M16—61no—
334M16—301no—
335M16—301no—
336C2×C8—61no—
337C2×C8—301no—
338C4⋊C4—52no—
339C4⋊C4—104no—
340C4⋊C4#120212yes—
341C22⋊C4—21no—
342C22⋊C4—101no—
343C22⋊C4—308no—
344C42—21no—
345C42—32no—
346C42—61no—
347C42—104no—
348C42—101no—
349C3×C5#188811yescharacteristic, complemented, Hall
350C22×C3—22no—
351C22×C3D5×C22⋊C411yescharacteristic
352C22×C3#402811yescharacteristic
353C22×C3#404011yescharacteristic
354C2×D3—204no—
355C2×D3—202no—
356C2×D3—402no—
357C2×D3—402no—
358C3×C4—42no—
359C3×C4—104no—
360C3×C4—202no—
361C3×C4C4×D2011yescharacteristic
362C3×C4#403011yescharacteristic
363C3×C4#403911yescharacteristic
364C3×C4#404111yescharacteristic
365Dic3—42no—
366C2×C5—21no—
367C2×C5—21no—
368C2×C5#521311yescharacteristic
369C2×C5#524512yes—
370D5—242no—
371C23—302no—
372C23—604no—
373C23—601no—
374C23—601no—
375C23D5×#6711yescharacteristic
376Q8—102no—
377Q8—102no—
378Q8—102no—
379Q8—102no—
380D4—302no—
381D4—302no—
382D4—601no—
383D4—601no—
384C2×C4—22no—
385C2×C4—54no—
386C2×C4—62no—
387C2×C4—102no—
388C2×C4—102no—
389C2×C4—102no—
390C2×C4—122no—
391C2×C4#677111yescharacteristic
392C2×C4#677311yescharacteristic
393C2×C4C4×D3×D511yescharacteristic
394C2×C4D5×D1211yescharacteristic
395C2×C4C2×#120211yescharacteristic
396C2×C4D5×#6711yescharacteristic
397C8—121no—
398C8—601no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3#6495411yescharacteristic
402C2×C3#6511612yes—
403D3—402no—
404C5#7166111yescharacteristic, complemented, Sylow
405C22—22no—
406C22—604no—
407C22—602no—
408C22—1202no—
409C22—1202no—
410C22#9134811yescharacteristic
411C22D5×(C2×D3)⋊C411yescharacteristic
412C22#9156411yescharacteristic
413C4—42no—
414C4—104no—
415C4—122no—
416C4—202no—
417C4C4×#120211yescharacteristic
418C4#9144411yescharacteristic
419C4#9144811yescharacteristic
420C4#9145011yescharacteristic
421C3#1059955911yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—1202no—
425C2#1174890711yescharacteristic
426C2#1174945712yes—
subgroupG/H
centerC22#91564
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5#120
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1888
LCSG ▷ C2×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×#65850C2semidirect
5. C2×#90961C2semidirect
6. C5×#4532C2semidirect
7. #11745974C2semidirect
21. #91374C4semidirect
26. #65850D3semidirect
72. #4532D5semidirect
349. C3×C5#1888semidirect
404. C5#71661semidirect
421. C3#10599559semidirect

Maximal subgroups

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

subgroupindex
C3×#658502
C2×#913742
#117511022
#117483922
C2×#909612
C5×#45322
#117459742
#105995593
#716615