Finite Group Databasebeta

← #14301600

Subgroups — #14301600

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
1C2×#91825C211yesmaximal, characteristic
2#11754213C211yesmaximal, characteristic, complemented
3C5×#5215C211yesmaximal, characteristic
4C3×#65504C211yesmaximal, characteristic
5C2×#90963C211yesmaximal, characteristic, complemented
6#11745975C211yesmaximal, characteristic, complemented
7#11745973C211yesmaximal, characteristic, complemented
8#10601118—31nomaximal, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C22×Dic30C2211yescharacteristic
11C2×#6719—22no—
12#91825—22no—
13#91825C414yescomplemented
14C2×C4×Dic15C2211yescharacteristic
15C5×#926C2211yescharacteristic
16C2×C5×#208C2211yescharacteristic
17C3×#4072C2211yescharacteristic
18C2×C3×#681C2211yescharacteristic
19#91532—22no—
20#91340—22no—
21#91333—22no—
22C5×#832—22no—
23C3×#3964—22no—
24#90963—22no—
25#70325—51nomaximal, Hall
26C2×#4082—31no—
27#65850—31no—
28C5×#418—31no—
29#65504D311yescharacteristic
30C2×#3956—31no—
31#64775—31no—
32#64752—31no—
33C24×C3×C5—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4C2×C412yes—
40C22×C3×C4×C5C2311yescharacteristic
41C22×Dic15—21no—
42C22×Dic15D412yes—
43C2×Dic30—21no—
44C2×Dic30—21no—
45C2×Dic30—21no—
46C2×Dic30—21no—
47C2×Dic30C2×C412yes—
48C2×Dic30D412yes—
49#6719—28no—
50C4×Dic15—21no—
51C4×Dic15C2×C412yes—
52C5×#217—21no—
53C5×C4⋊Dic3D412yes—
54C4×C5×Dic3—21no—
55C5×#208—21no—
56C5×#208—21no—
57C2×C5×C3⋊C8—21no—
58C3×C22⋊Dic5—21no—
59C3×C4⋊Dic5D412yes—
60C3×C4×Dic5—21no—
61C3×#681—21no—
62C3×#681—21no—
63C2×C3×#168—21no—
64#6669—24no—
65Dic3×Dic5—24no—
66C2×#936—51no—
67#5516—51no—
68C3×#418—51no—
69#5215D511yescharacteristic
70C2×#824—51no—
71#4555—51no—
72#4532—51no—
73C22×C5×D4C2×D311yescharacteristic
74C22×Dic10—31no—
75C2×C5×M16—31no—
76C5×#117—31no—
77C2×C22⋊Dic5—62no—
78#4082—34no—
79#4082—62no—
80#4072C2×D311yescharacteristic
81C2×C4×Dic5—31no—
82C2×#681C2×D311yescharacteristic
83#4033—62no—
84#4023—62no—
85C4×Dic10—62no—
86C5×#102—62no—
87#3964—22no—
88#3956—62no—
89#1888—151noSylow
90C23×C3×C5—22no—
91C23×C3×C5—44no—
92C23×C3×C5—41no—
93C23×C3×C5—41no—
94C23×C3×C5C2×D411yescharacteristic
95C3×C5×D4—22no—
96C3×C5×D4—22no—
97C3×C5×D4—41no—
98C3×C5×D4—41no—
99C2×C3×C4×C5C22×C411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×Dic15—22no—
106C2×Dic15—22no—
107C2×Dic15—22no—
108C2×Dic15—42no—
109C2×Dic15C22⋊C414yes—
110Dic30—22no—
111Dic30—22no—
112Dic30—22no—
113Dic30—22no—
114C2×C5×Dic3—22no—
115C2×C3×Dic5—22no—
116C3×#168—41no—
117C5×C3⋊C8—41no—
118C22×C3×D4C2×D511yescharacteristic
119C22×Dic6—51no—
120C2×C3×M16—51no—
121C3×#117—51no—
122C2×#217—102no—
123#936—54no—
124#936—102no—
125#926C2×D511yescharacteristic
126C2×C4×Dic3—51no—
127C2×#208C2×D511yescharacteristic
128#887—102no—
129#877—102no—
130C4×Dic6—102no—
131C3×#102—102no—
132#832—22no—
133#824—102no—
134C24×C5—21no—
135C2×C5×D4—21no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4C4×D312yes—
141C22×C4×C5C22×D311yescharacteristic
142C22×Dic5—32no—
143C22×Dic5—61no—
144C2×Dic10—32no—
145C2×Dic10—32no—
146C2×Dic10—61no—
147C2×Dic10—61no—
148C2×Dic10—61no—
149C2×Dic10—61no—
150C5×M16—61no—
151C5×M16—61no—
152C2×C5×C8—61no—
153C5×C4⋊C4—32no—
154C5×C22⋊C4—61no—
155C42×C5—61no—
156C22⋊Dic5—21no—
157C22⋊Dic5—68no—
158C4⋊Dic5#6712yes—
159#683—64no—
160C4×Dic5—21no—
161C4×Dic5—32no—
162C4×Dic5—64no—
163C4×Dic5—61no—
164#681—21no—
165#681—21no—
166C2×#168—21no—
167C2×#124—151no—
168#519—151no—
169#418—31no—
170#418—51no—
171C2×#101—151no—
172#339—151no—
173#339—151no—
174C22×C3×C5—22no—
175C22×C3×C5—44no—
176C22×C3×C5—42no—
177C22×C3×C5—82no—
178C22×C3×C5—82no—
179C22×C3×C5C2×C22⋊C411yescharacteristic
180C22×C3×C5#12011yescharacteristic
181C22×C3×C5#12311yescharacteristic
182C3×C4×C5C4×D411yescharacteristic
183C3×C4×C5C4×D411yescharacteristic
184C3×C4×C5#12111yescharacteristic
185C3×C4×C5#12111yescharacteristic
186Dic15—24no—
187Dic15—42no—
188C3×Dic5—42no—
189C5×Dic3—42no—
190C24×C3—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4C4×D512yes—
197C22×C3×C4C22×D511yescharacteristic
198C22×Dic3—52no—
199C22×Dic3—101no—
200C2×Dic6—52no—
201C2×Dic6—52no—
202C2×Dic6—101no—
203C2×Dic6—101no—
204C2×Dic6—101no—
205C2×Dic6—101no—
206C3×M16—101no—
207C3×M16—101no—
208C2×C3×C8—101no—
209C3×C4⋊C4—52no—
210C3×C22⋊C4—101no—
211C3×C42—101no—
212#217—21no—
213#217—108no—
214C4⋊Dic3C22⋊D512yes—
215Dic3⋊C4—104no—
216C4×Dic3—21no—
217C4×Dic3—52no—
218C4×Dic3—104no—
219C4×Dic3—101no—
220#208—21no—
221#208—21no—
222C2×C3⋊C8—21no—
223C23×C5—22no—
224C23×C5—44no—
225C23×C5—41no—
226C23×C5—41no—
227C23×C5D3×D411yescharacteristic
228C5×D4—22no—
229C5×D4—22no—
230C5×D4—41no—
231C5×D4—41no—
232C2×C4×C5—62no—
233C2×C4×C5C2×C4×D311yescharacteristic
234C2×C4×C5#23511yescharacteristic
235C2×C4×C5D3×D411yescharacteristic
236C2×C4×C5D3×D411yescharacteristic
237C2×C4×C5#23711yescharacteristic
238C2×C4×C5C2×#6711yescharacteristic
239C2×Dic5—22no—
240C2×Dic5—34no—
241C2×Dic5—62no—
242C2×Dic5—62no—
243C2×Dic5—62no—
244C2×Dic5—122no—
245Dic10—62no—
246Dic10—62no—
247Dic10—62no—
248Dic10—62no—
249C5×C8—121no—
250#168—41no—
251C22×Q8—151no—
252C22×D4D3×D511yescharacteristic
253C2×M16—31no—
254C2×M16—51no—
255#124—154no—
256#124—302no—
257C4×Q8—302no—
258C4×Q8—302no—
259#117—31no—
260#117—51no—
261#117—302no—
262C2×C22⋊C4—302no—
263C2×C42—151no—
264#102—62no—
265#102—102no—
266#101—302no—
267C2×C3×C5—21no—
268C2×C3×C5—21no—
269C2×C3×C5—82no—
270C2×C3×C5#38611yescharacteristic
271C2×C3×C5#45412yes—
272C23×C3—22no—
273C23×C3—44no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3D4×D511yescharacteristic
277C3×D4—22no—
278C3×D4—22no—
279C3×D4—41no—
280C3×D4—41no—
281C2×C3×C4—102no—
282C2×C3×C4C2×C4×D511yescharacteristic
283C2×C3×C4#70911yescharacteristic
284C2×C3×C4D4×D511yescharacteristic
285C2×C3×C4D4×D511yescharacteristic
286C2×C3×C4#71111yescharacteristic
287C2×C3×C4C2×C22⋊D511yescharacteristic
288C2×Dic3—22no—
289C2×Dic3—54no—
290C2×Dic3—102no—
291C2×Dic3—102no—
292C2×Dic3—102no—
293C2×Dic3—202no—
294Dic6—102no—
295Dic6—102no—
296Dic6—102no—
297Dic6—102no—
298C3×C8—201no—
299C3⋊C8—41no—
300C22×C5—22no—
301C22×C5—44no—
302C22×C5—42no—
303C22×C5—82no—
304C22×C5—82no—
305C22×C5D3×C22⋊C411yescharacteristic
306C22×C5#88311yescharacteristic
307C22×C5#93711yescharacteristic
308C4×C5—122no—
309C4×C5#67⋊C411yescharacteristic
310C4×C5#88411yescharacteristic
311C4×C5C4×#6711yescharacteristic
312C4×C5#93811yescharacteristic
313Dic5—42no—
314Dic5—64no—
315Dic5—122no—
316C24—21no—
317C2×Q8—152no—
318C2×Q8—152no—
319C2×Q8—301no—
320C2×Q8—301no—
321C2×Q8—301no—
322C2×Q8—301no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4#120012yes—
329C22×C4—152no—
330C22×C4—301no—
331C22×C4C2×D3×D511yescharacteristic
332M16—61no—
333M16—61no—
334M16—101no—
335M16—101no—
336C2×C8—61no—
337C2×C8—101no—
338C4⋊C4—32no—
339C4⋊C4—52no—
340C4⋊C4—304no—
341C22⋊C4—61no—
342C22⋊C4—101no—
343C22⋊C4—308no—
344C42—61no—
345C42—101no—
346C42—152no—
347C42—304no—
348C42—301no—
349C3×C5#188811yescharacteristic, complemented, Hall
350C22×C3—22no—
351C22×C3—44no—
352C22×C3—42no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3D5×C22⋊C411yescharacteristic
356C22×C3#402911yescharacteristic
357C22×C3#408311yescharacteristic
358C3×C4—202no—
359C3×C4#402711yescharacteristic
360C3×C4#403011yescharacteristic
361C3×C4C4×C22⋊D511yescharacteristic
362C3×C4#408411yescharacteristic
363Dic3—42no—
364Dic3—104no—
365Dic3—202no—
366C2×C5—21no—
367C2×C5—21no—
368C2×C5—82no—
369C2×C5#495611yescharacteristic
370C2×C5#520112yes—
371C23—22no—
372C23—44no—
373C23—41no—
374C23—41no—
375C23#679011yescharacteristic
376Q8—302no—
377Q8—302no—
378Q8—302no—
379Q8—302no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384C2×C4—62no—
385C2×C4—102no—
386C2×C4—154no—
387C2×C4—302no—
388C2×C4—302no—
389C2×C4—302no—
390C2×C4—602no—
391C2×C4#677911yescharacteristic
392C2×C4#678211yescharacteristic
393C2×C4C2×#120011yescharacteristic
394C2×C4D5×#6711yescharacteristic
395C2×C4D3×C22⋊D511yescharacteristic
396C2×C4#679011yescharacteristic
397C8—121no—
398C8—201no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—82no—
402C2×C3#6524511yescharacteristic
403C2×C3#6549012yes—
404C5#7032511yescharacteristic, complemented, Sylow
405C22—22no—
406C22—44no—
407C22—42no—
408C22—82no—
409C22—82no—
410C22#9156111yescharacteristic
411C22#9156211yescharacteristic
412C22#9157711yescharacteristic
413C4—122no—
414C4—202no—
415C4—304no—
416C4—602no—
417C4#9153311yescharacteristic
418C4#9155211yescharacteristic
419C4#9155611yescharacteristic
420C4#9155711yescharacteristic
421C3#1060111811yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—82no—
425C2#1174916612yes—
426C2#1174952011yescharacteristic
subgroupG/H
centerC22#91577
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
Fitting, prime radicalC22×C3×C5×D4C22
socleC22×C3×C5#120
cosocleC22×C3×C4×C5C23
hypercenterC22×D4D3×D5
hypocenterC3×C5#1888
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #11754213C2semidirect
5. C2×#90963C2semidirect
6. #11745975C2semidirect
7. #11745973C2semidirect
13. #91825C4semidirect
349. C3×C5#1888semidirect
404. C5#70325semidirect
421. C3#10601118semidirect

Maximal subgroups

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

subgroupindex
C2×#918252
#117542132
C5×#52152
C3×#655042
C2×#909632
#117459752
#117459732
#106011183
#703255