Finite Group Databasebeta

← #11754307

Subgroups — #11754307

Order 960.

584 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#709C211yesmaximal, characteristic, complemented
2C2×#6780C211yesmaximal, characteristic, complemented
3C2×#6712C211yesmaximal, characteristic, complemented
4C5×#929C211yesmaximal, characteristic, complemented
5#91571C211yesmaximal, characteristic, complemented
6#91559C211yesmaximal, characteristic, complemented
7#91537C211yesmaximal, characteristic, complemented
8#91447C211yesmaximal, characteristic, complemented
9#91440C211yesmaximal, characteristic, complemented
10#91410C211yesmaximal, characteristic, complemented
11#91408C211yesmaximal, characteristic, complemented
12#91407C211yesmaximal, characteristic, complemented
13D5×Dic3⋊C4C211yesmaximal, characteristic, complemented
14#91349C211yesmaximal, characteristic, complemented
15#91327C211yesmaximal, characteristic, complemented
16#65989—31nomaximal, Hall
17C22×C3×C4×C5C2211yescharacteristic
18C22×Dic15C2211yescharacteristic
19C22×Dic15C2211yescharacteristic
20C2×C5×#67C2211yescharacteristic
21C2×C3×C22⋊D5C2211yescharacteristic
22C2×C3×C22⋊D5C2211yescharacteristic
23C3×#709—24no—
24C2×C3×D20C2211yescharacteristic
25C2×C3×C4×D5C2211yescharacteristic
26C2×C3×C4×D5C2211yescharacteristic
27C2×C3×Dic10C2211yescharacteristic
28C2×#1204C2211yescharacteristic
29C2×#1201C2211yescharacteristic
30C2×#1201C2211yescharacteristic
31C2×D3×Dic5C2211yescharacteristic
32#6780—24no—
33C2×D5×Dic3C2211yescharacteristic
34C2×D5×Dic3C2211yescharacteristic
35#6712C2211yescharacteristic
36#6712C2211yescharacteristic
37#6712C2211yescharacteristic
38#6712C2211yescharacteristic
39C5×#217C2211yescharacteristic
40C5×(C2×D3)⋊C4C2211yescharacteristic
41C5×(C2×D3)⋊C4C2211yescharacteristic
42C5×Dic3⋊C4C2211yescharacteristic
43C5×Dic3⋊C4C2211yescharacteristic
44Dic3⋊Dic5C2211yescharacteristic
45Dic3⋊Dic5C2211yescharacteristic
46#6669C2211yescharacteristic
47#6668C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49#6665C2211yescharacteristic
50#6665C2211yescharacteristic
51#6665C2211yescharacteristic
52#6665C2211yescharacteristic
53Dic3×Dic5C2211yescharacteristic
54#5824—51nomaximal, Hall
55C2×#711—31no—
56C2×#709D311yescharacteristic, complemented
57C5×#123—31no—
58#4085—31no—
59C2×#683—31no—
60#4042—31no—
61#4040—31no—
62D5×C4⋊C4—31no—
63#4034—31no—
64#4032—31no—
65#4030—31no—
66#4029—31no—
67#4027—31no—
68#4027—31no—
69#4024—31no—
70C23×C3×C5C2311yescharacteristic
71C22×C5×D3C2311yescharacteristic
72C22×C3×D5C2311yescharacteristic
73C22×C3×D5C2311yescharacteristic
74C2×C3×C4×C5—22no—
75C2×C3×C4×C5C2311yescharacteristic
76C2×C3×C4×C5C2311yescharacteristic
77C2×Dic15—22no—
78C2×Dic15—22no—
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C2×Dic15C2311yescharacteristic
83C5×#67—22no—
84C2×C5×Dic3C2311yescharacteristic
85C2×C5×Dic3C2311yescharacteristic
86C2×C5×Dic3C2311yescharacteristic
87C3×C22⋊D5—22no—
88C3×C22⋊D5D414yescomplemented
89C2×C3×Dic5C2311yescharacteristic
90C2×C3×Dic5C2311yescharacteristic
91C3×D20—22no—
92C3×C4×D5—22no—
93C3×C4×D5—22no—
94C3×Dic10—22no—
95#1204—22no—
96#1201—22no—
97#1201—22no—
98D3×Dic5—22no—
99D5×Dic3—22no—
100D5×Dic3—22no—
101C2×C3×#41—51no—
102C2×#237—51no—
103#946—51no—
104#944—51no—
105#939—51no—
106#939—51no—
107#935—51no—
108#935—51no—
109D4×Dic3—51no—
110#929—51no—
111#929D511yescharacteristic, complemented
112C4×#67—51no—
113#924—51no—
114C2×Dic3⋊C4—51no—
115C2×Dic3⋊C4—51no—
116C2×C5×D4—31no—
117C22×C4×C5C2×D311yescharacteristic
118C2×C22⋊D5—31no—
119C2×C22⋊D5—31no—
120C2×C22⋊D5C2×D311yescharacteristic
121C2×C22⋊D5C2×D311yescharacteristic
122C22×Dic5—31no—
123C22×Dic5—31no—
124C22×Dic5—31no—
125#711—64no—
126#709—24no—
127C2×D20C2×D311yescharacteristic
128C2×C4×D5—31no—
129C2×C4×D5—31no—
130C2×C4×D5C2×D311yescharacteristic
131C2×C4×D5C2×D311yescharacteristic
132C2×Dic10—31no—
133C2×Dic10C2×D311yescharacteristic
134C5×C4⋊C4—31no—
135C5×C4⋊C4—31no—
136C5×C22⋊C4—31no—
137C5×C22⋊C4—31no—
138C5×C22⋊C4—31no—
139C22⋊Dic5—31no—
140#685—31no—
141#685—31no—
142#685—31no—
143#685—31no—
144C4⋊Dic5—31no—
145C4⋊Dic5—31no—
146#683—31no—
147#683—31no—
148#683—31no—
149#683—31no—
150#683—31no—
151#683—31no—
152C4×Dic5—31no—
153#547—151noSylow
154C22×C3×C5—21no—
155C22×C3×C5—21no—
156C22×C3×C5C2×D412yes—
157C22×C3×C5C2411yescharacteristic
158C2×C5×D3—21no—
159C2×C5×D3—21no—
160C2×C5×D3—21no—
161C2×C3×D5—21no—
162C2×C3×D5—21no—
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C2×C3×D5C2×D412yes—
167C3×C4×C5—21no—
168C3×C4×C5—21no—
169Dic15—21no—
170Dic15—21no—
171Dic15—21no—
172Dic15#4112yescomplemented
173C3×Dic5—21no—
174C3×Dic5C2×D412yes—
175C5×Dic3—21no—
176C5×Dic3—21no—
177C5×Dic3—21no—
178C3×#41—104no—
179C2×C3×Q8—51no—
180C2×C3×D4—51no—
181C2×C3×D4—51no—
182C2×C3×D4—51no—
183C22×C3×C4—51no—
184C22×C3×C4—51no—
185C22×C3×C4C2×D511yescharacteristic
186C2×#67—51no—
187C2×#67—51no—
188C2×#67C2×D511yescharacteristic
189C22×Dic3—51no—
190C22×Dic3—51no—
191C22×Dic3—51no—
192C22×Dic3—51no—
193#237—104no—
194C2×C4×D3—51no—
195C2×Dic6—51no—
196#217—51no—
197#217—51no—
198#217—51no—
199#217—51no—
200#217C2×D511yescharacteristic
201(C2×D3)⋊C4—51no—
202(C2×D3)⋊C4C2×D511yescharacteristic
203(C2×D3)⋊C4C2×D511yescharacteristic
204C4⋊Dic3—51no—
205Dic3⋊C4—51no—
206Dic3⋊C4—51no—
207Dic3⋊C4—51no—
208Dic3⋊C4—51no—
209Dic3⋊C4—51no—
210Dic3⋊C4—51no—
211Dic3⋊C4—51no—
212Dic3⋊C4C2×D511yescharacteristic
213Dic3⋊C4C2×D511yescharacteristic
214C4×Dic3—51no—
215C23×C5—31no—
216C23×C5C22×D311yescharacteristic
217C22×D5C22×D311yescharacteristic
218C22×D5C22×D311yescharacteristic
219C5×D4—62no—
220C2×C4×C5—22no—
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5C22×D311yescharacteristic
225C2×C4×C5C22×D311yescharacteristic
226C22⋊D5—22no—
227C22⋊D5—62no—
228C22⋊D5—62no—
229C22⋊D5#6714yescomplemented
230C2×Dic5—31no—
231C2×Dic5—31no—
232C2×Dic5—31no—
233C2×Dic5—31no—
234C2×Dic5—62no—
235C2×Dic5—62no—
236C2×Dic5—62no—
237C2×Dic5C22×D311yescharacteristic
238C2×Dic5C22×D311yescharacteristic
239D20—22no—
240C4×D5—22no—
241C4×D5—22no—
242C4×D5—62no—
243C4×D5—62no—
244Dic10—22no—
245Dic10—62no—
246C2×#41—51no—
247C2×#41—151no—
248C4⋊Q8—151no—
249#123—31no—
250#123—151no—
251#123—151no—
252#123—151no—
253C22⋊Q8—151no—
254C22⋊Q8—151no—
255#121—151no—
256#121—151no—
257C4×D4—151no—
258C4×D4—151no—
259C2×C4⋊C4—151no—
260C2×C4⋊C4—151no—
261C2×C3×C5—22no—
262C2×C3×C5C22×D411yescharacteristic
263C2×C3×C5C2×#4111yescharacteristic
264C2×C3×C52−1+411yescharacteristic
265C3×D5—22no—
266C3×D5—41no—
267C5×D3—41no—
268C23×C3—51no—
269C23×C3—51no—
270C23×C3C22×D511yescharacteristic
271C22×D3C22×D511yescharacteristic
272C3×Q8—102no—
273C3×D4—54no—
274C3×D4—102no—
275C3×D4—102no—
276C2×C3×C4—22no—
277C2×C3×C4—51no—
278C2×C3×C4—51no—
279C2×C3×C4—102no—
280C2×C3×C4—102no—
281C2×C3×C4C22×D511yescharacteristic
282C2×C3×C4C22×D511yescharacteristic
283#67—22no—
284#67—102no—
285#67—102no—
286C2×Dic3—51no—
287C2×Dic3—51no—
288C2×Dic3—51no—
289C2×Dic3—51no—
290C2×Dic3—102no—
291C2×Dic3—102no—
292C2×Dic3—102no—
293C2×Dic3—102no—
294C2×Dic3C22×D511yescharacteristic
295C2×Dic3C22×D511yescharacteristic
296C2×Dic3C22×D511yescharacteristic
297C4×D3—102no—
298Dic6—102no—
299C22×C5—21no—
300C22×C5—21no—
301C22×C5—61no—
302C22×C5—61no—
303C22×C5—61no—
304C22×C5C2×#6712yes—
305C22×C5C23×D311yescharacteristic
306C2×D5—21no—
307C2×D5—21no—
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—21no—
311C2×D5C2×#6712yes—
312C4×C5—21no—
313C4×C5—21no—
314C4×C5—61no—
315C4×C5—61no—
316C4×C5—61no—
317Dic5—21no—
318Dic5—32no—
319Dic5—61no—
320Dic5—61no—
321Dic5—61no—
322Dic5C2×#6712yes—
323#41—104no—
324#41—304no—
325C2×Q8—51no—
326C2×Q8—151no—
327C2×D4—31no—
328C2×D4—51no—
329C2×D4—51no—
330C2×D4—51no—
331C2×D4—151no—
332C2×D4—151no—
333C22×C4—51no—
334C22×C4—51no—
335C22×C4—151no—
336C22×C4—151no—
337C22×C4—151no—
338C22×C4—151no—
339C22×C4—151no—
340C22×C4D3×D511yescharacteristic
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—151no—
344C4⋊C4—151no—
345C4⋊C4—151no—
346C4⋊C4—151no—
347C4⋊C4—151no—
348C4⋊C4—151no—
349C4⋊C4—151no—
350C4⋊C4—151no—
351C22⋊C4—31no—
352C22⋊C4—31no—
353C22⋊C4—31no—
354C22⋊C4—151no—
355C22⋊C4—151no—
356C22⋊C4—151no—
357C22⋊C4—151no—
358C22⋊C4—151no—
359C42—151no—
360C3×C5#54711yescharacteristic, complemented, Hall
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—52no—
364C22×C3—101no—
365C22×C3—101no—
366C22×C3—101no—
367C22×C3—101no—
368C22×C3—101no—
369C22×C3D4×D512yes—
370C22×C3C23×D511yescharacteristic
371C2×D3—21no—
372C2×D3—21no—
373C2×D3—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—52no—
377C3×C4—101no—
378Dic3—21no—
379Dic3—21no—
380Dic3—21no—
381Dic3—52no—
382Dic3—101no—
383Dic3—101no—
384Dic3—101no—
385C2×C5—22no—
386C2×C5—121no—
387C2×C5D3×#4111yescharacteristic
388C2×C5#101011yescharacteristic
389C2×C5C22×#6711yescharacteristic
390D5—22no—
391D5—41no—
392C23—31no—
393C23—51no—
394C23—51no—
395C23C2×D3×D511yescharacteristic
396Q8—102no—
397Q8—302no—
398D4—54no—
399D4—62no—
400D4—102no—
401D4—102no—
402D4—302no—
403D4—302no—
404C2×C4—22no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—51no—
409C2×C4—51no—
410C2×C4—102no—
411C2×C4—102no—
412C2×C4—151no—
413C2×C4—151no—
414C2×C4—151no—
415C2×C4—151no—
416C2×C4—302no—
417C2×C4—302no—
418C2×C4—302no—
419C2×C4—302no—
420C2×C4—302no—
421C2×C4C2×D3×D511yescharacteristic
422C2×C4C2×D3×D511yescharacteristic
423C2×C3—22no—
424C2×C3—102no—
425C2×C3—201no—
426C2×C3C2×D4×D511yescharacteristic
427C2×C3D5×#4111yescharacteristic
428C2×C3#415011yescharacteristic
429D3—41no—
430C5#582411yescharacteristic, complemented, Sylow
431C22—21no—
432C22—21no—
433C22—52no—
434C22—61no—
435C22—61no—
436C22—61no—
437C22—101no—
438C22—101no—
439C22—101no—
440C22—101no—
441C22—101no—
442C22D5×#6712yes—
443C22C22×D3×D511yescharacteristic
444C4—21no—
445C4—21no—
446C4—52no—
447C4—61no—
448C4—61no—
449C4—61no—
450C4—101no—
451C4—152no—
452C4—301no—
453C4—301no—
454C4—301no—
455C3#6598911yescharacteristic, complemented, Sylow
456C2—22no—
457C2—102no—
458C2—121no—
459C2—201no—
460C2#9200111yescharacteristic
461C2#9201611yescharacteristic
462C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#547
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#709C2semidirect
2. C2×#6780C2semidirect
3. C2×#6712C2semidirect
4. C5×#929C2semidirect
5. #91571C2semidirect
6. #91559C2semidirect
7. #91537C2semidirect
8. #91447C2semidirect
9. #91440C2semidirect
10. #91410C2semidirect
11. #91408C2semidirect
12. #91407C2semidirect
13. D5×Dic3⋊C4C2semidirect
14. #91349C2semidirect
15. #91327C2semidirect
56. C2×#709D3semidirect
88. C3×C22⋊D5D4semidirect
111. #929D5semidirect
172. Dic15#41semidirect
229. C22⋊D5#67semidirect
360. C3×C5#547semidirect
430. C5#5824semidirect
455. C3#65989semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7092
C2×#67802
C2×#67122
C5×#9292
#915712
#915592
#915372
#914472
#914402
#914102
#914082
#914072
D5×Dic3⋊C42
#913492
#913272
#659893
#58245