Finite Group Databasebeta

← #11754789

Subgroups — #11754789

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×D5C211yesmaximal, characteristic, complemented
2C2×#6779C211yesmaximal, characteristic, complemented
3#91828C211yesmaximal, characteristic, complemented
4C5×#935C211yesmaximal, characteristic, complemented
5#91564C211yesmaximal, characteristic, complemented
6#91564C211yesmaximal, characteristic, complemented
7#91559C211yesmaximal, characteristic, complemented
8D5×#217C211yesmaximal, characteristic, complemented
9C2×#6665C211yesmaximal, characteristic, complemented
10#91534C211yesmaximal, characteristic, complemented
11#91533C211yesmaximal, characteristic, complemented
12#91444C211yesmaximal, characteristic, complemented
13#91409C211yesmaximal, characteristic, complemented
14#91407C211yesmaximal, characteristic, complemented
15#91349C211yesmaximal, characteristic, complemented
16#65988—31nomaximal, Hall
17C23×C3×D5C2211yescharacteristic
18C23×C3×D5C2211yescharacteristic
19C2×C3×C5×D4C2211yescharacteristic
20C2×#1220C2211yescharacteristic, complemented
21C2×#1220C2211yescharacteristic, complemented
22C22×Dic15C2211yescharacteristic, complemented
23C22×C5×Dic3C2211yescharacteristic, complemented
24C2×C3×C22⋊D5C2211yescharacteristic
25C2×C3×C22⋊D5C2211yescharacteristic
26C3×D4×D5—24no—
27C2×C3×D20C2211yescharacteristic
28C2×C3×C4×D5C2211yescharacteristic
29C2×#1204C2211yescharacteristic, complemented
30C2×#1202C2211yescharacteristic
31C2×#1202C2211yescharacteristic, complemented
32C2×#1202C2211yescharacteristic, complemented
33C2×#1200C2211yescharacteristic, complemented
34#6779—24no—
35C2×D5×Dic3C2211yescharacteristic, complemented
36C2×D5×Dic3C2211yescharacteristic, complemented
37#6719C2211yescharacteristic, complemented
38#6714C2211yescharacteristic, complemented
39#6712C2211yescharacteristic, complemented
40C5×#217C2211yescharacteristic, complemented
41C5×#217C2211yescharacteristic, complemented
42C5×#217C2211yescharacteristic, complemented
43C5×Dic3⋊C4C2211yescharacteristic, complemented
44C5×Dic3⋊C4C2211yescharacteristic, complemented
45#6668C2211yescharacteristic, complemented
46#6665C2211yescharacteristic
47#6665C2211yescharacteristic
48#6665C2211yescharacteristic
49#6665C2211yescharacteristic, complemented
50#6665C2211yescharacteristic, complemented
51#6665C2211yescharacteristic, complemented
52#6665C2211yescharacteristic, complemented
53Dic3×Dic5C2211yescharacteristic, complemented
54#5806—51nomaximal, Hall
55C2×D4×D5D311yescharacteristic, complemented
56C2×#709—31no—
57C5×#123—31no—
58#4085—31no—
59C2×#685—31no—
60#4042—31no—
61#4041—31no—
62#4040—31no—
63#4039—31no—
64#4031—31no—
65#4030—31no—
66#4028—31no—
67#4028—31no—
68#4027—31no—
69D5×C22⋊C4—31no—
70C23×C3×C5C2311yescharacteristic
71C23×C3×C5C2311yescharacteristic
72C22×D15C2311yescharacteristic
73C22×C3×D5—22no—
74C22×C3×D5—22no—
75C22×C3×D5—22no—
76C22×C3×D5—22no—
77C22×C3×D5—41no—
78C22×C3×D5—41no—
79C22×C3×D5C2311yescharacteristic
80C22×C3×D5C2311yescharacteristic
81C22×C3×D5C2311yescharacteristic
82C3×C5×D4—22no—
83C2×C3×C4×C5C2311yescharacteristic
84#1220—22no—
85#1220—41no—
86C2×Dic15—41no—
87C2×Dic15C2311yescharacteristic
88C2×Dic15C2311yescharacteristic
89C2×Dic15C2311yescharacteristic
90C2×C5×Dic3—22no—
91C2×C5×Dic3C2311yescharacteristic
92C2×C5×Dic3C2311yescharacteristic
93C2×C5×Dic3C2311yescharacteristic
94C2×C5×Dic3C2311yescharacteristic
95C3×C22⋊D5—22no—
96C3×C22⋊D5D414yescomplemented
97C2×C3×Dic5C2311yescharacteristic
98C3×D20—22no—
99C3×C4×D5—22no—
100#1204—22no—
101#1202—22no—
102#1202—22no—
103#1202—41no—
104#1200—22no—
105D5×Dic3—22no—
106D5×Dic3—41no—
107C22×C3×D4—51no—
108C2×#237—51no—
109#953—51no—
110#953—51no—
111C2×#217—51no—
112C2×#217—51no—
113#939—51no—
114#939—51no—
115#938—51no—
116#936—51no—
117#935—51no—
118#935D511yescharacteristic, complemented
119D4×Dic3—51no—
120C4×#67—51no—
121#924—51no—
122C23×D5C2×D311yescharacteristic
123C23×D5C2×D311yescharacteristic
124C2×C5×D4C2×D311yescharacteristic
125C22×C4×C5—31no—
126C2×C22⋊D5—31no—
127C2×C22⋊D5—31no—
128C2×C22⋊D5C2×D311yescharacteristic
129C2×C22⋊D5C2×D311yescharacteristic
130C22×Dic5—31no—
131D4×D5—24no—
132#709—64no—
133C2×D20—31no—
134C2×D20—31no—
135C2×D20—31no—
136C2×D20C2×D311yescharacteristic
137C2×C4×D5—31no—
138C2×C4×D5—31no—
139C2×C4×D5—31no—
140C2×C4×D5C2×D311yescharacteristic
141C2×Dic10—31no—
142C5×C4⋊C4—31no—
143C5×C4⋊C4—31no—
144C5×C22⋊C4—31no—
145C5×C22⋊C4—31no—
146C5×C22⋊C4—31no—
147C22⋊Dic5—31no—
148#685—31no—
149#685—31no—
150#685—31no—
151#685—31no—
152#685—31no—
153#685—31no—
154#685—31no—
155#685—31no—
156#683—31no—
157#683—31no—
158C4×Dic5—31no—
159#546—151noSylow
160C22×C3×C5—21no—
161C22×C3×C5—21no—
162C22×C3×C5—21no—
163C22×C3×C5—21no—
164C22×C3×C5—21no—
165C22×C3×C5C2×D412yes—
166C22×C3×C5C2411yescharacteristic
167C2×D15—21no—
168C2×D15—21no—
169C2×D15—21no—
170C2×C3×D5—21no—
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—21no—
177C2×C3×D5—44no—
178C2×C3×D5—44no—
179C2×C3×D5C2×D412yes—
180C2×C3×D5#4112yescomplemented
181C3×C4×C5—21no—
182Dic15—21no—
183Dic15—21no—
184Dic15—21no—
185C3×Dic5C2×D412yes—
186C5×Dic3—21no—
187C5×Dic3—21no—
188C5×Dic3—21no—
189C5×Dic3—21no—
190C24×C3—51no—
191C24×C3—51no—
192C2×C3×D4—51no—
193C2×C3×D4—51no—
194C2×C3×D4—51no—
195C2×C3×D4—104no—
196C2×C3×D4C2×D511yescharacteristic
197C22×C3×C4—51no—
198C2×#67—51no—
199C2×#67—51no—
200C2×#67—51no—
201C2×#67—51no—
202C2×#67—51no—
203C22×Dic3—51no—
204C22×Dic3—51no—
205C22×Dic3—51no—
206C22×Dic3C2×D511yescharacteristic, complemented
207#237—104no—
208C2×C4×D3—51no—
209C2×Dic6—51no—
210#217—51no—
211#217—51no—
212#217—51no—
213#217—51no—
214#217—51no—
215#217—51no—
216#217—51no—
217#217—51no—
218#217C2×D511yescharacteristic, complemented
219#217C2×D511yescharacteristic, complemented
220#217C2×D511yescharacteristic, complemented
221(C2×D3)⋊C4—51no—
222C4⋊Dic3—51no—
223Dic3⋊C4—51no—
224Dic3⋊C4C2×D511yescharacteristic, complemented
225Dic3⋊C4C2×D511yescharacteristic, complemented
226C4×Dic3—51no—
227C23×C5C22×D311yescharacteristic
228C23×C5C22×D311yescharacteristic
229C22×D5—22no—
230C22×D5—22no—
231C22×D5—22no—
232C22×D5—22no—
233C22×D5—31no—
234C22×D5—41no—
235C22×D5—41no—
236C22×D5C22×D311yescharacteristic
237C22×D5C22×D311yescharacteristic
238C22×D5C22×D311yescharacteristic
239C5×D4—22no—
240C2×C4×C5—31no—
241C2×C4×C5—31no—
242C2×C4×C5—31no—
243C2×C4×C5—31no—
244C2×C4×C5—62no—
245C2×C4×C5C22×D311yescharacteristic
246C22⋊D5—22no—
247C22⋊D5—62no—
248C22⋊D5—121no—
249C22⋊D5#6714yescomplemented
250C2×Dic5—31no—
251C2×Dic5—31no—
252C2×Dic5—31no—
253C2×Dic5—121no—
254C2×Dic5C22×D311yescharacteristic
255D20—22no—
256D20—62no—
257D20—62no—
258D20—121no—
259C4×D5—22no—
260C4×D5—62no—
261C4×D5—62no—
262C4×D5—121no—
263Dic10—62no—
264C2×#41—151no—
265C22×D4—51no—
266#124—151no—
267#123—31no—
268#123—151no—
269C22⋊Q8—151no—
270#121—151no—
271#121—151no—
272#121—151no—
273#120—151no—
274#120—151no—
275C4×D4—151no—
276C4×D4—151no—
277C2×C22⋊C4—151no—
278C2×C22⋊C4—151no—
279C2×C3×C5—22no—
280C2×C3×C5—41no—
281C2×C3×C5C22×D411yescharacteristic
282C2×C3×C5C2×#4111yescharacteristic
283C2×C3×C5H(5,𝔽2)11yescharacteristic
284D15—41no—
285C3×D5—22no—
286C3×D5—22no—
287C3×D5—41no—
288C23×C3—51no—
289C23×C3—51no—
290C23×C3—51no—
291C23×C3—102no—
292C23×C3—102no—
293C23×C3—102no—
294C23×C3—102no—
295C23×C3—201no—
296C23×C3—201no—
297C23×C3C22×D511yescharacteristic
298C23×C3C22×D511yescharacteristic
299C22×D3—51no—
300C3×D4—22no—
301C3×D4—54no—
302C3×D4—102no—
303C3×D4—102no—
304C2×C3×C4—51no—
305C2×C3×C4—102no—
306C2×C3×C4C22×D511yescharacteristic
307#67—102no—
308#67—102no—
309#67—102no—
310#67—201no—
311#67—201no—
312C2×Dic3—22no—
313C2×Dic3—51no—
314C2×Dic3—51no—
315C2×Dic3—51no—
316C2×Dic3—102no—
317C2×Dic3—201no—
318C2×Dic3—201no—
319C2×Dic3C22×D511yescharacteristic
320C2×Dic3C22×D511yescharacteristic
321C2×Dic3C22×D511yescharacteristic
322C2×Dic3C22×D511yescharacteristic
323C4×D3—102no—
324Dic6—102no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—21no—
328C22×C5—21no—
329C22×C5—21no—
330C22×C5C2×#6712yes—
331C22×C5C23×D311yescharacteristic
332C2×D5—21no—
333C2×D5—21no—
334C2×D5—21no—
335C2×D5—21no—
336C2×D5—21no—
337C2×D5—21no—
338C2×D5—21no—
339C2×D5—44no—
340C2×D5—44no—
341C2×D5—61no—
342C2×D5—61no—
343C2×D5—61no—
344C2×D5#23712yescomplemented
345C2×D5C2×#6712yes—
346C4×C5—21no—
347C4×C5—61no—
348C4×C5—61no—
349C4×C5—61no—
350C4×C5—61no—
351Dic5—61no—
352Dic5—61no—
353Dic5—61no—
354Dic5C2×#6712yes—
355C24—51no—
356C24—51no—
357#41—304no—
358C2×Q8—151no—
359C2×D4—51no—
360C2×D4—51no—
361C2×D4—51no—
362C2×D4—104no—
363C2×D4—151no—
364C2×D4—151no—
365C2×D4—151no—
366C2×D4—151no—
367C2×D4—151no—
368C2×D4D3×D511yescharacteristic
369C22×C4—31no—
370C22×C4—51no—
371C22×C4—151no—
372C22×C4—151no—
373C22×C4—151no—
374C22×C4—151no—
375C4⋊C4—31no—
376C4⋊C4—31no—
377C4⋊C4—151no—
378C4⋊C4—151no—
379C22⋊C4—31no—
380C22⋊C4—31no—
381C22⋊C4—31no—
382C22⋊C4—151no—
383C22⋊C4—151no—
384C22⋊C4—151no—
385C22⋊C4—151no—
386C22⋊C4—151no—
387C22⋊C4—151no—
388C22⋊C4—151no—
389C22⋊C4—151no—
390C22⋊C4—151no—
391C42—151no—
392C3×C5#54611yescharacteristic, complemented, Hall
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—52no—
399C22×C3—52no—
400C22×C3—101no—
401C22×C3—101no—
402C22×C3—101no—
403C22×C3—101no—
404C22×C3—101no—
405C22×C3—101no—
406C22×C3—101no—
407C22×C3—204no—
408C22×C3—204no—
409C22×C3D4×D512yes—
410C22×C3C23×D511yescharacteristic
411C2×D3—101no—
412C2×D3—101no—
413C2×D3—101no—
414C3×C4—21no—
415C3×C4—52no—
416Dic3—21no—
417Dic3—21no—
418Dic3—21no—
419Dic3—21no—
420Dic3—101no—
421Dic3—101no—
422Dic3—101no—
423C2×C5—22no—
424C2×C5—41no—
425C2×C5C2×#23711yescharacteristic
426C2×C5#100411yescharacteristic
427C2×C5C22×#6711yescharacteristic
428D5—22no—
429D5—22no—
430D5—41no—
431D5—121no—
432C23—51no—
433C23—51no—
434C23—51no—
435C23—102no—
436C23—102no—
437C23—102no—
438C23—102no—
439C23—151no—
440C23—201no—
441C23—201no—
442C23C2×D3×D511yescharacteristic
443C23C2×D3×D511yescharacteristic
444Q8—302no—
445D4—22no—
446D4—54no—
447D4—102no—
448D4—102no—
449D4—302no—
450D4—302no—
451D4—302no—
452D4—601no—
453D4—601no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—31no—
458C2×C4—51no—
459C2×C4—62no—
460C2×C4—102no—
461C2×C4—151no—
462C2×C4—151no—
463C2×C4—151no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—601no—
467C2×C4—601no—
468C2×C4C2×D3×D511yescharacteristic
469C2×C3—22no—
470C2×C3—41no—
471C2×C3—102no—
472C2×C3—102no—
473C2×C3—201no—
474C2×C3C2×D4×D511yescharacteristic
475C2×C3D5×#4111yescharacteristic
476C2×C3#414911yescharacteristic
477D3—201no—
478C5#580611yescharacteristic, complemented, Sylow
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—52no—
485C22—52no—
486C22—101no—
487C22—101no—
488C22—101no—
489C22—101no—
490C22—101no—
491C22—101no—
492C22—101no—
493C22—204no—
494C22—204no—
495C22—301no—
496C22—301no—
497C22—301no—
498C22D5×#6712yes—
499C22C22×D3×D511yescharacteristic
500C4—21no—
501C4—52no—
502C4—61no—
503C4—61no—
504C4—61no—
505C4—61no—
506C4—301no—
507C4—301no—
508C4—301no—
509C3#6598811yescharacteristic, complemented, Sylow
510C2—22no—
511C2—41no—
512C2—102no—
513C2—102no—
514C2—201no—
515C2—601no—
516C2D5×#23711yescharacteristic
517C2#9202611yescharacteristic
518C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#546
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×D5C2semidirect
2. C2×#6779C2semidirect
3. #91828C2semidirect
4. C5×#935C2semidirect
5. #91564C2semidirect
6. #91564C2semidirect
7. #91559C2semidirect
8. D5×#217C2semidirect
9. C2×#6665C2semidirect
10. #91534C2semidirect
11. #91533C2semidirect
12. #91444C2semidirect
13. #91409C2semidirect
14. #91407C2semidirect
15. #91349C2semidirect
20. C2×#1220C22semidirect
21. C2×#1220C22semidirect
22. C22×Dic15C22semidirect
23. C22×C5×Dic3C22semidirect
29. C2×#1204C22semidirect
31. C2×#1202C22semidirect
32. C2×#1202C22semidirect
33. C2×#1200C22semidirect
35. C2×D5×Dic3C22semidirect
36. C2×D5×Dic3C22semidirect
37. #6719C22semidirect
38. #6714C22semidirect
39. #6712C22semidirect
40. C5×#217C22semidirect
41. C5×#217C22semidirect
42. C5×#217C22semidirect
43. C5×Dic3⋊C4C22semidirect
44. C5×Dic3⋊C4C22semidirect
45. #6668C22semidirect
49. #6665C22semidirect
50. #6665C22semidirect
51. #6665C22semidirect
52. #6665C22semidirect
53. Dic3×Dic5C22semidirect
55. C2×D4×D5D3semidirect
96. C3×C22⋊D5D4semidirect
118. #935D5semidirect
180. C2×C3×D5#41semidirect
206. C22×Dic3C2×D5semidirect
218. #217C2×D5semidirect
219. #217C2×D5semidirect
220. #217C2×D5semidirect
224. Dic3⋊C4C2×D5semidirect
225. Dic3⋊C4C2×D5semidirect
249. C22⋊D5#67semidirect
344. C2×D5#237semidirect
392. C3×C5#546semidirect
478. C5#5806semidirect
509. C3#65988semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D52
C2×#67792
#918282
C5×#9352
#915642
#915592
D5×#2172
C2×#66652
#915342
#915332
#914442
#914092
#914072
#913492
#659883
#58065