Finite Group Databasebeta

← #11753236

Subgroups — #11753236

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×D5×#67C211yesmaximal, characteristic, complemented
2C2×#6771C211yesmaximal, characteristic, complemented
3#91772C211yesmaximal, characteristic, complemented
4C5×#884C211yesmaximal, characteristic, complemented
5C3×D5×C22⋊C4C211yesmaximal, characteristic, complemented
6#91563C211yesmaximal, characteristic, complemented
7#91557C211yesmaximal, characteristic, complemented
8#91525C211yesmaximal, characteristic, complemented
9#91475C211yesmaximal, characteristic, complemented
10#91473C211yesmaximal, characteristic, complemented
11#91449C211yesmaximal, characteristic, complemented
12#91447C211yesmaximal, characteristic, complemented
13#91444C211yesmaximal, characteristic, complemented
14#91441C211yesmaximal, characteristic, complemented
15#91422C211yesmaximal, characteristic, complemented
16#65946—31nomaximal, Hall
17C23×C3×D5C2211yescharacteristic
18C22×D3×D5C2211yescharacteristic
19C2×#1220C2211yescharacteristic
20C2×C4×D15C2211yescharacteristic, complemented
21C2×C5×#67C2211yescharacteristic
22C2×C5×#67C2211yescharacteristic, complemented
23C2×C5×D12C2211yescharacteristic, complemented
24C2×C4×C5×D3C2211yescharacteristic, complemented
25C2×C3×C4×D5C2211yescharacteristic, complemented
26C2×C3×C4×D5C2211yescharacteristic, complemented
27D5×#67—24no—
28C2×#1204C2211yescharacteristic, complemented
29C2×#1203C2211yescharacteristic, complemented
30C2×#1202C2211yescharacteristic
31C2×#1201C2211yescharacteristic
32C2×#1201C2211yescharacteristic
33C2×#1201C2211yescharacteristic, complemented
34C2×#1200C2211yescharacteristic, complemented
35C2×D5×Dic3C2211yescharacteristic
36#6771—24no—
37C3×C5×C22⋊C4C2211yescharacteristic, complemented
38#6719C2211yescharacteristic, complemented
39#6714C2211yescharacteristic, complemented
40C4⋊Dic15C2211yescharacteristic, complemented
41#6712C2211yescharacteristic, complemented
42C5×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C5×Dic3⋊C4C2211yescharacteristic, complemented
44C3×C22⋊Dic5C2211yescharacteristic, complemented
45C3×#685C2211yescharacteristic
46C3×#685C2211yescharacteristic
47Dic3⋊Dic5C2211yescharacteristic, complemented
48(C2×D3)⋊Dic5C2211yescharacteristic, complemented
49(C2×D3)⋊Dic5C2211yescharacteristic, complemented
50(C2×D3)⋊Dic5C2211yescharacteristic, complemented
51#6665C2211yescharacteristic
52#6665C2211yescharacteristic, complemented
53Dic3×Dic5C2211yescharacteristic, complemented
54#5490—51nomaximal, Hall
55C2×D4×D5—31no—
56C2×#709—31no—
57C5×#121—31no—
58#4084—31no—
59#4084—31no—
60#4083—31no—
61#4083—31no—
62#4082—31no—
63C4×C22⋊D5—31no—
64#4043—31no—
65#4039—31no—
66#4029—31no—
67#4029—31no—
68D5×C22⋊C4—31no—
69D5×C22⋊C4D311yescharacteristic, complemented
70C23×C3×C5C2311yescharacteristic
71C22×D15C2311yescharacteristic
72C22×C5×D3C2311yescharacteristic
73C22×C5×D3C2311yescharacteristic
74C22×C3×D5—22no—
75C22×C3×D5—22no—
76C22×C3×D5—41no—
77C22×C3×D5C2311yescharacteristic
78C22×C3×D5C2311yescharacteristic
79C2×D3×D5—22no—
80C2×D3×D5—22no—
81C2×D3×D5—41no—
82C2×C3×C4×C5C2311yescharacteristic
83C2×C3×C4×C5C2311yescharacteristic
84#1220—22no—
85C2×Dic15C2311yescharacteristic
86C2×Dic15C2311yescharacteristic
87C2×Dic15C2311yescharacteristic
88C4×D15—22no—
89C5×#67—22no—
90C5×#67—41no—
91C2×C5×Dic3C2311yescharacteristic
92C2×C5×Dic3C2311yescharacteristic
93C5×D12—41no—
94C4×C5×D3—22no—
95C2×C3×Dic5C2311yescharacteristic
96C2×C3×Dic5C2311yescharacteristic
97C3×C4×D5—22no—
98C3×C4×D5—41no—
99#1204—22no—
100#1203—22no—
101#1202D414yescomplemented
102#1201—22no—
103#1201—22no—
104#1201—22no—
105#1200—41no—
106D5×Dic3—22no—
107C22×#67—51no—
108C2×#235—51no—
109C2×C3×C22⋊C4—51no—
110#953—51no—
111#930—51no—
112#929—51no—
113C4×#67—51no—
114#885—51no—
115#884—51no—
116#884D511yescharacteristic, complemented
117#883—51no—
118#882—51no—
119#67⋊C4—51no—
120D3×C22⋊C4—51no—
121#878—51no—
122C23×D5—31no—
123C23×D5C2×D311yescharacteristic
124C2×C5×D4—31no—
125C2×C5×D4—31no—
126C2×C5×D4—31no—
127C22×C4×C5—31no—
128C2×C22⋊D5—31no—
129C2×C22⋊D5—31no—
130C2×C22⋊D5—31no—
131C2×C22⋊D5—31no—
132C2×C22⋊D5—31no—
133D4×D5—64no—
134#709—64no—
135C2×D20—31no—
136C2×C4×D5—31no—
137C2×C4×D5—31no—
138C2×C4×D5—31no—
139C2×C4×D5C2×D311yescharacteristic, complemented
140C2×C4×D5C2×D311yescharacteristic, complemented
141C2×Dic10—31no—
142C5×C4⋊C4—31no—
143C5×C22⋊C4—31no—
144C5×C22⋊C4C2×D311yescharacteristic, complemented
145C22⋊Dic5—31no—
146C22⋊Dic5—31no—
147C22⋊Dic5—31no—
148C22⋊Dic5—31no—
149C22⋊Dic5C2×D311yescharacteristic, complemented
150#685—31no—
151#685—31no—
152#685—31no—
153#685C2×D311yescharacteristic
154#685C2×D311yescharacteristic
155C4⋊Dic5—31no—
156C4⋊Dic5—31no—
157#683—31no—
158C4×Dic5—31no—
159#546—151noSylow
160C22×C3×C5—21no—
161C22×C3×C5—21no—
162C22×C3×C5—21no—
163C22×C3×C5C2411yescharacteristic
164C2×D15—21no—
165C2×D15—21no—
166C2×D15C2×D412yes—
167C2×C5×D3—21no—
168C2×C5×D3—21no—
169C2×C5×D3—21no—
170C2×C5×D3—21no—
171C2×C5×D3—21no—
172C2×C5×D3—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—21no—
177C2×C3×D5—44no—
178C2×C3×D5C2×D412yes—
179C2×C3×D5#4112yescomplemented
180D3×D5—44no—
181C3×C4×C5—21no—
182C3×C4×C5—21no—
183Dic15—21no—
184Dic15—21no—
185Dic15—21no—
186C3×Dic5—21no—
187C3×Dic5—21no—
188C5×Dic3—21no—
189C5×Dic3C2×D412yes—
190C24×C3—51no—
191C23×D3—51no—
192C22×C3×C4—51no—
193C22×C3×C4—51no—
194C2×#67—51no—
195C2×#67—51no—
196C2×#67—51no—
197C2×#67—51no—
198C2×#67—51no—
199C2×#67—104no—
200C2×#67C2×D511yescharacteristic
201C2×#67C2×D511yescharacteristic, complemented
202C22×Dic3—51no—
203#235—104no—
204C2×D12—51no—
205C2×D12C2×D511yescharacteristic, complemented
206C2×C4×D3—51no—
207C2×C4×D3—51no—
208C2×C4×D3C2×D511yescharacteristic, complemented
209C2×Dic6—51no—
210C3×C22⋊C4—51no—
211C3×C22⋊C4—51no—
212C3×C22⋊C4—51no—
213C3×C22⋊C4C2×D511yescharacteristic, complemented
214#217—51no—
215#217—51no—
216#217—51no—
217(C2×D3)⋊C4—51no—
218(C2×D3)⋊C4—51no—
219(C2×D3)⋊C4—51no—
220(C2×D3)⋊C4—51no—
221(C2×D3)⋊C4C2×D511yescharacteristic, complemented
222C4⋊Dic3—51no—
223Dic3⋊C4—51no—
224Dic3⋊C4—51no—
225Dic3⋊C4C2×D511yescharacteristic, complemented
226C4×Dic3—51no—
227C23×C5—31no—
228C23×C5—31no—
229C23×C5C22×D311yescharacteristic
230C22×D5—22no—
231C22×D5—22no—
232C22×D5—31no—
233C22×D5—41no—
234C22×D5—62no—
235C22×D5—62no—
236C22×D5—121no—
237C22×D5C22×D311yescharacteristic
238C22×D5C22×D311yescharacteristic
239C5×D4—62no—
240C5×D4—121no—
241C5×D4—121no—
242C2×C4×C5—31no—
243C2×C4×C5—31no—
244C2×C4×C5—62no—
245C2×C4×C5C22×D311yescharacteristic
246C2×C4×C5C22×D311yescharacteristic
247C22⋊D5—62no—
248C22⋊D5—62no—
249C22⋊D5—62no—
250C22⋊D5—62no—
251C22⋊D5—62no—
252C2×Dic5—31no—
253C2×Dic5—31no—
254C2×Dic5—31no—
255C2×Dic5C22×D311yescharacteristic
256C2×Dic5C22×D311yescharacteristic
257D20—34no—
258C4×D5—22no—
259C4×D5—41no—
260C4×D5—62no—
261C4×D5—62no—
262C4×D5—121no—
263Dic10—62no—
264C2×#41—151no—
265C22×D4—151no—
266#124—151no—
267#123—151no—
268#123—151no—
269C22⋊Q8—151no—
270#121—31no—
271#121—151no—
272#121—151no—
273#120—151no—
274#120—151no—
275C4×D4—151no—
276C4×D4—151no—
277C2×C22⋊C4—51no—
278C2×C22⋊C4—151no—
279C2×C3×C5—41no—
280C2×C3×C5C22×D411yescharacteristic
281C2×C3×C5C2×#4111yescharacteristic
282C2×C3×C5H(5,𝔽2)11yescharacteristic
283D15—22no—
284C3×D5—22no—
285C3×D5—22no—
286C5×D3—41no—
287C5×D3—41no—
288C23×C3—51no—
289C23×C3—51no—
290C23×C3—102no—
291C23×C3—102no—
292C23×C3—201no—
293C23×C3C22×D511yescharacteristic
294C22×D3—51no—
295C22×D3—102no—
296C22×D3—102no—
297C22×D3—201no—
298C22×D3C22×D511yescharacteristic
299C22×D3C22×D511yescharacteristic
300C2×C3×C4—51no—
301C2×C3×C4—51no—
302C2×C3×C4—102no—
303C2×C3×C4—201no—
304C2×C3×C4C22×D511yescharacteristic
305C2×C3×C4C22×D511yescharacteristic
306#67—22no—
307#67—41no—
308#67—54no—
309#67—102no—
310#67—102no—
311#67—102no—
312#67—102no—
313C2×Dic3—51no—
314C2×Dic3—51no—
315C2×Dic3—51no—
316C2×Dic3—102no—
317C2×Dic3C22×D511yescharacteristic
318C2×Dic3C22×D511yescharacteristic
319D12—41no—
320D12—102no—
321C4×D3—22no—
322C4×D3—102no—
323C4×D3—201no—
324Dic6—102no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—21no—
328C22×C5—61no—
329C22×C5—61no—
330C22×C5—61no—
331C22×C5—61no—
332C22×C5—61no—
333C22×C5—61no—
334C22×C5C23×D311yescharacteristic
335C2×D5—21no—
336C2×D5—21no—
337C2×D5—21no—
338C2×D5—21no—
339C2×D5—32no—
340C2×D5—44no—
341C2×D5—61no—
342C2×D5—61no—
343C2×D5—124no—
344C2×D5#23512yescomplemented
345C2×D5D3×D412yes—
346C4×C5—21no—
347C4×C5—21no—
348C4×C5—32no—
349C4×C5—61no—
350Dic5—21no—
351Dic5—21no—
352Dic5—61no—
353Dic5—61no—
354Dic5—61no—
355C24—51no—
356C24—151no—
357#41—304no—
358C2×Q8—151no—
359C2×D4—31no—
360C2×D4—31no—
361C2×D4—31no—
362C2×D4—151no—
363C2×D4—151no—
364C2×D4—151no—
365C2×D4—151no—
366C2×D4—151no—
367C2×D4—151no—
368C2×D4—304no—
369C22×C4—31no—
370C22×C4—51no—
371C22×C4—51no—
372C22×C4—151no—
373C22×C4—151no—
374C22×C4—151no—
375C4⋊C4—31no—
376C4⋊C4—151no—
377C4⋊C4—151no—
378C4⋊C4—151no—
379C22⋊C4—31no—
380C22⋊C4—51no—
381C22⋊C4—51no—
382C22⋊C4—51no—
383C22⋊C4—151no—
384C22⋊C4—151no—
385C22⋊C4—151no—
386C22⋊C4—151no—
387C22⋊C4—151no—
388C22⋊C4—151no—
389C22⋊C4—151no—
390C22⋊C4D3×D511yescharacteristic, complemented
391C42—151no—
392C3×C5#54611yescharacteristic, complemented, Hall
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—52no—
397C22×C3—52no—
398C22×C3—101no—
399C22×C3—101no—
400C22×C3—101no—
401C22×C3—101no—
402C22×C3—204no—
403C22×C3C23×D511yescharacteristic
404C2×D3—21no—
405C2×D3—21no—
406C2×D3—21no—
407C2×D3—21no—
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—52no—
411C2×D3—101no—
412C2×D3—101no—
413C2×D3—204no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—101no—
417C3×C4—101no—
418Dic3—21no—
419Dic3—101no—
420Dic3—101no—
421Dic3—101no—
422Dic3D4×D512yes—
423C2×C5—41no—
424C2×C5—121no—
425C2×C5—121no—
426C2×C5C2×#23511yescharacteristic
427C2×C5C2×D3×D411yescharacteristic
428C2×C5#100411yescharacteristic
429D5—22no—
430D5—22no—
431D5—62no—
432C23—31no—
433C23—31no—
434C23—51no—
435C23—51no—
436C23—102no—
437C23—102no—
438C23—151no—
439C23—201no—
440C23—302no—
441C23—302no—
442C23—601no—
443C23C2×D3×D511yescharacteristic
444Q8—302no—
445D4—62no—
446D4—121no—
447D4—121no—
448D4—154no—
449D4—302no—
450D4—302no—
451D4—302no—
452D4—302no—
453D4—302no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—51no—
457C2×C4—51no—
458C2×C4—62no—
459C2×C4—102no—
460C2×C4—151no—
461C2×C4—151no—
462C2×C4—151no—
463C2×C4—201no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—601no—
467C2×C4C2×D3×D511yescharacteristic
468C2×C4C2×D3×D511yescharacteristic
469C2×C3—41no—
470C2×C3—102no—
471C2×C3—102no—
472C2×C3C2×D4×D511yescharacteristic
473C2×C3#414411yescharacteristic
474C2×C3D5×#4111yescharacteristic
475D3—41no—
476D3—41no—
477D3—102no—
478C5#549011yescharacteristic, complemented, Sylow
479C22—21no—
480C22—21no—
481C22—21no—
482C22—52no—
483C22—52no—
484C22—61no—
485C22—61no—
486C22—61no—
487C22—61no—
488C22—61no—
489C22—61no—
490C22—101no—
491C22—101no—
492C22—101no—
493C22—101no—
494C22—152no—
495C22—204no—
496C22—301no—
497C22—301no—
498C22—604no—
499C22C22×D3×D511yescharacteristic
500C4—21no—
501C4—21no—
502C4—32no—
503C4—61no—
504C4—101no—
505C4—101no—
506C4—301no—
507C4—301no—
508C4—301no—
509C3#6594611yescharacteristic, complemented, Sylow
510C2—41no—
511C2—102no—
512C2—102no—
513C2—121no—
514C2—121no—
515C2—302no—
516C2D5×#23511yescharacteristic
517C2D3×D4×D511yescharacteristic
518C2#9202511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#546
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×D5×#67C2semidirect
2. C2×#6771C2semidirect
3. #91772C2semidirect
4. C5×#884C2semidirect
5. C3×D5×C22⋊C4C2semidirect
6. #91563C2semidirect
7. #91557C2semidirect
8. #91525C2semidirect
9. #91475C2semidirect
10. #91473C2semidirect
11. #91449C2semidirect
12. #91447C2semidirect
13. #91444C2semidirect
14. #91441C2semidirect
15. #91422C2semidirect
20. C2×C4×D15C22semidirect
22. C2×C5×#67C22semidirect
23. C2×C5×D12C22semidirect
24. C2×C4×C5×D3C22semidirect
25. C2×C3×C4×D5C22semidirect
26. C2×C3×C4×D5C22semidirect
28. C2×#1204C22semidirect
29. C2×#1203C22semidirect
33. C2×#1201C22semidirect
34. C2×#1200C22semidirect
37. C3×C5×C22⋊C4C22semidirect
38. #6719C22semidirect
39. #6714C22semidirect
40. C4⋊Dic15C22semidirect
41. #6712C22semidirect
42. C5×(C2×D3)⋊C4C22semidirect
43. C5×Dic3⋊C4C22semidirect
44. C3×C22⋊Dic5C22semidirect
47. Dic3⋊Dic5C22semidirect
48. (C2×D3)⋊Dic5C22semidirect
49. (C2×D3)⋊Dic5C22semidirect
50. (C2×D3)⋊Dic5C22semidirect
52. #6665C22semidirect
53. Dic3×Dic5C22semidirect
69. D5×C22⋊C4D3semidirect
101. #1202D4semidirect
116. #884D5semidirect
139. C2×C4×D5C2×D3semidirect
140. C2×C4×D5C2×D3semidirect
144. C5×C22⋊C4C2×D3semidirect
149. C22⋊Dic5C2×D3semidirect
179. C2×C3×D5#41semidirect
201. C2×#67C2×D5semidirect
205. C2×D12C2×D5semidirect
208. C2×C4×D3C2×D5semidirect
213. C3×C22⋊C4C2×D5semidirect
221. (C2×D3)⋊C4C2×D5semidirect
225. Dic3⋊C4C2×D5semidirect
344. C2×D5#235semidirect
390. C22⋊C4D3×D5semidirect
392. C3×C5#546semidirect
478. C5#5490semidirect
509. C3#65946semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#672
C2×#67712
#917722
C5×#8842
C3×D5×C22⋊C42
#915632
#915572
#915252
#914752
#914732
#914492
#914472
#914442
#914412
#914222
#659463
#54905