Finite Group Databasebeta

← #12963834

Subgroups — #12963834

Order 1280.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3455C211yesmaximal, characteristic
2#10615060C211yesmaximal, characteristic, complemented
3#10615059C211yesmaximal, characteristic, complemented
4#10615007C211yesmaximal, characteristic, complemented
5#10615006C211yesmaximal, characteristic, complemented
6#10614858C218yesmaximal, complemented
7#10614857C218yesmaximal, complemented
8#10614648C212yesmaximal, complemented
9#10614647C212yesmaximal, complemented
10#10614245C212yesmaximal, complemented
11C4×#4094C212yesmaximal, complemented
12#10613940C211yesmaximal, characteristic, complemented
13C2×#65286C211yesmaximal, characteristic, complemented
14C5×#542C414yes—
15C5×#541C412yes—
16C5×#540C412yes—
17C5×#539C414yes—
18C5×#538C414yes—
19C5×#533C2211yescharacteristic
20C5×#532C2211yescharacteristic
21C5×#529C2218yes—
22C5×#525C2212yes—
23C4×C5×#41C2212yes—
24C2×C5×#117C2211yescharacteristic
25C2×#4094C2212yes—
26C2×#4094C2212yes—
27C22×#681C2211yescharacteristic
28C23×#168C2211yescharacteristic
29#65523C2212yes—
30#65523C2212yes—
31#65496C2212yes—
32#65313C2214yescomplemented
33#65313C2218yescomplemented
34Q8×#168C2214yescomplemented
35#65301C2218yes—
36#65301C2214yescomplemented
37#65300C2218yes—
38#65300C2218yes—
39#65300C2218yescomplemented
40D4×#168C2218yes—
41D4×#168C2214yescomplemented
42#65289C2218yes—
43#65289C2218yescomplemented
44#65288C2212yes—
45#65288C2212yes—
46#65286C2212yes—
47#65286C2212yes—
48#65286C2214yes—
49#65286C2214yes—
50#65286C2214yescomplemented
51#65286C2214yescomplemented
52#65215C2218yescomplemented
53#65213C2211yescharacteristic
54#65213C2211yescharacteristic
55C4×#681C2212yes—
56C4×#681C2214yescomplemented
57C2×#3935C2211yescharacteristic
58C2×#3935C2211yescharacteristic
59C2×C4×#168C2211yescharacteristic
60C2×C4×#168C2211yescharacteristic
61C2×C4×#168C2214yescomplemented
62#60206—51nomaximal, Sylow
63C2×C5×#41C2312yes—
64C23×C4×C5C2311yescharacteristic
65C5×C4⋊Q8C2×C411yescharacteristic
66C5×C4⋊Q8C2×C412yes—
67C5×#127C2×C411yescharacteristic
68C5×#126C2×C418yes—
69C5×#126C2×C418yes—
70C5×#125C2×C418yes—
71C5×#125C2×C412yes—
72C5×#124C2×C418yes—
73C5×#124C2×C412yes—
74C5×#123C2×C414yes—
75C5×#123C2×C418yes—
76C5×#123C2×C418yes—
77C5×C22⋊Q8C2×C414yes—
78C5×C22⋊Q8C2×C418yes—
79C5×C22⋊Q8C2×C414yes—
80C5×#121C2×C414yes—
81C5×#121C2×C418yes—
82C5×#121C2×C414yes—
83C5×#120C2×C414yes—
84C4×C5×Q8C2×C414yes—
85C4×C5×Q8C2314yes—
86C4×C5×D4C2×C418yes—
87C4×C5×D4C2×C414yes—
88C4×C5×D4C2318yes—
89C4×C5×D4C2314yes—
90C5×#117C2×C412yes—
91C5×#117C2×C412yes—
92C5×#117C2312yes—
93C5×#117C2314yes—
94C5×#117C2314yes—
95C2×C5×C4⋊C4C2×C412yes—
96C2×C5×C22⋊C4C2×C412yes—
97C2×C42×C5C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C2×C42×C5C2314yes—
100#4094—28no—
101#4094—28no—
102C2×#681—24no—
103C2×#681C2314yes—
104C2×#681C2312yes—
105C2×#681C2312yes—
106C2×#681C2314yes—
107C2×#681C2314yes—
108C22×#168—24no—
109C22×#168C2312yes—
110C22×#168C2312yes—
111C22×#168C2314yes—
112C22×#168C2314yes—
113C22×#168C2311yescharacteristic
114C22×#168C2311yescharacteristic
115C22×#168C2311yescharacteristic
116C22×#168C2311yescharacteristic
117C22⋊#168C2318yes—
118C22⋊#168C2318yes—
119C22⋊#168C2318yes—
120C4⋊#168C2318yes—
121C4⋊#168C2318yes—
122C4⋊#168C2314yes—
123C4⋊#168C2314yes—
124#3935C2314yes—
125#3935C2314yes—
126#3935C2314yes—
127#3935C2314yes—
128#3935C2314yes—
129C4×#168C2312yes—
130C4×#168C2312yes—
131C4×#168C2312yes—
132C4×#168C2312yes—
133C4×#168C2314yes—
134C4×#168C2314yes—
135C4×#168C2314yes—
136#3455D511yescharacteristic
137#2967—58no—
138#2966—58no—
139#2961—51no—
140#2960—51no—
141#2932—51no—
142#2931—51no—
143#2930—52no—
144#2929—52no—
145#2923—52no—
146#2921—51no—
147C4×#131—52no—
148C2×#405—51no—
149C24×C5C22×C411yescharacteristic
150C5×#41—24no—
151C5×#41—24no—
152C2×C5×Q8C22×C412yes—
153C2×C5×Q8C22×C414yes—
154C2×C5×D4C22×C414yes—
155C2×C5×D4C22×C412yes—
156C2×C5×D4C22×C418yes—
157C2×C5×D4C22×C414yes—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5C22×C411yescharacteristic
161C22×C4×C5C22×C412yes—
162C22×C4×C5C22×C412yes—
163C22×C4×C5C22×C412yes—
164C22×C4×C5C22×C418yes—
165C22×C4×C5C2414yes—
166C22×C4×C5C2412yes—
167C22×C4×C5C2411yescharacteristic
168C5×C4⋊C4C22×C414yes—
169C5×C4⋊C4C22×C414yes—
170C5×C4⋊C4C22×C414yes—
171C5×C4⋊C4C22×C418yes—
172C5×C4⋊C4C22×C414yes—
173C5×C4⋊C4C22×C418yes—
174C5×C4⋊C4C22×C418yes—
175C5×C4⋊C4C22×C414yes—
176C5×C22⋊C4C22×C414yes—
177C5×C22⋊C4C22×C414yes—
178C5×C22⋊C4C22×C414yes—
179C5×C22⋊C4C22×C418yes—
180C5×C22⋊C4C22×C418yes—
181C5×C22⋊C4C22×C418yes—
182C5×C22⋊C4C22×C418yes—
183C42×C5C22×C412yes—
184C42×C5C22×C411yescharacteristic
185C42×C5C22×C411yescharacteristic
186C42×C5C22×C418yes—
187C42×C5C2412yes—
188C42×C5C2412yes—
189C42×C5C2414yes—
190#681—24no—
191#681—24no—
192#681—28no—
193#681—28no—
194#681—28no—
195C2×#168—22no—
196C2×#168—22no—
197C2×#168—24no—
198C2×#168—24no—
199C2×#168—28no—
200C2×#168—28no—
201C2×#168#4114yes—
202C2×#168#4114yes—
203C2×#168C2412yes—
204C2×#168C2414yes—
205C2×#168C2414yes—
206C2×#168C2412yes—
207C2×#168C2411yescharacteristic
208C2×#168C2412yes—
209C2×#168C2411yescharacteristic
210C2×#131—52no—
211C2×#131—52no—
212C22×M16—51no—
213C23×C8—51no—
214#542Dic514yes—
215#541Dic512yes—
216#540Dic512yes—
217#539Dic514yes—
218#538Dic514yes—
219#533C2×D511yescharacteristic
220#532C2×D511yescharacteristic
221#529C2×D518yes—
222#525C2×D512yes—
223C4×#41C2×D512yes—
224C2×#117C2×D511yescharacteristic
225#446—54no—
226#446—58no—
227C8×Q8—54no—
228#436—58no—
229#436—54no—
230#435—58no—
231#435—58no—
232#435—58no—
233C8×D4—58no—
234C8×D4—54no—
235#433—58no—
236#433—58no—
237#432—58no—
238#424—52no—
239#424—52no—
240#423—51no—
241#423—51no—
242#408—52no—
243#408—52no—
244#407—52no—
245#405—54no—
246#405—54no—
247#405—52no—
248#405—52no—
249#405—54no—
250#405—54no—
251C4×M16—52no—
252C4×M16—54no—
253C2×#97—51no—
254C2×#97—51no—
255C2×C4×C8—51no—
256C2×C4×C8—51no—
257C2×C4×C8—54no—
258C23×C5—22no—
259C23×C5—42no—
260C23×C5C23×C412yes—
261C23×C5C23×C414yes—
262C23×C5C23×C411yescharacteristic
263C5×Q8—24no—
264C5×Q8—44no—
265C5×D4—28no—
266C5×D4—24no—
267C5×D4—44no—
268C5×D4—48no—
269C2×C4×C5—22no—
270C2×C4×C5—24no—
271C2×C4×C5—22no—
272C2×C4×C5—24no—
273C2×C4×C5—24no—
274C2×C4×C5—22no—
275C2×C4×C5—42no—
276C2×C4×C5—48no—
277C2×C4×C5—42no—
278C2×C4×C5C23×C414yes—
279C2×C4×C5C23×C414yes—
280C2×C4×C5C23×C412yes—
281C2×C4×C5C23×C418yes—
282C2×C4×C5C23×C412yes—
283C2×C4×C5C23×C412yes—
284C2×C4×C5C23×C411yescharacteristic
285C2×C4×C5C2×#4112yes—
286C2×C4×C5C2×#4112yes—
287C2×C4×C5C2511yescharacteristic
288#168—22no—
289#168—22no—
290#168—24no—
291#168—24no—
292#168C2×#4112yes—
293#168C2×#4112yes—
294#168C2×#4114yes—
295C2×#41C22×D512yes—
296C23×C4C22×D511yescharacteristic
297#131—108no—
298#131—108no—
299C2×M16—54no—
300C2×M16—52no—
301C2×M16—52no—
302C2×M16—54no—
303C2×M16—54no—
304C2×M16—104no—
305C22×C8—51no—
306C22×C8—51no—
307C22×C8—51no—
308C22×C8—52no—
309C22×C8—52no—
310C22×C8—51no—
311C22×C8—54no—
312C22×C8—54no—
313C22×C8—104no—
314C4⋊Q8C2×Dic511yescharacteristic
315C4⋊Q8C2×Dic512yes—
316#127C2×Dic511yescharacteristic
317#126C2×Dic518yes—
318#126C2×Dic518yes—
319#125C2×Dic518yes—
320#125C2×Dic512yes—
321#124C2×Dic518yes—
322#124C2×Dic512yes—
323#123C2×Dic514yes—
324#123C2×Dic518yes—
325#123C2×Dic518yes—
326C22⋊Q8C2×Dic514yes—
327C22⋊Q8C2×Dic518yes—
328C22⋊Q8C2×Dic514yes—
329#121C2×Dic514yes—
330#121C2×Dic518yes—
331#121C2×Dic514yes—
332#120C2×Dic514yes—
333C4×Q8C2×Dic514yes—
334C4×Q8C22×D514yes—
335C4×D4C2×Dic518yes—
336C4×D4C2×Dic514yes—
337C4×D4C22×D518yes—
338C4×D4C22×D514yes—
339#117C2×Dic512yes—
340#117C2×Dic512yes—
341#117C22×D514yes—
342#117C22×D512yes—
343#117C22×D514yes—
344C2×C4⋊C4C2×Dic512yes—
345C2×C22⋊C4C2×Dic512yes—
346C2×C42C22×D511yescharacteristic
347C2×C42C22×D511yescharacteristic
348C2×C42C22×D514yes—
349C4⋊C8—54no—
350C4⋊C8—54no—
351C4⋊C8—58no—
352C4⋊C8—58no—
353C22⋊C8—58no—
354C22⋊C8—58no—
355C22⋊C8—58no—
356#97—54no—
357#97—54no—
358#97—54no—
359#97—54no—
360#97—54no—
361C4×C8—52no—
362C4×C8—52no—
363C4×C8—54no—
364C4×C8—52no—
365C4×C8—54no—
366C4×C8—52no—
367C4×C8—54no—
368C22×C5—22no—
369C22×C5—22no—
370C22×C5—24no—
371C22×C5—28no—
372C22×C5—24no—
373C22×C5—44no—
374C22×C5C4×#4112yes—
375C22×C5C24×C411yescharacteristic
376C4×C5—22no—
377C4×C5—24no—
378C4×C5—24no—
379C4×C5—22no—
380C4×C5—28no—
381C4×C5—22no—
382C4×C5C4×#4112yes—
383C4×C5C22×#4111yescharacteristic
384C4×C5C22×#4111yescharacteristic
385C24C22×Dic511yescharacteristic
386#41—24no—
387#41—24no—
388C2×Q8C22×Dic512yes—
389C2×Q8C22×Dic514yes—
390C2×D4C22×Dic514yes—
391C2×D4C22×Dic512yes—
392C2×D4C22×Dic518yes—
393C2×D4C22×Dic514yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4C22×Dic511yescharacteristic
397C22×C4C22×Dic512yes—
398C22×C4C22×Dic512yes—
399C22×C4C22×Dic512yes—
400C22×C4C22×Dic518yes—
401C22×C4C23×D514yes—
402C22×C4C23×D512yes—
403C22×C4C23×D511yescharacteristic
404M16—108no—
405M16—108no—
406M16—108no—
407M16—104no—
408M16—104no—
409C2×C8—54no—
410C2×C8—54no—
411C2×C8—51no—
412C2×C8—52no—
413C2×C8—51no—
414C2×C8—54no—
415C2×C8—54no—
416C2×C8—52no—
417C2×C8—52no—
418C2×C8—104no—
419C2×C8—104no—
420C2×C8—108no—
421C2×C8—108no—
422C2×C8—102no—
423C2×C8—102no—
424C4⋊C4C22×Dic514yes—
425C4⋊C4C22×Dic514yes—
426C4⋊C4C22×Dic514yes—
427C4⋊C4C22×Dic518yes—
428C4⋊C4C22×Dic514yes—
429C4⋊C4C22×Dic518yes—
430C4⋊C4C22×Dic518yes—
431C4⋊C4C22×Dic514yes—
432C22⋊C4C22×Dic514yes—
433C22⋊C4C22×Dic514yes—
434C22⋊C4C22×Dic514yes—
435C22⋊C4C22×Dic518yes—
436C22⋊C4C22×Dic518yes—
437C22⋊C4C22×Dic518yes—
438C22⋊C4C22×Dic518yes—
439C42C22×Dic512yes—
440C42C22×Dic511yescharacteristic
441C42C22×Dic511yescharacteristic
442C42C22×Dic518yes—
443C42C23×D512yes—
444C42C23×D512yes—
445C42C23×D514yes—
446C2×C5—22no—
447C2×C5—44no—
448C2×C5—42no—
449C2×C5C2×C4×#4111yescharacteristic
450C2×C5#357312yes—
451C23—22no—
452C23—42no—
453C23C23×Dic512yes—
454C23C23×Dic514yes—
455C23C23×Dic511yescharacteristic
456Q8—24no—
457Q8—44no—
458D4—28no—
459D4—24no—
460D4—44no—
461D4—48no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4D5×#4112yes—
472C2×C4D5×#4112yes—
473C2×C4C23×Dic514yes—
474C2×C4C23×Dic514yes—
475C2×C4C23×Dic512yes—
476C2×C4C23×Dic518yes—
477C2×C4C23×Dic512yes—
478C2×C4C23×Dic512yes—
479C2×C4C23×Dic511yescharacteristic
480C2×C4C24×D511yescharacteristic
481C8—52no—
482C8—52no—
483C8—54no—
484C8—102no—
485C8—104no—
486C8—104no—
487C8—102no—
488C5#6020611yescharacteristic, complemented, Sylow
489C22—22no—
490C22—22no—
491C22—24no—
492C22—28no—
493C22—24no—
494C22—44no—
495C22#41×Dic512yes—
496C22C24×Dic511yescharacteristic
497C4—22no—
498C4—24no—
499C4—24no—
500C4—22no—
501C4—28no—
502C4—22no—
503C4#41×Dic512yes—
504C4C2×D5×#4111yescharacteristic
505C4C2×D5×#4111yescharacteristic
506C2—22no—
507C2—42no—
508C2—44no—
509C2C2×#41×Dic511yescharacteristic
510C2#1061919712yes—
subgroupG/H
center, FrattiniC2×C4C24×D5
commutator, socleC22×C5C24×C4
Fitting, prime radicalC5×#3455C2
cosocleC2×C4×C5C25
hypercenter#3455D5
hypocenterC5#60206
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ #3455

Complements

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

NH ≅ G/Ntype
2. #10615060C2semidirect
3. #10615059C2semidirect
4. #10615007C2semidirect
5. #10615006C2semidirect
6. #10614858C2semidirect
7. #10614857C2semidirect
8. #10614648C2semidirect
9. #10614647C2semidirect
10. #10614245C2semidirect
11. C4×#4094C2semidirect
12. #10613940C2semidirect
13. C2×#65286C2semidirect
32. #65313C22semidirect
33. #65313C22semidirect
34. Q8×#168C22semidirect
36. #65301C22semidirect
39. #65300C22semidirect
41. D4×#168C22semidirect
43. #65289C22semidirect
50. #65286C22semidirect
51. #65286C22semidirect
52. #65215C22semidirect
56. C4×#681C22semidirect
61. C2×C4×#168C22semidirect
488. C5#60206semidirect

Maximal subgroups

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

subgroupindex
C5×#34552
#106150602
#106150592
#106150072
#106150062
#106148582
#106148572
#106146482
#106146472
#106142452
C4×#40942
#106139402
C2×#652862
#602065