Finite Group Databasebeta

← #14324363

Subgroups — #14324363

Order 1920.

1032 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×AGL(1,𝔽5)C211yesmaximal, characteristic
2C2×AGL(1,𝔽5)×Dic6C211yesmaximal, characteristic
3D5×#944C211yesmaximal, characteristic
4#11752449C211yesmaximal, characteristic
5#11751835C214yesmaximal
6#11751752C211yesmaximal, characteristic
7#11751751C211yesmaximal, characteristic
8#11751748C211yesmaximal, characteristic
9#11751745C214yesmaximal
10#10617386—31nomaximal, Hall
11C2×C3×Q8×D5C2211yescharacteristic
12C2×D5×Dic6C2211yescharacteristic
13C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
14C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
15C3×Q8×AGL(1,𝔽5)—24no—
16C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
17C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
18C2×C3×C4×AGL(1,𝔽5)C2212yes—
19C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
20C2×Dic3⋊AGL(1,𝔽5)C2212yes—
21C2×Dic3×AGL(1,𝔽5)C2212yes—
22AGL(1,𝔽5)×Dic6—24no—
23#91833C411yescharacteristic, complemented
24C5×#944C411yescharacteristic, complemented
25D5×Dic3⋊C4C2214yes—
26C4×D5×Dic3C2211yescharacteristic
27#91389C411yescharacteristic, complemented
28#91385C411yescharacteristic, complemented
29#91327C414yescomplemented
30#91235C2214yes—
31(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
32(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
33(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
34(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
35#82456—51nomaximal, Hall
36C2×Q8×AGL(1,𝔽5)—31no—
37C2×Q8×AGL(1,𝔽5)D311yescharacteristic
38D5×C4⋊Q8—31no—
39(C2×AGL(1,𝔽5))⋊Q8—31no—
40(C2×AGL(1,𝔽5))⋊Q8—31no—
41#65711—34no—
42#65711—34no—
43#65689—31no—
44#65686—31no—
45C22×ΓL(1,𝔽16)C2312yes—
46C22×ΓL(1,𝔽16)C2312yes—
47C22×C3×AGL(1,𝔽5)C2312yes—
48C22×C3×AGL(1,𝔽5)C2311yescharacteristic
49C22×C3×AGL(1,𝔽5)C2311yescharacteristic
50C2×C3×C5×Q8C2×C411yescharacteristic
51C2×Dic30C2×C411yescharacteristic
52C2×C5×Dic6C2×C411yescharacteristic
53C3×Q8×D5—22no—
54C3×Q8×D5—22no—
55C2×C3×C4×D5C2312yes—
56C2×C3×C4×D5C2311yescharacteristic
57C2×C3×Dic10C2×C412yes—
58C2×C3×Dic10C2×C411yescharacteristic
59C2×#1204C2×C412yes—
60C2×D5×Dic3C2312yes—
61C2×D5×Dic3C2312yes—
62D5×Dic6—22no—
63D5×Dic6—22no—
64C4⋊ΓL(1,𝔽16)—22no—
65C4×ΓL(1,𝔽16)—22no—
66C3×C4⋊AGL(1,𝔽5)—24no—
67C3×C4⋊AGL(1,𝔽5)—22no—
68C3×C4×AGL(1,𝔽5)—24no—
69C3×C4×AGL(1,𝔽5)D414yes—
70Dic3⋊AGL(1,𝔽5)—24no—
71Dic3×AGL(1,𝔽5)—24no—
72#6712C2×C414yes—
73C4×Dic15C2×C411yescharacteristic
74C5×Dic3⋊C4C2×C414yes—
75C4×C5×Dic3C2×C411yescharacteristic
76Dic3⋊Dic5C2×C414yes—
77#6669C2×C414yes—
78Dic3×Dic5C2×C412yes—
79C2×C3×C4×Q8—51no—
80C2×#944—51no—
81C2×C4×Dic6—51no—
82#5229—51no—
83#4978—54no—
84#4977—51no—
85#4973—51no—
86#4933—54no—
87#4932—51no—
88C2×Q8×D5—31no—
89C2×Q8×D5C2×D311yescharacteristic
90Q8×AGL(1,𝔽5)—24no—
91Q8×AGL(1,𝔽5)—64no—
92C2×C4⋊AGL(1,𝔽5)—32no—
93C2×C4⋊AGL(1,𝔽5)—31no—
94C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
95C2×C4⋊AGL(1,𝔽5)C2×D312yes—
96C2×C4×AGL(1,𝔽5)—32no—
97C2×C4×AGL(1,𝔽5)—31no—
98C2×C4×AGL(1,𝔽5)C2×D312yes—
99C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
100C5×C4⋊Q8—31no—
101#4090—31no—
102#4090—31no—
103D5×C4⋊C4—34no—
104#4034—34no—
105C42×D5—31no—
106#4015—31no—
107(C2×AGL(1,𝔽5))⋊C4—32no—
108(C2×AGL(1,𝔽5))⋊C4—32no—
109(C2×AGL(1,𝔽5))⋊C4—34no—
110(C2×AGL(1,𝔽5))⋊C4—34no—
111(C2×AGL(1,𝔽5))⋊C4—34no—
112#2327—151noSylow
113C22×C3×D5C2411yescharacteristic
114C2×ΓL(1,𝔽16)—22no—
115C2×ΓL(1,𝔽16)—22no—
116C2×ΓL(1,𝔽16)—24no—
117C2×ΓL(1,𝔽16)—22no—
118C2×ΓL(1,𝔽16)—22no—
119C2×C3×AGL(1,𝔽5)—22no—
120C2×C3×AGL(1,𝔽5)—22no—
121C2×C3×AGL(1,𝔽5)—22no—
122C2×C3×AGL(1,𝔽5)—24no—
123C2×C3×AGL(1,𝔽5)C2×D412yes—
124C2×C3×AGL(1,𝔽5)C2×D412yes—
125C3×C5×Q8—22no—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5C22×C411yescharacteristic
128C2×Dic15C22×C412yes—
129C2×Dic15C22×C412yes—
130Dic30—22no—
131C2×C5×Dic3C22×C412yes—
132C2×C5×Dic3C22×C412yes—
133C5×Dic6—22no—
134C2×C3×Dic5C22×C412yes—
135C2×C3×Dic5C22×C411yescharacteristic
136C3×C4×D5—22no—
137C3×C4×D5—22no—
138C3×C4×D5C2×D412yes—
139C3×C4×D5#4112yes—
140C3×Dic10—22no—
141C3×Dic10—24no—
142#1204—24no—
143D5×Dic3—22no—
144D5×Dic3—22no—
145D5×Dic3—22no—
146D5×Dic3—22no—
147C22×C3×Q8—51no—
148C22×Dic6—51no—
149C3×C4×Q8—104no—
150C2×C3×C4⋊C4—51no—
151C2×C3×C4⋊C4—52no—
152C2×C3×C42—51no—
153C2×C3×C42—52no—
154#944—51no—
155#944—51no—
156#944—54no—
157#944—51no—
158#944AGL(1,𝔽5)11yescharacteristic, complemented
159C2×C4⋊Dic3—51no—
160C2×Dic3⋊C4—54no—
161C2×Dic3⋊C4—52no—
162C2×C4×Dic3—51no—
163C2×C4×Dic3—52no—
164C2×C4×Dic3—51no—
165C4×Dic6—104no—
166C3⋊#95—52no—
167C3⋊#95—52no—
168C3⋊#95—54no—
169C3⋊#95—54no—
170C3⋊#95—54no—
171C22×AGL(1,𝔽5)—32no—
172C22×AGL(1,𝔽5)—32no—
173C22×AGL(1,𝔽5)C22×D312yes—
174C22×AGL(1,𝔽5)C22×D311yescharacteristic
175C22×AGL(1,𝔽5)C22×D311yescharacteristic
176C2×C5×Q8—31no—
177C2×C5×Q8C4×D311yescharacteristic
178Q8×D5—22no—
179Q8×D5—22no—
180Q8×D5—62no—
181Q8×D5—62no—
182C2×C4×D5—32no—
183C2×C4×D5—32no—
184C2×C4×D5C22×D312yes—
185C2×C4×D5C22×D311yescharacteristic
186C2×Dic10—31no—
187C2×Dic10—32no—
188C2×Dic10C4×D311yescharacteristic
189C2×Dic10C4×D312yes—
190C4⋊AGL(1,𝔽5)—24no—
191C4⋊AGL(1,𝔽5)—22no—
192C4⋊AGL(1,𝔽5)—64no—
193C4⋊AGL(1,𝔽5)—62no—
194C4×AGL(1,𝔽5)—24no—
195C4×AGL(1,𝔽5)—64no—
196C4×AGL(1,𝔽5)—62no—
197C4×AGL(1,𝔽5)#6714yes—
198C5×C4⋊C4—34no—
199C42×C5—31no—
200C4⋊Dic5—34no—
201#683—34no—
202#683—34no—
203C4×Dic5—31no—
204C4×Dic5—32no—
205C2×C4⋊Q8—151no—
206C2×C4×Q8—51no—
207C2×C4×Q8—151no—
208#391—151no—
209#391—151no—
210#387—154no—
211#387—154no—
212#382—151no—
213#376—151no—
214C22×C3×C5C23×C411yescharacteristic
215C2×C3×D5C22×D411yescharacteristic
216C2×C3×D5C2×#4111yescharacteristic
217C2×C3×D52−1+412yes—
218C2×C3×D52−1+412yes—
219ΓL(1,𝔽16)—42no—
220ΓL(1,𝔽16)—42no—
221C3×AGL(1,𝔽5)—22no—
222C3×AGL(1,𝔽5)—22no—
223C3×AGL(1,𝔽5)—42no—
224C3×C4×C5—22no—
225C3×C4×C5C4×D412yes—
226Dic15—22no—
227Dic15—22no—
228C3×Dic5—22no—
229C3×Dic5C4×D412yes—
230C5×Dic3—22no—
231C5×Dic3—22no—
232C2×C3×Q8—52no—
233C2×C3×Q8—51no—
234C2×C3×Q8—102no—
235C2×C3×Q8—102no—
236C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
237C22×C3×C4—51no—
238C22×C3×C4—51no—
239C22×C3×C4—51no—
240C22×C3×C4—52no—
241C22×C3×C4—52no—
242C22×Dic3—52no—
243C22×Dic3—52no—
244C22×Dic3—52no—
245C22×Dic3—52no—
246C2×Dic6—51no—
247C2×Dic6—52no—
248C2×Dic6—102no—
249C2×Dic6—102no—
250C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
251C3×C4⋊C4—102no—
252C3×C4⋊C4—104no—
253C3×C42—54no—
254C3×C42—104no—
255C4⋊Dic3—102no—
256Dic3⋊C4—54no—
257Dic3⋊C4—54no—
258Dic3⋊C4—54no—
259Dic3⋊C4—104no—
260Dic3⋊C4C2×AGL(1,𝔽5)14yes—
261C4×Dic3—51no—
262C4×Dic3—52no—
263C4×Dic3—104no—
264C4×Dic3—102no—
265C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
266C22×D5C23×D311yescharacteristic
267C2×AGL(1,𝔽5)—22no—
268C2×AGL(1,𝔽5)—22no—
269C2×AGL(1,𝔽5)—24no—
270C2×AGL(1,𝔽5)—22no—
271C2×AGL(1,𝔽5)—62no—
272C2×AGL(1,𝔽5)—64no—
273C2×AGL(1,𝔽5)—62no—
274C2×AGL(1,𝔽5)—62no—
275C2×AGL(1,𝔽5)—62no—
276C2×AGL(1,𝔽5)C2×#6712yes—
277C2×AGL(1,𝔽5)C2×#6712yes—
278C5×Q8—22no—
279C5×Q8—62no—
280C2×C4×C5—32no—
281C2×C4×C5—32no—
282C2×C4×C5C2×C4×D312yes—
283C2×C4×C5C2×C4×D311yescharacteristic
284C2×Dic5—32no—
285C2×Dic5—32no—
286C2×Dic5C2×C4×D312yes—
287C2×Dic5C2×C4×D311yescharacteristic
288C4×D5—22no—
289C4×D5—22no—
290C4×D5—62no—
291C4×D5—62no—
292C4×D5—62no—
293C4×D5—62no—
294C4×D5#23512yes—
295C4×D5C2×#6712yes—
296Dic10—22no—
297Dic10—24no—
298Dic10—62no—
299Dic10—64no—
300C22×Q8—51no—
301C22×Q8—151no—
302C4⋊Q8—31no—
303C4⋊Q8—151no—
304C4⋊Q8—154no—
305C4⋊Q8—151no—
306C4⋊Q8—151no—
307C4×Q8—104no—
308C4×Q8—304no—
309C2×C4⋊C4—51no—
310C2×C4⋊C4—52no—
311C2×C4⋊C4—154no—
312C2×C4⋊C4—151no—
313C2×C4⋊C4—152no—
314C2×C42—51no—
315C2×C42—52no—
316C2×C42—151no—
317C2×C42—152no—
318C2×C42—151no—
319#95—152no—
320#95—152no—
321#95—154no—
322#95—154no—
323#95—154no—
324C2×C3×C5C2×C4×D411yescharacteristic
325C2×C3×C5#51912yes—
326C3×D5#53611yescharacteristic
327C3×D5#53611yescharacteristic
328C3×D5#54112yes—
329C23×C3—51no—
330C3×Q8—22no—
331C3×Q8—104no—
332C3×Q8—102no—
333C2×C3×C4—51no—
334C2×C3×C4—52no—
335C2×C3×C4—52no—
336C2×C3×C4—52no—
337C2×C3×C4—52no—
338C2×C3×C4—52no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4—102no—
342C2×C3×C4—102no—
343C2×C3×C4—102no—
344C2×C3×C4—104no—
345C2×C3×C4C22×AGL(1,𝔽5)12yes—
346C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
347C2×Dic3—52no—
348C2×Dic3—52no—
349C2×Dic3—102no—
350C2×Dic3—102no—
351C2×Dic3—102no—
352C2×Dic3—102no—
353C2×Dic3—102no—
354C2×Dic3—102no—
355C2×Dic3—102no—
356C2×Dic3—104no—
357C2×Dic3—102no—
358C2×Dic3C22×AGL(1,𝔽5)12yes—
359C2×Dic3C22×AGL(1,𝔽5)12yes—
360Dic6—22no—
361Dic6—104no—
362Dic6—102no—
363C22×C5C22×C4×D311yescharacteristic
364C2×D5C2×#23511yescharacteristic
365C2×D5#100712yes—
366C2×D5#101012yes—
367C2×D5C22×#6711yescharacteristic
368AGL(1,𝔽5)—22no—
369AGL(1,𝔽5)—22no—
370AGL(1,𝔽5)—42no—
371AGL(1,𝔽5)—122no—
372AGL(1,𝔽5)—122no—
373C4×C5—22no—
374C4×C5—62no—
375C4×C5—62no—
376C4×C5C4×#6712yes—
377Dic5—22no—
378Dic5—62no—
379Dic5—62no—
380Dic5C4×#6712yes—
381C2×Q8—31no—
382C2×Q8—52no—
383C2×Q8—51no—
384C2×Q8—102no—
385C2×Q8—102no—
386C2×Q8—152no—
387C2×Q8—151no—
388C2×Q8—302no—
389C2×Q8—302no—
390C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
391C22×C4—51no—
392C22×C4—51no—
393C22×C4—51no—
394C22×C4—52no—
395C22×C4—52no—
396C22×C4—152no—
397C22×C4—152no—
398C22×C4—152no—
399C22×C4—152no—
400C4⋊C4—34no—
401C4⋊C4—102no—
402C4⋊C4—104no—
403C4⋊C4—154no—
404C4⋊C4—154no—
405C4⋊C4—154no—
406C4⋊C4—304no—
407C4⋊C4—302no—
408C42—31no—
409C42—54no—
410C42—104no—
411C42—152no—
412C42—151no—
413C42—304no—
414C42—302no—
415C3×C5#232711yescharacteristic, complemented, Hall
416C22×C3—51no—
417C22×C3—51no—
418C22×C3—52no—
419C22×C3—52no—
420C22×C3C23×AGL(1,𝔽5)11yescharacteristic
421C3×C4—22no—
422C3×C4—52no—
423C3×C4—102no—
424C3×C4—102no—
425C3×C4—102no—
426C3×C4—202no—
427C3×C4D4×AGL(1,𝔽5)12yes—
428Dic3—22no—
429Dic3—22no—
430Dic3—102no—
431Dic3—102no—
432Dic3—202no—
433Dic3—202no—
434C2×C5#557212yes—
435C2×C5C2×C4×#6711yescharacteristic
436D5#558312yes—
437D5#581611yescharacteristic
438D5#582511yescharacteristic
439C23—51no—
440Q8—22no—
441Q8—62no—
442Q8—104no—
443Q8—102no—
444Q8—304no—
445Q8—302no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—51no—
449C2×C4—52no—
450C2×C4—52no—
451C2×C4—52no—
452C2×C4—52no—
453C2×C4—52no—
454C2×C4—102no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—102no—
458C2×C4—102no—
459C2×C4—104no—
460C2×C4—152no—
461C2×C4—152no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—302no—
468C2×C4—302no—
469C2×C4—302no—
470C2×C4—304no—
471C2×C4C2×D3×AGL(1,𝔽5)12yes—
472C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
473C2×C3—51no—
474C2×C3—52no—
475C2×C3—51no—
476C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
477C2×C3#6626212yes—
478C5#8245611yescharacteristic, complemented, Sylow
479C22—52no—
480C22—52no—
481C22—51no—
482C22—51no—
483C22C22×D3×AGL(1,𝔽5)11yescharacteristic
484C4—22no—
485C4—52no—
486C4—62no—
487C4—62no—
488C4—102no—
489C4—102no—
490C4—102no—
491C4—202no—
492C4—302no—
493C4—302no—
494C4—602no—
495C4—602no—
496C4AGL(1,𝔽5)×#6712yes—
497C3#1061738611yescharacteristic, complemented, Sylow
498C2—51no—
499C2—51no—
500C2—52no—
501C2#1175684712yes—
502C2C2×AGL(1,𝔽5)×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×Q8C2×C4
prime radical, cosocleC22×C3×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#2327
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
23. #91833C4semidirect
24. C5×#944C4semidirect
27. #91389C4semidirect
28. #91385C4semidirect
29. #91327C4semidirect
158. #944AGL(1,𝔽5)semidirect
415. C3×C5#2327semidirect
478. C5#82456semidirect
497. C3#10617386semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×AGL(1,𝔽5)2
C2×AGL(1,𝔽5)×Dic62
D5×#9442
#117524492
#117518352
#117517522
#117517512
#117517482
#117517452
#106173863
#824565