Finite Group Databasebeta

← #14295287

Subgroups — #14295287

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×M16C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic
3D5⋊#830C211yesmaximal, characteristic
4C2×#91221C211yesmaximal, characteristic, complemented
5C2×C8×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
6#11746763C211yesmaximal, characteristic, complemented
7#11746763C211yesmaximal, characteristic, complemented
8#10604757—31nomaximal, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C2×#6763C412yes—
11C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
12C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
13C2×D5⋊(C3⋊C8)C2211yescharacteristic
14C2×D5⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C5×M16C411yescharacteristic, complemented
16C2×C3×#681C411yescharacteristic, complemented
17C3×D5×M16—22no—
18C3×D5×M16—22no—
19C2×C3×#676C411yescharacteristic, complemented
20C2×C3×#676C411yescharacteristic, complemented
21C2×C3×C8×D5C2211yescharacteristic
22C2×C3×C8×D5C2211yescharacteristic
23#91241C412yescomplemented
24#91235C412yes—
25#91232C412yescomplemented
26#91221—22no—
27C8×ΓL(1,𝔽16)—22no—
28#71983—51nomaximal, Hall
29C2×D5×M16D311yescharacteristic
30C4×C22⋊AGL(1,𝔽5)—31no—
31D5⋊(C22⋊C8)—31no—
32C2×#3992—31no—
33C2×C8×AGL(1,𝔽5)—31no—
34(C2×AGL(1,𝔽5))⋊C8—31no—
35(C2×AGL(1,𝔽5))⋊C8—31no—
36C23×C3×D5C2×C411yescharacteristic
37C22×ΓL(1,𝔽16)C2×C412yes—
38C22×ΓL(1,𝔽16)C2×C412yes—
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×Dic5C2×C411yescharacteristic
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5C2×C411yescharacteristic
46C2×C3×C4×D5C2×C411yescharacteristic
47C2×C3×C4×D5C2×C411yescharacteristic
48C2×C3×C4×D5C2311yescharacteristic
49#6763—24no—
50C2×C3⋊#170C2×C412yes—
51C2×C3⋊#170C2×C412yes—
52C4×ΓL(1,𝔽16)—21no—
53C4×ΓL(1,𝔽16)—21no—
54C4×ΓL(1,𝔽16)—21no—
55C4×ΓL(1,𝔽16)—21no—
56D5⋊(C3⋊C8)—21no—
57D5⋊(C3⋊C8)—21no—
58D5⋊(C3⋊C8)—21no—
59D5⋊(C3⋊C8)—21no—
60C3×C5×M16—22no—
61C2×C3×C5×C8C2×C411yescharacteristic
62C2×C3×C5×C8C2×C411yescharacteristic
63C3×#681—22no—
64C2×C3×#168C2×C411yescharacteristic
65C2×C3×#168C2×C411yescharacteristic
66C3×#676—22no—
67C3×#676—22no—
68C3×C8×D5—21no—
69C3×C8×D5—21no—
70C3×C8×D5D412yes—
71C3×C8×D5D412yes—
72C22×C3×M16—51no—
73C4×#217—51no—
74C2×#830—51no—
75C2×#815—51no—
76C2×C8×Dic3—51no—
77#4550—51no—
78#4550—51no—
79C22×C4×D5C2×D311yescharacteristic
80C2×C22⋊AGL(1,𝔽5)—32no—
81C2×C4×AGL(1,𝔽5)—31no—
82C2×C4×AGL(1,𝔽5)—31no—
83C2×D5⋊C8—31no—
84C2×D5⋊C8—31no—
85C2×C5×M16Dic311yescharacteristic, complemented
86C2×#681Dic311yescharacteristic, complemented
87D5×M16—22no—
88D5×M16—22no—
89C2×#676Dic311yescharacteristic, complemented
90C2×#676Dic311yescharacteristic, complemented
91C2×C8×D5C2×D311yescharacteristic
92C2×C8×D5C2×D311yescharacteristic
93C22⋊#170—32no—
94(C2×AGL(1,𝔽5))⋊C4—32no—
95#4003—32no—
96#3992—62no—
97C8×AGL(1,𝔽5)—62no—
98#1753—151noSylow
99C23×C3×C5C4211yescharacteristic
100C22×C3×D5—21no—
101C22×C3×D5—21no—
102C22×C3×D5—22no—
103C22×C3×D5—22no—
104C22×C3×D5C4211yescharacteristic
105C22×C3×D5C22×C411yescharacteristic
106C2×ΓL(1,𝔽16)—22no—
107C2×ΓL(1,𝔽16)—22no—
108C2×ΓL(1,𝔽16)—22no—
109C2×ΓL(1,𝔽16)—22no—
110C2×ΓL(1,𝔽16)—22no—
111C2×ΓL(1,𝔽16)M1614yescomplemented
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5C4211yescharacteristic
115C2×C3×C4×C5C22×C411yescharacteristic
116C2×C3×Dic5—21no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5C4211yescharacteristic
119C2×C3×Dic5C22×C411yescharacteristic
120C3×C4×D5—21no—
121C3×C4×D5—21no—
122C3×C4×D5—21no—
123C3×C4×D5—21no—
124C3×C4×D5—21no—
125C3×C4×D5—21no—
126C3×C4×D5C2×D411yescharacteristic
127C3×C4×D5C2×D411yescharacteristic
128C3×C4×D5#4111yescharacteristic
129C3×C4×D5#4111yescharacteristic
130C3⋊#170—22no—
131C3⋊#170—22no—
132C3×C5×C8—21no—
133C3×C5×C8C22⋊C412yescomplemented
134C3×#168—21no—
135C3×#168C22⋊C412yescomplemented
136C23×C3×C4—51no—
137C2×C3×M16—51no—
138C2×C3×M16—51no—
139C2×C3×M16—51no—
140C2×C3×M16—102no—
141C2×C3×M16—102no—
142C2×C3×M16AGL(1,𝔽5)11yescharacteristic, complemented
143C22×C3×C8—51no—
144C22×C3×C8—51no—
145C2×#217—52no—
146C2×C4×Dic3—51no—
147C2×C4×Dic3—51no—
148C22×C3⋊C8—51no—
149C22×C3⋊C8—51no—
150C3⋊#95—52no—
151#830—52no—
152#830—52no—
153#815—102no—
154C8×Dic3—102no—
155C23×D5C4×D311yescharacteristic
156C22×AGL(1,𝔽5)—32no—
157C22×AGL(1,𝔽5)—32no—
158C22×C4×C5C2×Dic311yescharacteristic
159C22×Dic5C4×D311yescharacteristic
160C2×C4×D5—21no—
161C2×C4×D5—21no—
162C2×C4×D5—21no—
163C2×C4×D5—21no—
164C2×C4×D5C4×D311yescharacteristic
165C2×C4×D5C4×D311yescharacteristic
166C2×C4×D5C2×Dic311yescharacteristic
167C2×C4×D5C22×D311yescharacteristic
168C22⋊AGL(1,𝔽5)—64no—
169C2×#170—32no—
170C2×#170—32no—
171C4×AGL(1,𝔽5)—61no—
172C4×AGL(1,𝔽5)—61no—
173C4×AGL(1,𝔽5)—61no—
174C4×AGL(1,𝔽5)—61no—
175D5⋊C8—61no—
176D5⋊C8—61no—
177D5⋊C8—61no—
178D5⋊C8—61no—
179C5×M16—22no—
180C2×C5×C8C2×Dic311yescharacteristic
181C2×C5×C8C2×Dic311yescharacteristic
182#681—22no—
183C2×#168C2×Dic311yescharacteristic
184C2×#168C2×Dic311yescharacteristic
185#676—22no—
186#676—22no—
187C8×D5—21no—
188C8×D5—21no—
189C8×D5#6712yes—
190C8×D5#6712yes—
191C22×M16—51no—
192C2×C22⋊C8—151no—
193C2×#97—151no—
194C2×C4×C8—151no—
195C4×C22⋊C4—151no—
196#336—151no—
197#336—151no—
198C22×C3×C5—21no—
199C22×C3×C5—21no—
200C22×C3×C5—21no—
201C22×C3×C5C2×C4211yescharacteristic
202C2×C3×D5—21no—
203C2×C3×D5—21no—
204C2×C3×D5—21no—
205C2×C3×D5—42no—
206C2×C3×D5—42no—
207C2×C3×D5C4×D411yescharacteristic
208C2×C3×D5C4×D411yescharacteristic
209C2×C3×D5C2×M1612yes—
210C2×C3×D5#13112yes—
211ΓL(1,𝔽16)—24no—
212ΓL(1,𝔽16)—42no—
213C3×C4×C5—21no—
214C3×C4×C5C2×C22⋊C411yescharacteristic
215C3×C4×C5#11711yescharacteristic
216C3×Dic5—21no—
217C3×Dic5C4×D411yescharacteristic
218C3×Dic5C4×D411yescharacteristic
219C24×C3—51no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4—51no—
223C22×C3×C4—51no—
224C22×C3×C4—51no—
225C22×C3×C4—101no—
226C22×C3×C4—101no—
227C22×C3×C4—101no—
228C22×C3×C4—101no—
229C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
230C22×Dic3—52no—
231C22×Dic3—52no—
232C3×M16—22no—
233C3×M16—102no—
234C3×M16—102no—
235C3×M16—102no—
236C2×C3×C8—51no—
237C2×C3×C8—52no—
238C2×C3×C8—52no—
239C2×C3×C8—51no—
240C2×C3×C8—101no—
241C2×C3×C8—101no—
242C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
243C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
244#217—104no—
245C4×Dic3—101no—
246C4×Dic3—101no—
247C4×Dic3—101no—
248C4×Dic3—101no—
249C2×C3⋊C8—52no—
250C2×C3⋊C8—52no—
251C2×C3⋊C8—101no—
252C2×C3⋊C8—101no—
253C2×C3⋊C8—101no—
254C2×C3⋊C8—101no—
255C23×C5C4×Dic311yescharacteristic
256C22×D5—22no—
257C22×D5—22no—
258C22×D5—21no—
259C22×D5—21no—
260C22×D5C4×Dic311yescharacteristic
261C22×D5C2×C4×D311yescharacteristic
262C2×AGL(1,𝔽5)—34no—
263C2×AGL(1,𝔽5)—62no—
264C2×AGL(1,𝔽5)—62no—
265C2×AGL(1,𝔽5)—62no—
266C2×AGL(1,𝔽5)—62no—
267C2×AGL(1,𝔽5)—62no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5C4×Dic311yescharacteristic
271C2×C4×C5C22×Dic311yescharacteristic
272C2×Dic5—21no—
273C2×Dic5—21no—
274C2×Dic5C4×Dic311yescharacteristic
275C2×Dic5C2×C4×D311yescharacteristic
276C4×D5—21no—
277C4×D5—21no—
278C4×D5—21no—
279C4×D5—21no—
280C4×D5—21no—
281C4×D5—21no—
282C4×D5#23511yescharacteristic
283C4×D5#23511yescharacteristic
284C4×D5C2×#6711yescharacteristic
285C4×D5C2×#6711yescharacteristic
286#170—62no—
287#170—62no—
288C5×C8—21no—
289C5×C8#21712yescomplemented
290#168—21no—
291#168#21712yescomplemented
292C23×C4—51no—
293C2×M16—51no—
294C2×M16—51no—
295C2×M16—51no—
296C2×M16—102no—
297C2×M16—102no—
298C2×M16ΓL(1,𝔽16)11yescharacteristic, complemented
299C22×C8—51no—
300C22×C8—51no—
301C22×C8—151no—
302C22×C8—151no—
303C2×C22⋊C4—152no—
304C2×C42—151no—
305C2×C42—151no—
306C22⋊C8—152no—
307C22⋊C8—152no—
308#97—302no—
309C4×C8—302no—
310#95—152no—
311C2×C3×C5—41no—
312C2×C3×C5C4×C22⋊C411yescharacteristic
313C2×C3×C5C4×M1611yescharacteristic
314C2×C3×C5#40511yescharacteristic
315C3×D5—41no—
316C3×D5#43512yes—
317C3×D5#43612yes—
318C23×C3—51no—
319C23×C3—51no—
320C23×C3—102no—
321C23×C3—101no—
322C23×C3—101no—
323C23×C3—102no—
324C23×C3C4×AGL(1,𝔽5)11yescharacteristic
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—51no—
328C2×C3×C4—51no—
329C2×C3×C4—51no—
330C2×C3×C4—51no—
331C2×C3×C4—51no—
332C2×C3×C4—51no—
333C2×C3×C4—101no—
334C2×C3×C4—101no—
335C2×C3×C4—101no—
336C2×C3×C4—101no—
337C2×C3×C4—101no—
338C2×C3×C4—101no—
339C2×C3×C4—101no—
340C2×C3×C4—101no—
341C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
342C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
343C2×Dic3—54no—
344C2×Dic3—102no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347C2×Dic3—102no—
348C2×Dic3—102no—
349C3×C8—21no—
350C3×C8—52no—
351C3×C8—101no—
352C3×C8C22⋊AGL(1,𝔽5)12yescomplemented
353C3⋊C8—102no—
354C3⋊C8—102no—
355C22×C5—21no—
356C22×C5—21no—
357C22×C5—21no—
358C22×C5C2×C4×Dic311yescharacteristic
359C2×D5—21no—
360C2×D5—21no—
361C2×D5—21no—
362C2×D5—42no—
363C2×D5—42no—
364C2×D5D3×M1612yes—
365C2×D5#90712yes—
366C2×D5C4×#6711yescharacteristic
367C2×D5C4×#6711yescharacteristic
368AGL(1,𝔽5)—64no—
369AGL(1,𝔽5)—122no—
370C4×C5—21no—
371C4×C5#92611yescharacteristic
372C4×C5C2×#21711yescharacteristic
373Dic5—21no—
374Dic5C4×#6711yescharacteristic
375Dic5C4×#6711yescharacteristic
376C24—51no—
377C22×C4—51no—
378C22×C4—51no—
379C22×C4—51no—
380C22×C4—51no—
381C22×C4—51no—
382C22×C4—101no—
383C22×C4—101no—
384C22×C4—101no—
385C22×C4—101no—
386C22×C4—152no—
387C22×C4—152no—
388C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
389M16—22no—
390M16—102no—
391M16—102no—
392M16—102no—
393C2×C8—51no—
394C2×C8—52no—
395C2×C8—52no—
396C2×C8—51no—
397C2×C8—101no—
398C2×C8—101no—
399C2×C8—152no—
400C2×C8—152no—
401C2×C8—301no—
402C2×C8—301no—
403C2×C8—301no—
404C2×C8—301no—
405C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
406C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
407C22⋊C4—304no—
408C42—301no—
409C42—301no—
410C42—301no—
411C42—301no—
412C3×C5#175311yescharacteristic, complemented, Hall
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—52no—
417C22×C3—52no—
418C22×C3—51no—
419C22×C3—51no—
420C22×C3—101no—
421C22×C3—101no—
422C22×C3—101no—
423C22×C3—202no—
424C22×C3—202no—
425C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
426C3×C4—21no—
427C3×C4—51no—
428C3×C4—51no—
429C3×C4—101no—
430C3×C4#413011yescharacteristic
431C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
432Dic3—104no—
433Dic3—202no—
434C2×C5—41no—
435C2×C5Dic3×M1611yescharacteristic
436C2×C5#512211yescharacteristic
437C2×C5C4×#21711yescharacteristic
438D5—41no—
439D5#512712yes—
440D5#512812yes—
441C23—51no—
442C23—51no—
443C23—102no—
444C23—101no—
445C23—101no—
446C23—102no—
447C23C4×ΓL(1,𝔽16)11yescharacteristic
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—51no—
451C2×C4—51no—
452C2×C4—51no—
453C2×C4—51no—
454C2×C4—51no—
455C2×C4—51no—
456C2×C4—101no—
457C2×C4—101no—
458C2×C4—101no—
459C2×C4—101no—
460C2×C4—101no—
461C2×C4—101no—
462C2×C4—101no—
463C2×C4—101no—
464C2×C4—154no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—302no—
468C2×C4—302no—
469C2×C4—302no—
470C2×C4C4×ΓL(1,𝔽16)11yescharacteristic
471C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
472C8—21no—
473C8—52no—
474C8—101no—
475C8—302no—
476C8—302no—
477C8#676312yescomplemented
478C2×C3—41no—
479C2×C3—52no—
480C2×C3—52no—
481C2×C3—201no—
482C2×C3M16×AGL(1,𝔽5)11yescharacteristic
483C2×C3#6572811yescharacteristic
484C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
485C5#7198311yescharacteristic, complemented, Sylow
486C22—21no—
487C22—21no—
488C22—21no—
489C22—52no—
490C22—52no—
491C22—51no—
492C22—51no—
493C22—101no—
494C22—101no—
495C22—101no—
496C22—202no—
497C22—202no—
498C22C2×C4×ΓL(1,𝔽16)11yescharacteristic
499C4—21no—
500C4—51no—
501C4—51no—
502C4—101no—
503C4—304no—
504C4—602no—
505C4#9198911yescharacteristic
506C4C2×#676311yescharacteristic
507C3#1060475711yescharacteristic, complemented, Sylow
508C2—41no—
509C2—52no—
510C2—52no—
511C2—201no—
512C2M16×ΓL(1,𝔽16)11yescharacteristic
513C2#1175238711yescharacteristic
514C2C4×#676311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C2×C42
FittingC2×C3×C5×M16C4
prime radicalC23×C3×D5C2×C4
cosocleC2×C3×C4×D5C23
hypercenterC2×M16ΓL(1,𝔽16)
hypocenterC3×C5#1753
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. C2×#91221C2semidirect
5. C2×C8×ΓL(1,𝔽16)C2semidirect
6. #11746763C2semidirect
7. #11746763C2semidirect
15. C2×C3×C5×M16C4semidirect
16. C2×C3×#681C4semidirect
19. C2×C3×#676C4semidirect
20. C2×C3×#676C4semidirect
23. #91241C4semidirect
25. #91232C4semidirect
85. C2×C5×M16Dic3semidirect
86. C2×#681Dic3semidirect
89. C2×#676Dic3semidirect
90. C2×#676Dic3semidirect
111. C2×ΓL(1,𝔽16)M16semidirect
133. C3×C5×C8C22⋊C4semidirect
135. C3×#168C22⋊C4semidirect
142. C2×C3×M16AGL(1,𝔽5)semidirect
289. C5×C8#217semidirect
291. #168#217semidirect
298. C2×M16ΓL(1,𝔽16)semidirect
352. C3×C8C22⋊AGL(1,𝔽5)semidirect
412. C3×C5#1753semidirect
477. C8#6763semidirect
485. C5#71983semidirect
507. C3#10604757semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D5×M162
C4×#67632
D5⋊#8302
C2×#912212
C2×C8×ΓL(1,𝔽16)2
#117467632
#106047573
#719835