Finite Group Databasebeta

← #14300539

Subgroups — #14300539

Order 1920.

752 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
1C22×#6758C211yesmaximal, characteristic, complemented
2C3×#65940C211yesmaximal, characteristic
3C2×#91237C212yesmaximal, complemented
4C4×#6763C211yesmaximal, characteristic, complemented
5#11746746C212yesmaximal, complemented
6#10604495—31nomaximal, Hall
7C2×C3×#711C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763—21no—
10C22×C3⋊#170—21no—
11C2×C4×ΓL(1,𝔽16)C2212yes—
12C2×#6758—21no—
13C2×#6758—24no—
14C2×#6758C2212yes—
15C3×C5×#121C411yescharacteristic, complemented
16C3×#4084C411yescharacteristic, complemented
17C3×#4083—21no—
18C3×D4×Dic5—21no—
19C3×#4070C411yescharacteristic, complemented
20C3×#4043C411yescharacteristic, complemented
21C3×#4038C2211yescharacteristic
22C3×#4029—21no—
23C3×#4027—21no—
24#91237—24no—
25#91235—21no—
26#71713—51nomaximal, Hall
27C22×#700—31no—
28#65940D311yescharacteristic
29C2×#4008—32no—
30C4×C22⋊AGL(1,𝔽5)—31no—
31#64875—32no—
32C23×C3×D5D411yescharacteristic
33C22×ΓL(1,𝔽16)—22no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4C2×C411yescharacteristic
36C22×C3×C4×C5C2×C411yescharacteristic
37C2×C3×C22⋊D5—21no—
38C2×C3×C22⋊D5—21no—
39C22×C3×Dic5—21no—
40C22×C3×Dic5D411yescharacteristic
41C3×#711—22no—
42C3×#711—22no—
43C2×C3×C4×D5—22no—
44C2×C3×C4×D5—22no—
45C2×C3×C4×D5C2×C411yescharacteristic
46C2×C3×C4×D5D411yescharacteristic
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×C4×D5C2311yescharacteristic
49C2×C3×Dic10C2×C411yescharacteristic
50#6763—24no—
51C2×C3⋊#170—24no—
52C2×C3⋊#170—22no—
53C4×ΓL(1,𝔽16)—22no—
54C4×ΓL(1,𝔽16)—22no—
55#6758—24no—
56#6758—22no—
57#6758D414yescomplemented
58C3×C5×C4⋊C4C2×C411yescharacteristic
59C3×C5×C22⋊C4—21no—
60C3×C22⋊Dic5—21no—
61C3×C22⋊Dic5—21no—
62C3×#685—21no—
63C3×#685—21no—
64C3×C4⋊Dic5—21no—
65C3×C4⋊Dic5C2×C411yescharacteristic
66C3×#683—21no—
67C3×C4×Dic5—21no—
68C3×#525—51no—
69C22×#208—51no—
70C2×#837—52no—
71C4×#217—51no—
72#4532—52no—
73C2×#711C2×D311yescharacteristic
74C22×C4×D5C2×D311yescharacteristic
75C2×C22⋊AGL(1,𝔽5)—61no—
76C22×#170—61no—
77C2×C4×AGL(1,𝔽5)—32no—
78C2×#700—32no—
79C2×#700—64no—
80C2×#700—61no—
81C5×#121Dic311yescharacteristic, complemented
82#4084Dic311yescharacteristic, complemented
83#4083—21no—
84D4×Dic5—21no—
85#4070Dic311yescharacteristic, complemented
86#4043Dic311yescharacteristic, complemented
87#4038C2×D311yescharacteristic
88#4029—21no—
89#4027—21no—
90#4008—64no—
91(C2×AGL(1,𝔽5))⋊C4—61no—
92#1871—151noSylow
93C23×C3×C5—21no—
94C23×C3×C5C22⋊C411yescharacteristic
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5—41no—
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×ΓL(1,𝔽16)—24no—
101C2×ΓL(1,𝔽16)—42no—
102C2×ΓL(1,𝔽16)—42no—
103C3×C5×D4—22no—
104C3×C5×D4—42no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5C22⋊C411yescharacteristic
108C2×C3×C4×C5C22×C411yescharacteristic
109C3×C22⋊D5—42no—
110C3×C22⋊D5—42no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—21no—
115C2×C3×Dic5—41no—
116C2×C3×Dic5—41no—
117C2×C3×Dic5C22⋊C411yescharacteristic
118C2×C3×Dic5C2×D411yescharacteristic
119C3×C4×D5—22no—
120C3×C4×D5—22no—
121C3×C4×D5—21no—
122C3×C4×D5—21no—
123C3×C4×D5C2×D412yes—
124C3×C4×D5#4112yes—
125C3×Dic10—22no—
126C3⋊#170—24no—
127C2×C3×#41—51no—
128C23×C3×C4—51no—
129C3×#123—101no—
130C3×C22⋊Q8—51no—
131C3×C22⋊Q8—51no—
132C3×#121—51no—
133C3×#121AGL(1,𝔽5)11yescharacteristic, complemented
134C3×#120—101no—
135C3×C4×D4—101no—
136C3×C4×D4—101no—
137C3×#117—51no—
138C2×#217—101no—
139C2×C4×Dic3—52no—
140C2×#208—52no—
141C2×#208—104no—
142C2×#208—101no—
143C22×C3⋊C8—101no—
144#837—104no—
145C3⋊#95—101no—
146C23×D5#6711yescharacteristic
147C22×AGL(1,𝔽5)—62no—
148C2×C5×D4—21no—
149C2×C5×D4C2×Dic311yescharacteristic
150C22×C4×C5C2×Dic311yescharacteristic
151C2×C22⋊D5—21no—
152C2×C22⋊D5—21no—
153C22×Dic5—21no—
154C22×Dic5#6711yescharacteristic
155#711—22no—
156#711—22no—
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5C2×Dic311yescharacteristic
160C2×C4×D5#6711yescharacteristic
161C2×C4×D5#6711yescharacteristic
162C2×C4×D5C22×D311yescharacteristic
163C2×Dic10C2×Dic311yescharacteristic
164C22⋊AGL(1,𝔽5)—64no—
165C2×#170—64no—
166C2×#170—62no—
167C4×AGL(1,𝔽5)—62no—
168C4×AGL(1,𝔽5)—62no—
169#700—34no—
170#700—64no—
171#700—62no—
172C5×C4⋊C4C2×Dic311yescharacteristic
173C5×C22⋊C4—21no—
174C22⋊Dic5—21no—
175C22⋊Dic5—21no—
176#685—21no—
177#685—21no—
178C4⋊Dic5—21no—
179C4⋊Dic5C2×Dic311yescharacteristic
180#683—21no—
181C4×Dic5—21no—
182C22×M16—151no—
183#525—51no—
184C2×#104—152no—
185C4×C22⋊C4—151no—
186#339—152no—
187C22×C3×C5—21no—
188C22×C3×C5—21no—
189C22×C3×C5—41no—
190C22×C3×C5—41no—
191C22×C3×C5—41no—
192C22×C3×C5#10412yescomplemented
193C22×C3×C5C2×C22⋊C411yescharacteristic
194C2×C3×D5—21no—
195C2×C3×D5—21no—
196C2×C3×D5—21no—
197C2×C3×D5—21no—
198C2×C3×D5—21no—
199C2×C3×D5—44no—
200C2×C3×D5#12112yes—
201ΓL(1,𝔽16)—44no—
202C3×C4×C5—21no—
203C3×C4×C5—41no—
204C3×C4×C5C4×D412yes—
205C3×Dic5—21no—
206C3×Dic5—41no—
207C3×Dic5—41no—
208C3×Dic5#12011yescharacteristic
209C3×Dic5#12311yescharacteristic
210C24×C3—51no—
211C3×#41—102no—
212C3×#41—102no—
213C2×C3×Q8—51no—
214C2×C3×D4—21no—
215C2×C3×D4—101no—
216C2×C3×D4—101no—
217C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4—51no—
223C22×C3×C4—102no—
224C22×C3×C4—102no—
225C22×C3×C4—101no—
226C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
227C22×Dic3—102no—
228C3×C4⋊C4—51no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
232C3×C22⋊C4—21no—
233C3×C22⋊C4—101no—
234C3×C22⋊C4—101no—
235C3×C22⋊C4—101no—
236C3×C22⋊C4—101no—
237C3×C42—101no—
238#217—104no—
239C4×Dic3—102no—
240C4×Dic3—102no—
241#208—54no—
242#208—102no—
243#208—104no—
244C2×C3⋊C8—102no—
245C2×C3⋊C8—104no—
246C23×C5—21no—
247C23×C5#21711yescharacteristic
248C22×D5—22no—
249C22×D5—22no—
250C22×D5—41no—
251C22×D5#21711yescharacteristic
252C22×D5C2×#6711yescharacteristic
253C2×AGL(1,𝔽5)—64no—
254C2×AGL(1,𝔽5)—122no—
255C2×AGL(1,𝔽5)—122no—
256C5×D4—22no—
257C5×D4—42no—
258C2×C4×C5—22no—
259C2×C4×C5—21no—
260C2×C4×C5#21711yescharacteristic
261C2×C4×C5C22×Dic311yescharacteristic
262C22⋊D5—42no—
263C22⋊D5—42no—
264C2×Dic5—21no—
265C2×Dic5—21no—
266C2×Dic5—21no—
267C2×Dic5—21no—
268C2×Dic5—41no—
269C2×Dic5—41no—
270C2×Dic5#21711yescharacteristic
271C2×Dic5C2×#6711yescharacteristic
272C4×D5—22no—
273C4×D5—22no—
274C4×D5—21no—
275C4×D5—21no—
276C4×D5D3×D412yes—
277C4×D5#23712yes—
278Dic10—22no—
279#170—64no—
280C2×#41—51no—
281C23×C4—51no—
282C2×M16—152no—
283C2×M16—304no—
284C2×M16—301no—
285C22×C8—301no—
286#123—101no—
287C22⋊Q8—51no—
288C22⋊Q8—51no—
289#121—51no—
290#121ΓL(1,𝔽16)11yescharacteristic, complemented
291#120—101no—
292C4×D4—101no—
293C4×D4—101no—
294#117—51no—
295C2×C22⋊C4—301no—
296C2×C42—152no—
297#104—304no—
298#95—301no—
299C2×C3×C5—22no—
300C2×C3×C5—81no—
301C2×C3×C5#38611yescharacteristic
302C2×C3×C5C2×#10411yescharacteristic
303C2×C3×C5#42111yescharacteristic
304C3×D5—22no—
305C3×D5—41no—
306C23×C3—21no—
307C23×C3—51no—
308C23×C3—51no—
309C23×C3—102no—
310C23×C3—102no—
311C23×C3—201no—
312C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
313C3×Q8—102no—
314C3×D4—22no—
315C3×D4—42no—
316C3×D4—202no—
317C3×D4—202no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—51no—
321C2×C3×C4—52no—
322C2×C3×C4—52no—
323C2×C3×C4—51no—
324C2×C3×C4—101no—
325C2×C3×C4—101no—
326C2×C3×C4—102no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—102no—
330C2×C3×C4—101no—
331C2×C3×C4—101no—
332C2×C3×C4—201no—
333C2×C3×C4—201no—
334C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
335C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
336C2×Dic3—104no—
337C2×Dic3—202no—
338C2×Dic3—202no—
339C3⋊C8—104no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5—41no—
345C22×C5#83712yescomplemented
346C22×C5C2×#21711yescharacteristic
347C2×D5—21no—
348C2×D5—21no—
349C2×D5—21no—
350C2×D5—21no—
351C2×D5—21no—
352C2×D5—44no—
353C2×D5#93912yes—
354AGL(1,𝔽5)—124no—
355C4×C5—21no—
356C4×C5—41no—
357C4×C5D4×Dic312yes—
358Dic5—21no—
359Dic5—41no—
360Dic5—41no—
361Dic5#93511yescharacteristic
362Dic5#93711yescharacteristic
363C24—51no—
364#41—102no—
365#41—102no—
366C2×Q8—51no—
367C2×D4—21no—
368C2×D4—101no—
369C2×D4—101no—
370C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—102no—
377C22×C4—102no—
378C22×C4—101no—
379C22×C4—302no—
380C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
381M16—154no—
382M16—302no—
383M16—304no—
384C2×C8—302no—
385C2×C8—304no—
386C4⋊C4—51no—
387C4⋊C4—101no—
388C4⋊C4—101no—
389C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
390C22⋊C4—21no—
391C22⋊C4—101no—
392C22⋊C4—101no—
393C22⋊C4—101no—
394C22⋊C4—101no—
395C22⋊C4—304no—
396C42—101no—
397C42—302no—
398C42—302no—
399C3×C5#187111yescharacteristic, complemented, Hall
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—52no—
406C22×C3—101no—
407C22×C3—101no—
408C22×C3—101no—
409C22×C3—101no—
410C22×C3—101no—
411C22×C3—204no—
412C22×C3#400812yescomplemented
413C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—51no—
417C3×C4—51no—
418C3×C4—101no—
419C3×C4—201no—
420C3×C4—201no—
421C3×C4D4×AGL(1,𝔽5)12yes—
422Dic3—204no—
423C2×C5—22no—
424C2×C5—81no—
425C2×C5#522011yescharacteristic
426C2×C5C2×#83711yescharacteristic
427C2×C5#523611yescharacteristic
428D5—22no—
429D5—41no—
430C23—21no—
431C23—51no—
432C23—51no—
433C23—102no—
434C23—102no—
435C23—201no—
436C23#676311yescharacteristic
437Q8—102no—
438D4—22no—
439D4—42no—
440D4—202no—
441D4—202no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—51no—
445C2×C4—52no—
446C2×C4—52no—
447C2×C4—51no—
448C2×C4—101no—
449C2×C4—101no—
450C2×C4—102no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—102no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—201no—
457C2×C4—201no—
458C2×C4—304no—
459C2×C4—602no—
460C2×C4—602no—
461C2×C4#676311yescharacteristic
462C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
463C8—304no—
464C2×C3—22no—
465C2×C3—81no—
466C2×C3—102no—
467C2×C3—201no—
468C2×C3C2×#400811yescharacteristic
469C2×C3#6577911yescharacteristic
470C2×C3#6579511yescharacteristic
471C5#7171311yescharacteristic, complemented, Sylow
472C22—21no—
473C22—21no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—52no—
478C22—101no—
479C22—101no—
480C22—101no—
481C22—101no—
482C22—101no—
483C22—204no—
484C22#9123712yescomplemented
485C22C2×#676311yescharacteristic
486C4—21no—
487C4—41no—
488C4—51no—
489C4—51no—
490C4—101no—
491C4—201no—
492C4—201no—
493C4—604no—
494C4D4×ΓL(1,𝔽16)12yes—
495C3#1060449511yescharacteristic, complemented, Sylow
496C2—22no—
497C2—81no—
498C2—102no—
499C2—201no—
500C2C2×#9123711yescharacteristic
501C2#1175243811yescharacteristic
502C2#1175245411yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#121C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65940C2
cosocleC2×C3×C4×D5C23
hypercenter#121ΓL(1,𝔽16)
hypocenterC3×C5#1871
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×#6758C2semidirect
3. C2×#91237C2semidirect
4. C4×#6763C2semidirect
5. #11746746C2semidirect
15. C3×C5×#121C4semidirect
16. C3×#4084C4semidirect
19. C3×#4070C4semidirect
20. C3×#4043C4semidirect
57. #6758D4semidirect
81. C5×#121Dic3semidirect
82. #4084Dic3semidirect
85. #4070Dic3semidirect
86. #4043Dic3semidirect
133. C3×#121AGL(1,𝔽5)semidirect
192. C22×C3×C5#104semidirect
290. #121ΓL(1,𝔽16)semidirect
345. C22×C5#837semidirect
399. C3×C5#1871semidirect
412. C22×C3#4008semidirect
471. C5#71713semidirect
484. C22#91237semidirect
495. C3#10604495semidirect

Maximal subgroups

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

subgroupindex
C22×#67582
C3×#659402
C2×#912372
C4×#67632
#117467462
#106044953
#717135