Finite Group Databasebeta

← #14326585

Subgroups — #14326585

Order 1920.

1208 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91989C212yesmaximal
2C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
3#11752424C211yesmaximal, characteristic
4#11752423C212yesmaximal
5C4×#6763C211yesmaximal, characteristic
6#11752371C212yesmaximal, complemented
7#11752370C214yesmaximal, complemented
8C4⋊C4×ΓL(1,𝔽16)C212yesmaximal, complemented
9#10617103—31nomaximal, Hall
10C22×C3×C4×D5C2212yes—
11C22×C3×C4×D5C2211yescharacteristic
12C2×#6763C2212yes—
13C2×#6763C2211yescharacteristic
14C2×#6763C2211yescharacteristic
15#91989—28no—
16C2×C4⋊ΓL(1,𝔽16)C2212yes—
17C2×C4⋊ΓL(1,𝔽16)C2214yes—
18C2×C4×ΓL(1,𝔽16)C2214yes—
19C2×C4×ΓL(1,𝔽16)C2212yes—
20C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
21C2×C3×C4⋊Dic5C411yescharacteristic, complemented
22C2×C3×#683C412yescomplemented
23C3×D5×C4⋊C4C411yescharacteristic, complemented
24C3×D5×C4⋊C4C411yescharacteristic, complemented
25C3×D5×C4⋊C4C412yescomplemented
26C3×D5×C4⋊C4C2211yescharacteristic
27C3×D5×C4⋊C4C2212yes—
28C3×D5×C4⋊C4C2211yescharacteristic
29#91235C2214yes—
30#91235C2212yes—
31#91235C2212yes—
32#91235C2214yes—
33#82159—51nomaximal, Hall
34C2×#4130—32no—
35C2×D5×C4⋊C4D311yescharacteristic
36#65765—31no—
37#65764—32no—
38C4×C22⋊AGL(1,𝔽5)—31no—
39#65712—32no—
40#65711—34no—
41C4⋊C4×AGL(1,𝔽5)—32no—
42C23×C3×D5C2311yescharacteristic
43C22×ΓL(1,𝔽16)C2314yes—
44C22×ΓL(1,𝔽16)C2312yes—
45C22×ΓL(1,𝔽16)C2312yes—
46C22×C3×C4×C5C2×C412yes—
47C22×C3×C4×C5C2×C411yescharacteristic
48C22×C3×Dic5C2×C411yescharacteristic
49C22×C3×Dic5C2×C412yes—
50C2×C3×C4×D5—24no—
51C2×C3×C4×D5—24no—
52C2×C3×C4×D5—24no—
53C2×C3×C4×D5C2×C412yes—
54C2×C3×C4×D5C2×C412yes—
55C2×C3×C4×D5C2×C411yescharacteristic
56C2×C3×C4×D5C2×C411yescharacteristic
57C2×C3×C4×D5C2312yes—
58C2×C3×C4×D5C2312yes—
59C2×C3×C4×D5C2311yescharacteristic
60C2×C3×C4×D5C2311yescharacteristic
61#6763—24no—
62#6763—22no—
63#6763D414yes—
64C4⋊ΓL(1,𝔽16)—28no—
65C4⋊ΓL(1,𝔽16)—24no—
66C4×ΓL(1,𝔽16)—28no—
67C4×ΓL(1,𝔽16)—24no—
68C3×C5×C4⋊C4C2×C412yes—
69C3×C5×C4⋊C4C2×C411yescharacteristic
70C3×C5×C4⋊C4C2×C411yescharacteristic
71C3×C4⋊Dic5C2×C412yes—
72C3×C4⋊Dic5C2×C411yescharacteristic
73C3×C4⋊Dic5C2×C411yescharacteristic
74C3×#683C2×C412yes—
75C3×#683C2×C412yes—
76C3×#683C2×C412yes—
77C3×#683C2×C412yes—
78C22×C3×C4⋊C4—51no—
79C2×#926—52no—
80#5212—51no—
81#5211—52no—
82C4×#217—51no—
83#4984—52no—
84#4980—54no—
85Dic3×C4⋊C4—52no—
86C22×C4×D5C2×D312yes—
87C22×C4×D5C2×D311yescharacteristic
88C2×C22⋊AGL(1,𝔽5)—31no—
89C2×C22⋊AGL(1,𝔽5)—32no—
90C2×C22⋊AGL(1,𝔽5)—31no—
91#4130—68no—
92C2×C4⋊AGL(1,𝔽5)—32no—
93C2×C4⋊AGL(1,𝔽5)—34no—
94C2×C4×AGL(1,𝔽5)—34no—
95C2×C4×AGL(1,𝔽5)—32no—
96C2×C5×C4⋊C4Dic311yescharacteristic, complemented
97C2×C4⋊Dic5Dic311yescharacteristic, complemented
98C2×#683Dic312yescomplemented
99D5×C4⋊C4Dic311yescharacteristic, complemented
100D5×C4⋊C4Dic311yescharacteristic, complemented
101D5×C4⋊C4Dic312yescomplemented
102D5×C4⋊C4C2×D311yescharacteristic
103D5×C4⋊C4C2×D312yes—
104D5×C4⋊C4C2×D311yescharacteristic
105(C2×AGL(1,𝔽5))⋊C4—34no—
106(C2×AGL(1,𝔽5))⋊C4—32no—
107(C2×AGL(1,𝔽5))⋊C4—32no—
108(C2×AGL(1,𝔽5))⋊C4—34no—
109#2344—151noSylow
110C23×C3×C5C22×C411yescharacteristic
111C22×C3×D5—22no—
112C22×C3×D5—22no—
113C22×C3×D5C22×C411yescharacteristic
114C22×C3×D5C2×D412yes—
115C22×C3×D5#4112yes—
116C22×C3×D5C2411yescharacteristic
117C2×ΓL(1,𝔽16)—22no—
118C2×ΓL(1,𝔽16)—22no—
119C2×ΓL(1,𝔽16)—24no—
120C2×ΓL(1,𝔽16)—22no—
121C2×ΓL(1,𝔽16)—24no—
122C2×ΓL(1,𝔽16)—22no—
123C2×ΓL(1,𝔽16)—24no—
124C2×ΓL(1,𝔽16)—22no—
125C2×ΓL(1,𝔽16)C2×D414yes—
126C2×C3×C4×C5—24no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5C22×C412yes—
129C2×C3×C4×C5C22×C412yes—
130C2×C3×C4×C5C22×C411yescharacteristic
131C2×C3×C4×C5C22×C411yescharacteristic
132C2×C3×Dic5—24no—
133C2×C3×Dic5—22no—
134C2×C3×Dic5C22×C411yescharacteristic
135C2×C3×Dic5C22×C412yes—
136C2×C3×Dic5C22×C412yes—
137C2×C3×Dic5C22×C411yescharacteristic
138C3×C4×D5—24no—
139C3×C4×D5—24no—
140C3×C4×D5—22no—
141C3×C4×D5—22no—
142C3×C4×D5—22no—
143C3×C4×D5—22no—
144C3×C4×D5—22no—
145C3×C4×D5—22no—
146C3×C4×D5—22no—
147C3×C4×D5#4114yes—
148C23×C3×C4—51no—
149C23×C3×C4—52no—
150C2×C3×C4⋊C4—52no—
151C2×C3×C4⋊C4—51no—
152C2×C3×C4⋊C4—51no—
153C2×C3×C4⋊C4—52no—
154C2×C3×C4⋊C4—51no—
155C2×C3×C4⋊C4—52no—
156C2×C3×C4⋊C4—51no—
157C2×C3×C4⋊C4—51no—
158C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
159C2×#217—51no—
160C2×#217—51no—
161C2×#217—52no—
162#926—108no—
163C2×C4⋊Dic3—52no—
164C2×C4⋊Dic3—54no—
165C2×C4×Dic3—52no—
166C2×C4×Dic3—54no—
167C3⋊#95—52no—
168C3⋊#95—54no—
169C3⋊#95—54no—
170C3⋊#95—52no—
171C23×D5C22×D311yescharacteristic
172C22×AGL(1,𝔽5)—32no—
173C22×AGL(1,𝔽5)—34no—
174C22×AGL(1,𝔽5)—32no—
175C22×C4×C5C2×Dic312yes—
176C22×C4×C5C2×Dic311yescharacteristic
177C22×Dic5C2×Dic311yescharacteristic
178C22×Dic5C2×Dic312yes—
179C2×C4×D5—24no—
180C2×C4×D5—24no—
181C2×C4×D5—24no—
182C2×C4×D5C2×Dic312yes—
183C2×C4×D5C2×Dic312yes—
184C2×C4×D5C2×Dic311yescharacteristic
185C2×C4×D5C2×Dic311yescharacteristic
186C2×C4×D5C22×D312yes—
187C2×C4×D5C22×D312yes—
188C2×C4×D5C22×D311yescharacteristic
189C2×C4×D5C22×D311yescharacteristic
190C22⋊AGL(1,𝔽5)—34no—
191C22⋊AGL(1,𝔽5)—64no—
192C22⋊AGL(1,𝔽5)—62no—
193C4⋊AGL(1,𝔽5)—68no—
194C4⋊AGL(1,𝔽5)—64no—
195C4×AGL(1,𝔽5)—68no—
196C4×AGL(1,𝔽5)—64no—
197C5×C4⋊C4C2×Dic312yes—
198C5×C4⋊C4C2×Dic311yescharacteristic
199C5×C4⋊C4C2×Dic311yescharacteristic
200C4⋊Dic5C2×Dic312yes—
201C4⋊Dic5C2×Dic311yescharacteristic
202C4⋊Dic5C2×Dic311yescharacteristic
203#683C2×Dic312yes—
204#683C2×Dic312yes—
205#683C2×Dic312yes—
206#683C2×Dic312yes—
207C2×#117—152no—
208C22×C4⋊C4—51no—
209#389—152no—
210#387—154no—
211#381—152no—
212#380—151no—
213C4×C4⋊C4—152no—
214C4×C22⋊C4—151no—
215C22×C3×C5—21no—
216C22×C3×C5—21no—
217C22×C3×C5C4×D412yes—
218C22×C3×C5C23×C411yescharacteristic
219C2×C3×D5—21no—
220C2×C3×D5—21no—
221C2×C3×D5—24no—
222C2×C3×D5—24no—
223C2×C3×D5C4×D412yes—
224C2×C3×D5C22×D411yescharacteristic
225C2×C3×D5C2×#4112yes—
226C2×C3×D5C2×#4111yescharacteristic
227C2×C3×D52−1+412yes—
228ΓL(1,𝔽16)—24no—
229ΓL(1,𝔽16)—44no—
230ΓL(1,𝔽16)—42no—
231C3×C4×C5—21no—
232C3×C4×C5—22no—
233C3×C4×C5—22no—
234C3×C4×C5#11712yescomplemented
235C3×Dic5—21no—
236C3×Dic5—22no—
237C3×Dic5—22no—
238C3×Dic5#11712yescomplemented
239C24×C3—51no—
240C22×C3×C4—51no—
241C22×C3×C4—51no—
242C22×C3×C4—52no—
243C22×C3×C4—51no—
244C22×C3×C4—52no—
245C22×C3×C4—52no—
246C22×C3×C4—51no—
247C22×C3×C4—52no—
248C22×C3×C4—52no—
249C22×C3×C4—51no—
250C22×C3×C4—104no—
251C22×C3×C4—104no—
252C22×C3×C4—104no—
253C22×C3×C4C2×AGL(1,𝔽5)12yes—
254C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
255C22×Dic3—52no—
256C22×Dic3—52no—
257C22×Dic3—54no—
258C3×C4⋊C4—52no—
259C3×C4⋊C4—52no—
260C3×C4⋊C4—52no—
261C3×C4⋊C4—52no—
262C3×C4⋊C4—52no—
263C3×C4⋊C4—51no—
264C3×C4⋊C4—51no—
265C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
266C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
267C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
268#217—54no—
269#217—104no—
270#217—102no—
271C4⋊Dic3—108no—
272C4⋊Dic3—104no—
273C4×Dic3—108no—
274C4×Dic3—104no—
275C23×C5C22×Dic311yescharacteristic
276C22×D5—22no—
277C22×D5—22no—
278C22×D5D3×D412yes—
279C22×D5#23712yes—
280C22×D5C22×Dic311yescharacteristic
281C22×D5C23×D311yescharacteristic
282C2×AGL(1,𝔽5)—34no—
283C2×AGL(1,𝔽5)—62no—
284C2×AGL(1,𝔽5)—62no—
285C2×AGL(1,𝔽5)—62no—
286C2×AGL(1,𝔽5)—64no—
287C2×AGL(1,𝔽5)—64no—
288C2×AGL(1,𝔽5)—62no—
289C2×AGL(1,𝔽5)—64no—
290C2×AGL(1,𝔽5)—62no—
291C2×C4×C5—24no—
292C2×C4×C5—22no—
293C2×C4×C5C22×Dic312yes—
294C2×C4×C5C22×Dic312yes—
295C2×C4×C5C22×Dic311yescharacteristic
296C2×C4×C5C22×Dic311yescharacteristic
297C2×Dic5—24no—
298C2×Dic5—22no—
299C2×Dic5C22×Dic311yescharacteristic
300C2×Dic5C22×Dic312yes—
301C2×Dic5C22×Dic312yes—
302C2×Dic5C22×Dic311yescharacteristic
303C4×D5—24no—
304C4×D5—24no—
305C4×D5—22no—
306C4×D5—22no—
307C4×D5—22no—
308C4×D5—22no—
309C4×D5—22no—
310C4×D5—22no—
311C4×D5—22no—
312C4×D5#23514yes—
313C23×C4—52no—
314C23×C4—51no—
315#117—308no—
316C2×C4⋊C4—52no—
317C2×C4⋊C4—51no—
318C2×C4⋊C4—51no—
319C2×C4⋊C4—52no—
320C2×C4⋊C4—51no—
321C2×C4⋊C4—52no—
322C2×C4⋊C4—51no—
323C2×C4⋊C4—51no—
324C2×C4⋊C4—154no—
325C2×C4⋊C4—152no—
326C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
327C2×C22⋊C4—151no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—151no—
330C2×C42—154no—
331C2×C42—152no—
332#95—152no—
333#95—154no—
334#95—152no—
335#95—154no—
336C2×C3×C5—22no—
337C2×C3×C5C2×#11711yescharacteristic
338C2×C3×C5C2×C4×D411yescharacteristic
339C2×C3×C5#51911yescharacteristic
340C3×D5—22no—
341C3×D5#54712yes—
342C3×D5#55212yes—
343C23×C3—51no—
344C23×C3—52no—
345C23×C3—52no—
346C23×C3—51no—
347C23×C3—102no—
348C23×C3—102no—
349C23×C3C22×AGL(1,𝔽5)11yescharacteristic
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—51no—
353C2×C3×C4—54no—
354C2×C3×C4—51no—
355C2×C3×C4—52no—
356C2×C3×C4—52no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4—102no—
360C2×C3×C4—102no—
361C2×C3×C4—102no—
362C2×C3×C4—102no—
363C2×C3×C4—104no—
364C2×C3×C4—104no—
365C2×C3×C4—102no—
366C2×C3×C4—104no—
367C2×C3×C4—102no—
368C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
369C2×C3×C4C22×AGL(1,𝔽5)12yes—
370C2×C3×C4C22×AGL(1,𝔽5)12yes—
371C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
372C2×Dic3—54no—
373C2×Dic3—102no—
374C2×Dic3—102no—
375C2×Dic3—104no—
376C2×Dic3—102no—
377C2×Dic3—104no—
378C2×Dic3—102no—
379C2×Dic3—102no—
380C2×Dic3—104no—
381C22×C5—21no—
382C22×C5—21no—
383C22×C5D4×Dic312yes—
384C22×C5C23×Dic311yescharacteristic
385C2×D5—21no—
386C2×D5—21no—
387C2×D5—24no—
388C2×D5—24no—
389C2×D5D4×Dic312yes—
390C2×D5C2×#23512yes—
391C2×D5C2×D3×D411yescharacteristic
392C2×D5C2×#23711yescharacteristic
393C2×D5#100712yes—
394AGL(1,𝔽5)—64no—
395AGL(1,𝔽5)—124no—
396AGL(1,𝔽5)—122no—
397C4×C5—21no—
398C4×C5—22no—
399C4×C5—22no—
400C4×C5#92612yescomplemented
401Dic5—21no—
402Dic5—22no—
403Dic5—22no—
404Dic5#92612yescomplemented
405C24—51no—
406C22×C4—51no—
407C22×C4—51no—
408C22×C4—52no—
409C22×C4—51no—
410C22×C4—52no—
411C22×C4—52no—
412C22×C4—51no—
413C22×C4—52no—
414C22×C4—52no—
415C22×C4—51no—
416C22×C4—104no—
417C22×C4—104no—
418C22×C4—104no—
419C22×C4—152no—
420C22×C4—152no—
421C22×C4—154no—
422C22×C4C2×ΓL(1,𝔽16)12yes—
423C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
424C4⋊C4—52no—
425C4⋊C4—52no—
426C4⋊C4—52no—
427C4⋊C4—52no—
428C4⋊C4—52no—
429C4⋊C4—51no—
430C4⋊C4—51no—
431C4⋊C4—304no—
432C4⋊C4—308no—
433C4⋊C4C2×ΓL(1,𝔽16)12yes—
434C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
435C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
436C22⋊C4—154no—
437C22⋊C4—304no—
438C22⋊C4—302no—
439C42—304no—
440C42—308no—
441C3×C5#234411yescharacteristic, complemented, Hall
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—52no—
445C22×C3—51no—
446C22×C3—51no—
447C22×C3—52no—
448C22×C3—52no—
449C22×C3—104no—
450C22×C3—104no—
451C22×C3—101no—
452C22×C3—101no—
453C22×C3D4×AGL(1,𝔽5)12yes—
454C22×C3C23×AGL(1,𝔽5)11yescharacteristic
455C3×C4—21no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—52no—
459C3×C4—101no—
460C3×C4—102no—
461C3×C4—102no—
462C3×C4#413012yescomplemented
463Dic3—104no—
464Dic3—204no—
465Dic3—202no—
466C2×C5—22no—
467C2×C5C2×#92611yescharacteristic
468C2×C5C2×D4×Dic311yescharacteristic
469C2×C5#581211yescharacteristic
470D5—22no—
471D5#550712yes—
472D5#551312yes—
473C23—51no—
474C23—52no—
475C23—52no—
476C23—51no—
477C23—102no—
478C23—102no—
479C23C22×ΓL(1,𝔽16)11yescharacteristic
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—51no—
483C2×C4—54no—
484C2×C4—51no—
485C2×C4—52no—
486C2×C4—52no—
487C2×C4—102no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4—102no—
492C2×C4—102no—
493C2×C4—104no—
494C2×C4—104no—
495C2×C4—102no—
496C2×C4—104no—
497C2×C4—102no—
498C2×C4—154no—
499C2×C4—302no—
500C2×C4—302no—
501C2×C4—304no—
502C2×C4—302no—
503C2×C4—302no—
504C2×C4—304no—
505C2×C4—302no—
506C2×C4—304no—
507C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
508C2×C4C22×ΓL(1,𝔽16)12yes—
509C2×C4C22×ΓL(1,𝔽16)12yes—
510C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
511C2×C3—22no—
512C2×C3—52no—
513C2×C3—52no—
514C2×C3—102no—
515C2×C3C2×#413011yescharacteristic
516C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
517C2×C3#6626211yescharacteristic
518C5#8215911yescharacteristic, complemented, Sylow
519C22—21no—
520C22—21no—
521C22—52no—
522C22—51no—
523C22—51no—
524C22—52no—
525C22—52no—
526C22—104no—
527C22—104no—
528C22—101no—
529C22—101no—
530C22D4×ΓL(1,𝔽16)12yes—
531C22C23×ΓL(1,𝔽16)11yescharacteristic
532C4—21no—
533C4—22no—
534C4—22no—
535C4—52no—
536C4—101no—
537C4—102no—
538C4—102no—
539C4—304no—
540C4—604no—
541C4—602no—
542C4#9198912yescomplemented
543C3#1061710311yescharacteristic, complemented, Sylow
544C2—22no—
545C2—52no—
546C2—52no—
547C2—102no—
548C2C2×#9198911yescharacteristic
549C2C2×D4×ΓL(1,𝔽16)11yescharacteristic
550C2#1175699111yescharacteristic
subgroupG/H
center, FrattiniC22C23×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC2×C4⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#2344
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #11752371C2semidirect
7. #11752370C2semidirect
8. C4⋊C4×ΓL(1,𝔽16)C2semidirect
20. C2×C3×C5×C4⋊C4C4semidirect
21. C2×C3×C4⋊Dic5C4semidirect
22. C2×C3×#683C4semidirect
23. C3×D5×C4⋊C4C4semidirect
24. C3×D5×C4⋊C4C4semidirect
25. C3×D5×C4⋊C4C4semidirect
96. C2×C5×C4⋊C4Dic3semidirect
97. C2×C4⋊Dic5Dic3semidirect
98. C2×#683Dic3semidirect
99. D5×C4⋊C4Dic3semidirect
100. D5×C4⋊C4Dic3semidirect
101. D5×C4⋊C4Dic3semidirect
158. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
234. C3×C4×C5#117semidirect
238. C3×Dic5#117semidirect
326. C2×C4⋊C4ΓL(1,𝔽16)semidirect
400. C4×C5#926semidirect
404. Dic5#926semidirect
441. C3×C5#2344semidirect
462. C3×C4#4130semidirect
518. C5#82159semidirect
542. C4#91989semidirect
543. C3#10617103semidirect

Maximal subgroups

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

subgroupindex
C2×#919892
C2×C3×D5×C4⋊C42
#117524242
#117524232
C4×#67632
#117523712
#117523702
C4⋊C4×ΓL(1,𝔽16)2
#106171033
#821595