Finite Group Databasebeta

← #14279964

Subgroups — #14279964

Order 1920.

1224 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×ΓL(1,𝔽16)C211yesmaximal, characteristic
2C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
3C2×#91235C212yesmaximal
4C2×#91235C211yesmaximal, characteristic
5C2×#91235C212yesmaximal
6#10604343—31nomaximal, Hall
7C23×ΓL(1,𝔽16)C2212yes—
8C23×ΓL(1,𝔽16)C2212yes—
9C22×C3×C4×D5C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C22×C3×C4×D5C2211yescharacteristic
12C2×C4×ΓL(1,𝔽16)—24no—
13C2×C4×ΓL(1,𝔽16)—22no—
14C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
15C2×C3×C4⋊Dic5C411yescharacteristic, complemented
16C2×C3×#683C411yescharacteristic, complemented
17C2×C3×#683C411yescharacteristic, complemented
18C3×D5×C4⋊C4—22no—
19C3×D5×C4⋊C4—22no—
20#91235—24no—
21#91235—22no—
22#91235C418yescomplemented
23#71564—51nomaximal, Hall
24C22×C4×AGL(1,𝔽5)—31no—
25C2×D5×C4⋊C4D311yescharacteristic
26C2×(C2×AGL(1,𝔽5))⋊C4—32no—
27C2×(C2×AGL(1,𝔽5))⋊C4—32no—
28C2×(C2×AGL(1,𝔽5))⋊C4—31no—
29C23×C3×D5C2311yescharacteristic
30C22×ΓL(1,𝔽16)—22no—
31C22×ΓL(1,𝔽16)—22no—
32C22×ΓL(1,𝔽16)—22no—
33C22×ΓL(1,𝔽16)—22no—
34C22×ΓL(1,𝔽16)—22no—
35C22×ΓL(1,𝔽16)—22no—
36C22×ΓL(1,𝔽16)—22no—
37C22×ΓL(1,𝔽16)—22no—
38C22×ΓL(1,𝔽16)—22no—
39C22×ΓL(1,𝔽16)—22no—
40C22×ΓL(1,𝔽16)C2×C414yes—
41C22×ΓL(1,𝔽16)C2×C414yes—
42C22×ΓL(1,𝔽16)D414yes—
43C22×ΓL(1,𝔽16)Q814yes—
44C22×C3×C4×C5C2×C411yescharacteristic
45C22×C3×C4×C5C2×C411yescharacteristic
46C22×C3×C4×C5C2×C411yescharacteristic
47C22×C3×Dic5C2×C411yescharacteristic
48C22×C3×Dic5C2×C411yescharacteristic
49C22×C3×Dic5C2×C411yescharacteristic
50C2×C3×C4×D5—22no—
51C2×C3×C4×D5—21no—
52C2×C3×C4×D5—21no—
53C2×C3×C4×D5—22no—
54C2×C3×C4×D5—22no—
55C2×C3×C4×D5—22no—
56C2×C3×C4×D5—21no—
57C2×C3×C4×D5—21no—
58C2×C3×C4×D5—22no—
59C2×C3×C4×D5C2×C414yes—
60C2×C3×C4×D5D412yes—
61C2×C3×C4×D5D412yes—
62C4×ΓL(1,𝔽16)—28no—
63C3×C5×C4⋊C4—22no—
64C3×C4⋊Dic5—22no—
65C3×#683—22no—
66C3×#683—22no—
67C22×C3×C4⋊C4—51no—
68C22×C4×Dic3—51no—
69C2×C3⋊#95—52no—
70C2×C3⋊#95—52no—
71C2×C3⋊#95—51no—
72C23×AGL(1,𝔽5)—32no—
73C23×AGL(1,𝔽5)—32no—
74C22×C4×D5C2×D311yescharacteristic
75C22×C4×D5C2×D311yescharacteristic
76C22×C4×D5C2×D311yescharacteristic
77C2×C4×AGL(1,𝔽5)—64no—
78C2×C4×AGL(1,𝔽5)—62no—
79C2×C5×C4⋊C4Dic311yescharacteristic, complemented
80C2×C4⋊Dic5Dic311yescharacteristic, complemented
81C2×#683Dic311yescharacteristic, complemented
82C2×#683Dic311yescharacteristic, complemented
83D5×C4⋊C4—22no—
84D5×C4⋊C4—22no—
85(C2×AGL(1,𝔽5))⋊C4—38no—
86(C2×AGL(1,𝔽5))⋊C4—64no—
87(C2×AGL(1,𝔽5))⋊C4—62no—
88#1436—151noSylow
89C23×C3×C5C22×C411yescharacteristic
90C22×C3×D5C22×C412yes—
91C22×C3×D5C2×D412yes—
92C22×C3×D5C2×D411yescharacteristic
93C22×C3×D5C2×D411yescharacteristic
94C22×C3×D5C2×Q812yes—
95C22×C3×D5#4112yes—
96C22×C3×D5#4112yes—
97C22×C3×D5#4112yes—
98C2×ΓL(1,𝔽16)—24no—
99C2×ΓL(1,𝔽16)—24no—
100C2×ΓL(1,𝔽16)—24no—
101C2×ΓL(1,𝔽16)—24no—
102C2×ΓL(1,𝔽16)—24no—
103C2×ΓL(1,𝔽16)—24no—
104C2×ΓL(1,𝔽16)—24no—
105C2×ΓL(1,𝔽16)—24no—
106C2×ΓL(1,𝔽16)—24no—
107C2×ΓL(1,𝔽16)—42no—
108C2×ΓL(1,𝔽16)—42no—
109C2×ΓL(1,𝔽16)—42no—
110C2×ΓL(1,𝔽16)—42no—
111C2×ΓL(1,𝔽16)C4⋊C418yescomplemented
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5C4212yescomplemented
118C2×C3×C4×C5C22⋊C412yes—
119C2×C3×Dic5—21no—
120C2×C3×Dic5—21no—
121C2×C3×Dic5—22no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5—22no—
124C2×C3×Dic5C4212yescomplemented
125C2×C3×Dic5C22⋊C412yes—
126C3×C4×D5—24no—
127C3×C4×D5—24no—
128C3×C4×D5—22no—
129C3×C4×D5—24no—
130C3×C4×D5—22no—
131C3×C4×D5—41no—
132C3×C4×D5—42no—
133C3×C4×D5—41no—
134C23×C3×C4—51no—
135C23×C3×C4—51no—
136C23×C3×C4—51no—
137C23×Dic3—52no—
138C23×Dic3—52no—
139C2×C3×C4⋊C4—51no—
140C2×C3×C4⋊C4—51no—
141C2×C3×C4⋊C4—51no—
142C2×C3×C4⋊C4—102no—
143C2×C3×C4⋊C4—102no—
144C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
145C2×C4×Dic3—104no—
146C2×C4×Dic3—102no—
147C3⋊#95—58no—
148C3⋊#95—104no—
149C3⋊#95—102no—
150C23×D5C22×D311yescharacteristic
151C22×AGL(1,𝔽5)—34no—
152C22×AGL(1,𝔽5)—34no—
153C22×AGL(1,𝔽5)—34no—
154C22×AGL(1,𝔽5)—34no—
155C22×AGL(1,𝔽5)—62no—
156C22×AGL(1,𝔽5)—62no—
157C22×AGL(1,𝔽5)—62no—
158C22×AGL(1,𝔽5)—62no—
159C22×AGL(1,𝔽5)—62no—
160C22×AGL(1,𝔽5)—62no—
161C22×AGL(1,𝔽5)—62no—
162C22×AGL(1,𝔽5)—62no—
163C22×AGL(1,𝔽5)—62no—
164C22×AGL(1,𝔽5)—62no—
165C22×C4×C5C2×Dic311yescharacteristic
166C22×C4×C5C2×Dic311yescharacteristic
167C22×C4×C5C2×Dic311yescharacteristic
168C22×Dic5C2×Dic311yescharacteristic
169C22×Dic5C2×Dic311yescharacteristic
170C22×Dic5C2×Dic311yescharacteristic
171C2×C4×D5—22no—
172C2×C4×D5—22no—
173C2×C4×D5—22no—
174C2×C4×D5—21no—
175C2×C4×D5—21no—
176C2×C4×D5—22no—
177C2×C4×D5—22no—
178C2×C4×D5—21no—
179C2×C4×D5—21no—
180C2×C4×D5C4×D314yes—
181C2×C4×D5#6712yes—
182C2×C4×D5#6712yes—
183C4×AGL(1,𝔽5)—68no—
184C5×C4⋊C4—22no—
185C4⋊Dic5—22no—
186#683—22no—
187#683—22no—
188C22×C4⋊C4—51no—
189C22×C42—151no—
190C2×#95—152no—
191C2×#95—151no—
192C2×#95—152no—
193C22×C3×C5C2×C4211yescharacteristic
194C22×C3×C5C2×C22⋊C411yescharacteristic
195C22×C3×C5#11711yescharacteristic
196C22×C3×C5C4×D412yes—
197C22×C3×C5C4×Q812yes—
198C2×C3×D5C2×C4⋊C412yes—
199C2×C3×D5#11712yes—
200C2×C3×D5C4×D412yes—
201C2×C3×D5C4×D412yes—
202C2×C3×D5C4×D412yes—
203C2×C3×D5C4×Q812yes—
204C2×C3×D5#12011yescharacteristic
205C2×C3×D5#12112yes—
206C2×C3×D5C22⋊Q812yes—
207C2×C3×D5C22⋊Q812yes—
208C2×C3×D5#12312yes—
209C2×C3×D5#12311yescharacteristic
210C2×C3×D5#12411yescharacteristic
211C2×C3×D5#12512yes—
212C2×C3×D5#12612yes—
213C2×C3×D5C4⋊Q811yescharacteristic
214ΓL(1,𝔽16)—28no—
215ΓL(1,𝔽16)—44no—
216C3×C4×C5—22no—
217C3×C4×C5—22no—
218C3×C4×C5—41no—
219C3×Dic5—22no—
220C3×Dic5—22no—
221C3×Dic5—41no—
222C24×C3—51no—
223C22×C3×C4—51no—
224C22×C3×C4—51no—
225C22×C3×C4—54no—
226C22×C3×C4—52no—
227C22×C3×C4—52no—
228C22×C3×C4—51no—
229C22×C3×C4—102no—
230C22×C3×C4—101no—
231C22×C3×C4—101no—
232C22×C3×C4—102no—
233C22×C3×C4—102no—
234C22×C3×C4—102no—
235C22×C3×C4—101no—
236C22×C3×C4—101no—
237C22×C3×C4—102no—
238C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
239C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
240C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
241C22×Dic3—54no—
242C22×Dic3—54no—
243C22×Dic3—54no—
244C22×Dic3—54no—
245C22×Dic3—102no—
246C22×Dic3—102no—
247C22×Dic3—102no—
248C22×Dic3—102no—
249C22×Dic3—102no—
250C22×Dic3—102no—
251C22×Dic3—102no—
252C22×Dic3—102no—
253C22×Dic3—102no—
254C22×Dic3—102no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—102no—
257C3×C4⋊C4—102no—
258C3×C4⋊C4—102no—
259C4×Dic3—108no—
260C23×C5C22×Dic311yescharacteristic
261C22×D5C2×C4×D312yes—
262C22×D5#23512yes—
263C22×D5D3×D412yes—
264C22×D5#23712yes—
265C22×D5D3×Q812yes—
266C22×D5#23912yes—
267C22×D5C2×#6711yescharacteristic
268C22×D5C2×#6711yescharacteristic
269C2×AGL(1,𝔽5)—38no—
270C2×AGL(1,𝔽5)—64no—
271C2×AGL(1,𝔽5)—64no—
272C2×AGL(1,𝔽5)—64no—
273C2×AGL(1,𝔽5)—64no—
274C2×AGL(1,𝔽5)—64no—
275C2×AGL(1,𝔽5)—64no—
276C2×AGL(1,𝔽5)—64no—
277C2×AGL(1,𝔽5)—64no—
278C2×AGL(1,𝔽5)—64no—
279C2×AGL(1,𝔽5)—122no—
280C2×AGL(1,𝔽5)—122no—
281C2×AGL(1,𝔽5)—122no—
282C2×AGL(1,𝔽5)—122no—
283C2×C4×C5—21no—
284C2×C4×C5—22no—
285C2×C4×C5—21no—
286C2×C4×C5—22no—
287C2×C4×C5—21no—
288C2×C4×C5C4×Dic312yescomplemented
289C2×C4×C5#21712yes—
290C2×Dic5—21no—
291C2×Dic5—22no—
292C2×Dic5—21no—
293C2×Dic5—22no—
294C2×Dic5—21no—
295C2×Dic5C4×Dic312yescomplemented
296C2×Dic5#21712yes—
297C4×D5—24no—
298C4×D5—24no—
299C4×D5—22no—
300C4×D5—22no—
301C4×D5—24no—
302C4×D5—41no—
303C4×D5—41no—
304C4×D5—42no—
305C23×C4—51no—
306C23×C4—51no—
307C23×C4—51no—
308C23×C4—152no—
309C23×C4—152no—
310C2×C4⋊C4—51no—
311C2×C4⋊C4—51no—
312C2×C4⋊C4—51no—
313C2×C4⋊C4—102no—
314C2×C4⋊C4—102no—
315C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
316C2×C42—302no—
317C2×C42—304no—
318#95—158no—
319#95—302no—
320#95—304no—
321C2×C3×C5C4×C22⋊C411yescharacteristic
322C2×C3×C5C4×C4⋊C412yes—
323C2×C3×C5#38611yescharacteristic
324C2×C3×C5#38712yes—
325C2×C3×C5#39111yescharacteristic
326C3×D5#38512yes—
327C3×D5#38712yes—
328C3×D5#38812yes—
329C3×D5#38912yes—
330C23×C3—52no—
331C23×C3—52no—
332C23×C3—52no—
333C23×C3—52no—
334C23×C3—52no—
335C23×C3—52no—
336C23×C3—51no—
337C23×C3—51no—
338C23×C3C22×AGL(1,𝔽5)11yescharacteristic
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—52no—
345C2×C3×C4—52no—
346C2×C3×C4—102no—
347C2×C3×C4—104no—
348C2×C3×C4—104no—
349C2×C3×C4—104no—
350C2×C3×C4—102no—
351C2×C3×C4—101no—
352C2×C3×C4—102no—
353C2×C3×C4—101no—
354C2×C3×C4—102no—
355C2×C3×C4—101no—
356C2×C3×C4—201no—
357C2×C3×C4—202no—
358C2×C3×C4—201no—
359C2×C3×C4C4×AGL(1,𝔽5)12yescomplemented
360C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
361C2×Dic3—58no—
362C2×Dic3—104no—
363C2×Dic3—104no—
364C2×Dic3—104no—
365C2×Dic3—104no—
366C2×Dic3—104no—
367C2×Dic3—104no—
368C2×Dic3—104no—
369C2×Dic3—104no—
370C2×Dic3—104no—
371C2×Dic3—202no—
372C2×Dic3—202no—
373C2×Dic3—202no—
374C2×Dic3—202no—
375C22×C5C2×C4×Dic311yescharacteristic
376C22×C5#92611yescharacteristic
377C22×C5D4×Dic312yes—
378C22×C5Q8×Dic312yes—
379C22×C5C2×#21711yescharacteristic
380C2×D5#88712yes—
381C2×D5#88912yes—
382C2×D5D3×C4⋊C412yes—
383C2×D5#89212yes—
384C2×D5#89312yes—
385C2×D5#89412yes—
386C2×D5#89612yes—
387C2×D5#89812yes—
388C2×D5C4×#6712yes—
389C2×D5C4×#6712yes—
390C2×D5#93511yescharacteristic
391C2×D5#93711yescharacteristic
392C2×D5#93912yes—
393C2×D5#94411yescharacteristic
394C2×D5#94612yes—
395C2×D5#94711yescharacteristic
396AGL(1,𝔽5)—68no—
397AGL(1,𝔽5)—124no—
398C4×C5—22no—
399C4×C5—22no—
400C4×C5—41no—
401Dic5—22no—
402Dic5—22no—
403Dic5—41no—
404C24—51no—
405C22×C4—51no—
406C22×C4—51no—
407C22×C4—54no—
408C22×C4—52no—
409C22×C4—52no—
410C22×C4—51no—
411C22×C4—102no—
412C22×C4—101no—
413C22×C4—101no—
414C22×C4—102no—
415C22×C4—102no—
416C22×C4—102no—
417C22×C4—101no—
418C22×C4—101no—
419C22×C4—102no—
420C22×C4—154no—
421C22×C4—154no—
422C22×C4—154no—
423C22×C4—154no—
424C22×C4—302no—
425C22×C4—302no—
426C22×C4—302no—
427C22×C4—302no—
428C22×C4—302no—
429C22×C4—302no—
430C22×C4—302no—
431C22×C4—302no—
432C22×C4—302no—
433C22×C4—302no—
434C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
435C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
436C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
437C4⋊C4—22no—
438C4⋊C4—102no—
439C4⋊C4—102no—
440C4⋊C4—102no—
441C42—308no—
442C3×C5#143611yescharacteristic, complemented, Hall
443C22×C3—52no—
444C22×C3—52no—
445C22×C3—52no—
446C22×C3—52no—
447C22×C3—52no—
448C22×C3—52no—
449C22×C3—52no—
450C22×C3—52no—
451C22×C3—52no—
452C22×C3—52no—
453C22×C3—51no—
454C22×C3—52no—
455C22×C3—51no—
456C22×C3—52no—
457C22×C3—51no—
458C22×C3—51no—
459C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
460C22×C3#413011yescharacteristic
461C22×C3D4×AGL(1,𝔽5)12yes—
462C22×C3Q8×AGL(1,𝔽5)12yes—
463C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—41no—
467C3×C4—102no—
468C3×C4—102no—
469C3×C4—201no—
470Dic3—108no—
471Dic3—204no—
472C2×C5Dic3×C4⋊C412yes—
473C2×C5#498012yes—
474C2×C5C4×#21711yescharacteristic
475C2×C5#522011yescharacteristic
476C2×C5#522911yescharacteristic
477D5#497612yes—
478D5#497812yes—
479D5#498912yes—
480D5#499012yes—
481C23—52no—
482C23—52no—
483C23—52no—
484C23—52no—
485C23—52no—
486C23—52no—
487C23—51no—
488C23—51no—
489C23C22×ΓL(1,𝔽16)11yescharacteristic
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—52no—
496C2×C4—52no—
497C2×C4—102no—
498C2×C4—104no—
499C2×C4—104no—
500C2×C4—104no—
501C2×C4—102no—
502C2×C4—101no—
503C2×C4—102no—
504C2×C4—101no—
505C2×C4—102no—
506C2×C4—101no—
507C2×C4—158no—
508C2×C4—201no—
509C2×C4—202no—
510C2×C4—201no—
511C2×C4—304no—
512C2×C4—304no—
513C2×C4—304no—
514C2×C4—304no—
515C2×C4—304no—
516C2×C4—304no—
517C2×C4—304no—
518C2×C4—304no—
519C2×C4—304no—
520C2×C4—602no—
521C2×C4—602no—
522C2×C4—602no—
523C2×C4—602no—
524C2×C4C4×ΓL(1,𝔽16)12yescomplemented
525C2×C4#676312yes—
526C2×C3—52no—
527C2×C3—52no—
528C2×C3—52no—
529C2×C3—52no—
530C2×C3C4⋊C4×AGL(1,𝔽5)12yes—
531C2×C3#6571112yes—
532C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
533C2×C3#6577911yescharacteristic
534C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
535C5#7156411yescharacteristic, complemented, Sylow
536C22—52no—
537C22—51no—
538C22—51no—
539C22—52no—
540C22—52no—
541C22—52no—
542C22—52no—
543C22—52no—
544C22—52no—
545C22—52no—
546C22—52no—
547C22—52no—
548C22—52no—
549C22—51no—
550C22—52no—
551C22—51no—
552C22C2×C4×ΓL(1,𝔽16)11yescharacteristic
553C22#9198911yescharacteristic
554C22D4×ΓL(1,𝔽16)12yes—
555C22Q8×ΓL(1,𝔽16)12yes—
556C22C2×#676311yescharacteristic
557C4—22no—
558C4—22no—
559C4—41no—
560C4—102no—
561C4—102no—
562C4—201no—
563C4—308no—
564C4—604no—
565C3#1060434311yescharacteristic, complemented, Sylow
566C2—52no—
567C2—52no—
568C2—52no—
569C2—52no—
570C2C4⋊C4×ΓL(1,𝔽16)12yes—
571C2#1175237012yes—
572C2C4×#676311yescharacteristic
573C2#1175243811yescharacteristic
574C2#1175244911yescharacteristic
subgroupG/H
center, FrattiniC23C22×ΓL(1,𝔽16)
commutatorC22×C3×C5C2×C42
FittingC2×C3×C5×C4⋊C4C4
socleC23×C3×C5C22×C4
prime radical, cosocleC23×C3×D5C23
hypercenterC2×C4⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#1436
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
14. C2×C3×C5×C4⋊C4C4semidirect
15. C2×C3×C4⋊Dic5C4semidirect
16. C2×C3×#683C4semidirect
17. C2×C3×#683C4semidirect
22. #91235C4semidirect
79. C2×C5×C4⋊C4Dic3semidirect
80. C2×C4⋊Dic5Dic3semidirect
81. C2×#683Dic3semidirect
82. C2×#683Dic3semidirect
111. C2×ΓL(1,𝔽16)C4⋊C4semidirect
117. C2×C3×C4×C5C42semidirect
124. C2×C3×Dic5C42semidirect
144. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
288. C2×C4×C5C4×Dic3semidirect
295. C2×Dic5C4×Dic3semidirect
315. C2×C4⋊C4ΓL(1,𝔽16)semidirect
359. C2×C3×C4C4×AGL(1,𝔽5)semidirect
442. C3×C5#1436semidirect
524. C2×C4C4×ΓL(1,𝔽16)semidirect
535. C5#71564semidirect
565. C3#10604343semidirect

Maximal subgroups

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

subgroupindex
C22×C4×ΓL(1,𝔽16)2
C2×C3×D5×C4⋊C42
C2×#912352
#106043433
#715645