Finite Group Databasebeta

← #14326664

Subgroups — #14326664

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×#91923C212yesmaximal
2C2×D5×C4⋊Dic3C211yesmaximal, characteristic
3#11752424C211yesmaximal, characteristic
4C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
5#11751984C212yesmaximal
6#11751859C212yesmaximal, complemented
7#11751839C214yesmaximal, complemented
8AGL(1,𝔽5)×C4⋊Dic3C212yesmaximal, complemented
9#10617103—31nomaximal, Hall
10C22×C3×C4×D5C2211yescharacteristic
11C22×D5×Dic3C2212yes—
12C2×#6763C2212yes—
13C2×C4⋊ΓL(1,𝔽16)C2212yes—
14C2×C3×C22⋊AGL(1,𝔽5)C2211yescharacteristic
15C2×C3×C22⋊AGL(1,𝔽5)C2211yescharacteristic
16C2×C3×C4×AGL(1,𝔽5)C2212yes—
17C2×Dic3⋊AGL(1,𝔽5)C2214yes—
18#91923—28no—
19C2×Dic3×AGL(1,𝔽5)C2214yes—
20C2×C4⋊Dic15C411yescharacteristic, complemented
21C2×C5×C4⋊Dic3C411yescharacteristic, complemented
22C2×#6668C412yescomplemented
23D5×C4⋊Dic3C411yescharacteristic, complemented
24D5×C4⋊Dic3C411yescharacteristic, complemented
25D5×C4⋊Dic3C412yescomplemented
26D5×C4⋊Dic3C2211yescharacteristic
27D5×C4⋊Dic3C2212yes—
28D5×C4⋊Dic3C2211yescharacteristic
29#91235C2212yes—
30C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
31(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
32(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
33#73456—51nomaximal, Hall
34C2×#4130—32no—
35C2×D5×C4⋊C4—31no—
36#65765—31no—
37#65764—32no—
38C4×C22⋊AGL(1,𝔽5)D311yescharacteristic
39#65712—32no—
40#65711—34no—
41C4⋊C4×AGL(1,𝔽5)—32no—
42C23×C3×D5C2311yescharacteristic
43C22×ΓL(1,𝔽16)C2314yes—
44C22×C3×AGL(1,𝔽5)C2312yes—
45C22×C3×AGL(1,𝔽5)C2312yes—
46C22×C3×C4×C5C2×C411yescharacteristic
47C22×Dic15C2×C412yes—
48C22×C5×Dic3C2×C412yes—
49C22×C3×Dic5C2×C411yescharacteristic
50C2×C3×C4×D5—24no—
51C2×C3×C4×D5C2×C411yescharacteristic
52C2×C3×C4×D5C2×C411yescharacteristic
53C2×C3×C4×D5C2311yescharacteristic
54C2×C3×C4×D5C2311yescharacteristic
55C2×D5×Dic3—24no—
56C2×D5×Dic3—24no—
57C2×D5×Dic3C2×C412yes—
58C2×D5×Dic3C2×C412yes—
59C2×D5×Dic3C2312yes—
60C2×D5×Dic3C2312yes—
61#6763—24no—
62C4⋊ΓL(1,𝔽16)—24no—
63C3×C22⋊AGL(1,𝔽5)—22no—
64C3×C22⋊AGL(1,𝔽5)D414yes—
65C3×C4×AGL(1,𝔽5)—24no—
66Dic3⋊AGL(1,𝔽5)—28no—
67Dic3×AGL(1,𝔽5)—28no—
68C4⋊Dic15C2×C412yes—
69C4⋊Dic15C2×C411yescharacteristic
70C4⋊Dic15C2×C411yescharacteristic
71C5×C4⋊Dic3C2×C412yes—
72C5×C4⋊Dic3C2×C411yescharacteristic
73C5×C4⋊Dic3C2×C411yescharacteristic
74#6668C2×C412yes—
75#6668C2×C412yes—
76#6668C2×C412yes—
77#6668C2×C412yes—
78C22×C4⋊Dic3—51no—
79C2×#877—52no—
80C3×C4×C22⋊C4—51no—
81#5212—51no—
82#4943—52no—
83C4×C4⋊Dic3—52no—
84#4928—52no—
85#4653—54no—
86C22×C4×D5—32no—
87C22×C4×D5C2×D311yescharacteristic
88C2×C22⋊AGL(1,𝔽5)—32no—
89C2×C22⋊AGL(1,𝔽5)C2×D311yescharacteristic
90C2×C22⋊AGL(1,𝔽5)C2×D311yescharacteristic
91#4130—68no—
92C2×C4⋊AGL(1,𝔽5)—34no—
93C2×C4⋊AGL(1,𝔽5)—32no—
94C2×C4×AGL(1,𝔽5)—34no—
95C2×C4×AGL(1,𝔽5)C2×D312yes—
96C2×C5×C4⋊C4—31no—
97C2×C4⋊Dic5—31no—
98C2×#683—32no—
99D5×C4⋊C4—31no—
100D5×C4⋊C4—31no—
101D5×C4⋊C4—32no—
102D5×C4⋊C4—31no—
103D5×C4⋊C4—32no—
104D5×C4⋊C4—31no—
105(C2×AGL(1,𝔽5))⋊C4—34no—
106(C2×AGL(1,𝔽5))⋊C4—34no—
107(C2×AGL(1,𝔽5))⋊C4—32no—
108(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
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)—24no—
118C2×ΓL(1,𝔽16)—24no—
119C2×ΓL(1,𝔽16)—24no—
120C2×C3×AGL(1,𝔽5)—22no—
121C2×C3×AGL(1,𝔽5)—22no—
122C2×C3×AGL(1,𝔽5)—22no—
123C2×C3×AGL(1,𝔽5)—22no—
124C2×C3×AGL(1,𝔽5)—22no—
125C2×C3×AGL(1,𝔽5)C2×D414yes—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5C22×C411yescharacteristic
128C2×C3×C4×C5C22×C411yescharacteristic
129C2×Dic15—24no—
130C2×Dic15C22×C412yes—
131C2×Dic15C22×C412yes—
132C2×C5×Dic3—24no—
133C2×C5×Dic3C22×C412yes—
134C2×C5×Dic3C22×C412yes—
135C2×C3×Dic5—22no—
136C2×C3×Dic5C22×C411yescharacteristic
137C2×C3×Dic5C22×C411yescharacteristic
138C3×C4×D5—22no—
139C3×C4×D5—22no—
140C3×C4×D5—22no—
141C3×C4×D5#4114yes—
142D5×Dic3—24no—
143D5×Dic3—24no—
144D5×Dic3—22no—
145D5×Dic3—22no—
146D5×Dic3—22no—
147D5×Dic3—22no—
148C23×C3×C4—51no—
149C23×Dic3—52no—
150C2×C3×C22⋊C4—51no—
151C2×C3×C22⋊C4—51no—
152C2×C3×C42—52no—
153C2×#217—52no—
154C2×C4⋊Dic3—51no—
155C2×C4⋊Dic3—52no—
156C2×C4⋊Dic3—52no—
157C2×C4⋊Dic3—51no—
158C2×C4⋊Dic3—51no—
159C2×C4⋊Dic3—52no—
160C2×C4⋊Dic3—51no—
161C2×C4⋊Dic3—51no—
162C2×C4⋊Dic3—52no—
163C2×C4⋊Dic3AGL(1,𝔽5)11yescharacteristic, complemented
164C2×Dic3⋊C4—54no—
165C2×C4×Dic3—54no—
166#877—108no—
167C3×#95—52no—
168C3⋊#95—54no—
169C3⋊#95—54no—
170C3⋊#95—52no—
171C23×D5C22×D311yescharacteristic
172C22×AGL(1,𝔽5)—34no—
173C22×AGL(1,𝔽5)C22×D312yes—
174C22×AGL(1,𝔽5)C22×D312yes—
175C22×C4×C5—32no—
176C22×C4×C5C4×D311yescharacteristic
177C22×Dic5—32no—
178C22×Dic5C4×D311yescharacteristic
179C2×C4×D5—24no—
180C2×C4×D5—32no—
181C2×C4×D5—32no—
182C2×C4×D5—32no—
183C2×C4×D5—32no—
184C2×C4×D5—64no—
185C2×C4×D5—64no—
186C2×C4×D5C4×D311yescharacteristic
187C2×C4×D5C4×D311yescharacteristic
188C2×C4×D5C22×D311yescharacteristic
189C2×C4×D5C22×D311yescharacteristic
190C22⋊AGL(1,𝔽5)—22no—
191C22⋊AGL(1,𝔽5)—64no—
192C22⋊AGL(1,𝔽5)D1214yes—
193C4⋊AGL(1,𝔽5)—68no—
194C4⋊AGL(1,𝔽5)—64no—
195C4×AGL(1,𝔽5)—24no—
196C4×AGL(1,𝔽5)—68no—
197C5×C4⋊C4—32no—
198C5×C4⋊C4—31no—
199C5×C4⋊C4—31no—
200C4⋊Dic5—32no—
201C4⋊Dic5—31no—
202C4⋊Dic5—31no—
203#683—32no—
204#683—32no—
205#683—32no—
206#683—32no—
207C2×#117—152no—
208C22×C4⋊C4—151no—
209#389—152no—
210#387—154no—
211#381—152no—
212#380—151no—
213C4×C4⋊C4—152no—
214C4×C22⋊C4—51no—
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×#4111yescharacteristic
226C2×C3×D5C2×#4112yes—
227C2×C3×D52−1+412yes—
228ΓL(1,𝔽16)—44no—
229C3×AGL(1,𝔽5)—24no—
230C3×AGL(1,𝔽5)—42no—
231C3×C4×C5—21no—
232C3×C4×C5#11712yescomplemented
233Dic15—22no—
234Dic15—22no—
235C3×Dic5—21no—
236C3×Dic5#11712yescomplemented
237C5×Dic3—22no—
238C5×Dic3—22no—
239C24×C3—51no—
240C22×C3×C4—51no—
241C22×C3×C4—52no—
242C22×C3×C4—51no—
243C22×C3×C4—51no—
244C22×C3×C4—51no—
245C22×C3×C4—51no—
246C22×C3×C4—52no—
247C22×C3×C4—104no—
248C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
249C22×Dic3—52no—
250C22×Dic3—52no—
251C22×Dic3—52no—
252C22×Dic3—52no—
253C22×Dic3—52no—
254C22×Dic3—54no—
255C22×Dic3—104no—
256C22×Dic3—104no—
257C22×Dic3C2×AGL(1,𝔽5)12yes—
258C3×C22⋊C4—54no—
259C3×C22⋊C4—102no—
260C3×C42—104no—
261#217—104no—
262C4⋊Dic3—52no—
263C4⋊Dic3—51no—
264C4⋊Dic3—51no—
265C4⋊Dic3—52no—
266C4⋊Dic3—52no—
267C4⋊Dic3—52no—
268C4⋊Dic3—52no—
269C4⋊Dic3—104no—
270C4⋊Dic3C2×AGL(1,𝔽5)12yes—
271C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
272C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
273Dic3⋊C4—108no—
274C4×Dic3—108no—
275C23×C5C2×C4×D311yescharacteristic
276C22×D5—22no—
277C22×D5—22no—
278C22×D5C2×C4×D311yescharacteristic
279C22×D5C2×D1212yes—
280C22×D5#23512yes—
281C22×D5C23×D311yescharacteristic
282C2×AGL(1,𝔽5)—22no—
283C2×AGL(1,𝔽5)—22no—
284C2×AGL(1,𝔽5)—22no—
285C2×AGL(1,𝔽5)—22no—
286C2×AGL(1,𝔽5)—22no—
287C2×AGL(1,𝔽5)—64no—
288C2×AGL(1,𝔽5)—64no—
289C2×AGL(1,𝔽5)—64no—
290C2×AGL(1,𝔽5)C2×D1214yes—
291C2×C4×C5—22no—
292C2×C4×C5—32no—
293C2×C4×C5—32no—
294C2×C4×C5—64no—
295C2×C4×C5C2×C4×D311yescharacteristic
296C2×C4×C5C2×C4×D311yescharacteristic
297C2×Dic5—22no—
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5—64no—
301C2×Dic5C2×C4×D311yescharacteristic
302C2×Dic5C2×C4×D311yescharacteristic
303C4×D5—22no—
304C4×D5—22no—
305C4×D5—22no—
306C4×D5—64no—
307C4×D5—64no—
308C4×D5—62no—
309C4×D5—62no—
310C4×D5—62no—
311C4×D5—62no—
312C4×D5#23714yes—
313C23×C4—51no—
314C23×C4—152no—
315#117—308no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—151no—
318C2×C4⋊C4—152no—
319C2×C4⋊C4—151no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4—151no—
323C2×C4⋊C4—152no—
324C2×C4⋊C4—151no—
325C2×C4⋊C4—154no—
326C2×C4⋊C4—152no—
327C2×C22⋊C4—51no—
328C2×C22⋊C4—51no—
329C2×C22⋊C4—152no—
330C2×C42—52no—
331C2×C42—154no—
332#95—52no—
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—22no—
351C2×C3×C4—51no—
352C2×C3×C4—54no—
353C2×C3×C4—54no—
354C2×C3×C4—51no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
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—102no—
364C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
365C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
366C2×Dic3—24no—
367C2×Dic3—52no—
368C2×Dic3—52no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3—102no—
372C2×Dic3—102no—
373C2×Dic3—104no—
374C2×Dic3—104no—
375C2×Dic3—104no—
376C2×Dic3—104no—
377C2×Dic3—104no—
378C2×Dic3—104no—
379C2×Dic3C22×AGL(1,𝔽5)12yes—
380C2×Dic3C22×AGL(1,𝔽5)12yes—
381C22×C5—21no—
382C22×C5—21no—
383C22×C5C4×D1212yes—
384C22×C5C22×C4×D311yescharacteristic
385C2×D5—21no—
386C2×D5—21no—
387C2×D5—24no—
388C2×D5—24no—
389C2×D5C4×D1212yes—
390C2×D5C22×D1211yescharacteristic
391C2×D5C2×#23511yescharacteristic
392C2×D5C2×#23712yes—
393C2×D5#101012yes—
394AGL(1,𝔽5)—24no—
395AGL(1,𝔽5)—42no—
396AGL(1,𝔽5)—124no—
397C4×C5—21no—
398C4×C5—62no—
399C4×C5—62no—
400C4×C5#87712yescomplemented
401Dic5—21no—
402Dic5—62no—
403Dic5—62no—
404Dic5#87712yescomplemented
405C24—51no—
406C22×C4—32no—
407C22×C4—51no—
408C22×C4—52no—
409C22×C4—51no—
410C22×C4—51no—
411C22×C4—51no—
412C22×C4—51no—
413C22×C4—52no—
414C22×C4—104no—
415C22×C4—152no—
416C22×C4—152no—
417C22×C4—152no—
418C22×C4—152no—
419C22×C4—152no—
420C22×C4—154no—
421C22×C4—304no—
422C22×C4—304no—
423C22×C4D3×AGL(1,𝔽5)11yescharacteristic
424C4⋊C4—32no—
425C4⋊C4—31no—
426C4⋊C4—31no—
427C4⋊C4—152no—
428C4⋊C4—152no—
429C4⋊C4—152no—
430C4⋊C4—152no—
431C4⋊C4—152no—
432C4⋊C4—151no—
433C4⋊C4—151no—
434C4⋊C4—308no—
435C4⋊C4—304no—
436C22⋊C4—54no—
437C22⋊C4—102no—
438C22⋊C4—304no—
439C42—104no—
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—52no—
457C3×C4—104no—
458C3×C4—101no—
459C3×C4—202no—
460C3×C4#413012yescomplemented
461Dic3—22no—
462Dic3—22no—
463Dic3—102no—
464Dic3—102no—
465Dic3—204no—
466C2×C5—22no—
467C2×C5C2×C4×D1211yescharacteristic
468C2×C5C2×#87711yescharacteristic
469C2×C5#551611yescharacteristic
470D5—22no—
471D5#554112yes—
472D5#555512yes—
473C23—51no—
474C23—52no—
475C23—52no—
476C23—51no—
477C23—102no—
478C23—102no—
479C23C2×D3×AGL(1,𝔽5)11yescharacteristic
480C2×C4—22no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—51no—
484C2×C4—54no—
485C2×C4—54no—
486C2×C4—51no—
487C2×C4—64no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4—102no—
492C2×C4—102no—
493C2×C4—102no—
494C2×C4—102no—
495C2×C4—102no—
496C2×C4—102no—
497C2×C4—152no—
498C2×C4—152no—
499C2×C4—302no—
500C2×C4—302no—
501C2×C4—302no—
502C2×C4—302no—
503C2×C4—304no—
504C2×C4—304no—
505C2×C4—304no—
506C2×C4—304no—
507C2×C4—304no—
508C2×C4—304no—
509C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
510C2×C4C2×D3×AGL(1,𝔽5)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#7345611yescharacteristic, complemented, Sylow
519C22—21no—
520C22—21no—
521C22—52no—
522C22—51no—
523C22—51no—
524C22—52no—
525C22—52no—
526C22—104no—
527C22—104no—
528C22—101no—
529C22—101no—
530C22AGL(1,𝔽5)×D1212yes—
531C22C22×D3×AGL(1,𝔽5)11yescharacteristic
532C4—21no—
533C4—52no—
534C4—62no—
535C4—62no—
536C4—104no—
537C4—101no—
538C4—202no—
539C4—302no—
540C4—302no—
541C4—604no—
542C4#9192312yescomplemented
543C3#1061710311yescharacteristic, complemented, Sylow
544C2—22no—
545C2—52no—
546C2—52no—
547C2—102no—
548C2#1175681211yescharacteristic
549C2C2×AGL(1,𝔽5)×D1211yescharacteristic
550C2C2×#9192311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2344
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11751859C2semidirect
7. #11751839C2semidirect
8. AGL(1,𝔽5)×C4⋊Dic3C2semidirect
20. C2×C4⋊Dic15C4semidirect
21. C2×C5×C4⋊Dic3C4semidirect
22. C2×#6668C4semidirect
23. D5×C4⋊Dic3C4semidirect
24. D5×C4⋊Dic3C4semidirect
25. D5×C4⋊Dic3C4semidirect
163. C2×C4⋊Dic3AGL(1,𝔽5)semidirect
232. C3×C4×C5#117semidirect
236. C3×Dic5#117semidirect
400. C4×C5#877semidirect
404. Dic5#877semidirect
441. C3×C5#2344semidirect
460. C3×C4#4130semidirect
518. C5#73456semidirect
542. C4#91923semidirect
543. C3#10617103semidirect

Maximal subgroups

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

subgroupindex
C2×#919232
C2×D5×C4⋊Dic32
#117524242
C3×C4×C22⋊AGL(1,𝔽5)2
#117519842
#117518592
#117518392
AGL(1,𝔽5)×C4⋊Dic32
#106171033
#734565