Finite Group Databasebeta

← #14331004

Subgroups — #14331004

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×C22⋊Q8C211yesmaximal, characteristic
2#11752449C212yesmaximal, complemented
3#11752424C211yesmaximal, characteristic
4#11752423C211yesmaximal, characteristic
5#11752371C212yesmaximal, complemented
6#11752370C212yesmaximal, complemented
7#11752370C212yesmaximal, complemented
8#11752359C212yesmaximal
9#11752359C212yesmaximal
10#10617288—31nomaximal, Hall
11C2×C3×Q8×D5C2211yescharacteristic
12C22×C3×C4×D5C2211yescharacteristic
13C2×#6763C2212yes—
14C2×#6763C2211yescharacteristic
15C2×#6763C2211yescharacteristic
16C2×C4⋊ΓL(1,𝔽16)C2212yes—
17C2×C4⋊ΓL(1,𝔽16)C2212yes—
18C2×C4⋊ΓL(1,𝔽16)C2212yes—
19C2×C4×ΓL(1,𝔽16)C2212yes—
20C2×C4×ΓL(1,𝔽16)C2212yes—
21C3×C5×C22⋊Q8C411yescharacteristic, complemented
22C3×#4092C411yescharacteristic, complemented
23C3×#4070C411yescharacteristic, complemented
24C3×#4043C411yescharacteristic, complemented
25C3×#4042C412yescomplemented
26C3×D5×C4⋊C4C2211yescharacteristic
27C3×D5×C4⋊C4C2212yes—
28C3×D5×C22⋊C4C2212yes—
29C3×#4024C412yescomplemented
30#91235C2212yes—
31#91235C2212yes—
32#91235C2212yes—
33#91235C2212yes—
34#91235C2212yes—
35#91235C2212yes—
36#91235C2211yescharacteristic
37#91235C2211yescharacteristic
38#82750—51nomaximal, Hall
39D5×C22⋊Q8D311yescharacteristic
40(C2×AGL(1,𝔽5))⋊Q8—32no—
41#65765—31no—
42#65764—31no—
43#65712—32no—
44#65711—32no—
45#65711—32no—
46#65700—32no—
47#65700—32no—
48C23×C3×D5C2311yescharacteristic
49C22×ΓL(1,𝔽16)C2312yes—
50C22×ΓL(1,𝔽16)C2312yes—
51C22×ΓL(1,𝔽16)C2312yes—
52C22×ΓL(1,𝔽16)C2312yes—
53C2×C3×C5×Q8C2×C411yescharacteristic
54C22×C3×C4×C5C2×C411yescharacteristic
55C22×C3×Dic5C2×C411yescharacteristic
56C3×Q8×D5—42no—
57C2×C3×C4×D5—42no—
58C2×C3×C4×D5C2×C411yescharacteristic
59C2×C3×C4×D5C2×C411yescharacteristic
60C2×C3×C4×D5C2312yes—
61C2×C3×C4×D5C2312yes—
62C2×C3×C4×D5C2311yescharacteristic
63C2×C3×C4×D5C2311yescharacteristic
64C2×C3×Dic10C2×C412yes—
65C2×C3×Dic10C2×C411yescharacteristic
66#6763—41no—
67#6763—42no—
68#6763—41no—
69C4⋊ΓL(1,𝔽16)—42no—
70C4⋊ΓL(1,𝔽16)—42no—
71C4⋊ΓL(1,𝔽16)—42no—
72C4×ΓL(1,𝔽16)—42no—
73C4×ΓL(1,𝔽16)—42no—
74C3×C5×C4⋊C4C2×C412yes—
75C3×C5×C4⋊C4C2×C411yescharacteristic
76C3×C5×C22⋊C4C2×C412yes—
77C3×C22⋊Dic5C2×C412yes—
78C3×#685C2×C412yes—
79C3×#685C2×C412yes—
80C3×C4⋊Dic5C2×C411yescharacteristic
81C3×C4⋊Dic5C2×C412yes—
82C3×#683C2×C412yes—
83C3×#683C2×C412yes—
84C3×#683C2×C412yes—
85C2×C3×C22⋊Q8—51no—
86#5229—52no—
87#5212—51no—
88#5211—51no—
89#4984—52no—
90#4980—52no—
91#4980—52no—
92#4951—52no—
93#4951—52no—
94C2×Q8×D5C2×D311yescharacteristic
95C22×C4×D5C2×D311yescharacteristic
96C2×C22⋊AGL(1,𝔽5)—32no—
97C2×C22⋊AGL(1,𝔽5)—31no—
98C2×C22⋊AGL(1,𝔽5)—31no—
99C2×C4⋊AGL(1,𝔽5)—32no—
100C2×C4⋊AGL(1,𝔽5)—32no—
101C2×C4⋊AGL(1,𝔽5)—32no—
102C2×C4×AGL(1,𝔽5)—32no—
103C2×C4×AGL(1,𝔽5)—32no—
104C5×C22⋊Q8Dic311yescharacteristic, complemented
105#4092Dic311yescharacteristic, complemented
106#4070Dic311yescharacteristic, complemented
107#4043Dic311yescharacteristic, complemented
108#4042Dic312yescomplemented
109D5×C4⋊C4C2×D311yescharacteristic
110D5×C4⋊C4C2×D312yes—
111D5×C22⋊C4C2×D312yes—
112#4024Dic312yescomplemented
113(C2×AGL(1,𝔽5))⋊C4—32no—
114(C2×AGL(1,𝔽5))⋊C4—32no—
115(C2×AGL(1,𝔽5))⋊C4—32no—
116(C2×AGL(1,𝔽5))⋊C4—32no—
117(C2×AGL(1,𝔽5))⋊C4—31no—
118(C2×AGL(1,𝔽5))⋊C4—32no—
119(C2×AGL(1,𝔽5))⋊C4—32no—
120(C2×AGL(1,𝔽5))⋊C4—31no—
121#2378—151noSylow
122C23×C3×C5C22×C411yescharacteristic
123C22×C3×D5—21no—
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5—21no—
127C22×C3×D5C22×C411yescharacteristic
128C22×C3×D5C2411yescharacteristic
129C2×ΓL(1,𝔽16)—22no—
130C2×ΓL(1,𝔽16)—22no—
131C2×ΓL(1,𝔽16)—22no—
132C2×ΓL(1,𝔽16)—22no—
133C2×ΓL(1,𝔽16)—22no—
134C2×ΓL(1,𝔽16)—22no—
135C2×ΓL(1,𝔽16)—22no—
136C2×ΓL(1,𝔽16)—22no—
137C2×ΓL(1,𝔽16)—22no—
138C2×ΓL(1,𝔽16)—22no—
139C2×ΓL(1,𝔽16)—22no—
140C2×ΓL(1,𝔽16)—22no—
141C3×C5×Q8—41no—
142C2×C3×C4×C5—41no—
143C2×C3×C4×C5C22×C411yescharacteristic
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C22×C412yes—
146C2×C3×C4×C5C22×C411yescharacteristic
147C2×C3×Dic5—41no—
148C2×C3×Dic5C22×C412yes—
149C2×C3×Dic5C22×C411yescharacteristic
150C2×C3×Dic5C22×C411yescharacteristic
151C2×C3×Dic5C22×C412yes—
152C3×C4×D5—22no—
153C3×C4×D5—22no—
154C3×C4×D5—22no—
155C3×C4×D5—22no—
156C3×C4×D5—22no—
157C3×C4×D5—22no—
158C3×C4×D5—41no—
159C3×C4×D5—41no—
160C3×Dic10—42no—
161C3×Dic10—41no—
162C22×C3×Q8—51no—
163C23×C3×C4—51no—
164C3×C22⋊Q8—51no—
165C3×C22⋊Q8—51no—
166C3×C22⋊Q8—52no—
167C3×C22⋊Q8—51no—
168C3×C22⋊Q8—52no—
169C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
170C2×C3×C4⋊C4—51no—
171C2×C3×C4⋊C4—52no—
172C2×C3×C22⋊C4—52no—
173C2×#217—52no—
174C2×#217—51no—
175C2×#217—51no—
176C2×C4⋊Dic3—52no—
177C2×C4⋊Dic3—52no—
178C2×C4⋊Dic3—52no—
179C2×C4×Dic3—52no—
180C2×C4×Dic3—52no—
181C3⋊#95—52no—
182C3⋊#95—52no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—51no—
187C3⋊#95—51no—
188C3⋊#95—52no—
189C23×D5C22×D311yescharacteristic
190C22×AGL(1,𝔽5)—32no—
191C22×AGL(1,𝔽5)—32no—
192C22×AGL(1,𝔽5)—32no—
193C22×AGL(1,𝔽5)—32no—
194C2×C5×Q8C2×Dic311yescharacteristic
195C22×C4×C5C2×Dic311yescharacteristic
196C22×Dic5C2×Dic311yescharacteristic
197Q8×D5—42no—
198C2×C4×D5—42no—
199C2×C4×D5C2×Dic311yescharacteristic
200C2×C4×D5C2×Dic311yescharacteristic
201C2×C4×D5C22×D312yes—
202C2×C4×D5C22×D312yes—
203C2×C4×D5C22×D311yescharacteristic
204C2×C4×D5C22×D311yescharacteristic
205C2×Dic10C2×Dic312yes—
206C2×Dic10C2×Dic311yescharacteristic
207C22⋊AGL(1,𝔽5)—122no—
208C22⋊AGL(1,𝔽5)—121no—
209C22⋊AGL(1,𝔽5)—121no—
210C4⋊AGL(1,𝔽5)—122no—
211C4⋊AGL(1,𝔽5)—122no—
212C4⋊AGL(1,𝔽5)—122no—
213C4×AGL(1,𝔽5)—122no—
214C4×AGL(1,𝔽5)—122no—
215C5×C4⋊C4C2×Dic312yes—
216C5×C4⋊C4C2×Dic311yescharacteristic
217C5×C22⋊C4C2×Dic312yes—
218C22⋊Dic5C2×Dic312yes—
219#685C2×Dic312yes—
220#685C2×Dic312yes—
221C4⋊Dic5C2×Dic312yes—
222C4⋊Dic5C2×Dic311yescharacteristic
223#683C2×Dic312yes—
224#683C2×Dic312yes—
225#683C2×Dic312yes—
226C2×C22⋊Q8—51no—
227#391—152no—
228#389—152no—
229#388—152no—
230#388—152no—
231#387—152no—
232#387—152no—
233#381—151no—
234#380—151no—
235C22×C3×C5—21no—
236C22×C3×C5—21no—
237C22×C3×C5—21no—
238C22×C3×C5C23×C411yescharacteristic
239C2×C3×D5—21no—
240C2×C3×D5—21no—
241C2×C3×D5—21no—
242C2×C3×D5—42no—
243C2×C3×D5—42no—
244C2×C3×D5H(5,𝔽2)12yes—
245C2×C3×D5H(5,𝔽2)11yescharacteristic
246C2×C3×D52−1+412yes—
247C2×C3×D52−1+411yescharacteristic
248ΓL(1,𝔽16)—42no—
249ΓL(1,𝔽16)—42no—
250ΓL(1,𝔽16)—42no—
251ΓL(1,𝔽16)—42no—
252C3×C4×C5—21no—
253C3×C4×C5—22no—
254C3×C4×C5—21no—
255C3×C4×C5—22no—
256C3×Dic5—21no—
257C3×Dic5—22no—
258C3×Dic5—21no—
259C3×Dic5—22no—
260C24×C3—51no—
261C2×C3×Q8—51no—
262C2×C3×Q8—52no—
263C2×C3×Q8—202no—
264C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
265C22×C3×C4—51no—
266C22×C3×C4—51no—
267C22×C3×C4—52no—
268C22×C3×C4—52no—
269C22×C3×C4—51no—
270C22×C3×C4—51no—
271C22×C3×C4—51no—
272C22×C3×C4—202no—
273C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
274C22×Dic3—52no—
275C22×Dic3—52no—
276C22×Dic3—52no—
277C22×Dic3—52no—
278C3×C4⋊C4—52no—
279C3×C4⋊C4—52no—
280C3×C4⋊C4—52no—
281C3×C4⋊C4—51no—
282C3×C4⋊C4—52no—
283C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
284C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
285C3×C22⋊C4—52no—
286C3×C22⋊C4—52no—
287C3×C22⋊C4—52no—
288C3×C22⋊C4C2×AGL(1,𝔽5)12yes—
289#217—201no—
290#217—202no—
291#217—201no—
292C4⋊Dic3—202no—
293C4⋊Dic3—202no—
294C4⋊Dic3—202no—
295C4×Dic3—202no—
296C4×Dic3—202no—
297C23×C5C22×Dic311yescharacteristic
298C22×D5—22no—
299C22×D5—22no—
300C22×D5—21no—
301C22×D5—21no—
302C22×D5C22×Dic311yescharacteristic
303C22×D5C23×D311yescharacteristic
304C2×AGL(1,𝔽5)—62no—
305C2×AGL(1,𝔽5)—62no—
306C2×AGL(1,𝔽5)—62no—
307C2×AGL(1,𝔽5)—62no—
308C2×AGL(1,𝔽5)—62no—
309C2×AGL(1,𝔽5)—62no—
310C2×AGL(1,𝔽5)—62no—
311C2×AGL(1,𝔽5)—62no—
312C2×AGL(1,𝔽5)—62no—
313C2×AGL(1,𝔽5)—62no—
314C2×AGL(1,𝔽5)—62no—
315C2×AGL(1,𝔽5)—62no—
316C5×Q8—41no—
317C2×C4×C5—41no—
318C2×C4×C5C22×Dic311yescharacteristic
319C2×C4×C5C22×Dic312yes—
320C2×C4×C5C22×Dic312yes—
321C2×C4×C5C22×Dic311yescharacteristic
322C2×Dic5—41no—
323C2×Dic5C22×Dic312yes—
324C2×Dic5C22×Dic311yescharacteristic
325C2×Dic5C22×Dic311yescharacteristic
326C2×Dic5C22×Dic312yes—
327C4×D5—22no—
328C4×D5—22no—
329C4×D5—22no—
330C4×D5—22no—
331C4×D5—22no—
332C4×D5—22no—
333C4×D5—41no—
334C4×D5—41no—
335Dic10—42no—
336Dic10—41no—
337C22×Q8—51no—
338C23×C4—51no—
339C22⋊Q8—51no—
340C22⋊Q8—51no—
341C22⋊Q8—52no—
342C22⋊Q8—51no—
343C22⋊Q8—52no—
344C22⋊Q8ΓL(1,𝔽16)11yescharacteristic, complemented
345C2×C4⋊C4—51no—
346C2×C4⋊C4—52no—
347C2×C4⋊C4—152no—
348C2×C4⋊C4—152no—
349C2×C4⋊C4—152no—
350C2×C22⋊C4—52no—
351C2×C22⋊C4—151no—
352C2×C22⋊C4—151no—
353C2×C22⋊C4—152no—
354C2×C42—152no—
355C2×C42—152no—
356#95—152no—
357#95—151no—
358#95—152no—
359#95—151no—
360#95—152no—
361#95—152no—
362#95—152no—
363#95—152no—
364C2×C3×C5—41no—
365C2×C3×C5#51811yescharacteristic
366C2×C3×C5#51911yescharacteristic
367C2×C3×C5#52011yescharacteristic
368C3×D5—41no—
369C3×D5#56212yes—
370C3×D5#56312yes—
371C23×C3—51no—
372C23×C3—51no—
373C23×C3—101no—
374C23×C3—102no—
375C23×C3—102no—
376C23×C3—101no—
377C23×C3C22×AGL(1,𝔽5)11yescharacteristic
378C3×Q8—41no—
379C3×Q8—201no—
380C3×Q8—202no—
381C2×C3×C4—41no—
382C2×C3×C4—52no—
383C2×C3×C4—52no—
384C2×C3×C4—51no—
385C2×C3×C4—51no—
386C2×C3×C4—102no—
387C2×C3×C4—102no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—102no—
392C2×C3×C4—201no—
393C2×C3×C4—201no—
394C2×C3×C4—201no—
395C2×C3×C4C22×AGL(1,𝔽5)12yes—
396C2×C3×C4C22×AGL(1,𝔽5)12yes—
397C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
398C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
399C2×Dic3—102no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—102no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C2×Dic3—102no—
406C2×Dic3—102no—
407C2×Dic3—102no—
408C2×Dic3—102no—
409C2×Dic3—102no—
410C2×Dic3—102no—
411C22×C5—21no—
412C22×C5—21no—
413C22×C5—21no—
414C22×C5C23×Dic311yescharacteristic
415C2×D5—21no—
416C2×D5—21no—
417C2×D5—21no—
418C2×D5—42no—
419C2×D5—42no—
420C2×D5#100412yes—
421C2×D5#100712yes—
422C2×D5#100911yescharacteristic
423C2×D5#101011yescharacteristic
424AGL(1,𝔽5)—122no—
425AGL(1,𝔽5)—122no—
426AGL(1,𝔽5)—122no—
427AGL(1,𝔽5)—122no—
428C4×C5—21no—
429C4×C5—22no—
430C4×C5—21no—
431C4×C5—22no—
432Dic5—21no—
433Dic5—22no—
434Dic5—21no—
435Dic5—22no—
436C24—51no—
437C2×Q8—51no—
438C2×Q8—52no—
439C2×Q8—202no—
440C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
441C22×C4—51no—
442C22×C4—51no—
443C22×C4—52no—
444C22×C4—52no—
445C22×C4—51no—
446C22×C4—51no—
447C22×C4—51no—
448C22×C4—152no—
449C22×C4—152no—
450C22×C4—152no—
451C22×C4—152no—
452C22×C4—202no—
453C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
454C4⋊C4—52no—
455C4⋊C4—52no—
456C4⋊C4—52no—
457C4⋊C4—51no—
458C4⋊C4—52no—
459C4⋊C4—602no—
460C4⋊C4—602no—
461C4⋊C4—602no—
462C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
463C4⋊C4C2×ΓL(1,𝔽16)12yes—
464C22⋊C4—52no—
465C22⋊C4—52no—
466C22⋊C4—52no—
467C22⋊C4—601no—
468C22⋊C4—602no—
469C22⋊C4—601no—
470C22⋊C4C2×ΓL(1,𝔽16)12yes—
471C42—602no—
472C42—602no—
473C3×C5#237811yescharacteristic, complemented, Hall
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—52no—
478C22×C3—52no—
479C22×C3—51no—
480C22×C3—51no—
481C22×C3—101no—
482C22×C3—101no—
483C22×C3—101no—
484C22×C3—202no—
485C22×C3—202no—
486C22×C3C23×AGL(1,𝔽5)11yescharacteristic
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—101no—
492C3×C4—101no—
493C3×C4—102no—
494C3×C4—102no—
495Dic3—202no—
496Dic3—202no—
497Dic3—202no—
498Dic3—202no—
499C2×C5—41no—
500C2×C5#579811yescharacteristic
501C2×C5#581211yescharacteristic
502C2×C5#582711yescharacteristic
503D5—41no—
504D5#564612yes—
505D5#564712yes—
506C23—51no—
507C23—51no—
508C23—101no—
509C23—102no—
510C23—102no—
511C23—101no—
512C23C22×ΓL(1,𝔽16)11yescharacteristic
513Q8—41no—
514Q8—201no—
515Q8—202no—
516C2×C4—41no—
517C2×C4—52no—
518C2×C4—52no—
519C2×C4—51no—
520C2×C4—51no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—302no—
531C2×C4—302no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—302no—
535C2×C4—302no—
536C2×C4—302no—
537C2×C4—302no—
538C2×C4—302no—
539C2×C4—302no—
540C2×C4—302no—
541C2×C4—302no—
542C2×C4C22×ΓL(1,𝔽16)12yes—
543C2×C4C22×ΓL(1,𝔽16)12yes—
544C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
545C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
546C2×C3—41no—
547C2×C3—52no—
548C2×C3—52no—
549C2×C3—201no—
550C2×C3#6625811yescharacteristic
551C2×C3#6626211yescharacteristic
552C2×C3#6626611yescharacteristic
553C5#8275011yescharacteristic, complemented, Sylow
554C22—21no—
555C22—21no—
556C22—21no—
557C22—52no—
558C22—52no—
559C22—51no—
560C22—51no—
561C22—101no—
562C22—101no—
563C22—101no—
564C22—202no—
565C22—202no—
566C22C23×ΓL(1,𝔽16)11yescharacteristic
567C4—21no—
568C4—21no—
569C4—22no—
570C4—22no—
571C4—101no—
572C4—101no—
573C4—102no—
574C4—102no—
575C4—602no—
576C4—602no—
577C4—602no—
578C4—602no—
579C3#1061728811yescharacteristic, complemented, Sylow
580C2—41no—
581C2—52no—
582C2—52no—
583C2—201no—
584C2#1175698711yescharacteristic
585C2#1175699111yescharacteristic
586C2#1175699511yescharacteristic
subgroupG/H
center, FrattiniC22C23×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×C22⋊Q8C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC22⋊Q8ΓL(1,𝔽16)
hypocenterC3×C5#2378
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11752449C2semidirect
5. #11752371C2semidirect
6. #11752370C2semidirect
7. #11752370C2semidirect
21. C3×C5×C22⋊Q8C4semidirect
22. C3×#4092C4semidirect
23. C3×#4070C4semidirect
24. C3×#4043C4semidirect
25. C3×#4042C4semidirect
29. C3×#4024C4semidirect
104. C5×C22⋊Q8Dic3semidirect
105. #4092Dic3semidirect
106. #4070Dic3semidirect
107. #4043Dic3semidirect
108. #4042Dic3semidirect
112. #4024Dic3semidirect
169. C3×C22⋊Q8AGL(1,𝔽5)semidirect
344. C22⋊Q8ΓL(1,𝔽16)semidirect
473. C3×C5#2378semidirect
553. C5#82750semidirect
579. C3#10617288semidirect

Maximal subgroups

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

subgroupindex
C3×D5×C22⋊Q82
#117524492
#117524242
#117524232
#117523712
#117523702
#117523592
#106172883
#827505