Finite Group Databasebeta

← #14300065

Subgroups — #14300065

Order 1920.

596 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×AGL(1,𝔽5)C211yesmaximal, characteristic
2C2×#91238C211yesmaximal, characteristic
3#11751859C211yesmaximal, characteristic
4C2×#91160C211yesmaximal, characteristic
5D5×#835C211yesmaximal, characteristic
6#11746528C211yesmaximal, characteristic
7(C2×AGL(1,𝔽5))⋊(C3⋊C8)C211yesmaximal, characteristic
8#10603989—31nomaximal, Hall
9C2×C3×Q8×D5C2211yescharacteristic
10C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
11C2×D5⋊(C3⋊C8)C2211yescharacteristic
12C3×Q8×AGL(1,𝔽5)—22no—
13C3×Q8×AGL(1,𝔽5)—22no—
14C2×C3×C4⋊AGL(1,𝔽5)—21no—
15C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
16C2×C3×C4×AGL(1,𝔽5)—21no—
17C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
18C2×Dic3⋊AGL(1,𝔽5)—21no—
19D5×C4⋊Dic3C2211yescharacteristic
20C2×D5×C3⋊C8C2211yescharacteristic
21#91238—22no—
22(C2×AGL(1,𝔽5))⋊Dic3—21no—
23#91160—22no—
24#91119C411yescharacteristic, complemented
25C5×#835C411yescharacteristic, complemented
26#90971C411yescharacteristic, complemented
27#90970C411yescharacteristic, complemented
28#71628—51nomaximal, Hall
29C2×Q8×AGL(1,𝔽5)D311yescharacteristic
30C2×#4009—31no—
31C2×#3994—31no—
32#65712—31no—
33D5×#103—31no—
34(C2×AGL(1,𝔽5))⋊C8—31no—
35#64874—31no—
36C22×ΓL(1,𝔽16)—21no—
37C22×C3×AGL(1,𝔽5)—21no—
38C22×C3×AGL(1,𝔽5)D411yescharacteristic
39C22×C3×AGL(1,𝔽5)D411yescharacteristic
40C2×C3×C5×Q8C2×C411yescharacteristic
41C3×Q8×D5—21no—
42C3×Q8×D5—21no—
43C3×Q8×D5D412yes—
44C3×Q8×D5Q812yes—
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5C2311yescharacteristic
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C411yescharacteristic
49C2×D5×Dic3—21no—
50C2×C3⋊#170—21no—
51C4⋊ΓL(1,𝔽16)—21no—
52C4⋊ΓL(1,𝔽16)—21no—
53D5⋊(C3⋊C8)—21no—
54D5⋊(C3⋊C8)—21no—
55C3×C4⋊AGL(1,𝔽5)—24no—
56C3×C4⋊AGL(1,𝔽5)—21no—
57C3×C4⋊AGL(1,𝔽5)D412yes—
58C3×C4×AGL(1,𝔽5)—24no—
59C3×C4×AGL(1,𝔽5)—21no—
60C3×C4×AGL(1,𝔽5)—21no—
61Dic3⋊AGL(1,𝔽5)—42no—
62C4⋊Dic15C2×C411yescharacteristic
63C2×C3⋊#168C2×C411yescharacteristic
64C5×C4⋊Dic3C2×C411yescharacteristic
65C2×C5×C3⋊C8C2×C411yescharacteristic
66#6668—21no—
67D5×C3⋊C8—21no—
68D5×C3⋊C8—21no—
69C2×C3×C4×Q8—51no—
70C2×#835—51no—
71C2×#835—51no—
72C2×#807—51no—
73#4928—51no—
74#4529—51no—
75#4524—51no—
76C2×Q8×D5C2×D311yescharacteristic
77Q8×AGL(1,𝔽5)—22no—
78Q8×AGL(1,𝔽5)—22no—
79C2×C4⋊AGL(1,𝔽5)—21no—
80C2×C4⋊AGL(1,𝔽5)—31no—
81C2×C4⋊AGL(1,𝔽5)—61no—
82C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
83C2×C4×AGL(1,𝔽5)—21no—
84C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
85C2×D5⋊C8—31no—
86C2×C8×D5—31no—
87D5×C4⋊C4—31no—
88#4009—62no—
89(C2×AGL(1,𝔽5))⋊C4—61no—
90#3994—62no—
91C5×#103—31no—
92#3967—31no—
93#3948—31no—
94#3942—31no—
95#1863—151noSylow
96C22×C3×D5C2×D411yescharacteristic
97C2×ΓL(1,𝔽16)—41no—
98C2×ΓL(1,𝔽16)—41no—
99C2×ΓL(1,𝔽16)—41no—
100C2×C3×AGL(1,𝔽5)—21no—
101C2×C3×AGL(1,𝔽5)—21no—
102C2×C3×AGL(1,𝔽5)—21no—
103C2×C3×AGL(1,𝔽5)—22no—
104C2×C3×AGL(1,𝔽5)—21no—
105C2×C3×AGL(1,𝔽5)—21no—
106C2×C3×AGL(1,𝔽5)—41no—
107C2×C3×AGL(1,𝔽5)—41no—
108C2×C3×AGL(1,𝔽5)Q1612yes—
109C3×C5×Q8—21no—
110C3×C5×Q8C4⋊C412yescomplemented
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×Dic15—21no—
114C2×C5×Dic3—21no—
115C2×C3×Dic5—21no—
116C2×C3×Dic5#4111yescharacteristic
117C3×C4×D5—22no—
118C3×C4×D5—22no—
119C3×C4×D5C2×D411yescharacteristic
120C3×C4×D5C2×D411yescharacteristic
121C3×C4×D5C2×Q811yescharacteristic
122C3×C4×D5#4111yescharacteristic
123C3×Dic10—21no—
124C3×Dic10—22no—
125C3×Dic10—22no—
126C3×Dic10C4⋊C412yescomplemented
127D5×Dic3—41no—
128D5×Dic3—41no—
129C3⋊#170—41no—
130C3⋊#168—21no—
131C5×C3⋊C8—21no—
132C22×C3×Q8—51no—
133C3×C4×Q8—102no—
134C3×C4×Q8—102no—
135C2×C3×C4⋊C4—51no—
136C2×C3×C4⋊C4—101no—
137C2×C3×C42—51no—
138C2×C3×C42—101no—
139C2×C4⋊Dic3—51no—
140C2×C4⋊Dic3—51no—
141C2×Dic3⋊C4—101no—
142C22×C3⋊C8—51no—
143C22×C3⋊C8—51no—
144#835—51no—
145#835—51no—
146#835—51no—
147#835—102no—
148#835AGL(1,𝔽5)11yescharacteristic, complemented
149C3⋊#95—101no—
150#807—102no—
151C22×AGL(1,𝔽5)—21no—
152C22×AGL(1,𝔽5)—61no—
153C22×AGL(1,𝔽5)#6711yescharacteristic
154C22×AGL(1,𝔽5)#6711yescharacteristic
155C2×C5×Q8C4×D311yescharacteristic
156Q8×D5—21no—
157Q8×D5—21no—
158Q8×D5Dic612yes—
159Q8×D5#6712yes—
160C2×C4×D5—21no—
161C2×C4×D5—61no—
162C2×C4×D5C22×D311yescharacteristic
163C2×Dic10—21no—
164C2×Dic10C4×D311yescharacteristic
165C2×#170—61no—
166C4⋊AGL(1,𝔽5)—24no—
167C4⋊AGL(1,𝔽5)—21no—
168C4⋊AGL(1,𝔽5)—61no—
169C4⋊AGL(1,𝔽5)—61no—
170C4⋊AGL(1,𝔽5)—122no—
171C4⋊AGL(1,𝔽5)#6712yes—
172C4×AGL(1,𝔽5)—24no—
173C4×AGL(1,𝔽5)—21no—
174C4×AGL(1,𝔽5)—21no—
175D5⋊C8—61no—
176D5⋊C8—61no—
177C2×C5×C8—31no—
178C5×C4⋊C4—31no—
179C4⋊Dic5—31no—
180#683—61no—
181C2×#168—31no—
182C8×D5—61no—
183C8×D5—61no—
184C2×C4×Q8—51no—
185C2×#107—151no—
186C2×#103—151no—
187C2×#103—151no—
188#389—151no—
189#340—151no—
190#336—151no—
191C22×C3×C5C4×D411yescharacteristic
192C2×C3×D5#12011yescharacteristic
193C2×C3×D5C22⋊Q811yescharacteristic
194C2×C3×D5C2×Q1611yescharacteristic
195C2×C3×D5#13511yescharacteristic
196C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
197C2×C3×D5#13711yescharacteristic
198ΓL(1,𝔽16)—81no—
199C3×AGL(1,𝔽5)—22no—
200C3×AGL(1,𝔽5)—41no—
201C3×AGL(1,𝔽5)—42no—
202C3×C4×C5—22no—
203C3×C4×C5C2×C4⋊C411yescharacteristic
204C3×C4×C5C4×D411yescharacteristic
205Dic15—41no—
206C3×Dic5—22no—
207C3×Dic5C22⋊Q811yescharacteristic
208C3×Dic5#12311yescharacteristic
209C5×Dic3—41no—
210C2×C3×Q8—52no—
211C2×C3×Q8—52no—
212C2×C3×Q8—51no—
213C2×C3×Q8—101no—
214C2×C3×Q8—101no—
215C2×C3×Q8—101no—
216C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—101no—
221C22×C3×C4—101no—
222C22×Dic3—101no—
223C22×Dic3—101no—
224C3×C4⋊C4—52no—
225C3×C4⋊C4—101no—
226C3×C4⋊C4—104no—
227C3×C42—101no—
228C3×C42—101no—
229C3×C42—104no—
230C4⋊Dic3—51no—
231C4⋊Dic3—101no—
232C4⋊Dic3—101no—
233C4⋊Dic3—101no—
234C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
235Dic3⋊C4—202no—
236C2×C3⋊C8—51no—
237C2×C3⋊C8—101no—
238C2×C3⋊C8—101no—
239C2×C3⋊C8—101no—
240C2×C3⋊C8—101no—
241C2×C3⋊C8—101no—
242C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
243C22×D5C2×#6711yescharacteristic
244C2×AGL(1,𝔽5)—21no—
245C2×AGL(1,𝔽5)—21no—
246C2×AGL(1,𝔽5)—21no—
247C2×AGL(1,𝔽5)—21no—
248C2×AGL(1,𝔽5)—21no—
249C2×AGL(1,𝔽5)—22no—
250C2×AGL(1,𝔽5)—41no—
251C2×AGL(1,𝔽5)—41no—
252C2×AGL(1,𝔽5)—121no—
253C2×AGL(1,𝔽5)—121no—
254C2×AGL(1,𝔽5)—121no—
255C2×AGL(1,𝔽5)#21612yes—
256C5×Q8—21no—
257C5×Q8Dic3⋊C412yescomplemented
258C2×C4×C5—21no—
259C2×C4×C5—61no—
260C2×C4×C5C2×C4×D311yescharacteristic
261C2×Dic5—21no—
262C2×Dic5—61no—
263C2×Dic5#23511yescharacteristic
264C4×D5—22no—
265C4×D5—22no—
266C4×D5—121no—
267C4×D5—121no—
268C4×D5C2×Dic611yescharacteristic
269C4×D5#23511yescharacteristic
270C4×D5C2×#6711yescharacteristic
271C4×D5C2×#6711yescharacteristic
272Dic10—21no—
273Dic10—22no—
274Dic10—22no—
275Dic10Dic3⋊C412yescomplemented
276#170—121no—
277C5×C8—61no—
278#168—61no—
279C22×Q8—51no—
280C22×C8—151no—
281C22×C8—151no—
282C4×Q8—102no—
283C4×Q8—102no—
284C2×C4⋊C4—51no—
285C2×C4⋊C4—101no—
286C2×C4⋊C4—151no—
287C2×C4⋊C4—151no—
288C2×C4⋊C4—301no—
289C2×C42—51no—
290C2×C42—101no—
291#107—302no—
292#103—31no—
293#103—151no—
294#103—151no—
295#103—151no—
296#103—302no—
297#95—301no—
298C2×C3×C5#38511yescharacteristic
299C2×C3×C5C4×Q1611yescharacteristic
300C2×C3×C5#44011yescharacteristic
301C3×D5#44911yescharacteristic
302C3×D5#45111yescharacteristic
303C3×D5#47711yescharacteristic
304C3×D5#47911yescharacteristic
305C23×C3—51no—
306C3×Q8—21no—
307C3×Q8—52no—
308C3×Q8—102no—
309C3×Q8—102no—
310C3×Q8—101no—
311C3×Q8C4⋊AGL(1,𝔽5)12yescomplemented
312C2×C3×C4—21no—
313C2×C3×C4—51no—
314C2×C3×C4—51no—
315C2×C3×C4—51no—
316C2×C3×C4—51no—
317C2×C3×C4—51no—
318C2×C3×C4—52no—
319C2×C3×C4—102no—
320C2×C3×C4—102no—
321C2×C3×C4—101no—
322C2×C3×C4—101no—
323C2×C3×C4—101no—
324C2×C3×C4—101no—
325C2×C3×C4—102no—
326C2×C3×C4—101no—
327C2×C3×C4—101no—
328C2×C3×C4—201no—
329C2×C3×C4—201no—
330C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
331C2×Dic3—21no—
332C2×Dic3—101no—
333C2×Dic3—201no—
334C2×Dic3—201no—
335C2×Dic3—201no—
336C2×Dic3—201no—
337C2×Dic3—201no—
338C3⋊C8—21no—
339C3⋊C8—101no—
340C3⋊C8—201no—
341C22×C5C4×#6711yescharacteristic
342C2×D5#92411yescharacteristic
343C2×D5#94211yescharacteristic
344C2×D5C2×#21611yescharacteristic
345C2×D5#94911yescharacteristic
346C2×D5#95011yescharacteristic
347C2×D5#95311yescharacteristic
348AGL(1,𝔽5)—22no—
349AGL(1,𝔽5)—41no—
350AGL(1,𝔽5)—42no—
351AGL(1,𝔽5)—241no—
352C4×C5—22no—
353C4×C5—121no—
354C4×C5C2×Dic3⋊C411yescharacteristic
355C4×C5C4×#6711yescharacteristic
356Dic5—22no—
357Dic5—121no—
358Dic5#92411yescharacteristic
359Dic5#92911yescharacteristic
360C2×Q8—52no—
361C2×Q8—52no—
362C2×Q8—51no—
363C2×Q8—101no—
364C2×Q8—101no—
365C2×Q8—101no—
366C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
367C22×C4—51no—
368C22×C4—51no—
369C22×C4—51no—
370C22×C4—101no—
371C22×C4—101no—
372C22×C4—301no—
373C22×C4—301no—
374C2×C8—31no—
375C2×C8—151no—
376C2×C8—301no—
377C2×C8—301no—
378C2×C8—301no—
379C2×C8—301no—
380C2×C8—301no—
381C4⋊C4—31no—
382C4⋊C4—52no—
383C4⋊C4—101no—
384C4⋊C4—104no—
385C4⋊C4—151no—
386C4⋊C4—301no—
387C4⋊C4—301no—
388C4⋊C4—301no—
389C4⋊C4—602no—
390C42—101no—
391C42—101no—
392C42—104no—
393C3×C5#186311yescharacteristic, complemented, Hall
394C22×C3—51no—
395C22×C3—51no—
396C22×C3—51no—
397C22×C3—51no—
398C22×C3—51no—
399C22×C3—51no—
400C22×C3D4×AGL(1,𝔽5)11yescharacteristic
401C3×C4—22no—
402C3×C4—51no—
403C3×C4—51no—
404C3×C4—102no—
405C3×C4—102no—
406C3×C4—201no—
407C3×C4—202no—
408C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
409C3×C4D4×AGL(1,𝔽5)11yescharacteristic
410Dic3—41no—
411Dic3—201no—
412Dic3—401no—
413C2×C5#502611yescharacteristic
414C2×C5C4×#21611yescharacteristic
415C2×C5#521011yescharacteristic
416D5#502111yescharacteristic
417D5#502211yescharacteristic
418D5#522811yescharacteristic
419D5#523811yescharacteristic
420C23—51no—
421Q8—21no—
422Q8—52no—
423Q8—102no—
424Q8—102no—
425Q8—101no—
426Q8Dic3⋊AGL(1,𝔽5)12yescomplemented
427C2×C4—21no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—51no—
433C2×C4—52no—
434C2×C4—61no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—101no—
438C2×C4—101no—
439C2×C4—101no—
440C2×C4—102no—
441C2×C4—101no—
442C2×C4—101no—
443C2×C4—101no—
444C2×C4—201no—
445C2×C4—201no—
446C2×C4—301no—
447C2×C4—601no—
448C2×C4—601no—
449C2×C4—601no—
450C2×C4—601no—
451C2×C4—601no—
452C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
453C8—61no—
454C8—301no—
455C8—601no—
456C2×C3—51no—
457C2×C3—51no—
458C2×C3—51no—
459C2×C3—51no—
460C2×C3#6569911yescharacteristic
461C2×C3#6573511yescharacteristic
462C2×C3Q16×AGL(1,𝔽5)11yescharacteristic
463C5#7162811yescharacteristic, complemented, Sylow
464C22—51no—
465C22—51no—
466C22—51no—
467C22—51no—
468C22—51no—
469C22—51no—
470C22AGL(1,𝔽5)×#6711yescharacteristic
471C4—22no—
472C4—51no—
473C4—51no—
474C4—102no—
475C4—102no—
476C4—121no—
477C4—201no—
478C4—202no—
479C4—601no—
480C4—1201no—
481C4C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
482C4AGL(1,𝔽5)×#6711yescharacteristic
483C3#1060398911yescharacteristic, complemented, Sylow
484C2—51no—
485C2—51no—
486C2—51no—
487C2—51no—
488C2#1175196411yescharacteristic
489C2AGL(1,𝔽5)×#21611yescharacteristic
490C2#1175202711yescharacteristic
subgroupG/H
centerC22AGL(1,𝔽5)×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C4×D4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1863
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
24. #91119C4semidirect
25. C5×#835C4semidirect
26. #90971C4semidirect
27. #90970C4semidirect
110. C3×C5×Q8C4⋊C4semidirect
126. C3×Dic10C4⋊C4semidirect
148. #835AGL(1,𝔽5)semidirect
257. C5×Q8Dic3⋊C4semidirect
275. Dic10Dic3⋊C4semidirect
311. C3×Q8C4⋊AGL(1,𝔽5)semidirect
393. C3×C5#1863semidirect
426. Q8Dic3⋊AGL(1,𝔽5)semidirect
463. C5#71628semidirect
483. C3#10603989semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×AGL(1,𝔽5)2
C2×#912382
#117518592
C2×#911602
D5×#8352
#117465282
(C2×AGL(1,𝔽5))⋊(C3⋊C8)2
#106039893
#716285