Finite Group Databasebeta

← #14305951

Subgroups — #14305951

Order 1920.

616 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#944C211yesmaximal, characteristic
2C2×#91238C211yesmaximal, characteristic
3C2×C3×#4009C211yesmaximal, characteristic
4#11751788C211yesmaximal, characteristic
5#11751751C211yesmaximal, characteristic
6C2×#91154C211yesmaximal, characteristic
7C2×#91153C211yesmaximal, characteristic
8#10604673—31nomaximal, Hall
9C2×C3×Q8×D5C2211yescharacteristic
10C2×D5×Dic6C2211yescharacteristic
11C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
12C2×D5⋊(C3⋊C8)C2211yescharacteristic
13C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
14C2×C3×D5⋊C8C2211yescharacteristic
15#91833C411yescharacteristic, complemented
16C5×#944C411yescharacteristic, complemented
17D5×Dic3⋊C4—22no—
18C4×D5×Dic3C2211yescharacteristic
19#91389C411yescharacteristic, complemented
20#91385C411yescharacteristic, complemented
21#91327—22no—
22#91238—22no—
23C3×#4009—22no—
24#91169—22no—
25(C2×AGL(1,𝔽5))⋊Dic3—22no—
26#91154—22no—
27#91153—22no—
28#70228—51nomaximal, Hall
29D5×C4⋊Q8—31no—
30C2×#4009—31no—
31C2×#4009—31no—
32C2×#4009—31no—
33C2×#4009D311yescharacteristic
34#65689—31no—
35#65676—31no—
36C22×ΓL(1,𝔽16)—21no—
37C22×C3×AGL(1,𝔽5)—21no—
38C2×C3×C5×Q8C2×C411yescharacteristic
39C2×Dic30C2×C411yescharacteristic
40C2×C5×Dic6C2×C411yescharacteristic
41C3×Q8×D5—21no—
42C3×Q8×D5—21no—
43C3×Q8×D5—21no—
44C3×Q8×D5—21no—
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5C2311yescharacteristic
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C411yescharacteristic
49C2×#1204—21no—
50C2×D5×Dic3—21no—
51C2×D5×Dic3D412yes—
52D5×Dic6—21no—
53D5×Dic6—21no—
54D5×Dic6—21no—
55D5×Dic6—21no—
56C2×C3⋊#170—21no—
57C4⋊ΓL(1,𝔽16)—21no—
58C4⋊ΓL(1,𝔽16)—21no—
59D5⋊(C3⋊C8)—21no—
60D5⋊(C3⋊C8)D412yes—
61C2×C3×#170—21no—
62C3×C4⋊AGL(1,𝔽5)—21no—
63C3×C4⋊AGL(1,𝔽5)—21no—
64C3×D5⋊C8—21no—
65C3×D5⋊C8D412yes—
66#6712—22no—
67C4×Dic15C2×C411yescharacteristic
68C5×Dic3⋊C4—22no—
69C4×C5×Dic3C2×C411yescharacteristic
70Dic3⋊Dic5—22no—
71#6669—22no—
72Dic3×Dic5D412yes—
73C2×#944—51no—
74C2×C3×#103—51no—
75C2×#835—51no—
76C2×#816—51no—
77C2×#815—51no—
78#4977—51no—
79C2×#810—51no—
80C2×Q8×D5—31no—
81C2×Q8×D5C2×D311yescharacteristic
82C2×C4⋊AGL(1,𝔽5)—31no—
83C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
84C2×D5⋊C8—31no—
85C2×D5⋊C8C2×D311yescharacteristic
86C5×C4⋊Q8—31no—
87#4090—31no—
88#4090—31no—
89D5×C4⋊C4—62no—
90#4034—62no—
91C42×D5—31no—
92#4015—31no—
93#4009—22no—
94#4009—62no—
95#4009—62no—
96#4009—62no—
97(C2×AGL(1,𝔽5))⋊C4—62no—
98#4002—62no—
99#1960—151noSylow
100C22×C3×D5C2×D411yescharacteristic
101C2×ΓL(1,𝔽16)—41no—
102C2×ΓL(1,𝔽16)—42no—
103C2×C3×AGL(1,𝔽5)—41no—
104C2×C3×AGL(1,𝔽5)—42no—
105C3×C5×Q8—21no—
106C3×C5×Q8—21no—
107C2×C3×C4×C5—21no—
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×Dic15—21no—
110C2×Dic15C22⋊C412yes—
111Dic30—21no—
112Dic30—21no—
113C2×C5×Dic3—21no—
114C2×C5×Dic3C22⋊C412yes—
115C5×Dic6—21no—
116C5×Dic6—21no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5C2×D411yescharacteristic
119C3×C4×D5—41no—
120C3×C4×D5—41no—
121C3×C4×D5C2×D411yescharacteristic
122C3×C4×D5C2×D411yescharacteristic
123C3×C4×D5#4111yescharacteristic
124C3×C4×D5#4111yescharacteristic
125C3×Dic10—21no—
126C3×Dic10—21no—
127C3×Dic10—41no—
128C3×Dic10—41no—
129#1204—41no—
130#1204—41no—
131D5×Dic3—22no—
132D5×Dic3—22no—
133D5×Dic3—41no—
134D5×Dic3—41no—
135C3⋊#170—22no—
136C3×#170—22no—
137C22×C3×Q8—51no—
138C22×Dic6—51no—
139C22×C3×C8—51no—
140C2×C3×C4⋊C4—51no—
141#944—51no—
142#944—51no—
143#944—51no—
144#944—102no—
145#944AGL(1,𝔽5)11yescharacteristic, complemented
146C2×C4⋊Dic3—51no—
147C2×Dic3⋊C4—102no—
148C2×C4×Dic3—51no—
149C22×C3⋊C8—51no—
150C3×#103—102no—
151#835—102no—
152C3⋊#95—102no—
153#816—102no—
154#815—102no—
155#810—102no—
156C22×AGL(1,𝔽5)—21no—
157C22×AGL(1,𝔽5)—61no—
158C2×C5×Q8—31no—
159C2×C5×Q8C4×D311yescharacteristic
160Q8×D5—21no—
161Q8×D5—21no—
162Q8×D5—21no—
163Q8×D5—21no—
164Q8×D5—61no—
165Q8×D5—61no—
166Q8×D5—61no—
167Q8×D5—61no—
168C2×C4×D5—21no—
169C2×C4×D5—32no—
170C2×C4×D5—61no—
171C2×C4×D5C22×D311yescharacteristic
172C2×Dic10—21no—
173C2×Dic10—31no—
174C2×Dic10—61no—
175C2×Dic10C4×D311yescharacteristic
176C2×#170—21no—
177C2×#170—61no—
178C4⋊AGL(1,𝔽5)—21no—
179C4⋊AGL(1,𝔽5)—21no—
180C4⋊AGL(1,𝔽5)—61no—
181C4⋊AGL(1,𝔽5)—61no—
182D5⋊C8—21no—
183D5⋊C8—32no—
184D5⋊C8—61no—
185D5⋊C8#6712yes—
186C5×C4⋊C4—62no—
187C42×C5—31no—
188C4⋊Dic5—62no—
189#683—62no—
190#683—62no—
191C4×Dic5—31no—
192C4×Dic5—32no—
193C2×C4⋊Q8—151no—
194C2×#103—51no—
195C2×#103—151no—
196C2×#103—151no—
197C2×#103—151no—
198C2×#97—151no—
199#382—151no—
200C22×C3×C5C2×C22⋊C411yescharacteristic
201C2×C3×D5#12411yescharacteristic
202C2×C3×D5#12711yescharacteristic
203C2×C3×D5#13712yes—
204C2×C3×D5#13712yes—
205ΓL(1,𝔽16)—81no—
206C3×AGL(1,𝔽5)—81no—
207C3×C4×C5—41no—
208C3×C4×C5C4×D411yescharacteristic
209C3×C4×C5C4×D411yescharacteristic
210Dic15—22no—
211Dic15—41no—
212C3×Dic5—41no—
213C3×Dic5#12111yescharacteristic
214C3×Dic5#12111yescharacteristic
215C5×Dic3—22no—
216C5×Dic3—41no—
217C2×C3×Q8—51no—
218C2×C3×Q8—101no—
219C2×C3×Q8—101no—
220C2×C3×Q8—101no—
221C2×C3×Q8—101no—
222C2×C3×Q8—101no—
223C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
224C22×C3×C4—51no—
225C22×C3×C4—101no—
226C22×C3×C4—101no—
227C22×Dic3—52no—
228C22×Dic3—101no—
229C22×Dic3—101no—
230C2×Dic6—51no—
231C2×Dic6—101no—
232C2×Dic6—101no—
233C2×Dic6—101no—
234C2×Dic6—101no—
235C2×Dic6—101no—
236C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
237C2×C3×C8—52no—
238C2×C3×C8—101no—
239C2×C3×C8—101no—
240C3×C4⋊C4—101no—
241C3×C4⋊C4—101no—
242C4⋊Dic3—101no—
243C4⋊Dic3—101no—
244Dic3⋊C4—22no—
245Dic3⋊C4—102no—
246Dic3⋊C4—102no—
247Dic3⋊C4—102no—
248C4×Dic3—51no—
249C4×Dic3—52no—
250C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
251C2×C3⋊C8—52no—
252C2×C3⋊C8—101no—
253C2×C3⋊C8—101no—
254C22×D5D3×D411yescharacteristic
255C2×AGL(1,𝔽5)—42no—
256C2×AGL(1,𝔽5)—41no—
257C2×AGL(1,𝔽5)—122no—
258C2×AGL(1,𝔽5)—121no—
259C5×Q8—21no—
260C5×Q8—21no—
261C5×Q8—61no—
262C5×Q8—61no—
263C2×C4×C5—21no—
264C2×C4×C5—32no—
265C2×C4×C5—61no—
266C2×C4×C5C2×C4×D311yescharacteristic
267C2×Dic5—21no—
268C2×Dic5—32no—
269C2×Dic5—61no—
270C2×Dic5D3×D411yescharacteristic
271C4×D5—41no—
272C4×D5—41no—
273C4×D5—62no—
274C4×D5—62no—
275C4×D5—121no—
276C4×D5—121no—
277C4×D5#23511yescharacteristic
278C4×D5D3×D411yescharacteristic
279C4×D5#23711yescharacteristic
280C4×D5C2×#6711yescharacteristic
281Dic10—21no—
282Dic10—21no—
283Dic10—41no—
284Dic10—41no—
285Dic10—61no—
286Dic10—61no—
287Dic10—121no—
288Dic10—121no—
289#170—22no—
290#170—62no—
291C22×Q8—51no—
292C22×Q8—151no—
293C22×C8—51no—
294C22×C8—151no—
295C4⋊Q8—31no—
296C4⋊Q8—151no—
297C4⋊Q8—151no—
298C4⋊Q8—151no—
299C4⋊Q8—302no—
300C2×C4⋊C4—51no—
301C2×C4⋊C4—151no—
302C2×C4⋊C4—302no—
303C2×C42—151no—
304#103—102no—
305#103—302no—
306#103—302no—
307#103—302no—
308#97—302no—
309#95—302no—
310C2×C3×C5#39011yescharacteristic
311C2×C3×C5#41912yes—
312C3×D5#49112yes—
313C3×D5#49711yescharacteristic
314C3×D5#49711yescharacteristic
315C23×C3—51no—
316C3×Q8—21no—
317C3×Q8—21no—
318C3×Q8—101no—
319C3×Q8—101no—
320C3×Q8—201no—
321C3×Q8—201no—
322C2×C3×C4—21no—
323C2×C3×C4—51no—
324C2×C3×C4—51no—
325C2×C3×C4—51no—
326C2×C3×C4—51no—
327C2×C3×C4—51no—
328C2×C3×C4—101no—
329C2×C3×C4—201no—
330C2×C3×C4—201no—
331C2×C3×C4—201no—
332C2×C3×C4—202no—
333C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
334C2×Dic3—21no—
335C2×Dic3—52no—
336C2×Dic3—102no—
337C2×Dic3—102no—
338C2×Dic3—101no—
339C2×Dic3—201no—
340C2×Dic3—201no—
341C2×Dic3—201no—
342C2×Dic3—202no—
343C2×Dic3C22⋊AGL(1,𝔽5)12yes—
344Dic6—21no—
345Dic6—21no—
346Dic6—101no—
347Dic6—101no—
348Dic6—201no—
349Dic6—201no—
350C3×C8—102no—
351C3⋊C8—102no—
352C22×C5D3×C22⋊C411yescharacteristic
353C2×D5#88511yescharacteristic
354C2×D5#91512yes—
355C2×D5#91812yes—
356C2×D5#94011yescharacteristic
357AGL(1,𝔽5)—81no—
358AGL(1,𝔽5)—241no—
359C4×C5—41no—
360C4×C5—62no—
361C4×C5—121no—
362C4×C5#67⋊C411yescharacteristic
363C4×C5C4×#6711yescharacteristic
364Dic5—41no—
365Dic5—62no—
366Dic5—121no—
367Dic5#88411yescharacteristic
368Dic5#93911yescharacteristic
369C2×Q8—31no—
370C2×Q8—51no—
371C2×Q8—101no—
372C2×Q8—101no—
373C2×Q8—101no—
374C2×Q8—101no—
375C2×Q8—101no—
376C2×Q8—151no—
377C2×Q8—301no—
378C2×Q8—301no—
379C2×Q8—301no—
380C2×Q8—301no—
381C2×Q8—301no—
382C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
383C22×C4—51no—
384C22×C4—101no—
385C22×C4—101no—
386C22×C4—152no—
387C22×C4—301no—
388C22×C4—301no—
389C2×C8—52no—
390C2×C8—101no—
391C2×C8—101no—
392C2×C8—152no—
393C2×C8—301no—
394C2×C8—301no—
395C4⋊C4—62no—
396C4⋊C4—101no—
397C4⋊C4—101no—
398C4⋊C4—302no—
399C4⋊C4—302no—
400C4⋊C4—301no—
401C4⋊C4—301no—
402C4⋊C4—302no—
403C42—31no—
404C42—152no—
405C42—151no—
406C3×C5#196011yescharacteristic, complemented, Hall
407C22×C3—51no—
408C22×C3—51no—
409C22×C3—52no—
410C22×C3—52no—
411C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
412C3×C4—41no—
413C3×C4—51no—
414C3×C4—51no—
415C3×C4—201no—
416C3×C4—401no—
417C3×C4D4×AGL(1,𝔽5)11yescharacteristic
418C3×C4D4×AGL(1,𝔽5)11yescharacteristic
419Dic3—22no—
420Dic3—41no—
421Dic3—102no—
422Dic3—201no—
423Dic3—401no—
424C2×C5#480212yes—
425C2×C5#495711yescharacteristic
426D5#479712yes—
427D5#516711yescharacteristic
428D5#519011yescharacteristic
429C23—51no—
430Q8—21no—
431Q8—21no—
432Q8—61no—
433Q8—61no—
434Q8—101no—
435Q8—101no—
436Q8—201no—
437Q8—201no—
438Q8—301no—
439Q8—301no—
440Q8—601no—
441Q8—601no—
442C2×C4—21no—
443C2×C4—32no—
444C2×C4—51no—
445C2×C4—51no—
446C2×C4—51no—
447C2×C4—51no—
448C2×C4—51no—
449C2×C4—61no—
450C2×C4—101no—
451C2×C4—152no—
452C2×C4—201no—
453C2×C4—201no—
454C2×C4—201no—
455C2×C4—202no—
456C2×C4—302no—
457C2×C4—302no—
458C2×C4—301no—
459C2×C4—601no—
460C2×C4—601no—
461C2×C4—601no—
462C2×C4—602no—
463C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
464C8—102no—
465C8—302no—
466C2×C3—51no—
467C2×C3—51no—
468C2×C3—52no—
469C2×C3#6578011yescharacteristic
470C2×C3#6578212yes—
471C5#7022811yescharacteristic, complemented, Sylow
472C22—52no—
473C22—52no—
474C22—51no—
475C22—51no—
476C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
477C4—41no—
478C4—51no—
479C4—51no—
480C4—62no—
481C4—121no—
482C4—201no—
483C4—302no—
484C4—401no—
485C4—601no—
486C4—1201no—
487C4AGL(1,𝔽5)×#6711yescharacteristic
488C4#67⋊AGL(1,𝔽5)11yescharacteristic
489C3#1060467311yescharacteristic, complemented, Sylow
490C2—51no—
491C2—51no—
492C2—52no—
493C2#1175196212yes—
494C2#1175201911yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1960
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
15. #91833C4semidirect
16. C5×#944C4semidirect
19. #91389C4semidirect
20. #91385C4semidirect
145. #944AGL(1,𝔽5)semidirect
406. C3×C5#1960semidirect
471. C5#70228semidirect
489. C3#10604673semidirect

Maximal subgroups

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

subgroupindex
D5×#9442
C2×#912382
C2×C3×#40092
#117517882
#117517512
C2×#911542
C2×#911532
#106046733
#702285