Finite Group Databasebeta

← #14305797

Subgroups — #14305797

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 500 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#11751860C211yesmaximal, characteristic
5Dic3×D5⋊C8C211yesmaximal, characteristic
6C2×#91154C211yesmaximal, characteristic
7C2×#91153C211yesmaximal, characteristic
8#10604672—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
15C2×Dic3⋊AGL(1,𝔽5)—22no—
16#91833C411yescharacteristic, complemented
17C5×#944C411yescharacteristic, complemented
18D5×Dic3⋊C4—22no—
19C4×D5×Dic3C2211yescharacteristic
20#91389C411yescharacteristic, complemented
21#91385C411yescharacteristic, complemented
22#91327—22no—
23#91238—22no—
24C3×#4009—22no—
25Dic3×#170—22no—
26#91154—22no—
27#91153—22no—
28#70225—51nomaximal, Hall
29D5×C4⋊Q8—31no—
30C2×#4009—31no—
31C2×#4009—31no—
32C2×#4009—31no—
33C2×#4009D311yescharacteristic
34#65688—31no—
35C4×D5⋊C8—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—
66Dic3⋊AGL(1,𝔽5)—44no—
67#6712—22no—
68C4×Dic15C2×C411yescharacteristic
69C5×Dic3⋊C4—22no—
70C4×C5×Dic3C2×C411yescharacteristic
71Dic3⋊Dic5—22no—
72#6669—22no—
73Dic3×Dic5D412yes—
74C2×#944—51no—
75C2×C3×#103—51no—
76C2×#835—51no—
77C2×#816—51no—
78C2×C8×Dic3—51no—
79#4975—51no—
80C2×#810—51no—
81C2×Q8×D5—31no—
82C2×Q8×D5C2×D311yescharacteristic
83C2×C4⋊AGL(1,𝔽5)—31no—
84C2×C4⋊AGL(1,𝔽5)—62no—
85C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
86C2×D5⋊C8—31no—
87C2×D5⋊C8C2×D311yescharacteristic
88C5×C4⋊Q8—31no—
89#4090—31no—
90#4090—31no—
91D5×C4⋊C4—62no—
92#4034—62no—
93C42×D5—31no—
94#4015—31no—
95#4009—22no—
96#4009—62no—
97#4009—62no—
98#4009—62no—
99C4×#170—62no—
100#1956—151noSylow
101C22×C3×D5C2×D411yescharacteristic
102C2×ΓL(1,𝔽16)—41no—
103C2×ΓL(1,𝔽16)—42no—
104C2×C3×AGL(1,𝔽5)—41no—
105C2×C3×AGL(1,𝔽5)—42no—
106C3×C5×Q8—21no—
107C3×C5×Q8—21no—
108C2×C3×C4×C5—21no—
109C2×C3×C4×C5C22×C411yescharacteristic
110C2×Dic15—21no—
111C2×Dic15C22⋊C412yes—
112Dic30—21no—
113Dic30—21no—
114C2×C5×Dic3—21no—
115C2×C5×Dic3C22⋊C412yes—
116C5×Dic6—21no—
117C5×Dic6—21no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5C2×D411yescharacteristic
120C3×C4×D5—41no—
121C3×C4×D5—41no—
122C3×C4×D5C2×D411yescharacteristic
123C3×C4×D5C2×D411yescharacteristic
124C3×C4×D5#4111yescharacteristic
125C3×C4×D5#4111yescharacteristic
126C3×Dic10—21no—
127C3×Dic10—21no—
128C3×Dic10—41no—
129C3×Dic10—41no—
130#1204—41no—
131#1204—41no—
132D5×Dic3—41no—
133D5×Dic3—41no—
134D5×Dic3SD1614yes—
135D5×Dic3Q1614yes—
136C3⋊#170—22no—
137C3×#170—22no—
138C22×C3×Q8—51no—
139C22×Dic6—51no—
140C22×C3×C8—51no—
141C2×C3×C4⋊C4—51no—
142#944—51no—
143#944—51no—
144#944—51no—
145#944—102no—
146#944AGL(1,𝔽5)11yescharacteristic, complemented
147C2×C4⋊Dic3—51no—
148C2×Dic3⋊C4—102no—
149C2×Dic3⋊C4—102no—
150C2×C4×Dic3—51no—
151C22×C3⋊C8—51no—
152C3×#103—102no—
153#835—102no—
154#816—102no—
155C8×Dic3—102no—
156#810—102no—
157C22×AGL(1,𝔽5)—21no—
158C22×AGL(1,𝔽5)—61no—
159C2×C5×Q8—31no—
160C2×C5×Q8C4×D311yescharacteristic
161Q8×D5—21no—
162Q8×D5—21no—
163Q8×D5—21no—
164Q8×D5—21no—
165Q8×D5—61no—
166Q8×D5—61no—
167Q8×D5—61no—
168Q8×D5—61no—
169C2×C4×D5—21no—
170C2×C4×D5—32no—
171C2×C4×D5—61no—
172C2×C4×D5C22×D311yescharacteristic
173C2×Dic10—21no—
174C2×Dic10—31no—
175C2×Dic10—61no—
176C2×Dic10C4×D311yescharacteristic
177C2×#170—21no—
178C2×#170—61no—
179C4⋊AGL(1,𝔽5)—21no—
180C4⋊AGL(1,𝔽5)—21no—
181C4⋊AGL(1,𝔽5)—61no—
182C4⋊AGL(1,𝔽5)—61no—
183C4⋊AGL(1,𝔽5)—124no—
184D5⋊C8—21no—
185D5⋊C8—32no—
186D5⋊C8—61no—
187D5⋊C8#6712yes—
188C5×C4⋊C4—62no—
189C42×C5—31no—
190C4⋊Dic5—62no—
191#683—62no—
192#683—62no—
193C4×Dic5—31no—
194C4×Dic5—32no—
195C2×C4⋊Q8—151no—
196C2×#103—51no—
197C2×#103—151no—
198C2×#103—151no—
199C2×#103—151no—
200C2×C4×C8—151no—
201#384—151no—
202C22×C3×C5C2×C22⋊C411yescharacteristic
203C2×C3×D5#12411yescharacteristic
204C2×C3×D5#12711yescharacteristic
205C2×C3×D5C2×SD1612yes—
206C2×C3×D5C2×Q1612yes—
207ΓL(1,𝔽16)—81no—
208C3×AGL(1,𝔽5)—81no—
209C3×C4×C5—41no—
210C3×C4×C5C4×D411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212Dic15—41no—
213Dic15#10314yescomplemented
214C3×Dic5—41no—
215C3×Dic5#12111yescharacteristic
216C3×Dic5#12111yescharacteristic
217C5×Dic3—41no—
218C5×Dic3#10314yescomplemented
219C2×C3×Q8—51no—
220C2×C3×Q8—101no—
221C2×C3×Q8—101no—
222C2×C3×Q8—101no—
223C2×C3×Q8—101no—
224C2×C3×Q8—101no—
225C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
226C22×C3×C4—51no—
227C22×C3×C4—101no—
228C22×C3×C4—101no—
229C22×Dic3—52no—
230C22×Dic3—101no—
231C22×Dic3—101no—
232C2×Dic6—51no—
233C2×Dic6—101no—
234C2×Dic6—101no—
235C2×Dic6—101no—
236C2×Dic6—101no—
237C2×Dic6—101no—
238C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
239C2×C3×C8—52no—
240C2×C3×C8—101no—
241C2×C3×C8—101no—
242C3×C4⋊C4—101no—
243C3×C4⋊C4—101no—
244C4⋊Dic3—101no—
245C4⋊Dic3—101no—
246Dic3⋊C4—22no—
247Dic3⋊C4—102no—
248Dic3⋊C4—102no—
249Dic3⋊C4—102no—
250Dic3⋊C4—204no—
251C4×Dic3—51no—
252C4×Dic3—52no—
253C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
254C2×C3⋊C8—52no—
255C2×C3⋊C8—101no—
256C2×C3⋊C8—101no—
257C22×D5D3×D411yescharacteristic
258C2×AGL(1,𝔽5)—42no—
259C2×AGL(1,𝔽5)—41no—
260C2×AGL(1,𝔽5)—122no—
261C2×AGL(1,𝔽5)—121no—
262C5×Q8—21no—
263C5×Q8—21no—
264C5×Q8—61no—
265C5×Q8—61no—
266C2×C4×C5—21no—
267C2×C4×C5—32no—
268C2×C4×C5—61no—
269C2×C4×C5C2×C4×D311yescharacteristic
270C2×Dic5—21no—
271C2×Dic5—32no—
272C2×Dic5—61no—
273C2×Dic5D3×D411yescharacteristic
274C4×D5—34no—
275C4×D5—34no—
276C4×D5—41no—
277C4×D5—41no—
278C4×D5—121no—
279C4×D5—121no—
280C4×D5#23511yescharacteristic
281C4×D5D3×D411yescharacteristic
282C4×D5#23711yescharacteristic
283C4×D5C2×#6711yescharacteristic
284Dic10—21no—
285Dic10—21no—
286Dic10—41no—
287Dic10—41no—
288Dic10—61no—
289Dic10—61no—
290Dic10—121no—
291Dic10—121no—
292#170—22no—
293#170—62no—
294C22×Q8—51no—
295C22×Q8—151no—
296C22×C8—51no—
297C22×C8—151no—
298C4⋊Q8—31no—
299C4⋊Q8—151no—
300C4⋊Q8—151no—
301C4⋊Q8—151no—
302C4⋊Q8—302no—
303C2×C4⋊C4—51no—
304C2×C4⋊C4—151no—
305C2×C4⋊C4—302no—
306C2×C4⋊C4—302no—
307C2×C42—151no—
308#103—102no—
309#103—302no—
310#103—302no—
311#103—302no—
312C4×C8—302no—
313C2×C3×C5#39011yescharacteristic
314C2×C3×C5C2×#10312yes—
315C3×D5#48712yes—
316C3×D5#49211yescharacteristic
317C3×D5#49411yescharacteristic
318C23×C3—51no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—101no—
322C3×Q8—101no—
323C3×Q8—201no—
324C3×Q8—201no—
325C2×C3×C4—21no—
326C2×C3×C4—51no—
327C2×C3×C4—51no—
328C2×C3×C4—51no—
329C2×C3×C4—51no—
330C2×C3×C4—51no—
331C2×C3×C4—101no—
332C2×C3×C4—201no—
333C2×C3×C4—201no—
334C2×C3×C4—202no—
335C2×C3×C4—201no—
336C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
337C2×Dic3—21no—
338C2×Dic3—52no—
339C2×Dic3—54no—
340C2×Dic3—54no—
341C2×Dic3—101no—
342C2×Dic3—201no—
343C2×Dic3—201no—
344C2×Dic3—202no—
345C2×Dic3—201no—
346C2×Dic3C22⋊AGL(1,𝔽5)12yes—
347Dic6—21no—
348Dic6—21no—
349Dic6—101no—
350Dic6—101no—
351Dic6—201no—
352Dic6—201no—
353C3×C8—102no—
354C3⋊C8—102no—
355C22×C5D3×C22⋊C411yescharacteristic
356C2×D5#88511yescharacteristic
357C2×D5D3×SD1612yes—
358C2×D5D3×Q1612yes—
359C2×D5#94011yescharacteristic
360AGL(1,𝔽5)—81no—
361AGL(1,𝔽5)—241no—
362C4×C5—34no—
363C4×C5—41no—
364C4×C5—121no—
365C4×C5#67⋊C411yescharacteristic
366C4×C5C4×#6711yescharacteristic
367Dic5—34no—
368Dic5—41no—
369Dic5—121no—
370Dic5#88411yescharacteristic
371Dic5#93911yescharacteristic
372C2×Q8—31no—
373C2×Q8—51no—
374C2×Q8—101no—
375C2×Q8—101no—
376C2×Q8—101no—
377C2×Q8—101no—
378C2×Q8—101no—
379C2×Q8—151no—
380C2×Q8—301no—
381C2×Q8—301no—
382C2×Q8—301no—
383C2×Q8—301no—
384C2×Q8—301no—
385C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
386C22×C4—51no—
387C22×C4—101no—
388C22×C4—101no—
389C22×C4—152no—
390C22×C4—301no—
391C22×C4—301no—
392C2×C8—52no—
393C2×C8—101no—
394C2×C8—101no—
395C2×C8—152no—
396C2×C8—301no—
397C2×C8—301no—
398C4⋊C4—62no—
399C4⋊C4—101no—
400C4⋊C4—101no—
401C4⋊C4—302no—
402C4⋊C4—302no—
403C4⋊C4—301no—
404C4⋊C4—301no—
405C4⋊C4—302no—
406C4⋊C4—604no—
407C42—31no—
408C42—152no—
409C42—151no—
410C3×C5#195611yescharacteristic, complemented, Hall
411C22×C3—51no—
412C22×C3—51no—
413C22×C3—52no—
414C22×C3—52no—
415C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
416C3×C4—41no—
417C3×C4—51no—
418C3×C4—51no—
419C3×C4—201no—
420C3×C4—401no—
421C3×C4D4×AGL(1,𝔽5)11yescharacteristic
422C3×C4D4×AGL(1,𝔽5)11yescharacteristic
423Dic3—41no—
424Dic3—54no—
425Dic3—201no—
426Dic3—401no—
427Dic3#400914yescomplemented
428C2×C5D3×#10312yes—
429C2×C5#495711yescharacteristic
430D5#479212yes—
431D5#517511yescharacteristic
432D5#518311yescharacteristic
433C23—51no—
434Q8—21no—
435Q8—21no—
436Q8—61no—
437Q8—61no—
438Q8—101no—
439Q8—101no—
440Q8—201no—
441Q8—201no—
442Q8—301no—
443Q8—301no—
444Q8—601no—
445Q8—601no—
446C2×C4—21no—
447C2×C4—32no—
448C2×C4—51no—
449C2×C4—51no—
450C2×C4—51no—
451C2×C4—51no—
452C2×C4—51no—
453C2×C4—61no—
454C2×C4—101no—
455C2×C4—154no—
456C2×C4—154no—
457C2×C4—152no—
458C2×C4—201no—
459C2×C4—201no—
460C2×C4—201no—
461C2×C4—202no—
462C2×C4—301no—
463C2×C4—601no—
464C2×C4—601no—
465C2×C4—601no—
466C2×C4—602no—
467C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
468C8—102no—
469C8—302no—
470C2×C3—51no—
471C2×C3—52no—
472C2×C3—51no—
473C2×C3#6578011yescharacteristic
474C2×C3C2×#400912yes—
475C5#7022511yescharacteristic, complemented, Sylow
476C22—52no—
477C22—52no—
478C22—51no—
479C22—51no—
480C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
481C4—34no—
482C4—41no—
483C4—51no—
484C4—51no—
485C4—121no—
486C4—154no—
487C4—201no—
488C4—401no—
489C4—601no—
490C4—1201no—
491C4AGL(1,𝔽5)×#6711yescharacteristic
492C4#67⋊AGL(1,𝔽5)11yescharacteristic
493C3#1060467211yescharacteristic, complemented, Sylow
494C2—51no—
495C2—51no—
496C2—52no—
497C2D3×#400912yes—
498C2#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#1956
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
16. #91833C4semidirect
17. C5×#944C4semidirect
20. #91389C4semidirect
21. #91385C4semidirect
146. #944AGL(1,𝔽5)semidirect
213. Dic15#103semidirect
218. C5×Dic3#103semidirect
410. C3×C5#1956semidirect
427. Dic3#4009semidirect
475. C5#70225semidirect
493. C3#10604672semidirect

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
#117518602
Dic3×D5⋊C82
C2×#911542
C2×#911532
#106046723
#702255