Finite Group Databasebeta

← #14307470

Subgroups — #14307470

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753611C211yesmaximal, characteristic
2#11751824C211yesmaximal, characteristic
3#11751748C211yesmaximal, characteristic
4#11746746C211yesmaximal, characteristic
5C3×#64875C211yesmaximal, characteristic
6#11746526C211yesmaximal, characteristic
7#11746524C211yesmaximal, characteristic
8#10604588—31nomaximal, Hall
9C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
10C2×#6758C2211yescharacteristic
11C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
12C2×C3×#700C2211yescharacteristic
13#91777C411yescharacteristic, complemented
14C5×#890C411yescharacteristic, complemented
15C3×#4038C2211yescharacteristic
16#91470C411yescharacteristic, complemented
17#91431—21no—
18#91430—21no—
19C4×D5×Dic3C2211yescharacteristic
20#91381C411yescharacteristic, complemented
21#91358C2211yescharacteristic
22#91333—21no—
23#91323—21no—
24Dic3⋊#170—21no—
25Dic3⋊#170—21no—
26(C2×AGL(1,𝔽5))⋊Dic3—21no—
27(C2×AGL(1,𝔽5))⋊Dic3—21no—
28#68732—51nomaximal, Hall
29#66022—31no—
30#65686—31no—
31#65681—31no—
32#64875—31no—
33#64875—31no—
34#64875—31no—
35#64875D311yescharacteristic
36C22×ΓL(1,𝔽16)—21no—
37C22×C3×AGL(1,𝔽5)—21no—
38C2×Dic30—21no—
39C2×C3×C4×D5C2311yescharacteristic
40C2×#1204—21no—
41C2×D5×Dic3D411yescharacteristic
42C2×D5×Dic3D411yescharacteristic
43C2×C3⋊#170—21no—
44C4×ΓL(1,𝔽16)—21no—
45C4×ΓL(1,𝔽16)Q812yescomplemented
46#6758—21no—
47#6758—21no—
48C2×C3×#170—21no—
49C3×C4×AGL(1,𝔽5)—21no—
50C3×C4×AGL(1,𝔽5)Q812yescomplemented
51C3×#700—21no—
52C3×#700—21no—
53C3×C5×C4⋊C4C2×C411yescharacteristic
54C4⋊Dic15C2×C411yescharacteristic
55#6712—21no—
56C4×Dic15C2×C411yescharacteristic
57C5×C4⋊Dic3—21no—
58C5×C4⋊Dic3C2×C411yescharacteristic
59C5×Dic3⋊C4—21no—
60C4×C5×Dic3C2×C411yescharacteristic
61C3×#685—21no—
62C3×C4⋊Dic5C2×C411yescharacteristic
63C3×C4×Dic5—21no—
64#6669—21no—
65#6669—21no—
66#6668—21no—
67#6665—21no—
68Dic3×Dic5—21no—
69Dic3×Dic5D411yescharacteristic
70Dic3×Dic5D411yescharacteristic
71#5517—51no—
72#5118—51no—
73#4932—51no—
74C3×#339—51no—
75#4555—51no—
76#4532—51no—
77#4523—51no—
78C2×C4×AGL(1,𝔽5)—31no—
79C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
80C2×#700—31no—
81C2×#700C2×D311yescharacteristic
82C5×#125—31no—
83#4042—61no—
84#4042—61no—
85#4038—31no—
86#4038C2×D311yescharacteristic
87#4034—31no—
88#4034—31no—
89#4033—61no—
90#4033—61no—
91C42×D5—31no—
92#4016—31no—
93(C2×AGL(1,𝔽5))⋊C4—61no—
94(C2×AGL(1,𝔽5))⋊C4—61no—
95C4⋊#170—61no—
96C4⋊#170—61no—
97#1995—151noSylow
98C22×C3×D5C2×D411yescharacteristic
99C2×ΓL(1,𝔽16)—22no—
100C2×ΓL(1,𝔽16)—41no—
101C2×ΓL(1,𝔽16)—41no—
102C2×C3×AGL(1,𝔽5)—22no—
103C2×C3×AGL(1,𝔽5)—41no—
104C2×C3×AGL(1,𝔽5)—41no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5C22×C411yescharacteristic
107C2×Dic15—21no—
108C2×Dic15C22⋊C411yescharacteristic
109C2×Dic15C22⋊C411yescharacteristic
110Dic30—42no—
111C2×C5×Dic3—21no—
112C2×C5×Dic3C22⋊C411yescharacteristic
113C2×C5×Dic3C22⋊C411yescharacteristic
114C2×C3×Dic5—21no—
115C2×C3×Dic5C2×D411yescharacteristic
116C3×C4×D5C2×Q811yescharacteristic
117C3×C4×D5C2×Q811yescharacteristic
118C3×C4×D5#4111yescharacteristic
119C3×C4×D5#4111yescharacteristic
120#1204—42no—
121D5×Dic3—41no—
122D5×Dic3—41no—
123C3⋊#170—41no—
124C3×#170—41no—
125C2×C3×M16—51no—
126C3×#117—51no—
127C2×C3×C42—51no—
128#926—51no—
129C2×C4×Dic3—51no—
130C2×C4×Dic3—51no—
131C2×#208—51no—
132#890—51no—
133#890AGL(1,𝔽5)11yescharacteristic, complemented
134#888—51no—
135#888—51no—
136#878—101no—
137#878—101no—
138C4×Dic6—101no—
139C4×Dic6—101no—
140C3⋊#95—101no—
141C3⋊#95—101no—
142Dic3⋊C8—101no—
143Dic3⋊C8—101no—
144C22×AGL(1,𝔽5)—21no—
145C22×AGL(1,𝔽5)—61no—
146C2×C4×D5—31no—
147C2×C4×D5—31no—
148C2×C4×D5C22×D311yescharacteristic
149C2×Dic10—61no—
150C2×Dic10—61no—
151C2×#170—21no—
152C2×#170—61no—
153C4×AGL(1,𝔽5)—21no—
154C4×AGL(1,𝔽5)—32no—
155C4×AGL(1,𝔽5)—61no—
156C4×AGL(1,𝔽5)Dic612yescomplemented
157#700—21no—
158#700—21no—
159#700—61no—
160#700—61no—
161C5×C4⋊C4—31no—
162C5×C4⋊C4—61no—
163C5×C4⋊C4—61no—
164C5×C4⋊C4C4×D311yescharacteristic
165C42×C5—31no—
166#685—21no—
167#685—61no—
168C4⋊Dic5—31no—
169C4⋊Dic5C4×D311yescharacteristic
170#683—61no—
171#683—61no—
172#683—61no—
173#683—61no—
174C4×Dic5—21no—
175C4×Dic5—31no—
176C4×Dic5—31no—
177C4×Dic5—31no—
178C4×Dic5—61no—
179#533—151no—
180#423—151no—
181#376—151no—
182#339—51no—
183#339—151no—
184#339—151no—
185#339—151no—
186C22×C3×C5C2×C22⋊C411yescharacteristic
187C2×C3×D5—21no—
188C2×C3×D5—21no—
189C2×C3×D5C22⋊Q811yescharacteristic
190C2×C3×D5C22⋊Q811yescharacteristic
191ΓL(1,𝔽16)—42no—
192C3×AGL(1,𝔽5)—42no—
193C3×C4×C5—41no—
194C3×C4×C5C4×Q811yescharacteristic
195C3×C4×C5C4×Q811yescharacteristic
196Dic15—21no—
197Dic15—21no—
198Dic15—41no—
199C3×Dic5—41no—
200C3×Dic5#12411yescharacteristic
201C3×Dic5C4⋊Q811yescharacteristic
202C5×Dic3—21no—
203C5×Dic3—21no—
204C5×Dic3—41no—
205C22×C3×C4—51no—
206C22×C3×C4—101no—
207C22×Dic3—51no—
208C22×Dic3—51no—
209C22×Dic3—101no—
210C2×Dic6—101no—
211C2×Dic6—101no—
212C3×M16—101no—
213C3×M16—101no—
214C2×C3×C8—101no—
215C3×C4⋊C4—51no—
216C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
217C3×C22⋊C4—101no—
218C3×C42—52no—
219C3×C42—101no—
220C3×C42—101no—
221#217—101no—
222C4⋊Dic3—21no—
223C4⋊Dic3—51no—
224C4⋊Dic3—101no—
225C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
226Dic3⋊C4—21no—
227Dic3⋊C4—101no—
228Dic3⋊C4—101no—
229Dic3⋊C4—101no—
230C4×Dic3—51no—
231C4×Dic3—51no—
232C4×Dic3—51no—
233C4×Dic3—52no—
234C4×Dic3—101no—
235C4×Dic3—101no—
236C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
237#208—101no—
238#208—101no—
239C2×C3⋊C8—101no—
240C22×D5D3×D411yescharacteristic
241C2×AGL(1,𝔽5)—22no—
242C2×AGL(1,𝔽5)—41no—
243C2×AGL(1,𝔽5)—41no—
244C2×AGL(1,𝔽5)—62no—
245C2×AGL(1,𝔽5)—121no—
246C2×AGL(1,𝔽5)—121no—
247C2×C4×C5—21no—
248C2×C4×C5—31no—
249C2×C4×C5—31no—
250C2×C4×C5—61no—
251C2×C4×C5C2×C4×D311yescharacteristic
252C2×Dic5—21no—
253C2×Dic5—31no—
254C2×Dic5—31no—
255C2×Dic5—61no—
256C2×Dic5D3×D411yescharacteristic
257C4×D5—121no—
258C4×D5—121no—
259C4×D5C2×Dic611yescharacteristic
260C4×D5#23511yescharacteristic
261C4×D5#23711yescharacteristic
262C4×D5D3×Q811yescharacteristic
263Dic10—122no—
264Dic10—122no—
265#170—41no—
266#170—121no—
267C2×M16—51no—
268C2×M16—151no—
269C4⋊Q8—151no—
270C4⋊Q8—151no—
271#125—31no—
272#125—151no—
273C22⋊Q8—301no—
274C22⋊Q8—301no—
275C4×Q8—301no—
276C4×Q8—301no—
277#117—51no—
278#117—151no—
279C2×C42—51no—
280C2×C42—151no—
281C2×C42—151no—
282C4⋊C8—301no—
283C4⋊C8—301no—
284#95—301no—
285#95—301no—
286C2×C3×C5#39111yescharacteristic
287C2×C3×C5#42111yescharacteristic
288C2×C3×C5#42111yescharacteristic
289C3×D5—21no—
290C3×D5—21no—
291C23×C3—51no—
292C2×C3×C4—21no—
293C2×C3×C4—51no—
294C2×C3×C4—51no—
295C2×C3×C4—51no—
296C2×C3×C4—51no—
297C2×C3×C4—51no—
298C2×C3×C4—102no—
299C2×C3×C4—101no—
300C2×C3×C4—201no—
301C2×C3×C4—201no—
302C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
303C2×Dic3—21no—
304C2×Dic3—51no—
305C2×Dic3—51no—
306C2×Dic3—102no—
307C2×Dic3—101no—
308C2×Dic3—201no—
309C2×Dic3—201no—
310C2×Dic3—201no—
311C2×Dic3—201no—
312C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
313C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
314Dic6—202no—
315Dic6—202no—
316C3×C8—201no—
317C3⋊C8—201no—
318C22×C5D3×C22⋊C411yescharacteristic
319C2×D5—21no—
320C2×D5—21no—
321C2×D5#87811yescharacteristic
322C2×D5#89611yescharacteristic
323AGL(1,𝔽5)—42no—
324AGL(1,𝔽5)—122no—
325C4×C5—41no—
326C4×C5—61no—
327C4×C5—61no—
328C4×C5—121no—
329C4×C5C4×Dic611yescharacteristic
330C4×C5#88711yescharacteristic
331Dic5—41no—
332Dic5—61no—
333Dic5—61no—
334Dic5—121no—
335Dic5#88511yescharacteristic
336Dic5#88811yescharacteristic
337C2×Q8—301no—
338C2×Q8—301no—
339C22×C4—51no—
340C22×C4—101no—
341C22×C4—151no—
342C22×C4—151no—
343C22×C4—301no—
344M16—101no—
345M16—101no—
346M16—301no—
347M16—301no—
348C2×C8—101no—
349C2×C8—301no—
350C4⋊C4—31no—
351C4⋊C4—51no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—151no—
355C4⋊C4—301no—
356C4⋊C4—301no—
357C4⋊C4—301no—
358C4⋊C4—301no—
359C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
360C22⋊C4—101no—
361C22⋊C4—301no—
362C42—31no—
363C42—52no—
364C42—101no—
365C42—101no—
366C42—151no—
367C42—151no—
368C42—152no—
369C42—151no—
370C42—301no—
371C42—301no—
372C3×C5#199511yescharacteristic, complemented, Hall
373C22×C3—51no—
374C22×C3—51no—
375C22×C3—101no—
376C22×C3—101no—
377C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
378C3×C4—41no—
379C3×C4—51no—
380C3×C4—51no—
381C3×C4—202no—
382C3×C4—201no—
383C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
384C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
385Dic3—21no—
386Dic3—21no—
387Dic3—41no—
388Dic3—101no—
389Dic3—101no—
390Dic3—202no—
391Dic3—201no—
392C2×C5#464611yescharacteristic
393C2×C5#482111yescharacteristic
394C2×C5#482111yescharacteristic
395D5—21no—
396D5—21no—
397C23—51no—
398Q8—602no—
399Q8—602no—
400C2×C4—21no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—51no—
404C2×C4—51no—
405C2×C4—51no—
406C2×C4—51no—
407C2×C4—51no—
408C2×C4—61no—
409C2×C4—102no—
410C2×C4—101no—
411C2×C4—151no—
412C2×C4—151no—
413C2×C4—201no—
414C2×C4—201no—
415C2×C4—302no—
416C2×C4—301no—
417C2×C4—601no—
418C2×C4—601no—
419C2×C4—601no—
420C2×C4—601no—
421C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
422C8—201no—
423C8—601no—
424C2×C3—101no—
425C2×C3—101no—
426C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
427C2×C3#6579511yescharacteristic
428C2×C3#6579511yescharacteristic
429C5#6873211yescharacteristic, complemented, Sylow
430C22—51no—
431C22—51no—
432C22—101no—
433C22—101no—
434C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
435C4—41no—
436C4—51no—
437C4—51no—
438C4—61no—
439C4—61no—
440C4—121no—
441C4—202no—
442C4—201no—
443C4—301no—
444C4—301no—
445C4—602no—
446C4—601no—
447C4AGL(1,𝔽5)×Dic611yescharacteristic
448C4ΓL(1,𝔽16)⋊Q811yescharacteristic
449C3#1060458811yescharacteristic, complemented, Sylow
450C2—101no—
451C2—101no—
452C2#1175184211yescharacteristic
453C2#1175194911yescharacteristic
454C2#1175197311yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1995
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
13. #91777C4semidirect
14. C5×#890C4semidirect
16. #91470C4semidirect
20. #91381C4semidirect
45. C4×ΓL(1,𝔽16)Q8semidirect
50. C3×C4×AGL(1,𝔽5)Q8semidirect
133. #890AGL(1,𝔽5)semidirect
156. C4×AGL(1,𝔽5)Dic6semidirect
372. C3×C5#1995semidirect
429. C5#68732semidirect
449. C3#10604588semidirect

Maximal subgroups

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

subgroupindex
#117536112
#117518242
#117517482
#117467462
C3×#648752
#117465262
#117465242
#106045883
#687325