Finite Group Databasebeta

← #14283142

Subgroups — #14283142

Order 1920.

692 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#930C211yesmaximal, characteristic
2#11752414C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4#11746497C212yesmaximal, complemented
5#11746496C212yesmaximal, complemented
6#10604468—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×D5×#67—21no—
9C2×D5×D12C411yescharacteristic, complemented
10C2×#6762—21no—
11C2×C3×C22⋊AGL(1,𝔽5)—21no—
12C2×C3×C4×AGL(1,𝔽5)—21no—
13#91819C411yescharacteristic, complemented
14C5×#930C411yescharacteristic, complemented
15#91566—21no—
16D5×(C2×D3)⋊C4—21no—
17#91450C2211yescharacteristic
18#91449C2211yescharacteristic
19#91448—21no—
20D5×C4⋊Dic3C411yescharacteristic, complemented
21#91232—21no—
22#91232C2212yescomplemented
23C3×(C2×AGL(1,𝔽5))⋊C4C2212yescomplemented
24#69444—51nomaximal, Hall
25D5×#121—31no—
26C4×C22⋊AGL(1,𝔽5)D311yescharacteristic, complemented
27#65755—31no—
28#64920—32no—
29#64868—32no—
30C23×C3×D5D411yescharacteristic
31C22×D3×D5—21no—
32C22×C3×AGL(1,𝔽5)—22no—
33C22×C3×C4×C5C2×C411yescharacteristic
34C2×#1220—21no—
35C2×D60C2×C411yescharacteristic
36C2×C5×#67—21no—
37C2×C5×D12C2×C411yescharacteristic
38C22×C3×Dic5D411yescharacteristic
39C2×C3×C4×D5—42no—
40C2×C3×C4×D5C2×C411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5C2311yescharacteristic
44D5×#67—44no—
45C2×#1203—21no—
46C2×#1202—21no—
47C2×#1201—21no—
48C2×D5×Dic3—21no—
49D5×D12—42no—
50#6762—42no—
51C2×C3⋊#170—22no—
52C3×C22⋊AGL(1,𝔽5)—24no—
53C3×C4×AGL(1,𝔽5)—42no—
54#6714—21no—
55C4⋊Dic15C2×C411yescharacteristic
56C5×(C2×D3)⋊C4—21no—
57C5×C4⋊Dic3C2×C411yescharacteristic
58#6668—21no—
59#6667—21no—
60(C2×D3)⋊Dic5—21no—
61C2×#930—51no—
62C3×C4×C22⋊C4—51no—
63#5207—51no—
64#4451—52no—
65#4451—52no—
66C2×D4×D5—31no—
67C2×D4×D5—61no—
68C22×C4×D5C2×D311yescharacteristic
69C2×C22⋊AGL(1,𝔽5)—21no—
70C2×#704—61no—
71C2×C4×AGL(1,𝔽5)—21no—
72C5×#121—31no—
73#4085—61no—
74#4084—31no—
75#4076—31no—
76#4041—31no—
77D5×C4⋊C4—31no—
78#4030—61no—
79D5×C22⋊C4—61no—
80(C2×AGL(1,𝔽5))⋊C4C2×D312yescomplemented
81#4003—32no—
82#4003—61no—
83#1507—151noSylow
84C23×C3×C5C22⋊C411yescharacteristic
85C22×D15—21no—
86C22×C5×D3—21no—
87C22×C3×D5—22no—
88C22×C3×D5—22no—
89C22×C3×D5—41no—
90C22×C3×D5C22⋊C411yescharacteristic
91C22×C3×D5C2×D411yescharacteristic
92C2×D3×D5—42no—
93C2×D3×D5—42no—
94C2×D3×D5—42no—
95C2×C3×AGL(1,𝔽5)—24no—
96C2×C3×AGL(1,𝔽5)—42no—
97C2×C3×AGL(1,𝔽5)—42no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C2×C3×C4×C5C22×C411yescharacteristic
101#1220—42no—
102C2×Dic15—21no—
103D60—41no—
104C5×#67—42no—
105C2×C5×Dic3—21no—
106C5×D12—41no—
107C2×C3×Dic5—22no—
108C2×C3×Dic5C22⋊C411yescharacteristic
109C2×C3×Dic5C2×D411yescharacteristic
110C3×C4×D5—22no—
111C3×C4×D5—22no—
112C3×C4×D5—41no—
113C3×C4×D5—41no—
114#1203—42no—
115#1202—42no—
116#1201—42no—
117D5×Dic3—42no—
118C3⋊#170—42no—
119C23×C3×C4—51no—
120C22×#67—101no—
121C22×D12—51no—
122C2×C3×C22⋊C4—101no—
123C2×C3×C42—101no—
124#930—51no—
125#930—51no—
126#930—51no—
127#930—101no—
128#930—101no—
129#930AGL(1,𝔽5)11yescharacteristic, complemented
130C2×(C2×D3)⋊C4—101no—
131C2×C4⋊Dic3—51no—
132C2×#208—101no—
133C3×#95—52no—
134#830—52no—
135#830—101no—
136C23×D5—61no—
137C23×D5#6711yescharacteristic
138C22×AGL(1,𝔽5)—22no—
139C2×C5×D4—31no—
140C2×C5×D4—61no—
141C22×C4×C5C4×D311yescharacteristic
142C2×C22⋊D5—61no—
143C2×C22⋊D5—61no—
144C2×C22⋊D5—61no—
145C22×Dic5D1211yescharacteristic
146D4×D5—122no—
147D4×D5—124no—
148C2×D20—31no—
149C2×D20—61no—
150C2×C4×D5—42no—
151C2×C4×D5—61no—
152C2×C4×D5C4×D311yescharacteristic
153C2×C4×D5D1211yescharacteristic
154C2×C4×D5#6711yescharacteristic
155C2×C4×D5C22×D311yescharacteristic
156C22⋊AGL(1,𝔽5)—24no—
157#704—122no—
158C2×#170—62no—
159C4×AGL(1,𝔽5)—42no—
160C5×C4⋊C4—31no—
161C5×C22⋊C4—61no—
162C22⋊Dic5—61no—
163#685—61no—
164#685—61no—
165C4⋊Dic5—31no—
166#683—61no—
167C2×#121—151no—
168#407—151no—
169C4×C22⋊C4—51no—
170#327—152no—
171#327—152no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5C2×C22⋊C411yescharacteristic
176C2×D15—41no—
177C2×D15—41no—
178C2×D15—41no—
179C2×C5×D3—41no—
180C2×C5×D3—41no—
181C2×C5×D3—41no—
182C2×C3×D5—21no—
183C2×C3×D5—21no—
184C2×C3×D5—21no—
185C2×C3×D5—21no—
186C2×C3×D5—44no—
187C2×C3×D5#10412yes—
188C2×C3×D5AGL(1,ℤ/8)12yes—
189D3×D5—84no—
190C3×AGL(1,𝔽5)—44no—
191C3×C4×C5—21no—
192C3×C4×C5—21no—
193Dic15—41no—
194C3×Dic5—21no—
195C3×Dic5#9912yes—
196C5×Dic3—41no—
197C24×C3—51no—
198C23×D3—101no—
199C22×C3×C4—51no—
200C22×C3×C4—51no—
201C22×C3×C4—51no—
202C22×C3×C4—51no—
203C22×C3×C4—51no—
204C22×C3×C4—102no—
205C22×C3×C4—202no—
206C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
207C2×#67—21no—
208C2×#67—101no—
209C2×#67—101no—
210C2×#67—101no—
211C2×#67—204no—
212C22×Dic3—101no—
213C2×D12—51no—
214C2×D12—101no—
215C2×D12—202no—
216C2×D12C2×AGL(1,𝔽5)11yescharacteristic
217C3×C22⋊C4—104no—
218C3×C42—202no—
219(C2×D3)⋊C4—21no—
220(C2×D3)⋊C4—101no—
221(C2×D3)⋊C4—101no—
222(C2×D3)⋊C4—101no—
223C4⋊Dic3—51no—
224C4⋊Dic3—101no—
225C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
226#208—202no—
227C2×C3⋊C8—102no—
228C23×C5—61no—
229C23×C5(C2×D3)⋊C411yescharacteristic
230C22×D5—22no—
231C22×D5—22no—
232C22×D5—41no—
233C22×D5—61no—
234C22×D5—122no—
235C22×D5—122no—
236C22×D5—122no—
237C22×D5(C2×D3)⋊C411yescharacteristic
238C22×D5C2×#6711yescharacteristic
239C2×AGL(1,𝔽5)—24no—
240C2×AGL(1,𝔽5)—42no—
241C2×AGL(1,𝔽5)—42no—
242C5×D4—121no—
243C5×D4—122no—
244C2×C4×C5—41no—
245C2×C4×C5—61no—
246C2×C4×C5(C2×D3)⋊C411yescharacteristic
247C2×C4×C5C2×C4×D311yescharacteristic
248C22⋊D5—122no—
249C22⋊D5—122no—
250C22⋊D5—122no—
251C2×Dic5—22no—
252C2×Dic5—61no—
253C2×Dic5(C2×D3)⋊C411yescharacteristic
254C2×Dic5C2×D1211yescharacteristic
255D20—121no—
256D20—122no—
257C4×D5—22no—
258C4×D5—22no—
259C4×D5—41no—
260C4×D5—41no—
261C4×D5—122no—
262#170—122no—
263C22×D4—151no—
264C22×D4—301no—
265C23×C4—51no—
266C2×M16—301no—
267#121—31no—
268#121—151no—
269#121—151no—
270#121—151no—
271#121—301no—
272#121—301no—
273C2×C4⋊C4—151no—
274C2×C22⋊C4—101no—
275C2×C22⋊C4—301no—
276C2×C42—101no—
277C22⋊C8—152no—
278C22⋊C8—301no—
279#95—52no—
280C2×C3×C5—41no—
281C2×C3×C5C2×#9911yescharacteristic
282C2×C3×C5#41811yescharacteristic
283C2×C3×C5C2×#10411yescharacteristic
284D15—81no—
285C3×D5—22no—
286C3×D5—22no—
287C5×D3—81no—
288C23×C3—51no—
289C23×C3—51no—
290C23×C3—102no—
291C23×C3—102no—
292C23×C3—201no—
293C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
294C22×D3—21no—
295C22×D3—101no—
296C22×D3—202no—
297C22×D3—202no—
298C22×D3—202no—
299C2×C3×C4—41no—
300C2×C3×C4—51no—
301C2×C3×C4—51no—
302C2×C3×C4—102no—
303C2×C3×C4—102no—
304C2×C3×C4—102no—
305C2×C3×C4—104no—
306C2×C3×C4—201no—
307C2×C3×C4—201no—
308C2×C3×C4—202no—
309C2×C3×C4—202no—
310C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
311C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
312#67—42no—
313#67—202no—
314#67—202no—
315#67—202no—
316C2×Dic3—21no—
317C2×Dic3—101no—
318C2×Dic3—202no—
319D12—41no—
320D12—201no—
321D12—202no—
322C3⋊C8—202no—
323C22×C5—21no—
324C22×C5—21no—
325C22×C5—21no—
326C22×C5—121no—
327C22×C5—121no—
328C22×C5—121no—
329C22×C5C2×(C2×D3)⋊C411yescharacteristic
330C2×D5—21no—
331C2×D5—21no—
332C2×D5—21no—
333C2×D5—21no—
334C2×D5—44no—
335C2×D5—121no—
336C2×D5—121no—
337C2×D5—121no—
338C2×D5—244no—
339C2×D5#80512yes—
340C2×D5#94912yes—
341AGL(1,𝔽5)—44no—
342C4×C5—21no—
343C4×C5—21no—
344C4×C5—121no—
345Dic5—21no—
346Dic5—121no—
347Dic5#80612yes—
348C24—51no—
349C24—301no—
350C2×D4—31no—
351C2×D4—61no—
352C2×D4—151no—
353C2×D4—301no—
354C2×D4—301no—
355C2×D4—301no—
356C2×D4—301no—
357C2×D4—602no—
358C2×D4—604no—
359C22×C4—51no—
360C22×C4—51no—
361C22×C4—51no—
362C22×C4—51no—
363C22×C4—51no—
364C22×C4—102no—
365C22×C4—202no—
366C22×C4—301no—
367C22×C4D3×AGL(1,𝔽5)11yescharacteristic
368M16—602no—
369C2×C8—302no—
370C4⋊C4—31no—
371C4⋊C4—151no—
372C4⋊C4—301no—
373C22⋊C4—61no—
374C22⋊C4—104no—
375C22⋊C4—301no—
376C22⋊C4—301no—
377C22⋊C4—301no—
378C42—202no—
379C3×C5#150711yescharacteristic, complemented, Hall
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—52no—
384C22×C3—52no—
385C22×C3—101no—
386C22×C3—101no—
387C22×C3—101no—
388C22×C3—101no—
389C22×C3—204no—
390C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
391C2×D3—41no—
392C2×D3—41no—
393C2×D3—41no—
394C2×D3—201no—
395C2×D3—201no—
396C2×D3—201no—
397C2×D3—404no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—52no—
401C3×C4—101no—
402C3×C4—204no—
403Dic3—41no—
404Dic3—201no—
405C2×C5—41no—
406C2×C5—241no—
407C2×C5C2×#80511yescharacteristic
408C2×C5C2×#80611yescharacteristic
409C2×C5#500111yescharacteristic
410D5—22no—
411D5—22no—
412D5—241no—
413C23—51no—
414C23—51no—
415C23—61no—
416C23—102no—
417C23—102no—
418C23—201no—
419C23—301no—
420C23—602no—
421C23—602no—
422C23—602no—
423C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
424D4—121no—
425D4—122no—
426D4—601no—
427D4—602no—
428D4—602no—
429D4—602no—
430D4—602no—
431C2×C4—41no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—61no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—102no—
438C2×C4—104no—
439C2×C4—201no—
440C2×C4—201no—
441C2×C4—202no—
442C2×C4—202no—
443C2×C4—301no—
444C2×C4—602no—
445C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
446C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
447C8—602no—
448C2×C3—41no—
449C2×C3—102no—
450C2×C3—102no—
451C2×C3#6576711yescharacteristic
452C2×C3D5⋊#9911yescharacteristic
453C2×C3D5⋊#10411yescharacteristic
454D3—81no—
455D3—401no—
456C5#6944411yescharacteristic, complemented, Sylow
457C22—21no—
458C22—21no—
459C22—21no—
460C22—52no—
461C22—52no—
462C22—101no—
463C22—101no—
464C22—101no—
465C22—101no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22—204no—
470C22—601no—
471C22—601no—
472C22—601no—
473C22—1204no—
474C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
475C4—21no—
476C4—21no—
477C4—52no—
478C4—101no—
479C4—121no—
480C4—204no—
481C4—601no—
482C3#1060446811yescharacteristic, complemented, Sylow
483C2—41no—
484C2—102no—
485C2—102no—
486C2—241no—
487C2—1201no—
488C2#1175170611yescharacteristic
489C2D5⋊#80511yescharacteristic
490C2D5⋊#80611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalD5×#930C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1507
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752414C2semidirect
3. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
4. #11746497C2semidirect
5. #11746496C2semidirect
9. C2×D5×D12C4semidirect
13. #91819C4semidirect
14. C5×#930C4semidirect
20. D5×C4⋊Dic3C4semidirect
22. #91232C22semidirect
23. C3×(C2×AGL(1,𝔽5))⋊C4C22semidirect
26. C4×C22⋊AGL(1,𝔽5)D3semidirect
80. (C2×AGL(1,𝔽5))⋊C4C2×D3semidirect
129. #930AGL(1,𝔽5)semidirect
379. C3×C5#1507semidirect
456. C5#69444semidirect
482. C3#10604468semidirect

Maximal subgroups

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

subgroupindex
D5×#9302
#117524142
C3×C4×C22⋊AGL(1,𝔽5)2
#117464972
#117464962
#106044683
#694445