Finite Group Databasebeta

← #14319705

Subgroups — #14319705

Order 1920.

1392 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91923C212yesmaximal
2C2×D5×Dic3⋊C4C211yesmaximal, characteristic
3C4×#6763C211yesmaximal, characteristic
4C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
5Dic3×C22⋊AGL(1,𝔽5)C212yesmaximal
6#11751833C214yesmaximal, complemented
7AGL(1,𝔽5)×Dic3⋊C4C214yesmaximal, complemented
8#10617098—31nomaximal, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C22×D5×Dic3C2211yescharacteristic
11C22×D5×Dic3C2211yescharacteristic
12C2×#6763C2212yes—
13C2×C4×ΓL(1,𝔽16)C2212yes—
14C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
15C2×C3×C4×AGL(1,𝔽5)C2212yes—
16C2×Dic3⋊AGL(1,𝔽5)C2214yes—
17#91923C4116yescomplemented
18C2×Dic3×AGL(1,𝔽5)C2214yes—
19C2×Dic3×AGL(1,𝔽5)C2214yes—
20C2×#6712C411yescharacteristic, complemented
21C2×C5×Dic3⋊C4C411yescharacteristic, complemented
22C2×Dic3⋊Dic5C411yescharacteristic, complemented
23C2×#6669C411yescharacteristic, complemented
24D5×Dic3⋊C4C412yescomplemented
25D5×Dic3⋊C4C412yescomplemented
26D5×Dic3⋊C4C2212yes—
27D5×Dic3⋊C4C2212yes—
28#91235C2212yes—
29C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
30(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
31#73435—51nomaximal, Hall
32C2×#4130—32no—
33C2×D5×C4⋊C4—31no—
34C4×C22⋊AGL(1,𝔽5)—31no—
35C4×C22⋊AGL(1,𝔽5)—32no—
36C4×C22⋊AGL(1,𝔽5)D311yescharacteristic
37C4⋊C4×AGL(1,𝔽5)—34no—
38C4⋊C4×AGL(1,𝔽5)—34no—
39C23×C3×D5C2311yescharacteristic
40C22×ΓL(1,𝔽16)C2314yes—
41C22×C3×AGL(1,𝔽5)C2314yes—
42C22×C3×C4×C5C2×C411yescharacteristic
43C22×Dic15C2×C411yescharacteristic
44C22×Dic15C2×C411yescharacteristic
45C22×C5×Dic3C2×C411yescharacteristic
46C22×C5×Dic3C2×C411yescharacteristic
47C22×C3×Dic5C2×C411yescharacteristic
48C2×C3×C4×D5—42no—
49C2×C3×C4×D5C2×C411yescharacteristic
50C2×C3×C4×D5C2×C411yescharacteristic
51C2×C3×C4×D5C2311yescharacteristic
52C2×C3×C4×D5C2311yescharacteristic
53C2×D5×Dic3—42no—
54C2×D5×Dic3C2×C412yes—
55C2×D5×Dic3C2×C412yes—
56C2×D5×Dic3C2×C418yes—
57C2×D5×Dic3C2312yes—
58C2×D5×Dic3C2312yes—
59#6763C2×C418yes—
60C4×ΓL(1,𝔽16)—42no—
61C3×C22⋊AGL(1,𝔽5)C2×C418yes—
62C3×C4×AGL(1,𝔽5)—42no—
63Dic3⋊AGL(1,𝔽5)C2×C4116yescomplemented
64Dic3×AGL(1,𝔽5)—44no—
65Dic3×AGL(1,𝔽5)C2×C4116yescomplemented
66#6712C2×C412yes—
67#6712C2×C412yes—
68C5×Dic3⋊C4C2×C412yes—
69C5×Dic3⋊C4C2×C412yes—
70Dic3⋊Dic5C2×C412yes—
71Dic3⋊Dic5C2×C412yes—
72#6669C2×C412yes—
73#6669C2×C412yes—
74C22×Dic3⋊C4—51no—
75C2×#877—52no—
76C3×C4×C22⋊C4—51no—
77C4×#217—51no—
78Dic3×C22⋊C4—52no—
79C4×Dic3⋊C4—54no—
80#4649—54no—
81C22×C4×D5—31no—
82C22×C4×D5—31no—
83C22×C4×D5C2×D311yescharacteristic
84C2×C22⋊AGL(1,𝔽5)—32no—
85C2×C22⋊AGL(1,𝔽5)C2×D312yes—
86#4130—316no—
87C2×C4⋊AGL(1,𝔽5)—34no—
88C2×C4×AGL(1,𝔽5)—34no—
89C2×C4×AGL(1,𝔽5)—34no—
90C2×C4×AGL(1,𝔽5)—32no—
91C2×C4×AGL(1,𝔽5)C2×D312yes—
92C2×C5×C4⋊C4—31no—
93C2×C4⋊Dic5—31no—
94C2×#683—31no—
95C2×#683—31no—
96D5×C4⋊C4—32no—
97D5×C4⋊C4—32no—
98D5×C4⋊C4—32no—
99D5×C4⋊C4—32no—
100(C2×AGL(1,𝔽5))⋊C4—34no—
101(C2×AGL(1,𝔽5))⋊C4—32no—
102(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
103#2274—151noSylow
104C23×C3×C5C22×C411yescharacteristic
105C22×C3×D5—22no—
106C22×C3×D5—22no—
107C22×C3×D5C22×C411yescharacteristic
108C22×C3×D5C22×C414yes—
109C22×C3×D5C2411yescharacteristic
110C2×ΓL(1,𝔽16)—24no—
111C2×ΓL(1,𝔽16)—24no—
112C2×ΓL(1,𝔽16)C22×C418yes—
113C2×C3×AGL(1,𝔽5)—24no—
114C2×C3×AGL(1,𝔽5)—24no—
115C2×C3×AGL(1,𝔽5)C22×C418yes—
116C2×C3×C4×C5—41no—
117C2×C3×C4×C5C22×C411yescharacteristic
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×Dic15—41no—
120C2×Dic15C4214yescomplemented
121C2×Dic15C22×C412yes—
122C2×Dic15C22×C412yes—
123C2×C5×Dic3—41no—
124C2×C5×Dic3C4214yescomplemented
125C2×C5×Dic3C22×C412yes—
126C2×C5×Dic3C22×C412yes—
127C2×C3×Dic5—41no—
128C2×C3×Dic5C22×C411yescharacteristic
129C2×C3×Dic5C22×C411yescharacteristic
130C3×C4×D5—22no—
131C3×C4×D5—22no—
132C3×C4×D5—41no—
133C3×C4×D5—41no—
134D5×Dic3—24no—
135D5×Dic3—42no—
136D5×Dic3C4218yescomplemented
137D5×Dic3C22×C414yes—
138D5×Dic3C22×C414yes—
139C23×C3×C4—51no—
140C23×Dic3—51no—
141C23×Dic3—51no—
142C2×C3×C22⋊C4—52no—
143C2×C3×C42—52no—
144C2×#217—52no—
145C2×Dic3⋊C4—54no—
146C2×Dic3⋊C4—51no—
147C2×Dic3⋊C4—51no—
148C2×Dic3⋊C4—51no—
149C2×Dic3⋊C4—52no—
150C2×Dic3⋊C4—52no—
151C2×Dic3⋊C4—52no—
152C2×Dic3⋊C4—52no—
153C2×Dic3⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
154C2×C4×Dic3—54no—
155C2×C4×Dic3—54no—
156C2×C4×Dic3—52no—
157#877—516no—
158C3×#95—52no—
159C3⋊#95—54no—
160C3⋊#95—52no—
161C23×D5C22×D311yescharacteristic
162C22×AGL(1,𝔽5)—34no—
163C22×AGL(1,𝔽5)C22×D314yes—
164C22×C4×C5—31no—
165C22×C4×C5—31no—
166C22×C4×C5C4×D311yescharacteristic
167C22×Dic5—31no—
168C22×Dic5—31no—
169C22×Dic5C4×D311yescharacteristic
170C2×C4×D5—38no—
171C2×C4×D5—32no—
172C2×C4×D5—32no—
173C2×C4×D5—32no—
174C2×C4×D5—32no—
175C2×C4×D5—42no—
176C2×C4×D5—122no—
177C2×C4×D5C4×D311yescharacteristic
178C2×C4×D5C4×D311yescharacteristic
179C2×C4×D5C22×D311yescharacteristic
180C2×C4×D5C22×D311yescharacteristic
181C22⋊AGL(1,𝔽5)—38no—
182C22⋊AGL(1,𝔽5)C4×D318yes—
183C4⋊AGL(1,𝔽5)—316no—
184C4×AGL(1,𝔽5)—316no—
185C4×AGL(1,𝔽5)—42no—
186C4×AGL(1,𝔽5)—122no—
187C4×AGL(1,𝔽5)—124no—
188C5×C4⋊C4—32no—
189C5×C4⋊C4—32no—
190C4⋊Dic5—32no—
191C4⋊Dic5—32no—
192#683—32no—
193#683—32no—
194#683—32no—
195#683—32no—
196C2×#117—152no—
197C22×C4⋊C4—151no—
198C4×C4⋊C4—154no—
199C4×C4⋊C4—154no—
200C4×C22⋊C4—51no—
201C4×C22⋊C4—152no—
202C4×C22⋊C4—151no—
203C22×C3×C5—21no—
204C22×C3×C5—21no—
205C22×C3×C5C2×C4212yes—
206C22×C3×C5C23×C411yescharacteristic
207C2×C3×D5—21no—
208C2×C3×D5—21no—
209C2×C3×D5—28no—
210C2×C3×D5C2×C4212yes—
211C2×C3×D5C23×C412yes—
212C2×C3×D5H(5,𝔽2)12yes—
213C2×C3×D52−1+412yes—
214ΓL(1,𝔽16)—28no—
215C3×AGL(1,𝔽5)—28no—
216C3×C4×C5—21no—
217C3×C4×C5—21no—
218Dic15—22no—
219Dic15C2×C4214yes—
220C3×Dic5—21no—
221C3×Dic5—21no—
222C5×Dic3—22no—
223C5×Dic3C2×C4214yes—
224C24×C3—51no—
225C22×C3×C4—54no—
226C22×C3×C4—51no—
227C22×C3×C4—51no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—51no—
231C22×C3×C4—202no—
232C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
233C22×Dic3—51no—
234C22×Dic3—58no—
235C22×Dic3—52no—
236C22×Dic3—52no—
237C22×Dic3—54no—
238C22×Dic3—51no—
239C22×Dic3—52no—
240C22×Dic3—52no—
241C22×Dic3—202no—
242C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
243C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
244C3×C22⋊C4—58no—
245C3×C42—202no—
246#217—58no—
247Dic3⋊C4—516no—
248Dic3⋊C4—52no—
249Dic3⋊C4—52no—
250Dic3⋊C4—52no—
251Dic3⋊C4—52no—
252Dic3⋊C4—52no—
253Dic3⋊C4—52no—
254Dic3⋊C4C2×AGL(1,𝔽5)12yes—
255Dic3⋊C4C2×AGL(1,𝔽5)12yes—
256C4×Dic3—516no—
257C4×Dic3—204no—
258C4×Dic3—202no—
259C23×C5C2×C4×D311yescharacteristic
260C22×D5—22no—
261C22×D5—22no—
262C22×D5C2×C4×D311yescharacteristic
263C22×D5C2×C4×D314yes—
264C22×D5C23×D311yescharacteristic
265C2×AGL(1,𝔽5)—24no—
266C2×AGL(1,𝔽5)—24no—
267C2×AGL(1,𝔽5)—38no—
268C2×AGL(1,𝔽5)—64no—
269C2×AGL(1,𝔽5)—64no—
270C2×AGL(1,𝔽5)C2×C4×D318yes—
271C2×C4×C5—34no—
272C2×C4×C5—32no—
273C2×C4×C5—32no—
274C2×C4×C5—41no—
275C2×C4×C5—121no—
276C2×C4×C5C2×C4×D311yescharacteristic
277C2×C4×C5C2×C4×D311yescharacteristic
278C2×Dic5—34no—
279C2×Dic5—32no—
280C2×Dic5—32no—
281C2×Dic5—41no—
282C2×Dic5—121no—
283C2×Dic5C2×C4×D311yescharacteristic
284C2×Dic5C2×C4×D311yescharacteristic
285C4×D5—22no—
286C4×D5—22no—
287C4×D5—38no—
288C4×D5—34no—
289C4×D5—34no—
290C4×D5—41no—
291C4×D5—41no—
292C4×D5—64no—
293C4×D5—122no—
294C23×C4—51no—
295C23×C4—151no—
296C23×C4—151no—
297#117—1516no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—152no—
300C2×C4⋊C4—152no—
301C2×C4⋊C4—152no—
302C2×C4⋊C4—152no—
303C2×C4⋊C4—151no—
304C2×C4⋊C4—151no—
305C2×C4⋊C4—154no—
306C2×C4⋊C4—151no—
307C2×C22⋊C4—52no—
308C2×C22⋊C4—152no—
309C2×C42—52no—
310C2×C42—154no—
311C2×C42—154no—
312C2×C42—152no—
313#95—52no—
314#95—154no—
315#95—152no—
316C2×C3×C5—22no—
317C2×C3×C5C22×C4211yescharacteristic
318C2×C3×C5#51811yescharacteristic
319C2×C3×C5#51911yescharacteristic
320C3×D5—22no—
321C3×D5#52014yes—
322C23×C3—51no—
323C23×C3—54no—
324C23×C3—51no—
325C23×C3—102no—
326C23×C3—102no—
327C23×C3C22×AGL(1,𝔽5)11yescharacteristic
328C2×C3×C4—41no—
329C2×C3×C4—58no—
330C2×C3×C4—51no—
331C2×C3×C4—51no—
332C2×C3×C4—102no—
333C2×C3×C4—102no—
334C2×C3×C4—104no—
335C2×C3×C4—104no—
336C2×C3×C4—201no—
337C2×C3×C4—201no—
338C2×C3×C4—201no—
339C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
340C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
341C2×Dic3—41no—
342C2×Dic3—54no—
343C2×Dic3—52no—
344C2×Dic3—58no—
345C2×Dic3—54no—
346C2×Dic3—54no—
347C2×Dic3—58no—
348C2×Dic3—52no—
349C2×Dic3—104no—
350C2×Dic3—104no—
351C2×Dic3—104no—
352C2×Dic3—202no—
353C2×Dic3—201no—
354C2×Dic3C4×AGL(1,𝔽5)14yescomplemented
355C2×Dic3C22×AGL(1,𝔽5)12yes—
356C2×Dic3C22×AGL(1,𝔽5)12yes—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5C42×D312yes—
360C22×C5C22×C4×D311yescharacteristic
361C2×D5—21no—
362C2×D5—21no—
363C2×D5—28no—
364C2×D5C42×D312yes—
365C2×D5C22×C4×D312yes—
366C2×D5#100412yes—
367C2×D5#101012yes—
368AGL(1,𝔽5)—28no—
369AGL(1,𝔽5)—68no—
370C4×C5—21no—
371C4×C5—21no—
372C4×C5—34no—
373C4×C5—62no—
374Dic5—21no—
375Dic5—21no—
376Dic5—34no—
377Dic5—62no—
378C24—51no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—54no—
382C22×C4—51no—
383C22×C4—51no—
384C22×C4—51no—
385C22×C4—51no—
386C22×C4—51no—
387C22×C4—152no—
388C22×C4—152no—
389C22×C4—158no—
390C22×C4—152no—
391C22×C4—152no—
392C22×C4—151no—
393C22×C4—154no—
394C22×C4—151no—
395C22×C4—202no—
396C22×C4—602no—
397C22×C4D3×AGL(1,𝔽5)11yescharacteristic
398C4⋊C4—32no—
399C4⋊C4—32no—
400C4⋊C4—152no—
401C4⋊C4—152no—
402C4⋊C4—152no—
403C4⋊C4—152no—
404C4⋊C4—1516no—
405C4⋊C4—152no—
406C4⋊C4—152no—
407C22⋊C4—58no—
408C22⋊C4—158no—
409C42—1516no—
410C42—202no—
411C42—602no—
412C42—604no—
413C3×C5#227411yescharacteristic, complemented, Hall
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—52no—
417C22×C3—52no—
418C22×C3—52no—
419C22×C3—52no—
420C22×C3—108no—
421C22×C3—101no—
422C22×C3—101no—
423C22×C3C2×C4×AGL(1,𝔽5)12yes—
424C22×C3C23×AGL(1,𝔽5)11yescharacteristic
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—108no—
428C3×C4—101no—
429C3×C4—101no—
430Dic3—22no—
431Dic3—54no—
432Dic3—108no—
433Dic3—102no—
434Dic3C2×C4×AGL(1,𝔽5)14yes—
435C2×C5—22no—
436C2×C5C2×C42×D311yescharacteristic
437C2×C5#548611yescharacteristic
438C2×C5#551611yescharacteristic
439D5—22no—
440D5#554614yes—
441C23—51no—
442C23—54no—
443C23—51no—
444C23—102no—
445C23—102no—
446C23C2×D3×AGL(1,𝔽5)11yescharacteristic
447C2×C4—34no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—41no—
451C2×C4—58no—
452C2×C4—51no—
453C2×C4—51no—
454C2×C4—102no—
455C2×C4—102no—
456C2×C4—104no—
457C2×C4—104no—
458C2×C4—121no—
459C2×C4—154no—
460C2×C4—154no—
461C2×C4—158no—
462C2×C4—154no—
463C2×C4—152no—
464C2×C4—158no—
465C2×C4—152no—
466C2×C4—201no—
467C2×C4—201no—
468C2×C4—201no—
469C2×C4—304no—
470C2×C4—304no—
471C2×C4—304no—
472C2×C4—602no—
473C2×C4—601no—
474C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
475C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
476C2×C3—22no—
477C2×C3—54no—
478C2×C3—102no—
479C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
480C2×C3#6625811yescharacteristic
481C2×C3#6626211yescharacteristic
482C5#7343511yescharacteristic, complemented, Sylow
483C22—21no—
484C22—21no—
485C22—52no—
486C22—52no—
487C22—52no—
488C22—52no—
489C22—108no—
490C22—101no—
491C22—101no—
492C22C4×D3×AGL(1,𝔽5)12yes—
493C22C22×D3×AGL(1,𝔽5)11yescharacteristic
494C4—21no—
495C4—21no—
496C4—34no—
497C4—62no—
498C4—108no—
499C4—101no—
500C4—101no—
501C4—154no—
502C4—308no—
503C4—302no—
504C3#1061709811yescharacteristic, complemented, Sylow
505C2—22no—
506C2—54no—
507C2—102no—
508C2#1175681211yescharacteristic
509C2C2×C4×D3×AGL(1,𝔽5)11yescharacteristic
510C2#1175687111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutatorC2×C3×C5C22×C42
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C23×C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2274
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11751833C2semidirect
7. AGL(1,𝔽5)×Dic3⋊C4C2semidirect
17. #91923C4semidirect
20. C2×#6712C4semidirect
21. C2×C5×Dic3⋊C4C4semidirect
22. C2×Dic3⋊Dic5C4semidirect
23. C2×#6669C4semidirect
24. D5×Dic3⋊C4C4semidirect
25. D5×Dic3⋊C4C4semidirect
63. Dic3⋊AGL(1,𝔽5)C2×C4semidirect
65. Dic3×AGL(1,𝔽5)C2×C4semidirect
120. C2×Dic15C42semidirect
124. C2×C5×Dic3C42semidirect
136. D5×Dic3C42semidirect
153. C2×Dic3⋊C4AGL(1,𝔽5)semidirect
354. C2×Dic3C4×AGL(1,𝔽5)semidirect
413. C3×C5#2274semidirect
482. C5#73435semidirect
504. C3#10617098semidirect

Maximal subgroups

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

subgroupindex
C2×#919232
C2×D5×Dic3⋊C42
C4×#67632
C3×C4×C22⋊AGL(1,𝔽5)2
Dic3×C22⋊AGL(1,𝔽5)2
#117518332
AGL(1,𝔽5)×Dic3⋊C42
#106170983
#734355