Finite Group Databasebeta

← #14322600

Subgroups — #14322600

Order 1920.

1344 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×C22⋊C4C211yesmaximal, characteristic
2#11752459C212yesmaximal
3#11752424C212yesmaximal, complemented
4C4×#6763C212yesmaximal, complemented
5#11752359C218yesmaximal, complemented
6#10617070—31nomaximal, Hall
7C24×C3×D5C2211yescharacteristic
8C22×C3×C4×D5C2211yescharacteristic
9C22×C3×C4×D5C2211yescharacteristic
10C2×#6763C2214yes—
11C2×#6763C2214yes—
12C2×#6763C2214yes—
13C2×C4⋊ΓL(1,𝔽16)C2214yescomplemented
14C2×C4×ΓL(1,𝔽16)C2214yescomplemented
15C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
16C2×C3×C22⋊Dic5C411yescharacteristic, complemented
17C2×C3×#685C411yescharacteristic, complemented
18C2×C3×#685C411yescharacteristic, complemented
19C3×D5×C22⋊C4C414yescomplemented
20C3×D5×C22⋊C4C2214yes—
21#91235C2214yescomplemented
22#91235C2214yescomplemented
23#82093—51nomaximal, Hall
24C2×D5×C22⋊C4D311yescharacteristic
25#65800—32no—
26#65765—32no—
27C4×C22⋊AGL(1,𝔽5)—32no—
28#65700—38no—
29C24×C3×C5C2×C411yescharacteristic
30C23×C3×D5—24no—
31C23×C3×D5—42no—
32C23×C3×D5—42no—
33C23×C3×D5C2×C412yes—
34C23×C3×D5C2×C411yescharacteristic
35C23×C3×D5C2312yes—
36C23×C3×D5C2311yescharacteristic
37C22×ΓL(1,𝔽16)C2318yes—
38C22×C3×C4×C5C2×C411yescharacteristic
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×Dic5C2×C411yescharacteristic
41C22×C3×Dic5C2×C411yescharacteristic
42C2×C3×C4×D5—42no—
43C2×C3×C4×D5—42no—
44C2×C3×C4×D5C2×C412yes—
45C2×C3×C4×D5C2×C412yes—
46C2×C3×C4×D5C2312yes—
47C2×C3×C4×D5C2312yes—
48#6763—28no—
49#6763—44no—
50#6763—44no—
51C4⋊ΓL(1,𝔽16)—44no—
52C4×ΓL(1,𝔽16)—44no—
53C3×C5×C22⋊C4C2×C414yes—
54C3×C22⋊Dic5C2×C414yes—
55C3×#685C2×C414yes—
56C3×#685C2×C414yes—
57C22×C3×C22⋊C4—51no—
58#5247—52no—
59#5212—52no—
60C4×#217—52no—
61#4951—58no—
62C24×D5C2×D311yescharacteristic
63C22×C4×D5C2×D311yescharacteristic
64C22×C4×D5C2×D311yescharacteristic
65C2×C22⋊AGL(1,𝔽5)—34no—
66C2×C22⋊AGL(1,𝔽5)—34no—
67C2×C22⋊AGL(1,𝔽5)—34no—
68C2×C4⋊AGL(1,𝔽5)—34no—
69C2×C4×AGL(1,𝔽5)—34no—
70C2×C5×C22⋊C4Dic311yescharacteristic, complemented
71C2×C22⋊Dic5Dic311yescharacteristic, complemented
72C2×#685Dic311yescharacteristic, complemented
73C2×#685Dic311yescharacteristic, complemented
74D5×C22⋊C4Dic314yescomplemented
75D5×C22⋊C4C2×D314yes—
76(C2×AGL(1,𝔽5))⋊C4—34no—
77(C2×AGL(1,𝔽5))⋊C4—34no—
78#2312—151noSylow
79C23×C3×C5—22no—
80C23×C3×C5—41no—
81C23×C3×C5—41no—
82C23×C3×C5C22×C412yes—
83C23×C3×C5C22×C411yescharacteristic
84C22×C3×D5—22no—
85C22×C3×D5—22no—
86C22×C3×D5—22no—
87C22×C3×D5—24no—
88C22×C3×D5—24no—
89C22×C3×D5—24no—
90C22×C3×D5—24no—
91C22×C3×D5—42no—
92C22×C3×D5—42no—
93C22×C3×D5—41no—
94C22×C3×D5—41no—
95C22×C3×D5—416no—
96C22×C3×D5C22×C412yes—
97C22×C3×D5C22×C411yescharacteristic
98C22×C3×D5#4114yes—
99C22×C3×D5C2411yescharacteristic
100C2×ΓL(1,𝔽16)—28no—
101C2×ΓL(1,𝔽16)—28no—
102C2×ΓL(1,𝔽16)—28no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5C22×C412yes—
106C2×C3×C4×C5C22×C412yes—
107C2×C3×Dic5—41no—
108C2×C3×Dic5—41no—
109C2×C3×Dic5C22×C412yes—
110C2×C3×Dic5C22×C412yes—
111C3×C4×D5—24no—
112C3×C4×D5—24no—
113C3×C4×D5—42no—
114C3×C4×D5—42no—
115C25×C3—51no—
116C23×C3×C4—51no—
117C23×C3×C4—51no—
118C2×C3×C22⋊C4—51no—
119C2×C3×C22⋊C4—51no—
120C2×C3×C22⋊C4—51no—
121C2×C3×C22⋊C4—54no—
122C2×C3×C22⋊C4—54no—
123C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
124C2×#217—54no—
125C2×#217—54no—
126C2×#217—54no—
127C2×C4⋊Dic3—54no—
128C2×C4×Dic3—54no—
129C3⋊#95—54no—
130C3⋊#95—54no—
131C24×C5C2×Dic311yescharacteristic
132C23×D5—24no—
133C23×D5—42no—
134C23×D5—42no—
135C23×D5C2×Dic312yes—
136C23×D5C2×Dic311yescharacteristic
137C23×D5C22×D312yes—
138C23×D5C22×D311yescharacteristic
139C22×AGL(1,𝔽5)—38no—
140C22×C4×C5C2×Dic311yescharacteristic
141C22×C4×C5C2×Dic311yescharacteristic
142C22×Dic5C2×Dic311yescharacteristic
143C22×Dic5C2×Dic311yescharacteristic
144C2×C4×D5—42no—
145C2×C4×D5—42no—
146C2×C4×D5C2×Dic312yes—
147C2×C4×D5C2×Dic312yes—
148C2×C4×D5C22×D312yes—
149C2×C4×D5C22×D312yes—
150C22⋊AGL(1,𝔽5)—68no—
151C22⋊AGL(1,𝔽5)—124no—
152C22⋊AGL(1,𝔽5)—124no—
153C4⋊AGL(1,𝔽5)—124no—
154C4×AGL(1,𝔽5)—124no—
155C5×C22⋊C4C2×Dic314yes—
156C22⋊Dic5C2×Dic314yes—
157#685C2×Dic314yes—
158#685C2×Dic314yes—
159C22×C22⋊C4—51no—
160#388—158no—
161#380—152no—
162#379—152no—
163C4×C22⋊C4—152no—
164C22×C3×C5—21no—
165C22×C3×C5—21no—
166C22×C3×C5—22no—
167C22×C3×C5—22no—
168C22×C3×C5—22no—
169C22×C3×C5—44no—
170C22×C3×C5#11712yes—
171C22×C3×C5C23×C411yescharacteristic
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—28no—
175C2×C3×D5—22no—
176C2×C3×D5—22no—
177C2×C3×D5—22no—
178C2×C3×D5—48no—
179C2×C3×D5—44no—
180C2×C3×D5—48no—
181C2×C3×D5#11712yes—
182C2×C3×D5C2×#4112yes—
183C2×C3×D5H(5,𝔽2)12yes—
184C2×C3×D5H(5,𝔽2)12yes—
185ΓL(1,𝔽16)—48no—
186C3×C4×C5—22no—
187C3×C4×C5—22no—
188C3×Dic5—22no—
189C3×Dic5—22no—
190C24×C3—51no—
191C24×C3—52no—
192C24×C3—52no—
193C24×C3—51no—
194C24×C3—104no—
195C24×C3—202no—
196C24×C3—202no—
197C24×C3C2×AGL(1,𝔽5)11yescharacteristic
198C22×C3×C4—51no—
199C22×C3×C4—51no—
200C22×C3×C4—52no—
201C22×C3×C4—52no—
202C22×C3×C4—52no—
203C22×C3×C4—52no—
204C22×C3×C4—202no—
205C22×C3×C4—202no—
206C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
207C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
208C22×Dic3—58no—
209C3×C22⋊C4—54no—
210C3×C22⋊C4—54no—
211C3×C22⋊C4—54no—
212C3×C22⋊C4C2×AGL(1,𝔽5)14yes—
213#217—108no—
214#217—204no—
215#217—204no—
216C4⋊Dic3—204no—
217C4×Dic3—204no—
218C23×C5—22no—
219C23×C5—41no—
220C23×C5—41no—
221C23×C5C22×Dic312yes—
222C23×C5C22×Dic311yescharacteristic
223C22×D5—22no—
224C22×D5—22no—
225C22×D5—22no—
226C22×D5—24no—
227C22×D5—24no—
228C22×D5—24no—
229C22×D5—24no—
230C22×D5—42no—
231C22×D5—42no—
232C22×D5—41no—
233C22×D5—41no—
234C22×D5—416no—
235C22×D5#23514yes—
236C22×D5C22×Dic312yes—
237C22×D5C22×Dic311yescharacteristic
238C22×D5C23×D311yescharacteristic
239C2×AGL(1,𝔽5)—68no—
240C2×AGL(1,𝔽5)—68no—
241C2×AGL(1,𝔽5)—68no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5C22×Dic312yes—
245C2×C4×C5C22×Dic312yes—
246C2×Dic5—41no—
247C2×Dic5—41no—
248C2×Dic5C22×Dic312yes—
249C2×Dic5C22×Dic312yes—
250C4×D5—24no—
251C4×D5—24no—
252C4×D5—42no—
253C4×D5—42no—
254C25—51no—
255C23×C4—51no—
256C23×C4—51no—
257C2×C4⋊C4—154no—
258C2×C22⋊C4—51no—
259C2×C22⋊C4—51no—
260C2×C22⋊C4—51no—
261C2×C22⋊C4—54no—
262C2×C22⋊C4—54no—
263C2×C22⋊C4—154no—
264C2×C22⋊C4—154no—
265C2×C22⋊C4—154no—
266C2×C22⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
267C2×C42—154no—
268#95—154no—
269#95—154no—
270C2×C3×C5—22no—
271C2×C3×C5—42no—
272C2×C3×C5C2×#11711yescharacteristic
273C2×C3×C5#51811yescharacteristic
274C2×C3×C5#51811yescharacteristic
275C3×D5—22no—
276C3×D5—42no—
277C3×D5#53814yes—
278C23×C3—22no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—51no—
282C23×C3—54no—
283C23×C3—51no—
284C23×C3—52no—
285C23×C3—102no—
286C23×C3—102no—
287C23×C3—104no—
288C23×C3—104no—
289C23×C3—104no—
290C23×C3—104no—
291C23×C3—102no—
292C23×C3—201no—
293C23×C3—201no—
294C23×C3—202no—
295C23×C3—202no—
296C23×C3—2016no—
297C23×C3C22×AGL(1,𝔽5)12yes—
298C23×C3C22×AGL(1,𝔽5)11yescharacteristic
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—52no—
302C2×C3×C4—52no—
303C2×C3×C4—104no—
304C2×C3×C4—104no—
305C2×C3×C4—202no—
306C2×C3×C4—202no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309C2×C3×C4C22×AGL(1,𝔽5)12yes—
310C2×C3×C4C22×AGL(1,𝔽5)12yes—
311C2×Dic3—108no—
312C2×Dic3—108no—
313C2×Dic3—108no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—22no—
317C22×C5—22no—
318C22×C5—22no—
319C22×C5—44no—
320C22×C5#92612yes—
321C22×C5C23×Dic311yescharacteristic
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—28no—
325C2×D5—22no—
326C2×D5—22no—
327C2×D5—22no—
328C2×D5—48no—
329C2×D5—44no—
330C2×D5—48no—
331C2×D5#92612yes—
332C2×D5C2×#23512yes—
333C2×D5#100412yes—
334C2×D5#100412yes—
335AGL(1,𝔽5)—128no—
336C4×C5—22no—
337C4×C5—22no—
338Dic5—22no—
339Dic5—22no—
340C24—51no—
341C24—52no—
342C24—52no—
343C24—51no—
344C24—104no—
345C24—202no—
346C24—202no—
347C24C2×ΓL(1,𝔽16)11yescharacteristic
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—52no—
351C22×C4—52no—
352C22×C4—52no—
353C22×C4—52no—
354C22×C4—158no—
355C22×C4—202no—
356C22×C4—202no—
357C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
358C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
359C4⋊C4—604no—
360C22⋊C4—54no—
361C22⋊C4—54no—
362C22⋊C4—54no—
363C22⋊C4—308no—
364C22⋊C4—604no—
365C22⋊C4—604no—
366C22⋊C4C2×ΓL(1,𝔽16)14yes—
367C42—604no—
368C3×C5#231211yescharacteristic, complemented, Hall
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—44no—
375C22×C3—52no—
376C22×C3—52no—
377C22×C3—52no—
378C22×C3—52no—
379C22×C3—108no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3—102no—
383C22×C3—102no—
384C22×C3—102no—
385C22×C3—204no—
386C22×C3—208no—
387C22×C3—208no—
388C22×C3#413012yes—
389C22×C3C23×AGL(1,𝔽5)11yescharacteristic
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—102no—
393C3×C4—102no—
394Dic3—208no—
395C2×C5—22no—
396C2×C5—42no—
397C2×C5C2×#92611yescharacteristic
398C2×C5#579811yescharacteristic
399C2×C5#579811yescharacteristic
400D5—22no—
401D5—42no—
402D5#549414yes—
403C23—22no—
404C23—41no—
405C23—41no—
406C23—51no—
407C23—54no—
408C23—51no—
409C23—52no—
410C23—102no—
411C23—102no—
412C23—104no—
413C23—104no—
414C23—104no—
415C23—104no—
416C23—102no—
417C23—201no—
418C23—201no—
419C23—202no—
420C23—202no—
421C23—2016no—
422C23C22×ΓL(1,𝔽16)12yes—
423C23C22×ΓL(1,𝔽16)11yescharacteristic
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—52no—
427C2×C4—52no—
428C2×C4—104no—
429C2×C4—104no—
430C2×C4—202no—
431C2×C4—202no—
432C2×C4—201no—
433C2×C4—201no—
434C2×C4—308no—
435C2×C4—308no—
436C2×C4—308no—
437C2×C4C22×ΓL(1,𝔽16)12yes—
438C2×C4C22×ΓL(1,𝔽16)12yes—
439C2×C3—22no—
440C2×C3—42no—
441C2×C3—54no—
442C2×C3—102no—
443C2×C3—202no—
444C2×C3C2×#413011yescharacteristic
445C2×C3#6625811yescharacteristic
446C2×C3#6625811yescharacteristic
447C5#8209311yescharacteristic, complemented, Sylow
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—44no—
454C22—52no—
455C22—52no—
456C22—52no—
457C22—52no—
458C22—108no—
459C22—101no—
460C22—101no—
461C22—102no—
462C22—102no—
463C22—102no—
464C22—204no—
465C22—208no—
466C22—208no—
467C22#9198912yes—
468C22C23×ΓL(1,𝔽16)11yescharacteristic
469C4—22no—
470C4—22no—
471C4—102no—
472C4—102no—
473C4—608no—
474C3#1061707011yescharacteristic, complemented, Sylow
475C2—22no—
476C2—42no—
477C2—54no—
478C2—102no—
479C2—202no—
480C2C2×#9198911yescharacteristic
481C2#1175698711yescharacteristic
482C2#1175698711yescharacteristic
subgroupG/H
center, FrattiniC22C23×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C4
prime radicalC24×C3×D5C22
cosocleC22×C3×D5C24
hypercenterC2×C22⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#2312
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #11752424C2semidirect
4. C4×#6763C2semidirect
5. #11752359C2semidirect
13. C2×C4⋊ΓL(1,𝔽16)C22semidirect
14. C2×C4×ΓL(1,𝔽16)C22semidirect
15. C2×C3×C5×C22⋊C4C4semidirect
16. C2×C3×C22⋊Dic5C4semidirect
17. C2×C3×#685C4semidirect
18. C2×C3×#685C4semidirect
19. C3×D5×C22⋊C4C4semidirect
21. #91235C22semidirect
22. #91235C22semidirect
70. C2×C5×C22⋊C4Dic3semidirect
71. C2×C22⋊Dic5Dic3semidirect
72. C2×#685Dic3semidirect
73. C2×#685Dic3semidirect
74. D5×C22⋊C4Dic3semidirect
123. C2×C3×C22⋊C4AGL(1,𝔽5)semidirect
266. C2×C22⋊C4ΓL(1,𝔽16)semidirect
368. C3×C5#2312semidirect
447. C5#82093semidirect
474. C3#10617070semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D5×C22⋊C42
#117524592
#117524242
C4×#67632
#117523592
#106170703
#820935