Finite Group Databasebeta

← #14307489

Subgroups — #14307489

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4092C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic
3C4×#6763C211yesmaximal, characteristic
4#11746788C214yesmaximal
5#10604982—31nomaximal, Hall
6C22×C3×C5×Q8C411yescharacteristic, complemented
7C22×C3×C4×D5C2211yescharacteristic
8C2×#6763C2212yes—
9C2×#6763C2212yes—
10C2×C4⋊ΓL(1,𝔽16)—21no—
11C2×C4×ΓL(1,𝔽16)—21no—
12C3×#4092—21no—
13C3×#4092—21no—
14C3×#4092—21no—
15C3×#4092—21no—
16C2×C3×#685C2212yes—
17C2×C3×C4⋊Dic5C411yescharacteristic, complemented
18C2×C3×#683C412yescomplemented
19#91235—21no—
20#91235—21no—
21#71701—51nomaximal, Hall
22C2×#4092D311yescharacteristic
23#65765—31no—
24C4×C22⋊AGL(1,𝔽5)—31no—
25#64917—34no—
26C23×C3×D5C2311yescharacteristic
27C22×ΓL(1,𝔽16)—22no—
28C22×ΓL(1,𝔽16)—22no—
29C2×C3×C5×Q8—21no—
30C2×C3×C5×Q8—21no—
31C2×C3×C5×Q8—42no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×C4×C5C2×C412yes—
34C22×C3×Dic5C2×C412yes—
35C22×C3×Dic5C2×C411yescharacteristic
36C2×C3×C4×D5—42no—
37C2×C3×C4×D5D411yescharacteristic
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41#6763—22no—
42#6763—24no—
43#6763Q814yescomplemented
44C4⋊ΓL(1,𝔽16)—42no—
45C4×ΓL(1,𝔽16)—42no—
46C3×#685—22no—
47C3×#685—22no—
48C3×C4⋊Dic5—21no—
49C3×C4⋊Dic5—21no—
50C3×#683—22no—
51C3×#683—22no—
52C2×C3×C22⋊Q8—51no—
53#5212—51no—
54C4×#217—51no—
55#4526—54no—
56C22×C5×Q8Dic311yescharacteristic, complemented
57C22×C4×D5C2×D311yescharacteristic
58C2×C22⋊AGL(1,𝔽5)—32no—
59C2×C22⋊AGL(1,𝔽5)—32no—
60C2×C4⋊AGL(1,𝔽5)—61no—
61C2×C4×AGL(1,𝔽5)—61no—
62#4092—21no—
63#4092—21no—
64#4092—21no—
65#4092—21no—
66C2×#685C2×D312yes—
67C2×C4⋊Dic5Dic311yescharacteristic, complemented
68C2×#683Dic312yescomplemented
69(C2×AGL(1,𝔽5))⋊C4—61no—
70(C2×AGL(1,𝔽5))⋊C4—61no—
71#1996—151noSylow
72C23×C3×C5C22×C411yescharacteristic
73C22×C3×D5—41no—
74C22×C3×D5—41no—
75C22×C3×D5C2×D411yescharacteristic
76C22×C3×D5C2×D411yescharacteristic
77C22×C3×D5C2×Q812yes—
78C22×C3×D5#4112yes—
79C2×ΓL(1,𝔽16)—24no—
80C2×ΓL(1,𝔽16)—42no—
81C2×ΓL(1,𝔽16)—42no—
82C2×ΓL(1,𝔽16)—42no—
83C2×ΓL(1,𝔽16)—42no—
84C2×ΓL(1,𝔽16)—42no—
85C3×C5×Q8—42no—
86C3×C5×Q8—42no—
87C2×C3×C4×C5—22no—
88C2×C3×C4×C5—22no—
89C2×C3×C4×C5—42no—
90C2×C3×C4×C5C22⋊C411yescharacteristic
91C2×C3×C4×C5C22⋊C411yescharacteristic
92C2×C3×Dic5—22no—
93C2×C3×Dic5—41no—
94C2×C3×Dic5—42no—
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C22⋊C411yescharacteristic
97C3×C4×D5—22no—
98C3×C4×D5—22no—
99C3×C4×D5—41no—
100C3×C4×D5—41no—
101C22×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
102C23×C3×C4—51no—
103C3×C22⋊Q8—101no—
104C3×C22⋊Q8—101no—
105C3×C22⋊Q8—101no—
106C3×C22⋊Q8—101no—
107C2×C3×C4⋊C4—52no—
108C2×C3×C4⋊C4—51no—
109C2×C3×C22⋊C4—52no—
110C2×#217—52no—
111C2×#217—52no—
112C2×C4⋊Dic3—101no—
113C2×C4×Dic3—101no—
114C3⋊#95—101no—
115C3⋊#95—101no—
116C23×D5C22×D311yescharacteristic
117C22×AGL(1,𝔽5)—62no—
118C22×AGL(1,𝔽5)—62no—
119C2×C5×Q8—21no—
120C2×C5×Q8—21no—
121C2×C5×Q8—42no—
122C22×C4×C5C2×Dic312yes—
123C22×C4×C5C2×Dic311yescharacteristic
124C22×Dic5C2×Dic312yes—
125C22×Dic5C2×Dic311yescharacteristic
126C2×C4×D5—42no—
127C2×C4×D5#6711yescharacteristic
128C2×C4×D5#6711yescharacteristic
129C2×C4×D5#6711yescharacteristic
130C2×C4×D5#6711yescharacteristic
131C22⋊AGL(1,𝔽5)—34no—
132C22⋊AGL(1,𝔽5)—62no—
133C22⋊AGL(1,𝔽5)—64no—
134C4⋊AGL(1,𝔽5)—122no—
135C4×AGL(1,𝔽5)—122no—
136#685—22no—
137#685—22no—
138C4⋊Dic5—21no—
139C4⋊Dic5—21no—
140#683—22no—
141#683—22no—
142C2×C22⋊Q8—51no—
143#380—151no—
144C4×C22⋊C4—151no—
145#342—154no—
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5C2×C22⋊C411yescharacteristic
149C22×C3×C5C4×Q812yes—
150C2×C3×D5—21no—
151C2×C3×D5—21no—
152C2×C3×D5—21no—
153C2×C3×D5—21no—
154C2×C3×D5—42no—
155C2×C3×D5—42no—
156C2×C3×D5C22⋊Q812yes—
157C2×C3×D5#12411yescharacteristic
158C2×C3×D5C4⋊Q811yescharacteristic
159ΓL(1,𝔽16)—44no—
160ΓL(1,𝔽16)—82no—
161C3×C4×C5—21no—
162C3×C4×C5—42no—
163C3×C4×C5#9912yes—
164C3×Dic5—21no—
165C3×Dic5—21no—
166C3×Dic5—42no—
167C24×C3—51no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—42no—
171C22×C3×C4—51no—
172C22×C3×C4—51no—
173C22×C3×C4—51no—
174C22×C3×C4—51no—
175C22×C3×C4—51no—
176C22×C3×C4—52no—
177C22×C3×C4—202no—
178C22×C3×C4C2×AGL(1,𝔽5)12yes—
179C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
180C22×Dic3—102no—
181C22×Dic3—102no—
182C3×C4⋊C4—102no—
183C3×C4⋊C4—102no—
184C3×C4⋊C4—101no—
185C3×C4⋊C4—101no—
186C3×C22⋊C4—102no—
187C3×C22⋊C4—102no—
188#217—54no—
189#217—102no—
190#217—104no—
191C4⋊Dic3—202no—
192C4×Dic3—202no—
193C23×C5C22×Dic311yescharacteristic
194C22×D5—41no—
195C22×D5—41no—
196C22×D5D3×Q812yes—
197C22×D5#23912yes—
198C22×D5C2×#6711yescharacteristic
199C22×D5C2×#6711yescharacteristic
200C2×AGL(1,𝔽5)—64no—
201C2×AGL(1,𝔽5)—122no—
202C2×AGL(1,𝔽5)—122no—
203C2×AGL(1,𝔽5)—122no—
204C2×AGL(1,𝔽5)—122no—
205C2×AGL(1,𝔽5)—122no—
206C5×Q8—42no—
207C5×Q8—42no—
208C2×C4×C5—22no—
209C2×C4×C5—22no—
210C2×C4×C5—42no—
211C2×C4×C5#21711yescharacteristic
212C2×C4×C5#21711yescharacteristic
213C2×Dic5—22no—
214C2×Dic5—41no—
215C2×Dic5—42no—
216C2×Dic5#21711yescharacteristic
217C2×Dic5#21711yescharacteristic
218C4×D5—22no—
219C4×D5—22no—
220C4×D5—41no—
221C4×D5—41no—
222C22×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
223C23×C4—51no—
224C22⋊Q8—101no—
225C22⋊Q8—101no—
226C22⋊Q8—101no—
227C22⋊Q8—101no—
228C2×C4⋊C4—52no—
229C2×C4⋊C4—51no—
230C2×C4⋊C4—301no—
231C2×C22⋊C4—52no—
232C2×C22⋊C4—152no—
233C2×C22⋊C4—152no—
234C2×C42—301no—
235#95—301no—
236#95—301no—
237C2×C3×C5—22no—
238C2×C3×C5#39111yescharacteristic
239C2×C3×C5C2×#9911yescharacteristic
240C2×C3×C5#41011yescharacteristic
241C3×D5—22no—
242C3×D5—21no—
243C3×D5—21no—
244C23×C3—51no—
245C23×C3—51no—
246C23×C3—52no—
247C23×C3—52no—
248C23×C3—201no—
249C23×C3—201no—
250C23×C3C22×AGL(1,𝔽5)11yescharacteristic
251C3×Q8—42no—
252C3×Q8—42no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—42no—
256C2×C3×C4—51no—
257C2×C3×C4—51no—
258C2×C3×C4—102no—
259C2×C3×C4—102no—
260C2×C3×C4—102no—
261C2×C3×C4—201no—
262C2×C3×C4—201no—
263C2×C3×C4—201no—
264C2×C3×C4—202no—
265C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
266C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
267C2×Dic3—104no—
268C2×Dic3—202no—
269C2×Dic3—202no—
270C2×Dic3—202no—
271C2×Dic3—202no—
272C2×Dic3—202no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5Q8×Dic312yes—
276C22×C5C2×#21711yescharacteristic
277C2×D5—21no—
278C2×D5—21no—
279C2×D5—21no—
280C2×D5—21no—
281C2×D5—42no—
282C2×D5—42no—
283C2×D5#94411yescharacteristic
284C2×D5#94612yes—
285C2×D5#94711yescharacteristic
286AGL(1,𝔽5)—124no—
287AGL(1,𝔽5)—242no—
288C4×C5—21no—
289C4×C5—42no—
290C4×C5#83412yes—
291Dic5—21no—
292Dic5—21no—
293Dic5—42no—
294C24—51no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—42no—
298C22×C4—51no—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—51no—
303C22×C4—52no—
304C22×C4—202no—
305C22×C4—302no—
306C22×C4—302no—
307C22×C4C2×ΓL(1,𝔽16)12yes—
308C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
309C4⋊C4—102no—
310C4⋊C4—102no—
311C4⋊C4—101no—
312C4⋊C4—101no—
313C4⋊C4—602no—
314C22⋊C4—102no—
315C22⋊C4—102no—
316C22⋊C4—154no—
317C22⋊C4—302no—
318C22⋊C4—304no—
319C42—602no—
320C3×C5#199611yescharacteristic, complemented, Hall
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—52no—
324C22×C3—51no—
325C22×C3—51no—
326C22×C3—101no—
327C22×C3—101no—
328C22×C3—101no—
329C22×C3—101no—
330C22×C3—202no—
331C22×C3—202no—
332C22×C3Q8×AGL(1,𝔽5)12yes—
333C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
334C3×C4—21no—
335C3×C4—42no—
336C3×C4—101no—
337C3×C4—101no—
338C3×C4—202no—
339C3×C4#401112yes—
340Dic3—204no—
341Dic3—402no—
342C2×C5—22no—
343C2×C5C2×#83411yescharacteristic
344C2×C5#522911yescharacteristic
345C2×C5#524011yescharacteristic
346D5—22no—
347D5—21no—
348D5—21no—
349C23—51no—
350C23—51no—
351C23—52no—
352C23—52no—
353C23—201no—
354C23—201no—
355C23C22×ΓL(1,𝔽16)11yescharacteristic
356Q8—42no—
357Q8—42no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—51no—
362C2×C4—51no—
363C2×C4—102no—
364C2×C4—102no—
365C2×C4—102no—
366C2×C4—201no—
367C2×C4—201no—
368C2×C4—201no—
369C2×C4—202no—
370C2×C4—304no—
371C2×C4—602no—
372C2×C4—602no—
373C2×C4—602no—
374C2×C4—602no—
375C2×C4—602no—
376C2×C4#676311yescharacteristic
377C2×C4#676311yescharacteristic
378C2×C3—22no—
379C2×C3—102no—
380C2×C3—101no—
381C2×C3—101no—
382C2×C3#6578511yescharacteristic
383C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
384C2×C3C2×#401111yescharacteristic
385C5#7170111yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—52no—
389C22—51no—
390C22—51no—
391C22—101no—
392C22—101no—
393C22—101no—
394C22—101no—
395C22—202no—
396C22—202no—
397C22Q8×ΓL(1,𝔽16)12yes—
398C22C2×#676311yescharacteristic
399C4—21no—
400C4—42no—
401C4—101no—
402C4—101no—
403C4—202no—
404C4—604no—
405C4—1202no—
406C4#9124012yes—
407C3#1060498211yescharacteristic, complemented, Sylow
408C2—22no—
409C2—102no—
410C2—101no—
411C2—101no—
412C2#1175244411yescharacteristic
413C2#1175244911yescharacteristic
414C2C2×#9124011yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radical, cosocleC23×C3×D5C23
hypercenterC22×Q8ΓL(1,𝔽16)
hypocenterC3×C5#1996
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
6. C22×C3×C5×Q8C4semidirect
17. C2×C3×C4⋊Dic5C4semidirect
18. C2×C3×#683C4semidirect
43. #6763Q8semidirect
56. C22×C5×Q8Dic3semidirect
67. C2×C4⋊Dic5Dic3semidirect
68. C2×#683Dic3semidirect
101. C22×C3×Q8AGL(1,𝔽5)semidirect
222. C22×Q8ΓL(1,𝔽16)semidirect
320. C3×C5#1996semidirect
385. C5#71701semidirect
407. C3#10604982semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40922
#117524242
C4×#67632
#117467882
#106049823
#717015