Finite Group Databasebeta

← #14307243

Subgroups — #14307243

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×#125C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
3D5⋊(C4⋊(C3⋊C8))C211yesmaximal, characteristic
4#11746745C212yesmaximal
5#11746745C212yesmaximal
6#10604589—31nomaximal, Hall
7C2×C4⋊ΓL(1,𝔽16)C2212yes—
8C2×C4×ΓL(1,𝔽16)—21no—
9C2×D5⋊(C3⋊C8)C2212yes—
10C3×C5×#125C411yescharacteristic, complemented
11C3×D5×C4⋊C4—21no—
12C3×D5×C4⋊C4—21no—
13C3×D5×C4⋊C4C2211yescharacteristic
14C3×D5×C4⋊C4C2211yescharacteristic
15C3×#4036C411yescharacteristic, complemented
16C3×#4036C411yescharacteristic, complemented
17C3×#4035—21no—
18C3×#4035—21no—
19C3×C42×D5C2211yescharacteristic
20C3×#4016C411yescharacteristic, complemented
21#91235—21no—
22#91233—21no—
23C4⋊(C3⋊#170)—21no—
24#71738—51nomaximal, Hall
25D5×#125D311yescharacteristic
26C4×C4⋊AGL(1,𝔽5)—31no—
27D5⋊(C4⋊C8)—31no—
28#64874—32no—
29#64874—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×C3×C4×D5—21no—
32C2×C3×C4×D5—21no—
33C2×C3×C4×D5D411yescharacteristic
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5C2311yescharacteristic
36C2×C3⋊#170—22no—
37C4⋊ΓL(1,𝔽16)—22no—
38C4⋊ΓL(1,𝔽16)Q814yes—
39C4×ΓL(1,𝔽16)—42no—
40D5⋊(C3⋊C8)—24no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4C2×C411yescharacteristic
44C3×C5×C4⋊C4C2×C411yescharacteristic
45C3×C42×C5C2×C411yescharacteristic
46C3×C4⋊Dic5—21no—
47C3×C4⋊Dic5—21no—
48C3×C4⋊Dic5C2×C411yescharacteristic
49C3×C4⋊Dic5C2×C411yescharacteristic
50C3×#683—21no—
51C3×#683—21no—
52C3×#683—21no—
53C3×#683—21no—
54C3×#683—21no—
55C3×#683—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C3×C4×Dic5D411yescharacteristic
58C3×C4×Dic5D411yescharacteristic
59C2×C3×#125—51no—
60C4×C4⋊Dic3—51no—
61C2×C4⋊(C3⋊C8)—51no—
62#4529—52no—
63#4529—52no—
64C2×C4⋊AGL(1,𝔽5)—32no—
65C2×C4×AGL(1,𝔽5)—61no—
66C2×D5⋊C8—32no—
67C5×#125Dic311yescharacteristic, complemented
68D5×C4⋊C4—21no—
69D5×C4⋊C4—21no—
70D5×C4⋊C4C2×D311yescharacteristic
71D5×C4⋊C4C2×D311yescharacteristic
72#4036Dic311yescharacteristic, complemented
73#4036Dic311yescharacteristic, complemented
74#4035—21no—
75#4035—21no—
76C42×D5C2×D311yescharacteristic
77#4016Dic311yescharacteristic, complemented
78(C2×AGL(1,𝔽5))⋊C4—61no—
79#4004—61no—
80C4⋊#170—61no—
81#1989—151noSylow
82C22×C3×D5C2×D411yescharacteristic
83C2×ΓL(1,𝔽16)—24no—
84C2×ΓL(1,𝔽16)—42no—
85C2×ΓL(1,𝔽16)—42no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C2×C3×Dic5—21no—
92C2×C3×Dic5—21no—
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C2×D411yescharacteristic
96C3×C4×D5—22no—
97C3×C4×D5—21no—
98C3×C4×D5—21no—
99C3×C4×D5—42no—
100C3×C4×D5—41no—
101C3×C4×D5—41no—
102C3×C4×D5C2×Q812yes—
103C3×C4×D5#4112yes—
104C3⋊#170—42no—
105C3×#125—51no—
106C3×#125—51no—
107C3×#125—51no—
108C3×#125—101no—
109C3×#125—101no—
110C3×#125AGL(1,𝔽5)11yescharacteristic, complemented
111C2×C3×C4⋊C4—51no—
112C2×C3×C4⋊C4—51no—
113C2×C3×C4⋊C4—101no—
114C2×C3×C4⋊C4—101no—
115C2×C3×C42—51no—
116C2×C4⋊Dic3—52no—
117C2×C4×Dic3—101no—
118C22×C3⋊C8—52no—
119C3⋊#95—101no—
120C4⋊(C3⋊C8)—101no—
121C4⋊(C3⋊C8)—101no—
122C22×AGL(1,𝔽5)—62no—
123C2×C4×D5—21no—
124C2×C4×D5—21no—
125C2×C4×D5#6711yescharacteristic
126C2×C4×D5#6711yescharacteristic
127C2×C4×D5C22×D311yescharacteristic
128C2×#170—62no—
129C4⋊AGL(1,𝔽5)—34no—
130C4⋊AGL(1,𝔽5)—62no—
131C4×AGL(1,𝔽5)—122no—
132D5⋊C8—64no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4C2×Dic311yescharacteristic
136C5×C4⋊C4C2×Dic311yescharacteristic
137C42×C5C2×Dic311yescharacteristic
138C4⋊Dic5—21no—
139C4⋊Dic5—21no—
140C4⋊Dic5C2×Dic311yescharacteristic
141C4⋊Dic5C2×Dic311yescharacteristic
142#683—21no—
143#683—21no—
144#683—21no—
145#683—21no—
146#683—21no—
147#683—21no—
148C4×Dic5C2×Dic311yescharacteristic
149C4×Dic5#6711yescharacteristic
150C4×Dic5#6711yescharacteristic
151C2×#125—51no—
152C2×C4⋊C8—151no—
153C4×C4⋊C4—151no—
154#340—152no—
155#340—152no—
156C22×C3×C5C2×C22⋊C411yescharacteristic
157C2×C3×D5C22⋊Q812yes—
158C2×C3×D5#13512yes—
159C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
160C2×C3×D5#13711yescharacteristic
161ΓL(1,𝔽16)—44no—
162C3×C4×C5—21no—
163C3×C4×C5—21no—
164C3×C4×C5—41no—
165C3×C4×C5—41no—
166C3×C4×C5C4×Q812yes—
167C3×Dic5—21no—
168C3×Dic5—21no—
169C3×Dic5—41no—
170C3×Dic5—41no—
171C3×Dic5#12411yescharacteristic
172C3×Dic5C4⋊Q811yescharacteristic
173C22×C3×C4—51no—
174C22×C3×C4—51no—
175C22×C3×C4—51no—
176C22×C3×C4—101no—
177C22×C3×C4—101no—
178C22×Dic3—102no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—51no—
182C3×C4⋊C4—51no—
183C3×C4⋊C4—101no—
184C3×C4⋊C4—101no—
185C3×C4⋊C4—101no—
186C3×C4⋊C4—101no—
187C3×C4⋊C4—101no—
188C3×C4⋊C4—101no—
189C3×C4⋊C4—101no—
190C3×C4⋊C4—101no—
191C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
192C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
193C3×C42—51no—
194C3×C42—51no—
195C3×C42—51no—
196C3×C42C2×AGL(1,𝔽5)11yescharacteristic
197C4⋊Dic3—54no—
198C4⋊Dic3—102no—
199C4×Dic3—202no—
200C2×C3⋊C8—102no—
201C2×C3⋊C8—104no—
202C22×D5C2×#6711yescharacteristic
203C2×AGL(1,𝔽5)—64no—
204C2×AGL(1,𝔽5)—122no—
205C2×AGL(1,𝔽5)—122no—
206C2×C4×C5—21no—
207C2×C4×C5—21no—
208C2×C4×C5#21711yescharacteristic
209C2×C4×C5#21711yescharacteristic
210C2×C4×C5C22×Dic311yescharacteristic
211C2×Dic5—21no—
212C2×Dic5—21no—
213C2×Dic5#21711yescharacteristic
214C2×Dic5#21711yescharacteristic
215C2×Dic5C2×#6711yescharacteristic
216C4×D5—21no—
217C4×D5—22no—
218C4×D5—21no—
219C4×D5—41no—
220C4×D5—42no—
221C4×D5—41no—
222C4×D5D3×Q812yes—
223C4×D5#23912yes—
224#170—122no—
225C22×C8—152no—
226#125—51no—
227#125—51no—
228#125—51no—
229#125—101no—
230#125—101no—
231#125ΓL(1,𝔽16)11yescharacteristic, complemented
232C2×C4⋊C4—51no—
233C2×C4⋊C4—51no—
234C2×C4⋊C4—101no—
235C2×C4⋊C4—101no—
236C2×C4⋊C4—152no—
237C2×C42—51no—
238C2×C42—301no—
239C4⋊C8—301no—
240C4⋊C8—301no—
241#95—301no—
242C2×C3×C5#39111yescharacteristic
243C2×C3×C5#41611yescharacteristic
244C2×C3×C5#41711yescharacteristic
245C3×D5#47812yes—
246C3×D5#47912yes—
247C23×C3—51no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—51no—
251C2×C3×C4—52no—
252C2×C3×C4—52no—
253C2×C3×C4—51no—
254C2×C3×C4—51no—
255C2×C3×C4—101no—
256C2×C3×C4—101no—
257C2×C3×C4—102no—
258C2×C3×C4—101no—
259C2×C3×C4—101no—
260C2×C3×C4—201no—
261C2×C3×C4—201no—
262C2×C3×C4—202no—
263C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
264C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
265C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
266C2×Dic3—104no—
267C2×Dic3—202no—
268C2×Dic3—202no—
269C3⋊C8—202no—
270C22×C5C2×#21711yescharacteristic
271C2×D5#94612yes—
272C2×D5#94911yescharacteristic
273C2×D5#95012yes—
274C2×D5#95111yescharacteristic
275AGL(1,𝔽5)—124no—
276C4×C5—21no—
277C4×C5—21no—
278C4×C5—41no—
279C4×C5—41no—
280C4×C5Q8×Dic312yes—
281Dic5—21no—
282Dic5—21no—
283Dic5—41no—
284Dic5—41no—
285Dic5#94411yescharacteristic
286Dic5#94711yescharacteristic
287C22×C4—51no—
288C22×C4—51no—
289C22×C4—51no—
290C22×C4—101no—
291C22×C4—101no—
292C22×C4—302no—
293C2×C8—302no—
294C2×C8—304no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—51no—
298C4⋊C4—51no—
299C4⋊C4—101no—
300C4⋊C4—101no—
301C4⋊C4—101no—
302C4⋊C4—101no—
303C4⋊C4—101no—
304C4⋊C4—101no—
305C4⋊C4—101no—
306C4⋊C4—101no—
307C4⋊C4—154no—
308C4⋊C4—302no—
309C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
310C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
311C42—51no—
312C42—51no—
313C42—51no—
314C42—602no—
315C42C2×ΓL(1,𝔽16)11yescharacteristic
316C3×C5#198911yescharacteristic, complemented, Hall
317C22×C3—51no—
318C22×C3—51no—
319C22×C3—52no—
320C22×C3—52no—
321C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—51no—
327C3×C4—51no—
328C3×C4—101no—
329C3×C4—101no—
330C3×C4—201no—
331C3×C4—201no—
332C3×C4Q8×AGL(1,𝔽5)12yes—
333Dic3—204no—
334C2×C5#522911yescharacteristic
335C2×C5#523211yescharacteristic
336C2×C5#523311yescharacteristic
337D5#506312yes—
338D5#506612yes—
339C23—51no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—51no—
343C2×C4—52no—
344C2×C4—52no—
345C2×C4—51no—
346C2×C4—51no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—102no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—201no—
353C2×C4—201no—
354C2×C4—202no—
355C2×C4—304no—
356C2×C4—602no—
357C2×C4—602no—
358C2×C4#676311yescharacteristic
359C2×C4#676311yescharacteristic
360C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
361C8—602no—
362C2×C3—52no—
363C2×C3—52no—
364C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
365C2×C3#6579311yescharacteristic
366C2×C3#6579411yescharacteristic
367C5#7173811yescharacteristic, complemented, Sylow
368C22—52no—
369C22—51no—
370C22—51no—
371C22—52no—
372C22C2×#676311yescharacteristic
373C4—21no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—51no—
378C4—51no—
379C4—101no—
380C4—101no—
381C4—201no—
382C4—201no—
383C4—604no—
384C4Q8×ΓL(1,𝔽16)12yes—
385C3#1060458911yescharacteristic, complemented, Sylow
386C2—52no—
387C2—52no—
388C2#1175244911yescharacteristic
389C2#1175245211yescharacteristic
390C2#1175245311yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#125C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenter#125ΓL(1,𝔽16)
hypocenterC3×C5#1989
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
10. C3×C5×#125C4semidirect
15. C3×#4036C4semidirect
16. C3×#4036C4semidirect
20. C3×#4016C4semidirect
67. C5×#125Dic3semidirect
72. #4036Dic3semidirect
73. #4036Dic3semidirect
77. #4016Dic3semidirect
110. C3×#125AGL(1,𝔽5)semidirect
231. #125ΓL(1,𝔽16)semidirect
316. C3×C5#1989semidirect
367. C5#71738semidirect
385. C3#10604589semidirect

Maximal subgroups

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

subgroupindex
C3×D5×#1252
C4×C4⋊ΓL(1,𝔽16)2
D5⋊(C4⋊(C3⋊C8))2
#117467452
#106045893
#717385