Finite Group Databasebeta

← #14307137

Subgroups — #14307137

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66022C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
3C4×#6758C211yesmaximal, characteristic
4#11746746C212yesmaximal
5#11746746C212yesmaximal
6#10604592—31nomaximal, Hall
7C2×C4⋊ΓL(1,𝔽16)—21no—
8C2×C4×ΓL(1,𝔽16)C2212yes—
9C2×#6758C2212yes—
10C3×C5×#125C411yescharacteristic, complemented
11C3×#4042—21no—
12C3×#4042—21no—
13C3×#4038C2211yescharacteristic
14C3×#4038C2211yescharacteristic
15C3×#4034C411yescharacteristic, complemented
16C3×#4034C411yescharacteristic, complemented
17C3×#4033—21no—
18C3×#4033—21no—
19C3×C42×D5C2211yescharacteristic
20C3×#4016C411yescharacteristic, complemented
21#91235—21no—
22#91231—21no—
23C4×C3⋊#170—21no—
24#71841—51nomaximal, Hall
25#66022D311yescharacteristic
26C4×C4⋊AGL(1,𝔽5)—31no—
27C4×#700—31no—
28#64875—32no—
29#64875—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×C3×C4×D5D411yescharacteristic
32C2×C3×C4×D5D411yescharacteristic
33C2×C3×C4×D5C2311yescharacteristic
34C2×C3×Dic10—21no—
35C2×C3×Dic10—21no—
36C2×C3⋊#170—22no—
37C4⋊ΓL(1,𝔽16)—24no—
38C4×ΓL(1,𝔽16)—24no—
39#6758—22no—
40#6758Q814yescomplemented
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×#685—21no—
47C3×#685—21no—
48C3×C4⋊Dic5C2×C411yescharacteristic
49C3×C4⋊Dic5C2×C411yescharacteristic
50C3×#683—21no—
51C3×#683—21no—
52C3×#683—21no—
53C3×#683—21no—
54C3×C4×Dic5—21no—
55C3×C4×Dic5—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C3×C4×Dic5D411yescharacteristic
58C3×C4×Dic5D411yescharacteristic
59C3×#533—51no—
60C4×C4⋊Dic3—51no—
61C4×#208—51no—
62#4532—52no—
63#4532—52no—
64C2×C4⋊AGL(1,𝔽5)—61no—
65C2×C4×AGL(1,𝔽5)—32no—
66C2×#700—32no—
67C5×#125Dic311yescharacteristic, complemented
68#4042—21no—
69#4042—21no—
70#4038C2×D311yescharacteristic
71#4038C2×D311yescharacteristic
72#4034Dic311yescharacteristic, complemented
73#4034Dic311yescharacteristic, complemented
74#4033—21no—
75#4033—21no—
76C42×D5C2×D311yescharacteristic
77#4016Dic311yescharacteristic, complemented
78(C2×AGL(1,𝔽5))⋊C4—61no—
79#4002—61no—
80C4×#170—61no—
81#1986—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—41no—
98C3×C4×D5C2×Q812yes—
99C3×C4×D5#4112yes—
100C3×Dic10—42no—
101C3×Dic10—42no—
102C3⋊#170—24no—
103C3×C4⋊Q8—51no—
104C3×C4⋊Q8—51no—
105C3×#125—51no—
106C3×#125AGL(1,𝔽5)11yescharacteristic, complemented
107C3×C22⋊Q8—101no—
108C3×C22⋊Q8—101no—
109C3×C4×Q8—101no—
110C3×C4×Q8—101no—
111C3×#117—51no—
112C3×#117—51no—
113C2×C3×C42—51no—
114C2×C4⋊Dic3—101no—
115C2×C4×Dic3—52no—
116C2×#208—52no—
117C3⋊#95—101no—
118#803—101no—
119C4×C3⋊C8—101no—
120C22×AGL(1,𝔽5)—62no—
121C2×C4×D5#6711yescharacteristic
122C2×C4×D5#6711yescharacteristic
123C2×C4×D5C22×D311yescharacteristic
124C2×Dic10—21no—
125C2×Dic10—21no—
126C2×#170—62no—
127C4⋊AGL(1,𝔽5)—64no—
128C4×AGL(1,𝔽5)—64no—
129#700—34no—
130#700—62no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—21no—
133C5×C4⋊C4C2×Dic311yescharacteristic
134C5×C4⋊C4C2×Dic311yescharacteristic
135C42×C5C2×Dic311yescharacteristic
136#685—21no—
137#685—21no—
138C4⋊Dic5C2×Dic311yescharacteristic
139C4⋊Dic5C2×Dic311yescharacteristic
140#683—21no—
141#683—21no—
142#683—21no—
143#683—21no—
144C4×Dic5—21no—
145C4×Dic5—21no—
146C4×Dic5C2×Dic311yescharacteristic
147C4×Dic5#6711yescharacteristic
148C4×Dic5#6711yescharacteristic
149#533—51no—
150C4×M16—151no—
151C4×C4⋊C4—151no—
152#339—152no—
153#339—152no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C2×C3×D5—21no—
156C2×C3×D5—21no—
157C2×C3×D5#12411yescharacteristic
158C2×C3×D5C4⋊Q811yescharacteristic
159ΓL(1,𝔽16)—44no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162C3×C4×C5—41no—
163C3×C4×C5—41no—
164C3×C4×C5C4×Q812yes—
165C3×Dic5—21no—
166C3×Dic5—41no—
167C3×Dic5—41no—
168C3×Dic5#10412yes—
169C3×Dic5C22⋊Q812yes—
170C2×C3×Q8—101no—
171C2×C3×Q8—101no—
172C22×C3×C4—51no—
173C22×C3×C4—51no—
174C22×C3×C4—51no—
175C22×Dic3—102no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—51no—
179C3×C4⋊C4—51no—
180C3×C4⋊C4—101no—
181C3×C4⋊C4—101no—
182C3×C4⋊C4—101no—
183C3×C4⋊C4—101no—
184C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
185C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
186C3×C22⋊C4—101no—
187C3×C22⋊C4—101no—
188C3×C42—51no—
189C3×C42—51no—
190C3×C42—51no—
191C3×C42—101no—
192C3×C42—101no—
193C3×C42C2×AGL(1,𝔽5)11yescharacteristic
194C4⋊Dic3—104no—
195C4×Dic3—104no—
196#208—54no—
197#208—102no—
198C2×C3⋊C8—102no—
199C22×D5C2×#6711yescharacteristic
200C2×AGL(1,𝔽5)—64no—
201C2×AGL(1,𝔽5)—122no—
202C2×AGL(1,𝔽5)—122no—
203C2×C4×C5—21no—
204C2×C4×C5—21no—
205C2×C4×C5#21711yescharacteristic
206C2×C4×C5#21711yescharacteristic
207C2×C4×C5C22×Dic311yescharacteristic
208C2×Dic5—21no—
209C2×Dic5—21no—
210C2×Dic5#21711yescharacteristic
211C2×Dic5#21711yescharacteristic
212C2×Dic5C2×#6711yescharacteristic
213C4×D5—22no—
214C4×D5—41no—
215C4×D5D3×Q812yes—
216C4×D5#23912yes—
217Dic10—42no—
218Dic10—42no—
219#170—64no—
220C2×M16—152no—
221C4⋊Q8—51no—
222C4⋊Q8—51no—
223#125—51no—
224#125ΓL(1,𝔽16)11yescharacteristic, complemented
225C22⋊Q8—101no—
226C22⋊Q8—101no—
227C4×Q8—101no—
228C4×Q8—101no—
229#117—51no—
230#117—51no—
231C2×C4⋊C4—301no—
232C2×C42—51no—
233C2×C42—152no—
234#97—301no—
235C4×C8—301no—
236#95—301no—
237C2×C3×C5#39111yescharacteristic
238C2×C3×C5C2×#10411yescharacteristic
239C2×C3×C5#42111yescharacteristic
240C3×D5—21no—
241C3×D5—21no—
242C23×C3—51no—
243C3×Q8—202no—
244C3×Q8—202no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—51no—
248C2×C3×C4—51no—
249C2×C3×C4—52no—
250C2×C3×C4—52no—
251C2×C3×C4—51no—
252C2×C3×C4—102no—
253C2×C3×C4—101no—
254C2×C3×C4—101no—
255C2×C3×C4—201no—
256C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
257C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
258C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
259C2×Dic3—104no—
260C2×Dic3—202no—
261C2×Dic3—202no—
262C3⋊C8—104no—
263C22×C5C2×#21711yescharacteristic
264C2×D5—21no—
265C2×D5—21no—
266C2×D5#94411yescharacteristic
267C2×D5#94711yescharacteristic
268AGL(1,𝔽5)—124no—
269C4×C5—21no—
270C4×C5—21no—
271C4×C5—41no—
272C4×C5—41no—
273C4×C5Q8×Dic312yes—
274Dic5—21no—
275Dic5—41no—
276Dic5—41no—
277Dic5#83712yes—
278Dic5#94612yes—
279C2×Q8—101no—
280C2×Q8—101no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—302no—
285M16—154no—
286M16—302no—
287C2×C8—302no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—51no—
291C4⋊C4—51no—
292C4⋊C4—101no—
293C4⋊C4—101no—
294C4⋊C4—101no—
295C4⋊C4—101no—
296C4⋊C4—304no—
297C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
298C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
299C22⋊C4—101no—
300C22⋊C4—101no—
301C42—51no—
302C42—51no—
303C42—51no—
304C42—101no—
305C42—101no—
306C42—304no—
307C42C2×ΓL(1,𝔽16)11yescharacteristic
308C3×C5#198611yescharacteristic, complemented, Hall
309C22×C3—51no—
310C22×C3—51no—
311C22×C3—101no—
312C22×C3—101no—
313C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—52no—
319C3×C4—52no—
320C3×C4—101no—
321C3×C4—201no—
322C3×C4—201no—
323C3×C4Q8×AGL(1,𝔽5)12yes—
324Dic3—204no—
325C2×C5#522911yescharacteristic
326C2×C5C2×#83711yescharacteristic
327C2×C5#523611yescharacteristic
328D5—21no—
329D5—21no—
330C23—51no—
331Q8—202no—
332Q8—202no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—51no—
336C2×C4—51no—
337C2×C4—52no—
338C2×C4—52no—
339C2×C4—51no—
340C2×C4—102no—
341C2×C4—101no—
342C2×C4—101no—
343C2×C4—201no—
344C2×C4—304no—
345C2×C4—602no—
346C2×C4—602no—
347C2×C4#676311yescharacteristic
348C2×C4#676311yescharacteristic
349C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
350C8—304no—
351C2×C3—101no—
352C2×C3—101no—
353C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
354C2×C3D5⋊#10411yescharacteristic
355C2×C3#6579511yescharacteristic
356C5#7184111yescharacteristic, complemented, Sylow
357C22—51no—
358C22—51no—
359C22—101no—
360C22—101no—
361C22C2×#676311yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—52no—
367C4—52no—
368C4—101no—
369C4—201no—
370C4—201no—
371C4—604no—
372C4Q8×ΓL(1,𝔽16)12yes—
373C3#1060459211yescharacteristic, complemented, Sylow
374C2—101no—
375C2—101no—
376C2#1175244911yescharacteristic
377C2D5⋊#83711yescharacteristic
378C2#1175245411yescharacteristic
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#1986
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×#4034C4semidirect
16. C3×#4034C4semidirect
20. C3×#4016C4semidirect
40. #6758Q8semidirect
67. C5×#125Dic3semidirect
72. #4034Dic3semidirect
73. #4034Dic3semidirect
77. #4016Dic3semidirect
106. C3×#125AGL(1,𝔽5)semidirect
224. #125ΓL(1,𝔽16)semidirect
308. C3×C5#1986semidirect
356. C5#71841semidirect
373. C3#10604592semidirect

Maximal subgroups

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

subgroupindex
C3×#660222
C4×C4⋊ΓL(1,𝔽16)2
C4×#67582
#117467462
#106045923
#718415