Finite Group Databasebeta

← #14282768

Subgroups — #14282768

Order 1920.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66025C211yesmaximal, characteristic
2#11752345C211yesmaximal, characteristic, complemented
3#11752337C211yesmaximal, characteristic, complemented
4#11746740C212yesmaximal, complemented
5#11746739C212yesmaximal
6#10604573—31nomaximal, Hall
7C2×C4×ΓL(1,𝔽16)—21no—
8C3×C5×#125C411yescharacteristic, complemented
9C3×#4044—21no—
10C3×#4043C2211yescharacteristic
11C3×#4041C2211yescharacteristic
12C3×#4040—21no—
13C3×#4039—21no—
14C3×#4039C411yescharacteristic, complemented
15C3×#4038—21no—
16C3×#4033C411yescharacteristic, complemented
17C3×#4021C411yescharacteristic, complemented
18C3×C42×D5C2211yescharacteristic
19#91235—21no—
20#91235C2212yes—
21#91233—21no—
22#91232C2212yes—
23C4×C3⋊#170—21no—
24#71754—51nomaximal, Hall
25#66025D311yescharacteristic
26#65686—31no—
27#65678—31no—
28#64869—32no—
29#64868—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×C3×D20—21no—
32C2×C3×D20C2×C411yescharacteristic
33C2×C3×C4×D5—21no—
34C2×C3×C4×D5D411yescharacteristic
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5C2311yescharacteristic
37C2×C3×Dic10C2×C411yescharacteristic
38C2×C3⋊#170—22no—
39C4×ΓL(1,𝔽16)—24no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4C2×C411yescharacteristic
43C3×C5×C4⋊C4C2×C411yescharacteristic
44C3×C42×C5C2×C411yescharacteristic
45C3×#685—21no—
46C3×#685—21no—
47C3×#685—21no—
48C3×#685—21no—
49C3×#685—21no—
50C3×C4⋊Dic5—21no—
51C3×#683—21no—
52C3×C4×Dic5—21no—
53C3×C4×Dic5C2×C411yescharacteristic
54C3×C4×Dic5D411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×#529—51no—
57#4932—51no—
58#4925—51no—
59#4538—52no—
60#4537—52no—
61C2×C4×AGL(1,𝔽5)—61no—
62C5×#125Dic311yescharacteristic, complemented
63#4044—21no—
64#4043C2×D311yescharacteristic
65#4041C2×D311yescharacteristic
66#4040—21no—
67#4039—21no—
68#4039Dic311yescharacteristic, complemented
69#4038—21no—
70#4033Dic311yescharacteristic, complemented
71#4021Dic311yescharacteristic, complemented
72C42×D5C2×D311yescharacteristic
73(C2×AGL(1,𝔽5))⋊C4—32no—
74(C2×AGL(1,𝔽5))⋊C4—61no—
75#4004—61no—
76#4003—32no—
77C4×#170—61no—
78#1500—151noSylow
79C22×C3×D5—21no—
80C22×C3×D5C2×D411yescharacteristic
81C2×ΓL(1,𝔽16)—24no—
82C2×ΓL(1,𝔽16)—42no—
83C2×ΓL(1,𝔽16)—42no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22×C411yescharacteristic
89C2×C3×Dic5—21no—
90C2×C3×Dic5C22⋊C411yescharacteristic
91C2×C3×Dic5C22⋊C411yescharacteristic
92C2×C3×Dic5C2×D411yescharacteristic
93C3×D20—41no—
94C3×D20—42no—
95C3×C4×D5—22no—
96C3×C4×D5—22no—
97C3×C4×D5—41no—
98C3×C4×D5—42no—
99C3×Dic10—41no—
100C3⋊#170—42no—
101C3×#126—101no—
102C3×#125AGL(1,𝔽5)11yescharacteristic, complemented
103C3×#124—51no—
104C3×#123—101no—
105C3×C22⋊Q8—51no—
106C3×#121—51no—
107C3×C4×Q8—51no—
108C3×C4×D4—51no—
109C3×C4×D4—101no—
110C3×#117—101no—
111C2×C3×C42—51no—
112C2×C4×Dic3—101no—
113C3⋊#95—52no—
114C3⋊#95—101no—
115#830—52no—
116C4⋊(C3⋊C8)—101no—
117C4×C3⋊C8—101no—
118C22×AGL(1,𝔽5)—62no—
119C2×D20—21no—
120C2×D20C2×Dic311yescharacteristic
121C2×C4×D5—21no—
122C2×C4×D5#6711yescharacteristic
123C2×C4×D5#6711yescharacteristic
124C2×C4×D5C22×D311yescharacteristic
125C2×Dic10C2×Dic311yescharacteristic
126C2×#170—62no—
127C4×AGL(1,𝔽5)—64no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4C2×Dic311yescharacteristic
131C5×C4⋊C4C2×Dic311yescharacteristic
132C42×C5C2×Dic311yescharacteristic
133#685—21no—
134#685—21no—
135#685—21no—
136#685—21no—
137#685—21no—
138C4⋊Dic5—21no—
139#683—21no—
140C4×Dic5—21no—
141C4×Dic5C2×Dic311yescharacteristic
142C4×Dic5#6711yescharacteristic
143C4×Dic5#6711yescharacteristic
144#529—51no—
145#431—151no—
146#376—151no—
147#328—152no—
148#327—152no—
149C22×C3×C5C2×C22⋊C411yescharacteristic
150C2×C3×D5—21no—
151C2×C3×D5—21no—
152C2×C3×D5—41no—
153C2×C3×D5—41no—
154C2×C3×D5—41no—
155C2×C3×D5#13512yes—
156ΓL(1,𝔽16)—44no—
157C3×C4×C5—21no—
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160C3×C4×C5—41no—
161C3×C4×C5—41no—
162C3×Dic5—21no—
163C3×Dic5—21no—
164C3×Dic5—41no—
165C3×Dic5#10412yes—
166C2×C3×Q8—51no—
167C2×C3×D4—51no—
168C2×C3×D4—101no—
169C22×C3×C4—51no—
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4—101no—
173C22×Dic3—102no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—101no—
177C3×C4⋊C4—101no—
178C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
179C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
180C3×C22⋊C4—101no—
181C3×C22⋊C4—101no—
182C3×C22⋊C4—101no—
183C3×C22⋊C4—101no—
184C3×C22⋊C4—101no—
185C3×C42—51no—
186C3×C42—51no—
187C3×C42—51no—
188C3×C42—101no—
189C3×C42C2×AGL(1,𝔽5)11yescharacteristic
190C4×Dic3—104no—
191C2×C3⋊C8—102no—
192C22×D5—21no—
193C22×D5C2×#6711yescharacteristic
194C2×AGL(1,𝔽5)—64no—
195C2×AGL(1,𝔽5)—122no—
196C2×AGL(1,𝔽5)—122no—
197C2×C4×C5—21no—
198C2×C4×C5—21no—
199C2×C4×C5#21711yescharacteristic
200C2×C4×C5#21711yescharacteristic
201C2×C4×C5C22×Dic311yescharacteristic
202C2×Dic5—21no—
203C2×Dic5#21711yescharacteristic
204C2×Dic5#21711yescharacteristic
205C2×Dic5C2×#6711yescharacteristic
206D20—41no—
207D20—42no—
208C4×D5—22no—
209C4×D5—22no—
210C4×D5—41no—
211C4×D5—42no—
212Dic10—41no—
213#170—122no—
214#126—101no—
215#125ΓL(1,𝔽16)11yescharacteristic, complemented
216#124—51no—
217#123—101no—
218C22⋊Q8—51no—
219#121—51no—
220C4×Q8—51no—
221C4×D4—51no—
222C4×D4—101no—
223#117—101no—
224C2×C42—51no—
225C2×C42—301no—
226C4⋊C8—301no—
227C22⋊C8—152no—
228C4×C8—301no—
229#95—152no—
230#95—301no—
231C2×C3×C5#41011yescharacteristic
232C2×C3×C5#41611yescharacteristic
233C2×C3×C5C2×#10411yescharacteristic
234C3×D5—22no—
235C3×D5—81no—
236C23×C3—51no—
237C23×C3—101no—
238C3×Q8—201no—
239C3×D4—201no—
240C3×D4—202no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—51no—
244C2×C3×C4—51no—
245C2×C3×C4—51no—
246C2×C3×C4—102no—
247C2×C3×C4—102no—
248C2×C3×C4—101no—
249C2×C3×C4—201no—
250C2×C3×C4—202no—
251C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
252C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
253C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
254C2×Dic3—104no—
255C2×Dic3—202no—
256C2×Dic3—202no—
257C3⋊C8—202no—
258C22×C5C2×#21711yescharacteristic
259C2×D5—21no—
260C2×D5—21no—
261C2×D5—41no—
262C2×D5—41no—
263C2×D5—41no—
264C2×D5#95012yes—
265AGL(1,𝔽5)—124no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—21no—
269C4×C5—41no—
270C4×C5—41no—
271Dic5—21no—
272Dic5—21no—
273Dic5—41no—
274Dic5#83712yes—
275C2×Q8—51no—
276C2×D4—51no—
277C2×D4—101no—
278C22×C4—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—101no—
282C22×C4—302no—
283C2×C8—302no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—101no—
287C4⋊C4—101no—
288C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
289C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
290C22⋊C4—101no—
291C22⋊C4—101no—
292C22⋊C4—101no—
293C22⋊C4—101no—
294C22⋊C4—101no—
295C42—51no—
296C42—51no—
297C42—51no—
298C42—101no—
299C42—304no—
300C42C2×ΓL(1,𝔽16)11yescharacteristic
301C3×C5#150011yescharacteristic, complemented, Hall
302C22×C3—52no—
303C22×C3—101no—
304C22×C3—101no—
305C22×C3—201no—
306C22×C3—201no—
307C22×C3—201no—
308C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—52no—
315C3×C4—101no—
316C3×C4—101no—
317C3×C4—201no—
318Dic3—204no—
319C2×C5#523311yescharacteristic
320C2×C5C2×#83711yescharacteristic
321C2×C5#524011yescharacteristic
322D5—22no—
323D5—81no—
324C23—51no—
325C23—101no—
326Q8—201no—
327D4—201no—
328D4—202no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—51no—
332C2×C4—51no—
333C2×C4—51no—
334C2×C4—102no—
335C2×C4—102no—
336C2×C4—101no—
337C2×C4—201no—
338C2×C4—202no—
339C2×C4—304no—
340C2×C4—602no—
341C2×C4—602no—
342C2×C4#676311yescharacteristic
343C2×C4#676311yescharacteristic
344C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
345C8—602no—
346C2×C3—102no—
347C2×C3—401no—
348C2×C3#6578511yescharacteristic
349C2×C3D5⋊#10411yescharacteristic
350C2×C3#6579411yescharacteristic
351C5#7175411yescharacteristic, complemented, Sylow
352C22—52no—
353C22—101no—
354C22—101no—
355C22—201no—
356C22—201no—
357C22—201no—
358C22C2×#676311yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—52no—
365C4—101no—
366C4—101no—
367C4—201no—
368C4—604no—
369C3#1060457311yescharacteristic, complemented, Sylow
370C2—102no—
371C2—401no—
372C2#1175244411yescharacteristic
373C2D5⋊#83711yescharacteristic
374C2#1175245311yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#125C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#4041C22
cosocleC2×C3×C4×D5C23
hypercenter#125ΓL(1,𝔽16)
hypocenterC3×C5#1500
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
2. #11752345C2semidirect
3. #11752337C2semidirect
4. #11746740C2semidirect
8. C3×C5×#125C4semidirect
14. C3×#4039C4semidirect
16. C3×#4033C4semidirect
17. C3×#4021C4semidirect
62. C5×#125Dic3semidirect
68. #4039Dic3semidirect
70. #4033Dic3semidirect
71. #4021Dic3semidirect
102. C3×#125AGL(1,𝔽5)semidirect
215. #125ΓL(1,𝔽16)semidirect
301. C3×C5#1500semidirect
351. C5#71754semidirect
369. C3#10604573semidirect

Maximal subgroups

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

subgroupindex
C3×#660252
#117523452
#117523372
#117467402
#117467392
#106045733
#717545